What If Ether Frame Existed? Galilean-Invariant Maxwell Eqs.
Mark
div D = rho; div B = 0
curl H - dD/dt = J; curl E + dB/dt = 0
were the product of a long line of experimental investigation that
had taken place under a wide variety of circumstances ... and in
particular, in different frames of reference. If they did not hold
in these frames of reference, they would have never been discovered
in the first place.
However, with the constitutive relations (using e0, m0 to stand
respectively for epsilon_0, mu_0):
D = e0 E; H = m0 H
one is able to derive the wave equations:
(del^2 - (1/c)^2 d^2/dt^2) E = grad rho/e0 + m0 dJ/dt
(del^2 - (1/c)^2 d^2/dt^2) B = -m0 curl J
Given that the constitutive relations, themselves, hold in every
(inertial) frame, then it follows that the same wave equations hold in
every frame -- which immediately implies that c is a frame-independent
speed.
But this is inconsistent with the Galilean transformation, since the
only invariant speed in Newtonian physics is infinity (which is just
another way of saying that simultaneity is absolute).
So, the question naturally rises: what WOULD the Maxwell equations
look like, had they been invariant under Galilean transformations
with the following transformation properties:
E --> E - u x B; D --> D
H --> H + u x D; B --> B
J --> J + rho u; rho --> rho
d/dt --> d/dt - u.Del; Del --> Del
The answer: THE SAME!
div D = Del.D --> Del.D = div D = rho
div B = Del.B --> Del.B = div B = 0
curl H - dD/dt = DelxH - dD/dt
--> Delx(H + uxD) - (d/dt-u.Del)D
= DelxH - dD/dt + Delx(uxD) + u.Del D
= (DelxH - dD/dt) + (u Del.D - u.Del D) + u.Del D
= J + rho u
curl E + dB/dt = DelxE + dB/dt
--> Delx(E - uxB) + (d/dt-u.Del)B
= DelxE + dB/dt - Delx(uxB) - u.Del B
= (DelxE - dB/dt) - u Del.B
= 0
It's a little-known, but true, fact that the macroscopic Maxwell
equations are also Galilean invariant. Any difference, therefore,
has to reside in the constitutive relations.
So, what would those be? This, then, gets to the question: what
would Maxwell's equations actually look like, given the speculations
of the 19th century?
Assume the following:
(A) Axiom of Rest: There is a frame of absolute rest.
(B) Axiom of Inertia: Inertial frames are precisely those which are
in uniform motion with respect to the Ether Frame.
(C) Non-Invariance of Light Speed: Maxwell's equations (as we know them)
hold exclusively in the frame of absolute rest.
Let V be the velocity of the inertial frame with respect to the absolute
rest frame. Then, by (C) we have in the absolute rest frame:
D = e0 E; B = m0 H
Under a Galilean transformation, this becomes:
D = e0 (E - V x B); B = m0 (H + V x D)
which yields also the following relations:
H = B/m0 - e0 V x (E - V x B)
E = D/e0 + m0 V x (H + V x D)
or
D = (e0 (c^2 E - V V.E) - V x H)/(c^2 - V^2)
B = (m0 (c^2 H - V V.H) + V X E)/(c^2 - V^2)
The macroscopic Maxwell equations -- which are identical in both the
Galilean and Relativistic forms -- with these relations will yield the
wave equation for wave speed c in ONLY the Ether frame, and will reduce
to the microscopic Maxwell equations (as we know them) ONLY in this frame.
In other frames, the microscopic form becomes:
div (E - V x B) = rho/e0
curl B - (1/c)^2 curl (V x (E-VxB)) - (1/c)^2 d(E-VxB)/dt = m0 J
div B = 0
curl E + dB/dt = 0
The Galilean invariance of the macroscopic and microscopic forms
applies even when as add in magnetic source terms:
div B = sigma; curl E + dB/dt = -K
and both the Lorentz force density and power density:
F_E = rho E + J x B; P_E = J.E
F_H = sigma H - K x D; P_H = K.H
transform accordingly:
F_E --> F_E; P_E --> P_E + u.F_E
F_B --> F_B; P_B --> P_B + u.F_B
Moataz Emam
This is interesting. It reminds me of the famous Feynman derivation of
Maxwell's homogenous equations. He starts with space-momentum
commutation relations and F=ma. With an elaborate, yet straightforward
calculation, he manages to derive the Maxwell equations, never using
relativity at any point. The explanation was given that this is not as
paradoxical as it seems since the homogenous equations are indeed
invariant under Gallilean transformations.
See:
Am. J. Phys. 58(3) March 1990
Am. J. Phys. 59(1) January 1991
quant-ph/9810088
--
Moataz H. Emam
URL: http://continue.to/emam
DickT
I went to the above URL and studied the Feyman argument. It sems to
me the steps are all reversable, so could someone in the nineteenth
century have derived the commutation laws from div H = 0 ,Faraday's
induction law, and F = Ma?
Moataz Emam
> I went to the above URL and studied the Feyman argument. It sems to
> me the steps are all reversable, so could someone in the nineteenth
> century have derived the commutation laws from div H = 0 ,Faraday's
> induction law, and F = Ma?
Interesting question. Perhaps it would have been possible. But, as in
many situations, it seems possible to do with hindsight. If you do not
know what you are looking for it would probably require a great genius.
But that is indeed an interesting question. Seems that based on this
argument, quantum mechanics is apparently derivable from F=ma and
Maxwell's theory !!!! But if a 19th century physicist stumbled on the
commutation relations, would s/he have been able to understand their
applicability? Sounds to me that if this has happened, it would have
been beyond the scope of verification and hence the physicist would have
been in the same situation string theorists are in right now. Elegant
theory - no experimental proof or disproof.
Rob Salgado
In article <a60io5$m66$1...@uwm.edu>, Mark <whop...@alpha2.csd.uwm.edu> wrote:
>Maxwell's equations
>
[snip]
>
>It's a little-known, but true, fact that the macroscopic Maxwell
>equations are also Galilean invariant. Any difference, therefore,
>has to reside in the constitutive relations.
>
>So, what would those be? This, then, gets to the question: what
>would Maxwell's equations actually look like, given the speculations
>of the 19th century?
>
[snip]
You might enjoy the following related references:
"If Maxwell had worked between Ampere and Faraday:
An historical fable with a pedagogical moral"
Max Jammer and John Stachel
Am. J. Phy, v48, no 1, Jan 1980, pp 5-7
"Galilean Electromagnetism"
M. Le Bellac and J.M. Levy-Leblond
Il Nuovo Cimento, v14 B, no 2, April 1973, pp 217-233
"The fundamental equations of electromagnetism, independent
of metrical geometry"
D. van Danzig
Proc. Cambridge Phil. Soc., v 30, 1934a, pp 421-427
"Formal Structure of Electromagnetics:
General Covariance and Electromagnetics"
E. J. Post, 1962
ISBN: 0486654273 (Dover!)
Danny Ross Lunsford
responding to keep the discussion to the technical aspects and try to
avoid phrasing things in ways that can be construed as statement about
people rather than physics. -- KS]
"Moataz Emam" <em...@physics.umass.edu> wrote in message
news:a69dju$9dq$1...@blinky.its.caltech.edu...
> But that is indeed an interesting question. Seems that based on this
> argument, quantum mechanics is apparently derivable from F=ma and
> Maxwell's theory !!!!
I am flabbergasted by such a statement. Nothing could be farther from the
truth. The argument and everything surrounding it is obviously *wrong*.
This should be clear based on several viewpoints:
1) Feynman's argument is an accident - the spacetime basis that supports the
EM field also has commutation relations that look like those for the x and p
of mechanics, and allows EM to be cast in "Hamiltonian" form. In fact *any*
dynamical theory (at least as we know them) can be cast in Hamiltonian form.
2) Maxwell's eqns are nothing but the Schroedinger eqn for photons.
3) It is well known that the free EM field can be represented as a pair of
independent harmonic oscillators - again, the resulting entities look like
those that show up in mechanics. (The oscillators correspond to the
polarization freedom in an EM wave.)
4) *Any* purely tensorial theory (i.e. integral spin) is going to have a
Maxwell-like structure on some level (for example, gravitational waves do).
So reproducing a Maxwell-like structure is hardly the same thing as
"deriving" quantum mechanics from it.
Physics is not mathematics. One doesn't prove theorems or derive oranges
from apples. One figures out how the world is put together with the help of
mathematical insight, not mere blundering about with powerful tools.
-drl
Pertti Lounesto
>
>Mark <whop...@alpha2.csd.uwm.edu> wrote:
>
>>Maxwell's equations
>>
> [snip]
>
>>It's a little-known, but true, fact that the macroscopic Maxwell
>>equations are also Galilean invariant. Any difference, therefore,
>>has to reside in the constitutive relations.
>>
del G = J, both of which are covariant under the general
linear group GL(4,R), but which tranform differently.
Together dF = 0 and del G = J are covariant only under
the Lorentz group (or the conformal group, if you consider
differentiable transformations instead of GL(4,R)).
The constitutive relations are also covariant under GL(4,R).
John Baez
news:<a64p0j$o3l$1...@blinky.its.caltech.edu>...
> This is interesting. It reminds me of the famous Feynman derivation of
> Maxwell's homogenous equations. He starts with space-momentum
> commutation relations and F=ma. With an elaborate, yet straightforward
> calculation, he manages to derive the Maxwell equations, never using
> relativity at any point. The explanation was given that this is not as
> paradoxical as it seems since the homogenous equations are indeed
> invariant under Galilean transformations.
It's somewhat misleading to say that the homogenous Maxwell
equations are invariant under Galilean transformations. It's
perfectly true - but it's misleading nonetheless, because
the homogenous Maxwell equations are actually invariant under
ALL transformations. "Galilean" is a red herring which only
serves to distract attention from what's really going on.
In modern terminology: the homogenous Maxwell equations
dF = 0
are invariant under all diffeomorphisms of spacetime, since
the operator d commutes with the action of diffeomorphisms
on differential forms.
Of course the other Maxwell equations
d*F* = J
involve the Hodge star operator, and are not diffeomorphism-invariant.
John Baez
Moataz Emam <em...@physics.umass.edu> wrote:
>Seems that based on this
>argument, quantum mechanics is apparently derivable from F=ma and
>Maxwell's theory!!!!
What argument? Let me see it... I didn't see anyone give an
argument that does that!
(Positing Heisenberg's commutation relations between position
and momentum is cheating if one wants to "derive" quantum mechanics
from F = ma and Maxwell's theory.)
>But if a 19th century physicist stumbled on the
>commutation relations, would s/he have been able to understand their
>applicability?
Well, there was a 19th-century physicist named Poisson who stumbled on
some commutation relations between position and momentum:
{p,q} = 1
but these are different from the Heisenberg commutation relations,
basically because they involve a Poisson algebra instead of a
noncommutative algebra. Now we know that Poisson's commutations are a
kind of hbar -> 0 limit of Heisenberg's, but you can't really "derive"
Heisenberg's from Poisson's - not without making extra assumptions that
amount to assuming quantum mechanics is true.
greywolf42
news:eb8f12dd.02030...@posting.google.com...
Yes. A fellow named Maxwell, in 1861. "On Physical Lines of Force."
Except he also derived div H = 0, rather than assuming it. While on the way
to deriving "Maxwell's equations," and identifying light as an
electromagnetic wave. He started with Faraday's laws and Newton's laws of
motion.
greywolf42
ubi dubium ibi libertas.
Urs Schreiber
news:zYXh8.21433$106.1...@bgtnsc05-news.ops.worldnet.att.net...
[...]
> 2) Maxwell's eqns are nothing but the Schroedinger eqn for photons.
[...]
This statement made me go back and forth between thinking "No!" and
"Well, maybe in some sense." Common lore is that coherent states of
the quantized EM field, which describe Poisson distributions of photons,
correspond to classical Maxwell fields. How could one make precise
your statement above?
[Moderator's note: There were some illuminating discussions
about this confusing subject on s.p.r a few years ago.
John Baez kept some highlights of the discussion, edited with
corrections by Michael Weiss:
http://math.ucr.edu/home/baez/photon/schmoton.htm
-MM]
Alfred Einstead
>Mark <whop...@alpha2.csd.uwm.edu> wrote:
> >>It's a little-known, but true, fact that the macroscopic Maxwell
> >>equations are also Galilean invariant. Any difference, therefore,
> >>has to reside in the constitutive relations.
> >>
> No. There is a pair of Mawell equations, dF = 0 and
> del G = J, both of which are covariant under the general
> linear group GL(4,R), but which tranform differently.
The relevant point (which is unrelated to your 'rewording'
of it) was already established in the article you replied to.
More generally, the macroscopic Maxwell equations:
div D = rho; curl H - dD/dt = J
div B = sigma; curl E + dB/dt = -K
(with or without K = 0, sigma = 0), along with the force & power
laws:
P = J.E + K.K; F = rho E + J x B + sigma B - K x D
are invariant under a HUGE group, which includes all of
the following: Galilei, Poincare(c), Poincare(c') as
subgroups, where Poincare(c') is Poincare with an
invariant velocity c' other than c = 1/sqrt(m0 e0).
The most general homogenous, linear group that leaves
all these equations invariant. such that the sources
transform linearly, are:
x -> L x; J -> w L J; F -> w L F L^T; P = |w|^2 Q P
with:
(x^0,x^1,x^2,x^3) = (t,x,y,z)
(J^0,J^1,J^2,J^3) = (rho,Jx,Jy,Jz) + q (sigma,Kx,Ky,Kz)
F^10 = Dx + q Bx, etc.
F^21 = Hz - q Ez, etc.
(P_0, P_1, P_2, P_3) = (-P, Fx, Fy, Fz)
Q^b_a = 1/6 e_{almn} L^l_t L^m_s L^n_t e^{brst}
e_{0123} = 1 = e^{0123}, both anti-symmetric
L in GL(4,R).
w = A + Bq; |w|^2 = A^2 + B^2 |q|^2
which includes Lorentz(c), Lorentz(c') & Galilei as subgroups
as well as rotations in the (1,q) plane.
The requirement for the invariance of F_a dx^a is that
e_{abcd} L^a_r L^b_s L^c_t L^d_u = e_{rstu}/|w|^2
Under the subgroup Poincare(c), the relations
D = e0 E; B = m0 H
can be invariantly maintained. The 'little-group' for this set
of relations is just the invariance group of microscopic
Maxwell's equations (in the form that we know them as). More
generally, (Poincare(c') and Galilei), the relations will only
hold in at most one frame -- which would be the "Ether Frame".
This includes the interesting cases of Poincare(c') where
you have both a Relativistic theory, but still an "Ether
Frame" which is uniquely singled out by the wave equation.
The interesting case is Poincare(c') with
0 < |c' - c| < epsilon,
where epsilon is small enough to fly under the
Michelson-Morley radar.
The relations between (E,H) and (D,B) for the Galilei
invariance was worked out in the previous article.
It would be interesting to work out the cases Poincare(c')
with a distinguished frame (D = e0 E; B = m0 H), but with
c' != c, to see what the resulting relations look like in
terms of the absolute speed V.
Thomas Larsson
>
> Physics is not mathematics. One doesn't prove theorems or derive oranges
> from apples. One figures out how the world is put together with the help of
> mathematical insight, not mere blundering about with powerful tools.
>
But you *do* prove theorems in physics! The difference is that in math, axioms
are always right by definition, as long as they are internally consistent. In
physics, perfectly consistent axioms can still be wrong, if they lead to
predictions which disagree with experiments.
Danny Ross Lunsford
> But you *do* prove theorems in physics! The difference is that in math,
axioms
> are always right by definition, as long as they are internally consistent.
In
> physics, perfectly consistent axioms can still be wrong, if they lead to
> predictions which disagree with experiments.
I certainly didn't mean to incite what could be a war, but I don't agree at
all. Any theorem proving is the role of math. Physics is using the insight
gained from *useful* math to the real world. What is useful? "I may not know
what art is, but I know what I like!!"
-drl
Danny Ross Lunsford
news:u8g6qn8...@corp.supernews.com...
> Yes. A fellow named Maxwell, in 1861. "On Physical Lines of Force."
> Except he also derived div H = 0, rather than assuming it. While on the
way
> to deriving "Maxwell's equations," and identifying light as an
> electromagnetic wave. He started with Faraday's laws and Newton's laws of
> motion.
> ubi dubium ibi libertas.
Wow, nice signature!
You don't derive "div B = 0". Maxwell's eqns are
Fmn,n = 0 (4 eqns)
F[mn,p] = 0 (4 eqns - [ ] means, "totally antisymmetrize", that is, it's a
dual form.)
These lead to
curl B - E dot = 0
div E = 0
curl E + B dot = 0
div B = 0
These are 8 eqns for 6 unknowns. But, assuming the curl eqns are true,
d/dt (div E) = 0
d/dt (div B) = 0
and because of linearity, if true at t=t0, they are always true. So, that
div eqns are *initial conditions* for the 1st order system of PDEs that are
Maxwell's eqns. So, now we have 6 eqns for 6 fields and we get a
well-determined system - but only by including the initial conditions. This
is entirely analogous to the ongoing discussion here lately about "Dirac's
sea" and boundary conditions.
(BTW this also shows why a uniform background charge distribution is
compatible with Maxwell's eqns.)
The same thing carries over to higher dimensions. Going to 5 dimensions, we
have from the 1st set 1 new eqn, and from the second, binom(5,3) -
binom(4,3) = 6 new eqns, or seven new eqns for (5-1)=4 new fields. So now
there are 15 eqns for 10 fields, so 5 are redundant, and can be regarded as
initial data on the new coordinate. Explicitly, in 3+2 space Maxwell's eqns
are
1st set) Fab,b = 0 => curl B - d/dt E - d/du S = 0
div E - d/du s = 0
div S + d/dt s = 0 (5 eqns)
=> d/du (div S + d/dt s) = 0 (1 redundant)
2nd set) F[ab,c] = 0 => curl E + d/dt B = 0
curl S + d/du B = 0
-d/dt S - grad s + d/du E = 0
div B = 0 (10 eqns)
=> d/du(div B) = 0
d/du(curl E + d/dt B) = 0 (4 redundant)
so there are (15 eqns)
with - ( 5 redundant)
for = (10 fields)
and we get again a well-determined system. On up it goes. Going backwards,
we have 3-d Maxwell's eqns.
1st set) curl B = 0
2nd set) div B = 0
Here we run into Helmholtz' famous theorem - a field with both div and curl
= 0 everywhere is constant. So there is no non-trivial free field theory in
less than four dimensions. Oddly, the more complex gravitational field does
have a non-trivial, if rather boring, structure (1 curvature variable).
So, in N dimensions for N>3, there are
N + N(N-1)(N-2)/6 equations for
N(N-1)/2 fields with
1 + (N-1)(N-2)(N-3)/6 redundant equations.
The last funny number comes directly from counting the number of times the
operator d/dxi for a specific i appears in the eqns. There are N choices for
i - only one eqn in the first set lacks d/dxi, while in the second, we have
binom(N-1,3) eqns that do.
-drl
Danny Ross Lunsford
news:u8g6qn8...@corp.supernews.com...
...followup to a previous post...
One can ask if there is a consistent system a totally antisymmetric F like:
Fmnp has binom(N,3) components satisfying
Fmnp,p = 0 binom(N,2) equations
F[mnp,q] = 0 binom(N,4) equations
which will only work if N>4. For five dimensions, we again have 10 fields,
with 10 eqns in the 1st set, and 5 in the 2nd. The operator dxi for a
specific i is missing from binom(N,2)-binom(N-1,2) eqns in the 1st set (to
see this, note that in order for dxi to appear in the first set, neither
free index can be i, so there are binom(N-1,2) equations *with* dxi) - and
from binom(N-1,4) eqns in the 2nd. So there are
binom(N,2) + binom(N,4) equations total
binom(N,2) - binom(N-1,2) + binom(N-1,4) equations missing a specific dxi,
which are redundant
The difference is
binom(N,3)
which is just the number of fields! So yes, this is a well-determined system
and constitutes a natural generalization of Maxwell's equations for N>4. Of
course, one can then look at Fs with 4 totally antsymmetric indices, etc.
etc. We get a hierarchy of theories, only one of which exists for N=4. These
will be dual to each other in a specific dimension, and in even dimensions,
there will be a self-dual structure which should be very interesting to
study.
Specifically for N=5:
Fabc,c = 0 10 eqns
F[abc,d] = 0 5 eqns
with 5 redundant - so 10 eqns for 10 unknowns.
N=6: 20 fields, 1st set 15 eqns, 2nd set 15 eqns for 30 total, with
(6,2)-(5,2)+(5,4)=10 redundant.
What do you imagine the spin properties of these fields will look like?
There will be a potential theory with a totally antisymmetric potential with
for example Fabc = W[ab,c] and gauge invariance will be Wab -> Wab + Va,b -
Vb,a and in general Wa1..aN -> Wa1..aN + V[a1..aN-1,aN].
It also occurs to me you can lump all of them together for a particular
dimension under the aegis of the extant Clifford algebra, with the potential
coming from the even sub-algebra.
-drl
Paul Colby
> It's somewhat misleading to say that the homogenous Maxwell
> equations are invariant under Galilean transformations. It's
> perfectly true - but it's misleading nonetheless, because
> the homogenous Maxwell equations are actually invariant under
> ALL transformations. "Galilean" is a red herring which only
> serves to distract attention from what's really going on.
>
> In modern terminology: the homogenous Maxwell equations
>
> dF = 0
>
> are invariant under all diffeomorphisms of spacetime, since
> the operator d commutes with the action of diffeomorphisms
> on differential forms.
>
> Of course the other Maxwell equations
>
> d*F* = J
>
> involve the Hodge star operator, and are not diffeomorphism-invariant.
Is it also misleading to say that the F in dF=0 is the
*same* F as in d*F* = J? The first is in terms of E and B while
the second is H and D. These are not the same fields. Isn't it
that in free space mu0 H = B and D = epsilon0 E for all
inertial observers that implies special relativity?
Regards
Paul Colby
John Baez
Danny Ross Lunsford <antima...@worldnet.att.net> wrote:
>So yes, this is a well-determined system
>and constitutes a natural generalization of Maxwell's equations for N>4. Of
>course, one can then look at Fs with 4 totally antsymmetric indices, etc.
>etc. We get a hierarchy of theories, only one of which exists for N=4. These
>will be dual to each other in a specific dimension, and in even dimensions,
>there will be a self-dual structure which should be very interesting to
>study.
Unless I'm confused, you're talking here about "p-form electromagnetism",
which is important in string theory, and has also been the topic of
my quantum gravity seminar this quarter, where I've been talking about
its relation to n-categories.
In differential form notation, p-form electromagnetism goes like this:
You start with a "vector potential" A which is a p-form. You
form the "electromagnetic field"
F = dA
which is a (p+1)-form. This automatically satisfies
dF = 0
but you also impose the equations
*d*F = J
where J is a (p-1)-form called the "current".
For p = 1 this is just Maxwell's equations. For higher p you
get analogues of Maxwell's equations where the current is naturally
produced not by point particles but by (p-1)-branes, that is,
by (p-1)-dimensional surfaces which trace out p-dimensional
worldsheets in spacetime.
(I wish string theorists would call (p-1)-branes "p-branes" - it
would make the above paragraph more elegant - but above I'm using
their numbering conventions.)
There is a lot of cool stuff about how these higher analogues
of Maxwell's equations relate to n-category theory, and how they
generalize to higher analogues of Yang-Mills equations, but this
is top secret.
Danny Ross Lunsford
> Unless I'm confused, you're talking here about "p-form electromagnetism",
> which is important in string theory, and has also been the topic of
> my quantum gravity seminar this quarter, where I've been talking about
> its relation to n-categories.
>
> In differential form notation, p-form electromagnetism goes like this:
>
> You start with a "vector potential" A which is a p-form. You
> form the "electromagnetic field"
>
> F = dA
>
> which is a (p+1)-form. This automatically satisfies
>
> dF = 0
>
> but you also impose the equations
>
> *d*F = J
>
> where J is a (p-1)-form called the "current".
Right - forms are just totally antisymmmetric tensors. I find the latter
somewhat more natural to work with when a metric is at hand, because you can
see the structure by inspection.
> generalize to higher analogues of Yang-Mills equations, but this
> is top secret.
I've already started on that :)
-drl
Thomas Larsson
> In differential form notation, p-form electromagnetism goes like this:
>
> You start with a "vector potential" A which is a p-form. You
> form the "electromagnetic field"
>
> F = dA
>
> which is a (p+1)-form. This automatically satisfies
>
> dF = 0
>
> but you also impose the equations
>
> *d*F = J
>
> where J is a (p-1)-form called the "current".
>
> of Maxwell's equations relate to n-category theory, and how they
> generalize to higher analogues of Yang-Mills equations, but this
> is top secret.
Let me guess. You formulate Yang-Mills in terms of Wilson loops. It
can be formulated as a gauge theory with zero curvature in loop space.
Polyakov did something like this around 1980. 2-Yang-Mills is then a
loop space gauge theory with non-zero curvature, etc.
Urs Schreiber
[...]
> Unless I'm confused, you're talking here about "p-form electromagnetism",
> which is important in string theory, and has also been the topic of
> my quantum gravity seminar this quarter, where I've been talking about
> its relation to n-categories.
I have looked at the quantum gravity seminar notes by Miguel
Carrión Álvarez, but p-form electromagnetism seems not to have
been included yet. Will there be notes available online?
(Maybe I have not looked close enough?) I'd be interested.
> In differential form notation, p-form electromagnetism goes like this:
>
> You start with a "vector potential" A which is a p-form. You
> form the "electromagnetic field"
>
> F = dA
>
> which is a (p+1)-form. This automatically satisfies
>
> dF = 0
>
> but you also impose the equations
>
> *d*F = J
>
> where J is a (p-1)-form called the "current".
>
> For p = 1 this is just Maxwell's equations. For higher p you
> get analogues of Maxwell's equations where the current is naturally
> produced not by point particles but by (p-1)-branes, that is,
> by (p-1)-dimensional surfaces which trace out p-dimensional
> worldsheets in spacetime.
For vanishing current J I look at all p-form electromagnetism
as a special case of the massless Kaehler equation, i.e. of
the equation
(1) (d + *d*) F = 0
for F any section of the exterior bundle (i.e. an
inhomogeneous form).
Source-free p-form electromagnetism results from this general
equation by imposing a further restrictions on F, namely
(2) N F = p (where N is the number operator) .
But imposing this condition on the original Kaehler equation
destroys many of its nice features. For example, the Kaehler
equation on unrestricted form sectors allows a reformulation
as a time evolution equation, which may be straigtforwardly
integrated for given boundary conditions. This is not possible
when restricting it to p-forms. But there is a way out, which
solves this problem and at the same time has a powerful
generalization:
Instead of (2) impose
(3) (d - *d*) F = 0 .
in addition to (1). By adding and substracting (1) and (3) it
is immediate that
(4) d F = 0
(5) d*F = 0 .
Since d and *d* strictly map p-form sectors of different p
into each other, these two equations must hold *in every
p-form sector seperately*. Hence solving (1) and (3) gives
vacuum solutions to *all* p-form electromagnetism theories at
once! Before you complain that I have just reformulated the
problem of solving (4) and (5) in each sector, without
achieving any simplification to this task, let me point out
that there is indeed a major simplification:
Observation: Consider any theory defined by N operators D_i
that satisfy the relation
(6) {D_i, D_j} = 2 delta_ij L
where {.,.} is the anticommutator and (by definition)
L = (D_i)^2 , for i in {1,...,N},
and where physical states F are defined by
D_i F = 0 , for i in {1,...,N}.
>From any solution G to the *single* equation
D_1 G = 0
one obtains a solution to all N constraints by the following
algorithm:
(a) Choose any i in {1,...,N} such that D_i G <> 0. If there
is no such i we are done.
(b) Replace G by its image under D_i, i.e. set
G <- D_i G,
and continue with (a).
It is easlily seen, by using (6), that the resulting G of this
algorithm solves all the equations
D_i G = 0, i in {1,...,N} .
Now for p-form electromagnetism set
(7) D_1 = d + *d*
(8) D_2 = i(d - *d*),
which satisfies (6) for N=2. For simplicity assume that there
is a coordinate patch in which the connection coefficients are
time independent. Then one can write
(d + *d*) F = 0
<=> yn (@n + 1/4 omega_nab (ya yb - Ya Yb) ) F = 0
<=> @0 F = - y0 (ym @n - 1/4 omega_nab (ya yb - Ya Yb)) F
<=> @0 F = H F
<=> F = exp(x0 H) F(x0 = 0)
with
yn = (e/\)n - (e->)n
Yn = (e/\)n + (e->)n
omega_nab = the Levi-Civita connection components
H = - y0 (ym @n - 1/4 omega_nab (ya yb - Ya Yb)) .
Hence D_1 F = 0 is (formally) emmediatly integrated. It may
happen that the result
exp(x0 H) F(x0 = 0)
by accident also satisfies
D_2 exp(x0 H) F(x0 = 0).
If it does we are done. If not choose instead
G = D_2 exp(x0 H) F(x0 = 0) .
This will do since
D_2 G = (D_2)^2 F = L F = (D_1)^2 F = 0 .
Thus G solves (1) and (3) and, by the above consideration,
every p-form sector of G is a solution to the respective
(source free) p-form electromagnetism.
But one can do more:
Suppose there are more D_i around than just (7) and (8), so
that (6) is satisfied for N > 2. This happens if the
underlying geometry admits complex structures on the tangent
bundle that satisfy one of the normed division algebras (i.e.
Kaehler, Hyperkaehler,... geometry). Then, acting with any one
of these additional operators on the above solution G gives
yet another solution in every p-form sector.
In fact, one can do even more:
Suppose you have used up all the complex structures and want
still more solutions. Then look for Killing-Yano tensors on
spacetime that are not complex structures. These give even
more D_i operators, but now the algebra will no longer close
on L = (D_1)^2 but will have "central charges":
{D_i, D_j} = 2 delta_ij Z_i
with
Z_1 = L
and all Z_i are Casimir operators of the algebra. (The Z_i
will generically have the form c1 L + c2 L', where L' is the
exterior Laplace operator of a metric "dual" to the original
metric.)
Just like D_1 + D_2 = 2d gives a nilpotent operator one may
generically construct two nilpotent operators by lineasr
combinations of the above D_i. This allows to put the above
diagonal algebra in "polar" form, with nontrivial bracket
{d_i, del_j} = delta _ij Z_i.
Find simultaneous eigenstates
F = |z1,z2,...zN>
of the Casimirs. One can assume these to be "vacuua" with
respect to all del_i:
del_i |z1,z2,...zN> = 0,
because if they are not, one applies del_i to them until they
are.
This then gives us entire "supermultiplets" (I am still
talking about ordinary EM (well, ordinary p-form EM in
arbitrary dimensions)) of states by acting on these vacuua
with the "creation operators"
d_i1 di_2 ... |z1,z2,...zN> .
All of these with z1 = 0 are (again in every p-form sector)
solutions to p-form electromagnetism.
I realize that a possible objection to the above is, that I
merrily generate further and further solutions, but without
proper control on the boundary conditions. This deserves
further thinking on my part. But it is nevertheless
remarkable, I think, that all p-form electromagnetisms (on a
given spacetime) are intimitely related and that one can map
solutions from one to the other in many ways. This is why I,
personally, do not think in terms of different types of EM,
but simply in terms of the massless Kaehler equation and its
higher symmetric solutions, which subsumes all of it.
> There is a lot of cool stuff about how these higher analogues
> of Maxwell's equations relate to n-category theory, and how they
> generalize to higher analogues of Yang-Mills equations, but this
> is top secret.
I'd be interested to know if and how the Kaehler-equation
point of view carries over to n-Category theory.
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>John Baez wrote:
>> Unless I'm confused, you're talking here about "p-form electromagnetism",
>> which is important in string theory, and has also been the topic of
>> my quantum gravity seminar this quarter, where I've been talking about
>> its relation to n-categories.
>I have looked at the quantum gravity seminar notes by Miguel
>Carrión Álvarez, but p-form electromagnetism seems not to have
>been included yet. Will there be notes available online?
Probably not!
Production of online quantum gravity seminar notes fizzled out in the
middle of last year, before we even got to the exciting stuff: getting
the theory of 3d Riemannian quantum gravity from the category of
representations of the quantum group SU_q(2). This material will
eventually be put online, and eventually become part of our book,
"Notes on Quantum Gravity".
Since January I've been lecturing on categorified gauge theory and
p-form electromagnetism. I plan to keep doing this until classes end in
June. This will *not* become part of the quantum gravity book; instead,
some of my students will be doing theses on this topic. Alissa Crans is
already writing her thesis on Lie 2-algebras and Lie 2-groups, and I
hope to persuade Miguel to do his thesis on the categorified Yang-Mills
equations. He and Toby have already posted lots of articles on
categorified bundles, which are part of the mathematical infrastructure
of this subject.
Next year I'll probably start lecturing on general relativity, the
problem of time, and constrained dynamical systems. This should show
up online sometime, and will become part of the book.
I'll try to think about what you wrote and see how it connects to
n-categories. It may take a while... but thanks for writing it!
Urs Schreiber
news:a6u158$md6$1...@glue.ucr.edu...
[...]
> Since January I've been lecturing on categorified gauge theory and
> p-form electromagnetism. I plan to keep doing this until classes end in
> June. This will *not* become part of the quantum gravity book; instead,
> some of my students will be doing theses on this topic.
Does p-form electromagnetism by itself have a direct connection to
quantum gravity, or only via its n-category aspect? Where does
p-form electromagnetism turn up in "practice"? Probably in string
theory related settings, right? I seem to know one example, namely
11-d supergravity with vanishing Chern-Simons term. The bosonic
Lagrangian
L = * R + c1 F^*F + c2 A^F^F
looks like 11-d gravity coupled to 3-form electromagnetism when
A^F^F = 0.
[...]
> Next year I'll probably start lecturing on general relativity, the
> problem of time, and constrained dynamical systems. This should show
> up online sometime, and will become part of the book.
BTW, did you see my recent answer to your question about local
observables in canonical gravity? Was it too confused? I am in
a strange hybrid state with respect to these matters: I have read a
couple of things and thought about them but did not have sufficient
exchange with others to properly thermalize my ideas with the
academic environment.
> I'll try to think about what you wrote and see how it connects to
> n-categories. It may take a while... but thanks for writing it!
Well, thanks for reading it!
Danny Ross Lunsford
> For p = 1 this is just Maxwell's equations. For higher p you
> get analogues of Maxwell's equations where the current is naturally
> produced not by point particles but by (p-1)-branes, that is,
> by (p-1)-dimensional surfaces which trace out p-dimensional
> worldsheets in spacetime.
Is this necessary? In hydrodynamics one has similar issues, "velocity
potentials" - these generalize to higher dimensions as antisymmetric
"sources", and in fact one can refine it down to identifying the sources
with "vortex filaments" - however the focus is on the flow, and the flow is
still there even when the vorticity is zero. The idea of "stringy sources"
just seems artificial in any context I can think of, while the idea of a
generalized kind of vorticity doesn't seem that way - in fact a continuous
spin distribution would behave like that, no?
In fact for p=2, the field is
Fabc = W[ab,c]
with gauge invariance
Wab -> Wab + V[a,b]
and generalized Lorentz condition
Wab,b = 0
so that
d'Alembert (Wab) = 0
and the covariant gauges are restricted by
d'Alembert (Va) = (Vb,b),a
So the p=2 potential and its gauge, behave exactly like a p=1 free field
with an *unrestricted* gauge. If you in turn restrict this gauge by Vb,b = 0
you would get ultimately get fourth-order equations for the gauge vector Vb.
This suggests a kind of bootstrapping of these theories that ultimately
relate back to simple sources at base obeying higher order equations.
Something similar happens in the theory of elasticity, when looking at the
elastic properties of stressed tubes, shells, etc. - one doesn't look at
this as elastic "brane" theory, just ordinary elasticity on a more complex
geometry. Mabye I'm rambling now.
-drl
Aaron Bergman
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> Where does
> p-form electromagnetism turn up in "practice"? Probably in string
> theory related settings, right?
One might thing, but, it turns out the RR fields in string theory are
actually elements of K-theory (at the very least) and not just p-forms.
> I seem to know one example, namely
> 11-d supergravity with vanishing Chern-Simons term. The bosonic
> Lagrangian
>
> L = * R + c1 F^*F + c2 A^F^F
>
> looks like 11-d gravity coupled to 3-form electromagnetism when
> A^F^F = 0.
Except for the fact that the quantization condition for F = dA isn't
that it's in integral cohomology.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
Toby Bartels
>John Baez wrote:
>>In modern terminology: the homogenous Maxwell equations dF = 0
>>are invariant under all diffeomorphisms of spacetime, since
>>the operator d commutes with the action of diffeomorphisms
>>on differential forms.
>>Of course the other Maxwell equations d*F* = J
>>involve the Hodge star operator, and are not diffeomorphism-invariant.
>Is it also misleading to say that the F in dF=0 is the
>*same* F as in d*F* = J? The first is in terms of E and B while
>the second is H and D. These are not the same fields. Isn't it
>that in free space mu0 H = B and D = epsilon0 E for all
>inertial observers that implies special relativity?
One way to look at it is: It is the same F in both equations,
and the constitutive relations are hidden in the *.
An invariant equation involving D and H would be dM = J,
where D and H are combined to make M in a similar way
(more precisely, a dual way) as E and B combine to make F.
Then to say M = *F for a certain metric is the constitutive relations.
Then dM = J is also invariant under all transformations.
Of course, to make these equations dF = 0 and dM = J really work,
you need to treat things as differential forms, not vector fields.
An explanation of this appeared recently on this newsgroup
and is on the <sci.physics.research> archive at
<http://www.lns.cornell.edu/spr/2002-02/msg0039236.html>.
This post requires familiarity with forms to understand
and is actually just a small part of a large debate,
so there is a lot of extraneous material
that you'll have to strip out if you read it.
But it shows how the metric on spacetime --
and with it any restriction on the transformations
under which the equations are invariant --
can be relegated only to the constitutive relations.
-- Toby
to...@math.ucr.edu
John Baez
Danny Ross Lunsford <antima...@worldnet.att.net> wrote:
>John Baez wrote:
>> For p = 1 this is just Maxwell's equations. For higher p you
>> get analogues of Maxwell's equations where the current is naturally
>> produced not by point particles but by (p-1)-branes, that is,
>> by (p-1)-dimensional surfaces which trace out p-dimensional
>> worldsheets in spacetime.
>Is this necessary?
It's not necessary, but it follows the Tao of mathematics and leads to
very beautiful things. The interaction between a point particle and the
electromagnetic field is described by a term in the action which is the
integral of a 1-form over the 1-dimensional worldline of the particle -
the simplest sort of way to calculate a number from a curve. Replacing
1 by higher numbers, we get p-form electromagnetism interacting with
branes having p-dimensional worldsheets. I don't know if it's real-world
physics, but it's pretty enough that I think it's fun to see where it leads.
Urs Schreiber
[...]
> In fact for p=2, the field is
>
> Fabc = W[ab,c]
>
> with gauge invariance
>
> Wab -> Wab + V[a,b]
>
> and generalized Lorentz condition
>
> Wab,b = 0
>
> so that
>
> d'Alembert (Wab) = 0
>
> and the covariant gauges are restricted by
>
> d'Alembert (Va) = (Vb,b),a
>
> So the p=2 potential and its gauge, behave exactly like a p=1 free field
> with an *unrestricted* gauge.
I follow that, but I do not quite understand the final sentence. You
want to identify the 2-form W, being the potential of a 3-form field
strength, with a 2-form field strength, right? But d'Alembert(W) = 0
is not the proper free field equation when one thinks of W a Faraday
tensor: d'Alembert(W) = 0 does not imply W satisfies Maxwell's
equations dW = 0 and delW = 0 (not on *pseudo*-Riemannian
manifolds, at least, on Riemannian manifolds it does!).
> This suggests a kind of bootstrapping of these theories that ultimately
> relate back to simple sources at base obeying higher order equations.
> Something similar happens in the theory of elasticity, when looking at the
> elastic properties of stressed tubes, shells, etc. - one doesn't look at
> this as elastic "brane" theory, just ordinary elasticity on a more complex
> geometry. Maybe I'm rambling now.
I do not know enough about the theory of elasticity, but I agree with this
"bootstrapping" idea: all p-form EMs are intimately related for different p.
In fact, one can formulate p-form EM for all p at once as one single
theory (and solve it by solving one single first order equation):
Assume some d-dimensional pseudo-Riemannian manifold. Let F'
and j' be general *inhomogeneous* forms, i.e.
F' = F(0) + F(1)_m dx^m + F(2)_mn dx^m dx^n + ...
j' = j(0) + j(1)_m dx^m + j(2)_mn dx^m dx^n + ...
and impose the generalized Maxwell equations:
dF' = 0, del F' = j' .
Note that it is important here that one writes these as two different
equations, since (d+del)F' = j' is not equivalent to these two
equations when F' and j' are inhomogeneous forms. On the other
hand it is easily checked that the above generalized Maxwell
equations have to hold on each p-form sector seperately:
(1) d F(p) = 0, del F(p) = j(p-1) .
So what is the point of looking at inhomogeneous forms?
The point is that things simplify significantly when working with an
inhomogeneous potental A':
A' = A(0) + A(1)_m dx^m + A(2)_mn dx^m dx^n + ...
with
F' = d A' .
To see this let k' (also inhomogeneous) be the "current potential"
del k' = j' .
(Since j' is coclosed, i.e. del j' = 0, such a potential k' always exists on
starshaped regions by Poincare's lemma.)
I claim that all p-form EM expressed by equation (1) is solved by
a single inhomogeneous-form equation for the potential A', namely:
(2) (d+del) A' = k' .
Note that this equation cannot be taken apart, it mixes p-form
sectors among each other. In fact, this is the massive Kaehler
equation for variable mass k' .
The easy proof that (2) implies (1) is:
d F' = d d A' = 0
del F' = del d A' = del (k' - del A') = del k' = j' .
In the last line equation (2) has been used.
Hence, instead of solving the second order equation
(d+del)^2 A = d'Alembert A = 0
as often done in electromagnetism, it is sufficient to solve the *first
order* equation (2). Furthermore, this can just as well be done for all
p-form sectors a once!
Thomas Larsson
news:<4b8cc0a6.02031...@posting.google.com>...
> ba...@galaxy.ucr.edu (John Baez) wrote in message
news:<a6rnaf$t5b$1...@glue.ucr.edu>...
> > There is a lot of cool stuff about how these higher analogues
> > of Maxwell's equations relate to n-category theory, and how they
> > generalize to higher analogues of Yang-Mills equations, but this
> > is top secret.
>
> Let me guess. You formulate Yang-Mills in terms of Wilson loops. It
> can be formulated as a gauge theory with zero curvature in loop space.
> Polyakov did something like this around 1980. 2-Yang-Mills is then a
> loop space gauge theory with non-zero curvature, etc.
Once I invented a lattice version of 2-Yang-Mills theory:
T A Larsson
"p-cell gauge theories, manifold space and multi-dimensional
integrability"
Mod Phys Lett A 5 (1990) 255-264
Let me explain this model, because I still think it is rather cool,
although I never did anything with it. Recall that in ordinary
lattice gauge theory, the amplitude for parallel transport along
a link in the k:th direction is the matrix
U_k = exp(i a A_k),
where a is the lattice spacing and A_k is the k:th component of the
gauge potential. If you transport a particle along a link and then
back, nothing has happened, so the same link with the opposite
orientation has the matrix U_k^{-1}. The curvature is given by the
holonomy around a plaquette, which is the smallest loop you can
construct:
U_1 U_2 U_1^{-1} U_2^{-1} = exp(i a^2 F_12).
The action is
\sum_{plaquettes} tr UUUU + h.c. = const + a^2 \int F^2 + O(a^3).
We see that in the limit that the lattice spacing a --> 0, this
becomes the Yang-Mills action, at least formally.
There is a gauge symmetry associated to each vertex, and the gauge
invariant observables are Wilson loops, i.e. the product of matrices
around a closed loop.
Instead of putting matrices on links, it is natural to put four-index
quantites on plaquettes. Associate a vector space V to each link,
and an element U_ij \in V_i@V_j to a plaquette in the ij-plane.
U_ij is in fact an element in V@V@...@V = V^{@n} (n = #dimensions),
which acts as the identity except on the i:th and j:th factors.
In fact, and this is something which I missed in my paper, you need
to put two independent variables on each plaquette. You can interpret
U_ij as the amplitude for parallel transport of a string element
across the NE diagonal. It is then clear that you need four such
amplitudes, for the four different directed diagonals. However, only
two amplitudes are independent, since U_ij(SE) = U_ij^{-1}(NW) and
U_ij(SW) = U_ij^{-1}(NE).
The infinitesimal holonomy is
UUUUUU = U_12 U_13 U_23 U_12^{-1} U_13^{-1} U_23^{-1}
and the action reads
\sum_{elementary cubes} tr UUUUUU + 7 more terms.
The eight terms corresponds to the cube's eight directed diagonals.
Just as the two terms in ordinary lattice gauge theory can be
interpreted as parallel transport of a particle around the plaquette,
in the clockwise and counter-clockwise directions, the eight terms
here rotate a string piece around the cube's diagonals.
There is a gauge symmetry associated to each link, and the gauge
invariant observables are Wilson surfaces, i.e. the product of 4-index
objects around a closed, 2D surface. The natural continuum formulations
of this model are in terms of loop or membrane variables (non-zero and
zero curvature, respectively). However, locality is not manifest in
the continuum formulations, but it is clear on the lattice that the
model is perfectly local; the action is a sum over elementary cubes.
The zero-curvature condition is quite interesting. In the spatially
homogeneous case (objects depend on orientation but not on location),
zero 1-curvature becomes U_1 U_2 U_1^{-1} U_2^{-1} = 0, i.e. U_1 and
U_2 commute. zero 2-curvature becomes
U_12 U_13 U_23 = U_23 U_13 U_12,
which is the Yang-Baxter equation, of paramount importance to the
theory of integrable lattice models in 2D.
It is immediate how to generalize this model to p-Yang Mills theory
on the lattice. The zero p-curvature condition is known as the
p-simplex equation. It is sufficient to construct integrable lattice
models in p dimensions, but unfortunately no interesting solutions
to it are known. Well, some solutions are known, but you need a
continuum of solutions depending on variables like temperature or a
magnetic field to be able to compute critical exponents.
Urs Schreiber
news:abergman-2F242C...@news.bellatlantic.net...
> In article <a6uvqi$14k0$1...@rs04.hrz.uni-essen.de>,
> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> > Where does
> > p-form electromagnetism turn up in "practice"? Probably in string
> > theory related settings, right?
> One might think, but, it turns out the RR fields in string theory are
> actually elements of K-theory (at the very least) and not just p-forms.
Could you briefly indicate what properties distinguish RR fields in
string theory (or elements of K-theory for that matter) from p-forms?
> > I seem to know one example, namely
> > 11-d supergravity with vanishing Chern-Simons term. The bosonic
> > Lagrangian
> >
> > L = * R + c1 F^*F + c2 A^F^F
> >
> > looks like 11-d gravity coupled to 3-form electromagnetism when
> > A^F^F = 0.
>
> Except for the fact that the quantization condition for F = dA isn't
> that it's in integral cohomology.
Aha, so what is the correct quantization condition?
Squark
What do you mean? F = dA is exact and thus vanishes in the cohomology.
Are you referring to the case we have a nontrivial bundle and then the
formula F = dA isn't valid globally (if we consider the forms here as
having values in a trivial bundle)? I guess you're talking about
something rather more exotic, though, if F isn't in the integral
cohomolohy (the curvature of a connection has to be there, as far as I
remember)?
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and use "com" rather than
"exe")
Urs Schreiber
news:a74d8j$pl2$1...@glue.ucr.edu...
[...]
> Of course, to make these equations dF = 0 and dM = J really work,
> you need to treat things as differential forms, not vector fields.
> An explanation of this appeared recently on this newsgroup
> and is on the <sci.physics.research> archive at
> <http://www.lns.cornell.edu/spr/2002-02/msg0039236.html>.
In this archived post it reads:
[...]
> Similarly, the electric and magnetic fields
> are combined in the Faraday 2form F = E ^ dt + B.
> If a small body carrying a unit charge
> takes 2 different paths through spacetime with the same endpoints,
> then the direction of proper time, together with a choice
> of which path is first and which path is second,
> will orient any surface bounded by the 2 paths;
> you integrate F over this surface to get
> the difference in action between the 2 paths.
[...]
I very much like your discussion. I want to add a visualization
of F that is maybe a little more intuitively accessible than the above
characterization via the action, but it requires an additional law,
namely the Lorentz force law:
f = v-> F
where f is the Lorentz force, v is the proper velocity vector of
a charged particle, "->" is contraction, and I have suppressed a
proportionality factor.
This formula allows us to visualize F as a family of weighted 2d
surfaces in 4d spacetime that locally coincide with the surface
in which the proper velocity vector of a test particle rotates.
Pertti Lounesto
>Paul Colby wrote:
>
>>John Baez wrote:
>>
>>>In modern terminology: the homogenous Maxwell equations dF = 0
>>>are invariant under all diffeomorphisms of spacetime, since
>>>the operator d commutes with the action of diffeomorphisms
>>>on differential forms.
>>>
>>>Of course the other Maxwell equations d*F* = J
>>>involve the Hodge star operator, and are not diffeomorphism-invariant.
>>>
How is that? To me it seems that also *d*F = J is invariant
under diffeomorphisms. Only, the sum (d+*d*)F = J is not
diffeomorphism invariant, but only conformally invariant.
The reason is that dF = 0 and *d*F = J do not transform the
same way under diffeomorphisms, that is, *d*F = J has another
rule of invariance than dF = 0 (but still has one). The sum
(d+*d*)F = J then must obey both the rules, which restricts
us to conformal invariance.
>>Is it also misleading to say that the F in dF=0 is the
>>*same* F as in d*F* = J? The first is in terms of E and B while
>>the second is H and D. These are not the same fields. Isn't it
>>that in free space mu0 H = B and D = epsilon0 E for all
>>inertial observers that implies special relativity?
>>
>
>One way to look at it is: It is the same F in both equations,
>and the constitutive relations are hidden in the *.
>
This is the usual practice, but one can also separate the
constitutive relations, which take care of the properties of
the medium, and the Hodge dual, which takes care of
going to the dual with respect to the Lorentzian metric.
I think it is clearer to separate the two actions (although
one seldom does that).
Aaron Bergman
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> "Aaron Bergman" <aber...@princeton.edu> schrieb im Newsbeitrag
> news:abergman-2F242C...@news.bellatlantic.net...
>
> > In article <a6uvqi$14k0$1...@rs04.hrz.uni-essen.de>,
> > "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
>
> > > Where does
> > > p-form electromagnetism turn up in "practice"? Probably in string
> > > theory related settings, right?
>
> > One might think, but, it turns out the RR fields in string theory are
> > actually elements of K-theory (at the very least) and not just p-forms.
>
> Could you briefly indicate what properties distinguish RR fields in
> string theory (or elements of K-theory for that matter) from p-forms?
There is a map via the chern character from K-theory to even (you can
get odd with a little more effort) cohomology with rational
coefficients. Thus, the only difference is torsion classes.
> > > I seem to know one example, namely
> > > 11-d supergravity with vanishing Chern-Simons term. The bosonic
> > > Lagrangian
> > >
> > > L = * R + c1 F^*F + c2 A^F^F
> > >
> > > looks like 11-d gravity coupled to 3-form electromagnetism when
> > > A^F^F = 0.
> >
> > Except for the fact that the quantization condition for F = dA isn't
> > that it's in integral cohomology.
>
> Aha, so what is the correct quantization condition?
G/2pi - p_1 / 4 is in H^4(M,Z).
Arkadiusz Jadczyk
<antima...@worldnet.att.net> wrote:
>> But that is indeed an interesting question. Seems that based on this
>> argument, quantum mechanics is apparently derivable from F=ma and
>> Maxwell's theory !!!!
>
>I am flabbergasted by such a statement. Nothing could be farther from the
>truth. The argument and everything surrounding it is obviously *wrong*.
>
>This should be clear based on several viewpoints:
>
>1) Feynman's argument is an accident - the spacetime basis that supports the
>EM field also has commutation relations that look like those for the x and p
>of mechanics, and allows EM to be cast in "Hamiltonian" form. In fact *any*
>dynamical theory (at least as we know them) can be cast in Hamiltonian form.
Perhaps the word "derivable" is incorrect. But there is some interesting
observation concerning "consistency" of Maxwell equations with quantum
mechanics. Normally we take into account classical e-m field acting on
charged particles by adding F to the symplectic form Omega. Then
quantization requires that Omega is a curvature form, thus closed.
Therefore dF=0 is required for quantization.
This argument works as well in a Galileian general relativity. In this
case dOmega=0 implies further constraints on the curvature (postulated
also by Kuenzle)
ark
--
Arkadiusz Jadczyk
http://www.cassiopaea.org/quantum_future/homepage.htm
--
Arkadiusz Jadczyk
<em...@physics.umass.edu> wrote:
>This is interesting. It reminds me of the famous Feynman derivation of
>Maxwell's homogenous equations. He starts with space-momentum
>commutation relations and F=ma. With an elaborate, yet straightforward
>calculation, he manages to derive the Maxwell equations, never using
>relativity at any point.
If you are really really careful, if you understand that momentum
operators are unbounded, with a dense domain where they are
self-adjoint, when you exponentiate their commutation relations, so as
to get finite quantities - then you get cocycle-like relations which
tell you that Maxwell equations _can_ be violated on a set of measure
zero - thus you allow for for discrete magnetic monopoles. From the
cocycle relation you derive at the same time quantization of the
magnetic charges.
Aaron Bergman
fii...@yahoo.com (Squark) wrote:
> Aaron Bergman <aber...@princeton.edu> wrote in message
> news:<abergman-2F242C...@news.bellatlantic.net>...
> > In article <a6uvqi$14k0$1...@rs04.hrz.uni-essen.de>,
> > "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> > > I seem to know one example, namely
> > > 11-d supergravity with vanishing Chern-Simons term. The bosonic
> > > Lagrangian
> > >
> > > L = * R + c1 F^*F + c2 A^F^F
> > >
> > > looks like 11-d gravity coupled to 3-form electromagnetism when
> > > A^F^F = 0.
> >
> > Except for the fact that the quantization condition for F = dA isn't
> > that it's in integral cohomology.
>
> What do you mean? F = dA is exact and thus vanishes in the cohomology.
Only if A is globally defined. It isn't.
For the rest, see hep-th/9609122.
Urs Schreiber
It isn't in general globally defined, but it may be if the
first Pontryagin class of spacetime is divisble by 4, right?
From hep-th/0203061 I learn that this is always the case for
11d spacetimes having a factor X which is a spin seven-fold.
So in such a situation is seems ok to say that (for vanishing
AFF) the bosonic sector of 11d sugra is 11d gravity coupled to
3-form electromagnetism. The 3-form potential A couples to
2-branes with worldvolume V via L = \int_V A. How do the
5-branes couple to A? Via \int A^A ?
It makes me feel quite uneasy to do away with A as a well
defined 3-form in supergravity. A is related by supersymmetry
to the fermions, so what happens to them when A is ill
defined? Can one extend the fermionic action to 12 dimensions
the way Witten does (in section 4 of hep-th/9609122) with the
Chern-Simons term? Does this imply that one has to consider
M-theory as really (at least) 12 dimensional?
This reminds me of hep-th/9904063, where it is suggested that
M-theory lives in 13 dimensions with OSp(1/64) supersymmetry.
The authors make use of
SP(64) > SO(11,2) -> SO(10,1)xSO(1,1), which again reminds me
of what Danny Ross Lunsford has been saying about how he gets
the Poincare group from SO(3,2).
Paul Colby
> Paul Colby wrote:
>>John Baez wrote:
>>>In modern terminology: the homogenous Maxwell equations dF = 0
>>>are invariant under all diffeomorphisms of spacetime, since
>>>the operator d commutes with the action of diffeomorphisms
>>>on differential forms.
>>>Of course the other Maxwell equations d*F* = J
>>>involve the Hodge star operator, and are not diffeomorphism-invariant.
>>Is it also misleading to say that the F in dF=0 is the
>>*same* F as in d*F* = J? The first is in terms of E and B while
>>the second is H and D. These are not the same fields. Isn't it
>>that in free space mu0 H = B and D = epsilon0 E for all
>>inertial observers that implies special relativity?
> One way to look at it is: It is the same F in both equations,
> and the constitutive relations are hidden in the *.
> An invariant equation involving D and H would be dM = J,
> where D and H are combined to make M in a similar way
> (more precisely, a dual way) as E and B combine to make F.
> Then to say M = *F for a certain metric is the constitutive relations.
> Then dM = J is also invariant under all transformations.
As usual the point(s) I'm attempting to make dissolve into
semantic mush or worse, matters of taste. Yes, one may absorb
epsilon and mu into the units used for F in free space and perhaps
the definition of the Hodge * operator. The question I ask
is should one. The answer I feel is no. The Hodge * operator
is geometric and with the proper metric holds for all phenomena
electromagnetic or not. The connection between E and D and
H and B in general contain the physics of the polarizability
of materials. On the other hand the two-form M obeys dM = *J in *all*
cases whereas it seems d*F=*J does not. Could one incorporate
the tensor relation between E and D and B and H in an
anisotropic crystal using some redefinition of the * operator?
My guess is no. Should one try this for epsilon and mu tensors for
materials? No since these polarizabilities don't effect the
connection between forms and their duals. This is a purely geometric
connection effecting all phenomena equally.
Regards
Paul Colby
John Baez
Thomas Larsson <thomas....@hdd.se> wrote:
> John Baez whispered:
>> There is a lot of cool stuff about how these higher analogues
>> of Maxwell's equations relate to n-category theory, and how they
>> generalize to higher analogues of Yang-Mills equations, but this
>> is top secret.
>Let me guess. You formulate Yang-Mills in terms of Wilson loops.
I know that...
>It can be formulated as a gauge theory with zero curvature in loop space.
>Polyakov did something like this around 1980.
... but I don't know this, and it sounds strange to me. You're
saying that from a connection A on the spacetime M you can get a
connection B on the free loop space LM such that the Yang-Mills
equations for A translate into flatness for B? It sounds awfully
*similar* to some things I've seen, but also very *different* - and
more to the point, I don't see how it works.
Can you give me a precise reference or even better an explanation
of exactly how this trick is supposed to work?
>2-Yang-Mills is then a
>loop space gauge theory with non-zero curvature, etc.
In his book, Jean-Luc Brylinski describes a trick like this:
you take a "2-connection" A on M and turn it into a connection
B on LM. Actually he only considers the case where the gauge
group is U(1). In the very simplest case, where the 2-bundle
(aka "gerbe") on M is trivial, A is just a 2-form on M, and we
turn it into a 1-form on LM by a trick which amounts to "integrating
around the loop". The same trick (usually called "transgression")
can be used to turn arbitrary p-forms on M into (p-1)-forms on LM.
So yeah, there is a general strategy for trying to turn "n-gauge
theories" on M into "(n-1)-gauge theories" on LM, which we can
try to iterate - but this sounds slightly different from what you
were describing.
Gotta go to sleep now... early tomorrow I'm flying to Seoul, where
I'll visit my pal Minhyong Kim and also Graeme Segal at the KIAS!
Aaron Bergman
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> Aaron Bergman wrote:
> >
> > In article <939044f.02032...@posting.google.com>,
> > fii...@yahoo.com (Squark) wrote:
> >
> > > Aaron Bergman <aber...@princeton.edu> wrote in message
> > > news:<abergman-2F242C...@news.bellatlantic.net>...
> > > > In article <a6uvqi$14k0$1...@rs04.hrz.uni-essen.de>,
> > > > "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> > > > > I seem to know one example, namely
> > > > > 11-d supergravity with vanishing Chern-Simons term. The bosonic
> > > > > Lagrangian
> > > > >
> > > > > L = * R + c1 F^*F + c2 A^F^F
> > > > >
> > > > > looks like 11-d gravity coupled to 3-form electromagnetism when
> > > > > A^F^F = 0.
> > > >
> > > > Except for the fact that the quantization condition for F = dA isn't
> > > > that it's in integral cohomology.
> > >
> > > What do you mean? F = dA is exact and thus vanishes in the cohomology.
> >
> > Only if A is globally defined. It isn't.
> >
> > For the rest, see hep-th/9609122.
>
> It isn't in general globally defined, but it may be if the
> first Pontryagin class of spacetime is divisble by 4, right?
I don't see why.
> From hep-th/0203061 I learn that this is always the case for
> 11d spacetimes having a factor X which is a spin seven-fold.
> So in such a situation is seems ok to say that (for vanishing
> AFF) the bosonic sector of 11d sugra is 11d gravity coupled to
> 3-form electromagnetism. The 3-form potential A couples to
> 2-branes with worldvolume V via L = \int_V A. How do the
> 5-branes couple to A? Via \int A^A ?
Let F = dA and define (locally a set of) *F = dA'. A' is a 6 form and
couples to five branes.
> It makes me feel quite uneasy to do away with A as a well
> defined 3-form in supergravity.
Does it make you feel better that A isn't a well-defined one-form in
ordinary electromagnetism. This isn't a big deal.
Urs Schreiber
[...]
> > It isn't in general globally defined, but it may be if the
> > first Pontryagin class of spacetime is divisible by 4, right?
>
> I don't see why.
Because then F is allowed to be in integral cohomology.
I am just trying to understand under what conditions the bosonic sector
of 11-d sugra can be regarded as gravity plus 3-form electromagnetism.
You said it can,
> Except for the fact that the quantization condition for F = dA isn't
> that it's in integral cohomology.
I seem to understand from hep-th/9609122 that this is the case if
the first Pontryagin class of spacetime is divisible by 4, because then
\lambda is even and hence \lambda/2 is integral. Isn't that correct?
> Does it make you feel better that A isn't a well-defined one-form in
> ordinary electromagnetism. This isn't a big deal.
I can evaluate the action for ordinary EM without A being globally defined.
In 11d sugra there is this AFF term which I cannot straightforwardly integrate
when A is not well defined. To do so I have to follow Witten and extend
the 4-form field to 12 dimensions. But I see your point, maybe I am not
flexible enough in my thinking. I had considered the 11d sugra 3-form A to
be somehow more fundamental than the vector potential in ordinary EM.
Urs Schreiber
news:abergman-6C4F00...@news.bellatlantic.net...
[...]
> > The 3-form potential A couples to 2-branes with worldvolume
> > V via L = \int_V A. How do the 5-branes couple to A? Via \int A^A ?
>
> Let F = dA and define (locally a set of) *F = dA'. A' is a 6 form and
> couples to five branes.
Interesting. I rewrite that as
F = dA = *dA' = *d* *A' = del *A'
(up to possibly a sign). The potentials A and *A' exist because F is closed
and coclosed and hence locally exact and coexact. Adding both potentials
gives a single inhomogeneous-form potential B:
B = A - *A'
which satisfies the Kaehler equation
(d+del) B = 0
if A is given in Lorentz gauge
del A = 0
and *A' in "co-Lorentz" gauge.
d *A' = 0.
(Other gauges may be chosen while preserving the Kaehler equation
by adding suitable terms to B. These terms will then serve as potentials
for other p-form sectors.)
So to every p-form potential (p<d-1) there is a "dual" (d-(p+2))-form
potential. To every p'-brane (p'<d-3) coupled to p=(p'+1)-form EM there
is a "dual" (d-(p+2))-1=(d-((p'+1)+2))-1=(d-(p'+4))-brane. (Here "dual"
refers to the above relation.)
Is there a "dual" brane for ordinary EM in 4 dimensions? In ordinary EM the
brane is a point particle, a (p'=0)-brane, coupled to (p=1)-form
vectorpotential.
The "dual" degrees are (d-(p'+4)) = 0, (d-(p+2)) = 1. Hence ordinary EM
is "self-dual" in the above sense, there is only one kind of brane here.
Urs Schreiber
news:sZym8.3080$Tk.2...@newsread1.prod.itd.earthlink.net...
[...]
> Could one incorporate
> the tensor relation between E and D and B and H in an
> anisotropic crystal using some redefinition of the * operator?
That's an interesting question, I think. Given two forms F and M,
under what condition can one find a metric on the underlying
manifold such that the Hodge * with respect to that metric
satisfies
F = *M
?
This looks like an easy question, actually. But it takes me more time
to think about it than to write this post. :-)
I recall that, a while ago, Pertti Lounesto said something about not
using the Hodge to incorporate constitutive relations.
> My guess is no. Should one try this for epsilon and mu tensors for
> materials? No since these polarizabilities don't effect the
> connection between forms and their duals. This is a purely geometric
> connection effecting all phenomena equally.
Right, one would want two different metrics. One describing the
"true" geometry, i.e. the gravitational field, the other describing
the constitutive relations.
Aaron Bergman
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> Is there a "dual" brane for ordinary EM in 4 dimensions? In ordinary EM the
> brane is a point particle, a (p'=0)-brane, coupled to (p=1)-form
> vectorpotential.
> The "dual" degrees are (d-(p'+4)) = 0, (d-(p+2)) = 1. Hence ordinary EM
> is "self-dual" in the above sense, there is only one kind of brane here.
Not quite. The dual to the normal EM particle is a magnetic monopole.
When you have N=4 SYM, then it is believed that the theory exhibits
what's called S-duality (or Montonen-Olive duality for this particular
case) where the theory of magnetic monopoles is equivalent to the theory
of charged fundamental particles. There are also various versions of
something similar with fewer symmetries: Seiberg-Witten theory for N=2
and Seiberg duality for N=1.
Toby Bartels
>Toby Bartels wrote:
>>One way to look at it is: It is the same F in both equations,
>>and the constitutive relations are hidden in the *.
>>An invariant equation involving D and H would be dM = J,
>>where D and H are combined to make M in a similar way
>>(more precisely, a dual way) as E and B combine to make F.
>>Then to say M = *F for a certain metric is the constitutive relations.
>>Then dM = J is also invariant under all transformations.
>The Hodge * operator
>is geometric and with the proper metric holds for all phenomena
>electromagnetic or not. The connection between E and D and
>H and B in general contain the physics of the polarizability
>of materials. On the other hand the two-form M obeys dM = *J in *all*
>cases whereas it seems d*F=*J does not. Could one incorporate
>the tensor relation between E and D and B and H in an
>anisotropic crystal using some redefinition of the * operator?
>My guess is no.
The answer is yes, so long as D and H depend linearly on E and B.
But this redefined * is not the geometric Hodge operator,
but instead some other operator that depends on the medium.
Then we say that * is the Hodge star iff we're in a vacuum.
If you like, use some symbol other than *.
(As for JB, he usually deals only with transparent media,
so his * probably *was* the Hodge dual.)
-- Toby
to...@math.ucr.edu
Alfred Einstead
> Is it also misleading to say that the F in dF=0 is the
> *same* F as in d*F* = J? The first is in terms of E and B while
> the second is H and D. These are not the same fields. Isn't it
> that in free space mu0 H = B and D = epsilon0 E for all
> inertial observers that implies special relativity?
Right. Then you back to the original macroscopic equations
dG = J; dF = K = 0
and force law
P = Power-density dt - Force-density . dr
= epsilon_abcd dx^a (J^b F^cd - K^b G^cd)
The vacuum relations B = mu0 H, D = epsilon0 E above break the
general symmetry and give you G = *F*, and the conformal group
as its "little group".
Likewise, the relations corresponding to Galilei as the little
group would be (assuming the above relations hold in a distinguished
frame), B = mu0 (H + vxD), D = epsilon0 (E - vxB) as described.
The most general constitutive relations (which have the same
invariance group as the macroscopic equations, to boot) would
be
Omega = dD^i ^ dE_i + dB^i ^ dH_i = 0.
All the above are special cases of this.
On a somewhat diversionary note it's also interesting to note
that the vacuum relations -- which hold for the free fields
can NOT be consistently asserted in the classical theory in
the vicinity of point charges. For, in fact it's the naive
extrapolation of the free field relations to sources
that's the origin of the classical divergences!
It's easy to see how this is so, and a close examonation also
shows why the magnetic and electric source terms J, K cannot both
be non-zero at the same points!
>>From the equations div D = rho, div B = sigma; the D and B fields
will inherit whatever point singularity occurs in the electric
charge density (rho) or magnetic charge density (sigma) respectively.
But then, in order for the force density
F = rho E + J x B + sigma H - K x D
to be well-defined (as distributions), that means that E, K
cannot be singular wherever rho is; and that J, H cannot be
singular wherever sigma is (unless the rho E divergence
ends up cancelling the -KxD's; and sigma H's, the JxB's).
That rules out D = epsilon0 E; B = mu0 H respectively in
the presence of eletric or magnetic point sources.
But the other equations: curl H - dD/dt = J; curl E + dB/dt = -K
mean that D cannot be singular where sigma is (since J and H
can't be); nor can B be singular where rho is (since K and E
can't be). Therefore:
In the classical theory, in order for F to be divergence-free
no point source can possess both a non-zero electric and
magneitc charge.
Likewise, an analysis of the power density P = J.E + K.H shows
that in order for P to be well-defined (distributionally); J & E
cannot be singular at the same point; nor can K & H -- unless
as a result the two divergences cancel one another out.
This reinforces the conclusions made above.
For point sources: rho = sum qA delta(r-rA), the D field would
be the sum of the contributions DA, up to a homogeneous solution
D0. The corresponding E field away from the rA's would just be
D/epsilon0 -- the vacuum relations. But *AT* rA, the classical
theory only gives you:
E(rA) = (D0(rA) + sum DB(rA))/epsilon0 + KA
summed ONLY over B not equal to A. The result is well-defined
only up to a "self-force" term KA.
This is, in fact, the classical origin of renormalization in QED.
D, B, E
and H fields would be gi
[Moderator's note: Message ends there. -MM]
Thomas Larsson
> >It can be formulated as a gauge theory with zero curvature in loop space.
> >Polyakov did something like this around 1980.
>
> ... but I don't know this, and it sounds strange to me. You're
> saying that from a connection A on the spacetime M you can get a
> connection B on the free loop space LM such that the Yang-Mills
> equations for A translate into flatness for B? It sounds awfully
> *similar* to some things I've seen, but also very *different* - and
> more to the point, I don't see how it works.
>
> Can you give me a precise reference or even better an explanation
> of exactly how this trick is supposed to work?
>
My Xerox copies have disappeared, but I think these are the pointers:
A M Polyakov, Phys Lett B 82 (1979) 247;
--, Nucl Phys B 164 (1979) 171.
Another inspiration was a series of CERN preprints by JM Maillet and
F Nijhoff. The only proper reference that I have is to
Phys Lett A 134 (1989) 221.
Polyakov gives an abbreviated account of his construction in
chapter 7 of
A M Polyakov, Gauge fields and strings, Harwood 1987.
Let me copy the main formulas, although this is probably too brief
to be comprehensible.
If C is a loop with parametrization x(s). Psi(C) = Psi(x(s)) is
a parametrization-invariant functional of C if
dx^i/ds dPsi/dx^i(s) = 0.
The quantity
F_i(s,C) = dPsi(C)/dx^i(s) Psi^{-1}(C)
is the holonomy around a Wilson loop C which is concentrated at
x(s). It is essentially the field strength at x(s).
__
| |
___________| |___________
__________________________|
It can be shown that
dF_i(s)/dx^j(s') - dF_j(s)/dx^i(s') + [F_i(s), F_j(s'] = 0,
which is zero curvature in loop space.
Aaron Bergman
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> "Aaron Bergman" <aber...@princeton.edu> schrieb im Newsbeitrag
> news:abergman-6C4F00...@news.bellatlantic.net...
> > In article <3C9B2912...@uni-essen.de>,
> > Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>
> [...]
>
> > > It isn't in general globally defined, but it may be if the
> > > first Pontryagin class of spacetime is divisible by 4, right?
> >
> > I don't see why.
>
> Because then F is allowed to be in integral cohomology.
If 'A' is globally defined, then F is cohomologically trivial because
it's exact.
> I am just trying to understand under what conditions the bosonic sector
> of 11-d sugra can be regarded as gravity plus 3-form electromagnetism.
The condition above is probably sufficient. I't might be a little weird,
though. You might need p_1 = 0.
> You said it can,
> > Except for the fact that the quantization condition for F = dA isn't
> > that it's in integral cohomology.
>
> I seem to understand from hep-th/9609122 that this is the case if
> the first Pontryagin class of spacetime is divisible by 4, because then
> \lambda is even and hence \lambda/2 is integral. Isn't that correct?
Sounds reasonable.
> > Does it make you feel better that A isn't a well-defined one-form in
> > ordinary electromagnetism. This isn't a big deal.
>
> I can evaluate the action for ordinary EM without A being globally defined.
> In 11d sugra there is this AFF term which I cannot straightforwardly integrate
> when A is not well defined. To do so I have to follow Witten and extend
> the 4-form field to 12 dimensions.
Yep. That's the only way to define it. As section 4 of the above paper
makes clear, there are a number of subtleties. Witten (and others) have
written a remarkable number of papers based on keeping track of these
anomalies.
> But I see your point, maybe I am not
> flexible enough in my thinking. I had considered the 11d sugra 3-form A to
> be somehow more fundamental than the vector potential in ordinary EM.
'taint.
BTW, since I know one of the guys who wrote it, I should probably
mention hep-th/0203218 which just came out. I haven't yet decided if
it's deep or trivial yet, though.
Danny Ross Lunsford
> ......(d+d*)F=0.......
You almost always throw in this type of equation. Why don't you stick to
Dirac-like equations? What does it buy you? Eqns like dF=0 have simple
geometric meaning. Throwing in the dual seems to me to be muddying things,
becuase they no longer seem like integrability conditions. Please explain in
direct terms what it means. It won't be as easy as explaining (dd* + d*d).
Also, when a metric is present, isn't (d+d*)F just d(F + mumble *F) = dG? So
what did you gain?
-drl
Paul Colby
> "Paul Colby" <paulc...@earthlink.net> schrieb im Newsbeitrag
> news:sZym8.3080$Tk.2...@newsread1.prod.itd.earthlink.net...
>
> [...]
>
>> Could one incorporate
>> the tensor relation between E and D and B and H in an
>> anisotropic crystal using some redefinition of the * operator?
>
> That's an interesting question, I think. Given two forms F and M,
> under what condition can one find a metric on the underlying
> manifold such that the Hodge * with respect to that metric
> satisfies
>
> F = *M
>
> ?
Wouldn't they have to be related by an anti-involution since
**M = -M for 2-forms on R^4? This is ruled out for most
constituative relations found in nature (try free space
region containing an iron sphere with mu=2000 for example).
Regards
Paul Colby
Urs Schreiber
> In article <a7o174$120a$1...@rs04.hrz.uni-essen.de>,
> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
>
> > Is there a "dual" brane for ordinary EM in 4 dimensions? In ordinary EM the
> > brane is a point particle, a (p'=0)-brane, coupled to (p=1)-form
> > vectorpotential.
> > The "dual" degrees are (d-(p'+4)) = 0, (d-(p+2)) = 1. Hence ordinary EM
> > is "self-dual" in the above sense, there is only one kind of brane here.
>
> Not quite. The dual to the normal EM particle is a magnetic monopole.
Ah, yes, of course. I should have said that there are only branes of
one single degree here, namely 0-brane point particles. But there are two
types of these, electric and magnetic ones, related by Hodge duality.
Still, this is a degeneracy as compared to general p-form EM: Only when
dual p-brane sources are of the same degree p can one transform one
of the respective currents to zero, by duality rotations, when all p-brane
sources carry the same ratio of charge and dual charge.
> When you have N=4 SYM, then it is believed that the theory exhibits
> what's called S-duality (or Montonen-Olive duality for this particular
> case) where the theory of magnetic monopoles is equivalent to the theory
> of charged fundamental particles.
Why is it "believed"? What prevents one from really knowing it?
Also, to my ignorant ear, the sentence "the theory of magnetic
monopoles is equivalent to the theory of charged fundamental
particles" sounds just like electric/magnetic duality in ordinary EM.
What's the difference?
Now that I can see how p-form EM has its place in string theory
I am wondering about p-form YM theory. John Baez and Thomas
Larsson mentioned how (p-1)-YM in loop space gives rise to
p-YM on spacetime. Can one relate this to string theory? When
does a brane carry a non-abelian charge?
> There are also various versions of
> something similar with fewer symmetries: Seiberg-Witten theory for N=2
> and Seiberg duality for N=1.
Does "S" in "S-duality" stand for "Seiberg"?
Paul Colby
> Paul Colby wrote:
>
>>Toby Bartels wrote:
>
>>Could one incorporate the tensor relation between
>>E and D and B and H in an anisotropic crystal using
>>some redefinition of the * operator?
>>My guess is no.
>
> The answer is yes, so long as D and H depend linearly
> on E and B.
A truly good point. B is a non-linear function of
H for magnetic materials *and* dM=*J continues to
hold everywhere. This is why I have a really hard
time "equating" H with B as far as the physics is
concerned. Writing M = *F where * is now neither
an involution nor a linear relation stretches the
Hodge deal a bit far. I guess I still prefer to
view B and H as physically different fields describing
essentially different aspects of EM phenomena.
Regards
Paul Colby
Aaron Bergman
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> "Aaron Bergman" <aber...@princeton.edu> schrieb im Newsbeitrag
> news:abergman-83C01D...@news.bellatlantic.net...
> > In article <a7o174$120a$1...@rs04.hrz.uni-essen.de>,
> > "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> >
> > > Is there a "dual" brane for ordinary EM in 4 dimensions? In ordinary EM
> > > the
> > > brane is a point particle, a (p'=0)-brane, coupled to (p=1)-form
> > > vectorpotential.
> > > The "dual" degrees are (d-(p'+4)) = 0, (d-(p+2)) = 1. Hence ordinary EM
> > > is "self-dual" in the above sense, there is only one kind of brane here.
> >
> > Not quite. The dual to the normal EM particle is a magnetic monopole.
>
> Ah, yes, of course. I should have said that there are only branes of
> one single degree here, namely 0-brane point particles. But there are two
> types of these, electric and magnetic ones, related by Hodge duality.
> Still, this is a degeneracy as compared to general p-form EM: Only when
> dual p-brane sources are of the same degree p can one transform one
> of the respective currents to zero, by duality rotations, when all p-brane
> sources carry the same ratio of charge and dual charge.
I can't decipher this last sentence.
> > When you have N=4 SYM, then it is believed that the theory exhibits
> > what's called S-duality (or Montonen-Olive duality for this particular
> > case) where the theory of magnetic monopoles is equivalent to the theory
> > of charged fundamental particles.
>
> Why is it "believed"? What prevents one from really knowing it?
We don't know how to define Quantum Field Theory.
> Also, to my ignorant ear, the sentence "the theory of magnetic
> monopoles is equivalent to the theory of charged fundamental
> particles" sounds just like electric/magnetic duality in ordinary EM.
> What's the difference?
When you quantize EM, you get a coupling constant that flows. Thus g <->
1/g can't be a symmetry of the theory. N=4 SYM is conformal so this
difficulty is averted.
> Now that I can see how p-form EM has its place in string theory
> I am wondering about p-form YM theory. John Baez and Thomas
> Larsson mentioned how (p-1)-YM in loop space gives rise to
> p-YM on spacetime. Can one relate this to string theory? When
> does a brane carry a non-abelian charge?
This is just the idea of a gerbe. A lot of people have tried to relate
gerbes to D-branes and to try to come up with a theory of Non-abelian
gerbes, but I don't think anyone's succeeded. One of the problems is
that the p-forms in string theory are really just the map of a K-theory
element into cohomology.
>
> > There are also various versions of
> > something similar with fewer symmetries: Seiberg-Witten theory for N=2
> > and Seiberg duality for N=1.
>
> Does "S" in "S-duality" stand for "Seiberg"?
"Strong" or "strength" or something like that. The original idea that
there might be a duality between fundamental particles and magnetic
monopoles goes back (at least) to Montonen and Olive in the seventies, I
think.
Urs Schreiber
> In article <a7s8li$15hc$1...@rs04.hrz.uni-essen.de>,
> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
[...]
> > Only when
> > dual p-brane sources are of the same degree p can one transform one
> > of the respective currents to zero, by duality rotations, when all p-brane
> > sources carry the same ratio of charge and dual charge.
>
> I can't decipher this last sentence.
Sorry. I was thinking aloud of this well known fact:
A particle in 4D with proper velocity covector v carrying an electric charge
q *and* a magnetic charge p produces an electric current
j = q v
and a magnetic current
k = p * v .
Maxwell's equation with electric and magnetic sources then read
(d +del) F = q v + p *v
But for constant p and q these can always be duality rotated by
F -> F ' = exp(alpha *) F = cos(alpha) F + sin(alpha) *F
j+k -> (j+k)' = exp(alpha *)(j+k) = sqrt(q^2+p^2) v
alpha = -arctan(p/q)
to yield
(d+del)F' = sqrt(q2+p2) v .
I know that you know that, but for completeness sake I note that this
works because with F a solution to
(d+del)F = q v + p *v
one also has
*(d + del)* *F = -* (q v + p *v)
but
*(d + del)* = *(d + *d*)* = *d* + d = (d+del)
and so *F solves Maxwell's equations for the dual current
p v - q *v .
Superposing both solutions gives the above result.
Hence, as long as the ratio p/q of magnetic to electric charge of
a particle carrying both charges is constant, one of both charges
may be transformed away by duality rotations to yield the
*conventional*
(d+del) F = j .
With j a purely electric current.
In my above remark I was simply noting that, in the context of higher EM,
such a construction is only possible when a charge carrier may carry both
charges, electric and magnetic. This is only possible when branes and dual
branes coincide in their dimensionality. So by the formula
p' = D - 4 - p
for the dimension p' of the dual brane in dependence of the dimension p
of the original brane this is only possible when
p = (D-4)/2 , and p a non-negative integer ,
that is, in
4D for 0 branes
6D for 1 branes
8D for 2 branes
10D for 3 branes.
In light of our recent conversation about the 11D sugra action being
really defined only in 12 dimensions it might make sense to include
12D for 4 branes ,
but I am not sure if one can make sense of this.
> > Also, to my ignorant ear, the sentence "the theory of magnetic
> > monopoles is equivalent to the theory of charged fundamental
> > particles" sounds just like electric/magnetic duality in ordinary EM.
> > What's the difference?
>
> When you quantize EM, you get a coupling constant that flows. Thus g <->
> 1/g can't be a symmetry of the theory. N=4 SYM is conformal so this
> difficulty is averted.
I see. So N=4 SYM solves a problem that I wasn't aware of. But
actually I do not really see it. Why can g <-> 1/g not be consistent
with a running coupling constant? (I know, this question reveals that
I do not have the slightest clue what I am even talking about... :-)
> This is just the idea of a gerbe. A lot of people have tried to relate
> gerbes to D-branes and to try to come up with a theory of Non-abelian
> gerbes, but I don't think anyone's succeeded. One of the problems is
> that the p-forms in string theory are really just the map of a K-theory
> element into cohomology.
Do you know a *brief* introduction into or overview of K-theory that
could help me get the basic idea here?
Urs Schreiber
news:Sj6o8.7346$Eb5.6...@bgtnsc05-news.ops.worldnet.att.net...
> Urs Schreiber wrote
>
> > ......(d+d*)F=0.......
>
> You almost always throw in this type of equation. Why don't you stick to
> Dirac-like equations? What does it buy you? Eqns like dF=0 have simple
> geometric meaning. Throwing in the dual seems to me to be muddying things,
> direct terms what it means. It won't be as easy as explaining (dd* + d*d).
It's a change of basis. We have the polar algebra (1)
{d,d} = 0
{del,del}= 0
{d,del} = H
as well as its diagonal version
<=> {Di, Dj} = 2 delta_ij H .
Both are related by a linear transformation of basis elements:
D1 = (d+del)
D2 = i(d-del) .
Some things are more naturally expressed in one basis than in the
other, and some structures are manifest in one basis and invisible
in the other (2).
One advantage of the Di is that they have right inverses
(Green operators) Gi such that
Di Gi f = f (no sum)
That's one reason that makes people suggestively write
(d+del) F = J .
But the same in the polar basis,
dF = 0 && del F = J (with F a homogeneous form),
has advantages, too. It depends on your exact problem which one is
preferable.
My point, which I tried to make in previous posts in this thread, is
that the diagonal basis, Di, offers advantages when it comes to
generalizations of ordinary EM:
For any number N of diagonal generators
{Di,Dj} = 2 delta_ij H, i,j in {1,...,N}
there is a simple algorithm for finding a solution f to
Di f = 0, i in {1,...,N}.
Namely, one can solve any *one* of these equations and then apply
the remaining Di to that result to obtain a solution to all N equations.
This is a generalization of the "vector" potential technique in ordinary EM,
which is recovered for N=2 and f a homogeneous form. But it still works
for inhomogeneous f. This means that by solving one first order equation,
a solution to p-form EM in all p-form sectors is obtained at one stroke.
I had mentioned how sources can be incorporated into this framework:
Let k be the potential for the (possibly inhomogeneous) current form j:
del k = j .
Then the single first order equation for the field potential A
(which is also possibly an inhomogeneous form) is
D1 A = k
<=> (d+del) A = k .
The simple proof that
F = dA
solves the generalized p-form Maxwell equations (3)
dF = 0, del F = j
is
d F = d d A = 0
del F = del (d A) = del (k - del A) = del k - del del A = del k = j .
Hence the solution to general p-form EM is
A = G1 k
F = d A
for any (inhomogeneous form) source potential k . (G1 also depends
on the boundary conditions.)
If you wish, you may of course rewrite the central equation
(d+del) A = k
in the polar basis, e.g. as
d A = -del A + k .
I have learned from Aaron Bergman that this has a very neat interpretation
in terms of branes and dual branes carrying electric and magnetic charge,
respectively:
Let
k = 0
for a moment and consider a definite (p+2)-form component F_(p+2)
of F:
F_(p+2) = (F)_(p+2) .
F_(p+2) is the electromagnetic field strength for a p-brane which
couples to the (p+1) field potential
A_(p+1) = (A)_(p+1)
via
q int_V_(p+1) A_(p+1),
where q is the electric charge and V_(p+1) the world volume of the
p-brane. Generalizing the well known electric-magnetic duality
of ordinary EM one can look at the dual field strength (the
"magnetoelectric" field strength) *(F_(p+2)):
*(F_(p+2)) = (*F)_(n-p-2)
(where n is the dimension of spacetime). This should couple to
(magnetic) (n-p-4)-branes by means of an (n-p-3)-form potential
A'_(n-p-3):
d A'_(n-p-3) = (*F)_(n-p-2) .
Expressing this in terms of the original field strength
d A'_(n-p-3) = (*F)_(n-p-2) .
<=>
* d A'_(n-p-3) = F_(p+2)
<=>
del (* A')_(p+3) = F_(p+2) = d A_(p+1)
(up to the sign of *^2 which I ignore for convenience)
finally reveals the physical meaning of the formula
(d+del) A = 0
(which, again, is not equivalent to d A = del A = 0, since A is an
inhomogeneous form):
This formula inherently relates electric with dual magnetic field
potentials. To make this explicit note that A_(p+1) and
(* A')_(p+3) can be chosen to be in Lorentz and co-Lorentz gauge
del A_(p+1) = 0
d (* A')_(p+3) = 0
in which case the solution A to
(d+del) A = 0
is
A = A_(p+1) - (* A')_(p+3) .
Hence the inhomogeneous-form field potential consists of the electric
and the magnetic versions of the usual homogeneous-form field potential.
Alternatively, one can solve (d+del)A = 0 for any inhomogeneous
A (for instance by means of A = G1 0) and then look at the (p+1)- and
(p+3)-form components of A to read off the usual field potential and
its magnetic dual. When, furthermore, other components of A are non-
vanishing then these will be related to A_(p+1) and (* A')_(p+3) by
affecting the gauge in which these appear. Conversely, every (p'+1)-
and (p'+3)- components of A solve free EM for p'-branes and give the
respective field potential and its dual. The same holds true for non-vanishing
sources k.
> Also, when a metric is present,
All this assumes that a metric is present.
> isn't (d+d*)F just d(F + mumble *F) = dG?
Not in general. For instance not if F has a non-coclosed 1-form
component. Also note that the "integrability condition" of your
equation is
(d+del) F = dG => d(d+del)F = ddG => d del F = 0,
which is a non-empty.
> So what did you gain?
It depends. As I said, there are two different versions of the same
algebra. One might argue that the "diagonal" version is a little more
general. For instance, it makes sense when N=1, in which case there is
no polar analogue (the spinning particle). It also facilitates finding
further polar algebras. The above solution strategy is transparent only
in the diagonal basis. All in all, I find it helpful to uncover Dirac
operators when I see them. That's why I do not understand why you
write
> Why don't you stick to Dirac-like equations?
because that's exactly what I am doing.
----- footnotes:
(1) I write "del" instead of "d*" to better distinguish between d* and d * .
(2) This is completely analogous to the following well known construction:
The polar algebra CAR of canonically anticommutating creators and
annihilators
{a_i, a_j} = 0
{a*_i, a*_j} = 0
{a*_i, a_j} = delta_ij
i,j in {1,...n}
is isomorphic to the Clifford algebra
{y_n, y_m} = 2 delta_nm
n,m in {1,...,2n}
under the transformation
y_(2i-1) = a*_i + a_i
y_(2i) = i(a*_i - a_i) .
(3) Now that we are dealing with inhomogeneous forms it makes
a big difference if one writes
d F = 0 && del F = j
or
(d+del) F = j .
Since we want p-form EM in each p-form sector we have to choose
the former as the proper generalization of Maxwell's equations. This is
not an ad hoc choice: Only the former equation is the equation of motion
for p-form fields as they arise from usual Lagrangians (e.g. in string
theory).
Aaron Bergman
"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
> "Aaron Bergman" <aber...@princeton.edu> schrieb im Newsbeitrag
> news:abergman-1CC45A...@news.bellatlantic.net...
> In light of our recent conversation about the 11D sugra action being
> really defined only in 12 dimensions it might make sense to include
>
> 12D for 4 branes ,
>
> but I am not sure if one can make sense of this.
Only if you figure out what to do with the second time dimension. I
should note that the 12th dimension is only necessary to rigorously
define the CS-type term. Once you've done that, it's no longer necessary.
Whether or not there's a deeper meaning to the trick is left as an
exercise to the reader.
>
> > > Also, to my ignorant ear, the sentence "the theory of magnetic
> > > monopoles is equivalent to the theory of charged fundamental
> > > particles" sounds just like electric/magnetic duality in ordinary EM.
> > > What's the difference?
> >
> > When you quantize EM, you get a coupling constant that flows. Thus g <->
> > 1/g can't be a symmetry of the theory. N=4 SYM is conformal so this
> > difficulty is averted.
>
> I see. So N=4 SYM solves a problem that I wasn't aware of. But
> actually I do not really see it. Why can g <-> 1/g not be consistent
> with a running coupling constant? (I know, this question reveals that
> I do not have the slightest clue what I am even talking about... :-)
If the two theories are the same, then the coupling constant should flow
in the same way, but 1/g flows in the opposite way as g.
> > This is just the idea of a gerbe. A lot of people have tried to relate
> > gerbes to D-branes and to try to come up with a theory of Non-abelian
> > gerbes, but I don't think anyone's succeeded. One of the problems is
> > that the p-forms in string theory are really just the map of a K-theory
> > element into cohomology.
>
> Do you know a *brief* introduction into or overview of K-theory that
> could help me get the basic idea here?
Take a manifold and the set of vector bundles on it. Under direct sum,
this is a semi-group. There is a standard trick for turning (abelian? --
I'm too lazy to check) semigroups into groups: take ordered pairs and
identify by the following equivalence
(a,b) ~ (a+c,b+c)
In this way, to any manifold, you can associate an Abelian group called
K^0(M). In string theory, one can think of this as the fact that, to any
D-brane, there is a Chan-Paton bundle and one can always nucleate or
destroy brane/anti-brane pairs. In IIB, you can have space filling
9-branes, which gives you K^0(M). (To IIA, you get K^1(M) which I'll
leave undefined.). The chern character gives a map
ch: K^0(M) -> H^ev(M)
which is an isomorphism when you tensor with the rationals (ignore
torsion). See Witten's paper hep-th/9810188. In the past two years or
so, people have started thinking derived categories are cool. I talked a
bit about them in
<http://groups.google.com/groups?selm=slrn9p8lcr.rck.abergman%40phoenix.P
rinceton.EDU>.
Aaorn
Urs Schreiber
news:a8ak2j$15ng$1...@rs04.hrz.uni-essen.de...
> "Danny Ross Lunsford" <antima...@worldnet.att.net> schrieb im Newsbeitrag
> news:Sj6o8.7346$Eb5.6...@bgtnsc05-news.ops.worldnet.att.net...
> > Urs Schreiber wrote
> >
> > > ......(d+d*)F=0.......
> >
> > You almost always throw in this type of equation.
> > So what did you gain?
>
> It depends. As I said, there are two different versions of the same
I forgot to mention the strongest reason to stick to the diagonal algebra
D1 = (d+del)
D2 = i(d-del)
instead of to the polar form with d and del. As explained in
hep-th/9612205 page 9,10, the diagonal algebra actually requires
less structure than the polar one. In the terminology of the above
paper, the diagonal algebra defines N=(1,1) geometry. To lift this
to N=2 geometry, were D1^2 = D2^2 and hence d = D1-i D2 is
nilpotent, one needs extra structure, namely a Z-grading
(instead of merely a Z2-grading) of the algebra. Hence an
advantage of using d+del and i(d-del) instead of d and del in
one's theory is that this way everything can be generalized to
cases where [D1,N] =!= i D2, with N the number operator
that usually induces the Z grading. Such a case arises for example,
according to the authors of hep-th/961205, in the presence
of non-vanishing torsion:
Let T = T_ijk be the torsion tensor, assumed to be closed
dT = 0 .
Then one has
D1_(T) = D1 - ym T_mno (e/\)n (e/\)o
D2_(T) = D2 + Ym T_mno (e/\)n (e/\)o
with Di the torsion-free operators. It is easily seen that
[D1_(T),N] =!= i D2_(T)
because of the two wedging operators in the torsion term.
But I admit that I have to better understand such cases.
Urs Schreiber
> In article <a81i6i$i7c$1...@rs04.hrz.uni-essen.de>,
> "Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote:
[...]
> > In light of our recent conversation about the 11D sugra action being
> > really defined only in 12 dimensions it might make sense to include
> >
> > 12D for 4 branes ,
> >
> > but I am not sure if one can make sense of this.
>
> Only if you figure out what to do with the second time dimension.
I must have missed that point in Witten's paper. Do we need to
have a metric with two timelike eigenvalues in 12D? Why?
BTW, meanwhile I have read:
Thomas Mohaupt, Black Holes in Supergravity and String Theory,
hep-th/0004098 .
I am not studying string theory (yet), the little I know I have learned by
accident. I was trying to better understand extremal black hole dynamics in
moduli space, because this should be closely related to supergravity
cosmology. One thing that puzzled me is that Shiraishi singled out
a time parameter and obtained unconstrained time evolution in moduli
space for a collection of extremal black holes. I had expected to see
a Hamiltonian constraint. Anyway, to better understand that I was looking
at a couple of papers, among them the one above. This happens to have
a brief section on p-form EM in string theory, so it was interesting to
compare that with our discussion. I learned that the "self-dual" p-branes,
that I was talking about in my previous post, which satisfy
> p = (D-4)/2 , and p a non-negative integer ,
are called "dyonic". I just wanted to add that piece of terminology here.
Also, somewhere, I forget where, I have seen "fractional branes" that
apparently couple to form-potentials of only half of the expected degree.
Is there an easy way to explain why and how fractional branes arise?
> See Witten's paper hep-th/9810188. In the past two years or
> so, people have started thinking derived categories are cool. I talked a
> bit about them in
> <http://groups.google.com/groups?selm=slrn9p8lcr.rck.abergman%40phoenix.P
> rinceton.EDU>.
Thanks, I'll look at that.
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>A lot of people have tried to relate
>gerbes to D-branes and to try to come up with a theory of Non-abelian
>gerbes, but I don't think anyone's succeeded.
There does exist a theory of nonabelian gerbes, going back to
Giraud's 1971 paper on nonabelian cohomology. Larry Breen and
William Messing have a nice paper on the math archive entitled
"Differential geometry of gerbes" which treats the concept of
connection for nonabelian gerbes, and they mention the relation
to D-branes, but I'm pretty sure you're right that nobody has
clarified the relation of D-branes to nonabelian gerbes. Luckily
for me, Breen and Messing's paper is sufficiently abstract that
few physicists will succeed in understanding it before I finish
writing a paper on this subject. :-) I don't know D-branes from
beans, but there's a nice theory of categorified differential
geometry waiting to be developed here, which seems bound to have
*some* application to physics... and nonabelian gerbes are really
just the tip of the iceberg.
Aaron Bergman
wrote:
> In article <abergman-1CC45A...@news.bellatlantic.net>,
> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >A lot of people have tried to relate
> >gerbes to D-branes and to try to come up with a theory of Non-abelian
> >gerbes, but I don't think anyone's succeeded.
>
> There does exist a theory of nonabelian gerbes, going back to
> Giraud's 1971 paper on nonabelian cohomology.
Is this what Brylinski talks about? That didn't seem to do much beyond
3-forms and it looked like a struggle to get there.
> Larry Breen and
> William Messing have a nice paper on the math archive entitled
> "Differential geometry of gerbes" which treats the concept of
> connection for nonabelian gerbes, and they mention the relation
> to D-branes, but I'm pretty sure you're right that nobody has
> clarified the relation of D-branes to nonabelian gerbes. Luckily
> for me, Breen and Messing's paper is sufficiently abstract that
> few physicists will succeed in understanding it before I finish
> writing a paper on this subject. :-)
I certainly bounced off it pretty hard.
> I don't know D-branes from beans,
Feel free to ask.
> but there's a nice theory of categorified differential
> geometry waiting to be developed here, which seems bound to have
> *some* application to physics... and nonabelian gerbes are really
> just the tip of the iceberg.
I think that some of this might have been developed in the theory of
stringy orbifolds, but I don't claim to understand it.
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>In article <a8r68c$kfu$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
>wrote:
>> There does exist a theory of nonabelian gerbes, going back to
>> Giraud's 1971 paper on nonabelian cohomology.
>Is this what Brylinski talks about?
No, Brylinski talks only about abelian gerbes: U(1) gerbes, to
be precise. In case anyone is wondering what the hell those
are, the basic goal is to dream up some sort of geometrical
structure on a manifold M, a "U(1) gerbe", which will be
classified by elements of the cohomology group H^3(M,Z) in
the same way that U(1) bundles are classified by elements of
H^2(M,Z). This is not so hard - it's a piddling special case
of what Giraud did back in 1971. Brylinski's book then goes
on to show that *everything* you can do with U(1) bundles has
an analogue for U(1) gerbes. For example, just as a U(1)
bundle can be equipped with a connection A whose curvature is
a closed 2-form F whose cohomology class defines the element
of H^2(M,Z) we're talking about here, a U(1) gerbe can be
equipped with a "connective structure" A whose curvature is
a closed 3-form whose cohomology class defines an element of
H^3(M,Z). Just as the connection on a U(1) bundle is not
really a 1-form A with dA = F, but only *locally* something
like a 1-form, the connective structure A on a U(1) gerbe is
not really a 2-form, but only *locally* something like a 2-form.
Just as you can also get the element of H^2(M,Z) from a U(1)
bundle via Cech cohomology, ditto for the element of H^3(M,Z)
coming from a U(1) gerbe. And so on.
One reason this is all so interesting is that there's a map
from H^3(M,Z) to H^2(LM,Z) where LM is the space of loops in
M - something which shows up all the time in string theory.
Another reason it's interesting is that when M = G is a simple
Lie group, there is always a god-given nonzero element of
H^3(G,Z). So, it turns out that the theory of the "loop group"
LG is profoundly related not only to string theory but also to
the theory of U(1) gerbes on G.
There is really a lot of nice math here, intimately connected to
the other "stringy math" like quantum groups, conformal field theories
and the like. What primarily excited me, though, was that a U(1)
gerbe is really a *categorification* of the concept of U(1) bundle
(or more precisely, of the sheaf of sections of a complex line
bundle, which is just another way of tackling the same concept).
In particular, it makes very precise the sense in which categorifying
electromagnetism gives "2-form electromagnetism" - a term which
neglects the fact that a connection is only *locally* the same
thing as a differential form.
I talked about this ages ago in "week25", which I'll append below.
The case of nonabelian gerbes is a lot more interesting, because
they are geometrical gadgets which are classified not by cohomology
classes in
H^3(M,Z) = H^2(M,U(1)),
but by second cohomology classes with coefficients in a *nonabelian*
group. The very concept of a second cohomology group with nonabelian
coefficients seemed like a contradiction in terms until the work of
Giraud! For example, first Cech cohomology makes sense with coefficients
in a nonabelian group G, and classifies principal G-bundles, but
*second* Cech cohomology with coefficients in a nonabelian group makes
no sense.
Nonetheless Giraud figured out how to define nonabelian 2nd cohomology
and see that it classifies a categorified version of principal G-bundles -
so-called "nonabelian gerbes". What Breen and Messing do is figure out
the correct concept of connection and curvature on such a thing.
>> Luckily
>> for me, Breen and Messing's paper is sufficiently abstract that
>> few physicists will succeed in understanding it before I finish
>> writing a paper on this subject. :-)
>I certainly bounced off it pretty hard.
Yup, me too - until I talked to him. There's a lot of it which
turns out to be, well, gratuitously abstract. For example, they
do all their geometry over a scheme instead of a manifold, which
is not really necessary - though it's good for the applications
*they* have in mind. Also, they use a definition of differential
forms based on Kock's book "Synthetic Differential Geometry",
instead of one physicists like us might be more familiar with.
Finally, physicists are generally happier with bundles rather
than sheaves, but gerbes are really a generalization of sheaves
rather than bundles. So, part of my goal is to rewrite this
paper for physicists. It will still be incomprehensible, of
course - but much, much less so.
(To prevent my paper from being too easy to follow, I will also
generalize 2nd cohomology with cohomology in a nonabelian group
to 2nd cohomology with coefficients in a 2-group - the categorified
version of a group. This extra generality actually allows one
to formulate a much broader class of "categorified Yang-Mills
equations".)
>> I don't know D-branes from beans,
>Feel free to ask.
Thanks - I'll keep the offer in mind! The problem is, I have
my own weird way of understanding everything, and until I figure
out where D-branes might possibly hook on to this, it's sort of hard
for me to know what questions to ask. Segal's talk on them helped
me a lot, though, so maybe I'll be able to come up with something
in a while.
.......................................................................
This Week's Finds in Mathematical Physics (Week 25)
John Baez
Lately, many things give me the feeling that we're on the brink of
some deeper understanding of the relations between geometry, topology,
and category theory. It is very tantalizing to see the array of clues
pointing towards the fact that many seemingly disparate mathematical
phenomena are aspects of some underlying patters that we don't really
understand yet. Louis Crane expressed it well when he said that it's as
if we are a bunch of archeologists digging away at different sites,
and are all starting to find different parts of the skeleton of some
gigantic prehistoric creature, the full extent of which is still unclear.
I want to keep studying the following book until I understand it,
because I think it makes a lot of important connections... pardon the
pun:
1) Loop Spaces, Characteristic Classes and Geometric Quantization, by
Jean-Luc Brylinski, Birkhauser, Boston, 1993. ISBN 0-176-3644-7
The title of this book, while accurate, really does not convey the
*novelty* of the ideas it contains. All three subjects listed have been
intensively studied by many people for at least several decades, but
Brylinski's book is not so much a summary of what is understood about
these subjects, as a plan to raise the subjects to a whole new level.
I can't really describe the full contents of the book, since I haven't
had time to really absorb some of the most interesting parts, but let me
start by listing the contents, and then talk about it a bit.
1. Complexes of Sheaves and Their Hypercohomology
2. Line Bundles and Geometric Quantization
3. Kaehler Geometry of the Space of Knots
4. Degree 3 Cohomology - The Dixmier-Douady Theory
5. Degree 3 Cohomology - Sheaves of Groupoids
6. Line Bundles over Loop Spaces
7. The Dirac Monopole
It should be clear that while this is a very mathematical book, it is
informed by ideas from physics. As usual, the physical universe is
serving to goad mathematics to new heights!
The first two chapters are largely, but not entirely, "standard"
material. I put the word in quotes because while Brylinski's treatment
of it starts with the basics - the definition of sheaves, sheaf
cohomology, Cech cohomology, deRham theory and the like - even these
"basics" are rather demanding, and the slope of the ascent is rather
steep. Really, the reader should already be fairly familiar with these
ideas, since Brylinski is mainly introducing them in order to describe a
remarkable generalization of them in the next chapters.
Let me quickly give a thumbnail sketch of the essential ideas behind
this "standard" material. In classical mechanics the main stage is
the phase space of a physical system. Points in this space represent
physical states; smooth functions on it represent observables.
Time evolution acts on this space as a one-parameter group of
diffeomorphisms. The remarkable fact is that time evolution is
determined by an observable, the Hamiltonian, or energy function, by
means of a geometric structure on phase space called a symplectic
structure. This is a nondegenerate closed 2-form. The idea is that the
differential of the Hamiltonian is a 1-form; since the symplectic
structure is nondegenerate it sets up an isomorphism of the tangent and
cotangent bundles of phase space, allowing us to turn the differential
of the Hamiltonian into a vector field; this vector field generates the
1-parameter group of diffeomorphisms representing time evolution; and by
the magic of symplectic geometry, these diffeomorphisms automatically
preserve the symplectic structure.
This is the starting-point of the beautiful approach to quantum theory
known as geometric quantization, founded by Kostant in the early 1970's.
His first paper is still a good place to start:
2) Quantization and unitary representations, by Bertram Kostant, in
Lectures in Modern Analysis and Applications III, Springer-Verlag
Lecture Notes in Mathematics 170 (1970), 87-208.
Here the idea is to construct a Hilbert space of states of the *quantum*
system corresponding to the classical system, and turn time evolution
into a one-parameter group of unitary operators on this Hilbert space.
Extremely roughly, the idea is to first look at the space of all L^2
complex functions on phase space, and then use a "polarization" to cut
down this "prequantum" Hilbert space to "half the size," by which one
means something vaguely like how L^2(R^n) is "half the size" of
L^2(R^{2n}) - this being the classic example. But in fact, it turns out
one doesn't really want to use *functions* on phase space, but instead
sections of a certain complex line bundle. The point is that the
classification of line bundles fits in beautifully with symplectic
geometry. We can equip any line bundle with a hermitian connection; the
curvature of this connection is a closed 2-form; this determines an
element of the 2nd cohomology of phase space called the first Chern
class. An important theorem says this class is necessarily an
*integral* class, that is, it comes from an element of the 2nd
cohomology with integer coefficients; moreover, isomorphism classes of
line bundles over a manifold are in one-to-one correspondence with
elements of its 2nd cohomology with integer coefficients. The trick,
then, is to try to cook up a line bundle over phase space with a
connection whose curvature is the symplectic structure! This will be
possible precisely when the symplectic structure defines an integral
cohomology class. In fact, this integrality condition is nothing but
the old Bohr-Sommerfeld quantization condition dressed up in spiffy new
clothes (and made far more precise).
So: the moral I want to convey here is just that if the symplectic
structure on phase space defines an integral class in the 2nd cohomology
group, then we get a line bundle over phase space which helps us get
going with quantization. It then turns out that the one-parameter
group of diffeomorphisms defined by any Hamiltonian on phase space lifts
to a one-parameter group of transformations of this line bundle, which
allows us to get a unitary operator on the space of L^2 sections of the
line bundle. This is not the end of the quantization story; one still
needs to chop down this "prequantum" space to half the size, etc.; but
let me leave off here.
What Brylinski wants to do is to find analogs of all these phenomena
involving the THIRD cohomology groups of manifolds.
At first glance, this might seem to be a very artificial desire. Note
that importance of the SECOND cohomology group in the above story is
twofold: 1) symplectic structures give elements of the second
cohomology, 2) the curvature of a connection gives an element of the
second cohomology, and in fact 2') line bundles are classified by
elements of second cohomology. None of these beautiful things seem to
have analogs in third cohomology! Of course, one can use the curvature
of a connection to get, not just the first Chern class, but higher Chern
classes. But the nth Chern class is an element of the 2nth cohomology
group, so the odd cohomology groups don't play a major role here. Of
course, experts will immediately reply that there are also Chern-Simons
"secondary characteristic classes" that live in odd cohomology, at least
when one has a flat bundle around. And the same experts will
immediately guess that, because Chern-Simons theory has been near the
epicenter of the explosion of new mathematics relating quantum groups,
topological quantum field theories, conformal field theory and all that
stuff, I must be leading up to something along these lines.... Well,
there *must be* a relationship here, but actually it is not emphasized
in Brylinski's book! He takes a different tack, as follows.
The basic point is that given a manifold M, the space of loops in M, say
LM, is a space of great interest in its own right. It is infinite-
dimensional, but that should not deter us. When G is a Lie group, LG is
also a group (with pointwise operations); these are the famous loop
groups, which appear as groups of gauge transformations in conformal
field theory. When M is a 3-dimensional manifold, LM contains within it
the space of all knots in M; also, we may think of LM as the
configuration space for the simplest flavor of string theory in the
spacetime R x M. Loops also serve to define observables called "Wilson
loops" in gauge theories, and these are the basis of the loop
representation of quantum gravity. So there is a lot of interesting
mathematics and physics to be found in the loop space.
What does this have to do with the 3rd cohomology group of M? Well, LM
is a bundle over M, so according to algebraic topology there is a
natural map from the 3rd cohomology of M to the 2nd cohomology of LM!
The ramifications of this are multiple.
First, every compact simple Lie group G has 3rd cohomology equal to Z.
(In fact, Brylinski notes that the cohomology group is not merely
isomorphic to Z, but canonically so - and this extra nuance turns out to
be quite significant!) This gives rise to a special element in the 2nd
cohomology of LG. This then gives a line bundle over LG.
Alternatively, it gives a circle bundle over LG, in fact a central
extension of LG, that is, a bigger group LG^ and an exact sequence
1 -> S^1 -> LG^ -> LG -> 1
This group is called a Kac-Moody group, and these are well-loved by
string theorists since it turns out that when one wants to quantize a
gauge theory on the string worldsheet (a kind of conformal field theory)
one gets, not a representation of the gauge group LG on the Hilbert
space of quantum states, but merely a projective representation, or in
other words, a representation of the central extension LG^. Brylinski
also notes that in some sense the canonical element in the 3rd
cohomology of G is responsible for the existence of quantum groups; this
is probably the deep reason for the association between quantum group
representations and Kac-Moody group representations, but, alas, this is
still quite murky to me.
Second, we can do better if we restrict ourself to knots (possibly with
nice self-intersections) rather than loops. Namely, given a 3-manifold
M equipped with a 3-form, one gets, not just an element of the 2nd
cohomology of LM, but a symplectic structure on the space of knots in M,
say KM. It may seem odd to think of the space of knots as a physical
*phase* space, but Brylinski shows that this idea is related to the work
of Marsden and Weinstein on "vortex filaments," an idealization of fluid
dynamics in which all the fluid motion is concentrated along some
curves. Brylinski also notes that if M is equipped with a Riemannian
structure then KM inherits a Riemannian structure (this is easy), and
that if M has a conformal structure KM has an almost complex structure.
In fact, in the Riemannian case all these structures on KM fit together
to make it a sort of Kaehler manifold (although one must be careful,
since the almost complex structure is only integrable in a certain
formal sense). Brylinski hints that all this geometry may give a
nice approach to the study of knot invariants; I will have to look at
the following papers sometime:
3) Vortices in He II, current algebras and quantum knots, by M. Rasetti
and T. Regge, Physica 80A (1975) 217-233.
4) A geometric approach to quantum vortices, by V. Penna and M. Spera,
J. Math. Phys. 30 (1989), 2778-2784.
However, Brylinski's real goal is something much more radical! The
beauty of 2nd cohomology is that integer classes in the 2nd cohomology
of M correspond to line bundles on M; there is, in other words, a very
nice geometrical picture of 2nd cohomology classes. What is the natural
analog for 3rd cohomology? Instead of just working with LM, it would be
nice to have some sort of geometrical objects on M that correspond to
integer classes in 3rd cohomology. What should they be?
Brylinski gives two answers, one in Chapter 4 and another in Chapter 5.
The first one, due mainly to Dixmier and Douady, is very appealing for
a quantum field theorist such as myself. Just as elements of H^2(M,Z)
correspond to line bundles over M, elements of H^3(M,Z) correspond to
projective Hilbert space bundles over M! Recall that in physics two
vectors in a Hilbert space correspond to the same physical state if one
is a scalar multiple of the other; the space of equivalence classes
(starting with a countable-dimensional Hilbert space) is what I'm
calling "projective Hilbert space," and it is bundles of such rascals
that correspond to elements of H^3(M,Z). The reason is roughly this:
the structure group G for such bundles is the group Aut(H)/C*, that is,
invertible operators on the Hilbert space H, modulo invertible complex
numbers. In other words, we have an exact sequence
1 -> C* -> Aut(H) -> G -> 1
This gives an exact sequence of sheaves on M, which, combined with the
marvelous fact that Aut(H) is contractible, gives an isomorphism between
H^1(M,sh(G)) (the cohomology of the sheaf of smooth G-valued functions
on M) and H^2(M,sh(C*)). But the latter is isomorphic to H^3(M,Z).
Brylinski pushes the analogy to the line bundle case further by showing
how to realize the element of H^3(M,Z) starting from a connection on a
projective Hilbert space bundle. But in Chapter 5 he takes a more
abstract approach that I want to sketch very vaguely, since I don't
understand it very well yet. This approach is exciting because it
connects to recent work on 2-categories (and higher n-categories), which
I am convinced will play a role in unifying the wild profusion of
mathematics we are seeing in this tail end of the twentieth century.
Here the best way to see the analogy to the line bundle case is through
Cech cohomology. Recall that we can patch a line bundle together by
covering our manifold M with charts O(i) and assigning to each
intersection O(i) ^ O(j) (that's supposed to be the symbol for
"intersection") a C*-valued function g(ij). These "transition
functions" must satisfy the compatibility condition
g(ij) g(jk) g(ki) = 1
We say then that the functions g(ij) define a 1-cocycle in Cech
cohomology - think of this as just jargon, if you like. Note that we
will get an isomorphic line bundle if we take some C*-valued functions
f(i), one on each chart O(i), and multiply g(ij) by f(i)f(j)^{-1}. This
simply amounts to changing the trivialization of the bundle on each
chart. We say that the new Cech cocycle differs by a coboundary.
So line bundles are in 1-1 correspondence with the 1st Cech cohomology
with values in sh(C*). This turns out to be the same thing as H^2(M,Z),
as noted above.
Now, there is a marvelous thing called a gerbe, which is like a bundle,
but is pieced together using Cech 2-cocyles! These will be classified
by the 2nd Cech cohomology with values in sh(C*), which is nothing but
H^3(M,Z).
What are these gerbes? Well, I wish I really understood them. Let me
just say what I know. The basic idea is to boost everything up a notch
using category-theoretic thinking. When we were getting ready to define
bundles, we needed to have the concept of a group at our disposal (to
have a structure group.) For gerbes, we need something called a category
of torsors. What is a group? Well, it is a SET equipped with various MAPS
satisfying various properties. What is a category of torsors? Well, it is a
CATEGORY equipped with various FUNCTORS satisfying utterly analogous
properties. Note how we are "categorifying" here. We have more
structure, since while a set is just a bunch of naked points, a category
is a bunch of points, namely objects, which are connected by arrows,
namely morphisms. Given the group C* we can get a corresponding category
of torsors as follows: the category of all manifolds with a simply transitive
C*-action (which are called torsors). A nice account of why this category
looks so much like a group appears in
5) Higher algebraic structures and quantization, by Dan Freed,
preprint, December 18, 1992, available as hep-th/9212115.
which I already mentioned in week12.
Just as a group can act on a set, a category of torsors can act on a
category. If we "sheafify" this notion, we get the concept of a gerbe.
Clear? Well, part of why I am interested in these ideas is the way they
make me a bit dizzy, so don't feel bad if you are a bit dizzy too now.
I really think that overcoming this dizziness will be necessary for certain
advances in mathematics and physics, though.
Instead of actually coming clean and defining the concept of a gerbe, let
me finish by saying what Brylinski does next. He defines an analog of
connections on bundles, called "connective structures" on gerbes. And he
defines an analog of the curvature, the "curving" of a connective
structure. This turns out to give an element of H^3(M,Z) in a natural
way. He concludes in a blaze of glory by showing how the Dirac
monopole gives a gerbe on S^3 whose curving is the volume form. The
integrality condition turns out to be related to Dirac's original
argument for quantization of electric charge. Whew!
Squark
> ...
> Nonetheless Giraud figured out how to define nonabelian 2nd cohomology
> and see that it classifies a categorified version of principal G-bundles -
> so-called "nonabelian gerbes". What Breen and Messing do is figure out
> the correct concept of connection and curvature on such a thing.
Where can I read about those results? Maybe they can be related to the
non-Abelians solitons I've mentioned here, which lead to natural notion of
a 2nd Eilenberg-MacLane space for a non-Abelian group.
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and use "com" rather than
"exe")
Aaron Bergman
wrote:
> In article <abergman-75036C...@news.bellatlantic.net>,
> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >In article <a8r68c$kfu$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
> >wrote:
>
> >> There does exist a theory of nonabelian gerbes, going back to
> >> Giraud's 1971 paper on nonabelian cohomology.
>
> >Is this what Brylinski talks about?
>
> No, Brylinski talks only about abelian gerbes: U(1) gerbes, to
> be precise.
ISTR some mumbling about being able to extend the exact sequence for
cohomology with coefficients in a nonabelian group one more step, but I
don't have any more.
[...]
> Nonetheless Giraud figured out how to define nonabelian 2nd cohomology
> and see that it classifies a categorified version of principal G-bundles -
> so-called "nonabelian gerbes". What Breen and Messing do is figure out
> the correct concept of connection and curvature on such a thing.
Do they do higher degrees?
>
> >> Luckily
> >> for me, Breen and Messing's paper is sufficiently abstract that
> >> few physicists will succeed in understanding it before I finish
> >> writing a paper on this subject. :-)
>
> >I certainly bounced off it pretty hard.
>
> Yup, me too - until I talked to him. There's a lot of it which
> turns out to be, well, gratuitously abstract.
Just what is a torsor, anyways....
> For example, they
> do all their geometry over a scheme instead of a manifold, which
> is not really necessary - though it's good for the applications
> *they* have in mind. Also, they use a definition of differential
> forms based on Kock's book "Synthetic Differential Geometry",
> instead of one physicists like us might be more familiar with.
> Finally, physicists are generally happier with bundles rather
> than sheaves, but gerbes are really a generalization of sheaves
> rather than bundles.
We're pretty cool with sheaves these days.
[...]
> >> I don't know D-branes from beans,
>
> >Feel free to ask.
>
> Thanks - I'll keep the offer in mind! The problem is, I have
> my own weird way of understanding everything, and until I figure
> out where D-branes might possibly hook on to this, it's sort of hard
> for me to know what questions to ask.
D-branes are the secret to everything, somehow. If only we knew what
they are. A recent paper seems to suggest that maybe D-branes are
classified by the K-theory of the space of sections of an LE_8 bundle as
opposed to a PU(oo) bundle a la Dixmier-Douady.
Of course, pertubatively D-branes are just boundary states in a CFT. I
think Moore and Segal have been working on this for a while. I don't
know what the progress, but that might be what Segal talked about.
Thomas Larsson
> First, every compact simple Lie group G has 3rd cohomology equal to Z.
> (In fact, Brylinski notes that the cohomology group is not merely
> isomorphic to Z, but canonically so - and this extra nuance turns out to
> be quite significant!) This gives rise to a special element in the 2nd
> cohomology of LG. This then gives a line bundle over LG.
> Alternatively, it gives a circle bundle over LG, in fact a central
> extension of LG, that is, a bigger group LG^ and an exact sequence
>
> 1 -> S^1 -> LG^ -> LG -> 1
>
More generally, for any n-dimensional manifold M, the group Map(M, G)
of G-valued maps from M has a central extension MG^, i.e. there is an
exact sequence
1 -> Omega^1/ d Omega^1 -> MG^ -> Map(M, G) -> 1,
where Omega^1 = Lambda^(n-1) T*M is the bundle of dual one-forms over
M. In particular, when n=1 the abelian ideal of closed dual 1-forms is
isomorphic to S^1, so this is a genuine generalization of loop groups.
It is important that the diffeomorphism group Diff(M) acts on MG^.
Pressley and Segal remark that "from one point of view it is a
disappointing result, as it tells us that there are no 'interesting'
extensions of Map(M; g) when dim(M) > 1. More precisely, if f: S^1 -> M
is any smooth loop in M one can always obtain an extension of Map(M, g)
by pulling back the universal extension of Lg by f. Proposition (4.2.8)
asserts that any extension is a weighted linear combination of
extensions of this form." However, to proclaim that this cocycle is
uninteresting is misleading for three reasons:
1. Although the cocycle lives on a fixed loop in M, there are
representations that genuinely depend on dimensionality. Given a
G-bundle E -> M, the natural modules are built from LJ^pE, the loops
in the jet bundle J^pE. Only if p=0 is the module a pull-back from
an Lg module.
2. There is an analogous, 2-parameter, extension of Diff(M):
1 -> Omega^1/ d Omega^1 -> Vir(M) -> Diff(M) -> 1.
The cocycle commutes with itself but not with Diff(M), and hence this
extension is not central, only abelian. This is dramatically different
from the one-dimensional case.
3. Already the classical representations of Diff(M) (tensor densities)
depend crucially on dimension.
This is a good point to clarify my dispute with Prof. Distler on
anomalies last fall. Locally and infinitesimally, an extension of
Map(M, G) gives rise to a cocycle of the Lie algebra map(n, g). The
generators have the form X_a(x) J^a, where J^a in g and X_a(x) are
g-valued functions on R^n. Any extension of this algebra takes the form
[J_X, J_Y] = J_[X,Y] + c(X,Y).
There are two main types of cocycles.
1. The Mickelsson-Faddeev cocycle (when n=3)
c_MF(X,Y) = \int tr (dX dY A)
= d^abc e^ijk \int d^3x d_i X_a(x) d_j Y_b(x) A_ck(x),
where A is the gauge connection 1-form and d_i = d/dx^i.
2. The Kac-Moody cocycle
c_KM(X,Y) = \int dt \dot q^i(t) d_i X_a(q(t)) Y_b(q(t)) \delta^ab,
where q^i(t) describes the loop in R^n. This cocycle
was first described by Kassel in 1985 and its representations were
first studied by Rao, Moody and Yokonoma in 1990. The global group
analogue MG^ has also recently been studied by Karl-Herrmann Neeb of
TU Darmstadt.
Both cocycles admit an intertwining action of vect(n), the algebra of
infinitesimal diffeomorphisms = vector fields. c_KM also naturally
admits an action of vect(1), which describes reparametrizations of
the marked loop. Fock modules are of lowest-energy type w.r.t. the L_0
in this vect(1). As the generator of rigid time translations along
the observer's trajectory, it can be considered as a kind of Hamiltonian,
hence the word lowest-energy rather than lowest-weight.
Let us contrast these very different cocycles:
1. c_MF lives on the full 3D space, c_KM only on a loop, as
Pressley-Segal noted.
2. c_MF is non-zero only if d^abc != 0, and in particular it always
vanishes for g = su(2). c_KM is always non-zero if the Killing metric
\delta^ab is.
3. c_MF arises as an unwanted anomaly in the quantization of chiral
fermions coupled to a gauge field. c_KM arises when one constructs
lowest-energy modules of map(n,g).
4. The representations of the MF algebra are only half quantum:
quantum chiral fermions but a classical background gauge field. The
reps of the higher-dim KM algebra are fully quantum - all types of
fields are treated equally, and the energy (= vect(1) L_0 eigenvalue)
is manifestly bounded from below.
5. The MF cocycle can be generalized to arbitrary odd dimensions n >= 3.
c_MF(X,Y) = \int tr (dX dY A F ... F)
where there are (n-3)/2 factors F (the gauge field strength), to ensure
that the integrand is an n-form. There is also an analogous cocycle for
the diffeomorphism algebra which involves four derivatives (rather than
two) and the Levi-Civita connection. When Distler says that there are
no gravitational anomalies in four dimensions, he probably means that
(n-3)/2 is not an integer when n=4, so there is no natural realization
of c_MF, or rather its diffeomorphism analogue. In contrast, c_KM is
non-zero in any spacetime dimension.
So my conclusion is that c_MF is a bad cocycle; mathematically because
it lacks true Fock modules, and physically because the ABJ anomaly is
known to vanish. In contrast, c_KM is a good cocycle at least
mathematically, since it has Fock modules. The higher-dimensional KM
algebra is IMO the correct quantum form of gauge symmetries. That it
is nevertheless possible to treat gauge symmetries within QFT without
knowing about this algebra is something of a mystery for me. OTOH, that
QFT works at all, with its strange renormalization prescription, is a
mystery for many, and it is well-known that it cannot handle gravity.
Therefore, I hope that it may be possible to reformulate QFT so that
the Kac-Moody cocycle becomes explicit, renormalization becomes
transparent, and gravity becomes quantizable. That the higher-dimensional
KM cocycle exists is at least an undisputable mathematical fact.
> This group is called a Kac-Moody group, and these are well-loved by
> string theorists
This contrasts sharply with theoretical physicists' stubborn refusal to
learn anything about its higher-dimensional analogue.
Rogier Brussee
John Baez wrote
In article <abergman-75036C...@news.bellatlantic.net>,
Aaron Bergman <aber...@princeton.edu> wrote:
>>In article <a8r68c$kfu$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
>>wrote:
>
>>> There does exist a theory of nonabelian gerbes, going back to
>>> Giraud's 1971 paper on nonabelian cohomology.
>
>>Is this what Brylinski talks about?
>
>No, Brylinski talks only about abelian gerbes: U(1) gerbes, to
>be precise. In case anyone is wondering what the hell those
>are, the basic goal is to dream up some sort of geometrical
>structure on a manifold M, a "U(1) gerbe", which will be
>classified by elements of the cohomology group H^3(M,Z) in
>the same way that U(1) bundles are classified by elements of
>H^2(M,Z).
[snip]
There is a very nice and readable article by Nigel Hitchin that (among
other things) describes gerbes in a relatively pedestrian way. In
particular it explains how Spinors give rise to special gerbes.
quote:
What is a gerbe? The word gerbe is an odd one for English speakers:
it is not a furry little animal, but derives instead from a French word
in more
common use (look at Renoir's painting "Petite fille a la gerbe").
However, it has in
fact existed in the English language for some time. Here's what the
Oxford English
Dictionary gives:
Gerbe. 1698 [ - Fr. gerbe wheat-sheaf ] 1. A wheat-sheaf 1808. 2.
Something resembling a sheaf of wheat: esp. a kind of firework.
and something resembling a sheaf" is quite close to the mathematical
meaning of
the word.
See math.DG/9907034 ( http://arxiv.org/pdf/math.DG/9907034 )
Title: Lectures on Special Lagrangian Submanifolds
Authors: Nigel Hitchin
Comments: Lectures given at the ICTP School on Differential Geometry
April 1999
Subj-class: Differential Geometry; Algebraic Geometry
MSC-class: 53C25, 53C80, 14J30
Rogier Brussee
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>"John Baez" <ba...@galaxy.ucr.edu> schrieb im Newsbeitrag
>news:a6u158$md6$1...@glue.ucr.edu...
>> Since January I've been lecturing on categorified gauge theory and
>> p-form electromagnetism.
>Does p-form electromagnetism by itself have a direct connection to
>quantum gravity, or only via its n-category aspect?
p-form electromagnetism shows up in string theory, as do more general
theories involving connections on n-gerbes, but the main reason I'm
interested in all this stuff is that it seems like a fun way of using
n-categories to dig deeper into differential geometry. For example:
the Yang-Mills equations have interesting self-dual solutions in 4
dimensions, which turn out to say a lot about the topology of 4-manifolds,
via Donaldson theory. If you categorify the Yang-Mills equations once,
you get some equations which with a suitable ansatz have self-dual
solutions in 5 dimensions! I can't help but hope this will be interesting.
Of course it would also be great if Nature likes categorified gauge
theory, but I don't know if it does. I believe it more or less has
to if some analogy like
parallel transport of point particles ---> connections on bundles
parallel transport of p-branes ---> ??????????????????????
is operative.
>BTW, did you see my recent answer to your question about local
>observables in canonical gravity? Was it too confused?
Yes, I saw it... it wasn't confused; I've just been too busy to
answer it! The main thing I wanted to say is that perhaps you
underestimate the enormous gap between models in which one has
eliminated all but finitely many degrees of freedom by symmetry
or other assumptions, and full-fledged quantum gravity. In the
former case the problem of getting diffeomorphism-invariant
observables is fairly tractable; in the latter case it is not.
Not yet, anyway! And personally, I believe that the former case
only gives a *tiny* bit of insight into the latter case, which is
the one I'm mainly interested in.
On a different note, your remarks about combining all p-form
electromagnetism theories into one big theory were interesting.
I have no idea how this stuff is related to n-categories!
John Baez
Squark <fii...@yahoo.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<a90mtc$htc$1...@glue.ucr.edu>...
>> Nonetheless Giraud figured out how to define nonabelian 2nd cohomology
>> and see that it classifies a categorified version of principal G-bundles -
>> so-called "nonabelian gerbes". What Breen and Messing do is figure out
>> the correct concept of connection and curvature on such a thing.
>Where can I read about those results? Maybe they can be related to the
>non-Abelians solitons I've mentioned here, which lead to natural notion of
>a 2nd Eilenberg-MacLane space for a non-Abelian group.
I understand what a "2nd Eilenberg-MacLane space for a nonabelian group"
would be, i.e. a classifying space for nonabelian gerbes, so if you
paid me enough money I could just tell you about that.
However, if you can't afford that, try:
Lawrence Breen, William Messing
Differential Geometry of Gerbes
http://xxx.lanl.gov/abs/math.AG/0106083
which while far from easy to read, does have extensive references to
the previous literature on nonabelian gerbes.
Even for free, I'll tell you what the classifying space for nonabelian
G-gerbes is: it's really the a classifying space for a strict 2-group
whose group of objects is Aut(G) and whose group of morphisms is the
semidirect product of Aut(G) and G. A strict 2-group, you'll recall, a
strict 2-groupoid with one object, or if you prefer, a group object in
the category of categories.
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>In article <a90mtc$htc$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
>wrote:
>> Nonetheless Giraud figured out how to define nonabelian 2nd cohomology
>> and see that it classifies a categorified version of principal G-bundles -
>> so-called "nonabelian gerbes". What Breen and Messing do is figure out
>> the correct concept of connection and curvature on such a thing.
>Do they do higher degrees?
No. With their approach that would probably become very time-consuming.
>Just what is a torsor, anyways....
I understand this a lot better than when I wrote "week25" - back
then, my brain fried right around the point where I was going to
explain torsors.
Given a group G, a G-torsor is a space on which G acts freely and
transitively. The most obvious example is G acting on itself by left
translation. In fact, all other G-torsors are isomorphic to this
one! So you might think it a completely silly concept. But it's not,
really: the difference is that G has a distinguished point, the
"identity", while a G-torsor does not. In short, it's just the obvious
generalization of the distinction between a vector space and an affine
space, which is a "vector space that has forgotten its origin".
Every group G gives a G-torsor by "forgetting the identity", and
every G-torsor can be identified with G after we pick a point in it
and decree that to be the identity.
The category of U(1)-torsors plays the same role in the theory of U(1)
gerbes that the group U(1) plays in the theory of U(1) bundles.
So, what's this category like? It has a bunch of objects, but they are
all isomorphic, so we get an equivalent category by just picking one,
say U(1) itself, and working with that. Then there are lots of
morphisms from this object to itself: all the right translations.
(In the abelian case right and left translations are the same, but I'm
getting you warmed up for the nonabelian case.) So, we have a category
with one object and a bunch of automorphisms of this object, which form
the group U(1).
Hey, but a category with one object and only invertible morphisms is
just a group in disguise! So, the category of U(1)-torsors is just the
group U(1) in disguise - thought of as a category rather than a set.
>Of course, pertubatively D-branes are just boundary states in a CFT. I
>think Moore and Segal have been working on this for a while.
For now I'd be happy to understand that better.
>I don't know what the progress, but that might be what Segal talked about.
Yes, though only tangentially: he mainly talked about "topological
D-branes", which are boundary states in a 2d TQFT. The idea is very
related, just as 2d TQFTs are related to CFTs.
Aaron Bergman
wrote:
> In article <abergman-3FC21E...@news.bellatlantic.net>,
Take a CFT on the upper half plane. What boundary conditions are you
going to put on the real axis?
> >I don't know what the progress, but that might be what Segal talked about.
>
> Yes, though only tangentially: he mainly talked about "topological
> D-branes", which are boundary states in a 2d TQFT. The idea is very
> related, just as 2d TQFTs are related to CFTs.
Yeah. That was it. CFTs were too hard, I think, but topological strings
might be tractable.
Urs Schreiber
[...]
> For example:
> the Yang-Mills equations have interesting self-dual solutions in 4
> dimensions, which turn out to say a lot about the topology of 4-manifolds,
> via Donaldson theory.
It is intriguing how the number of 4 dimensions keeps turning out to be
special. Yes, I know, many other numbers of dimensions are special in
one sense or another, but it is often 4 or (1+3) dimensions where things
are particularly nice and simple without being trivial. In this thread I have
learned that 4 is the unique number of dimensions in which one can have
pointlike dyonic particles. Given that any fundamental constituent of the
world will look pointlike in first approximation this seems remarkable.
[...]
> >BTW, did you see my recent answer to your question about local
> >observables in canonical gravity? Was it too confused?
>
> Yes, I saw it... it wasn't confused; I've just been too busy to
> answer it! The main thing I wanted to say is that perhaps you
> underestimate the enormous gap between models in which one has
> eliminated all but finitely many degrees of freedom by symmetry
> or other assumptions, and full-fledged quantum gravity. In the
> former case the problem of getting diffeomorphism-invariant
> observables is fairly tractable; in the latter case it is not.
> Not yet, anyway! And personally, I believe that the former case
> only gives a *tiny* bit of insight into the latter case, which is
> the one I'm mainly interested in.
I see. Yes, you are right that I am mainly thinking in terms of the
truncated theory and I am extrapolating by means of wishful
thinking. I am trying to first fully understand the 'mini'-models (which
need not be that mini, after all), because that seems to be demanding
enough. For me, that is. On the other hand, I think it is fair to say that
experts much more knowledgeable than me also do struggle with the
central concepts even of the simplified models. For example, I had
heard a talk by Renate Loll where she reviewed her work on
Lorentzian path integrals in 1+1 dimensional gravity. After the talk
members of the audience expressed their concern that they believed
that Loll's solutions did not satisfy the Hamiltonian constraint. She
disagreed, of course, and there was no general consensus.
Sometimes I feel that maybe tackling full fledged quantum gravity
without first having carefully sorted out the concepts of the truncated
theory is like trying to find the spectral theorem without having
matrix diagonalization down cold.
John Baez
decided I'd prefer to reply to it publicly, with his kind
permission. He writes:
>Hi,
>Just tried to reread all the posts in this thread--- my head is spinning!
>But it certainly seems that I should learn more about gerbes. I can tell
>you are very busy, but I hope you have time to answer a quick question,
>actually a really dumb question. I see you said:
>> first Cech cohomology makes sense with coefficients in a nonabelian
>> group G, and classifies principal G-bundles,
>What's the best place to read about how this works?*
Hmm... I don't know the BEST place. Perhaps Bott and Tu's
"Algebraic Topology and Differential Forms"? I know they
talk a lot about Cech cohomology there, and they'd have to
be perverse not to include this application.
Maybe someone else knows a good reference.
>I was very intrigued by this comment:
>> The case of nonabelian gerbes is a lot more interesting, because
>> >they are geometrical gadgets which are classified not by cohomology
>> classes in
>>
>> H^3(M,Z) = H^2(M,U(1)),
^^^
>Is this what you meant to write? I'm just checking that own all the
>confusion I'm feeling :-/
Yes, but this isn't quite so bad as it may seem: it's just
a consequence of the short exact sequence of abelian groups
0 -------> Z --------> R -------> U(1) ----> 0
which, as always, gives a long exact sequence in cohomology
.... -> H^n(M,Z) -> H^n(M,R) -> H^n(M,U(1)) -> H^{n+1}(M,Z) -> ....
Here we are taking advantage of the trickiest part of this long
exact sequence, the map H^n(M,U(1)) -> H^{n+1}(M,Z), which goes by
the charming name of "the Bockstein". This business works for
any exact sequence of abelian groups and any respectable cohomology
theory - but here we are using Cech cohomology, for which H^n(M,R)
always vanishes, giving us an exact sequence
.... -> H^n(M,Z) -> 0 -> H^n(M,U(1)) -> H^{n+1}(M,Z) -> 0 -> ....
and thus an isomorphism
H^2(M,U(1)) = H^3(M,Z)
Finally, we use the fact that the Cech cohomology H^n(M,Z) is
isomorphic to the good old singular cohomology H^n(M,Z), so we
can interpret the left-hand side in terms of that good old cohomology,
and forget all this Cech stuff!
For the exact same reasons we have
H^1(M,U(1)) = H^2(M,Z).
In an earlier thread on s.p.r. I got Toby Bartels to prove -
back when he was a lowly undergrad! - that H^2(M,Z) classifies
U(1) bundles (or equivalently, complex line bundles) on a manifold M.
The element of H^2(M,Z) corresponding to a given U(1) bundle is
called its "first Chern class", and there are lots of different
ways to compute it.
The business about using H^3(M,Z) to classify U(1) gerbes on M
is just a continuation of the same story.
>> ... but by second cohomology classes with coefficients in a *nonabelian*
>> group. The very concept of a second cohomology group with nonabelian
>> coefficients seemed like a contradiction in terms until the work of
>> Giraud!
>I wonder what happens if you try to "categorify" the cohomology with
>coefficients in an abelian group of an equivalence relation? I was going
>to try to post an explanation of cohomology of an equivalence relation
>last year, but my enthusiasm gave out. Maybe I'll try again next week.
Hmm, I don't know about this!
>*Defensive "explanation" for not knowing this already, which you can skip,
>since the only point is to express how foolish I feel for not having
>figured out where I can read all about it years and years ago-- I mean, I
>went to grad school so I wouldn't have to -ask- stupid questions, but it
>seems this plan didn't quite work...sigh...
I thought the point of going to grad school was to get a Ph.D so I
could ask stupid questions without feeling embarassed anymore: if anyone
says "that's really stupid" I just wave my diploma at them and say "maybe
so, but I have a *license* to ask stupid questions".
It's sort of like the scarecrow in the Wizard of Oz.
>About five years ago, when I was still a grad student I signed up for a
>summer seminar in quantum mechanics and representation theory--- after a
>few weeks I gave a presentation in which I claimed that state space is
>really -projective- Hilbert space. At that time I hadn't seen this stated
>anywhere, and I was really puzzled why none of the books I was looking at
>mentioned it. Even worse, -everyone- in the class, including the
>professor (who is an expert on representations but was using the seminar
>to try to teach himself about the physical applications) insisted I was
>-wrong-!!! Very frustrating to have learned years later that if only I'd
>known the right references I could have proven that they all really were
>missing my point--- instead, I just dropped the darned seminar. Story of
>my tiny life.
Well, by now I'm sure you know that you can extract useful information
from a course even if the professor doesn't know *everything* about
the subject.
>Last year I spent weeks studying some books on the topology of Lie groups
>specifically trying to learn about this, and wound up learning how to
>compute -equivariant- de Rham cohomology, but I didn't find what I was
>really looking for! Maybe this just means that I keep missing the point
>of what I'm reading...sigh...
I dunno. Equivariant de Rham cohomology is pretty cool, though
I could never stomach it until I learned other forms of equivariant
cohomology - it seemed like black magic.
Squark
> ...
> Even for free, I'll tell you what the classifying space for nonabelian
> G-gerbes is: it's really the a classifying space for a strict 2-group
> whose group of objects is Aut(G) and whose group of morphisms is the
> semidirect product of Aut(G) and G.
How much will I have to pay, then, so you tell me what's the groups of
morphisms of a strict 2-group is? :-) Hmm, maybe it's the group whose
elements are triples (a,b,f) where a and b are objects and f is in
Mor(a,b), with the product being (a, b, f)(c, d, g) = (ac, bd, fg),
where fg is of course defined, for instance because the "group
multiplication" is a functor. If I'm right, it only remains to check
whether it's the same space I had in mind...
Chris Hillman
John wrote:
> >> H^3(M,Z) = H^2(M,U(1)),
> ^^^
I, very confused, asked:
> >Is this what you meant to write? I'm just checking that own all the
^
I
> >confusion I'm feeling :-/
John explained:
> Yes, but this isn't quite so bad as it may seem: it's just
> a consequence of the short exact sequence of abelian groups
>
> 0 -------> Z --------> R -------> U(1) ----> 0
>
> which, as always, gives a long exact sequence in cohomology
>
> .... -> H^n(M,Z) -> H^n(M,R) -> H^n(M,U(1)) -> H^{n+1}(M,Z) -> ....
<foreheadsmack>
As it happens, I just reread a very nice introduction to homological
algebra in volume 2 of
author = {Nathan Jacobson},
title = {Basic Algebra},
note = {Two volumes},
edition = {Second},
publisher = {Freeman},
year = 1985}
This is a wonderful book! Every time I have opportunity to reread part of
it I am struck anew by how good the choice of topics and presentation
really is.
> Here we are taking advantage of the trickiest part of this long
> exact sequence, the map H^n(M,U(1)) -> H^{n+1}(M,Z), which goes by
> the charming name of "the Bockstein". This business works for
> any exact sequence of abelian groups and any respectable cohomology
> theory
The great thing about chapter on "homological algebra" in BAII is that
Jacobson manages to explain enough to taste the meat about a rather
general/abstract approach to homology and cohomology, but makes the
derived functors in particular seem downright friendly :-)
> - but here we are using Cech cohomology, for which H^n(M,R)
> always vanishes, giving us an exact sequence
>
> .... -> H^n(M,Z) -> 0 -> H^n(M,U(1)) -> H^{n+1}(M,Z) -> 0 -> ....
>
> and thus an isomorphism
>
> H^2(M,U(1)) = H^3(M,Z)
Excellent!!! Thanks much.
> In an earlier thread on s.p.r. I got Toby Bartels to prove - back when
> he was a lowly undergrad! - that H^2(M,Z) classifies U(1) bundles (or
> equivalently, complex line bundles) on a manifold M. The element of
> H^2(M,Z) corresponding to a given U(1) bundle is called its "first
> Chern class", and there are lots of different ways to compute it.
Toby, all this time I thought you were on the -faculty- at UCR!
Anyway, it took me much less time than I feared to brush up on homological
algebra, and I now intend to brush up on characteristic classes.
> The business about using H^3(M,Z) to classify U(1) gerbes on M
> is just a continuation of the same story.
I might ask again after I've had a chance to review some more background.
> >I wonder what happens if you try to "categorify" the cohomology with
> >coefficients in an abelian group of an equivalence relation? I was going
> >to try to post an explanation of cohomology of an equivalence relation
> >last year, but my enthusiasm gave out. Maybe I'll try again next week.
>
> Hmm, I don't know about this!
OK, I resolve to explain. I even have the additional motivation of
wanting to clarify the connection between this cohomology of G-sets and
group cohomology. Fortunately, the cohomology of an equivalence relation
is absolutely the simplest cohomology theory I've ever seen, bar none!
> I dunno. Equivariant de Rham cohomology is pretty cool, though I
> could never stomach it until I learned other forms of equivariant
> cohomology - it seemed like black magic.
For interested readers: I allege to have computed the equivariant de Rham
cohomology of the three-dimensional real Lie groups (as in Bianchi
classification) in two posts which you can find here:
http://www.math.washington.edu/~hillman/PUB/lietheory
However, this is nothing like an exposition. I toyed at the time of
posting an exposition based upon
author = {Werner Greub and Stephen Halperin and Ray Vanstone},
title = {Connections, Curvature, and Cohomology},
note = {Three volumes},
publisher = {Academic Press},
year = {1972--76}}
This book looks daunting but is actually very elegantly presented. I wish
I'd read the first few chapters of volume 1 -before- trying to learn a bit
about Grothendieck's much more abstract take. (Both approaches write long
exact sequences as -triangles-. This is actually a very good way to think
about it.)
BTW, I see from a post in sci.math.research that Vietoris has just died at
the age of 112. So this would be a good time for anyone thinking of
posting some exposition of some LES in algebraic topology to go for it!
Chris Hillman
Home page: http://www.math.washington.edu/~hillman/personal.html
Thomas Larsson
Inspired by some of the other posts in this thread, I looked up all
papers in hep-th and math-ph containing the word "gerbe" in the
abstract. Half of them were written by Jouko Mickelsson's group or
his Australian coworkers. I have a pretty good idea what Jouko is up
to, but I have never really penetrated the gerbe language. Among
the other half, I actually found a paper that was readable:
http://www.arxiv.org/abs/hep-th/0002074
p-Gerbes and Extended Objects in String Theory
Authors: Yonatan Zunger (Stanford University)
"p-Gerbes are a generalization of bundles that have (p+2)-form field
strengths. We develop their properties and use them to show that every
theory of p-gerbes can be reinterpreted as a gauge theory containing
p-dimensional extended objects. In particular, we show that every closed
(p+2)-form with integer cohomology is the field strength for a gerbe, and
that every p-gerbe is equivalent to a bundle with connection on the space
of p-dimensional submanifolds of the original space. We also show that
p-gerbes are equivalent to sheaves of (p-1)-gerbes, and use this to
define a K-theory of gerbes. This K-theory classifies the charges of
(p+1)-form connections in the same way that bundle K-theory classifies
1-form connections."
> Once I invented a lattice version of 2-Yang-Mills theory:
>
> T A Larsson
> "p-cell gauge theories, manifold space and multi-dimensional
> integrability"
> Mod Phys Lett A 5 (1990) 255-264
>
It thus seems that I studied (p-1)-gerbes on the lattice back in 1990;
the p-form connection and the (p-1)-dimensional submanifolds are there.
(My p = Zunger's p+1). It is also not surprising that I failed to find
a nice continuum formulation of my model, since gerbe theory was not
around at the time.
> The zero-curvature condition is quite interesting. In the spatially
> homogeneous case (objects depend on orientation but not on location),
> zero 1-curvature becomes U_1 U_2 U_1^{-1} U_2^{-1} = 0, i.e. U_1 and
> U_2 commute. zero 2-curvature becomes
>
> U_12 U_13 U_23 = U_23 U_13 U_12,
>
> which is the Yang-Baxter equation, of paramount importance to the
> theory of integrable lattice models in 2D.
This seems to indicate that the Yang-Baxter equation may have a natural
setting as flat connections on 1-gerbes, and more generally that the
p-simplex equation determines flat connections on (p-1)-gerbes. This is
both good and bad news.
The good news is that if one can use the gerbe picture to find solutions
to the tetrahedron (3-simplex) equation, one should be able to solve 3D
statistical models exactly. This is something that statistical physicists
have failed to do for for 75 years. The archetypal example is the Ising
model: Ising solved the 1D case in his 1925 thesis, Lars Onsager solved
the 2D model in 1944, and by now maybe 100 different ways to solve this
model exists. The 3D Ising model is still not solved, despite extensive
efforts. As a beginning graduate student in 1982, I witnessed a heated
discussion between two computer simulation groups over the fifth decimal
in the 3D Ising exponent gamma.
The bad news is that the failure to solve 3D models is probably not due
to incompetence of the statphys society, but rather to intrinsic
difficulties. These difficulties are somehow going to show up in gerbes
as well.
John Baez
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<a97mhm$r18$1...@glue.ucr.edu>...
>> ...
>> I know what the classifying space for
>> G-gerbes is: it's really the classifying space for a strict 2-group
>> whose group of objects is Aut(G) and whose group of morphisms is the
>> semidirect product of Aut(G) and G.
>How much will I have to pay, then, so you tell me what's the group of
>morphisms of a strict 2-group is? :-)
You just have to pay attention. :-)
>Hmm, maybe it's the group whose
>elements are triples (a,b,f) where a and b are objects and f is in
>Mor(a,b), with the product being (a, b, f)(c, d, g) = (ac, bd, fg),
>where fg is of course defined, for instance because the "group
>multiplication" is a functor.
That's exactly right; you've got it!
It's actually better to analyze strict 2-groups as follows.
The objects form a group H, the morphisms f: 1 -> b form a
group G - a subgroup of the group you just described. There
is a homomorphism t: G -> H sending each morphism f: 1 -> b
to its target b. Finally, there is an action A of H on G
given by
A(h)g = 1_h h (1_h)^{-1},
where the conjugation here is defined using the group
operation you just described (*not* using composition of
morphisms).
This quadruple (H,G,t,A) satisfies a few axioms making it
into what people call a "crossed module". And in fact,
any crossed module gives a strict 2-group. So strict
2-groups are really the same as crossed modules.
The crossed module corresponding to the strict 2-group
I was talking about has H = Aut(G), where A is the obvious
action of Aut(G) on G, and t: G -> Aut(G) send each
group element to the corresponding inner automorphism.
I'm writing a paper about this sort of thing, which is
why I'm so eager to talk about it even though nothing
I'm saying will make any sense except to people like
Squark who have thought about this stuff before. My
paper will explain things a lot better than I just did,
I promise!
Rogier Brussee
John Baez wrote
>Chris Hillman wrote me some email about this thread, and I
>decided I'd prefer to reply to it publicly, with his kind
>permission. He writes:
>
>>Hi,
>
>>Just tried to reread all the posts in this thread--- my head is spinning!
>>But it certainly seems that I should learn more about gerbes. I can tell
>>you are very busy, but I hope you have time to answer a quick question,
>>actually a really dumb question. I see you said:
>
>>> first Cech cohomology makes sense with coefficients in a nonabelian
>>> group G, and classifies principal G-bundles,
>
>>What's the best place to read about how this works?*
>
>Hmm... I don't know the BEST place. Perhaps Bott and Tu's
>"Algebraic Topology and Differential Forms"? I know they
>talk a lot about Cech cohomology there, and they'd have to
>be perverse not to include this application.
>
>Maybe someone else knows a good reference.
Galois Cohomology by Serre (Springer GTM ??)
However it is really a matter of rewriting the definitions:
a Cech 1 -Cochain is
a covering {U_i} together with a system
elements g__ij in G on U_j cap U_i (the intersection remembers the
order)
such that g_ii = 1 and g_ij = g_ji^{-1}.
A Cech cocycle satisfies in addition the (surprise !) cocycle condition
g_ij g_jk g_ki = 1 on U_i cap U_j cap U_k
and a Cech coboundary is of the form h_i h_j^{-1}. Two cochain's g_ij an
g'_ij are equivalent if their are h_i, such that
g'_ij = h_i g_ij h_j^{-1}
If you write that out, you see that the g_ij define the transition
funcions of a principal G-bundle and that changing by a coboundary gives
you an isomorphic bundle.
Clearly the construction works in the opposite direction too.
>>> H^3(M,Z) = H^2(M,U(1)),
> ^^^
>
>>Is this what you meant to write? I'm just checking that own all the
>>confusion I'm feeling :-/
>
>Yes, but this isn't quite so bad as it may seem: it's just
>a consequence of the short exact sequence of abelian groups
>
> 0 -------> Z --------> R -------> U(1) ----> 0
>
>which, as always, gives a long exact sequence in cohomology
>
> .... -> H^n(M,Z) -> H^n(M,R) -> H^n(M,U(1)) -> H^{n+1}(M,Z) -> ....
>
>Here we are taking advantage of the trickiest part of this long
>exact sequence, the map H^n(M,U(1)) -> H^{n+1}(M,Z), which goes by
>the charming name of "the Bockstein". This business works for
>any exact sequence of abelian groups and any respectable cohomology
>theory - but here we are using Cech cohomology, for which H^n(M,R)
>always vanishes,
Just to add to the confusion... The cohomology H^n(M,R) vanishes (for n
>0 ) only if we read this as cohomology of principal R bundles. The point is that Cech cohomology
H^n(M, A)
for abelian groups A usually means Cech cohomology with coefficients in
A which under very mild assumptions is isomorphic to ordinary cohomology
with coeffcients in A.
The best way to understand the difference (at least in my opinion) is to
think of Cech Cohomology with coeffcients in sheaves :
We have the sequences
(1) 0 -------> Z --------> R -------> U(1) ----> 0
of constant sheaves, but we also have the sequence
(2) 0 -------> Z -------->C^0(--, R) -------> C^0(--, U(1)) ---->
0
where the latter two terms are the sheaves of continuous R respectively
U(1) valued functions. Just as above we have the long exact sequence in
cohomology
.... -> H^n(M,Z) -> H^n(M,C^0(--,R) ) -> H^n(M,C^0(--, U(1))) ->
H^{n+1}(M,Z) -> ....
now for manifolds H^i(M, C^0(--, R)) = 0 for i >0. This is because we
can use partitions of unity in C^0(--,R) (we could have used C^\infty
functions just as well because they admit a partition of unity as well).
We conclude that
H^i(M, C^0(--,U(1) ) = H^{i+1}(M, Z).
More of less by definition
H^i(M, C^0(--,U(1) ) ) = H^i_{Principal bundle}(M, U(1))
On the other hand
H^i_sheaf (M, U(1) ) = H^i_{principal bundle}(M, U(1)_{discrete
topology})
classifies bundles with *locally constant* transition functions or
equivalently continuous transition function where U(1) has the discrete
topology. That is a different ballgame all together. We still get an
exact sequence
-> H^n(M,Z) -> H^n(M,R) -> H^n(M,U(1)) -> H^{n+1}(M,Z) -> ..
where the H^n's are H^n_sheaf's, but this time we conclude that
H^n_sheaf(M,U(1)) fits in an exact sequece
0 ----> H^n(M,R)/H^n(M,Z) ---> H^n(M, U(1) ) ---> Torsion(
H^{n+1}(M,Z))-->0
That is to say: H^n_sheaf(M,U(1)) is an extension of a torsion group
(typically finitely generated hence finite) by a torus of dimension b_n.
Ciao
Rogier Brussee
>
>
Thomas Larsson
> > Larry Breen and
> > William Messing have a nice paper on the math archive entitled
> > "Differential geometry of gerbes" which treats the concept of
> > connection for nonabelian gerbes, and they mention the relation
> > to D-branes, but I'm pretty sure you're right that nobody has
> > clarified the relation of D-branes to nonabelian gerbes. Luckily
> > for me, Breen and Messing's paper is sufficiently abstract that
> > few physicists will succeed in understanding it before I finish
> > writing a paper on this subject. :-)
>
> I certainly bounced off it pretty hard.
>
I spent some time looking at some gerbe-related papers, and a good
advice is: Don't start with Breen and Messing! The rest of the literature
is not an easy read, but B&M are two notches worse. You can e.g. try
Yonatan Zunger: "p-Gerbes and Extended Objects in String Theory",
http://www.arxiv.org/abs/hep-th/0002074
Marco Mackaay: "A note on the holonomy of connetions in twisted bundles",
http://www.arxiv.org/abs/math.DG/0106019
If you nevertheless have to read B&M (which is probably necessary
eventually), skip to page 60 and try to visualize the formulas there.
Then backtrack to the top of page 59 and note that the equation
(10.1.35) = (10.1.36) looks like the Yang-Baxter equation.
For whatever it is worth, here is what I have learnt so far.
1. Bundles
Let us first review the definition of an ordinary bundle over an
n-dimensional manifold M. Start with a good cover of M. Each neighborhood
U_a looks like R^n, and on the overlaps U_a /\ U_b we define transition
functions g_ab. We can illustrate each neighborhood by a * and the overlap,
or rather the transition function, as an arrow between stars:
U_a g_ab U_b
* -----------> *
The transition functions must satisfy the consistency conditions
1) g_ab g_ba = 1
which can be illustrated by the following diagram
U_a g_ab U_b
----------->
* *
<-----------
g_ba
2) g_ab g_bc g_ca = 1,
i.e. going round a triangle results in the unit operator.
U_a g_ab U_b
* ----------> *
< /
\ /
g_ca \ / g_bc
\ /
\ /
\ <
*
U_c
Two manifolds g_ab and g'_ab are equivalent if there exists functions
f_a on U_a such that
g'_ab = g_ab f_a^{-1} f_b
corresponding to the picture
f_a f_b
* -----------> *
This is already very reminiscent of a lattice gauge theory with
neighborhoods playing the role of points and transition functions the
role of amplitudes, and the equivalence relation is recognized as a
gauge transformation.
2. Gerbes
A gerbe (more precisely, 1-gerbe) is a generalizion of a bundle. On
every triple overlap U_a /\ U_b /\ U_c we define a function g_abc, which
can be illustrated by the oriented triangle
U_a U_b
* ----------> *
< /
\ g_abc /
\ /
\ /
\ /
\ <
*
U_c
The transition function satisfies
g_abc = g_bca = g_abc^{-1}
and the cocycle condition on quadruple overlaps
g_abc g_abd^{-1} g_acd g_bcd^{-1} = 1.
This condition corresponds to a tetrahedron diagram. Two gerbes g_abc and
g'_abc are equivalent if there are functions f_ab living on double
overlap (i.e. the edges of the triangle), such that
g'_abc = g_abc f_ab f_bc f_ca.
Again there is striking resemblance with the lattice 2-gauge theory I
described: the transition function g_abc lives on a plaquette, there is
a gauge invariance f_ab living on the link, and a curvature associated
to a 3-dimensional cell. The main difference is that I considered
square plaquettes whereas the gerbe picture gives rise to triangles.
It is now straightforward to extend the definitions to p-gerbes in terms
of functions living on (p+2)-fold overlaps and satisfying cocycle
conditions on (p+3)-fold overlaps. In particular, bundles = 0-gerbes and
gerbes = 1-gerbes.
3. Connection on abelian bundles
Define a 1-form A_a on U_a and a global 2-form F by
A_a - A_b = d log g_ab on U_a /\ U_b
F = dA_a on U_a
Two bundles g_ab and g'_ab with connections A_a and A'_a are equivalent
if g_ab ~ g'_ab with equivalence f_a and
A'_a = A_a + d log f_a on U_a
4. Connection on abelian gerbes
Define a 1-form A_ab = -A_ba on U_a /\ U_b, a 2-form 2-form F_a on U_a
and a global 3-form G.
A_ab + A_bc + A_ca = d log g_ab on U_a /\ U_b /\ U_c
F_b - F_a = dA_ab on U_a /\ U_b
G = dF_a on U_a
Two gerbes g_abc and g'_abc with gerbe-connections A_ab, F_a and
A'_ab, F'_a are equivalent if g_abc ~ g'_abc and
A'_ab = A_ab + B_b - B_a - d log f_ab on U_a /\ U_b
F'_a = F_a + dB_a on U_a
Note the trade-off between overlap and spacetime indices (form degree).
5. Connection on non-abelian gerbes
The treatment of abelian gerbes seems rather well established. However,
if one wants to identify the Yang-Baxter equation with a flatness
condition for gerbe connection, one must consider non-abelian gerbes;
the Yang-Baxter equation R_12 R_13 R_23 = R_23 R_13 R_12 is not very
interesting if R_ab is abelian.
Mackaay defines local one-forms A_a in U_a, valued in the Lie algebra of
G, such that
A_b - f_ba A_a f_ab - d log f_ab = A_ab on U_ab.
Two gerbes f_ab, g_abc and f'_ab, g'_abc with connections A_a, A_ab, F_a
and A'_a, A'_ab, F'_a are equivalent if there exists h_a such that
A'_a = h_a^{-1} A_a h_a + B_a + d log h_a.
I have no feeling for this definition, so I am not sure that it is the
right one. B&M also deal with non-abelian gerbes.
6. Description in local coordinates
A bundle is described locally as follows.
Choose local coordinates x \in R^n in U_a.
A section is a function f(x) valued in a representation of a group G.
The covariant derivative D = d + A, where A is a one-form, is a map
such that f and Df transform in the same way under G.
The curvature F = [D,D].
Problem: find an analogous local coordinate description of gerbes,
gerbe sections and gerbe connections. If anyone has an idea how to do
this, or has a reference, I would be very grateful. I for one will not
be able to visualize gerbes without such a local coordinate description.
Aaron Bergman
thomas....@hdd.se (Thomas Larsson) wrote:
[Gerbes]
> For whatever it is worth, here is what I have learnt so far.
[snip]
For this sort of approach, you can read Hitchin's paper "Lectures on
Special Lagrangian Submanifolds".
Buzurg Shagird
In message <4b8cc0a6.02042...@posting.google.com>
Thomas Larsson (thomas....@hdd.se) writes:
>Two gerbes f_ab, g_abc and f'_ab, g'_abc with connections A_a,
>A_ab, F_a and A'_a, A'_ab, F'_a are equivalent if there exists h_a such
>that
>
> A'_a = h_a^{-1} A_a h_a + B_a + d log h_a.
Sorry if this questions sounds really naive, but is this the same
as
A' = U A U^{-1} - dU U^{-1} + \Lambda ?
Meaning A transforms like a connection as well as shifts by a
one-form? Or is this something deeper than that?
-S.
_________________________________________________________________
Chat with friends online, try MSN Messenger: http://messenger.msn.com
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote:
>> Aaron Bergman <aber...@princeton.edu> wrote:
>> >Of course, pertubatively D-branes are just boundary states in a CFT.
>> For now I'd be happy to understand that better.
>Take a CFT on the upper half plane. What boundary conditions are you
>going to put on the real axis?
I haven't a clue. Of course, since D stands for "Dirichlet", I assume
that if my CFT is a nonlinear sigma model whose classical equation of
motion describes harmonic maps
f: H -> M
from the upper halfplane H to some target manifold M, the boundary
conditions we're interested in will be some generalization of
Dirichlet boundary conditions. But I don't understand exactly
what this generalization is, nor do I know what happens you wave
the magic wand of quantization over the whole story. Can we go
through it in an easy case, like when M is the real line?
And maybe you could also tell me the answer when M is a compact
Lie group and we're talking about the Wess-Zumino-Witten model?
Here I guess the answer is supposed to have something to do with
representations of the corresponding affine Lie algebra.
Chris Hillman
First, thanks to Thomas for the short introduction to gerbes. Now I have
a somewhat better idea what they are!
I have three questions for John and Thomas:
1. Any chance of a short exposition of the extension problem for gerbes
and how in particular the cohomology classes in H^3(*,_), for appropriate
objects in the two slots of the bifunctor, correspond bijectively to
equivalence classes of extensions? I know that this type of result seems
to require a lot of careful checking of all kinds of fine points, but a
sketch of the most important ideas would be much appreciated (especially
if I can then fill in the details following the model of the two standard
results for H^1 and H^2 in group cohomology).
2. In the classic (?) book Kenneth S. Brown, The Cohomology of Groups,
Springer-Verlag, GTM 87, 1982, there is an exposition of the following
facts (true at least for a finite group G):
(a) equivalence classes of -split- extensions E of G by A are in
bijection with the elements of
H^1(G,A)
(this should agree with the "crossed product" interpretation given in
Jacobson, Basic Algebra, but I haven't tried to work out the details
yet),
(b) equivalence classes of general extensions
0 ---> A --> E ---> G ---> 1
are in bijection with the elements of
H^2(G,A)
Here, A is an additive abelian group but G is written
multiplicatively; we define equivalence classes using isos
E
/ | \
/ | \
0 ---> A | G ---> 1
\ | /
\ | /
E'
(This is also proven in detail in Jacobson.)
Furthermore, there is a brief statement without proof of a result along
the following lines:
(c) for a generalized notion of extension which looks like this:
0 ---> H ---> N ---> E ---> G ---> 1
equivalence classes are in bijection with H^3(G,A). Here, to
define equivalence classes we have arrows N --> N', E--> E' and IIRC
a key point is now at least one of these need not be an iso. There
is a reference to some original papers coauthored by Mac Lane.
Furthermore, Brown mentions that there is an elaboration of this
idea works for all the higher cohomology groups; this is apparently
sketched in the papers by Mac Lane et al.
(BTW, our library copy is checked out and I have limited privileges, so I
can't easily consult Brown's textbook right now.) My question is whether
either John or Thomas happen to know (or can readily figure out) if
construction (c) above is equivalent to the gerbe construction.
3. I very quickly got very confused in reading Jean Renault, A Groupoid
Approach to C*-Algebras, Springer-Verlag, LTM 793, 1980, which
unfortunately uses an older traditional but IMO awful notation. It seems
clear that one can convert this notation into sensible categorical
notation (e.g. in which we consider a groupoid to be a small category in
which all arrows are isomorphisms), but as I think I said I seem to have
lost my notes, and discovered in trying to recreate them that I must be
getting less intelligent because I can't seem to translate Renault's
discussion of skew products or of the basic definition of the cohomology
of groupoids into sensible notation, although I had little trouble with
what goes before that. This is unfortunate, because the concept of a
groupoid is the common generalization of the concept of a group and of an
equivalence relation! Part of the trouble I am having may be due to the
fact that I think Renault is -en effet- writing composition left to right,
whereas in the translation I would write composition right to left, as is
standard in other areas of mathematics. But this in itself shouldn't be a
big problem, so there are other things going on. My question (3a) is
whether anyone else can carry out the translation I seem to be having
trouble with. My question (3b) concerns another fundamental issue which
seems to be obscured in at least the first chapter of Renault: what is the
notion which corresponds to "homotopy equivalence" in the sense of
abstract homological algebra?
BTW, if I didn't already mention this, for those who have seen homological
algebra in the concrete setting of R-modules, there is a nice discussion
in Mac Lane, Categories for the Working Mathematician, of how to convert
"element chasing arguments" to more abstract "arrow chases". This is
(AFAIK) essential for generalizing basic facts of homological algebra like
the Five Lemma to the context of possibly non-concrete abelian categories.
Mac Lane gives a concise statement and proof of some computational lemmata
which facilitate arrow chases in nonconcrete abelian categories.
Furthermore, in the textbook by M. Scott Osborne, Basic Homological
Algebra, Springer-Verlag, GTM 196, 2000, the author writes out in detail
side by side (!) element vs. arrow diagram chase proofs of the Five Lemma.
This gives an extensive and detailed example of how Mac Lane's lemmata are
used in practice. Mac Lane himself sketches the shorter arrow chase proof
of the simpler Three Lemma, IIRC.
Aaron Bergman
wrote:
> In article <abergman-6F103F...@news.bellatlantic.net>,
> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >ba...@galaxy.ucr.edu (John Baez) wrote:
>
> >> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >> >Of course, pertubatively D-branes are just boundary states in a CFT.
>
> >> For now I'd be happy to understand that better.
>
> >Take a CFT on the upper half plane. What boundary conditions are you
> >going to put on the real axis?
>
> I haven't a clue. Of course, since D stands for "Dirichlet", I assume
> that if my CFT is a nonlinear sigma model whose classical equation of
> motion describes harmonic maps
>
> f: H -> M
>
> from the upper halfplane H to some target manifold M, the boundary
> conditions we're interested in will be some generalization of
> Dirichlet boundary conditions. But I don't understand exactly
> what this generalization is, nor do I know what happens you wave
> the magic wand of quantization over the whole story. Can we go
> through it in an easy case, like when M is the real line?
Well, we can make life real easy. Just consider a CFT on the UHP with 26
X fields with the lagrangian
@^a_u X @^u_a X
where the metric on the target space is flat. It's easy to see that the
general solution has @\bar{@}X = 0, or
@X(z) and \bar{@}X(\bar{z})
It looks like the entire thing just factorizes into holomorphic and
anti-holomorphic parts. All we need to do is set boundary conditions on
the real line. The obvious choices are Neumann and Dirichlet. Neumann
means that the string can move freely, so there's no D-brane. Dirichlet
means that the endpoint of the string is fixed, so there is a D-brane.
The boundary conditions relate the left- and right-movers so the CFT
doesn't factorize into left- and right- moving parts.
One can also easily see how T-duality exchanges the boundary conditions.
The field X can be written as X(z,\bar{z}) = X_L(z) + X_R(\bar{z}). The
T-dual string is just X_L(z) - X_R(\bar{z}) which flips the boundary
conditions.
An interesting formalism is to map the upper-half-plane into complex
plane with a circle removed. One can shrink that circle to zero leaving
us with just a vertex operator in a closed string theory. This operator
will relate the left- and right- movers, leaving us with only one thing
in the end like the open string. So, one way to look for D-branes is to
try to classify all the boundary states. There are a number of
conditions one needs for them to satisfy in order to make sense as an
actual state in a string theory. Determining these can get rather
involved. It's not really something I've spent a lot of time on, but
hep-th/0202067 seems to do a lot of it. An introduction to boundary
states is 0201113.
> And maybe you could also tell me the answer when M is a compact
> Lie group and we're talking about the Wess-Zumino-Witten model?
> Here I guess the answer is supposed to have something to do with
> representations of the corresponding affine Lie algebra.
Everything in WZW madels has to do with representations of the
corersponding KM algebra.
Chris Hillman
On Sun, 21 Apr 2002, Rogier Brussee wrote:
> John Baez wrote
>
> >Chris Hillman wrote
[snip]
> >>> first Cech cohomology makes sense with coefficients in a nonabelian
> >>> group G, and classifies principal G-bundles,
> >
> >>What's the best place to read about how this works?
> >
> >Hmm... I don't know the BEST place. Perhaps Bott and Tu's
> >"Algebraic Topology and Differential Forms"? I know they
> >talk a lot about Cech cohomology there, and they'd have to
> >be perverse not to include this application.
> >
> >Maybe someone else knows a good reference.
>
> Galois Cohomology by Serre (Springer GTM ??)
Hmm... it looks a bit sketchy :-/ but I found the place you mean. Thanks
for the reference. (BTW, I just got hold of Bott and Tu, which John
recommended, but haven't had a chance to grok either Serre's notation or
look at Bott and Tu.)
> However it is really a matter of rewriting the definitions:
>
> a Cech 1 -Cochain is
>
> a covering {U_i} together with a system
>
> elements g__ij in G on U_j cap U_i (the intersection remembers the
> order) such that g_ii = 1 and g_ij = g_ji^{-1}.
I think this "intersection remembers order" business makes more sense if
you think in terms of sheaves... but I am sure you or John can explain
this better than I can!
> A Cech cocycle satisfies in addition the (surprise !) cocycle condition
>
> g_ij g_jk g_ki = 1 on U_i cap U_j cap U_k
Let X be a G-set and let X^(1) be the set of ordered pairs (x1,x0) such
that x1,x0 are in the same orbit under a given action by G. Let C^1 be
the space of functions from X^(1) into K some possibly nonabelian
multiplicative group. Then one can define 1-cocycles to the be the
1-cochains satisfying
f(x2,x0) = f(x2,x1) f(x1,x0)
Since in this theory f(x0,x1) = f(x1,x0)^(-1) is also true for 1-cocycles,
this definition is the same as the one you offered.
I think this cannot be a coincidence: any sensible cohomology theory of
etale groupoids should be closely connected to both cohomology of sheaves
and to the usual cohomology of groups.
> and a Cech coboundary is of the form h_i h_j^{-1}. Two cochain's g_ij an
> g'_ij are equivalent if their are h_i, such that
>
> g'_ij = h_i g_ij h_j^{-1}
Or in the above notation
f2(x,y) s(y) = s(x) f1(x,y)
The point here is that this notation is mnemonic. I can't remember if I
found this notation somewhere or cooked it up myself, but in any case this
is essentially the notion of cohomology which has been extensively used in
the context of ergodic theory by Schmidt and many others. I think its
origins may go back to Mackey and work on cohomology of equivalence
relations rather than to sheaf cohomology, which would be interesting if
true.
It really is unfortunate that I
(a) lost my notes on the cohomology of equivalence relations,
(b) forget almost entirely how it goes,
(c) can't remember the references I eventually found after working
out some of the theory on my own (the book by Renault and the books
by Schmidt don't contain the stuff I half-remember after all).
I -do- seem to still have the part of my notes dealing with 1-cocycles in
a possibly nonabelian group, so energy permitting tomorrow I'll try to at
least sketch this and explain an important construction in ergodic theory:
the skew product X |x_f Y defined by a 1-cocycle f in the sense above.
Then it is straightforward to show that if f~g are cohomologous
1-cocycles, then X |x_f Y is G-isomorphic to X |x_g Y. This result
intrigued me because of course usually in studying invariants the logic
works the other way: if any invariant gives different answers when
computed for two objects A,B in a particular category, then of course A,B
must be non-isomorphic.
Part of the problem I have in getting sufficient intuition for all this
stuff is that few books on the cohomology of groups (say) seem to trouble
to explain how to produce concrete examples of cocycles (I -do- know the
cohomology sequence for prime cyclic groups, however; all the books do
discuss this much), and while I've seen many books/papers which discuss
skew products in ergodic theory, again if any discuss how to produce a
plenitude of interesting ones, I've missed it. (The book by Olver on
symmetry etc. does give some general constructions, but in the context of
smooth actions). For those interested in explicit examples of things
cohomological, maybe I should point out that in a thread archived on my
website, I did compute (hopefully correctly) the equivariant de Rham
cohomology of the Bianchi groups. As everyone knows exhaust the three
dimensional real Lie groups; if I did the computations correctly, not very
surprisingly it turns out that this very tractable cohomology theory fails
to distinguish between all nine Bianchi groups, although it -does- yield
some interesting information.
> If you write that out, you see that the g_ij define the transition
> funcions of a principal G-bundle and that changing by a coboundary
> gives you an isomorphic bundle. Clearly the construction works in the
> opposite direction too.
Hmm... OK, thanks, this looks good but I'll have to think about the
details.
> >>> H^3(M,Z) = H^2(M,U(1)),
> > ^^^
> >
> >>Is this what you meant to write? I'm just checking that own all the
> >>confusion I'm feeling :-/
[snip more valuable stuff I've saved for the future]
Aaron Bergman
<Pine.OSF.4.33.020427...@goedel3.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> wrote:
> I think this "intersection remembers order" business makes more sense if
> you think in terms of sheaves... but I am sure you or John can explain
> this better than I can!
In fact, Cech cohomology for a sufficiently nice cover (or, more
generally, just take the direct limit in the space of covers) computes
sheaf cohomology.
Robert C. Helling
wrote:
>In article <abergman-6F103F...@news.bellatlantic.net>,
>Aaron Bergman <aber...@princeton.edu> wrote:
>>Take a CFT on the upper half plane. What boundary conditions are you
>>going to put on the real axis?
>I haven't a clue. Of course, since D stands for "Dirichlet", I assume
>that if my CFT is a nonlinear sigma model whose classical equation of
>motion describes harmonic maps
>
>f: H -> M
>
>from the upper halfplane H to some target manifold M, the boundary
>conditions we're interested in will be some generalization of
>Dirichlet boundary conditions. But I don't understand exactly
>what this generalization is, nor do I know what happens you wave
>the magic wand of quantization over the whole story. Can we go
>through it in an easy case, like when M is the real line?
You can avoid the question if you start with a quantum theory from the very
beginning. Then the procedure is roughly as follows: You already have a CFT
that lives on the Riemann-sphere, say. We will call it the "bulk theory" or
the "closed string theory" (for hopefull obvious resons). That means you
have the field operators (and their operator product expansion) and also the
chiral symmetry algebra (an algebra that contains the virasoro algebra) that
acts on the left (right) moving fields.
>And maybe you could also tell me the answer when M is a compact
>Lie group and we're talking about the Wess-Zumino-Witten model?
>Here I guess the answer is supposed to have something to do with
>representations of the corresponding affine Lie algebra.
Right. Your CFT could be a WZW-model (the real line in included in this case
as it is a U(1) WZW model). People have also looked at Gepner models and are
now also considering non-compact groups as targets. Furthermore you could take
cosets (gauged WZW models) or orbifolds.
Now you turn this model into a CFT on the upper half-plane. This of course
involves specifying a boundary condition at z = z-bar that couples left
moving to right moving fields. In order to preserve conformal invariance
(otherwise, you're completely lost) you demand that the left moving
stress-energy tensor (that generates Vir) equals the right moving stress
energy thensor at the boundary:
T(z) = T-bar(z-bar) at z=z-bar
Usually, you also want to preserve more (if not all) of your chiral symmetry
alegbra. For the preserved generators W(z) you impose:
W(z) = Omega W-bar(z-bar)
where Omega is some automorphism of the algebra. In the case of the real
line your field is X(z,z-bar) (strictly speaking, this is not a field because
the 2-point function contains a log. But the currents dX and d-bar X as well
as the vertex operator exp(ikX) are) there is the U(1) current dX on which
you impose
dX (z) = +/- d-bar X(z-bar)
If you express this condition not in light-cone variables z and z-bar but in
real part and imaginary part, you see that one choice of sign corresponds to
Neumann boundary conditions and the other to Dirichlet conditions.
Furthermore, the operators of the CFT have non-vanishing one-point functions
that have to be chosen in order to specify the boundary conditions:
<phi_{i, i-bar}(z,z-bar)> =
A_{i, i-bar} / |z - z-bar|^(h_i+h_i-bar) delta(i,Omega i-bar)
i and i-bar are labels for the specific field (with respect to the left and
right moving algebras) and the h's are the conformal weights. The only
freedom are the numbers A, the dependence on z is fixed by the conformal
weights.
Why are there one-point functions? One could imagine that the boundary
condition is implemented with mirror charges as one does in boundary value
problems in electro-magnetism: For a field in the upper half-plane the is a
mirror field in the lower half plane such that the combined effect obeys the
boundary conditions. Now, the one-point function in the theory with boundary
can be thought of as a two-point function of the field and the mirror field
in the bulk theory. If the OPE of the field with the mirror field contains
the vacuum, the expectation value does not vanish.
However, the choice for the A's is not completely arbitrary but there are
non-linear constraints: One comes from situations with two fields:
phi_2 x
x phi_1
----------------------------------------
There are two possibilities for evaluating the expectations value: One could
bring the two fields together, use the bulk OPE to express the result in
terms of one operator and use that operator's A or one could separate them,
use the cluster property to write the two point function as the product of
two one-point functions, that is two A's. Thus there is a relation involving
one A and the OPE coefficients on one side and two A's on the other side.
There is another constraint that comes from the famous string picture
/| /|
/ | / |
/ | / |
| _|______|_ |
| O__________O |
| | | |
| / | /
| / | /
|/ |/
There are two ways to view this picture: It's either a loop of open strings
streching between two D-branes or closed string that is emitted form one
brane and absorbed by the other. You can evaluate the amplitude from both
points of view and of course both have to agree.
The open strings amplitude is evalued in the boundary conformal field theory
using the data I just described as a trace over the Hilbert space of strings
strechting between the two boundaries (after Wick rotation, it's a thermal
correlation function).
On the closed string side, this is basically a partition function (nothing
to nothing amplitude). In this partition function various states of the
closed string can contribute. But they contribute with integer coefficients.
The fact, that these coefficients are integers gives very non-trivial
constraints on the open-string amplitude and thus on the A's.
The task of finding A's that satisfy the above constraints so far has only
been partially solved in most cases. However there is a very elegant
solution due to Cardy that uses the Verlinde formula that helps in many of
the most interesting cases. You have to look at the literature for an
explanation.
So far, I forgot to tell you about the physical meaning of the A's. Of
course, they describe the coupling of the closed strings to the branes. For
example, the A for the graviton vertex operator is proportional to the mass
(or tension) of the brane and the A for the gauge field (or RR-form field)
vertex operators encodes the charge of the brane. A closer look also tells
you about geometrical proberties (and especially: the position) of the
brane. Again: You can find more in the literature.
Too bad, I cannot give you _the_ reference for all these things. Volker
Schomerus has talked at various schools and conferences about this stuff,
but there are no lecture notes. You have to look at his papers (and also to
the papers of Andreas Recknagel, Matthias Gaberdiel, Juergen Fuchs,
Christoph Schweigert, just to name a few of the main players). I learned
most of this from Recknagels habilitation thesis, however this is not
officially published. But he sends out the file if requested by email.
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Institut fuer Physik
Humboldt-Universitaet zu Berlin
print "Just another Fon +49 30 2093 7964
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Chris Hillman
On Sun, 28 Apr 2002, Aaron Bergman wrote:
> Chris Hillman <hil...@math.washington.edu> wrote:
>
> > I think this "intersection remembers order" business makes more sense if
> > you think in terms of sheaves... but I am sure you or John can explain
> > this better than I can!
>
> In fact, Cech cohomology for a sufficiently nice cover (or, more
> generally, just take the direct limit in the space of covers) computes
> sheaf cohomology.
Yes, in fact, we (or at least John) have IIRC discussed this here before,
but I was suggesting a semi-detailed explanation or perhaps a repost.
Thomas Larsson
> First, thanks to Thomas for the short introduction to gerbes. Now I have
> a somewhat better idea what they are!
>
> I have three questions for John and Thomas:
>
The following new paper seems highly relevant:
Gerbes and Duality
Authors: M. I. Caicedo, I. Martin, A. Restuccia
http://www.arxiv.org/abs/hep-th/0205002
Maybe it will answer your questions.
John Baez
Robert C. Helling <hel...@atdotde.de> wrote:
>On Wed, 24 Apr 2002 19:59:20 +0000 (UTC), John Baez <ba...@galaxy.ucr.edu>
>wrote:
>>In article <abergman-6F103F...@news.bellatlantic.net>,
>>Aaron Bergman <aber...@princeton.edu> wrote:
>>>Take a CFT on the upper half plane. What boundary conditions are you
>>>going to put on the real axis?
>>I haven't a clue.
>>Can we go through it in an easy case, like when M is the real line?
>You can avoid the question if you start with a quantum theory from the very
>beginning.
I don't want to avoid the question - I'd like to know the answer!
Classically, boundary value problems are a wonderful subject about
which a lot is known - especially for elliptic PDE like the ones
we're dealing with here. So if we're gonna consider the quantum
version of these boundary conditions, it seems fun and enlightening
to start with the classical case, and then see how the quantum case
compares to that.
But anyway, I can try to go along with your strategy:
>You already have a CFT
>that lives on the Riemann-sphere, say. We will call it the "bulk theory" or
>the "closed string theory" (for hopefull obvious reasons). That means you
>have the field operators (and their operator product expansion) and also the
>chiral symmetry algebra (an algebra that contains the virasoro algebra) that
>acts on the left (right) moving fields.
>>And maybe you could also tell me the answer when M is a compact
>>Lie group and we're talking about the Wess-Zumino-Witten model?
>>Here I guess the answer is supposed to have something to do with
>>representations of the corresponding affine Lie algebra.
>Right. Your CFT could be a WZW-model (the real line in included in this case
>as it is a U(1) WZW model).
Actually there's a difference between R and U(1), but okay, I can pretend
to be a physicist and ignore that. :-)
>People have also looked at Gepner models and are
>now also considering non-compact groups as targets. Furthermore you could take
>cosets (gauged WZW models) or orbifolds.
Cool.
>Now you turn this model into a CFT on the upper half-plane. This of course
>involves specifying a boundary condition at z = z-bar that couples left
>moving to right moving fields. In order to preserve conformal invariance
>(otherwise, you're completely lost) you demand that the left moving
>stress-energy tensor (that generates Vir) equals the right moving stress
>energy thensor at the boundary:
>
>T(z) = T-bar(z-bar) at z=z-bar
Is this just a way to say that no energy-momentum falls off the
edge of the string - i.e., as a wiggle propagates down the string,
it bounces right back when it hits the end - or am I getting my signs
backwards?
>Usually, you also want to preserve more (if not all) of your chiral symmetry
>alegbra. For the preserved generators W(z) you impose:
>
>W(z) = Omega W-bar(z-bar)
>
>where Omega is some automorphism of the algebra. In the case of the real
>line your field is X(z,z-bar) (strictly speaking, this is not a field because
>the 2-point function contains a log. But the currents dX and d-bar X as well
>as the vertex operator exp(ikX) are) there is the U(1) current dX on which
>you impose
>
>dX (z) = +/- d-bar X(z-bar)
>
>If you express this condition not in light-cone variables z and z-bar but in
>real part and imaginary part, you see that one choice of sign corresponds to
>Neumann boundary conditions and the other to Dirichlet conditions.
Umm, okay, I can pretend to understand this. But I need stuff
that's much more elementary if I'm going to really grok this
stuff.
For starters, what's the physical difference between Dirichlet and
Neumann boundary conditions in terms of a wiggling string? Which
corresponds to a string whose ends are "nailed to the ground",
and which corresponds to one whose ends can wiggle around?
And why do people always talk about D-branes and never N-branes?
Hmm... here are my guesses. I guess that Neumann boundary conditions
would be where the string's ends are free to wiggle around, and people
never talk about "N-branes" because this is just the usual sort of open
string. That would mean the Dirichlet boundary conditions are where the
strings ends are "nailed to the ground". Right?
>Furthermore, the operators of the CFT have non-vanishing one-point functions
>that have to be chosen in order to specify the boundary conditions:
>
><phi_{i, i-bar}(z,z-bar)> =
> A_{i, i-bar} / |z - z-bar|^(h_i+h_i-bar) delta(i,Omega i-bar)
>
>i and i-bar are labels for the specific field (with respect to the left and
>right moving algebras) and the h's are the conformal weights. The only
>freedom are the numbers A, the dependence on z is fixed by the conformal
>weights.
Whoops - I can no longer pretend to understand! Big equations in
ASCII are hard for me to follow, especially when I don't see the
big picture yet. After I see the big picture I can sort of skim
over the equations without needing to read them, because they're
sort of obvious... but beforehand, I skim over them without understanding
anything.
I'd actually like to see something incredibly simple, like how this
stuff works for the scalar wave equation on the spacetime R x [0,1].
I get the impression that in general there's supposed to be a "Hilbert
space of boundary conditions". I guess these are different ways for
the string's ends to be "nailed to the ground" - i.e., different ways
for the field in question to take a "constant value" at the ends of the
interval [0,1]. Of course it's a quantum field; that's what makes
it tricky. So instead of taking just a *number* as a value, it must
be something cooler. What is it?
Aaron Bergman
ba...@galaxy.ucr.edu (John Baez) wrote:
> Umm, okay, I can pretend to understand this. But I need stuff
> that's much more elementary if I'm going to really grok this
> stuff.
Flat space is always much easier than WZW models.
> For starters, what's the physical difference between Dirichlet and
> Neumann boundary conditions in terms of a wiggling string?
Dirichlet boundary conditions mean that the value of X at the boundary
is fixed. Neumann means that the normal derivative of X at the boundary
is fixed which means that the ends are free.
> Which
> corresponds to a string whose ends are "nailed to the ground",
> and which corresponds to one whose ends can wiggle around?
> And why do people always talk about D-branes and never N-branes?
Because originally all the bdry conditions were taken to be Neumann and
you had freely moving strings in 26D spacetime (for the bosonic string).
When you gave some of the X's Dirichlet bondary conditions, that was
taken to be the existence of a brane. Really, having a Dp-brane in
spacetime means (p+1) X's with Neumann boundary conditions and 25-p X's
with Dirichlet boundary conditions. Thus, what was once the open string
should probably be thought of as a D25-brane (actually, 8192 D25 branes,
but that's a different story.)
> Hmm... here are my guesses. I guess that Neumann boundary conditions
> would be where the string's ends are free to wiggle around, and people
> never talk about "N-branes" because this is just the usual sort of open
> string. That would mean the Dirichlet boundary conditions are where the
> strings ends are "nailed to the ground". Right?
Yep, as I hope the above makes clear.
[...]
> I'd actually like to see something incredibly simple, like how this
> stuff works for the scalar wave equation on the spacetime R x [0,1].
> I get the impression that in general there's supposed to be a "Hilbert
> space of boundary conditions".
There are actually what are called boundary states.
> I guess these are different ways for
> the string's ends to be "nailed to the ground" - i.e., different ways
> for the field in question to take a "constant value" at the ends of the
> interval [0,1]. Of course it's a quantum field; that's what makes
> it tricky. So instead of taking just a *number* as a value, it must
> be something cooler. What is it?
I'm not sure I really see any problem with a number.
Robert C. Helling
On Tue, 14 May 2002 21:17:58 +0000 (UTC), John Baez <ba...@galaxy.ucr.edu> wrote:
>In article <slrnacf984....@ariel.aei-potsdam.mpg.de>,
>Robert C. Helling <hel...@atdotde.de> wrote:
>
>>On Wed, 24 Apr 2002 19:59:20 +0000 (UTC), John Baez <ba...@galaxy.ucr.edu>
>>wrote:
>
>>>In article <abergman-6F103F...@news.bellatlantic.net>,
>>>Aaron Bergman <aber...@princeton.edu> wrote:
>
>>>>Take a CFT on the upper half plane. What boundary conditions are you
>>>>going to put on the real axis?
>
>>>I haven't a clue.
>
>>>Can we go through it in an easy case, like when M is the real line?
>
>>You can avoid the question if you start with a quantum theory from the very
>>beginning.
>
>I don't want to avoid the question - I'd like to know the answer!
You're right. I should have discussed that first. At least for the case of a
flat background. Let us start with the classical action of the string (in
Polyakov form)
S = int d^2 z dX d-bar X
(z are the two coordinates on the world sheet, X is the coordinate
field, d is the 'holomorphic' partial derivative and d-bar the
antiholomorphic derivative). To get the Euler-Lagrange equations we have to
perform a partial integration. But if the worldsheet has a boundary (we are
talking about open strings), we pick up a boundary term, that is
proportional to the variation of X times the normal derivative of X at the
boundary.
This vanishes if either X is fixed at the boundary (endpoint of the string)
or the normal derivative vanishes at the end of the string. Thus we have
derived Dirichlet and Neumann boundary conditions. For D conditions, the
string end cannot move in the X direction and this is expressed as "the
string is attatched to a D-brane".
The rest of what I was saying connects to this classical analysis as
follows: Use a conformal transformation to have the complex upper halfplane
as your worldsheet. Then you automatically fulfill the boundary conditions
above if you pretend that the worldsheet is the full complex plane but the
field X has to fulfill
X(z) = +/- X(z-bar)
>From the worldsheet perspective, X is usually determined like the potential
in electro statics from a distribution of charges, that is it is harmonic
except for the points where the charges reside. Only that here the charges
arge called "vertex operators". And as in electro statics, to fulfill the
boundary conditions, with each charge at z we add a (possibly negative)
mirror charge at z-bar. The correlation functions I have been talking about
are just the correlation functions of these charges.
Usually the one point function of a charge vanishes, but as here with every
charge there comes the mirror charge the one point function on the upper
half plane is really a two point function on the full plane. And as usual,
by conformal invariance those are determined up to a constant. And I was
arguing that theses constants express the mass and charges of the brane
depending on the type of charge (vertex operator) that you are considering
(graviton = mass, fieldstrength = charge etc).
>Actually there's a difference between R and U(1), but okay, I can pretend
>to be a physicist and ignore that. :-)
Fine. If the circle is really big it's not much different from the line. On
the other hand, if you would really insist on a non-compact target space
things get quite messy. This is because what are discrete (and in most cases
finite) sums over representations in the compact case turn into integrals
in the non-compact case. In that more general case you have to care about
convergence, there is no more an operator-state correspondence and all that.
The case of R is still sufficiently simple but more general non-compact Lie
groups are just at the edge of what people can handle.
>>
>>T(z) = T-bar(z-bar) at z=z-bar
>
>Is this just a way to say that no energy-momentum falls off the
>edge of the string - i.e., as a wiggle propagates down the string,
>it bounces right back when it hits the end - or am I getting my signs
>backwards?
Hmm. I am not really sure. Maybe in the worldsheet sense. It definitely
says, that the boundary conditions do not break conformal invariance. In the
target space sense this is definitely not true. There the derivative of X is
the momentum (T is the square of that). The above relation is fulfilled for
both Neumann and Dirichlet conditions. But in Green, Schwarz, Witten you can
find the statement, that you have to use Neumann boundary conditions in
order not to have momentum flowing off the edge of the string (this it took
so long to take the D case seriously). And that is true: From the
perturbative point of view the D-brane is infinitely heavy and can absorb
momentum from the string (like a wall absorbs the momentum of a ball that is
thrown against it).
>Hmm... here are my guesses. I guess that Neumann boundary conditions
>would be where the string's ends are free to wiggle around, and people
>never talk about "N-branes" because this is just the usual sort of open
>string. That would mean the Dirichlet boundary conditions are where the
>strings ends are "nailed to the ground". Right?
Right.
>>that have to be chosen in order to specify the boundary conditions:
>>
>><phi_{i, i-bar}(z,z-bar)> =
>> A_{i, i-bar} / |z - z-bar|^(h_i+h_i-bar) delta(i,Omega i-bar)
>>
>>i and i-bar are labels for the specific field (with respect to the left and
>>right moving algebras) and the h's are the conformal weights. The only
>>freedom are the numbers A, the dependence on z is fixed by the conformal
>>weights.
>
>Whoops - I can no longer pretend to understand! Big equations in
>ASCII are hard for me to follow, especially when I don't see the
>big picture yet. After I see the big picture I can sort of skim
>over the equations without needing to read them, because they're
>sort of obvious... but beforehand, I skim over them without understanding
>anything.
I hope I explained that above: The one point functions of the vertex
operators are really two point functions of the vertex operator and its
image vertex operator.
>I'd actually like to see something incredibly simple, like how this
>stuff works for the scalar wave equation on the spacetime R x [0,1].
>I get the impression that in general there's supposed to be a "Hilbert
>space of boundary conditions". I guess these are different ways for
>the string's ends to be "nailed to the ground" - i.e., different ways
>for the field in question to take a "constant value" at the ends of the
>interval [0,1].
That's true, I have been simplifying above. There are more boundary
conditions. First of all, in the D case we only said that X is not allowed
to vary at the end of the string. But we haven't fixed it's value. This of
course is the position of the branes.
Furthermore there are mixtures of D and N conditions: As an additional
background field in space-time you can have a two from potential B. In the
simplest case, it is closed so its field strength vanishes and it doesn't
curve the background via Einstein's equations. To account for its influence
on the string you add a term to the action that is just the pull back of B
to the worldsheet.
As it is closed, we can use Stokes' theorem to rewrite it as an integral
over the boundary of the world sheet. There it influences the boundary
conditions for the X field. No formulas here, but take my word that the
effect is to form some kind of linear combination of Neumann and Dirchlet
conditions.
>Of course it's a quantum field; that's what makes
>it tricky. So instead of taking just a *number* as a value, it must
>be something cooler. What is it?
In the quantum theory, you never look at the actual values of X. But you can
probe the brane by throwing closed strings at it. In formulas this
corresponds to calculating correlation functions of closed string vertex
operators. And as I explaind above, you calculate those in the presence of
D-brane boundary conditions as correlation functions of the vertex operators
and their mirror vertex operators in a field theory on the full plane
without a boundary. The boundary condition is taken care of by the
particular choice of mirror vertex operators.
John Baez
>I have three questions for John and Thomas:
The main reason I haven't replied is that these questions are
too hard for me... but I guess I should say *something*.
>1. Any chance of a short exposition of the extension problem for gerbes
>and how in particular the cohomology classes in H^3(*,_), for appropriate
>objects in the two slots of the bifunctor, correspond bijectively to
>equivalence classes of extensions?
This may not be the right question to ask - I don't know.
The correct analogy table is:
group G 2-group C
G-bundle C-2-bundle (special case being a gerbe)
classifying space BG classifying space BC
A group is a category with one object and all morphisms invertible.
A 2-group is a 2-category (or alternatively, bicategory) with one
object and all morphisms and 2-morphisms invertible.
A G-bundle can be built from trivial G-bundles on the sets U_i in an open
cover using transition functions on double intersections U_i intersect U_j;
a C-2-bundle works similarly but you need to give object-valued
functions on double intersections and morphism-valued functions on
triple intersections. You won't find this in the literature until
I write it up; instead, you'll find a description of gerbes, which
correspond to C-2-bundles for a very special sort of 2-group C, which
I described earlier in this thread. Working with these special 2-groups
mainly has the effect of obscuring what's really going on!
There is a space BG, the classifying space of the group G, such that
all G-bundles over any space X can be obtained from maps f: X -> BG,
by pulling back a certain G-bundle EG -> BG. The cohomology of this
space BG is called "the cohomology of G" by people who do group cohomology,
but it's really important to realize that it's also the cohomology of
a *space*, not just some abstract algebraic nonsense.
Similarly, there is a space BC, the classifying space of the 2-group C,
such that all C-2-bundles over any space X can be obtained from maps
f: X -> BC, by pulling back a certain G-bundle EC -> BC. The cohomology
of BC is called "the cohomology of C" by people who do 2-group cohomology;
unfortunately it seems that very few such people exist. I did manage
while in Seattle recently to get some topologists to compute the
cohomology of BC in some simple examples, and I would like to get more
topologists involved in this, since I'm too lazy and incompetent to do
these calculations myself.
In group cohomology the low-dimensional groups like H^1 and H^2 have
fun interpretations, as you mention; something similar must be true
for 2-group cohomology, but I don't know how this works. Probably
the right person to ask is Larry Breen.
Luckily, we don't need to understand this to make some headway
on your next question:
> (b) equivalence classes of general extensions
>
> 0 ---> A --> E ---> G ---> 1
>
> are in bijection with the elements of
>
> H^2(G,A)
> (c) for a generalized notion of extension which looks like this:
>
> 0 ---> H ---> N ---> E ---> G ---> 1
>
> equivalence classes are in bijection with H^3(G,A). Here, to
> define equivalence classes we have arrows N --> N', E--> E' and IIRC
> a key point is now at least one of these need not be an iso. There
> is a reference to some original papers coauthored by Mac Lane.
> Furthermore, Brown mentions that there is an elaboration of this
> idea works for all the higher cohomology groups; this is apparently
> sketched in the papers by Mac Lane et al.
>
>(BTW, our library copy is checked out and I have limited privileges, so I
>can't easily consult Brown's textbook right now.) My question is whether
>either John or Thomas happen to know (or can readily figure out) if
>construction (c) above is equivalent to the gerbe construction.
I think it's a lot easier to start with H^2. Remember, the relation
between gerbes and cohomology is really all about H^2; H^3 only gets
into the act via a clever trick involving the Bockstein map, and this
trick only works for U(1)-gerbes, not general gerbes, much less general
2-bundles.
So here's the cute thing I think you secretly want to understand:
Suppose A is a discrete group. Then H^2(X,A) classifies A-gerbes
over the *space* X - here we are talking about cohomology of *spaces*.
On the other hand, H^2(G,A) classifies extensions involving the *group*
G - here we are talking about cohomology of *groups*. So, they are
somewhat different things. On the other hand, as I mentioned, there's
a very close relationship:
H^2(G,A) = H^2(BG,A)
This means that we should be able to turn an extension of A by G
into an A-gerbe over BG, and vice versa. They should in fact be
just two different ways of talking about the same thing: one more
"algebraic" in flavor, the other more "topological".
Now, I don't grok this just yet, but it shouldn't be too hard,
and I bet Breen has written about this already, perhaps in his book
"On the classification of 2-gerbes and 2-stacks".
Hmm, yes - it's in the introduction, on page 9! I guess he regards
it as a basic fact about gerbes that he is going to generalize to
2-gerbes... which, by the way, are a special case of 3-bundles!
The ladder never ends....
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
>In article <a9v4k9$2pl$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
>wrote:
>> In article <abergman-6F103F...@news.bellatlantic.net>,
>> Aaron Bergman <aber...@princeton.edu> wrote:
>> >Take a CFT on the upper half plane. What boundary conditions are you
>> >going to put on the real axis?
>> I haven't a clue.
>> Can we go through it in an easy case, like when M is the real line?
>Well, we can make life real easy. Just consider a CFT on the UHP with 26
>X fields with the lagrangian
>
>@^a_u X @^u_a X
>
>where the metric on the target space is flat.
Just out of curiosity, do we really need 26 of them here? In other
words, is there some reason we can't define the concept of D-brane
for conformal field theories with arbitrary central charge?
But anyway, okay....
>It looks like the entire thing just factorizes into holomorphic and
>anti-holomorphic parts. All we need to do is set boundary conditions on
>the real line. The obvious choices are Neumann and Dirichlet. Neumann
>means that the string can move freely, so there's no D-brane. Dirichlet
>means that the endpoint of the string is fixed, so there is a D-brane.
>The boundary conditions relate the left- and right-movers so the CFT
>doesn't factorize into left- and right- moving parts.
I get it; great! By the way, at least classically there is a whole
bunch of boundary conditions of which Neumann and Dirichlet are
just two: we can impose conditions which relate the X field and its
derivative. Do string theorists look at these, or not? Is there some
reason they're ruled out here?
>One can also easily see how T-duality exchanges the boundary conditions.
>The field X can be written as X(z,\bar{z}) = X_L(z) + X_R(\bar{z}). The
>T-dual string is just X_L(z) - X_R(\bar{z}) which flips the boundary
>conditions.
I've never understood T-duality. What's the best way to think of
this geometrically? There must be some really easy way to understand
the meaning of switching the sign of the right-movers.
>> And maybe you could also tell me the answer when M is a compact
>> Lie group and we're talking about the Wess-Zumino-Witten model?
>> Here I guess the answer is supposed to have something to do with
>> representations of the corresponding affine Lie algebra.
>Everything in WZW madels has to do with representations of the
>corresponding KM algebra.
Yes, I wasn't sticking my neck out too far there. But from
stuff Segal was saying, it seems there should be some *really
simple* description of the Hilbert space of boundary states
in this case: something like the direct sum of all the irreps
of the affine Lie algebra, or....
Aaron Bergman
In article <acdtij$bnp$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
wrote:
> In article <abergman-5F8C1C...@news.bellatlantic.net>,
> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >In article <a9v4k9$2pl$1...@glue.ucr.edu>, ba...@galaxy.ucr.edu (John Baez)
> >wrote:
>
> >> In article <abergman-6F103F...@news.bellatlantic.net>,
> >> Aaron Bergman <aber...@princeton.edu> wrote:
>
> >> >Take a CFT on the upper half plane. What boundary conditions are you
> >> >going to put on the real axis?
>
> >> I haven't a clue.
> >> Can we go through it in an easy case, like when M is the real line?
>
> >Well, we can make life real easy. Just consider a CFT on the UHP with 26
> >X fields with the lagrangian
> >
> >@^a_u X @^u_a X
> >
> >where the metric on the target space is flat.
>
> Just out of curiosity, do we really need 26 of them here? In other
> words, is there some reason we can't define the concept of D-brane
> for conformal field theories with arbitrary central charge?
You can certainly define CFTs with different boundary conditions. As to
whether or not they can be interpreted as dynamic objects in some
nonperturbative theory, beats me. I don't know much about non-critical
strings. Of course, you can always add on another CFT to make up the
rest of the central charge and then everything's fine.
> But anyway, okay....
>
> >It looks like the entire thing just factorizes into holomorphic and
> >anti-holomorphic parts. All we need to do is set boundary conditions on
> >the real line. The obvious choices are Neumann and Dirichlet. Neumann
> >means that the string can move freely, so there's no D-brane. Dirichlet
> >means that the endpoint of the string is fixed, so there is a D-brane.
> >The boundary conditions relate the left- and right-movers so the CFT
> >doesn't factorize into left- and right- moving parts.
>
> I get it; great! By the way, at least classically there is a whole
> bunch of boundary conditions of which Neumann and Dirichlet are
> just two: we can impose conditions which relate the X field and its
> derivative. Do string theorists look at these, or not?
Yeah. There is a two-from B-field which has an effect on the boundary
conditions. Generally, it is
g_ij @_n X^j + 2\pi i \a' B_ij @_t X^j = 0
where the second term is evaluated on the boundary. There are probably
others you can write down, too. More generally, this stuff is done in
terms of boundary states which are, I think, the most general thing that
you can write down. The trick is figuring out which ones are real
boundary states and which are inconsistent with being part of a string
theory.
> Is there some
> reason they're ruled out here?
>
> >One can also easily see how T-duality exchanges the boundary conditions.
> >The field X can be written as X(z,\bar{z}) = X_L(z) + X_R(\bar{z}). The
> >T-dual string is just X_L(z) - X_R(\bar{z}) which flips the boundary
> >conditions.
>
> I've never understood T-duality. What's the best way to think of
> this geometrically? There must be some really easy way to understand
> the meaning of switching the sign of the right-movers.
There are a lot of ways. There's a construction called Abelian duality
where you gauge the isometry associated with circle. You can then show
by fixing a gauge and integrating out in different ways that the two
theories are equivalent. For more general toroidal compactifications,
there's a construction due to Narain where you look at the left and
right moving momenta. You can show that they must form an even self-dual
lattice. An O(k,k,Z) transformation preserves the lattice and
corresponds to a T-duality transformation.
Of course, eventually T-duality (well, three T-dualities actually) is
thought to be closely related to mirror symmetry.
>
> >> And maybe you could also tell me the answer when M is a compact
> >> Lie group and we're talking about the Wess-Zumino-Witten model?
> >> Here I guess the answer is supposed to have something to do with
> >> representations of the corresponding affine Lie algebra.
>
> >Everything in WZW madels has to do with representations of the
> >corresponding KM algebra.
>
> Yes, I wasn't sticking my neck out too far there. But from
> stuff Segal was saying, it seems there should be some *really
> simple* description of the Hilbert space of boundary states
> in this case: something like the direct sum of all the irreps
> of the affine Lie algebra, or....
He was probably talking about Cardy states or Ishibashi states. It's
easy to write down a lot of things. The hard part is that they have to
make sense in string amplitudes. In particular, they have to be
consistent with all possible worldsheets. Showing this is rather hard.
I'm not really up on this (but I know where to look if need be), so
hopefully Robert Helling can explain this stuff better.
Squark
> I've never understood T-duality. What's the best way to think of
> this geometrically? There must be some really easy way to understand
> the meaning of switching the sign of the right-movers.
Just a note I had to slip in: as I read in Nobuyoshi Oshta's
"Introduction to M-theory for relativists and cosmologists"
(gr-gc/0205036), T-duality actually switches the (quantized becaus of
obvious reasons) momentum along the compactified direction with the
winding number of the string (the appropriately signed number of times
it wraps around that compactified dimension). This seems to me quite
extraordinary - the symmetry switches between a Noetherian charge (the
momentum) and a topologically conserved number (the winding number).
Robert C. Helling
<aber...@princeton.edu> wrote:
> Though not cited, it was John Baez who wrote:
>> Though not cited, it was Aaron Bergman who wrote:
>> >Though not cited, it was John Baez who wrote:
>> >> And maybe you could also tell me the answer when M is a compact
>> >> Lie group and we're talking about the Wess-Zumino-Witten model?
>> >> Here I guess the answer is supposed to have something to do with
>> >> representations of the corresponding affine Lie algebra.
>> >Everything in WZW models has to do with representations of the
>> >corresponding KM algebra.
>> Yes, I wasn't sticking my neck out too far there. But from
>> stuff Segal was saying, it seems there should be some *really
>> simple* description of the Hilbert space of boundary states
>> in this case: something like the direct sum of all the irreps
>> of the affine Lie algebra, or....
>He was probably talking about Cardy states or Ishibashi states. It's
>easy to write down a lot of things. The hard part is that they have to
>make sense in string amplitudes. In particular, they have to be
>consistent with all possible worldsheets. Showing this is rather hard.
>I'm not really up on this (but I know where to look if need be), so
>hopefully Robert Helling can explain this stuff better.
As far as I know, restrictions come from consistency with the OPE and
the cluster property as I explained in my first post in this
thread. You kind of guess one point functions in the presence of a
boundary (thereby implicitly defining your boundary conditions) and
then check the consistency of higher point amplitudes.
People who studied this problem (like Schomerus) are more CFT people
rather than string theorists (if this distinction makes sense). Thus in
their setting usually the worldsheet is fixed and they don't sum
genera and integrate over the moduli space of Riemann surfaces. Most
of the time they treat just the upper half plane (i.e. the disc) and
the annulus.
I cannot help with "the space of boundary conditions". The constraints
I was talking about are non-linear equations so I would assume that
that space is not linear. From a stringy point of view this is "the
space of all (stable) D-branes" which can be quite complicated in
curved space (do I hear somebody say "the derived category of coherent
sheaves"?).
Aaron Bergman
In article <slrnaen0jb...@ariel.aei-potsdam.mpg.de>,
hel...@ariel.physik.hu-berlin.de (Robert C. Helling) wrote:
> On Tue, 21 May 2002 20:55:27 +0000 (UTC), Aaron Bergman
> <aber...@princeton.edu> wrote:
> > Though not cited, it was John Baez who wrote:
> >> Though not cited, it was Aaron Bergman who wrote:
> >> >Everything in WZW models has to do with representations of the
> >> >corresponding KM algebra.
> >> Yes, I wasn't sticking my neck out too far there. But from
> >> stuff Segal was saying, it seems there should be some *really
> >> simple* description of the Hilbert space of boundary states
> >> in this case: something like the direct sum of all the irreps
> >> of the affine Lie algebra, or....
> >He was probably talking about Cardy states or Ishibashi states. It's
> >easy to write down a lot of things. The hard part is that they have to
> >make sense in string amplitudes. In particular, they have to be
> >consistent with all possible worldsheets. Showing this is rather hard.
> >I'm not really up on this (but I know where to look if need be), so
> >hopefully Robert Helling can explain this stuff better.
> As far as I know, restrictions come from consistency with the OPE and
> the cluster property as I explained in my first post in this
> thread. You kind of guess one point functions in the presence of a
> boundary (thereby implicitly defining your boundary conditions) and
> then check the consistency of higher point amplitudes.
I know you definitely have to check higher genus worldsheets. There's a
theorem that modular invariance is enough for the close string to ensure
that all higher worldsheets are OK, but I think things are much harder
with open strings. ISTR something about associativity of OPEs or
something like that. That's what you're referring to, right?
> People who studied this problem (like Schomerus) are more CFT people
> rather than string theorists (if this distinction makes sense).
Gaberdiel is the name that comes to mind for me. Particularly,
hep-th/0201113 which looks very readable.
Dave Rusin
In article <Pine.OSF.4.33.020519...@goedel3.math.washington.edu>,
Chris Hillman <hil...@math.washington.edu> wrote:
>I must beg forgiveness in advance because I haven't been able to follow
>this thread
Likewise here; I have only been reading bits and pieces, but I may be
able to offer some assistance with the math. (Connections with physics
are not clear to me!)
>> So here's the cute thing I think you secretly want to understand:
>> Suppose A is a discrete group. Then H^2(X,A) classifies A-gerbes
[Don't know gerbes]
>> over the *space* X - here we are talking about cohomology of *spaces*.
>> On the other hand, H^2(G,A) classifies extensions involving the *group*
>> G - here we are talking about cohomology of *groups*. So, they are
>> somewhat different things.
>>
>> On the other hand, as I mentioned, there's a very close relationship:
>>
>> H^2(G,A) = H^2(BG,A)
>>
>> This means that we should be able to turn an extension of A by G
>> into an A-gerbe over BG, and vice versa. They should in fact be
>> just two different ways of talking about the same thing: one more
>> "algebraic" in flavor, the other more "topological".
The topologists like to compute H^2(BG,A) with different groups A
because that allows them to focus on particular features of the
cohomology -- p-torsion, free rank, action of Steenrod algebra, etc.
But in most cases the group A just sits there. Not so with the
algebraists: it is of much more importance to them to compute
cohomology H^2(G,A) where A is a G-_module_. That action of
G on A is what makes cohomology useful to them in the first place
(usually).
Also I have to say I'm only accustomed to discrete groups. If G
itself has a topology (e.g. if G is a Lie group) then I have this
dim memory that rather than being isomorphic, H^*(G,A) and H^*(BG,A)
are involved in a long exact sequence with possibly non-zero intervening
terms. I don't know if I can even reconstruct what these functors _are_
in that case, so I'll just stick to discrete groups. (I think you
can generalize, though; read "fibration" for "covering space" below.)
If A is a trivial G-module, then yes, H^*(G,A) and H^*(BG,A)
are naturally isomorphic, and there's a very good reason. Do you know
what BG looks like? (Unfair question of course since BG is only
defined up to homotopy. But here is one model of BG.) First you
make EG , a CW complex, as follows. There's one vertex for each element
of G; one edge for each pair of elements of G, and the boundary of
the edge corresponding to (g1, g2) is {g1, g2}. Next there's a
two cell for every triple in G, etc. Glue these all together. That's
EG, which happens to be contractible. But G acts on EG by
permuting the 0-cells, the 1-cells, etc. by left multiplication.
That's a fixed-point-free action; let BG be the quotient space.
Then EG --> BG is a covering space with fibre equal to G.
So now think about how H^*(BG,A) and H^*(G,A) are defined. The
first can be computed using the chain complex having a copy of A
for each cell, i.e. for each equivalence class of tuples (g1, g2, ..., g_n).
The second is usually computed using the bar resolution, which
consists of, you guessed it, one copy of A for each equivalence
class of tuples of elements of G (traditionally written with bars:
[g1 | g2 | ... | g_n] -- get it? the "bar" construction).
By the way, if H is a subgroup of G, then H also acts on this
contractible space EG so the covering EG --> EG/H is a model
for the universal H-bundle as well as the G-bundle. So you don't have
to view the induced maps BH --> BG as some exotic thing; just
think about the big space EG being modded out by a couple of groups,
one containing the other.
>> Now, I don't grok this just yet, but it shouldn't be too hard,
>
>Well, if you learn how to do this from the book by Breen, I hope you'll
>mention the gist in a future Week. Even better, of course, if you can
>present some simple but nontrivial computational examples somewhere :-/
The canonical example is G = Z/2Z = {e, g}. That model I gave of EG
gives it two points e and g; two edges, (e,g) and (g,e) between them
(we can drop the "degenerate" cells (e,e) and so on); two 2-cells
with these as boundaries, etc. You're supposed to see the skeleta of
the final creation as being the chain of spheres:
S^0 \subset S^1 \subset S^2 ...
Now G has to act on this space, and it should act cellularly. Clearly
what it should do is map each point with its antipode (so e <--> g,
(e,g) <--> (g,e), and so on). The quotient space is then BG = projective
space RP^\infty, i.e. the union of the finite-dimensional projective spaces
RP^0 \subset RP^1 \subset RP^2 ...
each of which is the previous one plus one cell.
So the topologist now immediately sees H^n(RP^\infty, Z/2Z) = Z/2Z
for each n. But that's what the algebraist sees, too!
There are not many other groups with recognizable BG's. G=Z (circle)
and G = fundamental groups of Riemann surfaces come to mind. Also
the fact that PSL_2(Z) is just a free product Z/2Z * Z/3Z makes
it conceivable that one could visualize BPSL_2(Z) I guess.
dave
John Baez
Chris Hillman <hil...@math.washington.edu> wrote:
>Thanks for your answer to the first question; it seems helpful, but
>unfortunately I got stuck on a minor point:
>John Baez wrote:
>> The correct analogy table is:
>>
>> group G 2-group C
>> G-bundle C-2-bundle (special case being a gerbe)
>> classifying space BG classifying space BC
>[...] can you please extend this table to include one more line?
>
> (descriptor) EG (descriptor) EC
Sure: actually, I could write a table that was pages long
like this! But I'll spare you:
..............................................................
group G 2-group C
G-bundle C-2-bundle
classifying space BG classifying space BC
universal G-bundle EG -> BG universal C-2-bundle EC -> BC
..............................................................
Just as any G-bundle over X is obtained by pulling back the
universal G-bundle along a map f: X -> BG, any C-2-bundle
over X is obtained by pulling back the universal C-2-bundle
along a map f: X -> BC.
Of course we should be able to replace "2" by "n" in this
story, but I haven't actually worked out the details in general -
we don't know enough about n-categories to make that easy.
For more details try this:
http://math.ucr.edu/home/baez/gauge/
Dan Christensen
Maybe the following paper is relevant to this discussion?
Title: Gerbes and Homotopy Quantum Field Theories
Authors: Ulrich Bunke, Paul Turner and Simon Willerton
Subj-class: Algebraic Topology; Quantum Algebra
http://arXiv.org/abs/math/0201116
Dan
--
Dan Christensen
jdc+...@uwo.ca