Supergravity Cosmological Billards and the BIG group
Urs Schreiber
interesting observation that generic supergravities close to
a spacelike cosmological singularity can be described by
chaotic billiard motion in configuration space, e.g.
Damour & Henneuax & Nicolai, Cosmological Billiards,
hep-th/0212256
For supergravity theories that correspond to string theories
these billiards are related to interesting groups and the
hope/conjecture is that by generalizing these appropriately
one can guess/infer the BIG group behind it all, learning
something about M-theory.
Apparently there hasn't been much research into the quantization
of these cosmological billiards yet. I am currently at a seminar
on quantum chaos where everybody is talking about semiclassically
analyzing such chaotic billiards, so I began to wonder if some of
the results obtained here would pertain to, well, M-theory.
http://golem.ph.utexas.edu/string/archives/000342.html
In particular, I am wondering if the well known fact that
semiclassically a chaotic quantum system is well described
by random matrix theory has any relation to the fact that,
indeed, the cosmological billiards of Nicolai et al.
must also be related to a matrix theory, namely to that
of BFSS!
Apart from that speculation, I would be grateful if anyone
could provide me with some references to aspects of string
theory that can be studied using random matrix theory.
For example, apparently RMT also plays a role in describing
exact low-energy superpotentials of N=2 SYM theories broken
down to N=1.
n...@nw.com
> I would be grateful if anyone
> could provide me with some references to aspects of string
> theory that can be studied using random matrix theory.
Well, broadly interpreted, there's been a recent preprint math.CV/0402326
by Douglas, Schiffman & Zelditch that you might be interested in, -- I
mean, if you don't already know it. (BTW Zelditch is quite a big name in
mathematical quantum chaos, inverse spectral problem, etc; he has some
other papers in hep-th and many in math).
Related RMT & quantum chaos stuff (without stringy connections) are:
- Bogomolny's recent lectures on arithmetic quantum chaos
nlin.CD/0312061, again if you don't already know them.
- the papers of the Ulm group
<http://www.physik.uni-ulm.de/theo/qc/publications.html>.
Finally, have you seen that LQG paper citing the Nicolai et al. one which
somehow seems to contradict it -- at least for amateurs like me.
Urs Schreiber
> On Wed, 7 Apr 2004, Urs Schreiber wrote:
>
> > I would be grateful if anyone
> > could provide me with some references to aspects of string
> > theory that can be studied using random matrix theory.
> Finally, have you seen that LQG paper citing the Nicolai et al. one which
> somehow seems to contradict it -- at least for amateurs like me.
You mean gr-qc/0404039 ? I wouldn't say it contradicts anything.
Loop quantum cosmology is a speculation about what quantum cosmology is
which may be true or not
(see http://golem.ph.utexas.edu/string/archives/000299.html ).
The observation in hep-th/0212256 however is not a speculation,
but a fact, even if this fact naturally gives rise to many
interesting speculations.
The fact is that if you take the 1+0 dimensional sigma-model
of E10/K(E10), truncate it at order 30 (in the sense of hep-th/0207267)
thus obtaining a well defined object, then the dynamics of this
sigma model to this order is precisely that of 11d supergravity
truncated at a similar order in terms of spatial gradients
of its fields.
This is so far an observation pertaining to the classical
theory only. You may regard it as a way of introducing
"new variables" for (super)gravity in terms of which the
spatial diffeomorphism constraints are solved automatically
and only a single (not a familify of) Hamiltonian constraint
remains.
The most natural idea is to try to quantize the sigma model
on E10/K(E10), or one of its consistent truncations. In
principle this is straightforward, since quantizing 1+0 dim
theories just involves quantum mechanics. But of course there
are some thorny details. In any case, it hasn't been done yet.
But this in particular implies that it isn't certain,
as Martin Bojowald claims in gr-qc/0404039, that there
will still be the initial singularity in this model afer
quantization and after including all (or at least many)
of the degrees of freedom.
That's by the way the most intriguing aspect of this
business: One can show that the supergravity degrees of
freedom are only an infinitesimal fraction of all the
degrees of freedom described by the E10/K(10) model.
The natural guess is that the rest, which hasn't been
identified yet, corresponds to all the other excitations
of "M-theory".
Indeed, in hep-th/0402140 P. West shows how to get
several brane solutions using, not E10 but E11. I am
not sure if this clarifies the issue of the E10 model,
but it definitely points in the right direction.
More concretely, hep-th/0401053 claims to show that
the "imaginary roots" of E10 correspond to branes.
Looking at hep-th/0402090, where different
signatures of spacetime in the E11 context are discussed,
somehow makes one think of if there is any truth in
the analogy
E10 : E11 :: M-theory : F-theory,
maybe (but that's just a wild thought).
Urs Schreiber
> The most natural idea is to try to quantize the sigma model
> on E10/K(E10), or one of its consistent truncations. In
> principle this is straightforward, since quantizing 1+0 dim
> theories just involves quantum mechanics. But of course there
> are some thorny details. In any case, it hasn't been done yet.
Actually I now see that steps in this direction have been
done at least in
J. Brown, O. Ganor, C. Helfgott,
M-Theory on E10: Billiards, Branes, and Imaginary Roots,
hep-th/0401053 .
In section 7 the authors describe how to iteratively
construct the E10 Laplacian Delta (i.e. the "wave operator"
on the group G10). And they also discuss how certain
solutions of the conjectured "Wheeler-deWitt equation
of M-theory"
Delta |psi> = 0
should describe states containing brane/anti-brane pairs
as a pure quantum effect of this framework.
Furthermore they discuss a related "quantum test" of the
'Delta |psi> = 0'-conjecture related to the masses of the
branes.
One big open question is: What about fermions?
So far all these constructions seem to be purely bosonic,
even though on p.50 of the above paper the possibility
is mentioned that the fermions might be part of the E10
variables (in bosonized form).
But apparently one rather expects that a sort of
supersymmetrization of the E10 1+0d sigma model is
necessary.
I still think that it should be interesting to study the
deRahm operators d and del on E10. We know that every 1+0d
sigma model admits a straightforward N=2 susy extension
by replacing Delta by {d,del}. It seems to me that with
the technology presented in hep-th/0401053 it should be
easy to construct
d = d_h + d_0 + d_1 + d_2 + ...
and similarly for del in complete analogy to what is
done in section 7.2 of the above paper for
Delta = Delta_h + Delta_0 + Delta_1 + ... .
Hopefully I'll find the time to look into that
question. One should check if the fermions (differential
forms) introduced this way match the dynamics of
modes of the 11d sugra fermions, in analoggy to the bosonic
sector.
One important point at least seems to be immediate for
this kind of supersymmetrization of the E10 sigma model:
At the end of section 7.4 on p.53 of the above paper it
is mentioned that in the bosnic theory there is a problem
with zero-point energies which should better vanish. Of
course these do vanish automatically for the deRahm
model for harmonic forms of the respective d_i.
I see that Lubos is acknowledged by the above authors
for helpful discussion. Maybe he can comment on
the above issues.~
lumidek
I've been intrigued by Ori Ganor's proposals from the beginning.
Several years ago, he conjectured that the whole M-theory is encoded in
the Laplace equation defined on the E_{10} group manifold.
Partition sums of gravities, worldvolume theories and perhaps all other
things should be encoded in this allegedly unique function that
satisfies this Laplace equation on this huge manifold, see
http://tinyurl.com/2zuup http://arxiv.org/abs/hep-th/9910236
http://tinyurl.com/ys7jz http://arxiv.org/abs/hep-th/9903110
(Moderator's note: originally I only submitted the arxiv.org addresses,
but they were replaced by the tinyurl.com URLs, so I added the original
arxiv.org URLs again.)
This would have the advantage of making all U-dualities etc. manifest.
I think that the new paper by Ori et al. continues in this direction,
and they would also like to find a one-to-one map between the basis
vectors of the full M-theoretical Hilbert space on one side and the
roots of E_{10} on the other side, or something along
these lines.
We should perhaps ask Ori to write something interesting about it on
SPS.
Another purpose of this posting is to test LaTeX posted from
physicsforums.com. Click the link below to see the math version of this
post, including the nonsensical equation
E=mc^2 + \frac{\sqrt{1-x^2}}{\log(3)} + \int {\mathrm{d}} x\, \bar f(x)
g(x) + \dots
(Moderator's note: the commands (tex)...(/tex) were removed.)
All the best
Lubos
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Urs Schreiber
> I've been intrigued by Ori Ganor's proposals from the beginning.
Great to hear! :-)
BTW, do you know if the approach by Ganor et al. started independently
of the cosmological Billiard approach by Damour Henneaux and Nicolai
and later merged, or what is the historical development?
> Several years ago, he conjectured that the whole M-theory is encoded in
> the Laplace equation defined on the E_{10} group manifold.
Yes. That would be remarkable, if true. It is entertainingly
reminiscent of this famous saying, apparently due to Feynman,
that most of physics is about "Laplace phi = 0" in some
generalized sense.
> We should perhaps ask Ori to write something interesting about it on
> SPS.
Are you going to contact him or shall I?
By the way, while talking to Hermann Nicolai about these things
in Golm I learned about the interesting fact that E10 can be
regarded as sitting inside the vertex operator algebra of the
string. Of course that's long and well known (e.g. hep-th/9411188)
but I didn't fully appreciate before how already the symmetry
algebra of the single string knows about "M-theory" in this sense.
Right now I am visiting Ioannis Giannakis at Rockefeller University
and he also emphasizes this fact. As you know, I am here in order to
discuss deformations of superconformal worldsheet theories
(as in hep-th/0401175 and references given there) which in particular
include the symmetry transformations of the background (gauge
transformations, dualities, etc.). Ten years ago Evans, Giannakis and
Nanopoulos in
Evans, Giannakis & Nonopoulos,
An infinte dimensional symmetry algebra in string theory
hep-th/9401075
have already discussed the immense symmetry algebra that is
found by deforming the worldsheet CFT by similarity transformations
of the form
A -> exp(-W) A exp(W)
for W a general integrated vertex. I am not completely sure
what currently the state of the art is with respect to our
understanding of this algebra, though. I am being told that
physicists are currently better prepared to discuss E10
than mathematicians are.
Lubos Motl
thanks for your response! Ori et al. is writing another paper, so he is
pretty busy - and at least one of his collaborator is a student that is
finishing his/her PhD thesis, but I was told that eventually they will
write something about it on SPS.
> BTW, do you know if the approach by Ganor et al. started independently
> of the cosmological Billiard approach by Damour Henneaux and Nicolai
> and later merged, or what is the historical development?
My guess would be that it was independent, but they must know the answer
much better.
> Yes. That would be remarkable, if true.
Definitely. If true. ;-)
> It is entertainingly reminiscent of this famous saying, apparently due
> to Feynman, that most of physics is about "Laplace phi = 0" in some
> generalized sense.
Feynman also advertised the equation U=0 for the theory of everything
where U is Feynman's universal U function. Was the Laplacian comment more
serious?
> Are you going to contact him or shall I?
We have exchanged a couple of mails, but your mail might encourage him to
write something even more than one mail from me. ;-)
> By the way, while talking to Hermann Nicolai about these things
> in Golm I learned about the interesting fact that E10 can be
> regarded as sitting inside the vertex operator algebra of the
> string. Of course that's long and well known (e.g. hep-th/9411188)
This might be interesting, but let me ask a simple question. Are you/they
saying something beyond the simple claim that the E_{10} current algebra
can be represented by a compactified bosonic CFT?
> have already discussed the immense symmetry algebra that is
> found by deforming the worldsheet CFT by similarity transformations
> of the form
>
> A -> exp(-W) A exp(W)
>
> for W a general integrated vertex.
By a general integrated vertex, do you mean an integral of a (1,1) primary
field? Or the integral of an arbitrary field? Don't you just get the
algebra of all operators in the CFT? Although I used to think about
exactly these ideas in the past, today I don't quite know how big object
should I imagine when you talk about these intriguing concepts.
Moreover, it reminds me of some of our discussions about Thiemann's stuff.
Do you realize that exp(W).exp(-W) is not equal to identity if W is a
general enough operator in a quantum field theory? For example,
exp(i.k.X(z)) and exp(-i.k.X(z)) have huge short-distance singularities in
their OPEs, and so on. I hope you don't want to ignore the subtleties of
QFT and build something along these lines of Thiemann's papers - such an
approach contradicts all properties of quantum field theory that
distninguish it from simple quantum mechanics.
> I am not completely sure what currently the state of the art is with
> respect to our understanding of this algebra, though. I am being told
> that physicists are currently better prepared to discuss E10 than
> mathematicians are.
You are obviously a fan of the idea that the deep underlying principle
behind M-theory is some huge symmetry/group. A favorite idea of many great
people many decades ago, as well as of Thomas Larsson and others today.
Well, let me admit that my belief that the things are that simple has been
reduced significantly (all symmetries in string theory tend to be local
symmetries, and local symmetries describe a redundancy of your description
which is not quite a physical and measurable concept, and can differ
between different descriptions), and moreover the E_{10} group (or
hyperbolic algebra) just does not seem mysterious enough to me today. But
such exceptional groups must play *some* role.
Do you think that the existence of the E_9 and E_{10} symmetries of
M-theory on 9-dimensional and 10-dimensional tori is equally well
established as in the case of E_8 and lower groups?
All the best
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Urs Schreiber
> > Yes. That would be remarkable, if true.
>
> Definitely. If true. ;-)
Heh. Well, so it should be checked! I guess the task is to proceed as
in section 7 of hep-th/0401053 for higher orders
and try to identify the degrees of
freedom and their dynamical relations (e.g. as in section 7.4)
described by the various orders Delta_i of the Laplace operator.
At least for the pure supergravity degrees of freedom it
has been checked in
Damour, Henneaux, Nicolai
E10 and a "samll tension expansion" of M Theory
hep-th/0207267
that the E10 sigma-model reproduces these up to 30th order in
some appropriate expansion. So I guess there can be no doubt that
the E10 sigma-model does describe full 11d sugra plus lots of
other stuff. Wouldn't it be pretty weird if this other stuff is
_not_ that what we would hope it to be?
> Feynman also advertised the equation U=0 for the theory of everything
> where U is Feynman's universal U function. Was the Laplacian comment more
> serious?
I don't recall where I read it (must have been in the Feynman lectures,
somewhere) but, yes, in the context that I read it it was supposed to
be more serious than writing
0 = U = (F-ma)^2 + ... .
Personally I think the more refined version of the statement that I am
thinking about is the point of view expressed in
Froehlich, Grandjean, Recknagel,
Supersymmetric Quantum Theory, non-commutative geometry, and gravitation
hep-th/9706132
where it is emphasized that any system of susy QM can be viewed as
describing a particular (generalized) geometry. From this point of
view it wouldn't be surprising if a TOE is solved by "harmonic forms"
in some generalized sense (like e.g. harmonic functions on the group of
E10 in the case which we are discussing).
But I am digressing...
> This might be interesting, but let me ask a simple question. Are you/they
> saying something beyond the simple claim that the E_{10} current algebra
> can be represented by a compactified bosonic CFT?
I have barely begun learning about E10, so please bear with me.
First of all I am confused why you refer to the "E10 current algebra".
From hep-th/9411188 I seemed to learn that the whole point is that
E10 is not a current algebra, because of its indefinite Cartan
matrix. Only a Kac-Moody algebra with (semi)definite Cartan
matrix is a current algebra.
But what I was referring to was the fact that, according to
equation (0.11) of the above mentioned paper E10 sits inside the
Lie algebra of physical states of a completly compactified bosonic
string. Is that a "simple" claim?
If you feel that I am lacking some basic knowledge about E10,
please go ahead and educate me! :-)
> > have already discussed the immense symmetry algebra that is
> > found by deforming the worldsheet CFT by similarity transformations
> > of the form
> >
> > A -> exp(-W) A exp(W)
> >
> > for W a general integrated vertex.
>
> By a general integrated vertex, do you mean an integral of a (1,1) primary
> field? Or the integral of an arbitrary field? Don't you just get the
> algebra of all operators in the CFT? Although I used to think about
> exactly these ideas in the past, today I don't quite know how big object
> should I imagine when you talk about these intriguing concepts.
>
> Moreover, it reminds me of some of our discussions about Thiemann's stuff.
> Do you realize that exp(W).exp(-W) is not equal to identity if W is a
> general enough operator in a quantum field theory? For example,
> exp(i.k.X(z)) and exp(-i.k.X(z)) have huge short-distance singularities in
> their OPEs, and so on. I hope you don't want to ignore the subtleties of
> QFT and build something along these lines of Thiemann's papers - such an
> approach contradicts all properties of quantum field theory that
> distninguish it from simple quantum mechanics.
Ok, good point. The exponents used in hep-th/9401075 are supposed
to be normal orderd, but the full exponentials are not. So more explicitly
I was referring to
A -> exp(-:W:) A exp(:W:) .
(cf. eq. (2.5) of hep-th/9401075).
So this way exp(-:W:) is indeed the inverse of exp(:W:), at least
if one of them is well defined in the first place. The point of
all this is that given any stress-energy tensor T of the
worlsheet CFT you get a new worldsheet CFT by setting A=T in
the above formula. The new CFT can be interpreted as coming from
the old one by a generalized symmetry operation on the
background.
But, yes the W are supposed to be integrals over weight 1
fields, such that the sigma-reparameterization generator
T- bar T is left unaffected by the operation, as it must be.
The algebra of the :W: is huge and should have some close relation
to that discussed on pp3 of hep-th/9411188.
> > I am not completely sure what currently the state of the art is with
> > respect to our understanding of this algebra, though. I am being told
> > that physicists are currently better prepared to discuss E10 than
> > mathematicians are.
>
> You are obviously a fan of the idea that the deep underlying principle
> behind M-theory is some huge symmetry/group. A favorite idea of many great
> people many decades ago, as well as of Thomas Larsson and others today.
> Well, let me admit that my belief that the things are that simple has been
> reduced significantly (all symmetries in string theory tend to be local
> symmetries, and local symmetries describe a redundancy of your description
> which is not quite a physical and measurable concept, and can differ
> between different descriptions), and moreover the E_{10} group (or
> hyperbolic algebra) just does not seem mysterious enough to me today. But
> such exceptional groups must play *some* role.
I fully agree with what you are saying here.
> Do you think that the existence of the E_9 and E_{10} symmetries of
> M-theory on 9-dimensional and 10-dimensional tori is equally well
> established as in the case of E_8 and lower groups?
I don't know. I still know too little about this stuff to make an
educated guess. What I find impressive though is that E10 definitely
knows all about 11d sugra (at least of the bosonic sector, that is) while
still having way more variables. This alone seems to be very strong
evidence for the fact that E10 must say something about M theory,
I'd say.
Lubos Motl
> Heh. Well, so it should be checked!
Yes, it should be tried.
> that the E10 sigma-model reproduces these up to 30th order in
> some appropriate expansion.
I would probably believe this particular technical claim even with a small
number of orders than 30. ;-) I guess that you only talk about the bosonic
sector of SUGRA?
> So I guess there can be no doubt that
> the E10 sigma-model does describe full 11d sugra plus lots of
> other stuff. Wouldn't it be pretty weird if this other stuff is
> _not_ that what we would hope it to be?
If you ask me, my answer would be No, it would not be too weird.
Constructing the full covariant formulation of M-theory is a big task, and
it would seem more weird to me if someone suddenly found it using an
obscure and apparently a slightly ill-defined sigma-model, especially
because I don't expect M-theory to be *just* another string theory. :-)
Nevertheless, I am gonna look at these things because what you say sounds
highly nontrivial.
> First of all I am confused why you refer to the "E10 current algebra".
> From hep-th/9411188 I seemed to learn that the whole point is that
> E10 is not a current algebra, because of its indefinite Cartan
> matrix. Only a Kac-Moody algebra with (semi)definite Cartan
> matrix is a current algebra.
OK, so I wanted to say a more general word for the algebra that does not
have any such constraints of (semi)definiteness. Does it make a big
difference?
> But what I was referring to was the fact that, according to
> equation (0.11) of the above mentioned paper E10 sits inside the
> Lie algebra of physical states of a completly compactified bosonic
> string. Is that a "simple" claim?
I may misunderstand something, but what you say sounds as a simple claim.
If you read the paragraph below the equation (0.11) that you mentioned,
you will see that the nontrivial statement is not that g(A) is included in
g_\Lambda, but rather the fact that the inclusion is "proper", i.e. that
there are some missing states.
I thought that just like the bosons compactified on an even self-dual
lattice \Gamma_8 gives you an E_8 current algebra, the compactification on
another even self-dual lattice \Gamma_{9+1} gives you the E_{10}
hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra
(indefinite one, in this case) has vertex operators \partial{X^i}, while
the roots are represented by \exp(ik_iX_i(z)) where the momenta k_i belong
to the lattice, and they can be, in the E_{10} case, both time-like as
well as spacelike. Is there some huge difference that I missed?
> ...So this way exp(-:W:) is indeed the inverse of exp(:W:), at least
> if one of them is well defined in the first place.
No, the colons don't change anything about my claim. Let me repeat my
statements with the colons - which incidentally don't change the operators
in my example at all:
> > Do you realize that exp(:W:).exp(-:W:) is not equal to identity if W is a
> > general enough operator in a quantum field theory? For example,
> > exp(:i.k.X(z):) and exp(-:i.k.X(z):) have huge short-distance
> > singularities in their OPEs, and so on.
It is just not true that in quantum field theory you can write the inverse
operator to \exp(:W:) as \exp(-:W:) for a nontrivial operator W. The
subtlety is hidden in the OPE of W with itself. The exceptions where the
naive formula works are the operators W that are constructed as the
integrals of the (1,1) primary fields. The primary fields only have the
simple singularities with the stress energy tensor, and they lead to
"nonsingular" expressions after integration. Among these operators whose
exponentials are invertible, you will find the total momentum, the
generators of the Lorentz group (and its individual contributions, if
properly defined), and some others. But it is not correct to assume that
the exponential of a generic operator can be inverted in this classical
way.
> The point of all this is that given any stress-energy tensor T of the
> worlsheet CFT you get a new worldsheet CFT by setting A=T in the above
> formula. The new CFT can be interpreted as coming from the old one by
> a generalized symmetry operation on the background.
That's probably interesting, but I did not get it. ;-/
> The algebra of the :W: is huge and should have some close relation
> to that discussed on pp3 of hep-th/9411188.
Do you say that it is huge in a general case? The (1,1) primary fields are
certainly rare. For example, the Ising model CFT contains 3 of them.
> I fully agree with what you are saying here.
Nice to hear.
> I don't know. I still know too little about this stuff to make an
> educated guess. What I find impressive though is that E10 definitely
> knows all about 11d sugra (at least of the bosonic sector, that is) ...
The absence of fermions is another reason why I feel that this E_{10} is
unlikely to be far-reaching and even more unlikely to contain the whole
M-theory. The inclusion of the fermions would try to lead us to
superalgebras such as OSp(1|32) or something similar, which is very
different from E_{10}.
Cheers
Thomas Larsson
responded to the next message instead):
> You are obviously a fan of the idea that the deep underlying principle
> behind M-theory is some huge symmetry/group. A favorite idea of many great
> people many decades ago, as well as of Thomas Larsson and others today.
> Well, let me admit that my belief that the things are that simple has been
> reduced significantly (all symmetries in string theory tend to be local
> symmetries, and local symmetries describe a redundancy of your description
> which is not quite a physical and measurable concept, and can differ
> between different descriptions), and moreover the E_{10} group (or
> hyperbolic algebra) just does not seem mysterious enough to me today. But
> such exceptional groups must play *some* role.
Hmrph. It is hardly some random symmetry such as E_10 that I suggest. The
diffeomorphism group is *the* correct symmetry of GR (ask Rovelli!).
[Moderator's note: I hope that Rovelli is not necessary to answer basic
questions about classical general relativity. LM]
As Urs Schreiber has pointed out, LQG is not really canonical
quantization, which generically would give rise to lowest-energy reps of
the symmetry group - this is true even for gauge symmetries such as
worldsheet conformal symmetry. Thus, combining the symmetry principles
underlying GR and QM *and nothing else* - experimentally confirmed physics
- leads to LW reps of the diff group, and non-trivial such reps are
anomalous. That's just the way it is.
[Moderator's note: Sorry, I did not realize that you started to promote
another group as the theory of everything, beyond your original mb(3|8)
from http://arxiv.org/abs/hep-th/0208185 . I guess that E_{10} is closer
to the truth. LM]
It seems that the basic idea in West's approach is that M-theory and
11D SUGRA should for some reason be described by non-linear realization
of the Kac-Moody algebra E11. At least, this is what West writes in
http://www.arxiv.org/abs/hep-th/0104081, which seems to be the seminal
paper. A striking observation is that NO SUCH REALIZATION EXISTS! (if
the underlying space is finite-dimensional or mildly infinite-
dimensional). This follows immedately from the well-known fact that
non-affine Kac-Moody algebras like E11 are of expontential growth.
For the last 20 years at least, Victor Kac has emphasized that the lack
of a natural realization, linear or nonlinear, is a major problem for
non-affine Kac-Moody algebras; without a realization, these are somewhat
useless. Kac states this very explicitly in
http://www.arxiv.org/abs/q-alg/9709008:
"It is a well kept secret that the theory of Kac-Moody algebras has
been a disaster."
[Moderator's note: this disaster has not influenced physics too much,
and Kac-Moody algebras are very important and well functioning tool
to study conformal field theories and perhaps other realizations.
The users of physicsforums.com should probably click at "View the post
in the original ASCII form" because otherwise they might have problems
to see the equations and pictures in this post correctly.]
Thus, if West and others would find a nice realization of E_11 it would
be a major mathematical breakthrough, irrespective of its relevance to
M-theory (and the relevance of M-theory to physics). However, I have
glanced at quite a few of these E_11 papers, and I have never seen any
mentioning of exponential growth. This suggests that string theorists
do not realize that this is a major difficultly, which has stalled
Victor and other mathematicians for decades.
Nevertheless, West's original paper contains some beautiful math. First
recall that nonlinear realizations are the same thing as gradings.
E.g., the conformal realization of so(n+1,1) is a 3-grading:
so(n+1,1) = g_-1 + g_0 + g_1,
where g_-1 = translations, g_0 = rotations and dilatations, and
g_1 = conformal boosts. The grading is specified by defining the
grading operator, which here is the dilatation - the degree is the
dilatation eiginvalue. I had a nice discussion with John Baez and Tony
Smith about 16 months ago. The spr thread "Structures preserved by e8"
should be available from some archive.
Wests starts from a 7-grading of E8, eqs (4.1-11)
E8 = g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3.
If you formulate E8 in Cartan-Weil basis, H^i, E^\alpha, a grading
operator is of the form Z = alpha_i H^i. There is thus a 1-1 correspondence
between gradings and roots alpha. The relevant root here is
*
|
o-o-o-o-o-o-o
which gives
E8 = V* + A^6 V + A^3 V + (sl(8)+C) + A^3 V* + A^6 V* + V.
248 = 8 + 28 + 56 + 64 + 56 + 28 + 8
Here V is the 8-dim vector rep of sl(8) and V* its dual and A^n V is
the n:th anti-symmetric power. Note that g_0 = sl(8)+C and you get the
Dynkin diagram of sl(8) by removing the starred root - this is a general
phenomenon. This grading corresponds to a realization on the dual of
g_- = g_-3 + g_-2 + g_-1, according to the following table:
g_-1 x_abc R^abc = d/dx_abc + ...
g_-2 y_abcdef R^abcdef = d/dy_abcdef + .....
g_-3 z^a S_a = d/dz^a
Aside: other E8 gradings have also occurred in the literature. The one
corresponding to
o
|
o-o-o-o-o-o-*
(g_0 = E7) was discussed in a 2000 CMP paper by Gunaydin-Koepsell-Nicolai
and
o
|
*-o-o-o-o-o-o
(g_0 = so(14)) is implicit in GSW, (6.A.1-3).
However, E8 cannot be right for M-theory, because the vector rep of g_0
= sl(8) is 8-dimensional and describes translations in 8D rather than
11D. Therefore, West seems to propose that the E11 grading corresponding to
*
|
o-o-o-o-o-o-o-o-o-o
should be relevant to M-theory. Alas, this grading extends to infinity
in both directions,
E11 = ... + g_-4 + g_-3 + g_-2 + g_-1 + g_0 + g_1 + g_2 + g_3 + ... .
Moreover, dim g_k ~ exp(ck) for some constant c, i.e. E11 is of
exponential growth. This means that E11 is HUGE, much bigger than any
local symmetry. Local symmetries, e.g. diff and Yang-Mills in n dimensions,
grow polynomially, dim g_k ~ k^(n-1), i.e. there are finitely many
symmetry generators per spacetime point.
Urs Schreiber
> As Urs Schreiber has pointed out, LQG is not really canonical
> quantization, which generically would give rise to lowest-energy reps of
> the symmetry group - this is true even for gauge symmetries such as
> worldsheet conformal symmetry. Thus, combining the symmetry principles
> underlying GR and QM *and nothing else* - experimentally confirmed physics
> - leads to LW reps of the diff group, and non-trivial such reps are
> anomalous. That's just the way it is.
BTW, do we now agree that this fact is not in contradiction to any
known fact in string theory? You seemed to claim that the fact that
gravitational anomalies for chiral fermions occur in
4k + 2 dimensions is in contradiction with the above somehow. But
clearly it is not, since what you write above has nothing to do
with chiral fermions.
> For the last 20 years at least, Victor Kac has emphasized that the lack
> of a natural realization, linear or nonlinear, is a major problem for
> non-affine Kac-Moody algebras; without a realization, these are somewhat
> useless. Kac states this very explicitly in
> http://www.arxiv.org/abs/q-alg/9709008:
>
> "It is a well kept secret that the theory of Kac-Moody algebras has
> been a disaster."
>
> [Moderator's note: this disaster has not influenced physics too much,
> and Kac-Moody algebras are very important and well functioning tool
> to study conformal field theories and perhaps other realizations.
Lubos, please note that Larsson (and Kac for that matter) referred to
_non-affine_ Kac-Moody algebras, while it seems to me what you have
in mind are current algebras, i.e. _affine_ KM algebras.
KM algebras fall into 3 classes:
1) Cartan matrix positive definite --> finite algebra
2) Cartan matrix positive semi-definite --> affine algebra = current
algebra in 2d
3) Cartan matrix indefinite --> exponential growth, desaster and all that
(paraphrased from hep-th/9411188)
> Thus, if West and others would find a nice realization of E_11 it would
> be a major mathematical breakthrough, irrespective of its relevance to
> M-theory (and the relevance of M-theory to physics). However, I have
> glanced at quite a few of these E_11 papers, and I have never seen any
> mentioning of exponential growth. This suggests that string theorists
> do not realize that this is a major difficultly, which has stalled
> Victor and other mathematicians for decades.
This is at least not true for all physicist. In
Gebert, Nicolai,
E_10 for beginners,
hep-th/9411188
it is emphasized quite strongly that there is exponential growth and
that things are pretty problematic concerning E_10. But I am being
told that the point is that one can study consistent finite truncations
which can be handled and in terms of which we can do physics.
Urs Schreiber
> > that the E10 sigma-model reproduces these up to 30th order in
> > some appropriate expansion.
>
> I would probably believe this particular technical claim even with a small
> number of orders than 30. ;-) I guess that you only talk about the bosonic
> sector of SUGRA?
Yes. But I was told that using some appropriate susy extension (I am not
sure which one, though) you can similarly check that also the fermionic
degrees of freedom of sugra are there. Aparently this is not published
yet, though.
> If you ask me, my answer would be No, it would not be too weird.
> Constructing the full covariant formulation of M-theory is a big task, and
> it would seem more weird to me if someone suddenly found it using an
> obscure and apparently a slightly ill-defined sigma-model, especially
> because I don't expect M-theory to be *just* another string theory. :-)
Hm, the sigma model wouldn't be "another string theory", somehow, I think.
It would rather be a vast extension of what in quantum cosmology is
sometimes called "miDi superspace", i.e. the configuration space of all
modes of all objects in the theory.
I must say that I find the claim that such a 1+0 dim sigma model of all
of M-theory exists (BTW, this is not a 1+1d sigma model describing string
motion in some background!) plausible, because it is quite similar to
the M=Matrix proposal, where the claim is, too, that all the degrees of
freedom can be parameterized by a 1+0 dimensional quantum mechanics.
Indeed, I have a little private speculation how one could maybe make
a possible relation between the two 1+0d models which we know to
reduce to 11d sugra in some limit, namely the E_10 model and BFSS.
I have talked about that at the Coffee Table:
http://golem.ph.utexas.edu/string/archives/000342.html .
The rough idea is the following: We know that close to a
spacelike singularity (super)gravity decouples in the sense that
nearby points on spatial hyperslices no longer interact because
they are outside each other's backward light cones and that
in this limit the dynamics of each of the thus disconnected
small causal patches of space are described in config space by
the billiard with walls given by the Weyl chamber of E_10
(in the case of 11d sugra). This billiard motion is chaotic,
a generalization of the old BKL "mixmaster" idea.
Now the "first law" of quantum chaos is that the universal
behaviour of any classically chaotic system (i.e. that
behaviour which remains when you slightly coarse-grain the
dynamics so as to get rid of system-dependent details of the
dynamics on small scales) is described by Random Matrix Theory,
i.e. by a theory where you pick a random ensemble of Hamiltonians
and average over their properties (see the above link for more on
this). In particular, the ensemble is in general Gaussian in the
sense that the probability to find a particular u(N) matrix M
in the ensemble is proportional to
exp(- tr(M^2)) .
Taking this well known fact in the context of the above
cosmological Billiard tells us that 11d sugra in the limit
where inteaction are negligible is well approximated by
a canonical matrix ensemble exp(- tr(M^2)).
But isn't that suggestive? Take the BFSS canonical ensemble
in the limit where the interaction terms are negected. It,
too, looks like exp(- tr P^2).
> Nevertheless, I am gonna look at these things because what you say sounds
> highly nontrivial.
Great. I hope you'll let us know what you come up with! :-)
> > First of all I am confused why you refer to the "E10 current algebra".
> > From hep-th/9411188 I seemed to learn that the whole point is that
> > E10 is not a current algebra, because of its indefinite Cartan
> > matrix. Only a Kac-Moody algebra with (semi)definite Cartan
> > matrix is a current algebra.
>
> OK, so I wanted to say a more general word for the algebra that does not
> have any such constraints of (semi)definiteness. Does it make a big
> difference?
Yes, apparently. I am no expert yet on this stuff, but according
to the introductory article hep-th/9411188 it makes a huge
difference. Non-affine KM algebras with indefinite Cartan matrices
are apparently mind-bogglingly huge beasts which greatly
exceed the complexity of the affine KM algebras. Hermann Nicolai
has emphasized that to me over and over again. In order to
emphasize this point he had shown me files of piles of pages
of computer output where order-by-order the affine KM algebras contained
in E_10, as well as their irreps and the multiplicity with which
these appear, were listed. He said that mathematicians don't
understand this huge algebra well yet, but that string theorists
are slowly beginning to make progress in these matters.
> I thought that just like the bosons compactified on an even self-dual
> lattice \Gamma_8 gives you an E_8 current algebra, the compactification on
> another even self-dual lattice \Gamma_{9+1} gives you the E_{10}
> hyperbolic Kac-Moody algebra. The 10-dimensional Cartan subalgebra
> (indefinite one, in this case) has vertex operators \partial{X^i}, while
> the roots are represented by \exp(ik_iX_i(z)) where the momenta k_i belong
> to the lattice, and they can be, in the E_{10} case, both time-like as
> well as spacelike. Is there some huge difference that I missed?
Hm, no, probably not. It is just important that E_10 is not
a current algebra, as far as I understand.
> > ...So this way exp(-:W:) is indeed the inverse of exp(:W:), at least
> > if one of them is well defined in the first place.
>
> No, the colons don't change anything about my claim. Let me repeat my
The point of the colons was to distinguish
:exp(W):
from
exp(:W:) .
Due to OPEs we have
:exp(W): :exp(-W): = possibly lots of correction terms .
But on the other hand
exp(:W:)exp(-:W:) = exp(:W:-:W:) = 1 .
Ah, now I see what your are perhaps thinking: It is important
that here W is the same object everywhere, i.e. what I wrote is
_not_ supposed to be a shorthand for
exp(:W:(z)) exp(-:W:(w))
with z \neq w . In fact, W is not supposed to depend on any coordinate
anymore, because it is an integrated object.
> > The point of all this is that given any stress-energy tensor T of the
> > worlsheet CFT you get a new worldsheet CFT by setting A=T in the above
> > formula. The new CFT can be interpreted as coming from the old one by
> > a generalized symmetry operation on the background.
>
> That's probably interesting, but I did not get it. ;-/
There are really to statements here:
1) Take the Virasor algebra generated by L_m satisfying
[L_m,L_n] = (m-n)L_{m+n} + anomaly .
Now define
L'_m := exp(-W) L_m exp(W)
with the exponentials defined as discussed above so that indeed
exp(W)exp(-W) = 1 . Then obviously we have defined an algebra
isomorphism and the new generators still satisfy the Virasoro
algebra:
[L'_m, L'_n] = (m-n)L_{m+n} + anomaly .
2) The second statement is that one can indeed interpret the
new L'_m as the generators corresponding to a different background
of the string theory. Since the states annihilated by the L'_m
are just exp(-W) times the states annihilated by the original
L_m and the spectrum is the same and everything, the new
background must be related to the original one by a symmetry
of the background theory.
> > The algebra of the :W: is huge and should have some close relation
> > to that discussed on pp3 of hep-th/9411188.
>
> Do you say that it is huge in a general case? The (1,1) primary fields are
> certainly rare. For example, the Ising model CFT contains 3 of them.
I say that it is huge in genral. It is the full algebra of DDF invariants,
i.e. of polynomials in the A_n^mu where
A_n^mu \propto \int dz \partial X^mu exp(in k \cdot X)
(for the bosonic string).
> The absence of fermions is another reason why I feel that this E_{10} is
> unlikely to be far-reaching and even more unlikely to contain the whole
> M-theory. The inclusion of the fermions would try to lead us to
> superalgebras such as OSp(1|32) or something similar, which is very
> different from E_{10}.
As I said above, there is apparently a susy extension of E_10 which does
correctly reproduce the fermionic degrees of freedom of 11d sugra
(among other things). As far as I understood the corresponding
publications are in preparation.
Another (mayb equivalent) straighforward inclusion of fermions
is the promotion of the 1+0d E_10 model to the respective
deRahm model, as I have mentioned before.
Urs Schreiber
> I say that it is huge in genral. It is the full algebra of DDF invariants,
> i.e. of polynomials in the A_n^mu where
>
> A_n^mu \propto \int dz \partial X^mu exp(in k \cdot X)
>
> (for the bosonic string).
Let me make that more precise:
Pick any physical state by applying lots of the A_n^\mu to the
tachyon state. By the state/operator correspondence the resulting
state comes from an integrated weight 1 vertex (the V(\psi,z) in
equation (0.10) of hep-th/9411188).
Urs Schreiber
> 1) Take the Virasor algebra generated by L_m satisfying
> [L_m,L_n] = (m-n)L_{m+n} + anomaly .
> Now define
>
> L'_m := exp(-W) L_m exp(W)
>
> with the exponentials defined as discussed above so that indeed
> exp(W)exp(-W) = 1 . Then obviously we have defined an algebra
> isomorphism and the new generators still satisfy the Virasoro
> algebra:
>
> [L'_m, L'_n] = (m-n)L_{m+n} + anomaly .
In my discussions with Ioannis Giannakis I have come up
with an alternative description in terms of string
field theory, which might be instructive:
Consider the equation of motion of open bosonic cubic string
field theory:
Q \phi + \phi \star \phi = 0
(setting the constant factor to unity). Here Q is the BRST operator
for a string and \star is the star product which
merges two string states. \phi is the string field.
Now suppose we want to study single strings in non-trivial
backgrounds. We could do that by trying to find a consistent
worldsheet CFT. But the above equation should give us
an alternative prescription:
Pick any solution \phi to the above string field equations of motion.
Now perturb this solution by adding an infinitesimal "test field"
\psi, namely the field of a single "test string" probing the background
described by phi:
\phi \to \phi + \psi .
By inserting this ansatz into the above equation of motion, using the
fact that the original \phi is a solution all by itself and that
\psi is infinitesimal, we get the equation
Q \psi + \phi \star \psi + \psi \star \phi = 0 .
If I introduce the notation
\{\phi, \cdot\} = \psi \star \cdot + \cdot \star \phi
this becomes
(Q \psi + \{\phi, \cdpt\}) \psi = 0
which should be the quantum consraint for the single string \psi
in the background described by the string field \phi .
Is this interpretation consistent?
If it is, it should be possible ot interpret
Q^\prime := Q + \{\phi , \cdot \}
as the BRST operator of the string in the background \phi.
Can that be? Is Q^\prime nilpotent?
Hm, maybe I am confused. Comments are welcome. Anyway, what I
wanted to say is that if the excitation \phi is a symmetry of the
theory in the sense that turning on \phi does not really change
the physics, then we would expect that
Q^\prime = \exp(-W) Q \exp(W) .
This are symmetries of the type studied in
Ioannis Giannkis,
String in nontrivial gravitino and RR backgrounds
hep-th/0205219
Lubos Motl
> Yes. But I was told that using some appropriate susy extension (I am not
> sure which one, though) you can similarly check that also the fermionic
> degrees of freedom of sugra are there. Aparently this is not published
> yet, though.
I would like to see it first, to avoid building on someone's wishful
thinking. Be sure that I will be among the first believers if such a
theory really works, including the fermions. ;-)
> I must say that I find the claim that such a 1+0 dim sigma model of all
> of M-theory exists (BTW, this is not a 1+1d sigma model describing string
> motion in some background!) plausible, because it is quite similar to
> the M=Matrix proposal, where the claim is, too, that all the degrees of
> freedom can be parameterized by a 1+0 dimensional quantum mechanics.
Sure, why not. :-) But there is usually a long path from these big dreams
to actual results.
> ...dynamics so as to get rid of system-dependent details of the
> dynamics on small scales) is described by Random Matrix Theory,
Do you want to reproduce M-theory amplitudes approximately, or exactly?
Is this connection with Random Matrix Theory a trick to get the right
results? Or is it evidence that this description cannot work exactly?
> But isn't that suggestive? Take the BFSS canonical ensemble
> in the limit where the interaction terms are negected. It,
> too, looks like exp(- tr P^2).
There are also many systems that I don't like where you can obtain
\exp(-\tr P^2), too. :-) This exponential is not such a difficult concept
for us to be shocked if we see it somewhere.
> Yes, apparently. I am no expert yet on this stuff, but according
> to the introductory article hep-th/9411188 it makes a huge
> difference. Non-affine KM algebras with indefinite Cartan matrices
> are apparently mind-bogglingly huge beasts which greatly...
Well, I understand why E_9 is the affine E_8, and why the hyperbolic
algebra E_{10} is even "bigger". There are many generators for a single
root, and infinitely many roots organized in a Minkowski-like Cartan
space, and so forth. Using the words "mind-boggling" for such things does
not look quite appropriate to me. These objects are simply large, which is
both a good as well as a bad message. I don't see anything amazing about
an object only because it is "large"; and it makes such an object less
ready for various operations. For example, we don't want to build
Yang-Mills theories based on infinite-dimensional non-compact groups, at
least so far. :-)
What I still think is straightforward is that one can obtain such
algebras, with the correct commutation relations, from a compactified
bosonic CFT, and the maths justifying this fact is as trivial as in the
E_8 case. Am I wrong? This is about the very definition of E_{10}. The
compactified bosonic CFT simply reproduces (and allows you to check) the
defining axioms of E_{10} step by step. If there is something nontrivial
about these things, you will have to be more specific to explain it.
> The point of the colons was to distinguish
> :exp(W):
> from
> exp(:W:) .
OK, understood. But then one must note that the non-ordered exp(:W:)
contains a lot of hard-to-interpret singular terms which make such an
operator useless and, in fact, also ill-defined if W is a local operator.
> Ah, now I see what your are perhaps thinking: It is important
> that here W is the same object everywhere, i.e. what I wrote is
> _not_ supposed to be a shorthand for
>
> exp(:W:(z)) exp(-:W:(w))
>
> with z \neq w .
That's exactly one of the problems. If W(z) is a local operator, there is
no working easy way to define W(z)^2. The only way to do so is to consider
W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of
singularities that you must deal with, subtract, normal-order, and so on,
and these operations will destroy the naive identities that you are
thinking about.
> In fact, W is not supposed to depend on any coordinate
> anymore, because it is an integrated object.
I just don't see anything sharp about these objects. Of course, you can
define the U(\infty) group acting on the whole Hilbert space - which is
the only thing I can imagine where you are led if you try to define an
algebra that contains more than just a "couple" of operators. But every
quantum mechanical theory has this U(\infty) "symmetry" group and we use
it extensively all the time e.g. when we switch from one basis to another
(it is only a symmetry of the norm, not a symmetry of dynamics). I don't
precisely understand whether you want to define something more useful,
constrained, and original than this universal and well-known U(\infty),
and if you want, what it is. Depending on your constraints what the "W"
operators can be, you can obtain a couple of algebras, but because you
have not described any clear constraints so far, it seems that you talk
about the U(\infty) algebra.
> There are really to statements here:
>
> 1) Take the Virasor algebra generated by L_m satisfying
> [L_m,L_n] = (m-n)L_{m+n} + anomaly .
> Now define
>
> L'_m := exp(-W) L_m exp(W)
>
> with the exponentials defined as discussed above so that indeed
> exp(W)exp(-W) = 1 . Then obviously we have defined an algebra
> isomorphism and the new generators still satisfy the Virasoro
> algebra:
>
> [L'_m, L'_n] = (m-n)L_{m+n} + anomaly .
Well, that's a basic lesson of undergraduate linear algebra that
conjugation does not change the commutation relations. I guess that both
of us realize that every physicist must know these things, and that we are
using them all the time - and it is unlikely to make a revolutionary
discovery just by noting that conjugation preserves commutators.
> 2) The second statement is that one can indeed interpret the
> new L'_m as the generators corresponding to a different background
> of the string theory. Since the states annihilated by the L'_m
> are just exp(-W) times the states annihilated by the original
> L_m and the spectrum is the same and everything, the new
> background must be related to the original one by a symmetry
> of the background theory.
I find this statement physically vacuous. You have not constructed any new
theory at all; you just renamed the basis vectors of its Hilbert space in
a chaotic and arbitrary fashion. If you want to get a spacetime
interpretation of the "transformed" theory, you will have to undo this
unphysical exp(W) transformation, and return to the original operators
such as X^\mu(z) which encode the position of the string in spacetime -
instead of \exp(-W) X^\mu(z) \exp(W) - and you will finally obtain
*identical* physical results. You have not even changed any moduli in the
theory. It seems that what you did is nothing more than the U(\infty)
symmetry discussed above, and this whole "large" symmetry is not
physically useful unless you restrict it to something that preserves some
other structure of the theory beyond the norm on the Hilbert space.
> I say that it is huge in genral. It is the full algebra of DDF invariants,
OK, once again, it seems that you are talking about the full U(\infty)
acting on the Hilbert space which I find useless for physics. It is an
algebra that does not commute with the spacetime Hamiltonian, for example.
It does not even have any simple commutation relations with these
spacetime Poincare generators. I think that what you are talking about is
the algebra of everything - and everything is more or less isomorphic to
nothing. It is the sort of group that allows you to "prove" that a
harmonic oscillator is "equivalent" to the Standard Model: both of them
have an infinite-dimensional Hilbert space, and they can be mapped to one
another, and every operator in one theory can be identified with an
operator in the other theory. But I say that such a conclusion is
unphysical because the structures that physics requires are more than just
the norm. You need some sort of dynamics and interpretable operators - for
example the Hamiltonian - and the only interesting transformations are
those under which the Hamiltonian (...) is invariant, or at least
transforms in a controllable way.
> As I said above, there is apparently a susy extension of E_10 which does
> correctly reproduce the fermionic degrees of freedom of 11d sugra
> (among other things). As far as I understood the corresponding
> publications are in preparation.
I thought that there is a theorem that no useful superalgebra extending
E_{10} ( or at least E_8 ) exists.
> Another (mayb equivalent) straighforward inclusion of fermions
> is the promotion of the 1+0d E_10 model to the respective
> deRahm model, as I have mentioned before.
OK, but it is a random guessing, is not it? I sort of feel that you want
to say that if you construct anything huge enough, it must be M-theory.
Well, this is an assumption that I certainly don't share. But it is
probably still more rational than the assumption of Lee Smolin that
anything *simple* enough - e.g. the 3D Chern-Simons theory - must be
equivalent to M-theory. ;-)
Cheers,
Urs Schreiber
> Sure, why not. :-) But there is usually a long path from these big dreams
> to actual results.
Sure.
> > ...dynamics so as to get rid of system-dependent details of the
> > dynamics on small scales) is described by Random Matrix Theory,
>
> Do you want to reproduce M-theory amplitudes approximately, or exactly?
> Is this connection with Random Matrix Theory a trick to get the right
> results? Or is it evidence that this description cannot work exactly?
This was supposed to be evidence that at least in the discussed limit
the E_10 model is indeed equivalent to the BFSS model. Of course
the limit is pretty radical and you may argue that it doesn't allow
any conclusion about the equivalence of the two models
away from this limit.
> OK, understood. But then one must note that the non-ordered exp(:W:)
> contains a lot of hard-to-interpret singular terms which make such an
> operator useless and, in fact, also ill-defined if W is a local operator.
>
> > Ah, now I see what your are perhaps thinking: It is important
> > that here W is the same object everywhere, i.e. what I wrote is
> > _not_ supposed to be a shorthand for
> >
> > exp(:W:(z)) exp(-:W:(w))
> >
> > with z \neq w .
>
> That's exactly one of the problems. If W(z) is a local operator, there is
> no working easy way to define W(z)^2. The only way to do so is to consider
> W(z)W(w) for w approaching z. You will see that W(z)W(w) has a lot of
> singularities that you must deal with, subtract, normal-order, and so on,
> and these operations will destroy the naive identities that you are
> thinking about.
We are still not talking about the same constructions, it seems to me.
I am not considering things like W(z)^2. As I said, the W are integrated
objects W = \int dz something(z).
For instance the W which induces T-duality is of the form
W \propto \int d\sigma (exp(i \sqrt{2}k\cdot X) - h.c.) .
Compare equation (3.18) of hep-th/9511061 .
> I just don't see anything sharp about these objects. Of course, you can
> define the U(\infty) group acting on the whole Hilbert space - which is
> the only thing I can imagine where you are led if you try to define an
> algebra that contains more than just a "couple" of operators.
That's too large and indeed without physical content. But as I said
before, the restriction on the allowed W is that the similarity
transformation preserves T - \bar T. Only under this condition can the
resulting new algebra be interpreted as coming from a string in a certain
background.
> > [L'_m, L'_n] = (m-n)L_{m+n} + anomaly .
>
> Well, that's a basic lesson of undergraduate linear algebra that
Yes. :-)
> conjugation does not change the commutation relations. I guess that both
> of us realize that every physicist must know these things, and that we are
> using them all the time - and it is unlikely to make a revolutionary
> discovery just by noting that conjugation preserves commutators.
It is interesting to examine how different admissable W give
rise to certain symmetries/dualities of the string backgrounds.
You can use this also to find deformations which are not symmetries
but give new backgrounds, inequivalent to the one one started
from.
> I find this statement physically vacuous. You have not constructed any new
> theory at all; you just renamed the basis vectors of its Hilbert space in
True, but that's precisely what a symmetry/duality is supposed to do.
As you know a string state with winding number n and momentum number m
is sent to a state with winding m and momentum n and this is
a symmetry. So T-duality, too, could be described as just a
renaming of vectors in a Hilbert space, which it is. But of
course that does not mean that it is not interesting.
> I thought that there is a theorem that no useful superalgebra extending
> E_{10} ( or at least E_8 ) exists.
Aha, don't know about that. Do you have a reference?
> > is the promotion of the 1+0d E_10 model to the respective
> > deRahm model, as I have mentioned before.
>
> OK, but it is a random guessing, is not it?
Not fully random, but, yes, it is guessing. :-)
Lubos Motl
> This was supposed to be evidence that at least in the discussed limit
> the E_10 model is indeed equivalent to the BFSS model.
Does this proof explain why the BFSS model is the discrete light-cone
quantization of the covariant E_{10} model, and why "N" of BFSS is the
light-like momentum? (If it does not, it must be wrong.)
> U(\infty) is too large and indeed without physical content. But as I said
> before, the restriction on the allowed W is that the similarity
> transformation preserves T - \bar T.
T-\bar T ? Do you mean the different between T_{zz} and T_{\bar z \bar z} ?
That's very strange. You cannot just add different components of the
same tensor. More precisely, this difference is not Virasoro invariant.
The exact conformal symmetry is essential for string theory to work, at
least at the loop level, and the condition `` {T-\bar T} is invariant
under the elements of your group'' is not Virasoro invariant. So I guess
that you now understand that I consider the group preserving {T-\bar T} to
be as (un)natural as the group that preserves L_{2004}+2005 \tilde
L_{2006} or anything else. What's exactly your reason to think that there
is anything interesting about a group defined in this way, as opposed to
any other random combination of words and symbols that respect the
syntactic rules of our mathematical formalism?
> > I find this statement physically vacuous. You have not constructed any new
> > theory at all; you just renamed the basis vectors of its Hilbert space in
>
> True, but that's precisely what a symmetry/duality is supposed to do.
No, it's not. A duality (and similarly for a self-duality, which is
essentially the same thing as a symmetry) is a map redefining the
variables of a theory, a redefinition that transforms the known
observables in a first known theory AB exactly into some known observables
of another known theory CD (which may be the same thing as AB). For
example, if we work with the same theory AB and all the relevant operators
such as the Hamiltonian are *invariant* under the map, then we talk about
the symmetry of the theory AB.
On the other hand, your conjugation by a random operator exp(W) transforms
the quantities of a theory AB - for example, the Virasoro worldsheet
generators or the spacetime Lorentz generators - into some other,
different randomly modified operators in a theory EF whose form has never
been written before. The best way to solve the theory EF is to conjugate
them back, and realize that EF is just an awkward way to write the theory
AB; they are equivalent, and we don't earn anything by talking about a
"new" theory EF. Such a conjugation by exp(W) is not a symmetry of AB,
because the Hamiltonian and other operators are not invariant under it,
and it is also not a duality because it does not relate two useful
theories in a nontrivial way - it relates one theory to the same theory
written in awkward variables.
A duality or a symmetry are very priviliged words, and a random
redefinition of variables or a conjugation by a random operator certainly
don't deserve to be called a "duality" or a "symmetry". In CFT and
perturbative string theory, you should only use the word "symmetry" or
"duality" for the transformations that preserve the form of all Virasoro
generators, and be sure that the known spacetime symmetries and T-duality
are the only solutions.
> As you know a string state with winding number n and momentum number m
> is sent to a state with winding m and momentum n and this is
> a symmetry. So T-duality, too, could be described as just a
> renaming of vectors in a Hilbert space, which it is. But of
> course that does not mean that it is not interesting.
But there is a huge difference between T-duality, which is a textbook
example of a duality, on one side, and your generic conjugation on the
other side. T-duality is a map, namely X_L\to X_L, X_R\to -X_R, that
preserves all Virasoro generators, for example, and at the self-dual
radius, it even preserves the periodicities of all fields, and therefore
is a nontrivial symmetry from the Hilbert space to the same Hilbert space
(the same superselection sector). T-duality is a symmetry of string theory
mapping two backgrounds that were thought to be independent (such as the
Universe with radii R and 1/R of a circle) into one another. At the
self-dual radius, it is even a symmetry of the background - in fact, it is
an element of the SU(2)^2 enhanced gauge symmetry.
On the other hand, your generic conjugation is a meaningless redefinition
of fields that preserves neither the Virasoro algebra nor the spacetime or
other operators. It is an uninteresting change of the conventions
describing the same theory, not a map between two different theories or a
symmetry of a single theory. You just can't use the word "symmetry" or
"duality" for such contentless constructions.
> > I thought that there is a theorem that no useful superalgebra extending
> > E_{10} ( or at least E_8 ) exists.
>
> Aha, don't know about that. Do you have a reference?
Unfortunately I don't. I was asking the people who were telling me about
this theorem, but no one has given me any reference either.
> > > is the promotion of the 1+0d E_10 model to the respective
> > > deRahm model, as I have mentioned before.
> >
> > OK, but it is a random guessing, is not it?
>
> Not fully random, but, yes, it is guessing. :-)
OK, so if it is not fully random :-), do you have some explanation why you
said deRham model and not something else, for example the topological
B-model on the E_{10} group manifold? ;-)
Urs Schreiber
> T-\bar T ? Do you mean the different between T_{zz} and T_{\bar z \bar z} ?
> That's very strange. You cannot just add different components of the
> same tensor.
T - \bar T (in modes L_m - \bar L_{-m})
is always the generator of spatial reparameterizations
of the string at appropriately fixed worldsheet time. That it
must be preserved under a
change of background can be seen by either making an ADM-like analysis
of the worldsheet gravity theory or else by considering
deformations
T \to T + \delta T
\bar T \to \bar T + \delta \bar T
and noting that consistency requires
\delta T = \delta \bar T .
> L_{2006} or anything else. What's exactly your reason to think that there
> is anything interesting about a group defined in this way, as opposed to
It is precisely the group which gives you isomorphisms of the
left+right Virasoro algebra. It is possible to identify the group elements
which, for instance, correspond to background T-duality, background
diffeo transformations, background gauge transformations. (see below)
> > True, but that's precisely what a symmetry/duality is supposed to do.
>
> No, it's not. A duality (and similarly for a self-duality, which is
> essentially the same thing as a symmetry) is a map redefining the
> variables of a theory, a redefinition that transforms the known
> observables in a first known theory AB exactly into some known observables
> of another known theory CD (which may be the same thing as AB).
Sure. And the conjugation which I mentioned in my last post does
exactly that for T-duality.
> On the other hand, your generic conjugation is a meaningless redefinition
> of fields that preserves neither the Virasoro algebra
I don't understand why you are now saying that it does not preserve the
Virasoro algebra. In your last post you said that it is known to
every undergraduate that conjugation preserves the algebra.
> other operators. It is an uninteresting change of the conventions
> describing the same theory, not a map between two different theories or a
> symmetry of a single theory. You just can't use the word "symmetry" or
> "duality" for such contentless constructions.
Lubos, please have a look at
F. Lizzi, R. Szabo,
Duality symmetries and noncommutative geometry of string spacetimes,
hep-th/9707202
where it is shown in great detail that and how the duality symmetries
of the string are related to the conjugations in question.
> > > OK, but it is a random guessing, is not it?
> >
> > Not fully random, but, yes, it is guessing. :-)
>
> OK, so if it is not fully random :-), do you have some explanation why you
> said deRham model and not something else, for example the topological
> B-model on the E_{10} group manifold? ;-)
Yes, I have a reason for this guess:
I once checked that the supersymmetry generators of N=1 D=3+1 SUGRA
(or rather appropriate linear combinations of them) are
the exterior derivative d and its adjoint del on the configuration space
of the theory (really the Dolbeault operators \partial and \bar \partial
so that d = \partial + \bar \partial). The susy algebra of the constraints
of the theory are hence equivalent to the algebra which is schematically
of the form
{d,del} = H + ...
where H is the Hamiltonian constraint of the theory (and in particular
something like the Laplace-Beltrami operator on config space)
and the ellipsis indicates sa sum of patial diffeomorphism and
Lorentz rotation constraints.
It is to be expected that a similar statement applies to the
constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have
shown that the bosonic part of configuration space (which, to emphasize it
again, is the space on which the above operators d and del act) is
(a subspace of) the group manifold of E10/K(E10) and that the
(bosonic part of) the Hamiltonian constraint H is the Laplace
operator on this manifold. Hence by the above it is a reasonable
guess that the supersymmetric extension of this Hamiltonian
constraint is the Laplace-Beltrami operator {d,del} on the
exterior bundle over E10/K(E10) (or rather of finite truncations
of this object, since we don't know how to handle the full thing.)
Urs Schreiber
> On Sun, 18 Apr 2004, Urs Schreiber wrote:
>
> > This was supposed to be evidence that at least in the discussed limit
> > the E_10 model is indeed equivalent to the BFSS model.
>
> Does this proof explain why the BFSS model is the discrete light-cone
> quantization of the covariant E_{10} model, and why "N" of BFSS is the
> light-like momentum? (If it does not, it must be wrong.)
Thanks for asking this question! (And thanks, in fact, for all of this
rapid and very critical discussion.)
I have thought about precisely this type of question a lot in the last
couple of days. The point is: Can we, apart from noting that we have
ensembles of the form exp(-Tr(M^2)) on both sides in the given limit,
can we identify the physical meaning of the matrices on both sides
of the conjectured equivalence?
In order to answer the question we would of course first of all have
to figure out what the "meaning" of the matrices on the E_10 side
of the conjectured equivalence actually is.
Without any further work, all that we know is that the semiclassical
limit of 11d sugra close to a spacelike sinmgularity is, due to
chaoticity, equal to that of a theory of an ensemble of randomly
chosen NxN matrices for large N.
But what is the physical interpretation of N in this context?
In fact this is a very old question. It has been known for a long
time empirically (i.e. using numerics) that Random Matrix Theory (RMT)
does universally describe the semiclassical limit of all chaotic
quantum systems (i.e. the predictions obtained from the random
ensemble of matrices, for instance concerning the statistic of
the spectrum of the system's Hamiltonian, precisely coincide with
the predictions obtained from the origina theory).
But I am being told by specialists working on quantum chaos that
the reason for this "unreasonable effectiveness" of RMT in
quantum chaos has so far remained a mystery.
In fact, there is quite some excitement at my institution about
the recent results of one of our groups (S. Mueller, S. Heusler,
P. Braun, F. Haake et al.) who have solved the long-standing
problem of actually explicitly calculating the universal
semiclassical spectrum of general chaotic quantum systems and
checking equivalence with the resuts of RMT. So this finally
improves the comparison with numerical results to a formal
calculation. The prediction of RMT are now proven to be exactly
those found from a full semiclassical analysis of the true
systems.
So this is reassuring, because it shows that the RMT-conjecture
("every chaotic quantum system is described by RMT") is correct.
But unfortunately, at least as far as I can see, this still does
not tell us WHY the conjecture is correct, i.e. why this
ensemble of matrices describes single chaotic systems. It still
does not give us a physical interpretation of the large matrices
and of the parameter N, which is what we would need to answer
your question above.
For precisely this reason I have tried to figure out the answer
myself, recently. You can find my proposed solution as well as
some background information and discussion at the Coffee Table:
http://golem.ph.utexas.edu/string/archives/000342.html#c000927
I don't know if this works out as expected, but I think it looks
promising.
So my proposed solution is that we have to interpret the
ensemble of systems used in RMT as associated with the ensemble
of points in a "non-universal cell" of the config space
of the chaotic system.
More precisely, the classical chaoticity of the system
implies that in the semiclassical limit (which is dominated
by classical contributions to the path integral) we can
introduce a coarse-graining of the confuguration space
into cells, such that within each cell matrix elements of
the Hamiltonian are correlated, while outside they are not.
This reflects precisely the fact that on very small scales
the system will have non-universal behaviour while on
sufficiently large scales it will exhibt the universality of
ergodicity and chaos.
Please refer to the above links for some more details of
this, admittedly simple but apparently original, idea.
Let's assume this interpretation is approximately correct and
try to understand what it implies for the interpretation of the
matrices which would describe the rmt of 11d sugra close to
a spacelike singularity.
Now config space is the mini-superspace, and in particular the
Weyl-chamber of E10, where every point corresponds to a
specific set of values of the scale factors of the
sugra universe. The matrices would now describe transition
amplitudes between different "cells" of this config space
and N would be the total number of such cells.
Hence N would correspond roughly to a discretzation of the
scale factors, I'd think.
Ths does not look like it could have any relation to the
interpretation of N in the BFSS model, does it?
So maybe my conjecture is wrong. But maybe we should bea little more
careful before making this conclusion, because of various
reinterpretations which are possible.
For instance in order for my conjecture to make sense we would
really be looking at the canonical ensemble of BFSS theory
and hence at BFSS at _finite temperature_.
But we know that BFSS at finite temperature is nothing but
the IKKT model!
So it seems that we would rather have to identify the matrices
with those of the IKKT model.
There the above interpretation might make much more sense, since
it is known that we can interpret in the IKKT model the integer
N as the number of points in a discretization of spacetime.
(i assume that that's what you were referring to when mentioning
the relation of IKKT to "quantum foam".) So this would match
nicely with the interpretation of N as characterizing discrete
scale factors of the universe.
Of course I am aware that all this is rather vague and very
speculative. But I think it is a fun speculation and not
uninteresting. Maybe it is wrong. Maybe not. In fact, if there
is any truth to the E10 model then something along the lines
sketched above must be true.
Lubos Motl
> T - \bar T (in modes L_m - \bar L_{-m}) is always the generator of
> spatial reparameterizations of the string at appropriately fixed
> worldsheet time.
Right, that's the very correct answer. So anything that you will obtain
from such a special treatment of the "fixed" worldsheet time will be
non-invariant under the general conformal transformations of the 2D
worldsheet which can act nontrivially on the worldsheet time. These
general conformal symmetries are necessary for decoupling of ghosts, and
for calculations of the loop diagram, among many other things. If you take
the comment about the loop diagrams seriously, your group of spatial
reparameterizations does not even exist for loop diagrams. Such a group
only looks meaningful if you stare at the cylindrical worldsheet of a
single string, but it has no invariant meaning if you consider the
interactions. The whole conformal symmetry (and the Virasoro algebra) must
be treated as one group if you want to use its power - and believe me that
you need its full power to calculate the scattering amplitudes, especially
at the loop level.
> It is precisely the group which gives you isomorphisms of the
> left+right Virasoro algebra. It is possible to identify the group elements
> which, for instance, correspond to background T-duality, background
> diffeo transformations, background gauge transformations. (see below)
These transformations (T-duality etc.; the spacetime diffeomorphisms are
not the best example unless they are isometries, because the other
transformations *do* change the background i.e. they *do* change the form
of the worldsheet action and consequently of the Virasoro generators)
preserve the full left as well as the full right Virasoro group, and
therefore it is not surprising that they also preserve its subgroup or
subalgebra of the "diagonal spatial" Virasoro generators.
What I've been trying to say for some time is that the transformations
that preserve the whole Virasoro algebra are interesting (they are
symmetries of string theory, such as T-duality), while the remaining
generic transformations that only preserve a part of it (such as the
spatial diffeomorphisms) are not interesting (and are not physical) in
string theory, and they are not symmetries, much like any other
transformation or field redefinition that preserves nothing.
There is a huge difference between the words "field redefinition" and a
"symmetry" (or "duality"). Symmetries and dualities are much more special
and rare concepts!
> > No, it's not. A duality (and similarly for a self-duality, which is
> > essentially the same thing as a symmetry) is a map redefining the
> > variables of a theory, a redefinition that transforms the known
> > observables in a first known theory AB exactly into some known observables
> > of another known theory CD (which may be the same thing as AB).
>
> Sure. And the conjugation which I mentioned in my last post does
> exactly that for T-duality.
I don't quite understand. Is the conjugation that you talk about now the
conjugation by a generic exp(W)? T-duality is not a generic conjugation,
but a very specific transformation that preserves - unlike the conjugation
by a generic exp(W) - every single generator of the Virasoro algebra, and
as far as I know, it was not you who discovered T-duality. ;-)
> > On the other hand, your generic conjugation is a meaningless redefinition
> > of fields that preserves neither the Virasoro algebra
>
> I don't understand why you are now saying that it does not preserve the
> Virasoro algebra. In your last post you said that it is known to
> every undergraduate that conjugation preserves the algebra.
I guess that you only want to entertain us right now. What I wrote is that
every kid knows that every conjugation preserves the form of the
*commutation relations* between various generators, i.e. the form of the
identities of the form [A,B]=C which is transformed to [A',B']=C'. But
that's not enough for an operation to be called a symmetry. For a
transformation to be called "symmetry", the explicit form of every
Virasoro generator, in terms of the fundamental field, must be unchanged,
i.e. A=A',B=B',C=C' in the example above, or for example L_{m} must be
still equal to
\frac{1}{2}\sum_{n} \alpha_{m+n}^\mu \alpha_{-n}_\mu.
While the generic conjugation by \exp(W) preserves the form of the
*commutation relations*, it certainly does not preserve the generators
themselves (a generic transformation preserves neither L_m nor
\alpha^\mu_m) - even though you seem to be saying exactly this (!), and
therefore you cannot call it a "symmetry", and therefore it is also much
less interesting than the real symmetries such as T-duality.
> F. Lizzi, R. Szabo,
> Duality symmetries and noncommutative geometry of string spacetimes,
> hep-th/9707202
This paper offers a slightly non-standard approach, especially because of
their usage of the Dirac operators, but nothing against it. I am not
arguing against the fact that T-duality is important, interesting, and it
is a very special example of a transformation i.e. conjugation. ;-) On the
contrary, I was opposing your statement that you can extend the well-known
T-duality to a much larger group, and I claim that there exists nothing
beyond T-dualities (plus spacetime gauge symmetries and isometries) that
is a symmetry of perturbative string theory. T-duality is very special.
Some generic transformations that only preserve a subgroup of the Virasoro
algebra are physically uninteresting.
> Yes, I have a reason for this guess:
>
> I once checked that the supersymmetry generators of N=1 D=3+1 SUGRA
> (or rather appropriate linear combinations of them) are
> the exterior derivative d and its adjoint del on the configuration space
> of the theory (really the Dolbeault operators \partial and \bar \partial
> so that d = \partial + \bar \partial). The susy algebra of the constraints
> of the theory are hence equivalent to the algebra which is schematically
> of the form
>
> {d,del} = H + ...
>
> where H is the Hamiltonian constraint of the theory (and in particular
> something like the Laplace-Beltrami operator on config space)
> and the ellipsis indicates sa sum of patial diffeomorphism and
> Lorentz rotation constraints.
Lorentz rotation constraints? The RHS of the SUSY algebra in the flat
space should contain the momenta, but no Lorentz generators or other
diffeomorphisms. I also don't understand what you did with the spinor
indices of the supercharges. Your schematic formulae look very
Lorentz-non-invariant, also because of the special treatment of the
Hamiltonian, and it just seems difficult at this moment to imagine a
variation of these arguments that would be believable. Sorry, I am sure
that you will convince me later. ;-)
> It is to be expected that a similar statement applies to the
> constraints of 11d sugra. For 11d sugra Damour,Henneux&Nicolai have
> shown that the bosonic part of configuration space (which, to emphasize it
> again, is the space on which the above operators d and del act) is
> (a subspace of) the group manifold of E10/K(E10) and that the
> (bosonic part of) the Hamiltonian constraint H is the Laplace
> operator on this manifold. Hence by the above it is a reasonable
> guess that the supersymmetric extension of this Hamiltonian
> constraint is the Laplace-Beltrami operator {d,del} on the
> exterior bundle over E10/K(E10) (or rather of finite truncations
> of this object, since we don't know how to handle the full thing.)
OK, I think that some work is needed before others will see what this
proposal really means and why it is a reasonable guess. Good luck, LM
Lubos Motl
> > Does this proof explain why the BFSS model is the discrete light-cone
> > quantization of the covariant E_{10} model, and why "N" of BFSS is the
> > light-like momentum? (If it does not, it must be wrong.)
> ...
> In order to answer the question we would of course first of all have
> to figure out what the "meaning" of the matrices on the E_10 side
> of the conjectured equivalence actually is.
That's right. As you see, the problems in showing the equivalence seems to
be on the E_{10} side, because at least at this moment it is the weaker
and much less convincing side of this conjectured relation. The meaning of
"N" in the BFSS model is quite clear, and the Seiberg/Sen style of the
proof of the DLCQ conjecture makes it even more transparent. On the other
E_{10} side, we don't expect any special value of "N" to parameterize the
different models.
> But I am being told by specialists working on quantum chaos that
> the reason for this "unreasonable effectiveness" of RMT in
> quantum chaos has so far remained a mystery.
>
> In fact, there is quite some excitement at my institution about
> the recent results of one of our groups (S. Mueller, S. Heusler,
> P. Braun, F. Haake et al.) who have solved the long-standing
> problem of actually explicitly calculating the universal...
Interesting story. The main thing that I am missing is why you think that
this quantum chaos is directly related to M-theory, or why could it even
describe all of M-theory.
> Hence N would correspond roughly to a discretzation of the
> scale factors, I'd think.
Would this mean that only the large N limit should be given a physical
meaning? This is OK on the covariant side, but not enough on the DLCQ
side.
> Ths does not look like it could have any relation to the
> interpretation of N in the BFSS model, does it?
It certainly does not prove the relevance of the finite N model, and the
finite N models are the only ones we can really construct and the only
ones whose relation to M-theory we can prove by a universal formal
argument.
> For instance in order for my conjecture to make sense we would
> really be looking at the canonical ensemble of BFSS theory
> and hence at BFSS at _finite temperature_.
> But we know that BFSS at finite temperature is nothing but
> the IKKT model!
So far, I don't know that. BFSS at finite temperature is a SUSY-breaking
theory in 0+1 Euclidean dimensions, which tends to collapse all D0-branes
to one point, while the IKKT model is - at least formally - a
supersymmetric model in 0+0 dimensions with exact flat directions, giving
the spacetime interpretation. Could you list more details how this
hypothetical hotBFSS/IKKT map works?
> There the above interpretation might make much more sense, since
> it is known that we can interpret in the IKKT model the integer
> N as the number of points in a discretization of spacetime.
> (i assume that that's what you were referring to when mentioning
> the relation of IKKT to "quantum foam".)
Yes, something like that. Cumrun has one more sharp and new idea about it,
but I think that I can't publish it.
Good luck,
Lubos
Urs Schreiber
> > Sure. And the conjugation which I mentioned in my last post does
> > exactly that for T-duality.
>
> I don't quite understand. Is the conjugation that you talk about now the
> conjugation by a generic exp(W)?
No, it is the one that I have explicitly written down a couple of post
before, including the refernce where it is discussed in detail.
> For a
> transformation to be called "symmetry", the explicit form of every
> Virasoro generator, in terms of the fundamental field, must be unchanged,
> i.e. A=A',B=B',C=C' in the example above, or for example L_{m} must be
A symmetry may well change the form of the generators. Consider
for instance a B-field gauge transformation B -> B + dA . Since B
appears explicitly in the expression for the generators, this does
change their form, but not the physics.
Let's see if I find a reference... here for instance, pp. 7 of
hep-th/0209241.
I am not quite sure why we are arguing about this fact. Under
T-duality for instance both the metric and the B-field transform
and mix in general to become new, T-dual metrics and B-forms
which hence give T-dual expressions for the generators.
> > F. Lizzi, R. Szabo,
> > Duality symmetries and noncommutative geometry of string spacetimes,
> > hep-th/9707202
>
> This paper offers a slightly non-standard approach, especially because of
> their usage of the Dirac operators, but nothing against it.
Good. We shouldn't spend our time apparently disagreeing when actually
we agree.
> > where H is the Hamiltonian constraint of the theory (and in particular
> > something like the Laplace-Beltrami operator on config space)
> > and the ellipsis indicates sa sum of patial diffeomorphism and
> > Lorentz rotation constraints.
>
> Lorentz rotation constraints? The RHS of the SUSY algebra in the flat
> space should contain the momenta, but no Lorentz generators or other
> diffeomorphisms.
As I said, I am talking about the constraints. Take sugra and compute the
ADM constraints. They contain the Hamiltonian constraint, the spatial
diffeo constraints, the supersymmetry constraints and local Lorentz
constraints which ensure that no physical quantity depends on the
arbitrary choice of vielbein. You find the details reviewed in the book
P. D'Eath, Supersymmetric Quantum Cosmology, Cambridge (1996)
> I also don't understand what you did with the spinor
> indices of the supercharges.
Recall that I said the susy constraints of 3+1 sugra can be
linearly recombinated to yield the Dolbeault operators on config
space. This are four operator corresponding to the four
Q^\alpha and Q^{\dot \alpha} where the spinor indices
\alpha,\dot \alph \in {1,2} .
> Your schematic formulae look very
> Lorentz-non-invariant,
You need to pick some Lorentz frame to write down things. Writing
Q^\alpha also implies that you specify with respect to which frame
this component is supposed to be considered.
> also because of the special treatment of the
> Hamiltonian, and it just seems difficult at this moment to imagine a
> variation of these arguments that would be believable.
I hope you realize that in a canonical analysis of sugra the
Hamiltonian constraint is always special and that that's
the very reason why it features to prominently in the E10
sigma-model proposal. The Laplace operator appearing there
is precisely the Hamiltonian constraint.
> Sorry, I am sure that you will convince me later. ;-)
Of what? That the ADM constraints of sugra contain, among other
ingredients,a Hamiltonian
constraint and Lorentz constraints? If you don't believe me
in this point I won't try harder than pointing to some standard
literature.
If on the other hand you want to be convinced of the fact
that linear combinations of the susy constraints Q^alpha
and Q^{\dot \alpha} of sugra can be found which are nothing
but Dolbeault operators on config space, then I'd be happy
to do so. :-)
> OK, I think that some work is needed before others will see what this
> proposal really means and why it is a reasonable guess. Good luck, LM
A simpler example than 3+1 d sugra might convey the idea more easily:
Consider 1+1 d sugra, i.e. the superstring. Concentrate on
the bosonic version, first. Note that the Virasoro constraints
are equivalent to a Hamiltonian constraint T + \bar T and a
spatial reparameterization constraint T - \bar T and that
the Hamiltonian constraint is essentially the Laplace operator
on config space (loop space in this case) plus an extra term,
while the reparametrition constraint is essentially a Lie derivative
on loop space. Now lift this bosonic setup to the deRahm theory
on loop space by introducing exterior derivative d and its
adjoint del on loop space. This can be seen to be nothing but
the constraints of the superstring, where the supercurrents
are linear combinations of d and del:
T_F \propto i(d-del)
\bar T_F \propto d + del .
This has been realized long ago by Edward Witten as you can
see by looking at the second part of
E. Witten,
Supersymmetry and Morse Theory,
J. diff. Geom. 17 (1982) 661
and the second part of
E. Witten,
Global anomalies in string theory,
in
Symposium on anomalies, geometry and topology,
eds W. Bardeen and A. White
World Scientific
(1985)
pp. 61-99
Please, if you don't believe what I am telling you, look at these
papers.
What you cannot find in these papers but what I claim is true is that
the general mechanism concerning susy,->deRahm etc. that you can
see very explicitly in the case of 1+1d sugra carries over to
higher dimensional sugras as well.
You can also take the superstring as a nice example for how my
guess concerning the E10 model is at least not totally crazy
(although it is still a guess, otherwise I would write about it
to the archive instead of to sps :-):
Consder 1+1d sugra and in particular the RR sector. What would be
a cosmological model in this case? Right, it would be a restriction
to homogenous spatial modes, i.e. to center of mass modes of the string.
You see that the "minisuperspace" theory of the RR sector of sugra
is just that of the massless RR form quanta and is indeed governed
by the deRahm constraints
d|psi> = 0
del|psi> = 0
which are nothing but the susy constraints plus the
Hamiltonian constraint
H|psi> = {d,del}|psi> = 0
which of course follows automatically.
Now we csan easily do the analog of the E10 conjecture of
11d sugra for 1+1d sugra: We enlarge our "minisuperspace"
which just contained the homogeneous modes to contain all
further modes and we conjeture that essentially the Laplace
operator on this large space (loop space) should replace the
simple Laplace-Beltrami constraint in the center-of-mass
restriction. And indeed, as you can read in the above given
papers by Witten, this is how it works and it gives you the
full superstring.
My guess is simply that these construction generalize
to higher dimensions. You see (hopefully) that it is not a very
far fetched guess.
Urs Schreiber
> On Sun, 18 Apr 2004, Urs Schreiber wrote:
> > In order to answer the question we would of course first of all have
> > to figure out what the "meaning" of the matrices on the E_10 side
> > of the conjectured equivalence actually is.
>
> That's right. As you see, the problems in showing the equivalence seems to
> be on the E_{10} side, because at least at this moment it is the weaker
> and much less convincing side of this conjectured relation.
Yes. That's why we are interested in understanding it better. :-)
> Interesting story. The main thing that I am missing is why you think that
> this quantum chaos is directly related to M-theory, or why could it even
> describe all of M-theory.
Well, that's precisely the interesting result by Damour, Henneaux and
Nicolai. They showed that 11d sugra close to a spacelike singularity is
necessarily a chaotic system.
More precisely they show that the scale factors of 11d sugra close to a
spacelike singularity parameterize a chaotic geodesic motion
inside the Weyl chamber of E10.
Now, fully chaotic systems enjoy "universality". Many of their properties
are independent of the details of the system, like for instance the
co-correlation of energy eigenvalues. This universal behaviour is
described by random matrix theory.
> It certainly does not prove the relevance of the finite N model, and the
> finite N models are the only ones we can really construct and the only
> ones whose relation to M-theory we can prove by a universal formal
> argument.
Ok, good point.
> > But we know that BFSS at finite temperature is nothing but
> > the IKKT model!
>
> So far, I don't know that. BFSS at finite temperature is a SUSY-breaking
> theory in 0+1 Euclidean dimensions, which tends to collapse all D0-branes
> to one point, while the IKKT model is - at least formally - a
> supersymmetric model in 0+0 dimensions with exact flat directions, giving
> the spacetime interpretation. Could you list more details how this
> hypothetical hotBFSS/IKKT map works?
I have taken this observation from the paper
A. Connes, M. Douglas, A. Schwarz,
Noncommutative Geometry and Matrix Theory: Compactification on Tori
hep-th/9711162 .
See below equation (3.3).
(This paper was pointed out to me by Squark a while ago while we
were discussing matrix models on sci.physics.research.)
Lubos Motl
> Yes. That's why we are interested in understanding it better. :-)
the problem with this sentence is that it assumes that the relation
exists. I think that it is still much more likely that such a conjectured
duality is invalid. ;-)
> Well, that's precisely the interesting result by Damour, Henneaux and
> Nicolai. They showed that 11d sugra close to a spacelike singularity is
> necessarily a chaotic system.
You probably know that I've been thinking about these things many times,
and you've probably read our 1998 paper with Tom Banks and Willy Fischler
http://arxiv.org/abs/hep-th/9811194
cited by Nicolai et al. with the label [3] where we proved that infinitely
many copies of the fundamental M,IIA,IIB domains (and its U-dual copies)
completely and exactly cover the future light cone of the E_{10} Cartan
subalgebra - a portion of the moduli space of C_{MNP}=0 rectangular
toroidal compactifications of M-theory on T^{10}. So it's perhaps not
terribly unreasonable to guess that I know something about the Weyl
chamber of E_{10} because it was us who started this investigation. ;-)
Well, you can't treat these conclusions too seriously. It is a game.
Supergravity really breaks down near these singularities, and the
dualities must be applied very carefully because the system is highly
time-dependent. The moduli spaces don't exist for d<=2, and the analysis
of the moduli space is purely formal at this point.
> Now, fully chaotic systems enjoy "universality". Many of their properties
> are independent of the details of the system, like for instance the
> co-correlation of energy eigenvalues. This universal behaviour is
> described by random matrix theory.
This still seems as a one-way degeneration. If RMT describes a chaotic
limit of a system, such as the completely compactified SUGRA, it does not
matter that you can obtain the full non-chaotic and predictable M-theory
from RMT.
> > hypothetical hotBFSS/IKKT map works?
>
> I have taken this observation from the paper
> A. Connes, M. Douglas, A. Schwarz,
> hep-th/9711162 .
I think that this is a completely formal map, which uses the "compactified
IKKT" model, moreover with the antiperiodic boundary conditions for the
fermions. This is just a different way to say that you put BFSS on a
finite temperature circle, and you just forget about all the important
differences between BFSS and IKKT. The IKKT model itself does not have any
compact directions, and the compactification was just a trick. The paper
above was valuable as a textbook of Matrix theory for mathematicians, and
as an introduction of the concept of noncommutative geometry to this
subfield.
But the results of the paper hep-th/9711162 cited by you are obsolete;
they were still written assuming the old-fashioned misconception that the
different toroidal compactifications of Matrix theory should be treated as
the same model (which is not correct) and a more viable treatment of the
toroidal compactifications of Matrix theory was described by Sen, Seiberg,
and perhaps others.
http://arxiv.org/abs/hep-th/9709220
http://arxiv.org/abs/hep-th/9710009
Best wishes
Lubos Motl
> No, it is the one that I have explicitly written down a couple of post
> before, including the refernce where it is discussed in detail.
OK, so in that case I missed the whole point of your discussion about the
large symmetries. Can we summarize it by saying that you wanted to
emphasize that "T-duality exists"? ;-)
> A symmetry may well change the form of the generators. Consider
> for instance a B-field gauge transformation B -> B + dA . Since B
> appears explicitly in the expression for the generators, this does
> change their form, but not the physics.
You could not be more wrong. Physics of a CFT *is* the stress energy
tensor, and therefore you must preserve it by an operation that should be
called a "symmetry". If you tried to write down the actual worldsheet
stress energy tensor, you would see immediately that you will never be
able to write *any* term involving the B-field. The B-field does not
contribute anything to the stress-energy tensor. Note that the stress
energy tensor (assuming a constant dilaton) is proportional to
T_{zz} = \partial_z X^\mu \partial_z X^\nu g_{\mu\nu} X(z,\bar z)
and similarly for T_{\bar z\bar z}. The trace of the stress-energy tensor
i.e. the component T_{z\bar z} vanishes identically for CFTs. Note that if
you tried to add the "antisymmetric" part of the metric g_{\mu\nu}, namely
the B-field, you would obtain zero *identically* because
\partial_z X^\mu \partial_z X^\nu is \mu\nu symmetric and disappears
when contracted with an antisymmetric B_{\mu\nu}.
Alternatively, you can derive the stress energy tensor as the derivative
of the worldsheet Lagrangian with respect to the worldsheet metric. The
worldsheet metric does not enter the B-field term at all, because this
term is the integral of a 2-form (the pullback of the B-field)
\partial_\alpha X^\mu \partial_\beta X^\nu B_{\mu\nu}(X(z,bar z))
and therefore does not need any metric to be properly integrated over the
two-dimensional worldsheet. If you studied Polchinski's book, you would
also learn somewhere in Chapter 8 that the B-field only changes the
allowed lattices for the integrals of \partial X, and the relation between
\partial X and the canonical momenta, but the form of the stress energy
tensor in terms of \partial X is unchanged by *any* B-field, and it is
even more unchanged by a pure gauge B-field B=dA.
Therefore, the Virasoro generators, which are Fourier modes of the stress
energy tensor, are invariant under the transformation you mentioned, and
it must be so, otherwise it would not be a symmetry. Feel free to invent
another wrong example trying to support your misled statement, but I think
that it would be more reasonable to accept that the symmetries of a CFT
are either the conformal symmetries themselves, OR (times) some operations
that completely commute with the conformal group and therefore preserve
the form of every individual Virasoro generator. It's an example of the
Coleman-Mandula theorem that implies that the symmetries must factorize in
this way - and the worldsheet supersymmetry is again the only loophole
leading to a symmetry (generated by the superconformal generators) that is
NOT just the conformal symmetry, yet it does not commute with the
conformal symmetry.
> Let's see if I find a reference... here for instance, pp. 7 of
> hep-th/0209241.
You must know very well that no statement like "the B -> B+dA
transformation changes the Virasoro generators" is written in this paper -
such a statement would be pretty embarassing for Volker because he is not
a student anymore. ;-)
> I am not quite sure why we are arguing about this fact.
The reason why we are arguing about it is that you have written at least 5
posts that essentially claim that the symmetries don't have to preserve
anything, certainly not the Virasoro algebra ;-), and because I believe
that we could avoid many unproductive arguments and tens of incorrect
conclusions (like the conclusion about T_{zz} transforming under the
B-field gauge transformations) just by agreeing what the word "symmetry"
means, I am trying to convince you that the word "symmetry" means what it
means and moreover I hope that at the end, you will appreciate my effort.
> Under T-duality for instance both the metric and the B-field transform
> and mix in general to become new, T-dual metrics and B-forms which
> hence give T-dual expressions for the generators.
That's right. The radii and angles of the torus (and the components of the
B-field) transform under T-duality, but the Virasoro generators don't!
> > Lorentz rotation constraints? The RHS of the SUSY algebra in the flat
> > space should contain the momenta, but no Lorentz generators or other
> > diffeomorphisms.
>
> As I said, I am talking about the constraints. Take sugra and compute the
> ADM constraints. They contain the Hamiltonian constraint, the spatial...
Oh I see, you meant the local Lorentz symmetry acting on the vielbeins.
> Recall that I said the susy constraints of 3+1 sugra can be
> linearly recombinated to yield the Dolbeault operators on config
> space. This are four operator corresponding to the four
> Q^\alpha and Q^{\dot \alpha} where the spinor indices
> \alpha,\dot \alph \in {1,2} .
OK, so you suppressed this degeneracy previously.
> If on the other hand you want to be convinced of the fact
> that linear combinations of the susy constraints Q^alpha
> and Q^{\dot \alpha} of sugra can be found which are nothing
> but Dolbeault operators on config space, then I'd be happy
> to do so. :-)
The supercharges also contain terms with spacetime derivatives. By a
configuration space, do you mean the infinite-dimensional space of all
configurations in spacetime, which seems necessary, or the space at a
single point?
> Consider 1+1 d sugra, i.e. the superstring. Concentrate on
> the bosonic version, first. Note that the Virasoro constraints
> are equivalent to a Hamiltonian constraint T + \bar T and a
> spatial reparameterization constraint T - \bar T and that...
OK, this was very clear.
> What you cannot find in these papers but what I claim is true is that
> the general mechanism concerning susy,->deRahm etc.
Indeed, your repeated misspelling of deRham leads me to say that his name
is less German name than you think. ;-)
> that you can see very explicitly in the case of 1+1d sugra carries
> over to higher dimensional sugras as well.
Statement understood.
>... d|psi> = 0
> del|psi> = 0
>
> which are nothing but the susy constraints plus the
> Hamiltonian constraint
>
> H|psi> = {d,del}|psi> = 0
Clear, but I don't see any E_{10} in these simple formulae.
All the best
Lubosch
Urs Schreiber
> On Sun, 18 Apr 2004, Urs Schreiber wrote:
> able to write *any* term involving the B-field. The B-field does not
> contribute anything to the stress-energy tensor.
It does if you write it in term of the original free fields. Of course
if you rewrite everything in terms of the new interacting fields you
will see the B-field in the OPEs/commutators instead.
So, under e^(-W) T e^W the object T first changes its form. Then
you are free to redefine the notation such that T has the original
form again. But then the commutators of the objects \partial X
and \bar \partial X are different then they were before.
I don't think we really disagree about any technical aspect here.
> > I am not quite sure why we are arguing about this fact.
>
> The reason why we are arguing about it is that you have written at least 5
> posts that essentially claim that the symmetries don't have to preserve
> anything, certainly not the Virasoro algebra ;-), and because I believe
The Virasoro algebra is preserved under similarity transformations.
The form of the Virasoro generators in terms of the original fields
is not necessarily, though. As far as I remember you agreed with
what Lizzi and Szabo say so it is perhaps more productive if I
refer to statements made by them, since I am sensing a problem in
our communication here. So concerning the symmetry group of
string theory induced by conjugation automorphisms of the
algebra pleas see hep-th/9707202, for instance, where this
is discussed in detail. Note in particular section 7.
> The supercharges also contain terms with spacetime derivatives. By a
> configuration space, do you mean the infinite-dimensional space of all
> configurations in spacetime, which seems necessary,
Yes.
> > H|psi> = {d,del}|psi> = 0
>
> Clear, but I don't see any E_{10} in these simple formulae.
Good, because there wasn't any! :-)
See, my point was that in these examples the susy generators are
generalized exterior derivatives on configuration space. In the
context of the conjecture that the configuration space of M-theory
is that generated by E10/K(E10) it seems natural to guess that
the susy generators could again be simply exterior derivatives
on this space.
Lubos Motl
> It does if you write it in term of the original free fields. Of course
> if you rewrite everything in terms of the new interacting fields you
> will see the B-field in the OPEs/commutators instead.
No, you won't because X and \partial X and \bar\partial X (in the
worldsheet bulk) don't transform under the two-form gauge invariance at
all, and therefore it does not matter whether you use the "old" fields or
the "new" fields. If you think that they do - that there is any reasonable
direct action of the *spacetime* B-field gauge invariance on these
*worldsheet* fields - could you please write down which transformation
rule you mean? This is already your at least sixth post where you claim
that the symmetries don't have to preserve the dynamics, and you keep on
deriving increasingly incorrect conclusions from your incorrect
assumptions.
> So, under e^(-W) T e^W the object T first changes its form.
If it changes the form, then exp(W) is not a symmetry.
> Then you are free to redefine the notation such that T has the
> original form again. But then the commutators of the objects \partial
> X and \bar \partial X are different then they were before.
Having or transforming a B-field changes the relations between the
velocities and the momenta in the canonical formalism, but it does not
change the worldsheet stress energy tensor.
> I don't think we really disagree about any technical aspect here.
Once you write that what you were writing in the previous five posts was
incorrect, then we won't disagree about any technical aspect.
> > The reason why we are arguing about it is that you have written at least 5
> > posts that essentially claim that the symmetries don't have to preserve
> > anything, certainly not the Virasoro algebra ;-), and because I believe
>
> The Virasoro algebra is preserved under similarity transformations.
> The form of the Virasoro generators in terms of the original fields
> is not necessarily, though.
If your action (or in the CFT case, the stress energy tensor) is not
preserved by the operation XY, then XY is NOT a symmetry unless XY is a
conformal transformation itself. This is a simple fact but your opposition
has expanded it into an elephant.
> As far as I remember you agreed with
> what Lizzi and Szabo say
Frankly speaking I don't know who they are, but as far as I remember, no
article you were showing me contained the statement that the operators
defining the dynamics don't have to be preserved under the symmetries.
> our communication here. So concerning the symmetry group of
> string theory induced by conjugation automorphisms of the
> algebra pleas see hep-th/9707202, for instance, where this
> is discussed in detail. Note in particular section 7.
I saw it there. Do you want me to accept this definition of "symmetry"
because Mr. or Mrs. Lizzi and Szabo wrote it this way? Or perhaps the
reason to accept their unusual definitions of "symmetry" is that 15 of
their 23 citations are self-citations, and the rest is from you and Carlo
Rovelli? ;-) You have the amazing ability to choose the most controversial
and often murky articles and build on them.
> > Clear, but I don't see any E_{10} in these simple formulae.
>
> Good, because there wasn't any! :-)
It would be even more interesting if there was one. ;-)
> See, my point was that in these examples the susy generators are
> generalized exterior derivatives on configuration space.
These are just names, and the fact that the unbarred supercharges Q
satisfy {Q,Q}=0 in N=1 is obvious. What nontrivial property does your
statement imply?
Urs Schreiber
> > Well, that's precisely the interesting result by Damour, Henneaux and
> > Nicolai. They showed that 11d sugra close to a spacelike singularity is
> > necessarily a chaotic system.
>
> You probably know that I've been thinking about these things many times,
> and you've probably read our 1998 paper with Tom Banks and Willy Fischler
>
> http://arxiv.org/abs/hep-th/9811194
In that paper the 3-form is set to 0, that's why there are not
potential walls in moduli space and you only get the Kasner motion
without the bounces and hence no chaos.
> cited by Nicolai et al. with the label [3] where we proved that infinitely
> many copies of the fundamental M,IIA,IIB domains (and its U-dual copies)
> completely and exactly cover the future light cone of the E_{10} Cartan
> subalgebra - a portion of the moduli space of C_{MNP}=0 rectangular
> toroidal compactifications of M-theory on T^{10}. So it's perhaps not
> terribly unreasonable to guess that I know something about the Weyl
> chamber of E_{10} because it was us who started this investigation. ;-)
Actually I do assume that you know a whole lot of stuff! :-)
But you asked me why there is any relation to chaos so I gave
an answer.
Can the considerations of your above paper be extended to the
case where the 3-form is non-zero?
Urs Schreiber
> On Sun, 18 Apr 2004, Urs Schreiber wrote:
>
> > It does if you write it in term of the original free fields. Of course
> > if you rewrite everything in terms of the new interacting fields you
> > will see the B-field in the OPEs/commutators instead.
>
> No, you won't because X and \partial X and \bar\partial X (in the
Let's look at it in the example of the charged massless point particle
in a U(1) gauge field A . For A=0 the Hamiltonian constraint is
simply p^2. Now make a gauge transformation by conjugating with
exp(i L(x)). The momenta of course change as
p_m -> p'_m = p_m + d_m L .
So they do change their form, and yet this is of course a symmetry
In terms of the field dot x the Hamiltonian constraint will always
look like (dot x)^2, but in terms of the canonical data this
reads (p - A)^2 and does depend on A in this sense, even if
A is pure gauge.
For the string it works completely analogously. In particular
the background gauge transformation inducing B-> B + (1/T)dA
is induced by conjugating with exp(i int A_m X'^m).
> worldsheet bulk) don't transform under the two-form gauge invariance at
> all, and therefore it does not matter whether you use the "old" fields or
> the "new" fields. If you think that they do - that there is any reasonable
> direct action of the *spacetime* B-field gauge invariance on these
> *worldsheet* fields - could you please write down which transformation
> rule you mean? This is already your at least sixth post where you claim
> that the symmetries don't have to preserve the dynamics,
The dynamics is preserved, but not necessarily the form of the
constraints. So in the point particle example the tranformation
p^2 -> e^(-i L) p^2 e^(i L) = (p + dL)^2
changes the form of the constraint in this sense, but the
dynamics is of course preserved since the old solutions to
the constraint just have to be transformed by e^(-i L) to
give the new solutions.
Of course you know that. I am just saying this in an attempt
to bridge the communication problem between us.
> If it changes the form, then exp(W) is not a symmetry.
I hope the above point particle example clarifies this.
> > See, my point was that in these examples the susy generators are
> > generalized exterior derivatives on configuration space.
>
> These are just names, and the fact that the unbarred supercharges Q
> satisfy {Q,Q}=0 in N=1 is obvious. What nontrivial property does your
> statement imply?
That by lifting from the Laplace to the Laplace-Beltrami operator
you can straightforwardly get a susy extension of any bosonic
1+0d sigma model.
Thomas Larsson
> On Sun, 18 Apr 2004, Thomas Larsson wrote:
>
> > As Urs Schreiber has pointed out, LQG is not really canonical
> > quantization, which generically would give rise to lowest-energy reps of
> > the symmetry group - this is true even for gauge symmetries such as
> > worldsheet conformal symmetry. Thus, combining the symmetry principles
> > underlying GR and QM *and nothing else* - experimentally confirmed physics
> > - leads to LW reps of the diff group, and non-trivial such reps are
> > anomalous. That's just the way it is.
>
> BTW, do we now agree that this fact is not in contradiction to any
> known fact in string theory? You seemed to claim that the fact that
> gravitational anomalies for chiral fermions occur in
> 4k + 2 dimensions is in contradiction with the above somehow. But
> clearly it is not, since what you write above has nothing to do
> with chiral fermions.
My point was that these extensions are not part of string theory,
because if they were, GSW should have known about them.
They may of course become part of some future theory, which people
might refer to as string theory II. But this new theory might just as
well be called LQG III (since LQG II is already taken) or something
else, depending on who develops it. Either way, diff anomalies in 3+1D
are not part of neither string theory nor LQG as known today. It thus
seems appropriate to move this discussion to some other (and less
hostile) forum.
[Moderator's note: yes, string theory cancels all anomalies in local
symmetries, and the discussions about things that don't exist in string
theory strictly speaking don't belong to this newsgroup. LM
> This is at least not true for all physicist. In
>
> Gebert, Nicolai,
> E_10 for beginners,
> hep-th/9411188
>
> it is emphasized quite strongly that there is exponential growth and
> that things are pretty problematic concerning E_10. But I am being
> told that the point is that one can study consistent finite truncations
> which can be handled and in terms of which we can do physics.
A finite truncation of a graded Lie algebra violates the Jacobi identities,
no? E.g., if we have A, B in g_1 and C in g_-1, [[A,B],C] is in g_1 so it
should not be truncated, but it is because [A,B] is in g_2.
Urs Schreiber
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message news:<Pine.LNX.4.31.04041...@feynman.harvard.edu>...
> > BTW, do we now agree that this fact is not in contradiction to any
> > known fact in string theory? You seemed to claim that the fact that
> > gravitational anomalies for chiral fermions occur in
> > 4k + 2 dimensions is in contradiction with the above somehow. But
> > clearly it is not, since what you write above has nothing to do
> > with chiral fermions.
>
> My point was that these extensions are not part of string theory,
> because if they were, GSW should have known about them.
I think the point is that anomalies due to chiral fermions can be seen
already in the low energy effective action, so that we can say something
about them without knowing the full effective action/closed string field
theory.
> > This is at least not true for all physicist. In
> >
> > Gebert, Nicolai,
> > E_10 for beginners,
> > hep-th/9411188
> >
> > it is emphasized quite strongly that there is exponential growth and
> > that things are pretty problematic concerning E_10. But I am being
> > told that the point is that one can study consistent finite truncations
> > which can be handled and in terms of which we can do physics.
>
> A finite truncation of a graded Lie algebra violates the Jacobi identities,
> no? E.g., if we have A, B in g_1 and C in g_-1, [[A,B],C] is in g_1 so it
> should not be truncated, but it is because [A,B] is in g_2.
Sorry for expressing myself badly. What can be truncated is the expansion
of the sigma-model dynamics on the E10/K(E10) group. This truncation
is consistent in the sense of a perturbative truncation at some given
order is consistent in that higher order terms won't invalidate the
approximate result at lower order.
Of course you are perfectly right that there is no subalgebra of
a graded Lie algebra beyond level 0. But the truncation that I
was referring to is based on the fact that all elements of E10
(or any other KM algebra) at a given level transform under the
action of the level-0 subalgebra. Therefore every level can be
"understood" by decomposing it into irreps of the level-0
subalgebra.
In
Damour, Henneaux, Nicolai
E10 and the 'small tension expansion' of M Theory
hep-th/0207267
a grading of E10 is used which is based on the 'exceptional'
root (instead of on the over-extended root, in their
nomenclature). This way E10 is decomposed into irreps
of its SL(10) subalgebra, level by level. The important
point now is that these irreps are of course nothing
but tensors, and that these tensors can be identified with
spatial modes of tensor fields of the bosonic sector
of 11d sugra (namely the metric and the 3-form field).
So while the configuration point traces out a trajectory
on the E10/K(E10) 'manifold' we can, even lacking a complete
understanding of what this full group really is, understand
the sugra configuration that this trajectory describes
order by order in the above notion of level (which corresponds
to an expansion in terms of spatial gradients on the
sugra side).
That's what I was referring to when I said 'consistent
truncation'. But I still have to understand
many of the details involved.
Lubos Motl
> I think the point is that anomalies due to chiral fermions can be seen
> already in the low energy effective action, so that we can say something
> about them without knowing the full effective action/closed string field
> theory.
But we should also say that at the same moment, we can calculate the
anomalies both in the stringy language as well as in the effective field
theory, and the results agree: the anomalies cancel. In the stringy
language the cancellation is slightly more manifest. String theory is
guaranteed to cancel all anomalies in spacetime as a consequence of
the cancellation of various *worldsheet* anomalies - and the latter are
easier to deal with.
String theory (and I really don't mean "string field theory", but rather
the standard worldsheet S-matrix approach to string theory) automatically
sums the contributions of all "particles" and unifies them into one or
two neat stringy Feynman diagrams (Riemann's surfaces). Thomas Larsson
wrote that Green, Schwarz, and Witten should know something about these
anomalies. You bet that they know everything about it. ;-) Not only them.
It is a rather elementary thing in string theory, and students learn it
from the textbooks, e.g. volume 2 of Polchinski's "String Theory".
All the best
Lubos