Mathematics of string theory
Igor Khavkine
I am about to embark on my first course on string theory. At the moment
I feel similar to they way I felt before my first quantum mechanics course:
no intuition; don't have a big picture; heard things about it, but know nothing
specific. However, this time there isn't a mountain of experimental evidence
to suggest that what I'm learning is correct.
The way I try to approach learning new physics is to first try to get
a basic to reasonable understanding of the underlying mathematics beforehand
or concurrently. Also, rumor is that one of the reasons people keep working
on string theory, despite the fact that it's failing to produce any verifiable
predictions (at currently accessible energy levels), is that it involves a lot
of interesting mathematics.
In view of the above two reasons, I'd like to ask the following: what is the
math underlying string theory? Also, in physics courses, many mathematical
details (sometimes important ones) get swept under the rug, so what should I
watch out for?
Thanks.
Igor
Thomas Larsson
news:<f1ac2e6e.03011...@posting.google.com>...
> In view of the above two reasons, I'd like to ask the following: what is the
> math underlying string theory?
The most important math in old-fashioned perturbative string theory is
infinite-dimensional Lie algebras, in particular Kac-Moody and Virasoro
algebras. Goddard and Olive wrote a nice review in Int J Mod Phys A in 1986,
and I think that an expanded version is available as a book. You can pick up
a lot about finite-dimensional Lie algebras from that paper as well (I did).
This stuff is really cool and applicable in other parts of physics as well,
so learning it is a good investment irrespective of the ultimate fate of
string theory.
Lubos Motl
Hi Igor!
String theory is a theoretical framework that unifies General Relativity
with Quantum Field Theory. String theory is a very conservative approach
to physics; the only known natural extension of the Standard Model that
incorporates gravity. Nevertheless the unification of Quantum Field Theory
with General Relativity *forces us* to extend our imagination and
mathematical skills.
You must therefore learn all the basic (and advanced) stuff from Quantum
Field Theory: quantization, Feynman diagrams, gauge theories,
nonperturbative objects - instantons, solitons etc. You must also
understand all relevant classical physics and related maths: (Special and)
General Relativity, Riemannian geometry, gauge theory, group theory etc.
String theory, by definition, contains all good ideas in physics. The
ideas of Quantum Field Theory are used in many different ways by String
theory. String theory before the mid 1990s has been known as a
*perturbative S-matrix* theory: the only thing you could calculate were
the (asymptotic) Taylor expansions for scattering amplitudes, as a
function of the coupling constant. All these numbers were calculated from
two-dimensional Conformal Field Theory, usually in the path integral
formalism.
Today we know much more (but still much less than everything) and many
different Quantum Field Theories describe various aspects of string
theory: quantum mechanical matrix models describe M-theory in the infinite
momentum frame; Conformal Field Theories (theories where an overall
position-dependent rescaling of metric in unphysical - physics depends on
angles only) in various dimensions are equivalent to subsectors of string
theory on Anti De Sitter spaces (multiplied by various compact manifolds);
this is called AdS-CFT correspondence. String theory incorporates
holography; the property of any quantum theory containing gravity that
roughly implies that the information in a volume V can be encoded on its
surface.
Of course, one must know Quantum Field Theories also as a low-energy
approximation to String Theory. Not only standard gauge theories, Higgs
mechanism etc. is important; also more modern aspects of QFT, such as
supersymmetry, supergravity, anomalies, Renormalization Group, and
Kaluza-Klein theory, are important.
This was a partial list of some topics that are obviously important for
every imaginable theory that should replace the Standard Model and General
Relativity. String theory has however connected physics with many other
domains of mathematics.
Algebraic geometry - geometry of higher-dimensional (complex) manifolds -
especially Calabi-Yau manifolds - is very important to describe the extra
dimensions in string theory. One should know some topology; even K-theory
(for D-branes) and number theory for some special aspects of string
theory. Lie groups (including the representation theory) are omnipresent,
much like in most of theoretical physics.
String theory is full of mathematical miracles that indicate that the
whole structure can hardly be a collection of random coincidences. For
example, already the Standard Model is known to have no anomalies: if you
sum up the cubed charges of all the quarks and leptons, you obtain zero,
for example. String theory contains thousands of much more elaborate and
surprising miracles. If you compute something, you could worry that the
result will be a nonsense because of many reasons. However all the sick
features always cancel in a surprising way.
Unlike other approaches to describe Quantum Gravity, that were created by
Man, String theory smells by God. At every point, all your questions turn
out to have unique answers; the generalizations and extensions of the
parts of the theory that have been already found are determined by
internal mathematical consistency.
String theory connects many objects that are a priori unrelated. For
example, String theory implies that the ordinary geometry is just a
low-energy approximation to a more complicated mathematical framework that
generalizes geometry. A generator of translations can be shown to be
more-or-less equivalent to other operators - such as the winding number of
strings or branes - and they can transmute into each other as you change
the parameters. In String theory, mirror symmetry was discovered; it is an
equivalence between two apparently unrelated topologies of Calabi-Yau
(six-real-dimensional) manifolds: stringy physics with these shapes used
as hidden dimensions is equivalent. String theory allowed the people to
calculate many questions about higher-dimensional manifolds that
mathematicians could not have answered for decades.
String theory implies that black hole thermodynamics can be understood
without any modifications of quantum mechanics; the entropy of
(calculable) black holes always agrees with the Bekenstein-Hawking
(thermodynamic) prediction. Topology of space can change; new dimensions
of space can appear if one works with a large number of particle species;
all perturbative versions of string theory are equivalent.
I should stop at this moment. It is clear that many important things have
been omitted.
Among textbooks, you are recommended to read Joe Polchinski's "String
Theory" (two volumes) as well as an older book by Green, Schwarz, Witten,
"Superstring theory" (also two volumes).
Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
Urs Schreiber
> String theory implies that black hole thermodynamics can be understood
> without any modifications of quantum mechanics; the entropy of
> (calculable) black holes always agrees with the Bekenstein-Hawking
> (thermodynamic) prediction.
A while ago, I was delighted to learn that not only does
string theory give the exact thermodynamics of certain
supersymmetric and near-supersymmetric black holes, but that a
weaker result, namely the string/black-hole correspondence
priciple, which gives correct order-of-magnitude estimates,
applies to all kinds of black holes, including all the
realistic astrophysical ones. I know that this is an old hat
by now, but it may be worth saying it out loud once in a
while! :-) A particularly nice result in this context is that
presented in hep-th/9907030, where an old puzzle is resolved,
which was related to the fact that at the correspondence point
the string seemed to be too large in extent to correspond to
the BH of the given mass. The solution is, that when the
self-gravity of the massive string is taken into account, then
it is seen to shrink under its own attraction to precisely the
expected size.
> String theory connects many objects that are a priori unrelated.
[...]
> A generator of translations can be shown to be
> more-or-less equivalent to other operators - such as the winding number of
> strings or branes - and they can transmute into each other as you change
> the parameters.
I find this an intersting qualitative lesson learned from
string theory: It relativizes the concept of what one might be
willing to consider formally a spatial (or a spacetime-)
dimension: T-duality is about how an (almost) continuous
parameter (winding number) can look like spatial momentum
degree of freedom. Also, the heterotic string gives an example
of how spacetime and gauge degrees of freedoms can be on equal
footing, sort of, other than by, and in addition to, KK
reduction. Even if none of this should turn out to have
relevance for the world we measure, it shows that when looking
for TOEs one should perhaps not be too shy to mess with the
meaning of spacetime dimension.
Alfred Einstead
FIELDS.
Warren Siegel
Currently playing at:
insti.physics.sunysb.edu/~siegel/plan.html
And I do mean everything, including extensive coverage on
supersymmetry, string theory; coverage on general relativity;
on the different approaches to quantization, to gauge
theory; survey of the major GUT's, etc.
There's a separate book over there on string theory, too.
Charles Francis
Motl <mo...@feynman.harvard.edu> writes
>String theory is a very conservative approach
>to physics; the only known natural extension of the Standard Model that
>incorporates gravity.
I should have said string theory is a very extravagant approach, full of
wild claims to incorporate gravity, whereas actually all you have is the
thought of string theorists that it might incorporate gravity and the
failure of said string theorists to show that it does, despite huge
effort over the last twenty or so years by supposedly our brightest
theorists.
>String theory, by definition, contains all good ideas in physics.
Yes. It is quite typical of string theorists to define the word good
like this. ;-)
>I should stop at this moment. It is clear that many important things have
>been omitted.
Like mathematical rigour?
Regards
--
Charles Francis
[Moderator's note: readers wishing a big battle about string
theory are urged to use Google to read the one we already had. - jb]
Igor Khavkine
> Hi Igor!
>
> String theory is a theoretical framework that unifies General Relativity
> with Quantum Field Theory. String theory is a very conservative approach
> to physics; the only known natural extension of the Standard Model that
> incorporates gravity. Nevertheless the unification of Quantum Field Theory
> with General Relativity *forces us* to extend our imagination and
> mathematical skills.
I've already heard in the first few lectures that string theory gives
gravity "for free". And that Einstein's equations just "pop out" from the
dynamics of string theory somewher along the way. Suffice it to say that
"for free" and "pop out" are not very well defined concepts. Is this a
deep consequence of the fundamental assumptions of string theory, or is
it just an artefact of the way the theory is formulated?
> Of course, one must know Quantum Field Theories also as a low-energy
> approximation to String Theory. Not only standard gauge theories, Higgs
> mechanism etc. is important; also more modern aspects of QFT, such as
> supersymmetry, supergravity, anomalies, Renormalization Group, and
> Kaluza-Klein theory, are important.
I have to admit the shame of not knowing what supersymmetry is. I guess
this is as good a time as any to ask: what is supersymmetry? The only pieces
of information I have are that it turns fermions into bosons and vice versa,
enlarges the Poincare symmetry group of space time, and that "everyone" else
already knows what it is.
> String theory is full of mathematical miracles that indicate that the
> whole structure can hardly be a collection of random coincidences. For
> example, already the Standard Model is known to have no anomalies: if you
> sum up the cubed charges of all the quarks and leptons, you obtain zero,
> for example. String theory contains thousands of much more elaborate and
> surprising miracles. If you compute something, you could worry that the
> result will be a nonsense because of many reasons. However all the sick
> features always cancel in a surprising way.
What is an "anomaly", and why does the sum of the cubed charges of all
quarks show that the Standard Model has none? Is there a simple example
of such a miracle?
> Unlike other approaches to describe Quantum Gravity, that were created by
> Man, String theory smells by God. At every point, all your questions turn
> out to have unique answers; the generalizations and extensions of the
> parts of the theory that have been already found are determined by
> internal mathematical consistency.
I'd like to see an example that illustrates this.
> [...] String theory allowed the people to
> calculate many questions about higher-dimensional manifolds that
> mathematicians could not have answered for decades.
Once again, I'd like to see an example of this.
There seems to be a number of "cool" things about string theory that
people like to talk about in lay terms. I find these discussions,
entertaining, but hardly enlightening. I realise that often my backgound
knowledge is insufficient to follow a more enlightening discussion, but
at least I would like to know where my background knowledge should be headed.
Thanks.
Igor
John Devers
> There seems to be a number of "cool" things about string theory that
> people like to talk about in lay terms. I find these discussions,
> entertaining, but hardly enlightening. I realise that often my backgound
> knowledge is insufficient to follow a more enlightening discussion, but
> at least I would like to know where my background knowledge should be headed.
Hi Igor, have you had a look here yet?
I find the site faq and discussion extreamly helpful.
Urs Schreiber
> I've already heard in the first few lectures that string theory gives
> gravity "for free". And that Einstein's equations just "pop out" from the
> dynamics of string theory somewhere along the way. Suffice it to say that
> "for free" and "pop out" are not very well defined concepts. Is this a
> deep consequence of the fundamental assumptions of string theory, or is
> it just an artefact of the way the theory is formulated?
If by artefact you mean that it is just an illusion or something, then, no,
it's there.
The "for free"-part goes like this: On the superstring you have the
creators b^i and B^i of left and right moving fermionic modes. States of
the form b^i B^j|0> turn out to be massless and obviously transform as
spin-2 particles. There is a theorem that any massless spin-2 particle
must essentially be the graviton, though someone else than me has to tell
you more about this theorem.
The "pop-out" part goes like this: One way to study the string's
interaction is to look at the limit where a single string moves in a
"condensate" of other string states, which gives a classical background
field. The question is then which laws the string condensate obeys, since
we can't follow it on the individual worldsheet level. To find this, it
turns out to be sufficient to check the necessary condition that the
single string's motion in that background field is still consistent. This
can be done for instance by checking that the BRST operator is still
nilpotent or by checking that the world-sheet theory is still conformally
invariant (that the beta-functions vanish). Either way, one finds that the
background condensate has to obey the classical equations of motion of
supergravity (plus higher curvature corrections) coupled to the various
fields in the string spectrum. This confirms that the classical low energy
limit of that graviton state above is really general relativity.
> I have to admit the shame of not knowing what supersymmetry is. I guess
> this is as good a time as any to ask: what is supersymmetry? The only
> pieces of information I have are that it turns fermions into Bosons and
> vice versa, enlarges the Poincare symmetry group of space time, and that
> "everyone" else already knows what it is.
Here is one way to think about it, essentially due to Witten [1]
The archetype of all equations of physics is probably
Laplace psi = 0,
where psi is some function and "Laplace" some generalized Laplace operator
(in general not elliptic but hyperbolic) on some manifold (in general
infinite dimensional if we think of field theory). It is quite convenient
to rewrite this as
<=> del d psi = 0
where d and del are the (accordingly generalized) exterior derivative and
its adjoint. This way the Laplace operator is expressed as the projection
of the full (generalized) Laplace-Beltrami operator
Delta = d del + del d
on 0-forms. Hence there is nothing more natural than removing this
restrictions and considering the full equation
(del d + d del) psi = 0,
where we now think of psi as a (in general inhomogeneous) differential
form. Delta is well known to have a very special symmetry: Since
Delta = (d + del)^2 = -(d-del)^2
we have
[ Delta , D_+ ] = 0 = [ Delta , D_- ],
where
D_\pm = d \pm del .
are Dirac operators. This symmetry is called "supersymmetry", since we are
in the age where the silliness of the name that we give an idea is a
measure of how fond we are of it. ;-) Since Delta has this symmetry, it's
natural to also impose it on the state psi and demand that
D_\pm psi = 0 .
The D_\pm are hence called supersymmetry generators. Consider the ordinary
Dirac operator on flat space. Then
[D , x^m] = gamma^m
{D, gamma_m} ~ partial_m ~ p_m .
This way supersymmetry exchanges Bosons (x^m) with fermions (gamma^m).
The first simple example is the massless Dirac particle. Here Delta is the
spinor Laplacian and D the ordinary Dirac operator. The Dirac particle
hence has "worldline supersymmetry", a term that comes from the more usual
treatment of supersymmetry in terms of Lagrangians, which I do not discuss
here. Obviously I have skipped over some details.
The next simple example is a system, which is governed by the above
constraints when Delta and D_\pm are actually the ordinary exterior
derivative operators on spacetime. This describes massless p-form fields.
It is actually a restriction of the full superstring constraints to the
massless RR sector. To handle the full superstring look at its bosonic
configuration space, which is the loop space over spacetime, go to the
superspace over this space, which is its 1-form bundle, and let states be
superfunctions on this superspace, i.e. inhomogeneous differential forms.
Next look at the deformed exterior derivatives over loop space
d_K = d + i K/\
del_K = del -i K->
where
K = K^(m,s) \partial/\partial X^m(s)
is a Killing vector field over loop space which generates loop
reparametrizations, and where K/\ and K-> denote exterior and interior
multiplication, respectively, by K. This deformation is one of the
simplest possibilities that makes the equations
d_K psi = 0 = del_k psi
meaningful in the infinite dimensional setting. Since
d_K^2 = i L_K
del_K^2 = i L_K ,
where L_K = {d,K/\} is the Lie derivative along K, we have
D_+^2 = Delta + i L_K
D_-^2 = -Delta + i L_K .
It can be seen that Delta here generates "time" evolution on the string's
worldsheet whereas L_K generates "spatial" worldsheet transformations.
Hence if we think of D_+ and D_- as two components of a 2-dimensional
spinor the above algebra is of the form
{D_a , D_b} = (gamma C)_ab^m P_m , (1)
where P_0 = Delta, P_1 = i L_K are the translational Poincare generators on
the worldsheet. Equation (1), in the obvious way generalized to arbitrary
dimensions, gives the most important bracket of the super-Poincare
algebra, the supersymmetric extension of the ordinary Poincare algebra.
This is what most texts on supersymmetry start with, instead of the
unorthodox line of thought that I have outlined above. The idea of making
such a supersymmetric extension is to take any old Lie algebra and promote
it to a Z2-graded algebra by adding odd graded generators and brackets
(supercommutators) that respect this grading. Under some sensible
assumptions the super-Poincare algebra is the unique such extension. When
looking for Lagrangians that have this as a symmetry, one finds
Lagrangians that contain bosonic and fermionic fields and they being
supersymmetric refers to a certain invariance of those under a certain
exchange of bosonic and fermionic fields.
I think I got that completely reversed with respect to the historical
development, which started with somebody adding fermionic fields to the
Polyakov string and somebody else noting that this gives a strange new
symmetry and still somebody else jumping on and generalizing everything to
higher dimensions and lastly somebody trying to bring all this in a
coherent form by introducing superfield formalism and finally somebody
noting that in Schroedinger rep all this is about exterior differential
geometry and then receiving the fields medal. :-)
If we are lucky, then the unbalanced picture I painted will make somebody
want to add some helpful comments.
[1]
author = {E. Witten},
title = {Supersymmetry and Morse Theory},
year = {1982},
journal = {J. Diff. Geom.},
volume = {17},
pages = {661-692}
More details on the second half of the above paper are given in the second
half of the following paper:
author = {E. Witten},
title = {Global anomalies in string theory},
booktitle = {Symposium on Anomalies, Geometry, Topology},
pages = {61-99},
publisher = {World Scientific},
year = {1985},
editor = {W. Bardeen and A. White},
Thomas Larsson
>
> I have to admit the shame of not knowing what supersymmetry is. I guess
> this is as good a time as any to ask: what is supersymmetry? The only pieces
> of information I have are that it turns fermions into bosons and vice versa,
> enlarges the Poincare symmetry group of space time, and that "everyone" else
> already knows what it is.
>
A supersymmetry is a Lie superalgebra (or group) with a Poincare
subalgebra (or subgroup), to be identified with spacetime transformations.
A Lie superalgebra, or Z_2-graded Lie algebra, is like a Lie algebra
except that you put in minus signs at the right places and prefix
everything in sight with "super". So in addition to commutators
[A,B] = AB - BA, as in an ordinary Lie algebra, we also have anticommutators
{A,B} = AB + BA.
Unlike supersymmetries, Lie superalgebras are incredibly cool and of
proven relevance to physics. Suffice it to mention that a Clifford algebra
is a Lie superalgebra with one bosonic element (the unit operator).
Clifford algebras are necessary to describe spinors and fermions, whose
existence are proven beyond reasonable doubt.
The Poincare subalgebra condition has three important properties:
1. It makes it possible to circumvent the Coleman-Mandula theorem, which
roughly states that internal and space-time symmetries cannot be united
with only Lie algebras.
2. It implies the existence of tons of supersymmetric partners, none of
which has been seen in experiments.
3. It apparently lacks intrinsic mathematical interest.
However, sometimes the distinction between supersymmetry and superalgebra is
blurred, either due to lack of knowledge or as a marketing ploy;
I thank Lubos Motl for emphasizing the difference to me. E.g., it has been
claimed that supersymmetry has been observed in some kind of gold nuclei,
but this is incorrect. The spectrum falls into representations of some
superalgebra, but it is still an internal symmetry, so there are no
susy partners and no supersymmetry here.
Urs Schreiber
> A supersymmetry is a Lie superalgebra (or group) with a Poincare
> subalgebra (or subgroup), to be identified with spacetime transformations.
[...]
> Unlike supersymmetries, Lie superalgebras are incredibly cool and of
> proven relevance to physics.
However, the Poincare subgroup need not be that of spacetime for supersymmetry
to be useful! For instance, the ordinary massless Dirac particle has
*worldline* supersymmetry - and it does exist. The relevant Poincare algebra
is the trivial 1-dimensional algebra of reparametrizations of the particle's
worldline.
And it's useful, too, to recognize the worldline supersymmetry. For instance
one can use it [1] to get a deeper understanding of the relation between the
global anomaly in the path-integral quantization of that particle and the
existence or non-existence of a spin structure on the spacetime it propagates
in.
And didn't supersymmetric methods allow a nice proof of the positive mass
theorem?
> Suffice it to mention that a Clifford algebra
> is a Lie superalgebra with one bosonic element (the unit operator).
Hm, this is probably just an issue of terminology, but wouldn't we want to
call every even graded element of the Clifford algebra a bosonic element?
[1]
author = {E. Witten},
Thomas Larsson
news:b1svqo$1674go$1...@ID-168578.news.dfncis.de...
> Thomas Larsson wrote:
> > A supersymmetry is a Lie superalgebra (or group) with a Poincare
> > subalgebra (or subgroup), to be identified with spacetime
> > transformations.
> > Unlike supersymmetries, Lie superalgebras are incredibly cool and
> > of proven relevance to physics.
> However, the Poincare subgroup need not be that of spacetime for
> supersymmetry to be useful! For instance, the ordinary massless
> Dirac particle has *worldline* supersymmetry - and it does
> exist. The relevant Poincare algebra is the trivial 1-dimensional
> algebra of reparametrizations of the particle's worldline.
Hm. If this kind of worldline supersymmetry describes a Dirac particle, it
can hardly imply susy partners, can it? So the distinction between susy
and superalgebra is maybe even more blurred than I thought.
A good operational definition of supersymmetry might be that it implies
susy partners. At least this is the main consequence that I, and probably
many others, think of when I hear the word supersymmetry.
Another place where supersymmetry perhaps has been found is in the
tricritical Ising model in 2D. This is a lattice model of spins with
vacancies. By fine-tuning the coupling constant and the fugacity for
vacancies, you find a tricritical point described by a CFT with c = 7/10,
which is the first model in the unitary discrete series of the super-
Virasoro algebra. So this is a spacetime superalgebra symmetry, but it is
unclear to me whether there are any susy partners. At any rate, they are
not evident from the lattice formulation.
But even if the tricritical Ising model is an experimental example of
susy (it has not been marketed as such, though), it is not really relevant
to HEP. Besides, all other multicritical models are non-supersymmetric.
> And didn't supersymmetric methods allow a nice proof of the positive mass
> theorem?
I am not really familiar with this result, although I have heard about it.
Can you elaborate?
> > Suffice it to mention that a Clifford algebra
> > is a Lie superalgebra with one bosonic element (the unit operator).
> Hm, this is probably just an issue of terminology, but wouldn't we want to
> call every even graded element of the Clifford algebra a bosonic element?
I am probably guilty of a misnomer here. What I meant by Clifford algebra
was the Lie superalgebra with generators y_i and e, and brackets
{y_i, y_j} = g_ij e,
[y_i, e] = [e, e] = 0,
where g_ij = g_ji are structure constants. So in this case e is the only
bosonic element. However, the correct terminology seems to be that the
Clifford algebra is the associative universal envelope of this algebra
(modulo relations ex = xe = x for all x). Anyway, both the Lie and
associative variants are relevant for spinors and fermions and thus
proven physics, which was my point.
Urs Schreiber
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message
> news:b1svqo$1674go$1...@ID-168578.news.dfncis.de...
>> However, the Poincare subgroup need not be that of spacetime for
>> supersymmetry to be useful! For instance, the ordinary massless
>> Dirac particle has *worldline* supersymmetry - and it does
>> exist. The relevant Poincare algebra is the trivial 1-dimensional
>> algebra of reparametrizations of the particle's worldline.
>
> Hm. If this kind of worldline supersymmetry describes a Dirac particle, it
> can hardly imply susy partners, can it?
Worldline supersymmetry implies "worldline superpartners". :-) One may think
of the bosonic relativistic particle as a 1+0 dimensional field theory with
x^m being coordinate fields taking values in a target space - spacetime. Now
susy this 1+0 dim field theory by introducing fermionic superpartners psi^m
to the coordinate fields. (One version of the resulting action is given in
the article that I had mentioned.) The conserved supersymmetry charge is
D = psi^m d/dtau x_m .
Quantizing everything gives the anticommutator
{psi^m,psi^n} = g^mn,
where g^mn is the target space metric. Furthermore one gets
d/dtau x_m -> partial_m + omega_mab [psi^a,psi^b]/4 .
Hence D is the Dirac operator on spacetime. The superpartners of the
coordinate fields are hence the Clifford generators on spacetime.
This still does not give spacetime superpartners, of course. Not directly at
least. But there should be some relation (the details of which presently
escape me) of the particle with worldline supersymmetry, that I talked about
here, to what is ordinarily called the "superparticle", which has target
space supersymmetry and which is often used as a warmup for the GS
superstring. Now I know that you are not fond of strings, but they at least
serve here as a neat bookkeeping device for these theoretical concepts: The
spacetime supersymmetry of the GS superstring is well known to be related to
the worldsheet supersymmetry of the NSR string. The point particle limit of
the GS superstring gives the (target space supersymmetric) superparticle, and
the point particle limit of the NS (or R-NS) sector of the NSR string gives
the particle with worldline susy, namely the Dirac spinor particle.
I have read somewhere (I forget where) that our inability to have base space
supersymmetry and target space supersymmetry in one single formalism is due
to our poor understanding of embeddings of one supermanifold into another
supermanifold, which is what all this is really about. The notorious "kappa
symmetry" of the GS string and the supermembrane can apparently be traced
back to a gauge fixing effect related to such an embedding. But at this point
I don't really know what I am talking about. Maybe Robert Helling will chime
in and offer some insight.
> So the distinction between susy and superalgebra is maybe even more blurred
> than I thought.
>
> A good operational definition of supersymmetry might be that it implies
> susy partners. At least this is the main consequence that I, and probably
> many others, think of when I hear the word supersymmetry.
I'd say that with sentences like "We have not yet any indication of
supersymmetry." or "Supersymmetry may help solve the hierarchy problem." one
is implicitly referring to _spacetime supersymmetry_. For instance there are
plenty of examples of ordinary physical systems that have some sort
1-dimensional supersymmetry, namely those that are described by an ordinary
Hamiltonian which factors as H = D^2. These systems exhibit a special case of
the general supersymmetry algebra but are in no way related to spacetime
supersymmetry or any supersymmetric extension of the standard model. They all
have some sort of "superpartners", though, but not necessarily in the sense
of fields. Take any operator Op on the system's Hilbert space. It's
superpartner is sOp = [D,A], where the bracket denotes the supercommutator.
For instance in the above example of the Dirac particle we have psi^m =
[D,x^m] and p_m ~ partial_m = [D,psi_m].
> Another place where supersymmetry perhaps has been found is in the
> tricritical Ising model in 2D. This is a lattice model of spins with
> vacancies. By fine-tuning the coupling constant and the fugacity for
> vacancies, you find a tricritical point described by a CFT with c = 7/10,
> which is the first model in the unitary discrete series of the super-
> Virasoro algebra. So this is a spacetime superalgebra symmetry, but it is
> unclear to me whether there are any susy partners.
How does the operator that generates susy transformations look like in this
model?
>> And didn't supersymmetric methods allow a nice proof of the positive mass
>> theorem?
>
> I am not really familiar with this result, although I have heard about it.
> Can you elaborate?
Sorry, I am not really familiar with it either! :-) (About all I know is that
it makes use of the fact that, roughly and schematically, from {Q,Q} ~ P it
follows that P is non-negative.) But I have bookmarked this as one example
for the usefulness of the concept of supersymmetry in physics even in the
absence of observed superpartners in particle physics.
Thomas Larsson
> > Another place where supersymmetry perhaps has been found is in the
> > tricritical Ising model in 2D. This is a lattice model of spins with
> > vacancies. By fine-tuning the coupling constant and the fugacity for
> > vacancies, you find a tricritical point described by a CFT with c = 7/10,
> > which is the first model in the unitary discrete series of the super-
> > Virasoro algebra. So this is a spacetime superalgebra symmetry, but it is
> > unclear to me whether there are any susy partners.
>
> How does the operator that generates susy transformations look like in this
> model?
The Hilbert space carries a representation of the super-Virasoro
algebra, with generators L_m and G_r and brackets
[L_m, L_n] = (m-n) L_m+n + c/12 (m^3 - m) delta_m+n
[L_m, G_r] = (m/2 - r) G_m+r
{G_r, G_s} = 2 L_r+s + something x delta_r+s
I have forgotten exactly how the second central term looks, but it is
completely given by c. This algebra has a discrete series of unitary
irreps which starts at c = 7/10 and ends at c = 3/2. Again I have forgotten
the exact formula, but the corresponding series for Virasoro proper is
c = 1 - 6/m(m+1), m integer.
which starts at c = 1/2 (m=3) and ends at c = 1 (m = oo). Note that c = 7/10
corresponds to m=4. Anyway, for each c there are finitely many irreps
characterized by the L_0 eigenvalue h, which is related to critical
exponents in the lattice model. More exactly, we have two Virasoro algebras
corresponding to analytical and anti-analytical transformations, with L_0
eigenvalues h and h'. The scaling dimension x = h+h' and the spin s = h-h'.
Hmm. The tricritical model is in the A-A series. I don't really want to
explain what this is (look it up in the big yellow French bible on CFT), but
it means that the spin is always zero. So this is probably not related to
susy, although the algebra above is a superalgebra.
> >> And didn't supersymmetric methods allow a nice proof of the
> >> positive mass theorem?
> > I am not really familiar with this result, although I have heard
> > about it. Can you elaborate?
> Sorry, I am not really familiar with it either! :-) (About all I
> know is that it makes use of the fact that, roughly and
> schematically, from {Q,Q} ~ P it follows that P is non-negative.)
> But I have bookmarked this as one example for the usefulness of the
> concept of supersymmetry in physics even in the absence of observed
> superpartners in particle physics.
We have no disagreement here, since, this example seems to fall into the
superalgebra cathegory. I cannot imagine that the positive energy
theorem relies on susy as a physical property - this would make the
theorem highly dubious since susy has not been observed. But susy
methods/superalgebras are definitely valuable, and a sufficient
motivation for taking a course in susy. One example is cohomology,
which is an important part mathematics with many applications to
confirmed physics. This subject is based on the superalgebra {d, d} = 0,
which is just a fancy way to write div curl = curl grad = 0.
Still, when embarking on a study of supersymmetry, it might be a good
idea to keep in mind that it is a purely hypothetical symmetry. This is
true for spacetime susy, which is what people look for at Fermilab and
CERN. In fact, of the two regions singled out by theoretical arguments,
small tan beta and large tan beta (don't ask me what this means,
though), the former has already been excluded by experiments.
Worldsheet susy is of course doubly hypothetical, since fundamental
worldsheets are not experimentally confirmed in the first place.
Jeffery
You should be warned that the forums on that site are Crackpot Heaven!
Jeffery
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>There is a theorem that any massless spin-2 particle
>must essentially be the graviton, though someone else than me has to tell
>you more about this theorem.
This is one of those physics "theorems" that people like to cite
without:
1) a precise statement,
2) a proof,
3) any references.
I believe there does exist some result along the lines you mention,
but in the form you've stated it - and not just you, either! - it's
clearly false, because there's a perfectly consistent classical
field theory of massless noninteracting spin-2 fields, and this
isn't general relativity. If we quantize this, we get a perfectly
consistent *quantum* field theory of massless noninteracting spin-2
fields, and this can't be quantum gravity.
(You can replace "2" by 0, 1/2, 1, 3/2, 2, 5/2,... in the above
paragraph, and it will still be true.)
Luckily there is better evidence for the relation between
string theory and gravity than the fact that perturbative
string theory contains massless spin-2 particles.
I would not get so grumpy if people said "any massless
spin-2 particle has a chance of acting like a graviton".
Urs Schreiber
John Baez wrote:
> In article <b18ep5$109dme$1...@ID-168578.news.dfncis.de>,
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>>There is a theorem that any massless spin-2 particle
>>must essentially be the graviton, though someone else than me has to tell
>>you more about this theorem.
> This is one of those physics "theorems" that people like to cite
> without:
>
> 1) a precise statement,
> 2) a proof,
> 3) any references.
>
> I believe there does exist some result along the lines you mention,
> but in the form you've stated it - and not just you, either! - it's
> clearly false, because there's a perfectly consistent classical
> field theory of massless noninteracting spin-2 fields, and this
> isn't general relativity.
OK, I should at least have added the word "interacting". My statement would
probably still be false, but not that clearly, maybe...
> Luckily there is better evidence for the relation between
> string theory and gravity than the fact that perturbative
> string theory contains massless spin-2 particles.
I hope I did not make the impression that my previous post was
supposed to be a comprehensive summary of why string theory contains
gravity.
> I would not get so grumpy if people said "any massless
> spin-2 particle has a chance of acting like a graviton".
Thanks, I'll say that next time! :-)
Unfortunately I can't find the reference that I would have liked to
cite at this point, but since the original poster might be interested
in more background information, I'll point him to the introduction of
Green, Schwarz, Witten and quote a few lines from p. 42, where it
says:
"Nothing in the discussion justifies any expectation that this
'graviton' will couple in a gauge invariant way, yet one might
conjecture that this will turn out to be true simply because general
covariance is one of the few possible ways to make a consistent theory
of a massless spin two particle.
The only other possibility seems to be to have a massless spin two
particle with /linear/ gauge invariance that couples via derivative
interactions only. This is analogous to a massless vector meson that
couples to neutral particles only via derivative interactions such as
magnetic moment couplings. Theories of massless spin two particles
with derivative couplings only seem to be unrenormalizable and to
suffer from certain other pathologies, but there is no general theorem
guaranteeing that there cannot be a consistent theory of this kind. "
The authors then go on to discuss why the massless spin-2 state of the
string is indeed the graviton.
Robert C. Helling
> In article <b18ep5$109dme$1...@ID-168578.news.dfncis.de>,
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>
>>There is a theorem that any massless spin-2 particle
>>must essentially be the graviton, though someone else than me has to tell
>>you more about this theorem.
>
> This is one of those physics "theorems" that people like to cite
> without:
>
> 1) a precise statement,
> 2) a proof,
> 3) any references.
>
> I believe there does exist some result along the lines you mention,
> but in the form you've stated it - and not just you, either! - it's
> clearly false, because there's a perfectly consistent classical
> field theory of massless noninteracting spin-2 fields, and this
> isn't general relativity. If we quantize this, we get a perfectly
> consistent *quantum* field theory of massless noninteracting spin-2
> fields, and this can't be quantum gravity.
>
> (You can replace "2" by 0, 1/2, 1, 3/2, 2, 5/2,... in the above
> paragraph, and it will still be true.)
I think, the more precise version of this reads "Any theory with an
interacting spin 2 particle where the interaction does not have more
than two derivatives contains the Einstein-Hilbert term in the action
and is thus gravity in some sense". You might in addition require the
theory to have some gauge invariance (or to be ghost free at the
quantum level) but this might already be inculded in the above statement.
To my knowledge this goes back to
a paper by Feynman but I don't have a reference.
IIRC, in http://arXiv.org/abs/hep-th/0103034 Wellmann (a student of
Scharf) rederives this result in the context of causal perturbation
theory. I would think this satisfies even higher standards for "proof"
than the average physicist's ones.
The folklore goes on (and might be at a similar level of robustness)
to say, that supergravity is the only theory with interacting spn 3/2
particles, you cannot have more than one spin 2 particle in each
interacting sector of the theory and with finitely many fields you
cannot have spin>2 and interactions. I don't have any references to
substanciate these claims but good places to start searching are the
papers about higher spin theories, Micha Vassiliev being one of the
experts on this.
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Department of Applied Mathematics and Theoretical Physics
University of Cambridge
print "Just another Phone: +44/1223/766870
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Charles Torre
[unncecessary quoted text deleted by angry gods]
> John Baez <ba...@galaxy.ucr.edu> wrote:
> > Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> >>There is a theorem that any massless spin-2 particle
> >>must essentially be the graviton, though someone else than me has to tell
> >>you more about this theorem.
> > I believe there does exist some result along the lines you mention,
> > but in the form you've stated it - and not just you, either! - it's
> > clearly false, [....]
> I think, the more precise version of this reads "Any theory with an
> interacting spin 2 particle where the interaction does not have more
> than two derivatives contains the Einstein-Hilbert term in the
> action and is thus gravity in some sense". You might in addition
> require the theory to have some gauge invariance (or to be ghost
> free at the quantum level) but this might already be inculded in the
> above statement. To my knowledge this goes back to a paper by
> Feynman but I don't have a reference.
A good, relatively recent article by Wald (so it *will* be precise and
rigorous) on spin 2, relativity, general covariance, etc. is at
R. Wald, Phys. Rev. D33: 3613 (1986).
For recent work on spin 2 coupled to spin 3/2, have a look at
S. Anco, Annals Phys. 270: 52-125 (1998).
charlie
Arnold Neumaier
>
> >There is a theorem that any massless spin-2 particle
> >must essentially be the graviton, though someone else than me has to tell
> >you more about this theorem.
>
> This is one of those physics "theorems" that people like to cite
> without:
>
> 1) a precise statement,
> 2) a proof,
> 3) any references.
>
> I believe there does exist some result along the lines you mention,
See:
Steven Weinberg,
PHOTONS AND GRAVITONS IN PERTUBATION THEORY: DERIVATION
OF MAXWELL'S AND EINSTEIN'S EQUATIONS.
Phys.Rev.138:B988-B1002,1965,
based on
S. Weinberg,
Feynman rules for any spin. I-III
Phys. Rev.181 (1969), 1893
Phys. Rev. 133 (1964), 1318
Phys. Rev. 134 (1964), 882
Weinberg makes some assumptions (interacting particle with a
Lorentz-invariant scattering theory, inverse square law, and
microcausality), and uses semirigorous mathematical arguments
(of much better quality than most that exists in quantum field
theory outside the axiomatic island).
Arnold Neumaier
Demian H.J. Cho
<snip>
>
> Unfortunately I can't find the reference that I would have liked to
> cite at this point,
<snip>
Hi Urs,
Some classic papers I know are
Robert Wald, PRD 33, 3613 (1986)
Steven Weinberg, PR 135, B1049 (1964)
PR 138, B988 (1965)
For those of you who are too young to know "PR" is
Physical Review, grand daddy of all physical review series.
To me beta function argument of getting GR is not really
surprising. After all, you can argue similary for scalar field
on curved background and ask for consistency condition.
This way you will get effective lagrangian including
Einstein-Hilbert with cosmological constant, and all the
higher order terms in Riemann and Ricci which will be ignored
in lower curvature limit. (Well, it's not completely fair to
string theory. I guess that beta function argument with
the fact that string mode contains spin2 state will probably
convince me.)
But, alas, what you will get from the above argument is not GR.
We have all other junks like dilaton, moduli, p-form fields etc.
Theoretical physicists may be pleased with all those since it
give them more toys to play with. But, unless somebody show
me how they are going to put those junks back into their
Pandora's box I am not convinced.
Demian
Urs Schreiber
Demian H.J. Cho wrote:
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message
news:<b2qfra$1e9vac$1...@ID-168578.news.dfncis.de>...
> <snip>
>>
>> Unfortunately I can't find the reference that I would have liked to
>> cite at this point,
>
> Some classic papers I know are
>
> Robert Wald, PRD 33, 3613 (1986)
> Steven Weinberg, PR 135, B1049 (1964)
> PR 138, B988 (1965)
Thanks! (Also thanks to other's who have provided related material). I knew
that I had seen a reference to Weinberg in this context somewhere, but now I
could not find anymore the reference that referenced this reference! :-)
> To me beta function argument of getting GR is not really
> surprising. After all, you can argue similary for scalar field
> on curved background and ask for consistency condition.
> This way you will get effective lagrangian including
> Einstein-Hilbert with cosmological constant, and all the
> higher order terms in Riemann and Ricci which will be ignored
> in lower curvature limit.
I don't know about this. Are you saying that there is no non-trivial solution
to a scalar field coupled to ordinary EH gravity? That one needs higher oder
curvature corrections instead?
> But, alas, what you will get from the above argument is not GR.
> We have all other junks like dilaton, moduli, p-form fields etc.
> Theoretical physicists may be pleased with all those since it
> give them more toys to play with. But, unless somebody show
> me how they are going to put those junks back into their
> Pandora's box I am not convinced.
What is the state of the art in trying to close this Pandora's box?
I seem to recall that sometimes it is said that the mechanism of the moduli
becoming massive might be related to the unknown mechanism which breaks
supersymmetry. Hence it is consistent that one does not know how either
works! :-/
This reminds me that some day I wanted to ask for a brief list of the
different proposed attempts on a realistic string phenomenology. The classic
is heterotic string theory on M^4 x CY^6, right? How does that compare with
M-theory on M^4 x X^7, where X is, ahm, a manifold of G2 holonomy? I forget.
More recently, intersecting p-branes seem to have attracted a lot of
attention. Even if all this turns out to have nothing to do with reality,
isn't it tantalizing what wealth of models one can obtain this way that all
*almost* look like the standard model?
Then, I have seen (not read!) these papers by Deo, talking about a realistic
string theory of some new sort, which lives in 4 dimensions. What's that?
R.X.
> Lubos Motl <mo...@feynman.harvard.edu> wrote in message
news:<Pine.LNX.4.31.0301192116020.32652-100000@feynman>...
> I've already heard in the first few lectures that string theory
> gives gravity "for free". And that Einstein's equations just "pop
> out" from the dynamics of string theory somewher along the
> way. Suffice it to say that "for free" and "pop out" are not very
> well defined concepts. Is this a deep consequence of the fundamental
> assumptions of string theory, or is it just an artefact of the way
> the theory is formulated?
Well, "pop out" is indeed not too a bad expression, as GR with all its
complexity simply pops out from a simple, even free 2d field theory.
That is has to be this way is guaranteed by Ward identities that stem
from the 2d conformal invariance, which is one of the basic
fundamental ingredients into the construction.
> > String theory is full of mathematical miracles that indicate that
> > the whole structure can hardly be a collection of random
> > coincidences. For example, already the Standard Model is known to
> > have no anomalies: if you sum up the cubed charges of all the
> > quarks and leptons, you obtain zero, for example. String theory
> > contains thousands of much more elaborate and surprising
> > miracles. If you compute something, you could worry that the
> > result will be a nonsense because of many reasons. However all the
> > sick features always cancel in a surprising way.
> What is an "anomaly", and why does the sum of the cubed charges of
> all quarks show that the Standard Model has none? Is there a simple
> example of such a miracle?
The chiral anomaly is a standard field theory concept, and it can be
proven that in any consistent (modular invariant) string theory, all
anomalies of that sort and of more intricate sorts, cancel.
[Moderator's note: I sure hope someone breaks down and tells Khavkine
what an anomaly *is*. - jb]
> > Unlike other approaches to describe Quantum Gravity, that were
> > created by Man, String theory smells by God. At every point, all
> > your questions turn out to have unique answers; the
> > generalizations and extensions of the parts of the theory that
> > have been already found are determined by internal mathematical
> > consistency.
> I'd like to see an example that illustrates this.
Again, absence of anomalies in space-time is guaranteed by internal
consistency.
> > [...] String theory allowed the people to
> > calculate many questions about higher-dimensional manifolds that
> > mathematicians could not have answered for decades.
> Once again, I'd like to see an example of this.
Just have a look e.g. into the book of Brian Greene, "The Elegant
Universe". That should answer most of your immediate questions.
Examples in the present context are Calabi-Yau manifolds, for which
string theoretic methods (mirror symmetry) have allowed to do
computations mathematicians had dreamed of - like counting spheres and
other 2d surfaces inside such CY spaces, etc. Many more examples exist
and the list is growing all the time.
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message
news:b1svqo$1674go$1...@ID-168578.news.dfncis.de...
>thomas....@hdd.se (Thomas Larsson) wrote in message news:<4b8cc0a6.03=
>02070148...@posting.google.com>...
>> A good operational definition of supersymmetry might be that it
>> implies susy partners. At least this is the main consequence that
>> I, and probably many others, think of when I hear the word
>> supersymmetry.
Supersymmetry can also be spontaneously broken - it is then still
there, but non-linearly realized and perhaps not so easy to pinpoint:
the superpartners will have different masses in this case.
>> Another place where supersymmetry perhaps has been found is in the
>> tricritical Ising model in 2D. This is a lattice model of spins
>> with vacancies. By fine-tuning the coupling constant and the
>> fugacity for vacancies, you find a tricritical point described by a
>> CFT with c = 7/10, which is the first model in the unitary discrete
>> series of the super- Virasoro algebra. So this is a spacetime
>> superalgebra symmetry, but it is unclear to me whether there are
>> any susy partners.
> How does the operator that generates susy transformations look like
> in this model?
It is composed out of the currents in this model. There is even a
realization of N=2 supersymmetry with a single free periodic boson,
where c=1; all what is required is that the boson has a particular
periodicity. Surely SUSY partners do exist in these models. However,
SUSY is accidental and doesn't play an important role here, so that's
why it isn't often mentioned.
Arnold Neumaier
> Some classic papers I know are
>
> Robert Wald, PRD 33, 3613 (1986)
> Steven Weinberg, PR 135, B1049 (1964)
> PR 138, B988 (1965)
Wald's analysis (of conditions under which massless spin 2 implies
gravitation) was extended by
K. Heiderich and W. Unruh
Phys. Rev. D 38, 490 (1988)
to include conformally invariant theories.
Arnold Neumaier
Thomas Larsson
> Examples in the present context are Calabi-Yau manifolds, for which
> string theoretic methods (mirror symmetry) have allowed to do
> computations mathematicians had dreamed of - like counting spheres and
> other 2d surfaces inside such CY spaces, etc. Many more examples exist
> and the list is growing all the time.
Hmm. I know quite a few mathematicians, but I doubt that any of them dream
about counting spheres inside CY spaces. But ok, *some* mathematicians may
have such dreams.
> >thomas....@hdd.se (Thomas Larsson) wrote in message news:<4b8cc0a6.03=
> >02070148...@posting.google.com>...
>
> >> A good operational definition of supersymmetry might be that it
> >> implies susy partners. At least this is the main consequence that
> >> I, and probably many others, think of when I hear the word
> >> supersymmetry.
>
> Supersymmetry can also be spontaneously broken - it is then still
> there, but non-linearly realized and perhaps not so easy to pinpoint:
> the superpartners will have different masses in this case.
I was attempting to make the distinction between supersymmetries and
superalgebras, such as Clifford algebras. The latter are necessary to describe
electrons, but the existence of electrons is not by itself any sign of susy.
Similarly, I very much doubt that the positive energy theorem either depends
on or implies supersymmetry, since that would make it a very dubious theorem
indeed given current experimental data, or rather lack thereof.
However, from what I read I gather that there is limit to what extent susy
can be broken. If no sparticles show up below 3 TeV, a region that should be
covered by LHC during this decade, supersymmetry seems like a pretty dead idea.
>
> >> Another place where supersymmetry perhaps has been found is in the
> >> tricritical Ising model in 2D. This is a lattice model of spins
> >> with vacancies. By fine-tuning the coupling constant and the
> >> fugacity for vacancies, you find a tricritical point described by a
> >> CFT with c = 7/10, which is the first model in the unitary discrete
> >> series of the super- Virasoro algebra. So this is a spacetime
> >> superalgebra symmetry, but it is unclear to me whether there are
> >> any susy partners.
>
> > How does the operator that generates susy transformations look like
> > in this model?
>
> It is composed out of the currents in this model. There is even a
> realization of N=2 supersymmetry with a single free periodic boson,
> where c=1; all what is required is that the boson has a particular
> periodicity. Surely SUSY partners do exist in these models. However,
> SUSY is accidental and doesn't play an important role here, so that's
> why it isn't often mentioned.
Some aspects of the multi-critical model with c = 1 - 6/m(m+1) can be
described by the Landau-Ginzburg action
S = int d^2z ( 1/2 (d phi)^2 + phi^2(m-1) ),
see Francesco-Mathieu-Senechal (7.115) & (7.116). In particular, the
tricritical model corresponds to m = 4, i.e. c = 7/10. Is there any
way to see why only m = 4 gives supersymmetry in this picture?
And exactly what is the difference between an accendental symmetry and
a non-accidental symmetry?
Robert C. Helling
<Urs.Sc...@uni-essen.de> wrote:
> This reminds me that some day I wanted to ask for a brief list of the
> different proposed attempts on a realistic string phenomenology. The classic
> is heterotic string theory on M^4 x CY^6, right?
Well, probably if you try to fit it with both a GUT and gravity you
will find that the heterotic string theory is strongly coupled. So
M-Theory on M^4 x CY3 x S^1/Z_2 is a better description.
> How does that compare with M-theory on M^4 x X^7, where X is, ahm, a
> manifold of G2 holonomy? I forget.
CY3 x S^1/Z_2 is a singular G2 manifold. It probably comes from one
corner of the moduli space of G2 manifolds. Another way to obtain G2
manifolds from Calabi-Yaus is to attach a circle and then mod out a
Z_2 that acts both on the circle and as complex conjugation on the
CY. This is probably a dual description of the above mentioned moduli
space that is more in a type I spirit.
> More recently, intersecting p-branes seem to have attracted a lot of
> attention.
True. Most people consider D6 branes. 3+1 directions cover the M^4 so
they are three-branes on the CY (or torus or orbi-/orientifold of a
torus). But D6-branes lift to KK-monopoles in M-Theory. Those are
everywhere regular configurations of the metric. So once again, the CY
with the D6-branes is a G2 manifold from the 11d perspective.
Once again, all different looking descriptions become unified once
they are lifted to M-Theory.
> Even if all this turns out to have nothing to do with reality, isn't
> it tantalizing what wealth of models one can obtain this way that
> all *almost* look like the standard model?
Depends on your definition of "all almost look like". To get a low
energy spectrum that is in some sense similar to that of the SM still
requires a lot of engeneering so it is not clear to me whether this
has any more meaning than the statement: "Using a field theory
Lagrangian (or a N=1 superspace Lagrangian) I can obtain a wealth of
models that *almost* look like the SM".
Sorry for that.
Stein A. Stromme
Thomas Larsson wrote:
| redlu...@wanadoo.fr (R.X.) wrote in message
| news:<7b5bf479.030308125= 2.110...@posting.google.com>...
| > Examples in the present context are Calabi-Yau manifolds, for
| > which string theoretic methods (mirror symmetry) have allowed to
| > do computations mathematicians had dreamed of - like counting
| > spheres and other 2d surfaces inside such CY spaces, etc. Many
| > more examples exist and the list is growing all the time.
| Hmm. I know quite a few mathematicians, but I doubt that any of them
| dream about counting spheres inside CY spaces. But ok, *some*
| mathematicians may have such dreams.
I can confirm that.
Starting in the fall of 1989, Geir Ellingsrud and I (MR=3D94d:14050)
worked with computing the number of twisted cubic curves (=3D rational
degree three complex curves, or "spheres") contained in a general
quintic projective hypersurface threefold (=3D a CY space), a question
put to us by H. Clemens. While we were still working on this, happily
unaware of string theory, P. Candelas & Co came along in 1990 and (in
conjectural form) computed such numbers of rational curves of (in
principle) all degrees. That computation certainly exceeded what _we_
were able to dream of at the time. (It also uncovered an error in our
computation, but that is a different story.)
The influence of string theory on intersection theory and enumerative
geometry cannot be underestimated.
SA
--
Stein Arild Str=F8mme +47 55584825, +47 95801887
Universitetet i Bergen Fax: +47 55589672
Matematisk institutt www.mi.uib.no/stromme
Johs Brunsg 12, N-5008 BERGEN str...@mi.uib.no
Igor Khavkine
> k_ig...@lycos.com (Igor Khavkine) wrote in message
> news:<f1ac2e6e.03012...@posting.google.com>...
>
> > Lubos Motl <mo...@feynman.harvard.edu> wrote in message
> news:<Pine.LNX.4.31.0301192116020.32652-100000@feynman>...
>
> > I've already heard in the first few lectures that string theory
> > gives gravity "for free". And that Einstein's equations just "pop
> > out" from the dynamics of string theory somewher along the
> > way. Suffice it to say that "for free" and "pop out" are not very
> > well defined concepts. Is this a deep consequence of the fundamental
> > assumptions of string theory, or is it just an artefact of the way
> > the theory is formulated?
>
> Well, "pop out" is indeed not too a bad expression, as GR with all its
> complexity simply pops out from a simple, even free 2d field theory.
> That is has to be this way is guaranteed by Ward identities that stem
> from the 2d conformal invariance, which is one of the basic
> fundamental ingredients into the construction.
So basically, since the string spectrum contains a spin two particle,
it must be the graviton. Right?
I have seen a footnote to this effect in the first volume of Weinberg's
Quantum Theory of Fields. In the same context he shows that any Lorentz
invariant causal vector field must obey Maxwell's equations (or a
generalized version involving a mass term). So if the same can be said
about a rank two tensor field and Einstein's equations, them I'm willing
to buy it. But I will have to investigate it further.
BTW, I have briefly bumped into a Ward identity while studying QFT. That
was just the case for covariantly quantizing the photon field. But what
do Ward identieties look like in general? And what is their significance?
> > What is an "anomaly", and why does the sum of the cubed charges of
> > all quarks show that the Standard Model has none? Is there a simple
> > example of such a miracle?
>
> The chiral anomaly is a standard field theory concept, and it can be
> proven that in any consistent (modular invariant) string theory, all
> anomalies of that sort and of more intricate sorts, cancel.
Are you saying that chirality (or, in plain language, lack of mirror
symmetry) has something to do with the sum of cubed charges of quarks?
If so, I'm completely stumped.
> [Moderator's note: I sure hope someone breaks down and tells Khavkine
> what an anomaly *is*. - jb]
I think I can handle this one myself now. If a classical theory
respects some symmetry but the quantized version of the same theory
doesn't, this phenomenon is called an anomaly.
Thanks.
Igor
Robert C. Helling
> redlu...@wanadoo.fr (R.X.) wrote in message news:<7b5bf479.03030...@posting.google.com>...
>
>> > all quarks show that the Standard Model has none? Is there a simple
>> > example of such a miracle?
>>
>> The chiral anomaly is a standard field theory concept, and it can be
>> proven that in any consistent (modular invariant) string theory, all
>> anomalies of that sort and of more intricate sorts, cancel.
>
> Are you saying that chirality (or, in plain language, lack of mirror
> symmetry) has something to do with the sum of cubed charges of quarks?
> If so, I'm completely stumped.
>
>> [Moderator's note: I sure hope someone breaks down and tells Khavkine
>> what an anomaly *is*. - jb]
>
> I think I can handle this one myself now. If a classical theory
> respects some symmetry but the quantized version of the same theory
> doesn't, this phenomenon is called an anomaly.
Right. And an anomaly of a local gauge invariance is lethal to a QFT
(unitarity is lost). Thus you have to avoid it.
With massless fermions in even dimensions the left handed particles
can be distinguished from the right handed ones. And in general this
leads to an anomaly in the quantum theory (the so called axial current
is no longer conserved). This anomaly in four dimensions comes from
Feynman diagrams in which a fermion runs in a loop and emits three
gauge bosons (in a peculiar way). As the coupling of the fermion to
each gauge boson is given by its charge the diagram is proportional to
the cube of the charge. The sum over all possible fermions in the loop
is therefore proportional to the sum of the cubes of the charges. If
that vanishes (as in the standard model) the coefficent of the anomaly
vanishes and the theory is well behaved as a quantum theory.
R.X.
well this ensures a lively discussion ;-)
thomas....@hdd.se (Thomas Larsson) wrote in message news:<4b8cc0a6.03022...@posting.google.com>...
> redlu...@wanadoo.fr (R.X.) wrote in message news:<7b5bf479.03030...@posting.google.com>...
>
> > Examples in the present context are Calabi-Yau manifolds, for which
> > string theoretic methods (mirror symmetry) have allowed to do
> > computations mathematicians had dreamed of - like counting spheres and
> > other 2d surfaces inside such CY spaces, etc. Many more examples exist
> > and the list is growing all the time.
>
> Hmm. I know quite a few mathematicians, but I doubt that any of them dream
> about counting spheres inside CY spaces. But ok, *some* mathematicians may
> have such dreams.
>
Well I also do know *some* of them and these are excited, as this kind
of problems is very hard to conquer with traditional methods. Only
very recently were the mathematicians (most influential here
Kontsevich) able to do these computations independent from
(string-inspired) mirror symmetry, reproducing the same results, of
course. These are algebraic geometers, clearly mathematicians of other
kinds may not be excited.
But this was one example, there are plenty of other examples where
string theory and mathematics have a fruitful overlap; eg, knot theory
in relation to Chern Simons field theory, here even a physicist has
gotten the Fields Medal, which is the math analog of the Nobel price.
So indeed *some* mathematicians must have been excited here, isn't it?
> >
> > Supersymmetry can also be spontaneously broken - it is then still
> > there, but non-linearly realized and perhaps not so easy to pinpoint:
> > the superpartners will have different masses in this case.
>
> I was attempting to make the distinction between supersymmetries and
> superalgebras, such as Clifford algebras. The latter are necessary to describe
> electrons, but the existence of electrons is not by itself any sign of susy.
> Similarly, I very much doubt that the positive energy theorem either depends
> on or implies supersymmetry, since that would make it a very dubious theorem
> indeed given current experimental data, or rather lack thereof.
>
> However, from what I read I gather that there is limit to what extent susy
> can be broken. If no sparticles show up below 3 TeV, a region that should be
> covered by LHC during this decade, supersymmetry seems like a pretty dead idea.
>
True. There aren't very strong theoretical reasons for SUSY at low
energies anyway. But that has little to do with string theory.
>
> > > How does the operator that generates susy transformations look like
> > > in this model?
> >
> > It is composed out of the currents in this model. There is even a
> > realization of N=2 supersymmetry with a single free periodic boson,
> > where c=1; all what is required is that the boson has a particular
> > periodicity. Surely SUSY partners do exist in these models. However,
> > SUSY is accidental and doesn't play an important role here, so that's
> > why it isn't often mentioned.
>
> Some aspects of the multi-critical model with c = 1 - 6/m(m+1) can be
> described by the Landau-Ginzburg action
>
> S = int d^2z ( 1/2 (d phi)^2 + phi^2(m-1) ),
>
> see Francesco-Mathieu-Senechal (7.115) & (7.116). In particular, the
> tricritical model corresponds to m = 4, i.e. c = 7/10. Is there any
> way to see why only m = 4 gives supersymmetry in this picture?
>
> And exactly what is the difference between an accendental symmetry and
> a non-accidental symmetry?
The susy appears only in the exact quantum theory at m=4, which can be
inferred from the algebraic properties of "minimal models". It is not
manifest and easy to pinpoint in the LG formulation above, which is
why it may be called "accidental" symmetry in this context.
>From the point of view a supersymmetric LG formulation, which is
manifestly written in superspace, the susy would be considered as
"built in" and not as accidental. The exact quantum theory of these
two models is of course identical.
R.X.
> redlu...@wanadoo.fr (R.X.) wrote in message news:<7b5bf479.03030...@posting.google.com>...
> > k_ig...@lycos.com (Igor Khavkine) wrote in message
> > news:<f1ac2e6e.03012...@posting.google.com>...
> >
> > > Lubos Motl <mo...@feynman.harvard.edu> wrote in message
> > news:<Pine.LNX.4.31.0301192116020.32652-100000@feynman>...
> >
> > > I've already heard in the first few lectures that string theory
> > > gives gravity "for free". And that Einstein's equations just "pop
> > > out" from the dynamics of string theory somewher along the
> > > way. Suffice it to say that "for free" and "pop out" are not very
> > > well defined concepts. Is this a deep consequence of the fundamental
> > > assumptions of string theory, or is it just an artefact of the way
> > > the theory is formulated?
> >
> > Well, "pop out" is indeed not too a bad expression, as GR with all its
> > complexity simply pops out from a simple, even free 2d field theory.
> > That is has to be this way is guaranteed by Ward identities that stem
> > from the 2d conformal invariance, which is one of the basic
> > fundamental ingredients into the construction.
>
> So basically, since the string spectrum contains a spin two particle,
> it must be the graviton. Right?
In a consistent theory, and if massless, yes.
> I have seen a footnote to this effect in the first volume of Weinberg's
> Quantum Theory of Fields. In the same context he shows that any Lorentz
> invariant causal vector field must obey Maxwell's equations (or a
> generalized version involving a mass term). So if the same can be said
> about a rank two tensor field and Einstein's equations, them I'm willing
> to buy it. But I will have to investigate it further.
This is easier seen from a string perspective. As said, it's the 2d
conformal symmetry on the string world-sheet that implies that the
effective theory in space-time is invariant under general coordinate
transformations (and gauge transformations, if we talk about gauge
symmetries). This implies the presence of the relevant gauge fields
(gravitons and gauge bosons, and gravitinos if you have SUSY).
> BTW, I have briefly bumped into a Ward identity while studying QFT. That
> was just the case for covariantly quantizing the photon field. But what
> do Ward identieties look like in general? And what is their significance?
Ward identities simply express certain properties of correlation
functions that reflect underlying symmetries of the theory. Morally
and very loosely speaking, they are conditions for the correlation
functions to be "invariant". For example, in the case at hand, 2d
conformal invariance implies, as a Ward identity, the decoupling of
the longitudinal modes of the graviton from physical scattering
processes (the theory wouldn't be consistent otherwise).
> > The chiral anomaly is a standard field theory concept, and it can be
> > proven that in any consistent (modular invariant) string theory, all
> > anomalies of that sort and of more intricate sorts, cancel.
>
> Are you saying that chirality (or, in plain language, lack of mirror
> symmetry)
> has something to do with the sum of cubed charges of quarks?
Mirror symmetry has nothing to do with that.
But yes, chirality has to do
with the sum of cubed charges of quarks, as the numerical coefficient
in front of the anomaly is given be the sum. If this sum wouldn't
vanish, there would be a non-vanishing chiral anomaly, and the
corresponding gauge symmetries wouldn't be preserved at the quantum
level.
One can prove that in string theories, spectra of chiral fermions come
out always such that all anomalies cancel. This nicely ties together
the mathematical notions of modular functions, elliptic genera and
characteristic classes, and is ultimately also a consequence of 2d
conformal invariance.
> I think I can handle this one myself now. If a classical theory
> respects some symmetry but the quantized version of the same theory
> doesn't, this phenomenon is called an anomaly.
Yupp.
Thomas Larsson
> course. These are algebraic geometers, clearly mathematicians of other
> kinds may not be excited.
Yup, I mainly know algebraists.
>
> But this was one example, there are plenty of other examples where
> string theory and mathematics have a fruitful overlap; eg, knot theory
> in relation to Chern Simons field theory, here even a physicist has
> gotten the Fields Medal, which is the math analog of the Nobel price.
> So indeed *some* mathematicians must have been excited here, isn't it?
You forgot the probably most important mathematical contribution from
string theory, namely conformal field theory. Actually, CFT is little but
the representation theory of Virasoro and other chiral algebras, thinly
veiled in physics language. Nobody has ever denied that string theorists
made very important contributions to this part of mathematics in the 70s
and 80s, nor that this math is incredibly important to real physics,
especially 2D statphys.
However, all this happened almost a decade or more ago. After about 1995
the flood of cool new math coming from string theory has apparently been
reduced to a trickle. At the same time, major discoveries have been made
by the algebraic community, generalizing important structures known from
string theory:
1. The Virasoro algebra has been generalized beyond one dimension, and it
turns out to have a natural and beautiful representation theory. To me
this seems relevant to quantum gravity because the centerless Virasoro
algebra in 4D is the correct symmetry of general relativity. The same
construction also applies to other infinite-dimensional Lie algebras and
superalgebras of growth > 1. This shows in particular that the
superconformal algebra underlying perturbative string theory is not an
exceptional structure (one in the short list of central extensions), but
rather a quite ordinary structure (one in the infinite list of
Virasoro-like extensions).
2. Simple infinite-dimensional Lie superalgebras of polynomial vector
fields have been classified. Of particular interest is the case of
consistent grading (consistent with Grassmann parity). Apart from the
contact superalgebras k(1|m), i.e. the centerless so(m) superconformal
algebras studied by string theorists since the 1970s, there are also four
exceptions, two of which are closely related to the standard model
symmetries.
Now, I know that a lot of people dislike that I keep coming back to these
discoveries. However, the existence of both the multi-dimensional Virasoro
algebra and the exceptional Lie superalgebras are eternal truths. Eternal
truths tend to be annoyingly persevering.
Demian H.J. Cho
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message news:<b354kn$1gvs4e$3...@ID-168578.news.dfncis.de>...
<snip>
> I don't know about this. Are you saying that there is no non-trivial solution
> to a scalar field coupled to ordinary EH gravity? That one needs higher oder
> curvature corrections instead?
All I am saying is if you get one loop effective lagrangian for quantum fields
on a classical curved background you will see Einstein-Hilbert term in leading
order. It can be shown easily on Heat-Kernel expansion. This is the idea behind
induced gravity by Sakharov (S. Adler has an excellent review on Rev. Mod.
Phys.). Gravity need not to be quantized in this sense. GR is an effective way
of describing how background spacetime behave as a action of quantized matter.
> What is the state of the art in trying to close this Pandora's box?
I am really not the person to answer this. But, I will try as much as
I know. Even for pure Kaluza-Klein theory stability of compact
dimensions is a problem. Earlier attemps by many people
(Candelas-Weinberg, Chodos-Myers, Detweiler, Toms- Awada etc.)
essentially failed. The idea was to use quantum effective potential to
stablize the vacua, which ,I believe, is still the technique that is used
by string theorists. Use of simple matters like free scalars and fermions
can stablize vacuum if numbers and the ration of species are correctly
chosen. But, if you consider high enough temparature, which is
more realistic situation for early universe, you can see thermal effect
will destroy the stability.
Case for pure graviton is in worse shape. Computation of one loop
effective action for any gauge field is highly gauge dependent.
There is a formalism to compute gauge independent effective action
called Vilkovisky-DeWitt action, but it is very defficult to compute this
even for simple case like static Minkowski times N-sphere. Also, the
action has non-local terms. The only case I know of is for 5-dimensional
Kaluza-Klein model.
(Now, Demian will talk about somethings which he is not so sure about.)
In SUGRA, due to supersymmetry, you can show
that all the quantum effect can be cancelled out, so if classical solution
is stable, then so is full quantum state (at least perturbatively)-Gibbons
and Nicolai. String dilaton has a flat potential usually. Therefore, it usually
roll down to infinity rather than confined into one point. As long as SUSY
is preserved this is the case for all orders in quantum theory, too.
I guess you can argue that SUSY breaking will introduce the potential
term that will fix moduli and dilatons. All I know about this is that there
were some attempts on this direction, but most of them are highly
model dependent, and they have some problem to produce what they
want (Look for papers by M. Dine and T. Banks.)
Also, I've seen some papers that will include flux tubes and other exotic
monsters from string theory to produce appropriate potential. But, I
shouldn't talk about them since I am nowhere near understanding them.
I will (without consent of author - I hope he will be OK for my action here.)
quote Jacque Distler from his blog on this.
Moduli-Fixing in M-theory
So I thought I'd say some more about the relation between Bobby
Acharya's paper on moduli-fixing in M-theory (which I've blogged
about before) and thework of Kachru et al that I wrote about here.
Recall that the latter proceed in three steps
1. The flux-induced superpotential (in the Type-IIB orientifold
description) ∫ M (F 3-É—H 3)∧? fixes the complex
structure and the string coupling, leaving the Kahler modulus, ɜ
(assume just one), as a flat direction.
2. They then guess at the structure of the the nonperturbative
superpotential for ɜ . With ɜ fixed, we end up with a supersymmetric
solution in 4D anti-de Sitter space.
3. They introduce supersymmetry-breaking in the form of anti-D3
brane(s). This contribution to the potential for ɜ has its coefficient
fine-tuned so as to raise the previous anti-de Sitter minimum to slightly
above zero, producing a non-supersymmetric metastable solution with
a small positive cosmological constant.
M-theory compactified on a manifold X of G 2 -holonomy also
has a flux-induced superpotential W 1 =1 8 <pi> 2∫
X( C 2 +iÉ" )∧G where É" is the G 2 structure. In addition,
Bobby argues that if X is fibered over a 3-manifold Q , with the
generic fiber having an ALE singularity corresponding to the
simply-laced gauge group G , there's a further contribution to
the superpotential that looks like a complex Chern-Simons
term W 2 =1 8 <pi> 2∫ QTr(𝒜∧d𝒜+2 3
𝒜∧𝒜∧𝒜) where 𝒜 =A+iB .
A is the G gauge connection on Q and B is a 1-form in the
adjoint of G (the twisted version of the 3 scalars in the 7D
gauge multiplet).
The critical points of W 2 are flat (complexified) G -connections
on Q and on the space of critical points, we can write W 2 =c 1+ic 2
for some constants c 1 ,2 . The combination W 1 +W 2 lifts all
the flat directions, producing, as above, a supersymmetric
solution in 4D anti-de Sitter space.
Bobby argues that the supergravity computation that led to
this is reliable provided c 2 is large. Unfortunately, this excludes
the familiar candidates for Q , like S 3 or S 3 /ℤ n
(which have "known" heterotic duals). Q must be a
hyperbolic 3-manifold (yuck!).
Anyway, we have achieved points 1 and 2 above with no fudging
whatsoever. This puts us in comparatively better shape to
understand step 3. If we can introduce supersymmetry-breaking
in the M-theory formulation, we might actually be able to say
something reliable about the resulting de Sitter vacuum.
Urs Schreiber
> True. Most people consider D6 branes. 3+1 directions cover the M^4 so
> they are three-branes on the CY (or torus or orbi-/orientifold of a
> torus). But D6-branes lift to KK-monopoles in M-Theory. Those are
> everywhere regular configurations of the metric. So once again, the CY
> with the D6-branes is a G2 manifold from the 11d perspective.
Oh, didn't know that. That's pretty.
What are the problems that remain with such intersecting
p-brane models, as in hep-ph/0212048?
Hm, I should just read that paper, I guess...
[...]
> Sorry for that.
Ah well, I'll get over it. :-)
Robert C. Helling
> What are the problems that remain with such intersecting
> p-brane models, as in hep-ph/0212048?
>
> Hm, I should just read that paper, I guess...
I haven't read the paper either but I think all models of this type on
the market have tadpoles (if they do not preserve susy). So they are
not stable and typically decay wihtin a Planck time. Furthermore, I
assume they have moduli (sizes and shapes of the tori and positions of
the branes) that would correspond to massless scalar fields in the
theory that are of course not observed. Finally, all these arguments
are at best semi-classical so quantum corrections could potentially
destroy a lot of the nice structure. All this said wihtout really
looking at the paper.
John Baez
Robert C. Helling <hel...@atdotde.de> wrote:
>On Fri, 21 Feb 2003 19:23:15 +0000 (UTC), Urs Schreiber
><Urs.Sc...@uni-essen.de> wrote:
>> This reminds me that some day I wanted to ask for a brief list of the
>> different proposed attempts on a realistic string phenomenology. The classic
>> is heterotic string theory on M^4 x CY^6, right?
>Well, probably if you try to fit it with both a GUT and gravity you
>will find that the heterotic string theory is strongly coupled.
Oh?? So what were all those people doing when they advocated
heterotic string theory as a potential "theory of everything"
back in the 80's? Was this flaw unnoticed back then, or just
unadvertised? They sure never told me.
(And why do you say "probably"?)
>So M-Theory on M^4 x CY3 x S^1/Z_2 is a better description.
Hmm. I never know with M-theory - is M-theory on this orbifold
something that people really understand to their heart's content,
or only in some limit?
Is the idea that adjusting the size of the S^2/Z_2 lets you adjust
the gravitational constant or something?
>> How does that compare with M-theory on M^4 x X^7, where X is, ahm, a
>> manifold of G2 holonomy? I forget.
>CY3 x S^1/Z_2 is a singular G2 manifold.
COOL! How do you see this? Is it related to how SU(3) is
the subgroup of G2 fixing a vector in its 7-dimensional irrep?
(G2 is the automorphism group of the octonions; its 7-dimensional
irrep is the imaginary octonions, and the subgroup fixing any imaginary
octonion is isomorphic to SU(3), which acts on the 6d space of
imaginary octonions orthogonal to the chosen one. Since your CY3
is a 6d manifold with SU(3) holonomy, I can't help but think this
is relevant somehow.)
>> Even if all this turns out to have nothing to do with reality, isn't
>> it tantalizing what wealth of models one can obtain this way that
>> all *almost* look like the standard model?
>Depends on your definition of "all almost look like". To get a low
>energy spectrum that is in some sense similar to that of the SM still
>requires a lot of engeneering so it is not clear to me whether this
>has any more meaning than the statement: "Using a field theory
>Lagrangian (or a N=1 superspace Lagrangian) I can obtain a wealth of
>models that *almost* look like the SM".
Ah, honesty is a wonderful thing.
Urs Schreiber
[...]
> I haven't read the paper either but I think all models of this type on
> the market have tadpoles (if they do not preserve susy). So they are
> not stable and typically decay wihtin a Planck time.
I see that this issue is addressed in hep-ph/0212048 as well as in
L. Ibanez, Standard Model Engineering with Intersecting Branes,
hep-ph/0109082 ,
on which the former paper is based. On p.2 of hep-ph/0212048 it says:
>>
In order to cancel anomalies the net number of U(2)_b doublets has to
equal that of anti-doublets, which in these models happens only
because the number of generations equals the number of colours. In
addition, one of the nicest features of these constructions is that
the proton is automatically stable since baryon number (U(1)_a) is a
gauged symmetry.
<<
I don't really understand many of the details involved. For instance
could you perhaps briefly explain this tadpole issue?
> Furthermore, I
> assume they have moduli (sizes and shapes of the tori and positions of
> the branes) that would correspond to massless scalar fields in the
> theory that are of course not observed. Finally, all these arguments
> are at best semi-classical so quantum corrections could potentially
> destroy a lot of the nice structure. All this said wihtout really
> looking at the paper.
I have now skimmed the paper. It's hard for me to extract any specific
problems, though, since the author does not dwell much on potential
problems. The basic impression that remains is that with sufficient
effort a lot can be "engineered" at will, as you said before.
Apart from the fact that some aspects of the standard model come out
rather nicely in these intersecting p-brane models, one peculiar
feature with all of them is that they don't have any gauge
unification. The coupling strenghts can be varied more or less freely
by changing the internal volumes on which the branes wrap. Since I
find it remarkable, I quote the last paragraph from hep-ph/0109082:
>>
The different sizes of the SM gauge couplings in these theories have
to do with the different volumes on which each different D-brane is
wrapping, and one can vary these volumes so that the observed gauge
couplings are reproduced. Thus the logarithmc unification of couplings
will be lost, like in any model with a low string scale. In any event,
we physicists cannot rely on a single piece of data as MSSM gauge
coupling unification is. It could well be that the nice agreement of
gauge coupling unification in the MSSM could be fortuitous.
Recall: The apparent size of the sun agrees with that of the moon with
a good precision, For centuries mankind has given special meaning to
this "size unification" which turns out to be just an accident in the
formation of the solar system. Perhaps we should learn the lesson.
<<
(L. Ibanez in hep-ph/0109082)
I think the author is telling us that we should not want to see
circles where there might be just ellipses. While this may be a wise
insight, I am getting the impression that Demian Cho is right that
none of these phenomenological models is yet very convincing. What do
you think?
R.X.
thomas....@hdd.se (Thomas Larsson) wrote in message news:<4b8cc0a6.03030...@posting.google.com>...
....
> Now, I know that a lot of people dislike that I keep coming back to these
> discoveries. However, the existence of both the multi-dimensional Virasoro
> algebra and the exceptional Lie superalgebras are eternal truths. Eternal
> truths tend to be annoyingly persevering.
In fact string physics has developed much beyond conformal field
theory and related mathematical aspects. That is, CFT is more or
less the same as perturbative string theory, and the recent progress
just was to go beyond perturbative constructions. Non-perturbative
dualities and equivalences, in particular, cannot be easily or at
all be seen in terms of CFT.
As an example, consider the duality between the heterotic string
and type II strings, compactified on suitable manifolds. The
world-sheet theories are different as they can be, they have
different number of supersymmetries and chirality, and the heterotic
string has an E8xE8 current algebra that the type II string is
lacking. Nevertheless the theories are non-perturbatively related,
something which can not be addresssed in CFT.
People had been pondering for years about a deeper significance of
E8xE8, which is in some sense (almost) unique, being related to
self-dual lattices and other things in number theory. However, now
this fact that just one corner of the parameter space has a
perturbative description in terms of E8xE8 current algrebra doesn't
seem any more particularly important.
Urs Schreiber
news:<729f0929.03030...@posting.google.com>...
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message
news:<b354kn$1gvs4e$3...@ID-168578.news.dfncis.de>...
> > I don't know about this. Are you saying that there is no non-trivial
> > solution to a scalar field coupled to ordinary EH gravity? That one needs
> > higher order curvature corrections instead?
> This is the idea behind
> induced gravity by Sakharov (S. Adler has an excellent review on Rev. Mod.
> Phys.). Gravity need not to be quantized in this sense.
Ah, now I get it. I didn't understand that you were talking about
induced gravity.
[...]
> I will (without consent of author - I hope he will be OK for my action here.)
> quote Jacques Distler from his blog on this.
>
> Moduli-Fixing in M-theory
>
> So I thought I'd say some more about the relation between Bobby
> Acharya's paper on moduli-fixing in M-theory (which I've blogged
> about before) and the work of Kachru et al that I wrote about here.
Hey, great! What about posting blogs like that to spr in the first
place?
> Recall that the latter proceed in three steps
>
> 1. The flux-induced superpotential (in the Type-IIB orientifold
> description) ∫ M (F 3-É?H 3)∧? fixes the complex
> structure and the string coupling, leaving the Kahler modulus, É?
> (assume just one), as a flat direction.
> 2. They then guess at the structure of the the nonperturbative
> superpotential for É? . With É? fixed, we end up with a supersymmetric
> solution in 4D anti-de Sitter space.
> 3. They introduce supersymmetry-breaking in the form of anti-D3
> brane(s). This contribution to the potential for É? has its coefficient
> fine-tuned so as to raise the previous anti-de Sitter minimum to slightly
> above zero, producing a non-supersymmetric metastable solution with
> a small positive cosmological constant.
Hm, let me guess what these equations may have looked like before they
were scrambled:
The flux-induced superpotential apparently comes from an RR 3-form
flux F_3 as well as the NSNS 3-form H_3, being equal to
integral over (F_3 - c H_3)
with c = "É?" some further field. In point 2 this field is fixed to a
constant value, apparently. Step 3 then addresses some further
"engineering" in order to break supersymmetry - somehow. Hm, I guess I
need to look at that paper...
With respect to the other things you wrote I am getting the following
impression: Most of what is known about moduli fixing in ordinary
field theory is that a whole bunch of possibilities don't work. Maybe
we haven't tried hard enough yet, maybe there is no solution at all to
that problem. But now, in string theory there is such an immense
increase in degrees of freedom (all these "exotic monsters") that any
general conclusions are even more difficult, but that a wealth of new
possibilities arises. This way string theorists can still hope to find
realistic phenomenological models somewhere in moduli space, while
sceptics can point point at a huge volume of the moduli "landscape"
that corresponds to no known physics.
This reminds me of the recent paper
L. Susskind, The Anthropic Landscape of String Theory, hep-th/0302219
which, among other things, discusses the vast non-uniqueness of
realistic string vacua (if they exist at all). I'll quote some
paragraphs of this paper because they seem relevant to our discussion:
"However the continuum of solutions in the supermoduli-space are all
supersymmetric with exact super-particle degeneracy and vanishing
cosmological constant. Furthermore they all have massless scalar
particles, the moduli themselves. Obviously none of these vacua can
possibly be our world. Therefore the string theorist must believe that
there are other discrete islands lying off the coast of the
supermoduli-space. The hope now is that a single non-supersymmetric
island or at most a small number of islands exist and that
non-supersymmetric physics will prove to be approximately unique. This
view is not inconsistent with present knowledge (indeed it is possible
that there are no such islands) but I find it completely implausible.
It is much more likely that the number of discrete vacua is
astronomical, measured not in the millions or billions but in googles
or googleplexes.
"[...] The incredible smallness and apparent fine tuning of the
cosmological constant makes it absurdly improbable to find a vacuum in
the observed range unless there are an enormous number of solutions
with almost every possible value of lambda. It seems to me inevitable
that if we find one such vacuum we will find a huge number of them. I
will from now on call the space of all such string theory vacua the
/landscape/.
"[...] Thus there will be configurations of string theory which are
not globally described by a single vacum but instead consist of many
domains seperated by domain walls. Accordingly, the landscape in field
space is reflected in a complicated terrain in real space."
He then goes on to explain how, for instance, following an idea by
Bousso and Polchinski, a small non-zero cosmological constant could
arise not by fine-tuning but because the volume of moduli space that
corresponds to such a value can be argued to be huge. hep-th/0301240
is cited at this point. Is this the work that Distler's blog referred
to?
Susskind then further says:
"In other words there is no evidence in string theory that a hoped
for but unknown mechanism will automatically force the cosmological
constant to zero. It seems very likely that all of the
non-supersymmetric vacua have finite lambda.
"The vacua in [hep-th/0301240, see above] are not at all simple. They
are jury-rigged, Rube Goldberg contraptions that could hardly have
fundamental significance. But in an anthropic theory simplicity and
elegance are not considerations. The only criteria for choosing a
vacuum is utility, i.e. does it have the necessary elements such as
galaxy formation and complex chemistry that are needed for life. That
together with a cosmology that guarantees a high probability that at
least one large patch of space will form with that vacuum structure is
all we need."
(Heh, "jury-rigged, Rube Goldberg contraptions"! I looked up
"contraption" and I can guess what "jury-rigged" means (sort of like
"tailor made"?), but who is Rube Goldberg?)
I am going to have a look a Kachur et al's paper now. Not that I
expect to understand many of the details. But maybe I learn more about
what kind of world-views are currently considered by string theorists.
[Moderator's note: Rube Goldberg is a cartoonist famous for
depicting hilariously elaborate mechanisms. See
http://www.rube-goldberg.com/html/gallery.htm
for a nice sample. - jb]
Robert C. Helling
> Oh?? So what were all those people doing when they advocated
> heterotic string theory as a potential "theory of everything"
> back in the 80's? Was this flaw unnoticed back then, or just
> unadvertised? They sure never told me.
Don't blame me, I was in elementary school in the 80's. If you just
use the tree level relations then you find if you try to match both
your GUT and gravity to heterotic string theory compactified on a CY
(N=1 susy assumed) that it only works if the coupling of the
stringtheory is >1. Not much but it makes you suspicious about the
perturbative expansion that you would like to use.
From the M/(R^4 x CY x S^1/Z_2) perspective this means that the radius
of the eleventh dimension is about an order of magnitude smaller than
the radius of the CY (sixth root of the volume). So there is no ten
dimensional regime at all. When you crank up your energy you first see
a fifth dimension opening up and then, a little later, six more
dimensions become visible. So the heterotic description of a ten
dimensional theory compactified on a six space is not exactly valid.
But note that it's only an order of magnitude. It could have been much
worse. This was what made people in the old days think that maybe they
could fudge things in a way that you can get a matching. One obvious
thing to play with is the geometry of the CY. It's not a sphere that
only has one radius, it could be degenerate and you might be able to
squeeze a factor of 10 from it.
> (And why do you say "probably"?)
"Probably" meant that I am not an expert on this stuff. I haven't done
the calculations myself. I only repeat what I read and what I hear
people saying.
>>CY3 x S^1/Z_2 is a singular G2 manifold.
> COOL! How do you see this?
Again, 'probably' applies here. The easiest way to argue would be that
M/CY x S^1 would be type IIA/CY and that preserves two supersymmetries
(i.e. there are two Killing spinors). Modding out by Z/2 kills one
combination of them so M/ CYxS1/Z2 has one Killing spinor. General
reasoning says if a sevenspace X (as here CY x S1/Z2) has exactly one
Killing spinor it is of G2 holonomy. From this spinor you could
construct a harmonic threeform via
phi^abc = psi_alpha Gamma^(abc) _(alpha beta) psi_beta
that has the octonionic structure constants as components in an ON
basis. Furthermore you can construc the metric from it. I am sure you
can express it in terms of J and Omega of the CY and dz of S^1 but I
don't know the exact expression.
> Is it related to how SU(3) is
> the subgroup of G2 fixing a vector in its 7-dimensional irrep?
I don't know. I don't even have my Slansky at hand. Are you telling me
somehow 7 = 3 + 3-bar + 1 as SU3 reps of a G2 rep?
> (G2 is the automorphism group of the octonions; its 7-dimensional
> irrep is the imaginary octonions, and the subgroup fixing any imaginary
> octonion is isomorphic to SU(3), which acts on the 6d space of
> imaginary octonions orthogonal to the chosen one. Since your CY3
> is a 6d manifold with SU(3) holonomy, I can't help but think this
> is relevant somehow.)
Maybe. Sounds like this is related to the 11th direction being
fixed. I would love to see this worked out.
Robert
PS: Sorry for this late follow-up. I had some computer problems.
Urs Schreiber
> Non-perturbative
> dualities and equivalences, in particular, cannot be easily or at
> all be seen in terms of CFT.
A related question: Does mirror symmetry count as a non-perturbative duality?
Robert C. Helling
> "Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote in message news:<slrnb6bh0s....@ariel.physik.hu-berlin.de>...
>>>
> In order to cancel anomalies the net number of U(2)_b doublets has to
> equal that of anti-doublets, which in these models happens only
> because the number of generations equals the number of colours. In
> addition, one of the nicest features of these constructions is that
> the proton is automatically stable since baryon number (U(1)_a) is a
> gauged symmetry.
><<
>
> I don't really understand many of the details involved. For instance
> could you perhaps briefly explain this tadpole issue?
Generically, these models have problems with the equations of motion
for the closed string fields (i.e. gravity and sep. the dilaton). In
the above paragraph the author more or less says that they managed to
solve the equations of motion for the open string fields which is
already quite good but not the full thing. (I should stress that I
wouldn't like to sound too critical to this approach, I am just saying
this is not yet the standard model and nothing else but nobody has
gotten any much closer).
> I have now skimmed the paper. It's hard for me to extract any specific
> problems, though, since the author does not dwell much on potential
> problems.
He doesn't reiterate the problems because they are shared by all the
models of this type.
> The coupling strenghts can be varied more or less freely
> by changing the internal volumes on which the branes wrap.
You can view this as an advantage or a disadvantage: You can tune it
to whatever you like but it doesn't make any predictions. Furthermore
if it can be adjusted (it is a modulus) it shows up as a massless
scalar field in the low energy theory (wherever you find alpha, the
fine structure constant in your theory you should read it as alpha(x)
and alpha is a massless field). Then you have to explain why it is so
weekly coupled that we don't observe it in experiments so far. Or you
wave your hands, mumble the magic words 'non-perturbative effects' and
claim that the value of the coupling will be fixed by quantum
corrections and the field becomes very massive around its correct value.
> Since I
> find it remarkable, I quote the last paragraph from hep-ph/0109082:
> The different sizes of the SM gauge couplings in these theories have
> to do with the different volumes on which each different D-brane is
> wrapping, and one can vary these volumes so that the observed gauge
> couplings are reproduced. Thus the logarithmc unification of couplings
> will be lost, like in any model with a low string scale. In any event,
> we physicists cannot rely on a single piece of data as MSSM gauge
> coupling unification is. It could well be that the nice agreement of
> gauge coupling unification in the MSSM could be fortuitous.
>
> Recall: The apparent size of the sun agrees with that of the moon with
> a good precision, For centuries mankind has given special meaning to
> this "size unification" which turns out to be just an accident in the
> formation of the solar system. Perhaps we should learn the lesson.
><<
> (L. Ibanez in hep-ph/0109082)
>
> I think the author is telling us that we should not want to see
> circles where there might be just ellipses.
I think he is telling us that coupling unification is a coincidence
and has no physical meaning (as it does not happen in his model). Of
course, so far this is a matter of belief but personally I find it
very sugesstive that there might be something going on at some higher
energy scale.
Aaron Bergman
wrote:
You can see mirror symmetry in the worldsheet theory. It's really
the statement that certain N=2 SCFTs can correspond to different
CYs depending on how you label some of the objects therein.
Aaron
R.X.
From the world-sheet point of view: yes, as it involves world-sheet
instanton effects.
From the space-time point of view, no, as it does not involve the
string coupling and thus holds already at string tree level. But
it is supposed to hold also in the full quantum theory, at all
loops as well as at the non-perturbative level, ie, for branes. So
one might say it is a perturbative duality that is realized also
in the non-perturbative sectors.
(Geometrically it can be interpreted as perturbative T-duality
acting on the fibers of a 3-torus fibration describing the Calabi-Yau,
see hep-th/9606040).
Thomas Larsson
> > R.X. <redlu...@wanadoo.fr> wrote in message news:7b5bf479.03030...@posting.google.com...
> >
> >
> > You forgot the probably most important mathematical contribution from
> > string theory, namely conformal field theory....
>
>
> ....
> > Now, I know that a lot of people dislike that I keep coming back to these
> > discoveries. However, the existence of both the multi-dimensional Virasoro
> > algebra and the exceptional Lie superalgebras are eternal truths. Eternal
> > truths tend to be annoyingly persevering.
>
>
> In fact string physics has developed much beyond conformal field
> theory and related mathematical aspects. That is, CFT is more or
> less the same as perturbative string theory, and the recent progress
> just was to go beyond perturbative constructions. Non-perturbative
> dualities and equivalences, in particular, cannot be easily or at
> all be seen in terms of CFT.
I'm sorry, but you don't seem to understand what I talked about. CFT is
about chiral algebras, i.e. infinite-dimensional Lie algebras acting on
spaces with *one* bosonic dimension. Thus the base manifold has complex
dimension 1; the dimension of the target manifold is not really important.
There are several ways to generalize this - do things globally on a Riemann
surface, go to the group level, etc. - but locally things boil down to chiral
algebras. I don't see dualities changing that.
What I talk about is generalizing the Virasoro algebras to several
complex dimensions. So this is something new (and essentially harder) even
locally and on the algebra level. As is well known, Virasoro is described
by the exact sequence
0 --> C --> Vir --> vect(1) --> 0,
where vect(1) is the algebra of vector fields in 1D (on the circle) and C
is the trivial module. Its n-dimensional analogue can similarly be
described as
0 --> M --> Vir(n) --> vect(n) --> 0,
where M is the module of closed (n-1)-forms. In particular, when n=1 a
closed zero-form is a constant function is the trivial module, so the
Virasoro extension is central when n=1 but not otherwise.
For a description in a Fourier basis, which is more common in the physics
literature, please consult section 2 of
http://www.arxiv.org/abs/math-ph/0103013 and references therein.
Although the Virasoro-like extension of the multi-dimensional diffeomorphism
algebra has not appeared in physics, the analogous Kac-Moody-like extension
of the multi-dimensional current algebra has been considered by string
theorists:
Authors: A. Losev, G. Moore, N. Nekrasov, S. Shatashvili
http://www.arxiv.org/abs/hep-th/9509151
Four-Dimensional Avatars of Two-Dimensional RCFT
http://www.arxiv.org/abs/hep-th/9511185
Central Extensions of Gauge Groups Revisited
In the first paper, this algebra appears as the symmetry of a 4D
generalization of the WZW model. Ironically, these authors lists
nonperturbative string theory as one potentially important application.
So maybe I'm handing you the tools to find just that.
R.X.
thomas....@hdd.se (Thomas Larsson) wrote in message news:<4b8cc0a6.0303...@posting.google.com>...
> redlu...@wanadoo.fr (R.X.) wrote in message news:<7b5bf479.03030...@posting.google.com>...
> > thomas....@hdd.se (Thomas Larsson) wrote in message news:<4b8cc0a6.03030...@posting.google.com>...
> > > R.X. <redlu...@wanadoo.fr> wrote in message news:7b5bf479.03030...@posting.google.com...
> > >
> >
> > theory and related mathematical aspects. That is, CFT is more or
> > less the same as perturbative string theory, and the recent progress
> > just was to go beyond perturbative constructions. Non-perturbative
> > dualities and equivalences, in particular, cannot be easily or at
> > all be seen in terms of CFT.
>
> I'm sorry, but you don't seem to understand what I talked about. CFT is
> about chiral algebras, i.e. infinite-dimensional Lie algebras acting on
> spaces with *one* bosonic dimension. Thus the base manifold has complex
> dimension 1; the dimension of the target manifold is not really important.
> There are several ways to generalize this - do things globally on a Riemann
> surface, go to the group level, etc. - but locally things boil down to chiral
> algebras. I don't see dualities changing that.
I do. They do. Chiral algebras are defined only in terms of rational
conformal field theory, an algebraic concept that is tied to isolated
points in the parameter spaces, typically associated with extra
symmetries. However, the progess in dualities was just possible
because one was considering continuous families of theories, for
which there isn't any notion of chiral algebras.
Considerung continuous families is in a sense opposite to rational
CFT (roughly it is geometry versus algebra), and the recent progress
in non-perturbative physics came from the study of geometrical
rather than of algebraic structures. (Eg in Seiberg-Witten theory,
it was the global structure of the parameter space that was the
key for solving it.)
>
> What I talk about is generalizing the Virasoro algebras to several
> complex dimensions.
....snip
I am familiar with multi-loop algebras and alike, but fail to see
their broad usefulness. All of this is algebraic stuff and typically
useful for solving certain discrete models (lots of applications
in stat mech theory and integrable systems, etc), but not for
gravity or string theory. There are indeed many beautiful algebraic
constructions, usually tied to some group theory (to name a few:
the Leech lattice and the monster group, octonions, Freudenthal
triples and other exceptional algebraic structures like E8, etc).
But none of them ever played, so far, any significant role in
attempted "real world theories".
People were dreaming about that the world would be described by a
mathematically distinguished or even unique structure, but to date
there is no sign of such. One may ponder to believe in the opposite, see eg
http://arxiv.org/pdf/hep-th/0211048 for an inspiration.
John Baez
In article <slrnb6op2d....@ariel.physik.hu-berlin.de>,
Robert C. Helling <hel...@atdotde.de> almost wrote:
>On Thu, 6 Mar 2003 00:20:18 +0000 (UTC), John Baez <ba...@galaxy.ucr.edu> wrote:
>> Oh?? So what were all those people doing when they advocated
>> heterotic string theory as a potential "theory of everything"
>> back in the 80's? Was this flaw unnoticed back then, or just
>> unadvertised? They sure never told me.
>Don't blame me, I was in elementary school in the 80's.
... busily learning quantum field theory! Don't worry, I wasn't
blaming you. I was a postdoc in the late 80's, and back then
everybody seemed to be trumpeting the virtues of heterotic
string theory. Hawking was saying we might have a Theory of
Everything by the turn of the century. Ah, we were younger then!
>If you just
>use the tree level relations then you find if you try to match both
>your GUT and gravity to heterotic string theory compactified on a CY
>(N=1 susy assumed) that it only works if the coupling of the
>stringtheory is >1. Not much but it makes you suspicious about the
>perturbative expansion that you would like to use.
Hmm. Since these perturbative expansions never really converge
I'm suspicious of all of them. But I suppose most physicists
are happier when the coupling constant is small and the series
starts out acting like it's gonna converge... even though it probably
diverges in the long run.
>From the M/(R^4 x CY x S^1/Z_2) perspective this means that the radius
>of the eleventh dimension is about an order of magnitude smaller than
>the radius of the CY (sixth root of the volume). So there is no ten
>dimensional regime at all. When you crank up your energy you first see
>a fifth dimension opening up and then, a little later, six more
>dimensions become visible. So the heterotic description of a ten
>dimensional theory compactified on a six space is not exactly valid.
Hmm, let's see why this is the 11d way of saying "the coupling of the
string theory is >1". I guess you're just saying this coupling constant
is the radius of the Calabi-Yau divided by the radius of the S^1/Z_2.
Okay, that sounds vaguely familiar.
>But note that it's only an order of magnitude. It could have been much
>worse. This was what made people in the old days think that maybe they
>could fudge things in a way that you can get a matching.
Yes, the trumpeting of the heterotic string did seem to be based on lots
of optimism and fudging... I don't think there was ever a precise
account of how nature was supposed to break E8 x E8 down to the
Standard Model gauge group, much less break supersymmetry... I just
never heard about this particular piece of fudgery.
>> Is it related to how SU(3) is
>> the subgroup of G2 fixing a vector in its 7-dimensional irrep?
>I don't know. I don't even have my Slansky at hand. Are you telling me
>somehow 7 = 3 + 3* + 1 as SU3 reps of a G2 rep?
Yes. It's very pretty. You can think of the octonions as
linear combinations of these 8 guys:
1
/|\
/ | \
/ | \ if you're viewing this correctly
/ | \ it looks like a cube!
e3 e6 e5
|\ / \ /|
| \ / \ / |
| \ / |
| / \ / \ |
e4 \ / e1
\ e2 /
\ | /
\ | /
\ | /
\|/
e7
namely the identity and 7 square roots of minus one, e1 through e7.
Now, hold the cube between your thumb and forefinger, with your finger
on the top and thumb on the bottom, and rotate it a third of a turn!
This doubles all the indices on e1,...,e7 (modulo seven), and it turns
out to act as a symmetry of the octonion multiplication table.
So, we get a Z_3 symmetry of the octonions.
The symmetry group of the octonions is called G2. So, the picture above
gives a way to stick Z_3 inside G2. All the symmetries in this
Z_3 have a special extra property: they preserve a chosen imaginary
octonion, namely e7.
In fact, all of SO(3) acts as symmetries of the octonions that
preserve a chosen imaginary octonion. Z_3 sits in this SO(3)
as the transformations that cyclically permute the x,y,z axes.
The different "rows" of the cube correspond to the different irreps
of this SO(3):
1 1
/|\
/ | \
/ | \
/ | \
3 e3 e6 e5
|\ / \ /|
| \ / \ / |
| \ / |
| / \ / \ |
3 e4 \ / e1
\ e2 /
\ | /
\ | /
\ | /
\|/
1 e7
So, as a rep of SO(3) the octonions decompose as 8 = 1 + 3 + 3 + 1.
In particular, the imaginary octonions decompose as 7 = 3 + 3 + 1.
But it gets better. In fact, the symmetries of the octonions that
preserve a chosen imaginary octonion form a group bigger than SO(3):
they form the group SU(3). As a rep of SU(3), the octonions decompose
as 8 = 1 + 3 + 1. The imaginary octonions decompose as 7 = 3 + 1.
Huh? Have I forgotten how to add?
No: the octonions are a real vector space, but the "3" rep of SU(3)
is complex... so it counts for 6 real dimensions. So we get
1d real rep 1
of SU(3) /|\
/ | \
......................................................................
/ | \
e3 e6 e5
|\ / \ /|
3d complex rep | \ / \ / |
of SU(3) | \ / |
| / \ / \ |
e4 \ / e1
\ e2 /
\ | /
......................................................................
\ | /
\|/
1d real rep e7
of SU(3)
You probably like complex reps better. If we complexify the
octonions we get an 8d complex algebra called the bioctonions.
Under SU(3), this decomposes as 8 = 1 + 3 + 3* + 1 where all reps
are now complex. This is because when we complexify an already
complex rep we get that rep plus its dual.
Finally, if we complexify the imaginary octonions we get the 7d
complex rep of G2. As a rep of G2 this decomposes as 7 = 3 + 3* + 1.
There's also a cute scenario where this 8 = 1 + 3 + 3* + 1 decomposition
of the bioctonions corresponds to lepton+quark+antiquark+antilepton
in one generation of the Standard Model. It's related to Mark Hopkin's
story about red, green, blue and "white" quarks, where the "white"
quarks, which don't feel the color force, are really leptons. But
that's another story for another day.
Robert C. Helling
<ba...@galaxy.ucr.edu> wrote:
> In article <slrnb6op2d....@ariel.physik.hu-berlin.de>,
> Robert C. Helling <hel...@atdotde.de> almost wrote:
>>On Thu, 6 Mar 2003 00:20:18 +0000 (UTC), John Baez
>><ba...@galaxy.ucr.edu> wrote:
> I was a postdoc in the late 80's, and back then
> everybody seemed to be trumpeting the virtues of heterotic
> string theory. Hawking was saying we might have a Theory of
> Everything by the turn of the century. Ah, we were younger then!
I thought Hawking's optimism counted on supergravity rather than
string-theory.
>
> 1
> /|\
> / | \
> / | \ if you're viewing this correctly
> / | \ it looks like a cube!
> e3 e6 e5
> |\ / \ /|
> | \ / \ / |
> | \ / |
> | / \ / \ |
> e4 \ / e1
> \ e2 /
> \ | /
> \ | /
> \ | /
> \|/
> e7
>
> namely the identity and 7 square roots of minus one, e1 through e7.
>
> Now, hold the cube between your thumb and forefinger, with your finger
> on the top and thumb on the bottom, and rotate it a third of a turn!
> This doubles all the indices on e1,...,e7 (modulo seven), and it turns
> out to act as a symmetry of the octonion multiplication table.
How do I see this? What is the relation of this cube to the magic
triangle that encodes the multiplication? Oh, let me guess: The magic
triangle somehow arises from the projective plane. [some googling for
'octonions projective plane...] Oh, looking at
http://math.ucr.edu/home/baez/Octonions/node4.html
and assuming your notation is consistent, I find that the triangle
arises from the cube by looking at it along the e7-1 diagonal. Great.
> So, we get a Z_3 symmetry of the octonions.
>
> The symmetry group of the octonions is called G2. So, the picture above
> gives a way to stick Z_3 inside G2. All the symmetries in this
> Z_3 have a special extra property: they preserve a chosen imaginary
> octonion, namely e7.
Ok.
Good, now I understand how to decompose the 7 of G2 as SU(3)
irreps. But how does this precisely translate into constant spinors
and constant forms? Unfortunately, a combination of tiredness and a
cold prevent me from clear thinking at this moment, but let me list
the ingredients: G2 and SU(3) are the holonomy groups of the seven and
the six-manifold. The 8 dimensional spinor rep of SO(7) (the holonomy
group of a generic 7 manifold) decomposes as 7+1 under G2 and the 1 is
the covariantly constant spinor. For 6 manifolds this works as SO(6)
is SU(4) and the 4 decomposes as 3+1 under SU(3).
If I have a constant spinor psi I can turn it into constant forms as
f_{i1,...,ip} = psi-bar Gamma_{i1,...,ip} psi
and f = f_{i1,...,ip} dx^i1 /\ ... /\ dx^ip.
But for some reasons that I don't remember (probably (anti)symmetry of
Gamma matrices) in the 7 dim case only a threeform is non-trivial and
in the 6 dim case a 2 and a 3 form. For the Calabi-Yau those are the
complex structure (in some coordinates J = sum_i dz^i /\ dz-bar^i) and
the holomorphic 'volume' Omega = dz^1 /\ dz^2 /\ dz^3. The threeform
has the octonionic structure constants as components in an orthnormal
basis.
On the circle that we need to connect the CY and the G2-manifold we
have dx, the volume form. But that is odd under the Z_2 that we are
going to mod out.
Hmm. Let me speculate a bit. If we let the Z_2 also act on the CY,
namely by complex conjugation (but I have no physical justification
for that in the Horava Witten context), then
dx /\ J + Omega + Omega-bar
is even under the Z_2 and is obviously closed. Furthermore its Hodge
dual is something like
J /\ J + dx /\ (Omega - Omega-bar)
which is also closed so the above 3-form is harmonic and non-zero. I
would think this is a natural candidate for what we are looking
for. But I am too lazy to check whether the components have anything
to do with octoniaos.
I will follow up on this later.
Robert
John Baez
R.X. <redlu...@wanadoo.fr> wrote:
>I am familiar with multi-loop algebras and alike, but fail to see
>their broad usefulness. All of this is algebraic stuff and typically
>useful for solving certain discrete models (lots of applications
>in stat mech theory and integrable systems, etc), but not for
>gravity or string theory. There are indeed many beautiful algebraic
>constructions, usually tied to some group theory (to name a few:
>the Leech lattice and the monster group, octonions, Freudenthal
>triples and other exceptional algebraic structures like E8, etc).
>But none of them ever played, so far, any significant role in
>attempted "real world theories".
I'm not sure if you consider superstring theory or M-theory an
"attempted "real world theory"". I could argue it either way.
But either way, the existence of the superstring Lagrangian
in 10 dimensions arises from relations between the octonions and
Lorentzian geometry in this dimension, while M-theory is
intimately connected with 11d supergravity, which has something
like a hidden E8 symmetry (see hep-th/0006034). So in fact,
octonions and the exceptional Lie groups spawned by them permeate
quite a bit of superstring theory and M-theory. Indeed, both
the E8 x E8 heterotic string and the compactifications of M-theory
using a 7-manifold whose holonomy group is G2 (the automorphism
group of the octonions) have often been discussed as starting-points
for realistic physical theories. However, there is a long way from
these lofty heights to realistic particle phenomenology!
(The Monster group and Leech lattice appear to be much less relevant
to physics, since they're connected to bosonic string theory rather
than superstring theory. Freudenthal triples are mainly all about
the 56-dimensional rep of E7, which plays a certain role in 11d
supergravity.)
If anyone wants to help me understand this business, this reference
seems like an okay place to start, if only to find other references:
Luis J. Boya
Octonions and M-theory
http://www.arXiv.org/abs/hep-th/0301037
We explain how structures related to octonions are ubiquitous in
M-theory. All the exceptional Lie groups, and the projective Cayley
line and plane appear in M-theory. Exceptional G2-holonomy manifolds
show up as compactifying spaces, and are related to the M2 Brane and
3-form. We review this evidence, which comes from the initial 11-dim
structures. Relations between these objects are stressed, when extant
and understood. We argue for the necessity of a better understanding
of the role of the octonions themselves (in particular non-associativity)
in M-theory.
R.X.
news:<b516dq$t0c$1...@glue.ucr.edu>...
> In article <7b5bf479.03031...@posting.google.com>,
> R.X. <redlu...@wanadoo.fr> wrote:
> >But none of them ever played, so far, any significant role in
> >attempted "real world theories".
>
> I'm not sure if you consider superstring theory or M-theory an
> "attempted "real world theory"".
Yupp
> I could argue it either way.
> But either way, the existence of the superstring Lagrangian
> in 10 dimensions arises from relations between the octonions and
> Lorentzian geometry in this dimension, while M-theory is
> intimately connected with 11d supergravity, which has something
> like a hidden E8 symmetry (see hep-th/0006034). So in fact,
> octonions and the exceptional Lie groups spawned by them permeate
> quite a bit of superstring theory and M-theory.
Well, first of all it is clear that such beautiful structures do
appear here and there in string theory (like most other concepts
which make sense ;-).
But I don't see a particular significance of them. Those algebraic
constructions typically appear in compactifications with many
supersymmetries. However, the kind of compactification is a matter of
choice, eg if one wants to preserve so-and-so many supersymmetries
in so-and-so many dimensions, then one has to choose manifolds with
certain properties (appropriate holonomy groups, that is).
In particular, exceptional groups arise as by-product in
totoidal compactifications with extended supersymmetries. Those pose strong
constraints and determine eg the geometry of the target space of
the scalar fields. Clearly, the more supersymmetries a theory has
(generic in higher dimensions), the more restricted it is and
the more "exceptional" its structure becomes (essentially, through
the appearence of spinor reps; I think that's all there is to it).
On the other hand, in theories with less SUSY, eg N=1 in d=4, most
of this structure evaporates; I am not aware of any role of octonions
and alike in such theories.
> (The Monster group and Leech lattice appear to be much less relevant
> to physics, since they're connected to bosonic string theory rather
> than superstring theory.
Yes, and they arise only if one compactifies the bosonic string to
2 dimensions, on a multi-torus given by the Leech lattice. But
there are plenty of other possibilities, so it's hardly an intrinsic
and significant feature of the bosonic string.
There is in fact a supersymmetric variant of this, which (I think) called
the little monster superalgebra, which is more tied to 10d.
> Freudenthal triples are mainly all about
> the 56-dimensional rep of E7, which plays a certain role in 11d
> supergravity.
Again, only in certain compactifications with extended supersymmetries.
> Indeed, both
> the E8 x E8 heterotic string and the compactifications of M-theory
> using a 7-manifold whose holonomy group is G2 (the automorphism
> group of the octonions) have often been discussed as starting-points
> for realistic physical theories.
Again, this only for a certain class of N=1 compactifications. For
others, especially for the perturbative heterotic string, 3-manifolds
with SU(3) holonomy are relevant instead - no significance of
octonions here.
There seems to be no such thing as "the" underlying theory, rather
than a multitude of perturbative descriptions in certain limits of
the parameter space. One of them (heterotic) may have an E8xE8
structure, another one an SO(32) structure, still another one (type
II) no structure of that sort. Indeed, the perturbative heterotic
string exists besides in the E8xE8 also in the SO(32) variety,
which has nothing to do with exceptional groups. The important
feature, as you know, is the self-duality of the weight lattice,
and not an exceptional structure.
As for the review, Octonions and M-theory,
http://www.arXiv.org/abs/hep-th/0301037: in the first part it
summarizes the well-known role of divison algebras in supersymmetric
situations, which as said above, originate from the spinor reps due
to supersymmetry. I doubt that the division algebras play any deeper
or more significant role other than as a kinematical consequence
of extended supersymmetry. As for the second part, it brings up some aspects
of manifolds of G2 holonomy, which is per se interesting and new
territory apparently also for mathematicians. But physically, they
are not more distinguished than manifolds of SU(3) holonomy, or
SU(4), or Spin(7) for that matter.
Punchline: while often mathematically distinguished and beautiful,
algebraic -and specifically exceptional- structures do not seem
particularly important for physics in 4d with few supersymmetries,
as far as I know. But that's no reason not to study them!
/bin/bash: line 1: : command not found
Urs Schreiber
> thomas....@hdd.se (Thomas Larsson) wrote in message
news:<4b8cc0a6.0303...@posting.google.com>...
> Chiral algebras are defined only in terms of rational
> conformal field theory, an algebraic concept that is tied to isolated
> points in the parameter spaces, typically associated with extra
> symmetries. However, the progess in dualities was just possible
> because one was considering continuous families of theories, for
> which there isn't any notion of chiral algebras.
How is this related to what Polchinski writes on p. 256 of the second
volume:
"It has been conjectured that [...] any CFT can be arbitrarily well
approximated by a rational theory."? Is that still true?
> I am familiar with multi-loop algebras and alike, but fail to see
> their broad usefulness.
Could you say, briefly, what a multi-loop algebra is? Probably one
where the fields propagate not on a circle but on an n-torus? What
about 2-loop algebras? Any relation to membrane theory?
> All of this is algebraic stuff and typically
> useful for solving certain discrete models (lots of applications
> in stat mech theory and integrable systems, etc), but not for
> gravity or string theory.
As far as I understand, Thomas Larsson proposes multi-dimensional
Virasoro algebras to replace the constraint algebra of canonically
quantized gravity, somehow. (If that's not the case, he schould please
correct me.)
> People were dreaming about that the world would be described by a
> mathematically distinguished or even unique structure, but to date
> there is no sign of such. One may ponder to believe in the opposite,
> see eg http://arxiv.org/pdf/hep-th/0211048 for an inspiration.
I found the radical non-uniqueness paradigm particularly emphasized in
the recent hep-th/0302219 by Susskind. In another thread John Baez
thinks a googleplex of possible sets of low-energy "laws of nature" is
horrible. Looks like the old conflict between the need to see patterns
in the irregular to make progress and the ability to generalize the
known patterns to less regular phenomena. Like switching from
geocentric circles to helio-focused ellipses.
Urs Schreiber
"Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote in message news:<slrnb76u98....@ariel.physik.hu-berlin.de>...
> I thought Hawking's optimism counted on supergravity rather than
> string-theory.
How much room is there to count on supergravity but not on
superstrings as a UV regulator at the same time?
<snip>
> If I have a constant spinor psi I can turn it into constant forms as
>
> f_{i1,...,ip} = psi-bar Gamma_{i1,...,ip} psi
>
> and f = f_{i1,...,ip} dx^i1 /\ ... /\ dx^ip.
>
> But for some reasons that I don't remember (probably (anti)symmetry of
> Gamma matrices) in the 7 dim case only a threeform is non-trivial and
> in the 6 dim case a 2 and a 3 form.
About what kind of manifolds are you talking here? General 6- and
7-dim ones? And there is a theorem that says that for every
covariantly constant spinor psi the p-form psi-bar Gamma_{i1,...,ip}
psi is non-trivial only for certain values of p?
> For the Calabi-Yau those are the
> complex structure (in some coordinates J = sum_i dz^i /\ dz-bar^i) and
Is there some general relation between complex structures and
covariantly constant spinors? For instance can every complex structure
J be written as
psi-bar Clifford-2-vector psi for psi a cov. constant spinor?
> the holomorphic 'volume' Omega = dz^1 /\ dz^2 /\ dz^3. The threeform
> has the octonionic structure constants as components in an orthnormal
> basis.
>
> On the circle that we need to connect the CY and the G2-manifold we
> have dx, the volume form. But that is odd under the Z_2 that we are
> going to mod out.
>
> Hmm. Let me speculate a bit. If we let the Z_2 also act on the CY,
> namely by complex conjugation (but I have no physical justification
> for that in the Horava Witten context), then
>
> dx /\ J + Omega + Omega-bar
>
> is even under the Z_2 and is obviously closed. Furthermore its Hodge
> dual is something like
>
> J /\ J + dx /\ (Omega - Omega-bar)
How did you get this? I've been staring at it, but I don't see how it
works. (Probably I should study some CY geometry.)
> which is also closed so the above 3-form is harmonic and non-zero. I
> would think this is a natural candidate for what we are looking
> for.
Could you remind me: What _are_ you looking for?
Sorry for so many stupid questions! :-)
John Baez
Robert C. Helling <hel...@atdotde.de> wrote:
>On Sat, 15 Mar 2003 09:15:23 +0000 (UTC), John Baez
><ba...@galaxy.ucr.edu> wrote:
>> I was a postdoc in the late 80's, and back then
>> everybody seemed to be trumpeting the virtues of heterotic
>> string theory. Hawking was saying we might have a Theory of
>> Everything by the turn of the century. Ah, we were younger then!
>I thought Hawking's optimism counted on supergravity rather than
>string-theory.
Yes, before people guessed it was nonrenormalizable. But as
Hawking joked at Hartle's sixtieth birthday party, 11d supergravity
is back in fashion under the new name of M-theory. :-)
Btw, when I say "guessed", it's because as far as I can tell,
the question has taken quite a while to settle in a definitive
way. For example, in his 1999 review article:
Nonrenormalizability of (Last Hope) D=11 Supergravity, with a
Terse Survey of Divergences in Quantum Gravities,
http://www.arXiv.org/abs/hep-th/9905017
Deser writes in the abstract of "the new result that D=11 supergravity
is 2-loop nonrenormalizable" - but in the body of the paper he's
more cautious, saying "the strong odds are against even this maximal
theory". A more recent review by Zvi Bern:
Perturbative Quantum Gravity and its Relation to Gauge Theory,
http://www.livingreviews.org/Articles/Volume5/2002-5bern/index.html
shows the subject is amazingly subtle: for example, it appears
that N = 8 supergravity in D=4 dimensions is first nonrenormalizable
at *five* loops, contrary to an earlier belief that it would fail
at three loops. But, he seems sure 11d supergravity is norenormalizable.
Anyway:
>>
>> 1
>> /|\
>> / | \
>> / | \ if you're viewing this correctly
>> / | \ it looks like a cube!
>> e3 e6 e5
>> |\ / \ /|
>> | \ / \ / |
>> | \ / |
>> | / \ / \ |
>> e4 \ / e1
>> \ e2 /
>> \ | /
>> \ | /
>> \ | /
>> \|/
>> e7
>>
>> namely the identity and 7 square roots of minus one, e1 through e7.
>What is the relation of this cube to the magic
>triangle that encodes the multiplication? Oh, let me guess: The magic
>triangle somehow arises from the projective plane.
Right! The cube above is the 3d vector space over the field
with two elements, F_2. If we view this cube from its e7 corner,
it looks like this:
e6
/ \
e4 e1
|\ /|
| e7 |
e3 | e5
\ |/
e2
If we flatten this out, we get the magic multiplication triangle
for the octonions:
e6
e4 e1
e7
e3 e2 e5
This is really just a picture of the corresponding projective plane,
i.e. the set of all lines through the origin in the 3d vector space
over F_2.
>Oh, looking at http://math.ucr.edu/home/baez/Octonions/node4.html
>and assuming your notation is consistent, I find that the triangle
>arises from the cube by looking at it along the e7-1 diagonal. Great.
Exactly! Drawing the picture as a weird triangle may be more confusing
than helpful. Anyone can imagine a cube.
>> In fact, all of SO(3) acts as symmetries of the octonions that
>> preserve a chosen imaginary octonion.
>> The different "rows" of the cube correspond to the different irreps
>> of this SO(3):
>>
>> 1 1
>> /|\
>> / | \
>> / | \
>> / | \
>> 3 e3 e6 e5
>> |\ / \ /|
>> | \ / \ / |
>> | \ / |
>> | / \ / \ |
>> 3 e4 \ / e1
>> \ e2 /
>> \ | /
>> \ | /
>> \ | /
>> \|/
>> 1 e7
>> But it gets better. In fact, the symmetries of the octonions that
>> preserve a chosen imaginary octonion form a group bigger than SO(3):
>> they form the group SU(3).
>> 1d real rep 1
>> of SU(3) /|\
>> / | \
>> ......................................................................
>> / | \
>> e3 e6 e5
>> |\ / \ /|
>> 3d complex rep | \ / \ / |
>> of SU(3) | \ / |
>> | / \ / \ |
>> e4 \ / e1
>> \ e2 /
>> \ | /
>> ......................................................................
>> \ | /
>> \|/
>> 1d real rep e7
>> of SU(3)
>> Finally, if we complexify the imaginary octonions we get the 7d
>> complex rep of G2. As a rep of G2 this decomposes as 7 = 3 + 3* + 1.
>Good, now I understand how to decompose the 7 of G2 as SU(3)
>irreps. But how does this precisely translate into constant spinors
>and constant forms?
Ah, this is another charming aspect of the story!
For this, it's best to note that Spin(8) has three irreducible
8-dimensional real representations:
the vector rep V
the left-handed spinor rep S+ - since these are real, chiral
spinor physicists call them
the right-handed spinor rep S- - Majorana-Weyl spinors!
All three of these reps are the octonions in disguise! The reason
is that you can build a vector from a left-handed spinor and a
right-handed spinor, so we have an intertwining operator:
f: S+ tensor S- -> V
And since these reps are their own duals, we can rewrite this as:
m: V tensor S+ -> S-
But the cool part is that this map tells us how to multiply octonions!
More precisely, suppose we pick a unit vector 1 in V and a unit vector
1' in S+. It turns out that multiplying by 1 defines an isomorphism
from S+ to S-. Also, multiplying by 1' gives an isomorphism from V to S-.
So, we can think of all three spaces as the same: THE OCTONIONS,
with m as the octonion product and 1 (or equivalently 1') as its unit.
The moral here is that we get the octonions by picking a unit
vector and a unit left-handed spinor in 8 dimensions.
This implies that the automorphism group of the octonions, G2, is
precisely the subgroup of Spin(8) that fixes a unit vector and a unit
left-handed spinor!
But the subgroup of Spin(8) that fixes a unit vector is just Spin(7),
and left-handed spinors in 8 dimensions restrict to plain
old spinors in 7 dimensions. So:
G2 is precisely the subgroup of Spin(7) that fixes a unit spinor!
But here's another way to say what's going on:
The space of Majorana spinors in 7 dimensions is 8-dimensional;
picking a unit spinor is enough to automatically give this space
the structure of an octonion algebra.
So, if we have a 7-dimensional spin manifold that has a covariantly
constant unit spinor field, the spinors at each point of this manifold
automatically get the structure of an octonion algebra... with a
covariantly constant multiplication!
Also: the space of vectors in 7 dimensions is 7-dimensional,
picking a unit spinor is enough to automatically give this space
the structure of the imaginary octonions.
So, if we have a 7-dimensional spin manifold that has a covariantly
constant unit spinor field, the tangent space at any point of
this manifold automatically gets the structure of the imaginary
octonions... with all the wonderful things this has, all covariantly
constant!
>Unfortunately, a combination of tiredness and a
>cold prevent me from clear thinking at this moment, but let me list
>the ingredients: G2 and SU(3) are the holonomy groups of the seven and
>the six-manifold. The 8 dimensional spinor rep of SO(7) (the holonomy
>group of a generic 7 manifold) decomposes as 7+1 under G2 and the 1 is
>the covariantly constant spinor.
I think I've sort of explained most of this, with a minimum of
calculation and a maximum of conceptual reasoning... I didn't get
around to going back and showing again how SU(3) fits into the picture
from this spinor point of view. But, you've probably had more than enough,
especially if you still have a cold!
There is other stuff you said, that I still need to absorb.
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>"Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote in message
>news:<slrnb76u98....@ariel.physik.hu-berlin.de>...
>> I thought Hawking's optimism counted on supergravity rather than
>> string-theory.
>How much room is there to count on supergravity but not on
>superstrings as a UV regulator at the same time?
He's talking about way back when you were in kindergarten,
before people invented the world-wide web, when people
didn't know supergravity was nonrenormalizable, and old geezers
like me were still in school.
(If you don't know what a "geezer" is, look it up. Btw, I hope
you looked at that website full of Rube Goldberg contraptions,
which I cited at the end of that post of yours where you asked
about them.)
>> If I have a constant spinor psi I can turn it into constant forms as
>>
>> f_{i1,...,ip} = psi-bar Gamma_{i1,...,ip} psi
>>
>> and f = f_{i1,...,ip} dx^i1 /\ ... /\ dx^ip.
>>
>> But for some reasons that I don't remember (probably (anti)symmetry of
>> Gamma matrices) in the 7 dim case only a threeform is non-trivial and
>> in the 6 dim case a 2 and a 3 form.
>About what kind of manifolds are you talking here? General 6- and
>7-dim ones? And there is a theorem that says that for every
>covariantly constant spinor psi the p-form psi-bar Gamma_{i1,...,ip}
>psi is non-trivial only for certain values of p?
I don't know all the stuff Robert does, but:
The really juicy theorem in the theory of G2 manifolds is that
every 7d spin manifold with a covariantly constant spinor has
the group G2 as its holonomy group. I explained why in a recent
post on this thread. Briefly, the covariantly constant spinor
is enough to give each tangent space the structure of the imaginary
octonions in a covariantly constant way. The symmetry group of
the imaginary octonions is G2, so the holonomy group of the metric
must lie in this group. Also, just like the imaginary quaternions,
the imaginary octonions have a "triple product":
Im(O) x Im(O) x Im(O) -> R
a,b,c |----------> (axb).c
Since every tangent space of our manifold has this structure,
this defines a covariantly constant 3-form on our manifold.
People interested in M-theory compactified on a G2 manifold
find this stuff useful. Right now Robert and I are trying to
figure out how this should be related to superstring theory
compactified on a Calabi-Yau manifold. The basic clue is that
a Calabi-Yau has holonomy group lying in SU(3), which is the
subgroup of G2 preserving a unit imaginary octonion.
I'm a bit perverse: Calabi-Yau manifolds are suddenly looking
more interesting now that I'm realizing they're related to the
octonions! But I'm sure you know how it works: if there's some
cute little structure you happen to like, it's always fun to
see it showing up in another new context.
Andy Neitzke
Urs Schreiber wrote:
> How is this related to what Polchinski writes on p. 256 of the second
> volume:
>
> "It has been conjectured that [...] any CFT can be arbitrarily well
> approximated by a rational theory."? Is that still true?
It's still true that it has been conjectured! But it is also true that it
has been conjectured that _not_ every CFT can be arbitrarily well
approximated by a rational theory: Gukov and Vafa recently put forward a
conjecture relating rationality of sigma models into Calabi-Yau manifolds
to a fancy algebro-geometric property called "complex multiplication."
(This notion is a good thing to mention in a thread entitled "mathematics
of string theory!") Namely, Gukov and Vafa say that only those Calabi-Yaus
which have complex multiplication give rational CFTs. Anyway, this
conjecture -- combined with some conjectures made by mathematicians who
study complex multiplication -- would imply that a sufficiently generic
Calabi-Yau sigma model cannot be obtained as a limit of rational CFTs.
The paper is at
http://arXiv.org/pdf/hep-th/0203213
--
Andy Neitzke
nei...@fas.harvard.edu
Robert C. Helling
<Urs.Sc...@uni-essen.de> wrote:
> "Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote in
message news:<slrnb76u98....@ariel.physik.hu-berlin.de>...
>> If I have a constant spinor psi I can turn it into constant forms as
>>
>> f_{i1,...,ip} = psi-bar Gamma_{i1,...,ip} psi
>>
>> and f = f_{i1,...,ip} dx^i1 /\ ... /\ dx^ip.
>>
>> But for some reasons that I don't remember (probably (anti)symmetry of
>> Gamma matrices) in the 7 dim case only a threeform is non-trivial and
>> in the 6 dim case a 2 and a 3 form.
>
> About what kind of manifolds are you talking here? General 6- and
> 7-dim ones?
Well, as I am still not able to produce the exact argument (even after
annoying several senior physicists of lunchtime conversations with
this) I cannot answer that question. Maybe this holds only for
manifolds with a constant spinor. And those are CY's and G2 holonomy
ones.
> And there is a theorem that says that for every
> covariantly constant spinor psi the p-form psi-bar Gamma_{i1,...,ip}
> psi is non-trivial only for certain values of p?
That's what I would expect. At least, I can make another group theory
argument: On a general manifold I could try to define a p-form field by
specifying it at one point and the using parallel transport to define
it at the other points. However, in general this will depend on the
paths I take for the transport and different paths will result in
p-forms differing by the action of the holonomy group.
In the case of restricted holonomy, we have to decompose the
representations of SO(d) on p-form fields with respect to the smaller
group (SU(3) or G2 here). It turns out, that for d=7 the p=3 and for
d=6 p=2 and 3 contain singlets of the holonomy group. So for those,
the above procedure turns out not to depend on the paths and give us
well defined p-form fields. For the other cases there cannot be
constant p-forms and so I would expect them to vanish.
>> For the Calabi-Yau those are the
>> complex structure (in some coordinates J = sum_i dz^i /\ dz-bar^i) and
> Is there some general relation between complex structures and
> covariantly constant spinors? For instance can every complex structure
> J be written as
> psi-bar Clifford-2-vector psi for psi a cov. constant spinor?
Don't know enough complex geometry to answer this question. In that
direction it seems unlikely to me: Manifolds with constant spinors are
very special: In 2d it's just the torus, in 4d it's the torus and K3
but there are manu complex manifolds. But maybe my notions of
'special' and 'many' are screwed.
>> the holomorphic 'volume' Omega = dz^1 /\ dz^2 /\ dz^3. The threeform
>> has the octonionic structure constants as components in an orthnormal
>> basis.
>>
>> On the circle that we need to connect the CY and the G2-manifold we
>> have dx, the volume form. But that is odd under the Z_2 that we are
>> going to mod out.
>>
>> Hmm. Let me speculate a bit. If we let the Z_2 also act on the CY,
>> namely by complex conjugation (but I have no physical justification
>> for that in the Horava Witten context), then
>>
>> dx /\ J + Omega + Omega-bar
>>
>> is even under the Z_2 and is obviously closed. Furthermore its Hodge
>> dual is something like
>>
>> J /\ J + dx /\ (Omega - Omega-bar)
>
> How did you get this? I've been staring at it, but I don't see how it
> works. (Probably I should study some CY geometry.)
Just as I said. The existence of J and Omega are characteristic for
Calabi Yau's. The expressions I wrote above of course only hold in
good coordinates, but you can define them in an invariant way not
using coordinates. One of their properties is that
J /\ J /\ J = Omega /\ Omega-bar = vol(CY) (maybe up to signs or twos)
and this gives you the information you need to work out the Hodge
dual.
Then I knew of
SUPERCONFORMAL FIELD THEORIES FOR COMPACT G(2) MANIFOLDS.
By Ralph Blumenhagen, Volker Braun (Humboldt U., Berlin). HUB-EP-01-47,
Oct 2001. 33pp. Published in JHEP 0112:006,2001
e-Print Archive: hep-th/0110232
There they use this construction of moding out a Z_2 that acts both on
the circle by x -> -x and on the CY by complex conjugation. Knowing
that, I guessed a threeform that uses J, Omega, and dx and is even
under this Z_2.
> Could you remind me: What _are_ you looking for?
I am looking for an argument why CY3 x (S^1/Z_2) has G2 holonomy.
I was trying to prove this by constructing the 3-form that is
characteristic of G2 holonomy.
Robert C. Helling
<ba...@galaxy.ucr.edu> wrote:
> In article <slrnb76u98....@ariel.physik.hu-berlin.de>,
> Robert C. Helling <hel...@atdotde.de> wrote:
>>On Sat, 15 Mar 2003 09:15:23 +0000 (UTC), John Baez
>><ba...@galaxy.ucr.edu> wrote:
> Exactly! Drawing the picture as a weird triangle may be more confusing
> than helpful. Anyone can imagine a cube.
Right. But I cannot see an easy way to read of the structure constants
from the cube directly. Anyway...
> For this, it's best to note that Spin(8) has three irreducible
> 8-dimensional real representations:
>
> the vector rep V
>
> the left-handed spinor rep S+ - since these are real, chiral
> spinor physicists call them
> the right-handed spinor rep S- - Majorana-Weyl spinors!
>
> All three of these reps are the octonions in disguise! The reason
> is that you can build a vector from a left-handed spinor and a
> right-handed spinor, so we have an intertwining operator:
>
> f: S+ tensor S- -> V
This is what I would call a SO(8) gamma matrix: It has one vector, one
dotted and one undotted index. And we know that the octonions sit in
the SO(8) gammas. The parts that you describe next are probably just
more index free versions of theses statements.
> This implies that the automorphism group of the octonions, G2, is
> precisely the subgroup of Spin(8) that fixes a unit vector and a unit
> left-handed spinor!
>
> But the subgroup of Spin(8) that fixes a unit vector is just Spin(7),
> and left-handed spinors in 8 dimensions restrict to plain
> old spinors in 7 dimensions. So:
>
> G2 is precisely the subgroup of Spin(7) that fixes a unit spinor!
Good. So 8 of SO(7) splits into 7+1 of G2.
Nice to see why that is the case. Since my last posting I managed to
print the KEK scanned versions of Slansky's review and learned how to
compute branchings using the LiE programm. Both sources agree with you
that 8=7+1.
> I think I've sort of explained most of this, with a minimum of
> calculation and a maximum of conceptual reasoning...
Thanks. That was very nice.
I could add that for the candidate three form
phi = J /\ dx + Omega + Omega-bar
for the case where Z_2 also acts as complex conjugation, it's easy to
see that the components give you octonionic structure constants: If we
write dz^i = dx^i + i dy^i then obviously
J = 2 i sum_i (dx^i /\ dy^i)
Furthermore, we need the real part of Omega. The combination is real
if there are three dx's or one dx and two dy's, so we find
phi /2i = (sum_i dx^i /\ dy^i) /\ dx + dx^1 /\ dx^2 /\ dx^3
- dx^1 /\ dy^2 /\ dy^3 - dy^1 /\ dx^2 /\ dy^3 - dy^1 /\ dy^2 /\ dx^3
These seven terms exactly correspond to the lines in the magic
triangle if i write it as
dy^1
dx^2 dx^3
dx
dy^3 dx^1 dy^2
As I learned from your previous post, the SU(3) comes from the Z_3
that rotates the triangle around the centre and thus are the
automorphisms (holonomies) that leave the 'special' direction dx
untouched.
I still don't know how this works if the Z_2 acts only on the S^1. I
asked Volker Braun and he mentioned that S^1/Z_2 is not orientable but
that G2 manifolds are always orientable. So there might be a
problem...
John Baez
>redlu...@wanadoo.fr (R.X.) wrote in message
>news:<7b5bf479.03031...@posting.google.com>...
>> People were dreaming about that the world would be described by a
>> mathematically distinguished or even unique structure, but to date
>> there is no sign of such. One may ponder to believe in the opposite,
>> see eg http://arxiv.org/pdf/hep-th/0211048 for an inspiration.
>I found the radical non-uniqueness paradigm particularly emphasized in
>the recent hep-th/0302219 by Susskind. In another thread John Baez
>thinks a googleplex of possible sets of low-energy "laws of nature" is
>horrible. Looks like the old conflict between the need to see patterns
>in the irregular to make progress and the ability to generalize the
>known patterns to less regular phenomena. Like switching from
>geocentric circles to helio-focused ellipses.
Hmm, I can't tell if you're comparing me to Ptolemy or not!
There's nothing more insulting than being compared to Ptolemy. :-)
Actually, you should be comparing me to the young Kepler,
who wanted a pretty model for the radii of planetary orbits -
as opposed to the old Kepler, who gave up on that and came up
with a formula for what the planets would do *given* some
extra experimental data. At the time, it have seemed like
"radical non-uniqueness" to give up on understanding why
the planets had exactly the orbits they do! But, it was
the right move.
Anyway, if it turns out the ultimate theory of physics really
*is* a beautiful structure at high energies which reduces to
a messy googleplex of effective laws of nature at low energies,
I would not complain. The universe is what it is, and I don't
see any point in whining about it!
But one has to reflect very carefully if, after decades of work
by hundreds of smart people on string theory, the upshot is
a scenario like the one Lenny Susskind describes. I'll summarize
it rather rudely like this:
There seem to be many possible ways our universe could look
that are consistent with experiment and string theory.
Actually, we're not sure there are *any*, but if there are,
there are probably lots. We now have to understand the set of
options, and decide which one fits what we see around us.
This is rather little predictive payoff for a lot of work.
Max Tegmark's theory is only marginally less useful, and it
takes a lot less work to describe: "all consistent mathematical
structures exist, and we're in one." Of course, these theories
could be true! But, they leave a lot of work for the poor sod
who wants to make predictions about what we actually see.
One must also remember the possibility that these theories are
wrong - i.e., there may be a rather elegant theory that can
actually make predictions about what we see around us! It would
be terrible to neglect this possibility.
R.X.
news:<206f2305.03031...@posting.google.com>...
> redlu...@wanadoo.fr (R.X.) wrote in message
> news:<7b5bf479.03031...@posting.google.com>...
> "It has been conjectured that [...] any CFT can be arbitrarily well
> approximated by a rational theory."? Is that still true?
Andy has already answered this to the contrary.
> Could you say, briefly, what a multi-loop algebra is? Probably one
> where the fields propagate not on a circle but on an n-torus? What
> about 2-loop algebras? Any relation to membrane theory?
You are right with the second question; I think there are plenty
of variants of this, though. They might be candidates to describe
membranes, but to my knowledge this hasn't worked. Rather, if I
remember this correctly, the relevant area preserving diffeomorphisms
are related to W_infinity algebras.
> As far as I understand, Thomas Larsson proposes multi-dimensional
> Virasoro algebras to replace the constraint algebra of canonically
> quantized gravity, somehow. (If that's not the case, he should please
> correct me.)
It would be interesting to see what comes out from that.
> I found the radical non-uniqueness paradigm particularly emphasized in
> the recent hep-th/0302219 by Susskind.
Indeed, inspiring as well.
> In another thread John Baez
> thinks a googleplex of possible sets of low-energy "laws of nature" is
> horrible.
... so ? He may not be the only one....
Urs Schreiber
"Robert C. Helling" wrote:
> And we know that the octonions sit in the SO(8) gammas.
Could you expand on that? How do the octonions sit in CL(8)? What about
associativity?
[...]
> I still don't know how this works if the Z_2 acts only on the S^1. I
> asked Volker Braun and he mentioned that S^1/Z_2 is not orientable but
> that G2 manifolds are always orientable. So there might be a
> problem...
Isn't S^1/Z2 just the interval?
[Moderator's note: the answer to the last question is "yes, but
treated as a 1-dimensional orbifold". - jb]
Squark
> In article <206f2305.03031...@posting.google.com>,
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> >How much room is there to count on supergravity but not on
> >superstrings as a UV regulator at the same time?
> He's talking about way back when you were in kindergarten,
> before people invented the world-wide web, when people
> didn't know supergravity was nonrenormalizable, and old geezers
> like me were still in school.
Why, aren't there alternatives like loop quantum supergravity?
Or is there a reason why it would be less of a UV regulator?
> I'm a bit perverse: Calabi-Yau manifolds are suddenly looking
> more interesting now that I'm realizing they're related to the
> octonions!
I think I'm perverse too, because now I want to know what
Calabi-Yau manifolds _are_! For me it's the combination though: if
they are relevant both to octonions _and_ string theory, they must
count for something...
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and change the
extension in the obvious way)
Aaron Bergman
fii...@yahoo.com (Squark) wrote:
> I think I'm perverse too, because now I want to know what
> Calabi-Yau manifolds _are_!
Well, that's easy enough. A Calabi-Yau manifold is a 6D (3 complex
dimensions, really) manifold with SU(3) holonomy. Yau proved Calabi's
conjecture that all 3D complex manifolds with vanishing first Chern
class admit such a metric. (They might have to be Kaehler -- I don't
remember.)
They're important in string theory because SU(3) holonomy is equivalent
to the existence of a covariantly constant spinor which means that they
preserve N=1 symmetry in 4 dimensions.
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
<http://aleph.blogspot.com>
Urs Schreiber
> In article <939044f.03032...@posting.google.com>,
> fii...@yahoo.com (Squark) wrote:
> > I think I'm perverse too, because now I want to know what
> > Calabi-Yau manifolds _are_!
> Well, that's easy enough. A Calabi-Yau manifold is a 6D (3 complex
> dimensions, really) manifold with SU(3) holonomy.
As far as I understand the dimension is not restricted to real 6. For
instance from Polchinski II p. 415 I learn that there is a unique CY1,
namely T^2 and precisely two CY2, namely flat T^4 and K3. From 6 real
D on there are a lot.
> Yau proved Calabi's
> conjecture that all 3D complex manifolds with vanishing first Chern
> class admit such a metric. (They might have to be Kaehler -- I don't
> remember.)
I think they are automatically Kaehler since Kaehler <> 2n real D
manifold with U(n) holonomy (e.g. Pol. II, p. 307).
Here are some equivalent characterizations of CYs (again, as far as I
understand):
CY <> admits metric of SU(N) holonomy
<> Kaehler and first Chern class vanishes
<> Kaehler and Ricci flat
<> complex and admits everywhere nonzero (N,0) form
<> complex and admits covariantly constant (N,0) form
Here D 2N is the real dimension of the manifold.
> They're important in string theory because SU(3) holonomy is equivalent
> to the existence of a covariantly constant spinor which means that they
> preserve N1 symmetry in 4 dimensions.
For my own benefit I'll try to summarize what I learn from GSW and
Polchinski:
In the beginning there is supergravity in 9+1 dimensions. One is looking
for compactifications that preserve classically (i.e. at tree level)
precisely one supersymmetry. In the simplest case (which in particular
means that the antisymmetric tensor field strength H vanishes) the
supersymmetry transformation of the gravitino psi_m is to first order
delta psi_m nabla^S_m epsilon .
Here nabla^S is the spinor covariant derivative and epsilon is the
spinor-valued transformation parameter. Precisely one classical
supersymmetry means that for precisely one "direction" of epsilon we
have
delta psi_m 0 <> nabla^S_m epsilon 0
<> epsilon is covariantly constant.
In the simplest case we are compactifying on Minkowski^4 times compact
6D. A little reflection then shows that what we need is a covariantly
constant spinor on the compact 6 dimensional internal manifold.
GSW nicely explain (15.1.3) why existence of a constant spinor in 6D is
equivalent to SU(3) holonomy: The (4 component) chiral parts of a spinor
in 6D transform under SU(4). We fix one component of the spinor rep as
our assumed covariantly constant spinor. The remaining three components
transform under SU(3). So if this one component is really to be constant
it must be acted on only by SU(3) < SO(6).
It's actually easy to see why this implies Ricci flatness (GSW
15.4.1): The spin connection of a complex manifold is U(N) valued and
splits into an SU(N) and a U(1) connection. With some index algebra
one finds that the field strength of the U(1)-part is determined by
the Ricci tensor. But for the mentric to be of SU(N) holonomy the
U(1) connection must obviously be trivial, so its field strength must
vanish and therefore the Ricci tensor must vanish, too.
I now also understand why the 3-form that Robert Helling was talking
about is important (GSW 15.4.2): Just as the ordinary volume form is
always covariantly constant becuase its indices transform under SO(2N),
an (N,0) form on a complex manifold is covariantly constant if its
indices transform under SU(N) instead of under U(N). So the existence of
such a form is equivalent to SU(N) holonomy.
(All this to the best of my knowledge. I may have made mistakes.)
Urs Schreiber
John Baez schrieb:
> (If you don't know what a "geezer" is, look it up. Btw, I hope
> you looked at that website full of Rube Goldberg contraptions,
> which I cited at the end of that post of yours where you asked
> about them.)
According to
we have "geezer" = "alter Knacker" :-)
Thanks for reminding me of Rube Goldberg. I did indeed not see your
moderator's note because I didn't read my own posting! Looking at that
website I now see that Rube-Goldberg-contraption-ness puts pretty tight
bounds on naturalness...
> The really juicy theorem in the theory of G2 manifolds is that
> every 7d spin manifold with a covariantly constant spinor has
> the group G2 as its holonomy group. I explained why in a recent
> post on this thread.
Yes, that was a beautiful explanation, thanks!
> Also, just like the imaginary quaternions,
> the imaginary octonions have a "triple product":
>
> Im(O) x Im(O) x Im(O) -> R
>
> a,b,c |----------> (axb).c
>
I understand how that triple product works for the quaternions, but I
can only guess how it reads explicitly for the octonions. Probably it is
defined by giving +-1 for any three distinct unit imaginary octonions if
they lie on one line of the magic triangle, and zero if they do not,
extended linearly to the whole space?
> Since every tangent space of our manifold has this structure,
> this defines a covariantly constant 3-form on our manifold.
OK, clear. Now how do I see that this 3-form is equal to
\bar psi gamma^abc psi
with psi the covariantly constant spinor? Oh, wait, it should be
obvious:
Fix any one point p=0 on the manifold and erect an orthonormal 7-bein
e^a there. Write the above 3-form at that point as
f(0) = f_abc e^a /\ e^b /\ e^c(0) = f_abc gamma^abc(0) |0> .
Next identify the "rotor" R, an element of the even subgroup of the
Clifford algebra, that is associated with the constant spinor psi, i.e.
psi(x) = R(x)psi(0).
This object is covariantly constant itself
nabla^S R = 0
and turns the vielbein at p=0 into a covariantly constant vielbein on
the entire 7-fold according to
e^a(x) = R(x) gamma^a bar R(x) |0> .
For the 3-form this means
f(x) = f_abc R(x) gamma^abc bar R(x) |0> .
On the other hand, if that's right then I fail even more to see why
Robert Helling says that other p-forms constructed this way are
necessarily trivial (vanish?). I must still be missing something.
> But I'm sure you know how it works: if there's some
> cute little structure you happen to like, it's always fun to
> see it showing up in another new context.
Don't know what you mean! :-o (That's a joke, I know what you mean.)
Urs Schreiber
John Baez schrieb:
[...]
> For this, it's best to note that Spin(8) has three irreducible
> 8-dimensional real representations:
>
> the vector rep V
>
> the left-handed spinor rep S+ - since these are real, chiral
> spinor physicists call them
> the right-handed spinor rep S- - Majorana-Weyl spinors!
>
> All three of these reps are the octonions in disguise! The reason
> is that you can build a vector from a left-handed spinor and a
> right-handed spinor, so we have an intertwining operator:
>
> f: S+ tensor S- -> V
>
> And since these reps are their own duals, we can rewrite this as:
>
> m: V tensor S+ -> S-
>
> But the cool part is that this map tells us how to multiply octonions!
>
> More precisely, suppose we pick a unit vector 1 in V and a unit vector
> 1' in S+. It turns out that multiplying by 1 defines an isomorphism
> from S+ to S-. Also, multiplying by 1' gives an isomorphism from V to S-.
> So, we can think of all three spaces as the same: THE OCTONIONS,
> with m as the octonion product and 1 (or equivalently 1') as its unit.
This isomorphism from V to S- is what lies behind the "first bosonize
the NSR fermions, then re-fermionize to get the GS fermions"-magic that
translates between the worldsheet supersymmetric and the spacetime
supersymmetric string (e.g. GSW, pp. 264), and which is possible due to
triality of SO(8) (which means a higher symmetry of the Dynkin diagram
and hence further automorphisms of the algebra). What GSW don't say is
that one can think of any of these 8-dim reps as the octonions with the
above product. Can one spell out the physical meaning of the product m
on the space of light-cone worldsheet fermions?
Thomas Larsson
Urs.Sc...@uni-essen.de (Urs Schreiber) wrote in message news:<206f2305.03031...@posting.google.com>...
>
> > useful for solving certain discrete models (lots of applications
> > in stat mech theory and integrable systems, etc), but not for
> > gravity or string theory.
>
> As far as I understand, Thomas Larsson proposes multi-dimensional
> Virasoro algebras to replace the constraint algebra of canonically
> quantized gravity, somehow. (If that's not the case, he schould please
> correct me.)
>
Well, not quite. I think of it as spacetime diffeomorphisms rather than
spatial diffeomorphisms, so it is Lagrangian rather than Hamiltonian.
The full spacetime diffeomorphism group is the correct Noether symmetry
of general relativity, or of any background free theory for that sake.
It is thus the branch of mathematics that deals with background
independence. I am quite convinced that understanding the mathematics of
background freedom is relevant if you wish to construct a background
free theory.
That I work locally and on the algebra level is just for simplicity
- things are hard enough anyway. In fact, it is so hard that I'm stuck
at the moment. By advertising here, I hoped that somebody would step
in and solve my problems, in particular to find the analog of Kac'
determinant formula in more than one dimension.
One reason why the Virasoro algebra in n > 1 dimensions is hard is that
it is impossible to even write down a BRST charge. It does not just fail
to be nilpotent except for a special value like c = 26, but it fails to
exist at all. The ghost transforms in the adjoint rep, i.e. a particular
kind of tensor field, but in order to construct Fock representations,
one cannot start from the tensor fields themselves; that would lead to
an infinite central extension, i.e. nonsense.
Jeffery
Aaron Bergman <aber...@princeton.edu> wrote in message news:<abergman-DA290F.15315621032003@localhost>...
> In article <939044f.03032...@posting.google.com>,
> fii...@yahoo.com (Squark) wrote:
>
> > I think I'm perverse too, because now I want to know what
> > Calabi-Yau manifolds _are_!
>
> Well, that's easy enough. A Calabi-Yau manifold is a 6D (3 complex
> dimensions, really) manifold with SU(3) holonomy. Yau proved Calabi's
> conjecture that all 3D complex manifolds with vanishing first Chern
> class admit such a metric. (They might have to be Kaehler -- I don't
> remember.)
>
> They're important in string theory because SU(3) holonomy is equivalent
> to the existence of a covariantly constant spinor which means that they
> preserve N=1 symmetry in 4 dimensions.
>
> Aaron
Yeah they are Kahler manifolds. These are 6D complex manifolds with
SU(3) holonomy or equivalently vanishing first Chern class. They have
three complex dimensions, meaning six dimensions all together, so you
can take 10d superstring theory, compactify 6 of the 10 dimensions on
a Calabi-Yau manifold, leaving four uncompactified dimensions of
spacetime. You can compactify E_8 x E_8 which you then break down to
E_8 x E_6 and this produces a particle spectrum similar to the
Standard Model. The number of generations is determined by the Euler
characteristic of the manifold.
Jeffery Winkler
Squark
Aaron Bergman <aber...@princeton.edu> wrote in message news:<abergman-DA290F.15315621032003@localhost>...
Sorry for being so damn stupid, but what does "SU(3) holonomy"
mean? Let me guess,
> Yau proved Calabi's
> conjecture that all 3D complex manifolds with vanishing first Chern
> class admit such a metric. (They might have to be Kaehler -- I don't
> remember.)
Well it has something to do with a metric, for one thing.
Possibly one demands all monodromies of the Levi-Cevita connection
form an SU(3) subgroup of SO(6) (where "SO(6)" is really the SO
group on the corresponding tangent space at a point)? Well, if
it's Kaehler then I guess we require them to preserve the complex
metric, which leaves us with U(3) and we just have to add they
got unit determinant.
Am I way off?
> They're important in string theory because SU(3) holonomy is equivalent
> to the existence of a covariantly constant spinor
I suppose because the subgroup fixing a spinor in 6D is SU(3).
> which means that they
> preserve N=1 symmetry in 4 dimensions.
Beats me what it has to do with it. Anyone who is not too tired
with me already is urged to enlighten me.
Urs Schreiber
Aaron Bergman schrieb:
> They [CYs] are important in string theory because SU(3) holonomy is equivalent
> to the existence of a covariantly constant spinor which means that they
> preserve N=1 symmetry in 4 dimensions.
What is the Goldberg-contraption index of CY compactifications? I mean,
how natural are they from the dynamical viewpoint? I understand that,
being Ricci flat, they have a good chance to solve the background
equations. On the other hand their topology may be highly non-trivial.
What reason would the universe have to compactify on such a topology?
Urs Schreiber
> Urs.Sc...@uni-essen.de (Urs Schreiber) wrote in message
news:<206f2305.03031...@posting.google.com>...
> > As far as I understand, Thomas Larsson proposes multi-dimensional
> > Virasoro algebras to replace the constraint algebra of canonically
> > quantized gravity, somehow. (If that's not the case, he schould please
> > correct me.)
> Well, not quite. I think of it as spacetime diffeomorphisms rather than
> spatial diffeomorphisms, so it is Lagrangian rather than Hamiltonian.
> The full spacetime diffeomorphism group is the correct Noether symmetry
> of general relativity, or of any background free theory for that sake.
Allow this potentially very dumb question: How do we get the
multi-dimensional Virasoro algebra from spacetime
diffeomorphisms? Does the multi-dimensional Virasoro algebra
still describe _conformal_ fields?
Aaron Bergman
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> Aaron Bergman wrote:
> > In article <939044f.03032...@posting.google.com>,
> > fii...@yahoo.com (Squark) wrote:
> > > I think I'm perverse too, because now I want to know what
> > > Calabi-Yau manifolds are !
> > Well, that's easy enough. A Calabi-Yau manifold is a 6D (3 complex
> > dimensions, really) manifold with SU(3) holonomy.
> As far as I understand the dimension is not restricted to real 6.
Sure. In general, a CY n-fold is a complex n-dimensional manifold with
SU(n) holonomy. Elliptically fibered CY 4-folds are relevant in F-theory
compactifications. I think this thread has pretty much focussed on the
3-fold case, though.
[...]
> > Yau proved Calabi's
> > conjecture that all 3D complex manifolds with vanishing first Chern
> > class admit such a metric. (They might have to be Kaehler -- I don't
> > remember.)
> I think they are automatically Kaehler since Kaehler <=> 2n real D
> manifold with U(n) holonomy (e.g. Pol. II, p. 307).
Probably right.
> Here are some equivalent characterizations of CYs (again, as far as I
> understand):
>
> CY <=> admits metric of SU(N) holonomy
> <=> Kaehler and first Chern class vanishes
> <=> Kaehler and Ricci flat
> <=> complex and admits everywhere nonzero (N,0) form
> <=> complex and admits covariantly constant (N,0) form
>
> Here D = 2N is the real dimension of the manifold.
Admitting a covariantly constant spinor is also a characterization.
Marc Nardmann
>>> I think I'm perverse too, because now I want to know what
>>> Calabi-Yau manifolds _are_!
Perverse, indeed :-).
DEFINITION. A *Calabi-Yau metric* on a complex manifold X is a
Ricci-flat K"ahler metric on X. A *Calabi-Yau manifold* is a compact
complex manifold equipped with a Calabi-Yau metric.
Here I assume that you know
(#) that every hermitian metric h on a complex manifold X has a
decomposition h = g+iw, where g is a Riemannian metric on X_R
(and X_R is the smooth manifold X without its complex structure),
and w is a 2-form on X_R;
(#) and that each of h,g,w determines the other two via the R-vector
bundle morphism J: T(X_R) --> T(X_R) given by Jv = iv (where the
holomorphic tangent bundle TX is canonically identified as a real
vector bundle with T(X_R)). E.g. g(v,w) = w(v,Jw) up to a sign that
depends on our definition of hermiticity.
The hermitian metric h = g+iw is K"ahler
if and only if w is closed,
if and only if J, viewed as a real (1,1)-tensor field on X, is
parallel with respect to the Levi-Civita connection of g.
The Ricci-flatness condition in the Calabi-Yau definition refers to the
Riemannian metric g.
Urs Schreiber wrote:
> Aaron Bergman wrote [replying to Squark]: [snip Squark quote]
>
>> Well, that's easy enough. A Calabi-Yau manifold is a 6D (3 complex
>> dimensions, really) manifold with SU(3) holonomy.
>
> As far as I understand the dimension is not restricted to real 6. For
> instance from Polchinski II p. 415 I learn that there is a unique CY1,
> namely T^2 and precisely two CY2, namely flat T^4 and K3. From 6 real
> D on there are a lot.
I've never read Polchinski (and never learned string theory). But his
statement needs some fine print:
(0) Nitpicking first: He assumes CYs to be nonempty & connected; that
might be included in his definition. He assumes CYs to be compact, as I
have done above; but I once met a string theorist who objected to this
condition and said that he needs a more general definition in his daily
work :-). (This string theorist did really mean noncompact *manifolds*;
one could also consider more general Calabi-Yau varieties, but let's not
get into this.)
(1) Polchinski wants to classify CYs up to *diffeomorphy* here (i.e. as
*real* manifolds). More natural things to do would be a classification
up to *biholomorphy* (i.e. to consider CYs as complex manifolds) or a
classification of CYs as *K"ahler manifolds* (i.e. with complex
structure and K"ahler metric).
Let's consider CYs as complex manifolds. Then the 1-dimensional ones are
known as elliptic curves. All elliptic curves are diffeomorphic to T^2,
but there are uncountably many of them. Every elliptic curve is the
quotient of the complex plane by a lattice of rank 2, and inherits a
flat K"ahler metric from the complex plane. By the way, this
construction can be generalized to higher dimensions: Every complex
torus, i.e. every complex manifold of the form C^n/G, where G is a
maximal lattice in C^n, admits a flat K"ahler metric.
As to K3s, there are also uncountably many of them, all diffeomorphic.
A K3 surface is by definition a simply-connected compact complex
2-manifold with (holomorphically) trivial canonical line bundle.
(The canonical line bundle of a complex manifold X of complex dimension
n is the n-th exterior power of the holomorphic cotangent bundle of X;
that's a holomorphic line bundle, i.e. the total space is a complex
manifold and the projection is holomorphic. Therefore "holomorphically
trivial".)
Every K3 surface admits a K"ahler metric (nontrivial fact) and thus, by
Yau's (highly nontrivial) theorem, a Ricci-flat K"ahler metric (since
the first Chern class of every K3 surface vanishes).
(Whenever I said "uncountably many", this statement can be considerably
refined by introducing suitable moduli spaces.)
(2) If X is a Calabi-Yau manifold, then there is a finite holomorphic &
metric covering X' --> X, where X' is the product (in the category of
K"ahler manifolds) of a complex torus and a simply connected CY
manifold. Clearly, in dimension 2, X' is either a complex torus or a K3
surface. Er, um, can we divide some complex torus or some K3 surface by
a finite group of K"ahler automorphisms in such a way that the quotient
is not a complex torus resp. K3 surface? Urs says that Polchinski claims
"no", and I'm too lazy to think about that.
>> Yau proved Calabi's
>> conjecture that all 3D complex manifolds with vanishing first Chern
>> class admit such a metric. (They might have to be Kaehler -- I don't
>> remember.)
> I think they are automatically Kaehler since Kaehler <=> 2n real D
> manifold with U(n) holonomy (e.g. Pol. II, p. 307).
Yau's theorem is about *compact* K"ahler manifolds. The special case we
are interested in here is as follows. The first *real* Chern class of a
complex manifold X is i(c_1(X)), where c_1(X) in H^2(X;Z) is the first
Chern class of X, and i: H^2(X;Z) --> H^2(X;R) is the canonical
homomorphism (which is in general neither surjective nor injective).
YAU'S THEOREM. Let X be a compact K"ahler manifold with K"ahler 2-form w
and vanishing first real Chern class. Then there is a unique Ricci-flat
K"ahler metric with K"ahler 2-form w' such that the closed forms w and
w' are in the same cohomology class.
A complex n-manifold with hermitian metric h = g+iw is K"ahler if and
only if the holonomy group of g is contained in U(n). (The holonomy
group of a K"ahler manifold might be even trivial, of course.)
If the holonomy group of a K"ahler manifold X of complex dimension n is
contained in SU(n), then the manifold is Ricci-flat. If X is
simply-connected, then the reverse holds as well. For general X,
Ricci-flatness implies only that the *restricted* holonomy group
(defined via contractible loops) is contained in SU(n).
(Stuff in [] inserted into Urs' text:)
> Here are some equivalent characterizations of CYs (again, as far as I
> understand):
>
> CY <=> admits metric of SU(N) holonomy [A0]
> <=> Kaehler and first Chern class vanishes [B0]
> <=> Kaehler and Ricci flat [C0]
> <=> complex and admits everywhere nonzero (N,0) form [D0]
> <=> complex and admits [everywhere nonzero] covariantly constant
> (N,0) form [E0]
>
> Here D = 2N is the real dimension of the manifold.
That's basically true, but let's make it really precise. Do we want to
say under which conditions a given complex manifold admits a Calabi-Yau
metric, or do we want to say whether a given K"ahler metric on some
complex manifold is a CY metric? Let's assume we want the latter. Then
our list becomes
[Z] is Calabi-Yau
[A] has holonomy contained in SU(n)
[B] has vanishing first real Chern class
[C] is Ricci-flat
[E] admits a covariantly constant (n,0) form
Item D0 does not quite fit. Note that not every complex manifold
admits a K"ahler metric, so this condition has to appear somewhere! The
"admits an everywhere nonzero (n,0) form" condition means just that the
canonical line bundle is (holomorphically) trivial. That does not
guarantee the existence of a K"ahler metric.
Without assuming compactness and simply-connectedness, we have by (my)
definition Z <=> C; moreover,
B <= C <= A <=> E .
With simply-connectedness, also C => A holds. Condition B implies
nothing because it does not involve the metric.
Now let's characterize when a given complex manifold admits a Calabi-Yau
metric. Our list is then
[Z'] admits a Calabi-Yau metric
[A'] admits a K"ahler metric whose holonomy is contained in SU(n)
[B'] has vanishing first real Chern class and admits a K"ahler metric
[C'] admits a Ricci-flat K"ahler metric
[D'] admits a K"ahler metric and an everywhere nonzero (n,0) form
[E'] admits a K"ahler metric which admits an everywhere nonzero
covariantly constant (n,0) form
Without assuming compactness and simply-connectedness, Z' <=> C' by
definition and
B' <= C' <= A' <=> E' => D'.
With simply-connectedness, C' => A'; with compactness, B' => C'.
When the canonical line bundle of X is holomorphically trivial, then it
is continuously trivial, i.e. its first Chern class vanishes. Then the
first Chern class of the holomorphic tangent bundle vanishes, too:
c_1(\Lambda^n T^*X) = c_1(T^*X) = -c_1(TX). Thus the first real Chern
class of the holomorphic tangent bundle vanishes. In other words,
D' => B' (without assuming compactness and simply-connectedness).
To everyone who has read up to this point, I should confess that I am
not an expert on this stuff, that it's been a while since I learned it,
that corrections are welcome, and that I'm probably not able to answer
follow-up questions. (Nor do I have the time to do that at the moment,
except maybe in a three-lines way.)
-- Marc Nardmann (posted March 24)
To reply, remove every occurrence of a certain letter from my e-mail
address.
Urs Schreiber
Also for D \neq 6?
Robert C. Helling
On 21 Mar 2003 17:04:44 GMT, Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>
> "Robert C. Helling" wrote:
>
>> And we know that the octonions sit in the SO(8) gammas.
>
> Could you expand on that? How do the octonions sit in CL(8)? What about
> associativity?
John has already given a nice explanation of this in this thread and
the latest TWFITP, but maybe you didn't notice because he was talking
about SO(8) and maps from the vectors times the chiral spinors to the
anti-chiral spinors. Of course, this is just an index free version of
talking about SO(8) gammas. For a version with many indices, see for
example
THE PARALLELIZING S(7) TORSION IN GAUGED N=8 SUPERGRAVITY.
By B. de Wit (NIKHEF, Amsterdam), H. Nicolai (CERN). NIKHEF-H-83-8, Jun 1983. 40pp.
Published in Nucl.Phys.B231:506,1984
where it is a footnote.
John Baez
In article <7b5bf479.03031...@posting.google.com>,
R.X. <redlu...@wanadoo.fr> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<b516dq$t0c$1...@glue.ucr.edu>...
>> [...] the existence of the superstring Lagrangian
>> in 10 dimensions arises from relations between the octonions and
>> Lorentzian geometry in this dimension, while M-theory is
>> intimately connected with 11d supergravity, which has something
>> like a hidden E8 symmetry (see hep-th/0006034). So in fact,
>> octonions and the exceptional Lie groups spawned by them permeate
>> quite a bit of superstring theory and M-theory.
>Well, first of all it is clear that such beautiful structures do
>appear here and there in string theory (like most other concepts
>which make sense ;-).
Yes - the mere fact that a particular piece of math shows up in
string theory mainly just means that some string theorist knows
this piece of math. :-)
Of course, the octonions happen to be a little hobby of mine, so
I'm glad to see them in string theory, and even happier if they
played a truly important role.
But, I'd rather know how important they really are, than fool
myself into thinking they're more important than they are!
Unfortunately, this task is complicated by the fact that I can't
completely trust what people say about this.
First of all, *nobody* fully understands string theory / M-theory.
So, various bits of math could be more important for string theory
than anyone currently realizes. Second of all, the octonions are
a great example of an "unloved" algebraic structure - something that
most people would rather avoid. So, people tend to underestimate
their importance.
Of course, you think I overestimate it. That's okay.
But anyway, let's think about this:
>But I don't see a particular significance of them. Those algebraic
>constructions typically appear in compactifications with many
>supersymmetries.
There's a good point here! If some "exceptionally beautiful algebra"
shows up only when we compactify string theory in an exceptionally
symmetrical way, one can argue the exceptional algebra is not really
central to understanding string theory - nor relevant to real-world
physics, where too much supersymmetry is generally a bad thing.
>In particular, exceptional groups arise as by-product in
>toroidal compactifications with extended supersymmetries.
Well, you know more about string theory than I do, so maybe you
could give me your opinion on these papers:
H. Nicolai
On Hidden Symmetries in d=11 Supergravity and Beyond
http://www.arXiv.org/abs/hep-th/9906106
He says "... the hidden exceptional symmetries of maximal
supergravity theories discovered long ago may provide important
clues as to where we should be looking. Support for this
strategy derives from the fact that some local symmetries of
the dimensionally reduced theories can be lifted back to
eleven dimensions." And it looks like he goes on to sketch
a way of organizing a bunch the fields in 11d supergravity into
something that transforms in the the 248-dimensional rep of E8.
More details appear here:
K. Koepsell, H. Nicolai, H. Samtleben
An exceptional geometry for d=11 supergravity?
http://www.arXiv.org/abs/hep-th/0006034
We analyze the algebraic constraints of the generalized
vielbein in SO(1,2) x SO(16) invariant d=11 supergravity,
and show that the bosonic degrees of freedom of d=11 supergravity,
which become the physical ones upon reduction to d=3, can be
assembled into an E_8-valued vielbein already in eleven
dimensions. A crucial role in the construction is played by
the maximal nilpotent commuting subalgebra of E_8, of
dimension 36, suggesting a partial unification of general
coordinate and tensor gauge transformations.
In short, they seem to be trying to transcend your objection
by taking symmetries that appear in certain compactifications
of 11d supergravity and lift them to the 11d theory itself.
>On the other hand, in theories with less SUSY, eg N=1 in d=4, most
>of this structure evaporates; I am not aware of any role of octonions
>and [the] like in such theories.
How about this paper:
Bobby S. Acharya,
M theory, G2 manifolds and four-dimensional physics,
Class. Quant. Grav. 19 (2002), 5619-5453.
I know, it's evil - it's not on the arXiv! - but anyway, it's
a review of attempts to get M-theory to give nice field theories
with N=1 supersymmetry in 4 spacetime dimensions by compactifying
on a 7-manifold with G2 holonomy. He seems to really be aiming
for realistic particle physics, e.g. the Standard Model gauge
group and 3 generations of fermions. The G2 manifold needs to
have conical singularities to give chiral fermions.
He only uses the octonions a bit in describing the geometry of
G2 manifold. I think he could use them much more if he wanted;
but my question to you is more whether this approach has a chance
of giving realistic physics.
>> Indeed, both
>> the E8 x E8 heterotic string and the compactifications of M-theory
>> using a 7-manifold whose holonomy group is G2 (the automorphism
>> group of the octonions) have often been discussed as starting-points
>> for realistic physical theories.
>Again, this only for a certain class of N=1 compactifications. For
>others, especially for the perturbative heterotic string, 3-manifolds
>with SU(3) holonomy are relevant instead - no significance of
>octonions here.
As you'll see from my discussions with Robert Helling, that may
not be true. SU(3) is the subgroup of G2 that fixes a unit imaginary
octonion, and this seems to explain why a 6-manifold with SU(3)
holonomy times S^1/Z_2 is a singular G2 manifold, which may help us
understand the relation between M-theory on M^4 x CY3 x S^1/Z_2 and
heterotic string theory on M^4 x CY3.
And this may be particularly nice if, as Robert suggests, sticking
in the 7th compactified dimension (the imaginary octonion!) solves
some problems with coupling constants in heterotic string theory.
>I doubt that the division algebras play any deeper
>or more significant role other than as a kinematical consequence
>of extended supersymmetry.
I'll add your doubt to my list of things to worry about, but
continue to think it's cool, and possibly important, that classical
superstring Lagrangians work in dimensions 3,4,6,10, exactly because
these are 2 more than the dimensions of the reals, complexes,
quaternions and octonions - with the last case being the most relevant
for physics!
>Punchline: while often mathematically distinguished and beautiful,
>algebraic -and specifically exceptional- structures do not seem
>particularly important for physics in 4d with few supersymmetries,
>as far as I know. But that's no reason not to study them!
I'll keep studying them and hoping they're important in physics.
They are so beautiful that I will feel sorry only for the physical
world, and not myself, if they turn out not to be!
Aaron Bergman
In article <3E7F569...@webx.de>,
Marc Nardmann <Marxc.N...@webx.de> wrote:
[...]
> (0) Nitpicking first: He assumes CYs to be nonempty & connected; that
> might be included in his definition. He assumes CYs to be compact, as I
> have done above; but I once met a string theorist who objected to this
> condition and said that he needs a more general definition in his daily
> work :-). (This string theorist did really mean noncompact *manifolds*;
> one could also consider more general Calabi-Yau varieties, but let's not
> get into this.)
Noncompact CYs are very common. The most famous is the conifold:
x^2 + y^2 + z^2 + w^2 = 0
in C^4. In the AdS/CFT conjecture, one often considers cones like the
above. The base of the cone turns out to be what's called an
Einstein-Sasaki manifold which is in term a (generalized) circle bundle
over a Kaehler-Eistein manifold.
[...]
> (2) If X is a Calabi-Yau manifold, then there is a finite holomorphic &
> metric covering X' --> X, where X' is the product (in the category of
> K"ahler manifolds) of a complex torus and a simply connected CY
> manifold. Clearly, in dimension 2, X' is either a complex torus or a K3
> surface. Er, um, can we divide some complex torus or some K3 surface by
> a finite group of K"ahler automorphisms in such a way that the quotient
> is not a complex torus resp. K3 surface? Urs says that Polchinski claims
> "no", and I'm too lazy to think about that.
As I remember it, you can find K3 by quotienting the torus by a Z_mumble
and then blowing up all the singularities.
> >> Yau proved Calabi's
> >> conjecture that all 3D complex manifolds with vanishing first Chern
> >> class admit such a metric. (They might have to be Kaehler -- I don't
> >> remember.)
>
> > I think they are automatically Kaehler since Kaehler <=> 2n real D
> > manifold with U(n) holonomy (e.g. Pol. II, p. 307).
>
> Yau's theorem is about *compact* K"ahler manifolds.
There's a theorem (Tian and Yau?) for noncompact manifolds, but I don't
remember the statement thereof.
Aaron Bergman
> Aaron Bergman <aber...@princeton.edu> wrote in message
> news:<abergman-DA290F.15315621032003@localhost>...
> > ...A Calabi-Yau manifold is a 6D (3 complex
> > dimensions, really) manifold with SU(3) holonomy.
>
> Sorry for being so damn stupid, but what does "SU(3) holonomy"
> mean? Let me guess,
>
> > Yau proved Calabi's
> > conjecture that all 3D complex manifolds with vanishing first Chern
> > class admit such a metric. (They might have to be Kaehler -- I don't
> > remember.)
>
> Well it has something to do with a metric, for one thing.
> Possibly one demands all monodromies of the Levi-Cevita connection
> form an SU(3) subgroup of SO(6) (where "SO(6)" is really the SO
> group on the corresponding tangent space at a point)? Well, if
> it's Kaehler then I guess we require them to preserve the complex
> metric, which leaves us with U(3) and we just have to add they
> got unit determinant.
> Am I way off?
Sounds good to me.
> > They're important in string theory because SU(3) holonomy is equivalent
> > to the existence of a covariantly constant spinor
>
> I suppose because the subgroup fixing a spinor in 6D is SU(3).
Yep.
> > which means that they
> > preserve N=1 symmetry in 4 dimensions.
>
> Beats me what it has to do with it. Anyone who is not too tired
> with me already is urged to enlighten me.
You write down the supersymmetry transformation and it just comes right
out. It's done in Polchinksi and GSW.
Aaron Bergman
In article <325dbaf1.03032...@posting.google.com>,
jeffery...@mail.com (Jeffery) wrote:
> You can compactify E_8 x E_8 which you then break down to
> E_8 x E_6 and this produces a particle spectrum similar to the
> Standard Model.
You get E_6 because the commutant of SU(3) in E_8 is E_6. This is for
the most obvious choice of gauge bundle for the compactification. You
can get other groups, too. E_6 is really quite large and it takes some
work to break it down to the standard model we all know and have
somewhat fond feelings for.
Robert C. Helling
On Wed, 26 Mar 2003 05:26:09 +0000 (UTC), John Baez
<ba...@galaxy.ucr.edu> wrote:
> In article <7b5bf479.03031...@posting.google.com>,
> R.X. <redlu...@wanadoo.fr> wrote:
> Well, you know more about string theory than I do, so maybe you
> could give me your opinion on these papers:
>
> H. Nicolai
> On Hidden Symmetries in d=11 Supergravity and Beyond
> http://www.arXiv.org/abs/hep-th/9906106
>
> An exceptional geometry for d=11 supergravity?
> http://www.arXiv.org/abs/hep-th/0006034
My understanding of these papers (and from sharing the office with two
of the authors of the second paper at the time when it was written) is
that the second indeed tries to work out what the first one is
announcing and that it doesn't quite work. They can group the fields
in E8 represenations (and the '248-Beins') but the action does not
transform properly under E8 unless you impose a further condition
which brings you back in the direction of dimensional reduction (you
have to impose that some fields are constant in 8 directions that are
singled out). So there are indications that E8 plays a role in 11d but
so far, it's not found to be a symmetry.
>>On the other hand, in theories with less SUSY, eg N=1 in d=4, most
>>of this structure evaporates; I am not aware of any role of octonions
>>and [the] like in such theories.
Exactly these theories, if you try to obtain them from M-Theory, require
manifolds of G2 holonomy as we discussed before - and G2 has an obvious
relation to octonions.
> And this may be particularly nice if, as Robert suggests, sticking
> in the 7th compactified dimension (the imaginary octonion!) solves
> some problems with coupling constants in heterotic string theory.
Today, I tried to back these claims with references to the
literature. I learned about this from lectures Jan Louis gave in
Trieste in '98 and there is a nice summary in the proceedings
PHENOMENOLOGICAL ASPECTS OF STRING THEORY.
By J. Louis (Martin Luther U., Halle-Wittenberg). Mar 1998.
Prepared for ICTP Spring School on Nonperturbative Aspects of String
Theory and Supersymmetric Gauge Theories, Trieste, Italy, 23-31 Mar
1998.
Published in *Trieste 1998, Nonperturbative aspects of strings, branes
and supersymmetry* 178-208
There is a version available online on his home page:
http://www.desy.de/~jlouis/lectures.html
Jim Graber
ba...@galaxy.ucr.edu (John Baez) wrote in message
> First of all, *nobody* fully understands string theory / M-theory.
M theory as a purely mathematical problem
I have wondered from time to time about the status of the "unknown"
M-theory as a purely Mathematical problem.
Suppose we define Mother theory as "the" or "a" minimal cover of the
six baby string theories.
Then it should be a simple matter of mathematics to prove that this
cover exists and is unique.
I would believe the Matrix theory approach would help solve the
existence part very easily.
I never seem to hear or read anything about this type of approach.
Probably this just means I don't know what's going on, although I did
just read through Smolin's 80 page review.
So I have a few related questions.
Is the existence of the six baby theories adequately established from
this point of view?
(Apparently, they are somewhat better established than the Mother
theory. Is this correct?)
Smolin seems to say that the matrix theory only serves to establish
the existence of another limit point of the Mother theory, not the
whole thing. What would be necessary to establish the existence of
the whole thing? (I am thinking here in terms of an existence proof,
rather than a construction. Is this a valid way to think, or is a
construction really necessary? Or is the existence proof already in
hand, at least to physical standards of rigor, and the explicit
construction really the only interesting outstanding problem? )
What if anything would be a problem with a cover which is not a
minimal cover?
That's enough for now.
Thanks for any illuminating responses.
Jim Graber
Robert C. Helling
On 25 Mar 2003 23:19:13 GMT, Marc Nardmann <Marxc.N...@webx.de> wrote:
> (0) Nitpicking first: He assumes CYs to be nonempty & connected; that
> might be included in his definition. He assumes CYs to be compact, as I
> have done above; but I once met a string theorist who objected to this
> condition and said that he needs a more general definition in his daily
> work :-). (This string theorist did really mean noncompact *manifolds*;
> one could also consider more general Calabi-Yau varieties, but let's not
> get into this.)
In the end, for string compactifications you probably want a compact
CY (not only because that is where 'compactification' comes from). But
restricting yourself to manifolds is probably not good enough. You
might want to include some singularities (for example to get
non-abelian gauge fields and chiral matter), so you should go for the
variety option. On the other hand, to study the physical effect of
these singularities, it often helps to study local models of them like
the conifold. And those are usually non-compact. But the idea is that
those are just local models and in the end you glue them together to
form a compact space.
> (1) Polchinski wants to classify CYs up to *diffeomorphy* here (i.e. as
> *real* manifolds). More natural things to do would be a classification
> up to *biholomorphy* (i.e. to consider CYs as complex manifolds) or a
> classification of CYs as *K"ahler manifolds* (i.e. with complex
> structure and K"ahler metric).
CY's can allow for small deformations still preserving the
CYishness. The space of these deformations is called the moduli space
(and it is know to factor into a 'Kaehler' and a 'complex structure'
part). The coordinates on this moduli space appear as massless scalar
fields in the low energy field theory. If they take vacuum expectation
values you should think of the CY as being deformed accordingly. So
all CY's in a connected component of the moduli space should be
considered equivalent when you include the dynamics of the field
theory.
It even turns out that some deformations that produce 'singular' CY's
are included in the dynamics ('flops') so that what appears like
different components of moduli space might be connected by such
transitions.
In the end, the appropriate equivalence relation is quite big. Ich
think it's called 'bi-rational equivalence' but abart from the physics
description I just gave I have no idea what these words mean.
It definitely connects all 2-tori as they are described by the
(complex structure) moduli space H/SL(2,Z) which has only one
component. I that sense there is only one (compact, connected) CY1.
Aaron Bergman
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> Aaron Bergman schrieb:
>
> > They [CYs] are important in string theory because SU(3) holonomy is
> > equivalent
> > to the existence of a covariantly constant spinor which means that they
> > preserve N=1 symmetry in 4 dimensions.
>
> What is the Goldberg-contraption index of CY compactifications?
There are a ----load of CY_3s/
> I mean,
> how natural are they from the dynamical viewpoint?
I'm not sure this is necessarily a question of dynamics. Rather, say,
vacuum selection.
> I understand that,
> being Ricci flat, they have a good chance to solve the background
> equations.
One can write down explicit CFTs that are believed to describe certain
CY. Look up Gepner models. One also expects nonlinear sigma models for
CYs to show up as the IR fixed point of certain linear sigma models. The
linear sigma model description is quite common and powerful.
> On the other hand their topology may be highly non-trivial.
> What reason would the universe have to compactify on such a topology?
Everything started out small. The better question to ask is: why did
some stuff get big?
Aaron Bergman
"Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote:
> In the end, the appropriate equivalence relation is quite big. Ich
> think it's called 'bi-rational equivalence' but abart from the physics
> description I just gave I have no idea what these words mean.
IIRC, birationally equivalent varieties are basically the same outside
of a subvariety. (More properly, I should probably be mumbling things
about invertible morphisms between open subsets in the Zariski
topology.). Thus things like blowdowns and blowups (in other words, a
flop, IIRC) lead to birationally equivalent CYs. I think the conifold
transition works, too. I don't know if there's always a path in moduli
space between any two birationally equivalent CYs, though.
> It definitely connects all 2-tori as they are described by the
> (complex structure) moduli space H/SL(2,Z) which has only one
> component. I that sense there is only one (compact, connected) CY1.
I think that it was shown that all complete intersections in weighted
projective space are linked. It was by Candelas and some others, IIRC.
Marc Nardmann
Robert C. Helling wrote:
> On 25 Mar 2003 23:19:13 GMT, Marc Nardmann wrote:
>
>> (0) Nitpicking first: He assumes CYs to be nonempty & connected; that
>> might be included in his definition. He assumes CYs to be compact, as
>> I have done above; but I once met a string theorist who objected to
>> this condition and said that he needs a more general definition in
>> his daily work :-). (This string theorist did really mean noncompact
>> *manifolds*; one could also consider more general Calabi-Yau
>> varieties, but let's not get into this.)
>
> In the end, for string compactifications you probably want a compact
> CY (not only because that is where 'compactification' comes from).
... but also because...?
> But restricting yourself to manifolds is probably not good enough. You
> might want to include some singularities (for example to get
> non-abelian gauge fields and chiral matter), so you should go for the
> variety option.
Well, I'm a mathematician, so I don't know why the restriction to
manifolds is not good enough in string theory as a physical theory;
in particular I have no idea about the relation between singularities
and non-abelian gauge theory (mathematical concepts that I understand).
I sort of know that manifolds are not good enough for the mathematical
theory called "mirror symmetry", which emerged from string theory.
> On the other hand, to study the physical effect of
> these singularities, it often helps to study local models of them like
> the conifold. And those are usually non-compact. But the idea is that
> those are just local models and in the end you glue them together to
> form a compact space.
Okay, that's good to know.
>> (1) Polchinski wants to classify CYs up to *diffeomorphy* here (i.e.
>> as *real* manifolds). More natural things to do would be a
>> classification up to *biholomorphy* (i.e. to consider CYs as complex
>> manifolds) or a classification of CYs as *K"ahler manifolds* (i.e.
>> with complex structure and K"ahler metric).
> CY's can allow for small deformations still preserving the
> CYishness. The space of these deformations is called the moduli space
> (and it is know to factor into a 'Kaehler' and a 'complex structure'
> part).
You mean "factor" in the sense that the projection of the corresponding
equivalence relation factors, right? (This deformation business is the
basis for mirror symmetry.) What I wrote in my previous post about
classification up to diffeomorphy is even true for classification up to
*deformation equivalence* with respect to the complex structure.
This might be the relation that Polchinski talks about (I don't have
access to his book).
> The coordinates on this moduli space appear as massless scalar
> fields in the low energy field theory. If they take vacuum expectation
> values you should think of the CY as being deformed accordingly. So
> all CY's in a connected component of the moduli space should be
> considered equivalent when you include the dynamics of the field
> theory.
Could you provide a poor mathematician with some basic intuition what
this means and why it's true? :-)
> It even turns out that some deformations that produce 'singular' CY's
> are included in the dynamics ('flops') so that what appears like
> different components of moduli space might be connected by such
> transitions.
Sounds interesting. :-)
> In the end, the appropriate equivalence relation is quite big. Ich
> think it's called 'bi-rational equivalence' but abart from the physics
> description I just gave I have no idea what these words mean.
Well, I sort of remember the definition of birational equivalence
(isomorphy in the category of projective algebraic varieties and
rational maps; by the way, all compact K"ahler manifolds are projective
algebraic), but the relation you described seems to be deformation
equivalence.
> It definitely connects all 2-tori as they are described by the
> (complex structure) moduli space H/SL(2,Z) which has only one
> component. I that sense there is only one (compact, connected) CY1.
Yes, and in this sense there are only two (compact nonempty connected
manifold) CY2s. Thanks for reminding me.
-- Marc Nardmann (posted March 27)
Urs Schreiber
I wrote:
>
> Aaron Bergman schrieb:
> > Admitting a covariantly constant spinor is also a characterization.
>
> Also for D \neq 6?
Sorry, I should have taken the time to think before asking this. Of
course the subgroup fixing a spinor will be bigger than SU(3) for D>6,
so a constant spinor won't be sufficient for higher dimensional CYs.
However it is sufficient to construct a covariantly constant (n/2,0)
form.
Further conditions should be necessary to guarantee that this form
is nontrivial.
Marc Nardmann
Aaron Bergman wrote:
Thanks for mentioning this. I've found a review of a Tian/Yau paper
which seems to deal with a version of Yau's theorem for noncompact
complete manifolds, but from the few sentences in the review it seems
to me that it's about K"ahler Einstein metrics with negative (not zero)
constant scalar curvature.
If we don't care about completeness of the metric, we have of course a
trivial corollary to Yau's theorem: If the noncompact complex manifold
M can be realized as a holomorphic cover of an open subset of a compact
complex manifold N with vanishing first real Chern class, then there is
a Ricci-flat K"ahler metric on M (obtained by pull-back of a Ricci-flat
K"ahler metric on N).
-- Marc Nardmann (posted March 27)
John Baez
In article <b174a6a6.0303...@posting.google.com>,
Jim Graber <jgr...@mailaps.org> wrote:
>Suppose we define Mother theory as "the" or "a" minimal cover of the
>six baby string theories.
>
>Then it should be a simple matter of mathematics to prove that this
>cover exists and is unique.
A "simple matter of mathematics", eh?
All the best string theorists have thought about this problem,
and so far they haven't succeeded in figuring out what M-theory really
is. The answer may seem simple when we finally get it - that's often
how it works - but right now it seems anything but simple.
Furthermore, it's not a well-posed mathematics problem. Your notion
of "cover" (or whatever we may prefer to call it) has no precise
definition. In *some sense* M-theory should reduce to the 5
superstring theories and 11d supergravity in various different
limits... but figuring out *what* sense is part of the problem,
and this will probably require a lot of physics intuition.
>I would believe the Matrix theory approach would help solve the
>existence part very easily.
Very easily, eh? Well then, go ahead!
>I never seem to hear or read anything about this type of approach.
Lots of people have tried to construct M-theory as a "matrix theory" -
a quantum field theory in which large NxN matrices replace the
algebra of functions on space. For two review articles that explain
the complicated problems this leads to, see:
Hermann Nicolai and Robert Helling, Supermembranes and M(atrix) theory,
http://xxx.lanl.gov/abs/hep-th/9809103
Washington Taylor, M(atrix) theory: matrix quantum mechanics as a
fundamental theory, http://xxx.lanl.gov/abs/hep-th/0101126
John Baez
In article <939044f.03032...@posting.google.com>,
Squark <fii...@yahoo.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<b5bese$f2e$1...@glue.ucr.edu>...
>> In article <206f2305.03031...@posting.google.com>,
>> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>> >How much room is there to count on supergravity but not on
>> >superstrings as a UV regulator at the same time?
>> [Helling is] talking about way back when you were in kindergarten,
>> before people invented the world-wide web, when people
>> didn't know supergravity was nonrenormalizable, and old geezers
>> like me were still in school.
>Why, aren't there alternatives like loop quantum supergravity?
There is now:
Yi Ling, Lee Smolin
Supersymmetric Spin Networks and Quantum Supergravity
Phys. Rev. D61 (2000) 044008
http://www.arXiv.org/abs/hep-th/9904016
But there wasn't when Urs Schreiber was in kindergarten.
>Or is there a reason why it would be less of a UV regulator?
Loop quantum supergravity could work just as well as loop quantum
gravity when it comes to eliminating ultraviolet divergences by
introducing a shortest length scale. Its Hamiltonian constraint
might even be better behaved than in ordinary loop quantum gravity -
nobody really knows.
A skeptic might instead argue that loop quantum supergravity
shares all the flaws of loop quantum gravity and perturbative
supergravity: as in loop quantum gravity, we don't yet understand
the dynamics, and as in supergravity, we get fields that aren't
observed in nature.
Graham Jones
[...]
>In the end, the appropriate equivalence relation is quite big. Ich
>think it's called 'bi-rational equivalence' but abart from the physics
>description I just gave I have no idea what these words mean.
I hope someone can explain this in more detail. I'm a mathematician, not
a physicist - I remember birational equivalence from algebraic geometry,
but don't understand much else you said. I know roughly what the words
mean, but I won't attempt an explanation since I'd surely get something
wrong.
Anyway, I hope you are right: if string theory is this algebraic, I
might even like it!
Graham
--
Graham Jones, author of SharpEye Music Reader
http://www.visiv.co.uk
21e Balnakeil, Durness, Lairg, Sutherland IV27 4PT, Scotland, UK
Thomas Larsson
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote in message news:<3E80973B...@uni-essen.de>...
The conformal algebra in 2D is isomorphic to vect(1)+vect(1), whose central
extension Vir+Vir is a subalgebra of Vir(2). So the conformal algebra acts
on everything on which the 2D Virasoro algebra acts. I haven't said anything
about *how* the algebras act, though. A conformal field can perhaps be
characterized as something on which the conformal groups acts in a particularly
simple way; Vir(2) may act in a complicated way, though (e.g. by the induced
representation).
BTW, my previous comment was misleading, and your original impreesion was
right. The constraint algebra of canonical gravity (ADM) is equivalent to the
space-time diff algebra, once the equations of motion are taken into account
(not otherwise). So in this sense the 4D Virasoro algebra without extension is
already the constraint algebra (and now everybody will undoubtedly shout
ANOMALY).
Aaron Bergman
Marc Nardmann <Marxc.N...@webx.de> wrote:
> Aaron Bergman wrote:
>
> > In article <3E7F569...@webx.de>,
> > Marc Nardmann <Marxc.N...@webx.de> wrote:
> >
> >>Yau's theorem is about *compact* K"ahler manifolds.
> >
> > There's a theorem (Tian and Yau?) for noncompact manifolds, but I
> > don't remember the statement thereof.
>
> Thanks for mentioning this. I've found a review of a Tian/Yau paper
> which seems to deal with a version of Yau's theorem for noncompact
> complete manifolds, but from the few sentences in the review it seems
> to me that it's about K"ahler Einstein metrics with negative (not zero)
> constant scalar curvature.
The ref is Inv. Math. 106 (1991) 27.
Urs Schreiber
> restricting yourself to manifolds is probably not good enough. You
> might want to include some singularities (for example to get
> non-abelian gauge fields and chiral matter)
I keep hearing that chiral matter requires additional "effort". Why
can't we just use IIB strings to get a chiral spectrum?
Andy Neitzke
Marc Nardmann wrote:
> Aaron Bergman wrote:
>
> > There's a theorem (Tian and Yau?) for noncompact manifolds, but I
> > don't remember the statement thereof.
>
> Thanks for mentioning this. I've found a review of a Tian/Yau paper
> which seems to deal with a version of Yau's theorem for noncompact
> complete manifolds, but from the few sentences in the review it seems
> to me that it's about K"ahler Einstein metrics with negative (not zero)
> constant scalar curvature.
Hey, I just happen to have (sketchy) notes on this! I think the theorem
Aaron is referring to is the one in
Tian, G. and Yau, Shing-Tung, "Complete Kahler manifolds with zero Ricci
curvature. I," J. Amer. Math. Soc. 3, 579-609 (1990).
The statement of the theorem is technical enough that I don't have it
written here -- but roughly, the idea seems to be that if you are given a
Kahler metric, with Kahler form omega, you can consider deforming it to
another Kahler form in the same cohomology class by adding d(dbar(phi)),
and ask that the resulting metric be Ricci flat. This leads you to a
pretty technical PDE and Tian and Yau show existence of solutions under
certain growth conditions (i.e. if the given omega behaves badly at
infinity you might be out of luck.) I don't think they say anything about
uniqueness, in contrast to the compact case, and I'm not sure how much is
known.
In particular, one case in which this works is if you're given a projective
variety which is not Calabi-Yau but has "positive curvature" (more
precisely, ample anticanonical class) -- then the theorem says that after
cutting out a locus which is Poincare dual to the anticanonical class, you
can put a complete Ricci-flat metric on the thing that's left behind.
--
Andy Neitzke
nei...@fas.harvard.edu
Robert C. Helling
> I keep hearing that chiral matter requires additional "effort". Why
> can't we just use IIB strings to get a chiral spectrum?
Without additional efford, you always get at least N=2 susy in D=4
when you compactify type IIB. On shell, there are only two different
N=2 multiplets (vector and hyper) and both do not contain chiral
matter. So the maximum susy you can have with chiral matter in D=4 is
N=1.
Maybe there are other reasons, too.
Robert C. Helling
<aber...@princeton.edu> wrote:
> In article <slrnb83bq5....@ariel.physik.hu-berlin.de>,
> "Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote:
> I think that it was shown that all complete intersections in weighted
> projective spaces are linked. It was by Candelas and some others, IIRC.
Are you talking about
ON THE CONNECTEDNESS OF MODULI SPACES OF CALABI-YAU MANIFOLDS.
By A.C. Avram, P. Candelas, D. Jancic, M. Mandelberg (Texas U.). UTTG-20-95,
Nov 1995. 15pp.
Published in Nucl.Phys.B465:458-472,1996
e-Print Archive: hep-th/9511230
How does this blend with Mike Douglas latest paper where in section
3.4 he cites a lower bound on the number of distict CY's as 30108?
There he also writes
"...mathematicians still debate whether there are finitely many or
infinitely many distinct M\footnote{More precisely, we want the number
of components of the moduli space of birational equivalence classes of
complex CY_3's}".
Or is the class of complete intersections in weighted projective space
very non-generic?
John Baez
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>Aaron Bergman schrieb:
>> In article <3E7CA56B...@uni-essen.de>,
>> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>> > Here are some equivalent characterizations of CYs (again, as far as I
>> > understand):
>> >
>> > CY <=> admits metric of SU(N) holonomy
>> > <=> Kaehler and first Chern class vanishes
>> > <=> Kaehler and Ricci flat
>> > <=> complex and admits everywhere nonzero (N,0) form
>> > <=> complex and admits covariantly constant (N,0) form
>> >
>> > Here D = 2N is the real dimension of the manifold.
I understand why U(N) holonomy <=> Kaehler.
You said that you could show SU(N) holonomy <=> Kaehler and Ricci flat
by a little index juggling, at least in the simply connected case.
Let's see how far I can get...
The curvature tensor of a Riemannian manifold is locally an
o(2N)-valued 2-form. If our manifold is Kaehler, this curvature tensor
is actually a u(N)-valued 2-form. This splits into an su(N)-valued
2-form and a u(1)-valued 2-form. The latter, say F, is just a plain
old 2-form.
If our manifold has SU(N) holonomy, then F must vanish. If our
manifold is simply connected, the converse is true too: F vanishing
implies SU(N) holonomy. (This can fail when our manifold is not
simply connnected, since when there are noncontractible loops,
curvature doesn't say everything about holonomies. But let's not
worry about this case.)
That's nice, but now let's try to show that F vanishes
if and only if the Ricci tensor vanishes.
The nicest way to do this would be to cook up a formula for the
Ricci tensor in terms of this curvature 2-form F. And since the
Ricci tensor should vanish if *and only if* F vanishes, this
formula should also let us describe F in terms of the Ricci tensor.
So, there should be a nice way to turn a 2-form (a skew-symmetric
bilinear form) into something like a Ricci tensor (a symmetric
bilinear form) and conversely, when our manifold is Kaehler.
If our manifold is Kaehler, there's a complex structure J on
each tangent space, i.e. an operator with J^2 = -1. But
there's a standard way to turn skew-symmetric bilinear forms
into symmetric ones when we have a complex structure around!
So maybe we should use this.
So, here's my guess: if F is the u(1)-valued curvature 2-form
and R is the Ricci tensor on a Kaehler manifold, they are related
by
F(v,w) = c R(v,Jw) for all tangent vectors v,w
where c is some nonzero constant that I can't determine.
This guess can only have a chance of working - F will only be
skew-symmetric - if we have
R(Jv,Jw) = R(v,w)
or equivalently
F(Jv,Jw) = F(v,w)
I'm not completely sure why these should be true on a Kaehler
manifold, but maybe someone else knows.
Or, maybe someone (e.g. Urs Schreiber) can just give an index-juggling
proof that
F(v,w) = c R(v,Jw)
for constant c.
>> Admitting a covariantly constant spinor is also a characterization.
>Also for D \neq 6?
It's hard to believe for arbitrary D = 2N, as this would seem to imply
that the subgroup of Spin(D) fixing a nonzero spinor is isomorphic to
SU(N) - and that's only true in certain special dimensions, like D = 6,
where we use the fact that Spin(6) = SU(4).
Urs Schreiber
> I keep hearing that chiral matter requires additional "effort". Why
> can't we just use IIB strings to get a chiral spectrum?
Hm, of course this is chiral in 10D. Don't know how that helps to get
chirality in 4D.
Aaron Bergman
"Robert C. Helling" <hel...@ariel.physik.hu-berlin.de> wrote:
> On Thu, 27 Mar 2003 23:30:38 +0000 (UTC), Aaron Bergman
> <aber...@princeton.edu> wrote:
> > I think that it was shown that all complete intersections in weighted
> > projective spaces are linked. It was by Candelas and some others, IIRC.
> Are you talking about
>
> ON THE CONNECTEDNESS OF MODULI SPACES OF CALABI-YAU MANIFOLDS.
> By A.C. Avram, P. Candelas, D. Jancic, M. Mandelberg (Texas U.). UTTG-20-95,
> Nov 1995. 15pp.
> Published in Nucl.Phys.B465:458-472,1996
> e-Print Archive: hep-th/9511230
Actually, I was thinking of something earlier which was only for
complete intersections in projective space. Yours is a better reference.
> How does this blend with Mike Douglas latest paper where in section
> 3.4 he cites a lower bound on the number of distict CY's as 30108?
> There he also writes
>
> "...mathematicians still debate whether there are finitely many or
> infinitely many distinct M\footnote{More precisely, we want the number
> of components of the moduli space of birational equivalence classes of
> complex CY_3's}".
I think the problem is that a conifold transition is not a birational
equivalence because and S^3 is blown up and that's not a complex
subvariety. The two tricks used to connect the moduli space up are flops
(which are a birational equivalence) and confiold transitions.
> Or is the class of complete intersections in weighted projective space
> very non-generic?
I think every known CY_3 is a hypersurface in a toric variety. Those are
what Douglas is talking about.
It's been a while since I've read about this sort of stuff, so I might
be forgetting some stuff. Brian Greene has a really nice set of lecture
notes on string theory on Calabi-Yaus.
Aaron Bergman
wrote:
> In article <3E80EC22...@uni-essen.de>,
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>
> >Aaron Bergman schrieb:
>
> >> In article <3E7CA56B...@uni-essen.de>,
> >> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>
> >> > Here are some equivalent characterizations of CYs (again, as far as I
> >> > understand):
> >> >
> >> > CY <=> admits metric of SU(N) holonomy
> >> > <=> Kaehler and first Chern class vanishes
> >> > <=> Kaehler and Ricci flat
> >> > <=> complex and admits everywhere nonzero (N,0) form
> >> > <=> complex and admits covariantly constant (N,0) form
> >> >
> >> > Here D = 2N is the real dimension of the manifold.
>
> I understand why U(N) holonomy <=> Kaehler.
>
> You said that you could show SU(N) holonomy <=> Kaehler and Ricci flat
> by a little index juggling, at least in the simply connected case.
> Let's see how far I can get...
Try taking the commutator of two covariant derivatives acting on the
covariantly constant spinor.
Marc Nardmann
Indeed. The Ricci tensor Ric is symmetric. The 2-form rho defined by
rho(v,u) = Ric(Jv,u) is called the Ricci form of our K"ahler metric. The
curvature R maps 2-forms to 2-forms, and can be applied to the
K"ahler 2-form w defined by w(v,u) = g(Jv,u). Then the formula
R(w) = rho
holds. Let's sketch the proof: We choose around any point a local
orthonormal frame (e1,...,eN,Je1,...,JeN); you might wish to prove that
this is possible. The K"ahler form is w = sum_{i=1}^N ei ^ Jei; check
this. Now evaluate R(w) and rho on our local vector fields to see that
they coincide; use the formula I'll write down below.
> So, here's my guess: if F is the u(1)-valued curvature 2-form
> and R is the Ricci tensor on a Kaehler manifold, they are related
> by
>
> F(v,w) = c R(v,Jw) for all tangent vectors v,w
>
> where c is some nonzero constant that I can't determine.
>
> This guess can only have a chance of working - F will only be
> skew-symmetric - if we have
>
> R(Jv,Jw) = R(v,w)
>
> or equivalently
>
> F(Jv,Jw) = F(v,w)
>
> I'm not completely sure why these should be true on a Kaehler
> manifold, but maybe someone else knows.
The formula to remember for K"ahler manifolds is
R(v,w)(Ju) = J(R(v,w)u).
It implies R(Jv,Jw) = R(v,w):
<R(Jv,Jw)u,z> = <R(u,z)Jv,Jw> = <J(R(u,z)v),Jw> = <R(u,z)v,w> =
<R(v,w)u,z>.
To understand the formula R(v,w)(Ju) = J(R(v,w)u), note that K"ahler
manifolds are characterized by the condition that the Levi-Civita
connection be parallel, i.e. nabla_u(Jv) = J(nabla_u(v)). The formula
follows immediately.
-- Marc Nardmann (posted April 1)