Combining Loop QG and String Theory

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Daryl McCullough

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Oct 31, 2003, 6:35:24 PM10/31/03
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As I understand it, loop quantum gravity, which is a theory of pure
gravity, can easily (?) be extended to a theory that includes a
variety of matter fields and interactions. Can loop quantum gravity be
formulated in which the matter fields and their interactions are
described by string theory?

I know that would be somewhat redundant, since both LQG and string
theory are competing approaches to a quantum theory of gravity, but I
was just wondering if anyone has looked at combining the two theories
in this way?

--
Daryl McCullough
Ithaca, NY


Urs Schreiber

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Oct 31, 2003, 6:58:39 PM10/31/03
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"Daryl McCullough" <da...@atc-nycorp.com> schrieb im Newsbeitrag
news:bnjtn...@drn.newsguy.com...

> As I understand it, loop quantum gravity, which is a theory of pure
> gravity, can easily (?) be extended to a theory that includes a
> variety of matter fields and interactions. Can loop quantum gravity be
> formulated in which the matter fields and their interactions are
> described by string theory?

No one really knows. But A. Ashtekar says that he has thought about how
perturbing about a semiclassical state in LQG could perhaps reproduce
excitations as known from string theory. That's quite plausible, in a vague
sense, since after all spin network fluctuations about a semiclassical state
would be fluctuations of objects which locally have 1 spatial dimensions.
Of course no semiclassical state is available yet. That's the most important
open question in LQG.

But this already addresses the point: Before one can compare the background
free LQG results and perturbative string results - or even perturbative
field theory results for that matter - one has to find LQG states that
approximate the backgrounds which are taken for granted in string theory.


> I know that would be somewhat redundant, since both LQG and string
> theory are competing approaches to a quantum theory of gravity,

A priori these approaches are not competing. It's mainly because
"quantization is a mystery" that not everybody agrees on the most natural
choices of focus when looking for a new quantum theory.

Lubos Motl

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Nov 2, 2003, 12:01:52 PM11/2/03
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On 31 Oct 2003, Daryl McCullough wrote:

> As I understand it, loop quantum gravity, which is a theory of pure
> gravity, can easily (?) be extended to a theory that includes a
> variety of matter fields and interactions. Can loop quantum gravity be
> formulated in which the matter fields and their interactions are
> described by string theory?

I am convinced that the answer is a definite "no". It is conceivable that
one can keep on adding new ideas and disorder to loop quantum gravity, and
in this way one can change one inconsistent theory into another.

But string theory is a consistent theory that does not admit any sort of
junk to be added if you want to keep it consistent. There is no way how
can you add new fields, new "links" or new interactions to string theory
in a given background - except the possibilities that are classified and
that are related to the massless states of the original background.

Once you know the background, string theory predicts the full spectrum of
objects and everything about their interactions - there is no way to
continuously deform the laws of physics.

Moreover, the metric is a *prediction* of string theory, it is not an
assumption. In other words, geometry is an "emergent" field that is on
equal footing with other fields. Although the coordinates and geometry may
play a special role for the *interpretation* of our theories (especially
the time coordinate), the metric is treated on equal footing with other
fields as far as the dynamical laws are concerned.

It would be therefore very surprising if some sort of discrete dual
description were possible for the full string theory. However if you read
the recent papers of Cumrun Vafa et al., you will see that topological
string theory does admit discretization of spacetime that is somewhat
similar to the situation of loop quantum gravity, but unlike loop quantum
gravity it is based on a well-defined calculational framework.

Look at it, it is beautiful mathematics (of MacMahon's functions and so
on), and new papers are on their way.

> I know that would be somewhat redundant, since both LQG and string
> theory are competing approaches to a quantum theory of gravity, but I
> was just wondering if anyone has looked at combining the two theories
> in this way?

You know, even string theory is rarely expressed as an infinite collection
of fields. This approach is called "string field theory" and it is
possible for perturbative calculations of open string phenomena, but it is
not the most popular or the most useful approach (at least so far).

As far as I know, none has tried to insert string field theory as an
infinite collection of fields into loop quantum gravity, and my guess is
that it would not lead to anything consistent or interesting. What the
people have tried was to start with a simple model of spin networks and
keep on hoping that all the complicated structure of string theory will
emerge spontaneously. Well, so far it hasn't. ;-)
______________________________________________________________________________
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.

Alfred Einstead

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Nov 3, 2003, 10:15:11 AM11/3/03
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Lubos Motl <mo...@feynman.harvard.edu> wrote:
> Moreover, the metric is a *prediction* of string theory, it is not an
> assumption.

Oh really? Last I checked, nobody's ever defined or formulated any
such thing as string theory on a generic differential manifold, much
less one that predicts that the manifold will have a metric, and
what metric it will have.

Squark

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Nov 4, 2003, 2:46:09 AM11/4/03
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Lubos Motl <mo...@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.03110...@feynman.harvard.edu>...

> Moreover, the metric is a *prediction* of string theory, it is not an
> assumption.

This seems to be an overstatement. The dynamic nature of metric
appears to a prediction of string theory, which is great, but not
the existance of metric itself. At least, I don't know a way of
writing an action for string theory in which the spacetime metric
isn't included in the first place.
If you wish, one might say the B-field is a "prediction" (though
it's not an empirical fact) as one can consider background with a
vanishing B-field. This, however, is not possible for the metric,
which has to be non-degenerate.

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

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Nov 4, 2003, 4:04:09 AM11/4/03
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In article <939044f.03110...@posting.google.com>,
fii...@yahoo.com (Squark) wrote:

> Lubos Motl <mo...@feynman.harvard.edu> wrote in message
> news:<Pine.LNX.4.31.03110...@feynman.harvard.edu>...
> > Moreover, the metric is a *prediction* of string theory, it is not an
> > assumption.
>
> This seems to be an overstatement. The dynamic nature of metric
> appears to a prediction of string theory, which is great, but not
> the existance of metric itself. At least, I don't know a way of
> writing an action for string theory in which the spacetime metric
> isn't included in the first place.

You should look into Gepner models. You can stick together a bunch of
known CFTs (minimal models) until you get the right amount of central
charge. You do some projection to get spacetime susy. The amazing thing
is that for a number of c=9 combinations, this seems to be exactly a
superconformal nonlinear sigma model with target space a Calabi-Yau
3-fold.

Aaron

Lubos Motl

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Nov 5, 2003, 1:50:54 AM11/5/03
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On 3 Nov 2003, Alfred Einstead wrote:

> Oh really? Last I checked, nobody's ever defined or formulated any
> such thing as string theory on a generic differential manifold,

If such a definition existed, it would be a disaster. A consistent quantum
gravity can only be able to define physics on spaces that satisfy the
equations of motion. We can define perturbative string theory on the
proper backgrounds using the CFT techniques - in fact all manifolds where
it is meaningful to talk about the S-matrix.

> much less one that predicts that the manifold will have a metric, and
> what metric it will have.

Of course the string theory predicts that there will be a dynamical metric
on any manifold, regardless of your starting point. This is the whole
point why we say that string theory predicts gravity.

String theory as we know it today does not constrain the metric
completely, but it constrains it significantly.

> Last I checked, ...

Let me recommend you to check more often, at least once per 20 years.

Lubos Motl

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Nov 5, 2003, 1:56:01 AM11/5/03
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On Tue, 4 Nov 2003, Squark wrote:

> This seems to be an overstatement. The dynamic nature of metric
> appears to a prediction of string theory, which is great, but not
> the existance of metric itself. At least, I don't know a way of
> writing an action for string theory in which the spacetime metric
> isn't included in the first place.

But this is just a question about the formalism (a "way", as you correctly
wrote), not a question about physics of the theory. String theory predicts
that whatever metric you have, it must be dynamical and its dynamics
follows the rules of general relativity.

If you want string theory to prove that *a* metric must always exist to
describe all of its degrees of freedom, without putting it in, then you
want string theory to prove an incorrect statement: string theory has also
taught us that there exist non-geometrical backgrounds (Gepner models, for
example) in which the usual notions of geometry break down (although they
can be thought of as manifolds with large, stringy curvature - or
equivalently small, stringy size).

> If you wish, one might say the B-field is a "prediction" (though
> it's not an empirical fact) as one can consider background with a
> vanishing B-field. This, however, is not possible for the metric,
> which has to be non-degenerate.

This is the physical reason why we - unlike the people in loop quantum
gravity - usually expand around a nonzero value of the metric. ;-) Loop
quantum gravity, on the other hand, tries to expand around g_{mn}=0, and
these colleagues of us think that already this may be called "background
independence", although they have no evidence that you can actually obtain
any nontrivial backgrounds.

The main difference between string theory and LQG is, in this respect,
that string theory expands around physically meaningful backgrounds, and
it can definitely get to other physically meaningful backgrounds. LQG
expands around g_{mn}=0 and it is unlikely that it can describe another,
second different smooth background. It is background-independent as long
as the background is bound to be g_{mn}=0 and all the distances vanish -
except for some chaotic Planckian fluctuations around zero that are
described by the spin networks.

What do you think is a more reasonable point to expand about?

Best
Lubos

Squark

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Nov 5, 2003, 2:01:26 AM11/5/03
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Aaron Bergman <aber...@physics.utexas.edu> wrote in message
news:<abergman-BBA709.01592704112003@localhost>...

> You should look into Gepner models. You can stick together a bunch of
> known CFTs (minimal models) until you get the right amount of central
> charge. You do some projection to get spacetime susy. The amazing thing
> is that for a number of c=9 combinations, this seems to be exactly a
> superconformal nonlinear sigma model with target space a Calabi-Yau
> 3-fold.

1) What are "minimal models"?

2) "Spacetime SUSY" implies you consider non-linear sigma models
in the first place, no?

3) What is so amazing?

Kris Kennaway

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Nov 5, 2003, 6:48:01 AM11/5/03
to
On 2003-11-05, Squark <fii...@yahoo.com> wrote:
> Aaron Bergman <aber...@physics.utexas.edu> wrote in message
> news:<abergman-BBA709.01592704112003@localhost>...
>
>> You should look into Gepner models. You can stick together a bunch of
>> known CFTs (minimal models) until you get the right amount of central
>> charge. You do some projection to get spacetime susy. The amazing thing
>> is that for a number of c=9 combinations, this seems to be exactly a
>> superconformal nonlinear sigma model with target space a Calabi-Yau
>> 3-fold.
>
> 1) What are "minimal models"?

CFTs with a finite number of primary fields. See
http://arxiv.org/abs/hep-th/9108028 or any introduction to conformal
field theory.

> 2) "Spacetime SUSY" implies you consider non-linear sigma models
> in the first place, no?

No, all we start with is a two-dimensional unitary superconformal
field theory with vanishing conformal anomaly. Non-linear sigma
models are particular examples of such CFTs; as Aaron says above and I
explained further in a previous message, Gepner models are not
non-linear sigma models, but you can perform a smooth marginal
deformation of the theory to a phase where the CFT becomes a NLSM.

> 3) What is so amazing?

You said:

> At least, I don't know a way of writing an action for string theory
> in which the spacetime metric isn't included in the first place.

You obtain the 10-dimensional target space metric (and GR) from the
two-dimensional string theory, without putting it in by hand. In
general CFTs are not defined from an action principle; a Gepner model
is an example of a CFT where you start with something that does not
have a geometrical interpretation (so no metric), but you can
nonetheless generate a Calabi-Yau manifold together with Ricci-flat
metric from this simple, non-geometrical CFT.

The target space metric in string theory is a derived quantity. You
don't always have a metric because the string doesn't always propagate
on a geometrical space, but when you do it is built up from fields on
the worldsheet and not put in by hand.

Kris

Kris Kennaway

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Nov 5, 2003, 12:50:46 PM11/5/03
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This isn't quite right..the Gepner point is a point in the enlarged
Kahler moduli space, and the nonlinear sigma model is a different
phase of this theory. While it's true that you have the marginal
operators corresponding to the Kahler and Complex Structure moduli in
the Gepner model spectrum, the "target space" of the Gepner model is a
single point, and I don't think it's known explicitly how to
continuously deform the CFT to the phase where you have a target space
with a metric.

The phase structure of the moduli space, i.e. the fact that Gepner
models *are* smoothly connected to nonlinear sigma models, was
elucidated by Witten using a non-conformal UV theory (gauged linear
sigma models) that interpolates them and flows to the CFT in the
infrared, but obtaining the IR theory for generic points in a
non-geometrical phase isn't easy.

So you can show that the Gepner model is smoothly connected to a
region in moduli space where the 2-dimensional CFT is a theory of
strings propagating on a Calabi-Yau (with the Ricci-flat target space
metric arising from worldsheet fields in the usual way), but the
Gepner model itself is not that.

Kris

Kris Kennaway

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Nov 5, 2003, 12:51:08 PM11/5/03
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Not on a generic manifold (because we don't understand generic
non-supersymmetric backgrounds), but on a Calabi-Yau manifold the
target space metric is proven to be the Ricci-flat metric, modified by
string worldsheet instanton corrections in a calculable way if the
manifold (specifically, if the holomorphic curves of the manifold) is
of finite size. This is a consequence of requiring the non-linear
sigma model beta function to vanish, as it must for a conformal field
theory.

Kris

Aaron Bergman

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Nov 6, 2003, 12:36:18 AM11/6/03
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In article <iuv8ob.8ib.ln@xor>,
Kris Kennaway <kk...@rot13.obsecurity.org> wrote:

> This isn't quite right..the Gepner point is a point in the enlarged
> Kahler moduli space, and the nonlinear sigma model is a different
> phase of this theory.

Yeah, and I'm embarrased to see that the relevant minimal models have
lagrangian descriptions, too.

Sigh.

Aaron

Squark

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Nov 6, 2003, 2:01:53 PM11/6/03
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Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message news:<85faob.27i.ln@xor>...

> On 2003-11-05, Squark <fii...@yahoo.com> wrote:
> > 2) "Spacetime SUSY" implies you consider non-linear sigma models
> > in the first place, no?
>
> No, all we start with is a two-dimensional unitary superconformal
> field theory with vanishing conformal anomaly.

The what is the meaning of "spacetime SUSY"? You
can only have worldsheet SUSY.

> You obtain the 10-dimensional target space metric (and GR) from the
> two-dimensional string theory, without putting it in by hand.

That is only a meaningful claim if the CFT is uniquely
defined by natural, in some sense, assumptions. What
are these assumptions in this case?

> The target space metric in string theory is a derived quantity. You
> don't always have a metric because the string doesn't always propagate
> on a geometrical space,

What do you mean? Are you talking about some non-geometrical
superselection sectors, or specific states which are, in some
sense, non-geometrical?

Kris Kennaway

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Nov 6, 2003, 5:29:28 PM11/6/03
to

On 2003-11-06, Squark <fii...@yahoo.com> wrote:
> Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message news:<85faob.27i.ln@xor>...
>> On 2003-11-05, Squark <fii...@yahoo.com> wrote:
>> > 2) "Spacetime SUSY" implies you consider non-linear sigma models
>> > in the first place, no?
>>
>> No, all we start with is a two-dimensional unitary superconformal
>> field theory with vanishing conformal anomaly.
>
> The what is the meaning of "spacetime SUSY"? You
> can only have worldsheet SUSY.

The would-be spacetime N=2 SUSY generators are linear combinations of
the worldsheet N=2 SUSY generators (in the left and right-moving
sector). You can do this whether or not your CFT has an
interpretation as a sigma model ("strings propagating on a space"),
it's just not very useful if you don't have the dual target space
interpretation.

>> You obtain the 10-dimensional target space metric (and GR) from the
>> two-dimensional string theory, without putting it in by hand.
>
> That is only a meaningful claim if the CFT is uniquely
> defined by natural, in some sense, assumptions. What
> are these assumptions in this case?

Perturbative string theory is a world-sheet theory. This means that
we mostly care about 2-dimensional conformal field theory. We start
by writing down a CFT that is unitary and anomaly-free. Anomaly
freedom and unitarity constrains us to start with one of the 5 famous
classes of superstring theory (type I, IIA, IIB, 2 types of
Heterotic). We then have to make the total conformal anomaly of the
CFT vanish (by balancing off positive contributions from the
"internal" part of the CFT with the negative contributions from the
non-unitary ghost sector). If we do this by using a sigma model for
the internal CFT then this is the constraint that tells us that the
sigma model must have a 6-dimensional target space, giving 4+6=10
dimensions.

But no-one forces us to choose the CFT to be a sigma model. There are
lots of other choices of CFT with the right conformal anomaly that are
not sigma models. See below:

>> The target space metric in string theory is a derived quantity. You
>> don't always have a metric because the string doesn't always propagate
>> on a geometrical space,
>
> What do you mean? Are you talking about some non-geometrical
> superselection sectors, or specific states which are, in some
> sense, non-geometrical?

In the worldsheet approach to string theory, the target space is a
derived quantity. If our CFT is a sigma model, then there are fields
living on the worldsheet that have a dual interpretation as
geometrical quantities of the target space. For example, the scalar
fields living on the worldsheet parametrise the embedding of the
worldsheet into the target space, so they are coordinates on the
target manifold. The worldsheet field with spacetime spin-2 gives you
the metric on the target space, and so on.

However, no-one forces you to make your CFT a sigma model. There are
lots of unitary CFTs with the right conformal anomaly to use them as
part of our worldsheet theory, but there is no sense in which they are
a sigma model.

For example, Gepner models have a finite number of primary fields, but
none of them make sense as coordinates parametrising the worldsheet
embedding into a target space. This is the sense in which there is no
6-dimensional target space geometry associated to the Gepner models.

What are they good for? Two things:

1) Gepner models are exactly solvable. You can compute all
correlation functions of the theory explicitly, because there are no
pesky details like a Ricci-flat metric that we know (by Yau's theorem)
exists, but we cannot write down exactly.

2) Gepner models are in fact smoothly connected (by marginal
deformations) to conformal field theories which *are* sigma models
onto a Calabi-Yau manifold. From the point of view, it's clear why
the Gepner model isn't a sigma model: running the process backwards
the entire Calabi-Yau collapses to a single point.

This illustrates the point I made above: in string theory, space-time
is a derived quantity. It emerges as a consequence of two-dimensional
world-sheet dynamics, and it is not a static and unchanging background
of the theory. The topology - and even dimension - of this derived
target space may fluctuate and change drastically. Dimensions may
dynamically compactify and decompactify. Cycles of the manifold may
dynamically shrink and be replaced by other cycles of different
dimension. One Calabi-Yau manifold may dynamically turn into another
Calabi-Yau manifold (in fact it is believed that all Calabi-Yau
manifolds are smoothly connected in string theory by these processes).

But this is all fine, because these processes are smooth on the
worldsheet. Whenever the target space is behaving semi-classically,
its derived dynamics reduce to those of General Relativity. When the
target space becomes non-classical, then string theory produces
corrections to General Relativity in just the right way to smooth out
singular processes like topology change (also naked singularities,
etc), so everything is always smooth and non-singular from the point
of view of the string.

Kris

Aaron Bergman

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Nov 6, 2003, 7:47:47 PM11/6/03
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In article <m3eeob.u4d1.ln@xor>, Kris Kennaway wrote:
>
> Perturbative string theory is a world-sheet theory. This means that
> we mostly care about 2-dimensional conformal field theory. We start
> by writing down a CFT that is unitary and anomaly-free. Anomaly
> freedom and unitarity constrains us to start with one of the 5 famous
> classes of superstring theory (type I, IIA, IIB, 2 types of
> Heterotic).

There are thousands more less famous ones, really. OA, OB,
SO(16)xSO(16) heterotic are standard examples. All are anomaly
free and unitary.

Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>

Jeffery

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Nov 7, 2003, 6:17:16 PM11/7/03
to
Aaron Bergman <aber...@princeton.edu> wrote in message
news:<slrnbqljaq....@cardinal3.Stanford.EDU>...

The five he listed are the only known perturbative superstrings. The
Type 0 strings are another name for the bosonic string, and thus not
superstrings. The Heterotic SO(32) and Heterotic E8 x E8 are related
by T-duality, and thus have must have a common subgroup for the gauge
groups, which is SO(16) x SO(16), but this is not a separate
perturbative superstring. Of course, there's an infinite number of
points in the moduli space of M-theory.

Jeffery Winkler

http://www.geocities.com/jefferywinkler

Lubos Motl

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Nov 7, 2003, 6:17:17 PM11/7/03
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On 5 Nov 2003, Squark wrote:

> 1) What are "minimal models"?

The minimal models are a family of specific two-dimensional and very
abstract conformal field theories, labeled by one or two integers (two if
you allow non-unitary models). One of the first members of the family is
the Ising model (its infrared limit). Note that the Ising model describes
physics of a single spin at each point. You don't see a hidden geometry.
These minimal models are important even in other parts of physics, for
example condensed matter physics.

> 2) "Spacetime SUSY" implies you consider non-linear sigma models
> in the first place, no?

Nope. The minimal models are not non-linear sigma models. Nevertheless,
there are simple ways to combine that and project them out so that the
result has a spacetime supersymmetry.

> 3) What is so amazing?

Aaron has to explain himself what *he* finds amazing about it. I find it
amazing because it shows a very clear example of the theorem saying that
"all roads lead to string theory". One starts with completely
non-geometrical two-dimensional theories that were used outside string
theory; he might even think that he discovered a new non-string theory;
he combines them so that the consistency conditions for the stringy
worldsheet are satisfied. Once again, it does not seem that we are adding
extra dimensions with a specific geometry. Nevertheless, a simple analysis
proves that what we have constructed is string theory on a specific
Calabi-Yau shape! Its size is usually small - comparable to the string
length - and it can have large fields on it. However it is still connected
to the Calabi-Yau moduli space; the Gepner models - how these combinations
of the minimal models are called - describe an expansion of the Calabi-Yau
physics around a concrete, pretty weird point. You may add deformations to
your theory, that was constructed as a group of several Ising-like models,
and soon or later you will see that it is the old string theory with the
total of 10 dimensions.

Best wishes
Lubos

Aaron Bergman

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Nov 8, 2003, 5:22:20 AM11/8/03
to

> Aaron Bergman <aber...@physics.utexas.edu> wrote in message
> news:<abergman-BBA709.01592704112003@localhost>...
>
> > You should look into Gepner models. You can stick together a bunch of
> > known CFTs (minimal models) until you get the right amount of central
> > charge. You do some projection to get spacetime susy. The amazing thing
> > is that for a number of c=9 combinations, this seems to be exactly a
> > superconformal nonlinear sigma model with target space a Calabi-Yau
> > 3-fold.
>
> 1) What are "minimal models"?

Particularly simple examples of conformal field theories. Ising models,
potts models and the like are examples. These theories aren't usually
constructed from a lagrangian, but rather by specifying all the possible
correlation functions. (roughly)

> 2) "Spacetime SUSY" implies you consider non-linear sigma models
> in the first place, no?

That was probably a bad way of stating it. You do need to do a
projection, but we're not considering nonlinear sigma models in any way
here.

> 3) What is so amazing?

That you can take this purely algebraic construction and then end up
with what (is conjectured to be) the CY nonlinear sigma model.

The CY wasn't put in there at all. The construction of the CFTs doesn't
know anything that a target space might exist or anything like that. So,
this is an example where we started with a string theory that had no
reason to have a geometric interpretation, and it ended up having one.

Aaron

Squark

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Nov 8, 2003, 5:27:58 AM11/8/03
to

> This is the physical reason why we - unlike the people in loop quantum


> gravity - usually expand around a nonzero value of the metric.

> ...


> What do you think is a more reasonable point to expand about?

There's no telling, really. "Realistic backgrounds" provide a
closer contact with quantum field theory, while g = 0 with
background invariance.

Squark

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Nov 8, 2003, 5:29:08 AM11/8/03
to
Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message
news:<m3eeob.u4d1.ln@xor>...

> Perturbative string theory is a world-sheet theory. This means that
> we mostly care about 2-dimensional conformal field theory. We start
> by writing down a CFT that is unitary and anomaly-free. Anomaly
> freedom and unitarity constrains us to start with one of the 5 famous
> classes of superstring theory (type I, IIA, IIB, 2 types of
> Heterotic).

Aaron pointed out this is not correct.

> We then have to make the total conformal anomaly of the
> CFT vanish (by balancing off positive contributions from the
> "internal" part of the CFT with the negative contributions from the
> non-unitary ghost sector). If we do this by using a sigma model for
> the internal CFT then this is the constraint that tells us that the
> sigma model must have a 6-dimensional target space, giving 4+6=10
> dimensions.

1) What is this 6 / 4 division there? Somehow
you are automatically constraining yourself
to Calabi-Yau compactifications?

2) Are you saying that any sigma model CFT which
is unitary and anomaly free is a string theory
(in the usual sense, i.e. one of the 5)?



> But no-one forces us to choose the CFT to be a sigma model. There are
> lots of other choices of CFT with the right conformal anomaly that are
> not sigma models.

So you're saying _any_ unitary and anomaly free CFT
represents a string theory? That on the non-perturbative
level all of them are connected? I understand how you can
change the Lagrangean by inserting vertex operators, but
you can't change the field content.

> Dimensions may dynamically compactify and decompactify.

This cannot be, at least not globally. Are you saying
some kind of a singular spacetime forms, with components
of different dimension?
Globally, different dimensions apparently correspond to
different superselection sectors. For asymptotically
flat string theory those are matrix models of different
dimension, for asymptotically AdS string theory those
are SCFTs of different dimension.

> When the
> target space becomes non-classical, then string theory produces
> corrections to General Relativity in just the right way to smooth out
> singular processes like topology change (also naked singularities,
> etc), so everything is always smooth and non-singular from the point
> of view of the string.

This "smoothization" I don't understand. Where can I read about
it?

Aaron Bergman

unread,
Nov 8, 2003, 5:30:19 AM11/8/03
to
In article <575262ce.03110...@posting.google.com>, Jeffery wrote:
> Aaron Bergman <aber...@princeton.edu> wrote in message
> news:<slrnbqljaq....@cardinal3.Stanford.EDU>...
>
>> In article <m3eeob.u4d1.ln@xor>, Kris Kennaway wrote:
>
>> > Perturbative string theory is a world-sheet theory. This means that
>> > we mostly care about 2-dimensional conformal field theory. We start
>> > by writing down a CFT that is unitary and anomaly-free. Anomaly
>> > freedom and unitarity constrains us to start with one of the 5 famous
>> > classes of superstring theory (type I, IIA, IIB, 2 types of
>> > Heterotic).
>
>> There are thousands more less famous ones, really. OA, OB,
>> SO(16)xSO(16) heterotic are standard examples. All are anomaly
>> free and unitary.
>
> The five he listed are the only known perturbative superstrings.

Nope.

> The Type 0 strings are another name for the bosonic string, and thus not
> superstrings.

No again. They don't have any spacetime fermions, but they are
still superstrings because they have worldsheet supersymmetry.

> The Heterotic SO(32) and Heterotic E8 x E8 are related
> by T-duality, and thus have must have a common subgroup for the gauge
> groups, which is SO(16) x SO(16), but this is not a separate
> perturbative superstring.

Yes, it is. Both SO(32) and E8xE8 heterotic are spacetime
supersymmetric. The SO(16)xSO(16) heterotic theory has spacetime
fermions but is not spacetime supersymmetric.

R.X.

unread,
Nov 9, 2003, 11:11:42 AM11/9/03
to
Aaron Bergman <aber...@physics.utexas.edu> wrote in message news:<abergman-BBE517.02051405112003@localhost>...

Right - and this point is a point of confusion almost everywhere,
especially for people who speak out against "extra dimensions".
A string model in 4d can have a lot of internal degrees of freedom,
like matter and gauge fields, and the precise field content and
their interactions depend largely of the particular vacuum state.
The point is that it is nowhere important that these degrees of
freedom stem from a compactification of higher dimensions; only in
certain limits there may be an interpretation of rolled up dimensions.
The generic case is non-geometric, which means that there is
no natural interpretation in terms of a manifold. Means sometimes also that
there is no energy scale above which the theory gains some
higher dimensional Lorentz-invariance - so it makes little sense to speak
about "compactification".

And even if there is one, a geometric interpretation is in general
highly ambigous, and thus non-universal: eg one and the same 4d
string theory may be interpreted in terms of an M-theory compactifcation
down from 11 dimensions, or equally well in terms of an F-theory
compactifications on a 4-fold, or a heterotic compactification on
a Calabi-Yau threefold, or a Type II string compactification on a
Calabi-Yau threefold with extra D-branes on top, or some non-geometrical
CFT made out of non-interacting free fields (the 1^9 Gepner model
tensor product is of the latter type, for example). Of course, a
particular one of these descriptions may be preferred for a given
parameter range, but that doesn't change the general point I want
to make: there is in general no concrete definite unambiguous
reality of any given background geometry, like a specific "D-brane
world" configuration. All these constructions are essentially different
ways to parametrize the theory, and perhaps visualize or emphasize
certain properties. And again, in the generic non-geometric phase,
there is no good geometrical description at all, at least in terms
of conventional geometry.

Punchline is that the antipathy against extra dimensions one has to read
all around here is basically pointless.

Aaron Bergman

unread,
Nov 9, 2003, 11:11:53 AM11/9/03
to

> Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message
> news:<m3eeob.u4d1.ln@xor>...
>
> > Perturbative string theory is a world-sheet theory. This means that
> > we mostly care about 2-dimensional conformal field theory. We start
> > by writing down a CFT that is unitary and anomaly-free. Anomaly
> > freedom and unitarity constrains us to start with one of the 5 famous
> > classes of superstring theory (type I, IIA, IIB, 2 types of
> > Heterotic).
>
> Aaron pointed out this is not correct.

But it's hardly a big deal/=,


>
> > We then have to make the total conformal anomaly of the
> > CFT vanish (by balancing off positive contributions from the
> > "internal" part of the CFT with the negative contributions from the
> > non-unitary ghost sector). If we do this by using a sigma model for
> > the internal CFT then this is the constraint that tells us that the
> > sigma model must have a 6-dimensional target space, giving 4+6=10
> > dimensions.
>
> 1) What is this 6 / 4 division there? Somehow
> you are automatically constraining yourself
> to Calabi-Yau compactifications?

4 noncompact dimensions which implies 6 other dimensions. The other 6
dimensions being a CY is one example, but there are others.

> 2) Are you saying that any sigma model CFT which
> is unitary and anomaly free is a string theory
> (in the usual sense, i.e. one of the 5)?

Once you start compactifying, dualities make it difficult to say which
string theory you're in. They're all part of the same moduli space,
after all.

> > But no-one forces us to choose the CFT to be a sigma model. There are
> > lots of other choices of CFT with the right conformal anomaly that are
> > not sigma models.
>
> So you're saying _any_ unitary and anomaly free CFT
> represents a string theory?

As long as you can cancel the conformal anomaly in the ghost (or
superghost) system.

> That on the non-perturbative
> level all of them are connected?

Some of them certainly are.

> I understand how you can
> change the Lagrangean by inserting vertex operators, but
> you can't change the field content.

Field content is not particularly well-defined, especially in 2
dimensions. For example, in bosonization, you can trade two fermions for
a boson. There are plenty of other examples.

> > Dimensions may dynamically compactify and decompactify.
>
> This cannot be, at least not globally. Are you saying
> some kind of a singular spacetime forms, with components
> of different dimension?
> Globally, different dimensions apparently correspond to
> different superselection sectors. For asymptotically
> flat string theory those are matrix models of different
> dimension, for asymptotically AdS string theory those
> are SCFTs of different dimension.

One hopes that everything can have a unified example. Others believe,
however, that the asymptotics of the spacetime are essential.

> > When the
> > target space becomes non-classical, then string theory produces
> > corrections to General Relativity in just the right way to smooth out
> > singular processes like topology change (also naked singularities,
> > etc), so everything is always smooth and non-singular from the point
> > of view of the string.
>
> This "smoothization" I don't understand. Where can I read about
> it?

Never heard of smoothization. You can read lots of fun stuff about all
of this in Brian Greene's lecture notes: hep-th/9702155 (which I
apparently don't remember as well as I ought to).

Aaron

Kris Kennaway

unread,
Nov 9, 2003, 10:30:58 PM11/9/03
to

On 2003-11-08, Squark <fii...@yahoo.com> wrote:

> Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message
> news:<m3eeob.u4d1.ln@xor>...

>> Perturbative string theory is a world-sheet theory. This means that
>> we mostly care about 2-dimensional conformal field theory. We start
>> by writing down a CFT that is unitary and anomaly-free. Anomaly
>> freedom and unitarity constrains us to start with one of the 5 famous
>> classes of superstring theory (type I, IIA, IIB, 2 types of
>> Heterotic).

> Aaron pointed out this is not correct.

OK, but it doesn't change my point.

>> We then have to make the total conformal anomaly of the
>> CFT vanish (by balancing off positive contributions from the
>> "internal" part of the CFT with the negative contributions from the
>> non-unitary ghost sector). If we do this by using a sigma model for
>> the internal CFT then this is the constraint that tells us that the
>> sigma model must have a 6-dimensional target space, giving 4+6=10
>> dimensions.

> 1) What is this 6 / 4 division there? Somehow
> you are automatically constraining yourself
> to Calabi-Yau compactifications?

You're constraining yourself to have 4 non-compact dimensions, because
that seems physically reasonable :-) You don't have to do this if you
don't want to..other possibilities are routinely considered.

> 2) Are you saying that any sigma model CFT which
> is unitary and anomaly free is a string theory
> (in the usual sense, i.e. one of the 5)?

More or less, with precise meanings of what those constraints are.
See e.g. Polchinski's book.

>> But no-one forces us to choose the CFT to be a sigma model. There are
>> lots of other choices of CFT with the right conformal anomaly that are
>> not sigma models.

> So you're saying _any_ unitary and anomaly free CFT
> represents a string theory? That on the non-perturbative
> level all of them are connected? I understand how you can
> change the Lagrangean by inserting vertex operators, but
> you can't change the field content.

Not every CFT comes from a Lagrangian. String theory is not always a
two dimensional CFT (e.g. M-Theory, which is the strong coupling limit
of IIA, is not believed to be described by a 2d CFT). In the 1980s it
was thought that 2d CFT was everything. Now we know that (because of
non-perturbative effects) this perturbative result is not true, and
understanding precisely what else it can be is a central problem in
string theory.

However, it's believed that a large class of string theory vacua
(e.g. all Calabi-Yau sigma models) - and perhaps all supersymmetric
vacua - are connected via smooth transitions in string theory. Some
of them, like topology-changing geometric transitions, are visible in
CFT. Some (e.g. those that connect "different" perturbative
superstring vacua like IIA and IIB, etc) are not, because you have to
go through regions in the string moduli space where it is no longer
described by a CFT.

>> Dimensions may dynamically compactify and decompactify.

> This cannot be, at least not globally. Are you saying
> some kind of a singular spacetime forms, with components
> of different dimension?

Because the target space of the string is a derived quantity, it is
not always classical. Indeed, it is only a classical manifold in
certain special limits (where all the scales of the manifold are large
compared to the string scale). When this is not true there is no
classical geometrical description of the manifold, and classical
geometry cannot be used to describe it. Fortunately there is a
well-developed mathematical framework (mirror symmetry) for analyzing
string theory away from the classical limits and making precise sense
of what the "quantum geometry" means.

> Globally, different dimensions apparently correspond to
> different superselection sectors. For asymptotically
> flat string theory those are matrix models of different
> dimension, for asymptotically AdS string theory those
> are SCFTs of different dimension.

You're still thinking in terms of quantum field theory on a manifold.
String theory is not a QFT, and it is particularly not a QFT on a
manifold, it just reduces to one in certain special limits. If you're
in a limit where the target space is classical, then yes, it behaves
like classical geometry :-)

>> When the
>> target space becomes non-classical, then string theory produces
>> corrections to General Relativity in just the right way to smooth out
>> singular processes like topology change (also naked singularities,
>> etc), so everything is always smooth and non-singular from the point
>> of view of the string.

> This "smoothization" I don't understand. Where can I read about
> it?

A good starting point is Brian Greene's lectures on Strings on CY
manifolds:

http://xxx.arxiv.org/abs/hep-th/9702155

It covers essentially everything we've been talking about in this
thread, so you should read it if you want to understand what we've
been talking about.

Kris

Lubos Motl

unread,
Nov 10, 2003, 1:45:46 AM11/10/03
to

On Sat, 8 Nov 2003, Squark wrote:

> > What do you think is a more reasonable point to expand about?
>
> There's no telling, really. "Realistic backgrounds" provide a
> closer contact with quantum field theory, while g = 0 with
> background invariance.

(Un)fortunately it has been showed - to the satisfaction of most
theoretical physicists - that your point of view is incorrect. The regime
g=0 is equivalent to the (sub)planckian regime - the distances are
essentially zero (at least, they are not much bigger than the Planck
scale) - and the laws and symmetries of classical general relativity are
not applicable in this regime.

It is not only string theory that teaches us that all these concepts
acquire heavy corrections in the Planckian regime, but string theory also
shows us that the general covariance is extended into a much more general
symmetry structure once the distance become very short.

All evidence that we have strongly suggests that the attempts to claim
that the principles and symmetries of GR hold at the Planck scale are
flawed. There is no evidence for GR at sub-20-micrometers scale, and
there is an overwhelming theoretical evidence that things become very
different at even shorter distance scales.

Loop quantum gravity practitioners try to artificially export the general
covariance into a regime where it is almost clearly non-applicable, but
their model fails to reproduce this symmetry - and the smooth space itself
- at all other distances scales (from the point of view of practical
reality, it fails at *all* distances scales). String theory works in the
opposite way. It reproduces the physics of GR (plus other stuff) that we
know - physics at realistic long-distance scales - but it also
quantitatively predicts that physics at very short distance scales is very
different than a naive extrapolation of everyday-life physics would lead
us to believe.

Lubos Motl

unread,
Nov 10, 2003, 1:59:56 AM11/10/03
to

On Sat, 8 Nov 2003, Squark wrote:

> Aaron pointed out this is not correct.

It depends on what you call "a superstring theory" - whether you require
spacetime supersymmetry, or worldsheet supersymmetry. Aaron uses the
old-fashioned terminology that involves worldsheet supersymmetry, and
therefore he sees many more theories than the five
spacetime-supersymmetric ones.

> 2) Are you saying that any sigma model CFT which
> is unitary and anomaly free is a string theory
> (in the usual sense, i.e. one of the 5)?

You should distinguish the words "string theory" and "a
solution/background of string theory". According to all the insights that
we have today, there is *one* string theory only that admits different
"classical solutions" or more precisely "vacua". Before the 2nd
superstring revolution, we thought that there were five
(spacetime-supersymmetric) superstring theories, but each of them still
admitted many solutions/backgrounds/vacua. After 1995, everything has been
unified.

Yes, any unitary and anomaly-free CFT (not necessarily a sigma model)
defines a perturbative expansion of (the same) string theory around
different points as long as you are also able to satisfy some rather mild
discrete conditions for the torus amplitudes. These conditions are called
"modular invariance" and they essentially demand that the theory (and its
spectrum) must be invariant not only under the worldsheet coordinate
transformations that are continuously connected to the identity, but also
under the "large diffeomorphisms" whose classes form the group SL(2,Z) in
the case of the torus.

> So you're saying _any_ unitary and anomaly free CFT
> represents a string theory?

Up to some technicalities, the answer is "yes, sure".

> That on the non-perturbative
> level all of them are connected?

Yes, they are. Either off-shell or on-shell - in some cases you don't even
need to invest energy to connect them.

> I understand how you can change the Lagrangean by inserting vertex
> operators, but you can't change the field content.

String theory allows you to change not only the field content, but also
topology of your space and anything like that. These different phases are
either completely equivalent and they describe the same portions of the
moduli space - and we can prove the exact equivalence - or they are
connected via critical transitions that we can also study very well. For
example, the topology of the internal Calabi-Yau space can change once a
3-cycle shrinks to zero volume (conifold transition, to be discussed
below). New nonperturbative states (wrapped D3-branes) become massless.
The perturbative description of the original theory that ignored these
objects breaks down. But one can count their effects explicitly and
everything works. A new branch of the moduli space - a new family of
shapes - emerges.

String theory can also change the number of other massless fields in
spacetime, much like those on the worldsheet. For example, it can suddenly
create 29 hypermultiplets in 6 dimensions, once a tensor multiplet
disappears. Hundreds of dualities between different descriptions of string
theory are known, and they imply that string theory is a unified
framework. Even though it is far from obvious, the Landau-Ginzburg models,
Gepner models, nonlinear sigma models on spaces with different topology
and a plenty of other things are unified under the same umbrella called
string/M-theory. (In the previous sentence, I was focusing on vacua that
have 4 large dimensions and an equivalent of 6 compactified ones - but the
statement holds generally.)

> This cannot be, at least not globally. Are you saying
> some kind of a singular spacetime forms, with components
> of different dimension?
> Globally, different dimensions apparently correspond to
> different superselection sectors.

That's correct as long as these two states differ by their asymptotic
behavior at infinity. This is not necessarily the case. In principle you
can create a bubble of a Universe - that would otherwise belong to a
different superselection sector - inside your Universe. (Tom Banks
discussed the practical limitations of such a procedure - often you would
need so much energy that a black hole would form - but morally it is still
true that a piece of a "different Universe" can be created inside our
Universe.)

> For asymptotically flat string theory those are matrix models of
> different dimension, for asymptotically AdS string theory those are
> SCFTs of different dimension.

That's right, all these things belong to different superselection sectors.
But one can't forget that the local physics described by N=4 d=4 SCFT is
still type IIB string theory, and one can study all the local
perturbations of type IIB stringy space that the theory predicts.

> This "smoothization" I don't understand. Where can I read about
> it?

If you mean why string theory makes topology change smooth, I recommend
you to start with The Elegant Universe by Brian Greene (chapters 11,13),
and only afterwards you should look at the technical papers that Brian
describes in popular terms. The papers related to the chapter 13 are

http://arxiv.org/abs/hep-th/9504090
http://arxiv.org/abs/hep-th/9504145

An easier topology transition is the flop (chapter 11 of The Elegant
Universe). It works even perturbatively. See

http://arxiv.org/abs/hep-th/9301043
http://arxiv.org/abs/hep-th/9309097
http://arxiv.org/abs/hep-th/9301042

The first papers use the tools of mirror symmetry - the physical
equivalence between two Calabi-Yau shapes whose Hodge diamonds differ by
a mirror reflection.

The last paper of Witten also explains the equivalence between Calabi-Yau
nonlinear sigma model and the Landau-Ginzburg models, and it explains why
worldsheet instantons - Brian's "protective stringy shields" - protect the
Universe against the disaster implied by the singularity that is
calculated classically. The worldsheet instantons exactly compensate all
the discontinuities that you would expect from a classical topology
transition.

Cheers,
Lubos

Squark

unread,
Nov 11, 2003, 2:39:37 PM11/11/03
to

Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message news:<neojob.agv1.ln@xor>...

> On 2003-11-08, Squark <fii...@yahoo.com> wrote:
>
> > Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message
> > news:<m3eeob.u4d1.ln@xor>...
> >> We then have to make the total conformal anomaly of the
> >> CFT vanish (by balancing off positive contributions from the
> >> "internal" part of the CFT with the negative contributions from the
> >> non-unitary ghost sector). If we do this by using a sigma model for
> >> the internal CFT then this is the constraint that tells us that the
> >> sigma model must have a 6-dimensional target space, giving 4+6=10
> >> dimensions.

So here you only say string theory predicts
10 spacetime dimensions, assumin spacetime
itself a priori. I knew that.



> > 2) Are you saying that any sigma model CFT which
> > is unitary and anomaly free is a string theory
> > (in the usual sense, i.e. one of the 5)?
>
> More or less, with precise meanings of what those constraints are.
> See e.g. Polchinski's book.

Umm, frankly it's kinda hard to believe. For instance,
any TQFT is also a CFT. Most of the TQFTs I know are
gauge theories, but gauge theories can also be thought
of as sigma models, the target space perhaps being
quite spooky. They certainly can be unitary and they
are anomaly-free more or less by definition.



> Not every CFT comes from a Lagrangian. String theory is not always a
> two dimensional CFT (e.g. M-Theory, which is the strong coupling limit
> of IIA, is not believed to be described by a 2d CFT). In the 1980s it
> was thought that 2d CFT was everything. Now we know that (because of
> non-perturbative effects) this perturbative result is not true, and
> understanding precisely what else it can be is a central problem in
> string theory.
>
> However, it's believed that a large class of string theory vacua
> (e.g. all Calabi-Yau sigma models) - and perhaps all supersymmetric
> vacua - are connected via smooth transitions in string theory.

This still doesn't explain how the metric "arises" in
string theory. Okay, so string theory has sectors without
any metric, that's not a point to the string theory score
by itself. If you could formulate elegant basic principles
for string theory which don't mention the notion metric
then your claim would be right.

> >> Dimensions may dynamically compactify and decompactify.
>
> > This cannot be, at least not globally. Are you saying
> > some kind of a singular spacetime forms, with components
> > of different dimension?
>
> Because the target space of the string is a derived quantity, it is
> not always classical.

> ...


> > Globally, different dimensions apparently correspond to
> > different superselection sectors. For asymptotically
> > flat string theory those are matrix models of different
> > dimension, for asymptotically AdS string theory those
> > are SCFTs of different dimension.
>
> You're still thinking in terms of quantum field theory on a manifold.
> String theory is not a QFT, and it is particularly not a QFT on a
> manifold, it just reduces to one in certain special limits.

Firstly, the AdS/CFT correspondence states string theory
_is_ the CFT, not just a certain limit of is. At least
the strong form of the AdS/CFT correspondence. The same,
I think, is valid for matrix theory.
Secondly, I don't see how anything of what you said
addresses my point of different dimensions corresponding
to different superselection sectors, which goes against
dynamical compactification / decompactification, at least
globally.

Aaron Bergman

unread,
Nov 11, 2003, 4:07:17 PM11/11/03
to

In article <939044f.03111...@posting.google.com>, Squark wrote:
>
> Umm, frankly it's kinda hard to believe. For instance,
> any TQFT is also a CFT. Most of the TQFTs I know are
> gauge theories, but gauge theories can also be thought
> of as sigma models, the target space perhaps being
> quite spooky. They certainly can be unitary and they
> are anomaly-free more or less by definition.

And there's something called topological string theory. It has a
number of interesting mathematical applications.

Kris Kennaway

unread,
Nov 11, 2003, 4:08:22 PM11/11/03
to

On 2003-11-11, Squark <fii...@yahoo.com> wrote:

> So here you only say string theory predicts
> 10 spacetime dimensions, assumin spacetime
> itself a priori. I knew that.

Congratulations :)

>> > 2) Are you saying that any sigma model CFT which
>> > is unitary and anomaly free is a string theory
>> > (in the usual sense, i.e. one of the 5)?
>>
>> More or less, with precise meanings of what those constraints are.
>> See e.g. Polchinski's book.
>
> Umm, frankly it's kinda hard to believe. For instance,
> any TQFT is also a CFT. Most of the TQFTs I know are
> gauge theories, but gauge theories can also be thought
> of as sigma models, the target space perhaps being
> quite spooky. They certainly can be unitary and they
> are anomaly-free more or less by definition.

Yes, many topological field theories are string theories. There's
extensive literature on this subject going back to the 1980s.

>> Not every CFT comes from a Lagrangian. String theory is not always a
>> two dimensional CFT (e.g. M-Theory, which is the strong coupling limit
>> of IIA, is not believed to be described by a 2d CFT). In the 1980s it
>> was thought that 2d CFT was everything. Now we know that (because of
>> non-perturbative effects) this perturbative result is not true, and
>> understanding precisely what else it can be is a central problem in
>> string theory.
>>
>> However, it's believed that a large class of string theory vacua
>> (e.g. all Calabi-Yau sigma models) - and perhaps all supersymmetric
>> vacua - are connected via smooth transitions in string theory.
>
> This still doesn't explain how the metric "arises" in
> string theory. Okay, so string theory has sectors without
> any metric, that's not a point to the string theory score
> by itself. If you could formulate elegant basic principles
> for string theory which don't mention the notion metric
> then your claim would be right.

I have no idea what you're asking for. I'm tempted to speculate that
you haven't really gone through some of the basics of string theory,
like how Einstein's equations emerge from the worldsheet formalism.

>> You're still thinking in terms of quantum field theory on a manifold.
>> String theory is not a QFT, and it is particularly not a QFT on a
>> manifold, it just reduces to one in certain special limits.
>
> Firstly, the AdS/CFT correspondence states string theory
> _is_ the CFT, not just a certain limit of is.

Yes, when we fix the string theory to be a sigma model onto a fixed,
classical target space. That's not in conflict with anything I've
said.

In other examples of gauge theory/gravity duality, precisely the kind
of topology changing transitions I've mentioned are crucial (see
e.g. http://arxiv.org/abs/hep-th/0103067 for one well-known example).

> The same,
> I think, is valid for matrix theory.

I don't know anything about matrix theory, so I can't speak to this.

> Secondly, I don't see how anything of what you said
> addresses my point of different dimensions corresponding
> to different superselection sectors, which goes against
> dynamical compactification / decompactification, at least
> globally.

I don't understand your objection (superselection sectors of what?) -
can you phrase it more precisely?

Kris

Squark

unread,
Nov 11, 2003, 8:19:55 PM11/11/03
to
Lubos Motl <mo...@feynman.harvard.edu> wrote in message
news:<Pine.LNX.4.31.03110...@klein.physics.harvard.edu>...

> It is not only string theory that teaches us that all these concepts
> acquire heavy corrections in the Planckian regime, but string theory also
> shows us that the general covariance is extended into a much more general
> symmetry structure once the distance become very short.

Unfortunatelly, the way string theory is done today fails to
provide us even with a description covariant even under "usual"
general covariance.

Lubos Motl

unread,
Nov 12, 2003, 1:37:59 PM11/12/03
to
On 11 Nov 2003, Squark wrote:

> Unfortunatelly, the way string theory is done today fails to
> provide us even with a description covariant even under "usual"
> general covariance.

String theorists are using effective field theory descriptions of stringy
vacua - and general covariance in these descriptions is completely
"usual". However the usual general covariance cannot define the symmetry
of the full theory at the fundamental scale. We have even indirect
experimental evidence for this fact - let me call it this way.

And string theory *does* correctly confirm that general covariance is
*not* usual at the fundamental scale. Without this virtue, string theory
could hardly be a consistent theory. If geometry and general covariance
were exact notions at the Planck scale in this theory, the theory would
most likely be as inconsistent as other failed attempts to quantize
gravity, for example loop quantum gravity.

The assumption that general covariance works at the Planck scale is a
totally naive, unjustified and illegitimate extrapolation of the laws
known from macroscopic (astronomical, in particular) scales to the
ultramicroscopic realm, and today we know beyond reasonable doubt that
this extrapolation is incorrect.

Squark

unread,
Nov 13, 2003, 4:13:49 AM11/13/03
to
Kris Kennaway <kk...@rot13.obsecurity.org> wrote in message news:<hdirob.g5r.ln@xor>...

> On 2003-11-11, Squark <fii...@yahoo.com> wrote:
>
> > So here you only say string theory predicts
> > 10 spacetime dimensions, assumin spacetime
> > itself a priori. I knew that.
>
> Congratulations :)

Thank you.

> Yes, many topological field theories are string theories. There's
> extensive literature on this subject going back to the 1980s.

How can a TQFT be a perturbative description of
a theory that contains something which is not
TQFT?

> I have no idea what you're asking for. I'm tempted to speculate that
> you haven't really gone through some of the basics of string theory,
> like how Einstein's equations emerge from the worldsheet formalism.

As always with you stringists, nothing but
speculations ;-) You said something like
that the metric in string theory arises
without having to be put in by hand. I claim
that you have a good case for the metric
being dynamical, but practically none for it
"arising" without having to be put in by hand.
String theory is perturbation theory (of
whatever) around that or other fixed metric
background, so metric is there from starters.
A claim like yours sounds as if string theory
provides a philosophical explanation for the
appearance of metric in general.



> > Firstly, the AdS/CFT correspondence states string theory
> > _is_ the CFT, not just a certain limit of is.
>
> Yes, when we fix the string theory to be a sigma model onto a fixed,
> classical target space.

I disagree. The only thing you do is limit yourself
to a subset of the superselction sectors by required
AdS spacetime asymptotics. Probably there is no
consistent way to perform a reduction on the full
non-perturbative string theory which is anything
besides fixing superselction rules and get something
interesting.
The string theory here is _not_ a sigma model onto
a fixed classical target space - it is
nonperturbative and has string theories with various
asymptotically AdS spacetimes as perturbative
expansions.

> > Secondly, I don't see how anything of what you said
> > addresses my point of different dimensions corresponding
> > to different superselection sectors, which goes against
> > dynamical compactification / decompactification, at least
> > globally.
>
> I don't understand your objection (superselection sectors of what?) -
> can you phrase it more precisely?

I claim that dimension can't compactify / decompactify
dynamically, since string theories with different number of large
spacetime dimensions correspond to different superselection
sectors of the nonperturbative theory ("M-theory"), and therefore,
no dynamical transition between them is possible. If something in
this spirit is possible, then only locally.

Squark

unread,
Nov 15, 2003, 12:35:24 PM11/15/03
to

"Lubos Motl" <mo...@feynman.harvard.edu> wrote in message
news:Pine.LNX.4.31.031112...@feynman.harvard.edu...

> String theorists are using effective field theory descriptions of stringy
> vacua - and general covariance in these descriptions is completely
> "usual".

This is not interesting. What is interesting is the (appropriate form
of) general covariance of the stringy (alpha') corrections.

> However the usual general covariance cannot define the symmetry
> of the full theory at the fundamental scale.

I guess you're saying that since there is evidence for
spacetime non-commutativity on the Placnk scale, and
since there are vacua without geometric interpretation?

> We have even indirect
> experimental evidence for this fact - let me call it this way.

No idea what are you talking about here.

> And string theory *does* correctly confirm that general covariance is
> *not* usual at the fundamental scale.

Ok, so give me "unusual" general covariance :-)
The point is that there are no known observables
in string theory which have clear physical meaning
except the S-matrix. It is not clear how to describe
processes that are localized in a finite region of
spacetime, like me writing this post.

Btw, AdS/CFT seems to be manifestly covariant,
with general covariance in the usual sense.
However, it suffers from the same problem.

Jason

unread,
Nov 16, 2003, 5:09:10 AM11/16/03
to
Squark wrote:

> but gauge theories can also be thought
> of as sigma models, the target space perhaps being
> quite spooky. They certainly can be unitary and they
> are anomaly-free more or less by definition.

Really? Can you spell out the details?


Thomas Larsson

unread,
Nov 19, 2003, 10:46:21 AM11/19/03
to
Jason <pri...@excite.com> wrote in message news:<bou49c$vs$1...@news.udel.edu>...

A gauge theory can be viewed as a sigma model in loop space. Polyakov
investigated this in 1979-80, and I think that it led him to invent the
Polyakov action for string theory in 1981. Basically the idea is that
if you have a gauge theory in Wilson loop space, the zero-curvature condition
means that the gauge potential, which lives on surfaces, can be expressed
in terms of fields living on loops. The analogous thing in ordinary gauge
theory is this: if F = 0, A = g^-1 dg, and we get a sigma model defined
in terms of the fields g living on points.

I can probably dig up a reference somewhere if you want to. There must also
be some kind of relation to gerbes and 2-groups and the like.

Aaron Bergman

unread,
Nov 19, 2003, 4:00:18 PM11/19/03
to
In article <24a23f36.03111...@posting.google.com>,
thomas_l...@hotmail.com (Thomas Larsson) wrote:

> Jason <pri...@excite.com> wrote in message news:<bou49c$vs$1...@news.udel.edu>...
> > Squark wrote:
> >
> > > but gauge theories can also be thought
> > > of as sigma models, the target space perhaps being
> > > quite spooky. They certainly can be unitary and they
> > > are anomaly-free more or less by definition.
> >
> > Really? Can you spell out the details?
>
> A gauge theory can be viewed as a sigma model in loop space.

Lots of people would like to reformulate gauge theories as string
theories, but it hasn't been done in generality, yet. The AdS/CFT
conjecture is really the best realization of this old idea.

Though, Witten just gave a talk at Rutgers entitled: "Yang-Mills
Perturbation Theory, Strings and Twistor Space" and has an upcoming talk
at Princeton entitled: "Perturbative Yang-Mills Theory As A String
Theory in Twistor Space". So, maybe there's more news.

Aaron

Peter Woit

unread,
Nov 20, 2003, 4:41:16 AM11/20/03
to
Jason wrote:

One way to do this is to use a Grassmanian as
target space. Think of it this way: the Grassmanian
G(k,n) of k-planes in n-space has a tautological vector
bundle of dim k above it (fiber above a point is the
plane specified by the point). You can use the standard
metric in n-space to relate neighboring fibers, so there
is a kind of canonical connection on this bundle.

Now, it turns out that every dim k vector bundle with
connection is a pull-back of this tautogical bundle with
connection, for some map into some G(k,n) with n large
enough. So, fixing n, using G(k,n) as your target space,
the space of all maps into it will give you some subset
of all the connections. For large enough n, you get them
all.

Problems: this map between connections (gauge fields)
and sigma model fields is not one-to-one, and the
path integral measure you would like to believe in for
connections is related in a complicated way to the one
you would like to believe in for sigma models. So its
not at all clear that this is a particularly useful idea.

Lubos Motl

unread,
Nov 24, 2003, 9:09:55 AM11/24/03
to
On 15 Nov 2003, Squark wrote:

> This is not interesting. What is interesting is the (appropriate form
> of) general covariance of the stringy (alpha') corrections.

Of course, if you express these corrections perturbatively, like the
Riemann tensor squared and contracted in some way, the theory will still
be generally covariant in the usual sense, because R^2 is still a tensor.
Is this what you were asking about?

> I guess you're saying that since there is evidence for
> spacetime non-commutativity on the Placnk scale, and
> since there are vacua without geometric interpretation?

You can call it "spacetime non-commutativity", although its precise nature
is not as simple as the word "non-commutativity" suggests. Yes, you
mentioned two examples of the arguments; there are many more - dualities
transforming geometries, dualities between geometry and other fields,
topology change and so on.

> > We have even indirect experimental evidence for this fact - let me
> > call it this way.

> No idea what are you talking about here.

We can determine Einstein's equations from the observations, and
extrapolate their effects to determine Planckian physics using the tools
of QFT. At any rate, we obtain a meaningless answer. Any local theory
based on the metric that exists at *any* scale, including the fundamental
scale, will require infinitely many terms to be fine-tuned, and therefore
it can't be predictive.

> Ok, so give me "unusual" general covariance :-)

OK. Not sure why your question ends with the smiling face.

String field theory has an "unusual" general covariance whose parameter is
a whole string field. If one works with the (slightly problematic) closed
string field theory, the parameters that determine the diffeomorphism will
be just one small part of an infinitely bigger family of parameters that
determine the symmetry transformation.

In this sense, a physicist who only wants to understand general covariance
forces himself to see just an infinitesimal small droplet of the truth.

> The point is that there are no known observables
> in string theory which have clear physical meaning
> except the S-matrix.

This is not the case of string field theory. In fact, it is also not the
case of the light-cone formulations of string/M-theory, namely Matrix
theory that also defines the light-cone Hamiltonian - and unlike string
field theory, it predicts the correct dynamics of all closed strings and
other objects.

Yes, if one does not want to fix the gauge, there will be no local
gauge-invariant operators in a gravitational theory. But this is a direct
consequence of general relativity - assuming that we know how to analyze
it - and of course, every consistent theory must confirm this fact: there
are no local gauge-invariant operators in a gravitational theory.

String theory, of course, confirms this fact, otherwise it would be
inconsistent.

> It is not clear how to describe
> processes that are localized in a finite region of
> spacetime, like me writing this post.

Such local phenomena can only be described approximately in a
gravitational theory. This is just another fact.

> Btw, AdS/CFT seems to be manifestly covariant,
> with general covariance in the usual sense.
> However, it suffers from the same problem.

The CFT description does not make the bulk gravity manifest, and therefore
it does not explain the origin of the general covariance. Of course, the
gravitational low-energy physics is *identical* to physics of a theory
that would be derived from general covariance. But this is the case not
only in the AdS/CFT correspondence; it is generally true in string theory.
Well, this is the reason why string theory is a theory of quantum gravity
and why we say that it *predicts* gravity. The low-energy physics can
always be described by an effective field theory that includes the metric
plus general covariance.

Best
Lubos

Florian Gmeiner

unread,
Dec 1, 2003, 11:24:07 AM12/1/03
to
"Jeffery" <jeffery...@mail.com> schrieb im Newsbeitrag
news:575262ce.03110...@posting.google.com...

It is true that type 0 strings do not contain fermions,
but this is *NOT* another name for the bosonic string!
They are more related to type II stringtheories (by
limits of twisted compactifications).
If you call theories only "different" if they are not related
by dualities then there is only one string theory.

Florian

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