What Has Happened So Far

March 27, 2008

Posted by Urs Schreiber

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The $n$ -Category Café has recently passed beyond $6 \cdot 10^2$ entries, $1.3 \cdot 10^3$ trackbacks and $1\cdot 10^4$ comments. Maybe a good time to look back at what has happened so far.

Our subtitle says “A blog on math, physics and philosophy”. For me, there is one major question sitting at the intersection of these three subjects. It is

The fundamental question of quantum physics: What is a $\Sigma$ -model, really?

I have been exclusively talking about this question ever since we started the blog. I started referring to it as the question of the QFT of the charged $n$ -particle #. I still think this is the more descriptive term, but it was rightly indicated to me that it is not politically advisable for somebody in my position to make up new terminology.

Since it was also pointed out to me ## that it may at times be hard to remember the big picture, let me recall:

The proposed answer to the fundamental question of quantum physics: Pull-push of nonabelian differential cocycles.

We are in the setting of general cohomology theory, where generalized/homotopy/ana-morphisms $X \stackrel{\nabla}{\to} \mathbf{B}G$ between “spaces” (usually # presheaves with values in a homotopy category) are “cocycles” encoding higher fiber bundles. And also higher fiber bundles with connection, which are addressed as (nonabelian) differential cocycles #.

Given a (nonabelian, differential) cocycle on $X$ , and given another “space” $\Sigma$ , there is a canonical way to obtain a cocycle on $\Sigma$ : we pull-push $\nabla$ through the correspondence $\array{ && hom(\Sigma,X)\otimes \Sigma \\ & {}^{ev}\swarrow && \searrow^{p_2} \\ X&&&& \Sigma \\ \nabla && \stackrel{\Gamma_\Sigma ev^*(-)}{\mapsto} && \Gamma_\Sigma( ev^* \nabla ) }\,.$

The pullback along $\mathrm{ev}$ (followed by the hom-adjunction) is transgression of the cocycle on $X$ to a cocycle on $hom(\Sigma,X)$ . The push-forward along $p_2$ is “taking sections” ## #.

Usually the push-forward along $p_2$ won’t exist. The chances that it exists increase when the original cocycle is pushed-forward along a representation $\rho : \mathbf{B} G \to n\mathrm{Vect} \,.$

In the context of quantum physics, $X$ is the target space in which an “( $n-1$ )-brane” (= $n$ -particle) with worldvolume # of shape $\Sigma$ propagates and is charged # # under a background field $\rho_* \nabla$ . The pull-push $\Gamma_{\Sigma} ev^*(-)$ is quantization in the extended/localized # sense of Freed ##. $\Gamma_{\Sigma} ev^*(\nabla)$ is the Schrödinger picture # propagation. Applying an endomorphism functor sends it to the Heisenberg picture # of AQFT #. Since quantization sends differential cocycles to differential cocycles, we can iterate. This is second quantization #.

While following through this program, we ran into one big puzzle, concerning the proper nature of $n$ -curvature: it turned out that a differential cocycle “with values in $\mathbf{B}G$ ” is actually a certain constrained generalized morphism into # $\mathbf{B E}G$ . Understanding that funny shift in dimension properly used up maybe 50 percent of my time here, and is probably the reason if the effort looked less than coherent at times.

Making recourse to the “rationalized” approximation of $L_\infty$ -connections # the pattern was finally understood, and now there are very nice relations emerging # between this question and major programs of my co-bloggers: higher topos theory and geometric representation theory/groupoidification.
There is one main class of examples which motivates all this effort: quantization of # (higher) Chern-Simons bundles with connection to Chern-Simons QFT ## and its holographic # #boundary theory. Indeed, the realization # that the known modular category theoretic formulation of 2-dimensional CFT # # was in fact secretly a differential cocycle was what originally lead to the proposed answer above. This is being worked out with Jens Fjeldstad #.

The hardest part of figuring out the pull-push of a given cocycle is in top dimension. This is no surprise, since there it must reproduce the “path integral”. But first consistency checks in simple toy examples suggest that it does work # # allright.

But with the big picture finally stabilizing, many details need to be worked out further.

Posted at March 27, 2008 10:47 AM UTC

TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1643

7 Comments & 2 Trackbacks

Re: What has happened so far

Dear Urs,

you posted on Peter Woit’s blog:

Bosonic 2d CFTs of central charge 26 correspond to effective target spaces which are 26-dimensional manifolds only in a tiny subset of the space of all such CFTs, namely those that are entirely of the naive sigma-model type with large flat dimensions.

Supersymmetric 2d CFTs of central charge 15 correspond to effective target spaces which are 10-dimensional manifolds only in a tiny subset of the space of all such CFTs, namely those that are entirely of the naive sigma-model type with large flat dimensions.

That there is and has been so much focus on such utterly over-simplistic CFTs as string backgrounds is the problem whose very point we are discussing: the space of all CFTs is little understood and everybody has been searching the key under the lamppost which illuminates only the most elementary examples.

Not that it is guaranteed that the key actually is somewhere else in the currently dark realms of CFT-land, but it is certainly premature to make statements about the shape of the key from what we can find under this tiny spotlight.

To appreciate the situation, it may help to simplify it drastically and marvel at how complicated it still is:

As Roggenkamp and Wendland show #, and especially Yan Soibelman describes # in a big (unfortunately not yet published) opus, a 2dCFT encodes a categorification (2-dimensional version) of a Connes spectral triple. The effective target space described by the CFT regarded as a string background is the spectral geometry encoded by the point particle limit of that spectral triple. So when we are talking about string backgrounds, we are talking about a vast generalization of ordinary spectral (”noncommutative”) geometry, which itself is a vast generalization of ordinary geometry.

Alain Connes picks # a certain spectral triple that encodes a target space which is a weird non-commutative space and argues that it comes close to encoding the standard model. Nobody complains that he picks that spectral triple from a huge “landscape”, namely from the space of all spectral triples. Remarkably, his spectral triple has K-theoretic dimension 10. Suppose this arises as the point-particle limit of a 2-spectral triple a la Soibelman, i.e. from a 2dSCFT of central charge 15. Then, clearly, this won’t be of sigma-model type and will not describe an effective target which is a 10-dimensional manifold.

All this mostly shows one thing: we know so shockingly little about the space of all 2dCFTs and yet are used to hearing so shockingly many claims about what it looks like.

So perturbative string theory does not predict that spacetime is 10-dimensional. What it does predict (essentially as its fundamental hypothesis!) is that spacetime is the effective target geometry of a 2dSCFT of central charge 15. That’s all.

10-dimensional manifolds appear here only in the most simple minded examples. Claiming that string theory predicts 10-dimensional spacetime is exactly like claiming that general relativity predicts flat empty Minkowski spacetime. No, it does not. This just happens to be the most simple solution that comes to mind.

Precisely similar comments apply to heterotic models which predict 496 gauge bosons.

I thought that the models that are uniquely said to be truely superstrings. Or at least that is what seems but taking a daily look at hep-th. Some trolle people even troll to say that those are uniquely the true ones. I mean, that’s the impression I take by looking everywhere…

At that time I thought ( in an aesthetical way, because we always think of this theories as pants and strings and sponges…) of the foam models and (perhaps) LQG could be adapted to string theories… But right now, on this post, you pointed to this, and I saw Eric’s post

http://golem.ph.utexas.edu/string/archives/000813.html

In a sense, are you and J. Baez trying to unify “non string theories” and “string theories” ?

Daniel de França

Posted by: Daniel de França MTd2 on March 27, 2008 2:38 PM | Permalink | Reply to this

Re: What has happened so far

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Daniel wrote:

I thought that…

Right, and I tried to point out that this is, while the impression one is likely to get by following much of the discussion, not the full story.

Let’s recapitulate:

Perturbative string theory starts with the following observation:

The amplitudes which are assigned to a given Feynman diagram when doing the pertubative quantization of ordinary quantum field theory as a sum over Feynman diagrams is itself the correlator of a 1-dimensional QFT on the worldline, i.e. on that diagram.

That worldline QFT comes from the action functional of the Klein-Gordon particle and its generalizations and can be rewritten equivalently in Polyakov form, where it looks like 1-dimensional gravity coupled to scalar and spinor fields.

The precise choice of the 1-dimensional QFT on the worldline (on the graphs) determines which “effective target space” the perturbation expansion is computing field theory amplitudes on. For instance if there are simply four free bosonic scalar fields in the worldline QFT, the effective target space will appear to be 4-dimensional Minskowski space.

That’s the state of affairs for ordinaty perturbative field theory. The basic premise of perturbative string theory is:

let’s see what happens to the perturbation expansion when we lift the dimension in this setup by one, replace graphs by 2-dimensional manifolds and assign amplitudes to these by computing the correlators of 2-dimensional gravity coupled to certain fields on these surfaces.

This is arguably a natural generalization, and it is natural to wonder what the outcome will be.

In two dimensions one can still rather easily make sense of the quantization of general relativity, so this is tractable. One notices that the Polyakov action in two dimensions is scale invariant and declares this to be a property which one wants to retain after quantization. This is another assumption: after allowing ourselves to generalize from 1-dimensional to 2-dimensional worldvolumes, we constrain ourselves next to only those worldvolume theories which retain the 2-dimensional scale invariance after quantization. (One could drop this assumption, and some people do. The result is called the “non-critical string” then. But let’s not get into that now.)

Then after working a bit, one finds that the theories of 2-dimensional gravity satisfying this constraint are determined by ordinary (non-gravitational) 2-dimensional conformal field theories which have

- central charge 26 if they are bosonic

- central charge 15 if they are supersymmetric.

The restriction on the central charge here is a consequence of the assumption that we require scale invariance in the quantum theory. The central charge is a measure for the failure of that scale invariance. But as one derives 2d conformal field theory from 2-d gravity as indicated above, one proceeds via a gauge fixing procedure which produces a certain transformation weight into the game (called the Fadeev-Popov determinant) which can itself be represented by a “ghost CFT”. That ghost CFT happens to have central charge -26 in the bosonic case and -15 in the supersymmetric case. Hence for the total central charge to vanish we need to require the above constraints.

The outcome of this observation is that for every 2d (S)CFT of central charge (15)26 one obtains one generalization of the Feynman perturbation series for quantum field theory from graphs to surfaces. This sum then is called the “string perturbation expansion” around the “background” given by that choice of 2d CFT.

The reason for addressing the choice of worldvolume QFT as a “background” here is as before in the $d=1$ -case: the nature of the worldvolume QFT induces an “effective target space” geometry.

In particular, if our 2d(S)CFT is that of

- 26 free bosons on the worldsheet in the non-supersymmetric case

- 10 free bosons and 10 free spinors on the worldsheet in the supersymmetric case

then it has the right central charge and describes an effective target space which is 26- respectively 10-dimensional Minkowski space.

Every free boson contributes 1 to the central charge. Every free fermion contributes $1/2$ to the central charge.

So the central charge of the 2dCFT, or $2/3$ of it in the supersymmetric case, is a measure for a “dimensionality” of target space. But then one finds that there are more intricate 2-dimensional (S)CFTs than those consisting just of free fields, and they correspond to “effective target space” which do not look like 26 or 10-dimensional manifolds. A popular example are CFTs containing direct summands which are “Gepner models”.

One knows in some special cases and imagines in general that these “non-geometric phases” of effective target space encoded in 2-d (S)CFTs can “continuously be connected” to 2dCFTs which do correspond to 26 or 10-dimensional target manifolds. One can see that the CFTs not of this type encode target spaces which are of some generalized form where some of the dimensions have been “compactified” (meaning here: there metric size has shrunk away) and left a “non-commutative” or other remnant.

This is very similar (and, as I tried to point out, actually a generalization of) a phenomenon in spectral “noncommutative” geometry:

there are spectral triples which from some perspective (namely from the perspective of vector bundles living over their effective target space) appear to be 10-dimensional, but which really are nothing like 10-dimensional manifolds (but rather, in this case, like a 4-dimensional manifold times a weird “noncommutative space of tiny size”).

Alain Connes found # one of these which comes very close to encoding the effective target space and its field content which we see around us.

I keep mentioning these Connes spectral triples because my feeling is that comparing perturbative string theory to them may be pedagogically useful in explaining how in string theory 2d(S)CFTs encode effective target spaces.

And in fact, this comparison is way more than a mere analogy, as I tried to indicate.

Posted by: Urs Schreiber on March 27, 2008 4:30 PM | Permalink | Reply to this

Re: What has happened so far

Hi Urs,

One can see that the CFTs not of this type encode target spaces which are of some generalized form where some of the dimensions have been “compactified” (meaning here: there metric size has shrunk away) and left a “non-commutative” or other remnant.”

Can you give us a small list of literature, or at least key words to use at google, to see examples of this generalization?

Thanks! :) :) :) :D

Posted by: Daniel de França MTd2 on March 27, 2008 5:20 PM | Permalink | Reply to this

Re: What has happened so far

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One can see that the CFTs not of this type encode target spaces which are of some generalized form where some of the dimensions have been “compactified” (meaning here: there metric size has shrunk away) and left a “non-commutative” or other remnant.

Can you give us a small list of literature, or at least key words to use at google, to see examples of this generalization?

The most famous example is related to the flop transition of Calabi-Yau manifolds:

starting with a $\Sigma$ -model which describes a string propagating on a Calabi-Yau, manifold, the size of parts of the Calabi-Yau may be “shrunk away”. It turns out that this geometrically discontinuous process actually corresponds to a continuous path in the space of 2dCFTs, passing through a “non-geometric phase” called a “Gepner model” CFT and emerging on the other side as a “Landau-Ginzburg model”.

You can find a review of that in

Brian Greene, String Theory on Calabi-Yau Manifolds .

On the other hand, this particular example, while very interesting, does not quite make my point as well, since one interesting aspect here is that, while the geometric interpretation first gets lost as we pass through the Gepner point, it then turns out that the resulting non-geometric CFT is equivalent to one which happens to have a geometric interpretation after all. That’s why it’s called a “flop transition”: the geometry does something geometrically impossible, sort of an “inside out flop” but reemerges as geometry.

Some general remarks on geometric and non-geometric background are in

Michael R. Douglas, Topics in D-geometry.

The second item in his list in the introduction has the point I was talking about

Perturbative string compactification can be defined non-geometrically, by specifying an appropriate internal CFT.

Saying “internal CFT” here means that we take the $c=15$ SCFT which we need as the direct sum of one “external” CFT with central charged $c_1$ whose effective target is a geometric target space, with an arbitrary abstractly (algebraically) defined CFT of central charge $c_2$ , the “internal” one, such that $c_1 + c_2 = 15 \,.$

Specifically concerning your interest in understanding how much of the “string theory landscpae”, i.e. how much of the space of all 2d SCFTs of central charge 15, consists of non-geometric theories the answer is that: supposedly many, but nobody really knows. And that was part of my point in the remark which you quoted.

You can find remarks on that issue in

Fernando Marchesano, Progress in D-brane model building

To quote from section 5.3:

As stated before, the mirror of a type IIB flux background may not have a geometric description. While intuitively less clear, one can still make sense of these backgrounds as string theory compactifications […]. Finally, duality arguments suggest that the different kinds of ‘non-geometric fluxes’ that one may introduce in type II theories form a much larger class than the geometric ones [140]. While our knowledge of these non-geometric constructions is still quite poor, and mainly based on mirrors of toroidal compactifications, the above results have led many to believe that non-geometric backgrounds correspond to the largest fraction of the ‘type II landscape’. A fraction which has so far been unexplored.

(my emphasis)

You can chase the references from there if you are interested. For instance [140] is

G. Aldazabal, P.G. Camara, A. Font, L.E. Ibanez, Progress in D-brane model building

From the abstract:

The chains of dualities relating type II orientifolds to heterotic and M-theory compactifications suggests the existence of yet further flux degrees of freedom. Restricting to a particular type IIA/IIB or heterotic compactification only some of these degrees of freedom have a simple perturbative and/or geometric interpretation.

Posted by: Urs Schreiber on March 27, 2008 6:22 PM | Permalink | Reply to this

Re: What has happened so far

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In a sense, are you and J. Baez trying to unify “non string theories” and “string theories” ?

I certainly can’t speak for John, who has lately been busy more with very fundamental fundamental questions (like the emergence of linear algebra and linear representation theory from categorical combinatorics called geometric representation theory; as well as a general theory of systems and processes).

But I’ll give you my answer:

An utterly crucial concept of modern theoretical physics is that of second quantization.

Given a quantum field theory defined on all $d$ -dimensional spaces, we can obtain from it another quantum field theory, or something similar, on some “effective target space”.

I tried to indicate that in the above comment:

Starting from a 1-dimensional QFT, second quantization yields ordinary field theory on an “effective target space”. We interpret that effective target space as “the world as seen by the (1-)particles whose worldvolume theory is the 1d QFT that we started with”.

$1d QFT on all worldlines \stackrel{second quantization}{\mapsto} QFT on effective target space \,.$

If we try to do the same but starting with a 2-dimensional QFT, we arrive at String field theory

$2d QFT on all worldsheets \stackrel{second quantization}{\mapsto} String field theory on effective target space \,.$

Therefore, it appears to be not a good idea not to try to discuss field theory and “string theory” on some same footing.

What I would like to understand is what quantum field theory really is, such that it allows me to take a differential cocycle as input (a Kalb-Ramond gerbe with connection, for instance) and produce another “quantized” differential cocycles as output (the WZW 2dCFT, for instance).

Any answer to that cannot be called successful if it does not also have something to say about “second quantization”, namely about iterating that procedure. (I indicated in my entry above how it should work.)

But once we know what it means to second quantize a $(d \gt 1)$ -dimensional QFT, we are entering the realms of “string theory”.

Posted by: Urs Schreiber on March 27, 2008 5:14 PM | Permalink | Reply to this

Re: What has happened so far

As for a sponge, i mean, a bath foam. If you look closely, you see that it is made of knoted strings.

http://www.riovistaproducts.com/dealers/press/natural%20sponge.jpg” title

Posted by: Daniel de França MTd2 on March 27, 2008 3:07 PM | Permalink | Reply to this

Re: What has happened so far

It’s certainly fascinating to me that we all keep bumping into each other. Maybe I shouldn’t been so surprised.

When I get a chance I mean to take a look at Jeremy Butterfield’s Some Aspects of Modality in Analytical Mechanics. As the author says,

The modal involvements of analytical mechanics turn out to be rich and subtle.

Perhaps it can provide further clues to linking mechanics to 2-geometry to modal logic.

Posted by: David Corfield on March 27, 2008 6:56 PM | Permalink | Reply to this

Read the post Limits and Push-Forward
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March 27, 2008