## May 19, 2006

### The FRS Theorem on RCFT

#### Posted by Urs Schreiber

I was asked to say more about the FRS theorem. Here is a rough account. For more details see the existing literature ($\to$).

Recall that an $n$-dimensional quantum field theory is (at least for our present purpose) a functor

(1)$\mathrm{QFT}:n\mathrm{Cob}\to \mathrm{Vect}$

from some $n$-dimensional cobordism category to some flavor of the category of vector spaces.

This means that QFT is a rule which assigns to each $\left(n-1\right)$-dimensional space, of sorts, a vector space - the space of incoming (outgoing) states of the QFT.

Moreover the rule assigns to each cobordism a linear map from the vector space assigned to the incoming map to the vector space assigned to the outgoing map - the propagator.

If the cobordisms are really just (diffeomorphism classes of) manifolds, without extra structure, then we call such a QFT a topological quantum field theory (TQFT $\to$).

Usually there is more structure around that we want to be respected. Instead of just diffeomorphism classes of manifolds, the cobordimsms might be equipped, for instance, with a conformal structure (CFT), or a metric, or with some kinds of bundles over them, possibly with connection (HQFT $\to$).

Moreover, the target category may be demanded to carry more structure. Often one considers topological vector spaces or Hilbert spaces.

In particular, for the case of CFT we want the functors to be only projective. Instead of taking the same value on all cobordisms with metric in a conformal class, they will assign propagators that differ by an overall factor depending on the central charge.

With the general definition of an $n$-dimensional QFT stated this way, we would like to construct examples.

For the purely topological case the situation is pretty well understood. Fukuma-Hosono-Kawai have shown how to construct a topological functor

(2)$2D-\mathrm{TFT}:2\mathrm{Cob}\to \mathrm{Vect}$

by the following procedure:

Pick a Frobenius algebra (in $\mathrm{Vect}$). For any cobordism, choose a dual triangulation (only trivalent vertices). To each edge associate a copy of the Frobenius algebra and to each vertex associate the product or coproduct (depending on the orientaiton of the coincicent edges). Evaluating the resulting diagram produces a functor of the desired form. Roughly.

The next simplest case, that of 2-dimensional conformal field theory is already much harder. One big insight is that, at least for the case of rational conformal field theory, the problem splits into a “complex analytic” problem and a “topological” problem.

(3)$\begin{array}{ccccc}\text{(R)CFT}& \simeq & \text{complex-analytic}& +& \text{topological}\\ & \simeq & \text{chiral vertex algebra}V& +& \text{OPEs}\\ & \simeq & C=\mathrm{Rep}\left(V\right)& +& A\in C\text{int. Frobenius algebra.}\end{array}$

The chiral vertex algebra encodes the local symmetries of the theory, like the local conformal symmetry (given by the Virasoro algebra) as well as other possibly present symmetries (given for instance by current algebras in the case of WZW models).

From the knowledge of the vertex algebra $V$ alone one computes what are called spaces of conformal blocks. In the above language, these are spaces which contain information about all those maps

(4)$2D-\mathrm{CFT}:2\mathrm{Cob}\to \mathrm{Vect}$

which satisfy a necessary (but insufficient) condition for being actual functors.

In more standard terms, every $n$-point correlator in the theory is an element of the corresponding space of conformal blocks. In order to define a well defined conformal field theory we need to assign to each surface of genus $g$, with given insertions, correlators in such a way that the sewing constraints are satisfied. These say essentially that it must be possible to compute correlators on a given surface by cutting that surface into little pieces, computing correlators on each of these pieces, and then composing the result to obtain the correlator of the full surface.

This is of course nothing but functoriality of $2D-\mathrm{CFT}$, in the above sense.

Therefore, in order to obtain a full 2D CFT, we need to do two things:

1) compute the conformal blocks for the underlying vertex algebra $V$

2) pick correlators from these in such a way that the sewing constraints are satisfied.

Given a solution of problem 1), there are in general different, inequivalent, solutions to problem 2). There are distinct 2D CFTs that share the same chiral algebra.

The FRS theorem says that the solutions to problems 1) and 2) are in bijection with Morita classes of special symmetric Frobenius algebra objects internal to $C=\mathrm{Rep}\left(V\right)$.

More precisely, that’s the statement for CFTs of orientable worldsheets. For non-oriented (“type I”) theories we need a special symmetric Frobenius algebras with what is called a Jandl structure ($\to$).

The way this is shown is a direct generalization of the construction of Fukuma-Hosono-Kawai for the purely topological case. The main difference is that where everything took place in the category $\mathrm{Vect}$ before, it now takes place internal to $C=\mathrm{Rep}\left(V\right)$, the representation category of $V$.

Spelled out, we arrive at a detailed “cooking recipe” for computing correlators given $A\in C$. That recipe tells us to (roughly)

$•$ pick a dual triangulation of the surface (really: of its complex double)

$•$ label boundaries by internal $A$-modules (these are the D-branes $\to$)

$•$ label boundary field insertions by internal module homomorphisms

$•$ label bulk field insertions by internal bimodule homomorphisms

$•$ label defect lines ($\to$) by internal bimodules .

Doing all this one obtains a certain ribbon graph (since $C$ is a ribbon category) drawn on our surface. In order to produce the correlator from that we need to work a little harder than in the topological case. We realize the complex double of our surface as the boundary of a 3-dimensional cobordism in which this ribbon graph is embedded and evaluate a 3-dimensional topological field theory functor on that extended 3-cobordism. This yields a vector in a vector space, which may be identified with the correlator in the space of conformal blocks.

This is a generalization of the classic insight by E. Witten, that 3-dimensional Chern-Simons TQFT computes the conformal blocks for 2D WZW models on its boundary.

There would be much more to say, but I’ll stop here.

However, in closing I cannot refrain from mentioning that I believe ($\to$) that the slightly baroque “cooking recipe”, which makes the FRS theorem work, appears automatically as the result of pulling back a 2-functor

(5)${P}_{2}\to {}_{C}\mathrm{Mod}$

along the injections

(6)$\Sigma \left(C\right)\to \mathrm{BiMod}\left(C\right)\to {}_{C}\mathrm{Mod}\phantom{\rule{thinmathspace}{0ex}}.$
Posted at May 19, 2006 11:44 AM UTC

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### Re: The FRS Theorem on RCFT

Hi Urs,

It might be blasphemous to say so, but your cookbook recipe seems to have some nontrivial similarities to spin foam models. Is that a coincidence? Am I just imagining things?

Eric

Posted by: Eric on May 20, 2006 7:57 AM | Permalink | Reply to this

### Re: The FRS Theorem on RCFT

Hi Eric,

we had a little discussion about this a while ago ($\to$). Certainly, graphs decorated in certain categories appear here and there in different contexts.

I know regrettably little spin foam literature. Apparently the concept is pretty general, so that it might well subsume all these constructions.

Whether that also means that the role played by these constructions in the FRS context is that intended by the inventors of spin foams I don’t know.

The similarity might actually be most pronounced viewed from the 3-dimensional TQFT, which is used in the FRS construction.

There is a large class of 3-D topological field theories which you obtain this way:

Fix some modular tensor category $C$ (a monoidal category with a bunch of further properties).

Consider 3-dimensional cobordisms which have ribbon graphs embedded into them.

A ribbon graph is like a graph, only that its edges behave as if they have a small transversal extension, like a ribbon does.

Let these ribbon graphs be decorated in $C$. So each of the ribbons is labled by an object in $C$ and each vertex by a morphism in $C$.

These extended 3D cobordisms form a category (essentially by glueing them in the obvious way).

The important point is that from the data provided by $C$, one can construct a functor from this category to vector spaces, i.e. a 3-dimensional “topological” field theory on these extended cobordisms.

This is the sort of TFT that I mentioned a at the end of the above text.

In any case, these TFTs look somewhat like something one would construct of spin networks with decoration in $C$ instead of in $\mathrm{Rep}\left(G\right)$.

But I am not quite sure to what extent this is a coincidence or really points to some connection between different ideas.

Posted by: urs on May 22, 2006 5:07 PM | Permalink | Reply to this

### Re: The FRS Theorem on RCFT

Your answer was a lot less vague than the question :) Thanks and sorry for the distraction.

Posted by: Eric on May 23, 2006 6:34 AM | Permalink | Reply to this

### Re: The FRS Theorem on RCFT

sorry for the distraction.

Actually, I am grateful for every distraction. At the moment I spent most of the day with work related to teaching duties.

For instance, I really wanted to write something about a talk Simon Willerton gave in Vienna - math.QA/0503266. He is thinking about Dijkgraaf-Witten theory ($\to$) using a higher categorical point of view inspired by Dan Freed’s work, e.g. hep-th/9212115.

In slight disguise, this is essentially based on the same idea that I am trying to apply to 2D CFT, namely to realize that there are $n$-Hilbert spaces associated to points, $\left(n-1\right)$-Hilbert spaces associated to edges, in general $\left(n-d\right)$-Hilbert spaces associated to $d$-dimensional subspaces of $n$-dimensional cobordisms.

This is closely related to the 3D TFTs we talked about above, and maybe makes the similarity to spin foam ideas even more pronounced. At least I gather from John Baez’s lectures that we can regard Dijkgraaf-Witten theory as a categorification of Fukuma-Hosono-Kawai topological strings, probably along the lines of Aaron Lauda’s ideas on topological membranes ($\to$).

You see, this is all on my to-do list. But no way. Have to do more boring things first.

Posted by: urs on May 23, 2006 8:21 AM | Permalink | Reply to this
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