The String Coffee Table

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

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 $(n-1)$-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

by the following procedure:

Pick a Frobenius algebra (in $\mathbf{Vect}$). 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.

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

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 $2D-CFT$, 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 $C = \mathrm{Rep}(V)$.

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 $\mathbf{Vect}$ before, it now takes place internal to $C = \mathrm{Rep}(V)$, 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)

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

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

$\bullet$ label boundary field insertions by internal module homomorphisms

$\bullet$ label bulk field insertions by internal bimodule homomorphisms

$\bullet$ 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

along the injections