The String Coffee Table

John Baez wrote:

[…] Following Moore and Segal, they also bring ‘closed strings’ into the game, which form a Frobenius algebra of their own, where the multiplication looks like an upside-down pair of pants: […]

I would like to make the following general comment on the meaning of Frobenius algebras in 2-dimensional quantum field theory.

Interestingly, noncommutative Frobenius algebras play a role even for closed strings, and even if the worldhseet theory is not purely topological.

The archetypical example for this is the class of 2D TFTs invented by Fukuma, Hosono and Kawai. There one has a non-commutative Frobenius algebra which describes not the splitting/joining of the entire worldsheet, but rather the splitting/joining of edges in any one of its dual triangulations. It is the center of (the Morita class of) the noncommutative Frobenius algebra decorating dual triangulations which is the commutative Frobenius algebra describing the closed 2D TFT.

One might wonder if it has any value to remember a non-commutative Frobenius algebra when only its center matters (in the closed case). The point is that the details of the non-commutative Frobenius algebra acting in the ‘interior’ of the world sheet affects the nature of ‘bulk field insertions’ that one can consider and hence affects the (available notions of) $n$-point correlators of the theory, for $n>0$.

This aspect, however, is pronounced only when one switches from 2D topological field theories to conformal ones.

The fascinating thing is that even 2D conformal field theories are governed by Frobenius algebras. The difference lies in different categorical internalization. The Frobenius algebras relevant for CFT don’t live in $\mathrm{Vect}$, but in some other (modular) tensor category, usually that of representations of some chiral vertex operator algebra. It is that ambient tensor category which ‘knows’ if the Frobenius algebra describes a topological or a conformal field theory (in 2D) - and which one.

Of course what I am referring to here is the work by Fjelstad, Froehlich, Fuchs, Runkel, Schweigert and others. I can recommend their most recent review which will appear in the Streetfest proceedings.

I. Runkel, J. Fjelstad, J. Fuchs, Ch. Schweigert
Topological and conformal field theory as Frobenius algebras
math.CT/0512076.

The main result is, roughly, that given any modular tensor category with certain properties, and given any (symmetric and special) Frobenius algebra object internal to that category, one can construct functions on surfaces that satisfy all the properties that one would demand of an $n$-point function of a 2D (conformal) field theory.

If we define a field theory to be something not given by an ill-defined path integral, but something given by its set of correlation functions, then this amounts to constructing a (conformal) field theory.

This result is achieved by first defining a somewhat involved procedure for generating certain classes of functions on marked surfaces, and then proving that the functions generated by this procedure do indeed satisfy all the required properties to qualify as correlations functions.

In broad terms, the prescription is to choose a dual triangulation of the marked worldsheet whose correlation function is to be computed, to decorate the edges of the dual triangulation with symmetric special Frobenius algebra objects in some modular tensor category, to decorate the vertices by product and coproduct morphisms of this algebra, to embed the whole thing in a certain 3-manifold in a certain way and for every boundary or bulk field insertion to add one or two threads labeled by simple objects of the tensor category which connect edges of the chosen triangulation with the boundary of that 3-manifold. Then you are to hit the resulting extended 3-manifold with the functor of a 3D TFT and hence obtain a vector in a certain vector space. This vector, finally, is claimed to encode the correlation function.

This procedure is deeply rooted in well-known relations between 3-(!)-dimensional topological field theory, modular functors and modular tensor categories and may seem very natural to people who have thought long enough about it. It is already indicated in Witten’s paper on the Jones polynomial, that 3D TFT (Chern-Simons field theory in that case) computes conformal blocks of conformal field theories on the boundaries of these 3-manifolds. To others, like me in the beginning, it may seem like a miracle that an involved and superficially ad hoc procedure like this has anything to do with correlations functions of conformaal field theory in the end.

In trying to understand the deeper ‘meaning’ of it all I played around with the idea that this prescription is really, to some extent at least, the ‘dual’ incarnation of the application of a certain 2-functor to the worldsheet. Namely a good part of the rough structure appearing here automatically drops out when a 2-functor applied to some 2-category of surfaces is ‘locally trivialized’. I claim that any local trivialization of a 2-functor on some sort of 2-category of surface elements gives rise to a dual triangulation of the surface whose edges are labeled by (possibly a generalization of) a Frobenius algebra object and whose vertices are labeled by (possibly a generalization of) product and coproduct operations. There is more data in a locally trivialized 2-functor, and it seems to correctly reproduce the main structure of bulk field insertions as appearing above. But of course there is a limit to what a two-functor can know about a structure that is inherently 3-dimensional.

I have begun outlining some of the details that I have in mind here.

This has grown out of a description of gerbes with connective structure in terms of transport 2-functors. Note that in what is called a bundle gerbe we also do have a certain product operation playing a decisive role. Bundle gerbes can be understood as ‘pre-trivializations’ of 2-functors to $\mathrm{Vect}$ and the product appearing is one of the Frobenius products mentioned above. For a bundle gerbe the coproduct is simply the inverse of the product, since this happens to be an isomorphism. The claim is that 2-functors to Vect more generally give rise to non-trivial Frobenius algebras when locally trivialized.

This is work in progress and will need to be refined. I thought I’d mention it here as an amplification of John’s general statements about how Frobenius algebras know about 2-dimensional physics. I am grateful for all kinds of comments.