Readers should be warned that the author is not an expert of CFT and, in fact, not even a novice in physics. What follows should be taken with a hefty pinch of salt.

Given any Riemann surface (as a target manifold) one is able to associate to it an or a super conformal field theory. The word super can just be construed as a -grading of the theory. This is a simplistic version as normally one should also take into account several other parameters like a B-field and so on. Within an SCFT there is a topological sector called a TQFT (topological quantum field theory) which is insensitive to the metric on the target space. There are axiomatic descriptions of this theory due to Atiyah and Segal and in its latest version possibly due to Costello. Roughly an -dimensional TQFT is a functor satisfying a lot of axioms from -manifolds with labelled boundaries (incoming and outgoing) to symmetric monoidal DG (differential graded) categories with some twisting.

Now people with a background on derived categories would like to see the triangulated structure of . Indeed, the homotopy category of a DG category resembles a triangulated category. It seems at this point it is possible to associate two models, namely, and to the theory after Witten. The model leads to a complicated Fukaya category (possibly as an -category) and the model leads to the derived (or DG) category of coherent sheaves on . There is a chiral ring that can be associated to each of these two categories. On the side it is the quantum cohomology ring and on the side it is its Hochschild homology . There is a Frobenius algebra structure on them; on at least when is Calabi-Yau.

The homological mirror symmetry conjecture by Kontsevich would predict the existence of an equivalence between the two categories arising out of the and models. This should naturally give rise to an isomorphism of Frobenius algebras after passing on to their chiral rings.

The construction of CFTs is a hard problem in general. Sometimes it is possible to get one’s hands on to the (possibly local) symmetries of a CFT, which are called chiral algebras and then one might be tempted to construct the CFT out of it. The chiral algebras as mathematical objects have the structure of a VOA (vertex operator algebra) . The construction of CFT from its chiral algebra seems to be possible if the representation category of the chiral algebra has finitely many irreducible objects (communicated by Liang Kong). This can also be phrased in the language of the partition function of the CFT as Gukov and Vafa do. Such a CFT is called *rational (abbreviated RCFT).* They enjoy some other very desirable properties, which make them particularly interesting. When the target space is a complex torus Gukov and Vafa argue that rationality of the CFT is related to the complex torus having complex multiplication. So in this case the RCFTs are plentiful but in higher dimensions they are supposed to be rather sparse (not dense in the moduli space).

Meng Chen studied the higher dimensional case of the same in her thesis and came up with her own geometric definition of the rationality of a CFT and related it to abelian varieties with large endomorphism rings. It is not clear if her geometric definition of rationality of a CFT is related to the, rather algebraic, notion of rationality presented above.

This is an intriguing connection between physics and number theory. In which direction the information would flow remains to be seen. There are some more interesting connections of this sort relating generating functions of some counting problems of (arithmetic) algebro-geometric nature to the partition functions of CFTs. If the author can weather this storm more postings on them will follow.

Sincere apologies for the inaccuracies and for straying from the main theme of the blog which is arithmetic algebraic geometry. The author just stumbled upon the thesis of Meng Chen recently, got fascinated and wanted to share this new-found knowledge.

## 2 comments

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August 5, 2007 at 12:45 am

A.J. TollandHi,

I hope you don’t mind if I offer a few comments/corrections.

1) It’s not true that the topological sector of a sigma model is independent of the metric on the target space; the A-model, for example, depends on the Kahler structure of the target. Topological field theories depend only on the topology of their spacetime, and in the case of a sigma model of maps , the spacetime is . (This can be a little confusing, because sometimes one is using the sigma model to describe a string theory whose spacetime is .)

2) With regard to axiomatic systems: Atiyah’s axioms describe QFTs which depend only on the topological structure of spacetime.

Segal’s axioms describe conformal field theories in 2d. Naively, one expects that correlation functions of a 2d quantum field theory are functions of the spacetime metric g. However, in conformal field theories, this dependence is very weak: the correlation functions only depend on the conformal class of g, which we may think of as the complex structure of our 2d spacetime. Consequently, correlation functions are functions on the stack of complex curves, or more generally sections of some bundle, when the central charge is non-zero or the fields in the correlation function transform non-trivially. In particular, correlation functions in supersymmetric theories can often be interpreted as differential forms on .

Of course, when one looks at only fields with correlation functions which are constant on , one gets an Atiyah-type QFT. But one can get more interesting information by trying to study sectors which have _closed_ differential forms for correlation functions. Costello’s axioms are an attempt to capture this situation, where the correlation functions can see more of the topology of the stack of curves. But, as far as I know, they can’t be used to describe full conformal field theories.

3) I’m not sure there’s really any misunderstanding here, but you should be aware that the term chiral algebra doesn’t necessarily refer to the symmetries of a CFT. Rather, in CFTs, the fields are expected to behave like harmonic functions on the 2d worldsheet. Harmonic functions locally are just the sum of a holomorphic and an anti-holomorphic function; analogously in CFT, the correlation functions, at least locally on the worldsheet, split into holomorphic and anti-holomorphic sectors. These sectors are CFTs in their own right, and are said to be chiral CFTs, vertex algebras, chiral algebras, etc,… CFTs are hard to construct because its difficult to sew the chiral sectors together, especially when the worldsheet topology is non-trivial.

August 8, 2007 at 10:45 am

Some truly outrageous! « Vivatsgasse 7[...] mysterious “field with one element” or better yet, an unpacking of our guest-blogger Sniggy Mahanta’s post on conformal field theories. (Thanks goes to AJ Tolland for pointing out some gross inaccuracies in that [...]