Higher genus closed string B-model

Question

The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which is then an example of a so-called Frobenius manifold. The mirror genus 0 B-model theory on the mirror manifold (or Landau-Ginzburg model) is usually described, mathematically, in terms of variation of Hodge structure data, or some generalization thereof.

Since the higher genus A-model has a nice mathematical description as higher genus GW invariants, I am wondering whether the higher genus B-model has a nice mathematical description as well. Costello's paper on TCFTs and Calabi-Yau categories gives a partial answer to this. One can say that GW theory is the study of algebras over the (homology) operad of compactified (Deligne-Mumford) moduli space; I say that Costello gives a partial answer because he only gives an algebra over the operad of uncompactified moduli space. Though, according to Kontsevich (see Kontsevich-Soibelman "Notes on A-infinity..." and Katzarkov-Kontsevich-Pantev), we can extend this to the operad of compactified moduli space given some assumptions (a version of Hodge-de Rham degeneration). There are also various results (e.g. Katzarkov-Kontsevich-Pantev, Teleman/Givental) which say that the higher genus theory is uniquely determined by the genus 0 theory. But --- despite these sorts of results, I still have not seen any nice mathematical description of the higher genus B-model which "stands on its own", as the higher genus GW invariants do. I have only seen the higher genus B-model described as some structure which is obtained formally from genus 0 data, or, as in the situation of Costello's paper, a Calabi-Yau category, e.g. derived category of coherent sheaves of a Calabi-Yau manifold, matrix factorizations category of a Landau-Ginzburg model, etc.

So, my questions are:

• Are there any mathematical descriptions of the higher genus closed string B-model which "stand on their own"? What I mean is something that can be defined without any reference to other structure, just as genus 47 GW invariants can be defined without any reference to genus 0 GW invariants or the Fukaya category.

• Genus 0 theory is essentially equivalent to the theory of Frobenius manifolds. What about higher genus theory? Is there any nice geometric structure behind, say, the genus 1 theory, analogous to the Frobenius manifold structure that we get out of genus 0 theory?

Answer 1

This is a great question I wish I understood the answer to better. I know two vague answers, one based on derived algebraic geometry and one based on string theory. The first answer, that Costello explained to me and I most likely misrepeat, is the following. The B-model on a CY X as an extended TFT can be defined in terms of DAG: we consider the worldsheet $\Sigma$ as merely a topological space or simplicial set (this is a reflection of the lack of instanton corrections in the B-model), and consider the mapping space $X^\Sigma$ in the DAG sense. For example for $\Sigma=S^1$ this is the derived loop space (odd tangent bundle) of $X$.. In this language it's very easy to say what the theory assigns to 0- and 1-manifolds: to a point we assign coherent sheaves on $X$, to a 1-manifold cobordism we assign the functor given by push-pull of sheaves between obvious maps of mapping spaces (see e.g. the last section here). For example for $S^1$ we recover Hochschild homology of $X$. Now for 2-manifold bordisms we want to define natural operations by push-pull of functions, but for that we need a measure -- and the claim is the Calabi-Yau structure (together with the appropriate DAG version of Grothendieck-Serre duality, which Kevin said Lurie provides) gives exactly this integration...

Anyway that gives a tentative answer to your question: the B-model assigns to a surface $\Sigma$ the "volume" of the mapping space $X^\Sigma$, defined in terms of the CY form. More concretely, you chop up $\Sigma$ into pieces, and use the natural operations on Hochschild homology, such as trace pairing and identification with Hochschild cohomology (and hence pair-of-pants multiplication).. of course this last sentence is just saying "use the Frobenius algebra structure on what you assigned to the circle" so doesn't really address your question - the key is to interpret the volume of $X^\Sigma$ correctly.

The second answer from string theory says that while genus 0 defines a Frobenius manifold you shouldn't consider other genera individually, but as a generating series -- i.e. the genus is paired with the (topological) string coupling constant, and together defines a single object, the topological string partition function, which you should try to interpret rather than term by term. (This is also the topic of Costello's paper on the partition function). BTW for genus one there is a concrete answer in terms of Ray-Singer torsion, but I don't think that extends obviously to higher genus.

As to how to interpret it, that's the topic of the famous BCOV paper - i.e. the Kodaira-Spencer theory of gravity. For one thing, the partition function is determined recursively by the holomorphic anomaly equation, though I don't understand that as "explaining" the higher genus contributions. But in any case there's a Chern-Simons type theory quantizing the deformation theory of the Calabi-Yau, built out of the Kodaira-Spencer dgla in a simple looking way, and that's what the B-model is calculating. A very inspiring POV on this is due to Witten, who interprets the entire partition function as the wave function in a standard geometric quantization picture for the middle cohomology of the CY (or more suggestively, of the moduli of CYs). This is also behind the Givental quantization formalism for the higher genus A-model, where the issue is not defining the invariants but finding a way to calculate them.

Anyway I don't know a totally satisfactory mathematical formalism for the meaning of this partition function (and have tried to get it from many people), so would love to hear any thoughts. But the strong message from physics is that we should try to interpret this entire partition function - in particular it is this function which appears in a million different guises under various dualities (eg in gauge theory, as solution to quantum integrable systems, etc etc...)

Answer 2

One thing missing from this discussion is the even-more-mysterious holomorphic ambiguity (not "anomaly"). BCOV is not deterministic, and should probably be thought of as part of a general schema for a B-model-type topological theory. What I mean is that there are other solutions to the BCOV equation which do not yield GW invariants. The reason is that BCOV determines the partition function only up to a holomorphic function at each genus. The choice of this function -- analogous to initial conditions -- is usually set by matching to GW invariants or by imposing some structure at the singularities of CY moduli space.

I don't think there is a comprehensive understanding of how this ambiguity is resolved, though there are cases where it is fixed by other symmetries or properties of the full partition function.