From classical Weierstraß elliptic functions to quantum invariants

I will talk about a joint work with Si Li on the computation of higher genus B-model for elliptic curves.

I will first formulate the Feynman amplitudes in the higher genus B-model (Kodaira-Spencer theory)
in terms of cohomological parings. Then I will discuss properties of the Feynman amplitudes, including
the origin of their quasi-modularity, the geometric Interpretation of their modular completions, etc.
Finally I explain the implication of the cohomological reformation in renormalization.

Our method mainly uses the basic theory of algebraic curves. It applies to a large class of two-dimensional
conformal field theories and can hopefully find application in Eynard-Orantin topological recursion as well.