Dimitri Zvonkine, Paris 6
Cohomological relations on the moduli space of stable curves
We construct a family of relations between tautological cohomology classes on the moduli space Mbar_{g,n} of stable curves. This family contains all relations known to this day and is expected to be complete and optimal. The construction uses the Frobenius manifold of the A_2 singularity, the 3-spin Witten class and the Givental-Teleman classification of semi-simple cohomological field theories (CohFTs). The plan of the three talks will be as follows.
1. An introduction to moduli space and its tautological cohomology ring; simplest examples of tautological relations.
2. Cohomological field theories and Witten's r-spin class. Witten's r-spin class is actually a family of cohomology classes on the space of stable maps, defined using the space of tensor r-th roots of the canonical line bundle. I will explain why these classes satisfy the axioms of a cohomological field theory (CohFT).
3. The Givental-Teleman classification of semi-simple CohFTs. I will explain the classification theorem, show how it applies to Witten's class and how one can deduce tautological relations from it. In the end I will compute several cohomological relations using our method.
This is a joint work with R. Pandharipande and A. Pixton.
