Cohomological relations on Mbar_{g,n} via 3-spin structures
We construct a family of relations between tautological cohomology classes on the moduli space Mbar_{g,n}. 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). I will start with a short introduction into the cohomology of moduli spaces and give simplest examples of tautological relations. Then I will proceed to define Witten's r-spin class, explain why it is a CohFT and how Teleman's classification applies to it. In the end I will compute several cohomological relations using our method. This is a joint work with R. Pandharipande and A. Pixton.
Lecture 1. Moduli spaces of curves, tautological rings, tautological relations.
M_{g,n}, examples of M_{0,n}, M_{1,1}, M_{2,0}. A few words about orbifolds. Stable curves, the Deligne-Mumford space Mbar_{g,n}, examples. Tautological classes: psi-classes, kappa-classes, boundary divisors. Some elementary computations. Stable graphs. Definition and examples of tautological relations.
Lecture 2. Cohomological field theories.
Axioms of a CohFT. Moduli space of r-spin curves, its compactification, Witten's r-spin class, check that it's a CohFT. Frobenius manifolds as genus 0 part of CohFT and as a differential geometric structure (family of fusion algebras). GW-potential of a Frobenius manifold. Shift of CohFTs. Semi-simple CohFTs.
Lecture 3. Givental's group action, Teleman's classification, Pixton's relations.
Givental's group actions on CohFTs. Teleman's classification of semisimple CohFTs. Getting Pixton's relations and an explicit expression for Witten's 3-spin class. Sample computations of tautological relations.