The tautological ring and cohomological field theories

Aaron Pixton

(Harvard University)

Let Mbar_{g,n} be the moduli space of stable curves of genus g with n marked points. The tautological ring is a subring of the Chow ring A^*(Mbar_{g,n}) consisting of the classes that arise naturally in geometry through forgetful and gluing morphisms. After reviewing the basic theory of the tautological ring, I will explain the concept of a cohomological field theory (CohFT), a family of tautological classes satisfying certain axioms.

I will discuss how to apply the theory of CohFTs developed by Givental and Teleman to obtain a very large family of tautological relations from Witten's r-spin class (this is joint work with Pandharipande and Zvonkine). These tautological relations generate all known relations. I'll conclude by briefly describing a conjectural formula for the double ramification cycle and its relationship to CohFTs.