Frobenius Manifolds near the Discriminant and Relations in the Tautological Ring

TL;DR

This paper establishes criteria for extending Frobenius manifolds to cohomological field theories near discriminant points and demonstrates that many tautological relations derive from Pixton's generalized Faber--Zagier relations.

Contribution

It provides a new criterion for extending Frobenius manifolds to cohomological field theories and links tautological relations to Pixton's relations in a broad setting.

Findings

Extension criteria for Frobenius manifolds near discriminant points.

Many tautological relations follow from Pixton's generalized Faber--Zagier relations.

Application to Givental--Teleman classification of cohomological field theories.

Abstract

We give a criterion for extending a generically semisimple (not necessarily conformal) Frobenius manifold locally near a smooth point of the discriminant to a cohomological field theory. As an application, we show that a large set of tautological relations related to the Givental--Teleman classification for any generically semisimple cohomological field theories follow from Pixton's generalized Faber--Zagier relations.