SL(2,C) group action on Cohomological field theories

TL;DR

This paper introduces an $ ext{SL}(2, ext{C})$ group action on the partition function of Cohomological Field Theories, providing explicit formulas and demonstrating its equivalence to known transformations in specific singularity cases.

Contribution

It defines a new $ ext{SL}(2, ext{C})$ action on CohFTs and connects it to existing transformations, enriching the understanding of symmetries in these theories.

Findings

Explicit formulas for the $ ext{SL}(2, ext{C})$ action on CohFT potentials.

Proof of equivalence between the introduced action and Milanov--Ruan transformation.

Application to simple elliptic singularities showing the action's relevance.

Abstract

We introduce the $\mathrm{SL}(2,\mathbb{C})$ group action on a partition function of a Cohomological field theory via the certain Givental's action. Restricted to the small phase space we describe the action via the explicit formulae on a CohFT genus $g$ potential. We prove that applied to the total ancestor potential of a simple elliptic singularity the action introduced coincides with the transformation of Milanov--Ruan changing the primitive form (cf. arXiv:1106.2321).