Givental group action on Topological Field Theories and homotopy Batalin--Vilkovisky algebras

TL;DR

This paper explores the Givental group action on Cohomological Field Theories using homotopical algebra, establishing a correspondence between stabilizers of Topological Field Theories and certain homotopy BV-algebras.

Contribution

It introduces a novel homotopical algebra framework to understand the Givental group action on Cohomological Field Theories and characterizes stabilizers via homotopy BV-algebras.

Findings

Stabilizers in genus 0 correspond to commutative homotopy BV-algebras.

Stabilizers in genera 0 and 1 correspond to wheeled commutative homotopy BV-algebras.

Provides a new algebraic perspective on the symmetries of Topological Field Theories.

Abstract

In this paper, we initiate the study of the Givental group action on Cohomological Field Theories in terms of homotopical algebra. More precisely, we show that the stabilisers of Topological Field Theories in genus 0 (respectively in genera 0 and 1) are in one-to-one correspondence with commutative homotopy Batalin--Vilkovisky algebras (respectively wheeled commutative homotopy BV-algebras).