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

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).