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

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).