$$mathrm{SL}(2, mathbb {C})$$ group action on cohomological field theories

We introduce the (mathrm {SL} (2,mathbb {C})) group action on a partition function of a cohomological field theory via a 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. Milanov and Ruan in Gromov–Witten theory of elliptic orbifold (mathbb {P}^{1}) and quasi-modular forms, arXiv:1106.2321, 2011).