The Eynard–Orantin recursion for the total ancestor potential

Abstract

It was proved recently that the correlation functions of a semisimple cohomological field theory satisfy the so-called local Eynard–Orantin topological recursion. We prove that in the settings of singularity theory, the local Eynard–Orantin recursion is equivalent to $N$ copies of Virasoro constraints for the total ancestor potential. The latter follow easily from some known properties of the period integrals in singularity theory. Our approach generalizes easily to an arbitrary semisimple cohomological field theory, which yields a simple proof of the local Eynard–Orantin recursion for an arbitrary semisimple cohomological field theory.