An Action of the Polishchuk Differential Operator via Punctured Surfaces

Abstract

Abstract For a family of Jacobians of smooth pointed curves, there is a notion of tautological algebra. There is an action of ${\mathfrak{s}}l_2$ on this algebra. We define and study a lifting of the Polishchuk operator, corresponding to ${\mathfrak{f}} \in{\mathfrak{s}}l_2$, on an algebra consisting of punctured Riemann surfaces. As an application, we compare a class of tautological relations on moduli of curves, discovered by Faber and Zagier and relations on the universal Jacobian. We prove that the so called top Faber–Zagier relations come from a class of relations on the Jacobian side.