Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations

Abstract

We derive residue formulas for the regularized integrals (introduced by Li and Zhou in Commun Math Phys 388:1403–1474, 2021) on configuration spaces of elliptic curves. Based on these formulas, we prove that the regularized integrals satisfy holomorphic anomaly equations, providing a mathematical formulation of the so-called contact term singularities. We also discuss residue formulas for the ordered A-cycle integrals and establish their relations with those for the regularized integrals.