Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations

TL;DR

This paper develops residue formulas for regularized integrals on elliptic curves, proving they satisfy holomorphic anomaly equations and clarifying their relation to contact term singularities and A-cycle integrals.

Contribution

It introduces residue formulas for regularized integrals on elliptic curves and establishes their connection with holomorphic anomaly equations and contact term singularities.

Findings

Regularized integrals satisfy holomorphic anomaly equations.

Residue formulas relate regularized and A-cycle integrals.

Mathematical formulation of contact term singularities.

Abstract

We derive residue formulas for the regularized integrals (introduced by Li-Zhou) 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.