Regularized Integrals on Riemann Surfaces and Modular Forms

TL;DR

The paper develops a geometric regularization method for integrals with singularities on Riemann surfaces, connecting them to modular forms and providing new insights into quantum field theory integrals.

Contribution

It introduces an intrinsic regularization scheme for singular integrals on Riemann surfaces, linking them to modular forms and quantum field theory applications.

Findings

Regularized integrals are almost-holomorphic modular forms.

Provides a geometric proof of quasi-modularity of A-cycle integrals.

Derives combinatorial formulas for components of different weights.

Abstract

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry. Applied to products of Riemann surfaces, this regularization scheme establishes an analytic theory for integrals over configuration spaces, including Feynman graph integrals arising from two dimensional chiral quantum field theories. Specializing to elliptic curves, we show such regularized graph integrals are almost-holomorphic modular forms that geometrically provide modular completions of the corresponding ordered $A$-cycle integrals. This leads to a simple geometric proof of the mixed-weight quasi-modularity of ordered A-cycle integrals, as well as novel combinatorial formulae for all the components of different weights.