L-functions for Meromorphic Modular Forms and Sum Rules in Conformal Field Theory

TL;DR

This paper introduces L-functions for meromorphic modular forms, establishing new links between number theory and conformal field theory, including sum rules for central charges and spectral data, through regularized Mellin transforms.

Contribution

It defines L-functions for meromorphic modular forms with poles away from cusps and applies them to derive novel relationships in number theory and conformal field theory.

Findings

New relationships between Hurwitz class numbers and traces of singular moduli.

Confirmation of T-reflection predictions in 2d CFT via L-functions.

Expression of CFT central charges as regularized sums over states.

Abstract

We define L-functions for meromorphic modular forms that are regular at cusps, and use them to: (i) find new relationships between Hurwitz class numbers and traces of singular moduli, (ii) establish predictions from the physics of T-reflection, and (iii) express central charges in two-dimensional conformal field theories (2d CFT) as a literal sum over the states in the CFTs spectrum. When a modular form has an order-$p$ pole away from cusps, its $q$-series coefficients grow as $n^{p-1} e^{2 \pi n t}$ for $t \geq \sqrt{3}/2$. Its L-function must be regularized. We define such L-functions by a deformed Mellin transform. We study the L-functions of logarithmic derivatives of modular forms.L-functions of logarithmic derivatives of Borcherds products reveal a new relationship between Hurwitz class numbers and traces of singular moduli. If we can write 2d CFT path integrals as infinite products, our L-functions confirm T-reflection predictions and relate central charges to regularized sums over the states in a CFTs spectrum. Equating central charges, which are a proxy for the number of degrees of freedom in a theory, directly to a sum over states in these CFTs is new and relies on our regularization of such sums that generally exhibit exponential (Hagedorn) divergences.