L-functions for meromorphic modular forms and sum rules in conformal field theory

• Main Author: David A. McGady
• Format: Article
• Language: English
• Published: SpringerOpen 2019-01-01
• Series: Journal of High Energy Physics
• Subjects: Conformal Field Theory; Anomalies in Field and String Theories; Space-Time Symmetries
• Online Access: http://link.springer.com/article/10.1007/JHEP01(2019)135

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 2πnt for t ≥ 3 2 $$ t\ge \frac{\sqrt{3}}{2} $$ . 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.