Journal of High Energy Physics

, 2019:135 | Cite as

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

  • David A. McGadyEmail author
Open Access
Regular Article - Theoretical Physics


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 np−1e2πnt for \( t\ge \frac{\sqrt{3}}{2} \). 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.


Conformal Field Theory Anomalies in Field and String Theories Space-Time Symmetries 


Open Access

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Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.The Niels Bohr International AcademyCopenhagen UniversityCopenhagenDenmark

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