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 np−1e2πnt for tge 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.
Highlights
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
We further show that the special value at s = 1 of the same L-function confirms a sumrule that was motivated by a recently noticed symmetry of path integrals in quantum field theory (QFT), when applied to 2d CFT path integrals with infinite product expansions
The motivation for this paper was to verify a conjecture of Treflection [14], in the physics-agnostic setting of the mathematics of modular forms [6, 7]
Summary
Three physics considerations are the driving motivation for this work. First, there is a persistent connection between the behavior of quantum chromodynamics (QCD) at low energies and the behavior of strings. We may expect the one-loop path integral for low-energy, low-temperature QCD to be modular invariant. D(E) ∼ Ep−1e+βHE [12, 13] This leads to poles in the path integral Z(β) when the inverse temperature, β nears βH : Z(β) =. Juxtaposing the expected modularity of the low-energy QCD path integral with the presence of Hagedorn poles suggests that the theory of meromorphic modular forms may give an interesting set of tools to study low-energy QCD. On a related note, L-functions for meromorphic modular forms may allow us to rewrite central charges for unitary CFTs in terms of a direct sum over states in the spectrum, even if the spectrum exhibits Hagedorn growth. We confirm these predictions from [6, 14]
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