An $L$-function Approach to Two-Dimensional Conformal Field Theory

Abstract

We introduce a framework for two-dimensional conformal field theory (CFT) in the language of analytic number theory. Attached to the torus partition function of every two-dimensional CFT is a self-dual, degree-4 $L$-function of root number $\varepsilon=1$, with a universal gamma factor determined by $SL(2,\mathbb{Z})$ and local conformal invariance. Due to the richness of CFT operator spectra, these are not, in general, standard $L$-functions. We explicate their analytic structure, exploring the interplay of the Hadamard product over non-trivial zeros with the generalized Dirichlet series over CFT scalar primary conformal dimensions. We derive a zero sum rule in terms of the spectrum, and a global zero density bound in terms of the spectral gap. Convergence of the series representation implies square root cancellation of the degeneracies; we relate this to random matrix behavior of high-energy level spacings. Random matrix universality of the CFT implies "Riemann zeta universality" of the $L$-function: an average relation between the $L$-function on the critical line and Riemann zeta on the 1-line. This in turn yields a subconvexity bound. For a compact free boson, the $L$-function is a product of Riemann zeta functions times an analytic factor. Extensions to correlator $L$-functions and spinning spectra are briefly discussed. In the course of this work, we are led to sharpen the notion of random matrix universality in two-dimensional CFTs. We formulate a precise version of the following standalone conjecture, logically independent of $L$-functions: in unitary, compact Virasoro CFTs with central charge $c>1$, fixed-spin primary spectra at high energy are asymptotically simple, with random extreme gap statistics.