Properties of scalar partition functions of 2d CFTs

TL;DR

This paper investigates the scalar primary operator spectrum in 2D conformal field theories, revealing a crossing equation and analyzing the high-temperature behavior of scalar partition functions, linking them to modular integrals, light states, and the Riemann zeta function.

Contribution

It extends previous work by deriving a crossing equation for scalars in 2D CFTs and explores the high-temperature behavior of scalar partition functions with connections to number theory.

Findings

Scalar operators obey a nontrivial crossing equation.

High-temperature differences relate to modular integrals, light states, and zeros of the Riemann zeta function.

Numerical examples illustrate the theoretical results.

Abstract

We study the spectrum of scalar primary operators in any two-dimensional conformal field theory. We show that the scalars alone obey a nontrivial crossing equation. This extends previous work that derived a similar equation for Narain conformal field theories. Additionally, we show that at high temperature, the difference between the true scalar partition function and the one predicted from a semiclassical gravity calculation is controlled by: the modular integral of the partition function, the light states of the theory, and an infinite series terms directly related to the nontrivial zeros of the Riemann zeta function. We give several numerical examples and compute their modular integrals.