Constraints on Flavored 2d CFT Partition Functions

TL;DR

This paper explores how modular invariance constrains the spectrum of charged states in 2d conformal field theories with flavor, improving bounds and applying the extremal functional method to identify specific theories.

Contribution

It provides a new proof of the modular transformation law for flavored partition functions and enhances bounds on charged states, enabling the extremal functional method to solve for certain partition functions.

Findings

Improved upper bounds on the mass-to-charge ratio of states.

Predictions for occupation numbers that are often integers.

Added flavor allows the extremal functional method to analyze more theories.

Abstract

We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e. when the partition functions are "flavored". We begin with a new proof of the transformation law for the modular transformation of such partition functions. Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory. We improve previous upper bounds on the state with the greatest "mass-to-charge" ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory. We apply the extremal functional method to theories that saturate such bounds, and in several cases we find the resulting prediction for the occupation numbers are precisely integers. Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise.