Non-chiral 2d CFT with integer energy levels

TL;DR

This paper demonstrates that all modular invariant partition functions in 2d CFTs can be related to holomorphic ones, revealing that certain energy levels are necessarily integer multiples and their degeneracies are uniquely determined.

Contribution

It introduces a method to map general modular invariant partition functions to holomorphic ones, extending understanding of energy level degeneracies in 2d CFTs.

Findings

Partition functions can be mapped to holomorphic functions using medium temperature expansion.

Left and right central charges are multiples of 4 for theories with half-integer weights.

Degeneracy of high-energy levels is uniquely determined by low-energy state degeneracies.

Abstract

The partition function of 2d conformal field theory is a modular invariant function. It is known that the partition function of a holomorphic CFT whose central charge is a multiple of 24 is a polynomial in the Klein function. In this paper, by using the medium temperature expansion we show that every modular invariant partition function can be mapped to a holomorphic partition function whose structure can be determined similarly. We use this map to study partition function of CFTs with half-integer left and right conformal weights. We show that the corresponding left and right central charges are necessarily multiples of 4. Furthermore, the degree of degeneracy of high-energy levels can be uniquely determined in terms of the degeneracy in the low energy states.