Quasi-Characters in $\widehat{su(2)}$ Current Algebra at Fractional Levels

TL;DR

This paper investigates the structure of even characters in $\, ext{su}(2)\, ext{hat}$ conformal field theories at fractional levels, identifying special classes, their properties, and introducing quasi-characters with unique modular features.

Contribution

It classifies admissible fractional levels with even characters, relates them to known current algebra characters, and introduces quasi-characters with distinctive modular and positivity properties.

Findings

Admissible even characters occur only at three special fractional levels.

Half-odd integer level characters relate to $\, ext{su}(2)\, ext{hat}$ at multiples of 4.

Quasi-characters are modular functions with non-positive integer coefficients.

Abstract

We study the even characters of $\widehat{su(2)}$ conformal field theories (CFTs) at admissible fractional levels obtained from the difference of the highest weight characters in the unflavoured limit. We show that admissible even character vectors arise only in three special classes of admissible fractional levels which include the threshold levels, the positive half-odd integer levels, and the isolated level at -$5/4$. Among them, we show that the even characters of the half-odd integer levels map to the difference of characters of $\widehat{su(2)}_{4N+4}$, with $N\in\mathbb{Z}_{>0}$, although we prove that they do not correspond to rational CFTs. The isolated level characters maps to characters of two subsectors with $\widehat{so(5)}_1$ and $\widehat{su(2)}_1$ current algebras. Furthermore, for the $\widehat{su(2)}_1$ subsector of the isolated level, we introduce discrete flavour fugacities. The threshold levels saturate the admissibility bound and their even characters have previously been shown to be proportional to the unflavoured characters of integrable representations in $\widehat{su(2)}_{4N}$ CFTs, where $N\in\mathbb{Z}_{> 0}$ and we reaffirm this result. Except at the three classes of fractional levels, we find special inadmissible characters called quasi-characters which are nice vector valued modular functions but with $q$-series coefficients violating positivity but not integrality.