Two-dimensional conformal field theories with matrix-valued level

TL;DR

This paper introduces a new class of two-dimensional conformal field theories extending WZW models with matrix-valued levels, leading to novel algebraic structures and modular properties.

Contribution

It develops a framework for conformal field theories with matrix-valued levels, including their algebraic, spectral, and modular characteristics.

Findings

Derived energy-momentum tensor and central charge for the new models

Constructed genus-1 characters satisfying modular group representations

Identified modular anomalies in the character transformations

Abstract

We introduce a new class of two dimensional conformal field theories by extending Wess-Zumino-Witten (WZW) models to chiral algebras with matrix-valued levels. The new CFTs are based on holomorphic currents with an operator product expansion characterized by a positive integer-valued matrix $K_{AB}$. We use the Sugawara construction to compute the energy-momentum tensor, the central charge, and the spectrum of conformal dimensions of the CFTs based on this algebra. We also construct a set of genus-$1$ characters and show that they fulfil a representation of the modular group $\text{SL}(2,\mathbb{Z})$ up to a modular anomaly.