Two-dimensional conformal field theories with matrix-valued level

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 conformal field theories (CFTs) are based on holomorphic currents with an operator product expansion characterized by a positive integer-valued matrix KAB. 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 SL(2, ℤ) up to a modular anomaly.