On Modular Invariant Partition Functions for Tensor Products of Conformal Field Theories

TL;DR

This paper investigates the construction of modular invariant partition functions for tensor products of conformal field theories, providing constraints based on individual invariants and methods to build asymmetric invariants.

Contribution

It introduces constraints on modular invariants for tensor products based on classified factors and demonstrates how to construct asymmetric invariants from known diagonal ones.

Findings

Classified all consistent theories for $SU(2)_{K_A} imes SU(2)_{K_B}$ with odd $K_A,K_B$.

Provided explicit examples of asymmetric modular invariants.

Showed how to use diagonal invariants to generate asymmetric invariants.

Abstract

We give two results concerning the construction of modular invariant partition functions for conformal field theories constructed by tensoring together other conformal field theories. First we show how the possible modular invariants for the tensor product theory are constrained if the allowed modular invariants of the individual conformal field theory factors have been classified. We illustrate the use of these constraints for theories of the type $SU(2)_{K_A}\otimes SU(2)_{K_B}$, finding all consistent theories for $K_A,K_B$ odd. Second we show how known diagonal modular invariants can be used to construct some inherently asymmetric ones where the holomorphic and anti-holomorphic theories do not share the same chiral algebra. Some explicit examples are given.