Non-Abelian T-duality and Modular Invariance

TL;DR

This paper explores the non-Abelian T-duality limit of certain coset conformal field theories, demonstrating that the resulting partition functions remain modular invariant, with detailed analysis for the case of (2) algebras.

Contribution

It establishes that the non-Abelian T-dual limit of coset CFTs preserves modular invariance, providing a concrete example with (2) algebras.

Findings

The non-Abelian T-dual limit of the coset CFT is well-defined.

The resulting partition function remains modular invariant.

Application to (2) case confirms the general result.

Abstract

Two-dimensional $\sigma$-models corresponding to coset CFTs of the type $ (\hat{\mathfrak{g}}_k\oplus \hat{\mathfrak{h}}_\ell )/ \hat{\mathfrak{h}}_{k+\ell}$ admit a zoom-in limit involving sending one of the levels, say $\ell$, to infinity. The result is the non-Abelian T-dual of the WZW model for the algebra $\hat{\mathfrak{g}}_k$ with respect to the vector action of the subalgebra $\mathfrak{h}$ of $ \mathfrak{g}$. We examine modular invariant partition functions in this context. Focusing on the case with $\mathfrak{g}=\mathfrak{h}=\mathfrak{su}(2)$ we apply the above limit to the branching functions and modular invariant partition function of the coset CFT, which as a whole is a delicate procedure. Our main concrete result is that such a limit is well defined and the resulting partition function is modular invariant.