Non-Abelian T-duality and Modular Invariance

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.

33 pages, this version published in Nuclear Physics B