Non-Abelian T-duality and modular invariance

Two-dimensional σ-models corresponding to coset CFTs of the type $({\hat{g}}_k\oplus {\hat{h}}_{\ell })/{\hat{h}}_{k+\ell }$ admit a zoom-in limit involving sending one of the levels, say ℓ, to infinity. The result is the non-Abelian T-dual of the WZW model for the algebra ${\hat{g}}_k$ with respect to the vector action of the subalgebra $h$ of $g$. We examine modular invariant partition functions in this context. Focusing on the case with $g=h=\mathrm{su}\left(2\right)$ 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.