Non-Abelian T-duality and modular invariance

Abstract

Two-dimensional σ-models corresponding to coset CFTs of the type (ĝ k ⊕ĥ )/ĥ k+ admit a zoom-in limit involving sending one of the levels, say , to infinity. The result is the non-Abelian T-dual of the WZW model for the algebraĝ 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 = 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.

Type II DFT solutions from Poisson–Lie $T$-duality/plurality

“…Various aspects have been studied in [12,[22][23][24][25][26][27][28][29][30][31][32][33][34][35], but unlike the Abelian T -duality, there are still many things to be clarified. For example, the partition function in the dual model is not the same as that of the original model (see [36] for a recent study), and NATD may rather be regarded as a map between two string theories. The global structure of the dual geometry is also not clearly understood [12].…”

“…The transformation rules for the Ramond-Ramond (R-R) fields and spacetime fermions were determined in [13][14][15][16].This well-established symmetry of string theory is called the Abelian T -duality since it relies on the existence of Killing vectors which commute with each other (see [17,18] for reviews).An extension of the T -duality to the case of non-commuting Killing vectors was explored in [19] (see [20,21] for earlier works), and this is known as the non-Abelian T -duality (NATD).Various aspects have been studied in [12,[22][23][24][25][26][27][28][29][30][31][32][33][34][35], but unlike the Abelian T -duality, there are still many things to be clarified. For example, the partition function in the dual model is not the same as that of the original model (see [36] for a recent study), and NATD may rather be regarded as a map between two string theories. The global structure of the dual geometry is also not clearly understood [12].…”

Doubled aspects of generalised dualities and integrable deformations

“…The partition sums of these theories need not match. This viewpoint dates back to [7] and was recently shown to be the case in a simple SU (2) non-Abelian T-dualisation [8].…”

D-branes (or not) in the non-Abelian T-dual of the SU(2) WZW model

“…At k large this function localizes around the conjugacy class ψ = (λ/k)π, as found in [9] 2 Let us now consider the large-k theory where we truncate to primaries whose weight λ scales like k 1/2 , as discussed in [6]. The closed string states, because they have wavelength ∼ k −1/2 times the linear size (∼ √ k) of the target manifold (in units of string length), are supported by a zoomed-in North pole geometry where we send…”

“…It was shown in [5], that the classical action for the non-Abelian T-dual of the SU(2) k WZW model with respect to the vector SU(2) isometry can be obtained as the limit of the coset model su(2) k ⊕ŝu(2) l su(2) k+ℓ (1.1) in which ℓ → ∞, and we must also zoom in close to the identity in the corresponding coset geometry, parametrizing the SU(2) ℓ group element as g 2 = 1+i v/ℓ. The recent paper [6] developed a corresponding modular invariant truncation of the exact coset CFT. The torus partition (namely, the closed string sector) was calculated, and the fusion rules were discussed.…”