T-Duality and Non-Local Supersymmetries

TL;DR

The paper addresses how extended worldsheet supersymmetry behaves under T-duality when the associated complex structure depends on the duality coordinate $\theta$. It extends T-duality to $N=1$ supersymmetric non-linear $\sigma$-models via a canonical transformation, deriving a non-local object $\widetilde{J}$ that replaces the $\theta$-dependent complex structure in the dual theory and establishing the corresponding invariance conditions. By relating worldsheet and target-space supersymmetry, it shows that the dual theory realizes extended SUSY non-locally through a non-local spinor parameter and generalized Killing-like equations, with locality recovered only in special self-dual cases. The results illuminate how T-duality and $O(d,d)$ deformations reshape the target-space SUSY structure, implying that dual string theories can be equivalent even when their low-energy SUSY realizations are non-local, and highlight potential subtleties with S-duality in related duality chains.

Abstract

We study the non-localization of extended worldsheet supersymmetry under T-duality, when the associated complex structure depends on the coordinate with respect to which duality is performed. First, the canonical transformation which implements T-duality is generalized to the supersymmetric non-linear $σ$-models. Then, we obtain the non-local object which replaces the complex structure in the dual theory and write down the condition it should satisfy so that the dual action is invariant under the non-local supersymmetry. For the target space, this implies that the supersymmetry transformation parameter is a non-local spinor. The analogue of the Killing equation for this non-local spinor is obtained. It is argued that in the target space, the supersymmetry is no longer realized in the standard way. The string theoretic origin of this phenomenon is briefly discussed.