T-Duality and Non-Local Supersymmetries

TL;DR

This paper investigates how T-duality affects extended worldsheet supersymmetry when the complex structure depends on the dualized coordinate, revealing non-local supersymmetries and their implications in string theory.

Contribution

It generalizes the T-duality transformation to supersymmetric non-linear sigma models and introduces a non-local complex structure that preserves supersymmetry in the dual theory.

Findings

The complex structure becomes non-local after T-duality.

Supersymmetry transformations involve non-local spinors.

Standard realization of supersymmetry is broken in the dual space.

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 $\sigma$-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.