Supersymmetric non-linear sigma-models with boundaries revisited

TL;DR

This paper revisits supersymmetric non-linear sigma-models with boundaries, providing a simplified off-shell formulation, analyzing boundary conditions, and exploring conditions for extended supersymmetry in a geometric framework.

Contribution

It offers a comprehensive, simplified off-shell formulation of N=1 supersymmetric sigma-models with boundaries and clarifies conditions for extended supersymmetry.

Findings

Derived the most general boundary conditions for non-supersymmetric case

Showed no additional conditions are needed for N=1 supersymmetry

Identified geometric conditions for second supersymmetry involving complex structures

Abstract

We study two-dimensional supersymmetric non-linear sigma-models with boundaries. We derive the most general family of boundary conditions in the non-supersymmetric case. Next we show that no further conditions arise when passing to the N=1 model. We present a manifest N=1 off-shell formulation. The analysis is greatly simplified compared to previous studies and there is no need to introduce non-local superspaces nor to go (partially) on-shell. Whether or not torsion is present does not modify the discussion. Subsequently, we determine under which conditions a second supersymmetry exists. As for the case without boundaries, two covariantly constant complex structures are needed. However, because of the presence of the boundary, one gets expressed in terms of the other one and the remainder of the geometric data. Finally we recast some of our results in N=2 superspace and discuss applications.