Pseudo-hyperkahler Geometry and Generalized Kahler Geometry

TL;DR

This paper explores the geometric structures underlying N=(2,2) supersymmetric sigma models, revealing conditions for twisted supersymmetry that lead to pseudo-hyperkähler geometries on the target space.

Contribution

It identifies the geometric conditions for twisted supersymmetry in semi-chiral sigma models, linking them to pseudo-hyperkähler structures, which was not previously established.

Findings

Twisted supersymmetry solutions correspond to pseudo-hyperkähler geometry.

Additional linear supersymmetry solutions are absent or trivial.

The target space admits a bi-hermitian metric of signature (2,2).

Abstract

We discuss the conditions for additional supersymmetry and twisted supersymmetry in N = (2, 2) supersymmetric non-linear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We find that additional linear supersymmetry has no interesting solution, whereas additional linear twisted supersymmetry has solutions with interesting geometrical properties. We solve the conditions for invariance of the action and show that these solutions correspond to a bi-hermitian metric of signature (2, 2) and a pseudo-hyperkaehler geometry of the target space.