$N=(2,2)$ superfields and geometry revisited

TL;DR

This paper revisits the connection between generalized Kähler geometry and $N=(2,2)$ supersymmetric sigma models, introducing a doubled target space formulation that offers new geometric insights and extensions.

Contribution

It introduces a doubled target space formulation combining original and dual superfields, providing a new geometric perspective on $N=(2,2)$ sigma models and their equations of motion.

Findings

Doubled geometry corresponds to Donaldson's deformation of the cotangent bundle.

Equations of motion are interpreted as intersections of Lagrangian submanifolds.

Extensions include quadrupling the geometry and exploring their geometric structures.

Abstract

We take a fresh look at the relation between generalised K\"ahler geometry and $N=(2,2)$ supersymmetric sigma models in two dimensions formulated in terms of $(2,2)$ superfields. Dual formulations in terms of different kinds of superfield are combined to give a formulation with a doubled target space and both the original superfield and the dual superfield. For K\"ahler geometry, we show that this doubled geometry is Donaldson's deformation of the holomorphic cotangent bundle of the original K\"ahler manifold. This doubled formulation gives an elegant geometric reformulation of the equations of motion. We interpret the equations of motion as the intersection of two Lagrangian submanifolds (or of a Lagrangian submanifold with an isotropic one) in the infinite dimensional symplectic supermanifold which is the analogue of phase space. We then consider further extensions of this formalism, including one in which the geometry is quadrupled, and discuss their geometry.