Generalized Kahler manifolds and off-shell supersymmetry
Ulf Lindstrom, Martin Rocek, Rikard von Unge, Maxim Zabzine
TL;DR
The paper resolves the long-standing problem of formulating general $N=(2,2)$ off-shell supersymmetric two-dimensional sigma models by showing that the target geometry is generalized Kahler and that the full local geometry is encoded in a single generalized Kahler potential $K$ constructed from chiral, twisted chiral, and semichiral superfields. Using three Poisson structures $oldsymbol{ p}_\pm$ and $oldsymbol{ ho}$, the authors develop natural coordinates adapted to their foliations and show that, locally, metric and $B$-field can be expressed nonlinearly through second derivatives of $K$, with $K$ serving as both the superspace Lagrangian and the generating function between $J_+$- and $J_-$-adapted coordinates. The work provides a complete geometric framework for off-shell $N=(2,2)$ supersymmetry in general settings, including the nontrivial kernel and cokernel cases, and connects generalized Kahler geometry to holomorphic symplectic and Calabi–Yau structures. This advances both the physical formulation and the mathematical understanding of generalized Kahler manifolds, enabling quotients, hyperkähler considerations, and further exploration via Poisson geometry.
Abstract
We solve the long standing problem of finding an off-shell supersymmetric formulation for a general N = (2, 2) nonlinear two dimensional sigma model. Geometrically the problem is equivalent to proving the existence of special coordinates; these correspond to particular superfields that allow for a superspace description. We construct and explain the geometric significance of the generalized Kahler potential for any generalized Kahler manifold; this potential is the superspace Lagrangian.