Generalized complex manifolds and supersymmetry

TL;DR

This work builds a worldsheet realization of Hitchin's generalized complex geometry by developing three two-dimensional models whose fields live in $T\oplus T^*$ and whose $N=(2,0)$ supersymmetry enforces a generalized complex structure ${\cal J}$. Through a sequence of first-order actions, topological models (with and without a WZ term), and a physical sigma model, the authors show that integrability of ${\cal J}$ with respect to the Courant bracket (and its twisted variant with $H$) is precisely the condition for extended supersymmetry. They derive algebraic and differential constraints on the tensor data that collapse to generalized complex (and generalized Kähler) structures in suitable limits, and they illuminate explicit solution classes and local Darboux forms. The results connect worldsheet supersymmetry, $SO(d,d)$ covariance, and generalized geometry, suggesting new avenues for string backgrounds, including nongeometric ones, and indicating rich directions for future work on boundaries and broader algebroid frameworks.

Abstract

We find a worldsheet realization of generalized complex geometry, a notion introduced recently by Hitchin which interpolates between complex and symplectic manifolds. The two-dimensional model we construct is a supersymmetric relative of the Poisson sigma model used in context of deformation quantization.