Geometry of D-branes for general N=(2,2) sigma models

TL;DR

This work develops a world-sheet description of D-branes in general $N=(2,2)$ sigma models using gluing conditions on $T{ m M}igoplus T^*{ m M}$ and places the analysis in the framework of (twisted) generalized Kähler geometry. It shows that B-type branes are (twisted) generalized complex submanifolds with respect to ${\cal J}_1$ and A-type branes with respect to ${\cal J}_2$, with brane geometry governed by coisotropic conditions and foliations tied to the Poisson structures $\omega_+^{-1}\mp\omega_-^{-1}$; in the Kähler or $H$-flux backgrounds these reduce to familiar objects such as complex submanifolds with $F$ of type $(1,1)$ and Lagrangian/coisotropic A-branes, with conjugacy-class examples on group manifolds providing explicit instances. Mirror symmetry exchanges ${\cal J}_1$ and ${\cal J}_2$, mapping B-branes to A-branes, consistent with generalized mirror symmetry beyond Calabi–Yau. The paper thus establishes a unified $T{ m M}igoplus T^*{ m M}$ perspective for $N=(2,2)$ D-branes and highlights the potential for topological twists and a broader notion of D-branes as generalized complex foliations.

Abstract

We give a world-sheet description of D-brane in terms of gluing conditions on T+T^*. Using the notion of generalized Kahler geometry we show that A- and B-types D-branes for the general N=(2,2) supersymmetric sigma model (including a non-trivial NS-flux) correspond to the (twisted) generalized complex submanifolds with respect to the different (twisted) generalized complex structures however.