Supersymmetric D-branes and calibrations on general N=1 backgrounds

TL;DR

This work develops a unified framework to analyze supersymmetric D-branes in general ${\cal N}=1$ flux backgrounds using two $O(6,6)$ pure spinors $\Psi^\pm$ that encode the internal geometry. It derives brane SUSY conditions from κ-symmetry, showing that a brane must wrap a generalised complex submanifold associated with the integrable pure spinor and must satisfy a stability condition linked to the non-integrable spinor; together these reduce to a generalised calibration expressed in terms of $\Psi^\pm$ and RR potentials. The results illuminate the geometric and stability data governing D-branes in flux backgrounds and reinforce a generalized mirror symmetry exchanging IIA and IIB data, including the corresponding calibrations. By specializing to ${\rm SU}(3)$-structure manifolds, the paper recovers familiar holomorphic and coisotropic brane cases while placing them in the broader $SU(3)\times SU(3)$ generalized geometry framework. Overall, the work provides precise, calibration-based criteria for constructing and studying SUSY D-branes in flux backgrounds with potential implications for phenomenology and holography.

Abstract

We study the conditions to have supersymmetric D-branes on general {\cal N}=1 backgrounds with Ramond-Ramond fluxes. These conditions can be written in terms of the two pure spinors associated to the SU(3)\times SU(3) structure on T_M\oplus T^\star_M, and can be split into two parts each involving a different pure spinor. The first involves the integrable pure spinor and requires the D-brane to wrap a generalised complex submanifold with respect to the generalised complex structure associated to it. The second contains the non-integrable pure spinor and is related to the stability of the brane. The two conditions can be rephrased as a generalised calibration condition for the brane. The results preserve the generalised mirror symmetry relating the type IIA and IIB backgrounds considered, giving further evidence for this duality.