Reformulating Supersymmetry with a Generalized Dolbeault Operator

TL;DR

Tomasiello reformulates the ${\cal N}=1$ SUSY conditions in type II supergravity using a generalized Dolbeault operator ${d^{\cal J}}$ tied to generalized complex geometry, removing explicit metric dependence from the equations. The key advance is expressing the RR flux as $F=-8 d_H^{\cal J_1}(e^{-3A}{\rm Im}\Phi_2)$, which linearizes the flux problem and enables a cohomological treatment of moduli, including a Hitchin-functional perspective for unobstructed deformations. The paper analyzes a subset of moduli controlled by deformations of $\Phi_2$, discusses the $dd^{\cal J}$-lemma and generalized Kodaira–Spencer theory as tools for existence results, and extends the framework to AdS$_4$ with orientifolds and branes. This metric-free reformulation aims to broaden the landscape of admissible vacua beyond Calabi–Yau geometries and to provide practical avenues for proving the existence of ${\cal N}=1$ backgrounds in flux compactifications.

Abstract

The conditions for N=1 supersymmetry in type II supergravity have been previously reformulated in terms of generalized complex geometry. We improve that reformulation so as to completely eliminate the remaining explicit dependence on the metric. Doing so involves a natural generalization of the Dolbeault operator. As an application, we present some general arguments about supersymmetric moduli. In particular, a subset of them are then classified by a certain cohomology. We also argue that the Dolbeault reformulation should make it easier to find existence theorems for the N=1 equations.