Supersymmetric sources, integrability and generalized-structure compactifications

TL;DR

This work shows that supersymmetric orientifold sources can be compatible with a dynamic ${\rm SU}(3)\times{\rm SU}(3)$-structure in type II flux compactifications, and it establishes integrability results asserting that, under mild assumptions, the Einstein and dilaton equations of motion follow from the supersymmetry conditions plus the form equations and Bianchi identities in the presence of calibrated sources. It provides explicit two-parameter T-duality constructions from a type ${\rm B}$ torus background to realize such structures on nilmanifolds, including one ${\cal N}=1$ example with an O5 plane. The paper also derives a calibration form for a space-filling NS5-brane and shows that the NSNS $H$-field equation of motion is implied by supersymmetry and Bianchi identities for Minkowski vacua. Together these results broaden the allowed landscape of generalized-structure flux vacua and deepen the link between bulk supersymmetry, calibrations, and back-reacting sources, with potential implications for AdS/CFT and phenomenology.

Abstract

In the context of supersymmetric compactifications of type II supergravity to four dimensions, we show that orientifold sources can be compatible with a generalized SU(3) x SU(3)-structure that is neither strictly SU(3) nor static SU(2). We illustrate this with explicit examples, obtained by suitably T-dualizing known solutions on the six-torus. In addition we prove the following integrability statements, valid under certain mild assumptions: (a) for general type II supergravity backgrounds with orientifold and/or D-brane generalized-calibrated sources, the source-corrected Einstein and dilaton equations of motion follow automatically from the supersymmetry equations once the likewise source-corrected form equations of motion and Bianchi identities are imposed; (b) in the special case of supersymmetric compactifications to four-dimensional Minkowski space, the equations of motion of all fields, including the NSNS three-form, follow automatically once the supersymmetry and the Bianchi identities of the forms are imposed. Both (a) and (b) are equally valid whether the sources are smeared or localized. As a byproduct we obtain the calibration form for a space-filling NS5-brane.