G-Structures and Wrapped NS5-Branes

TL;DR

Gauntlett et al. develop a geometric framework that ties supersymmetric type II backgrounds with NS flux to $G$-structures on seven-manifolds arising from NS5-branes wrapped on associative or SLAG three-cycles. They prove that a single $G_2$-structure or an $SU(3)$-structure, with specific torsion and flux constraints, is both necessary and sufficient for SUSY, enabling direct construction of explicit solutions and revealing a vanishing theorem for compact manifolds. The paper provides both a known (agk, GKMW) and a new non-compact associative-cycle solution, illustrating the method's power and its ability to recover known results in a unified framework. The approach generalizes to other type II backgrounds and to type I, highlighting the role of intrinsic torsion and generalized calibrations in wrapped-brane holography.

Abstract

We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form R^{1,9-n} x M_n with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d=7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M_7 admits a G_2-structure or an SU(3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M_7 is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity.