\(G\)-structures and wrapped NS\(5\)-branes. (English) Zbl 1061.81058

Gauntlett, Jerome P.; Martelli, Dario; Pakis, Stathis; Waldram, Daniel

Commun. Math. Phys. 247, No. 2, 421-445 (2004).

Summary: We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form \(\mathbb{R}^{1,9-n}\times M_n\) 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 \(\text{SU}(3)\)-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.

MSC:

81T30	String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E50	Supergravity
53C29	Issues of holonomy in differential geometry
83E30	String and superstring theories in gravitational theory