$Spin(n,n)\times\mathbb{R}^+$ Generalised Geometry and Consistent Truncations on Branes

TL;DR

This paper integrates known brane truncations into the framework of $Spin(n,n)\times\mathbb{R}^+$ generalised geometry, deriving new consistent truncations for various branes and supergravity theories.

Contribution

It demonstrates how brane truncations fit into exceptional generalised geometry and introduces new consistent truncations for specific branes.

Findings

Embedded brane solutions as torsion-free $Spin(n)$ structures.

Derived new truncation for IIA NS5-brane to 6D supergravity.

Established truncations for D6- and D7-branes to 7D and 8D supergravity.

Abstract

In this note we show how the consistent truncations on half-supersymmetric branes of Leung and Stelle and Lin, Skrzypek and Stelle fit into the general exceptional generalised geometry analysis of Cassani \emph{et al.}. Each solution defines a torsion-free $Spin(n)$ structure in the $Spin(n,n)\times \mathbb{R}^+$ generalised geometry introduced by Strickland--Constable, where $n$ is the dimension of the space transverse to the brane. Embedding this into the appropriate exceptional generalised geometry then defines the truncation. As a by-product we derive a new consistent truncation on the IIA NS5-brane to six-dimensional $\mathcal{N}=(2,0)$ supergravity coupled to a tensor mutliplet, and new consistent truncations on the D6- and D7-branes to seven- and eight-dimensional pure half-maximal supergravity respectively.