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

Jieming Lin, Kellogg S. Stelle, Daniel Waldram

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.

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

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 structure in the generalised geometry introduced by Strickland--Constable, where 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 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.

Paper Structure

This paper contains 17 sections, 1 theorem, 180 equations.

Table of Contents

  1. Introduction
  2. $Spin(n,n)$ geometry and consistent truncations
  3. Consistent truncations from generalised $G$-structures
  4. Spin(n,n) gg
  5. Branes and torsion-free $Spin(n)\subset Spin(n,n)$ structures
  6. Consistent truncation to supergravities on branes
  7. D3-brane
  8. D4-brane
  9. M5-brane
  10. Type II fivebranes
  11. D6-brane
  12. D7-brane
  13. Conclusion and outlook
  14. Conventions
  15. Exceptional generalised geometry
  16. ...and 2 more sections

Key Result

Theorem 1

Let $k=n$ and $k=n+1$ for eleven-dimensional and type II supergravity respectively on $Z\times M$. If $M$ admits a generalised $G_S$-structure with $G_S\subset H_k$, and $G_S$ has constant singlet intrinsic torsion, then there is a consistent truncation of the supergravity theory to $Z$.

Theorems & Definitions (1)

  • Theorem