# nLab exceptional generalized geometry

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Idea

A variant of the idea of generalized complex geometry given by passing from generalization of complex geometry to generalization of exceptional geometry. Instead of by reduction of structure groups along inclusions like $O(d)\times O(d) \to O(d,d)$ it is controled by inclusions of into split real forms of exceptional Lie groups.

This serves to neatly encode U-duality groups in supergravity as well as higher supersymmetry of supergravity compactifications.

## Examples

### Higher supersymmetry

Compactification of 11-dimensional supergravity on a manifold of dimension 7 preserves $N = 1$ supersymmetry precisely if its generalized tangent bundle has G-structure for the inclusion

$SU(7) \hookrightarrow E_{7(7)}$

of the special unitary group in dimension 7 into the split real form of E7. This is shown in (Pacheco-Waldram).

One dimension down, compactification of 10-dimensional type II supergravity on a 6-manifold $X$ preserves $N = 2$ supersymmetry precisely if the generalized tangent bundle $T X \otimes T^* X$ in the NS-NS sector admits G-structure for the inclusion

$SU(3) \times SU(3) \hookrightarrow O(6,6) \,.$

This is reviewed in (GLSW, section 2).

## References

### General

• David Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry (arXiv:1101.0856)

### In supergravity

Survey slides include

• David Baraglia, Exceptional generalized geometry and $N = 2$ backgrounds (pdf)

Reviewes include

• Daniel Persson, Arithmetic and Hyperbolic Structures in String Theory (arXiv:1001.3154)

• Nassiba Tabti, Kac-Moody algebraic structures in supergravity theories (arXiv:0910.1444)

Original articles include

E6,E7, E8-geometry is discussed in

• Machiko Hatsuda, Kiyoshi Kamimura, M5 algebra and $SO(5,5)$ duality (arXiv:1305.2258)