# nLab exceptional generalized geometry

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 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. See also at exceptional field theory for more on this.

## 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 08).

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

• David Baraglia, Leibniz algebroids, twistings and exceptional generalized geometry, Journal of Geometry and Physics 62 (2012), pp. 903-934 (arXiv:1101.0856)

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

• Christian Hillmann, Generalized E(7(7)) coset dynamics and D=11 supergravity, JHEP 0903 (2009) 135 (arXiv:0901.1581)

• Hadi Godazgar, Mahdi Godazgar, Hermann Nicolai, Generalised geometry from the ground up (arXiv:1307.8295)

• Olaf Hohm, Henning Samtleben, Exceptional Form of $D=11$ Supergravity, Phys. Rev. Lett. 111, 231601 (2013) (arXiv:1308.1673)

The E10-geometry of 11-dimensional supergravity compactified to the line is discussed in

The E11-geometry of 11-dimensional supergravity compactified to the point is discussed in

The generalized-U-duality+diffeomorphism invariance in 11d is discussed in

For the worldvolume theory of the M5-brane this is discussed in

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

Relation to Borcherds superalgebras is surveyed and discussed in

• Jakob Palmkvist, Exceptional geometry and Borcherds superalgebras (arXiv:1507.08828)

black branes in the exotic spacetime are discussed in

The string- and membrane sigma-models on exceptional spacetime (the “exceptional sigma models”) are discussed in

Last revised on April 24, 2018 at 11:57:59. See the history of this page for a list of all contributions to it.