group theory

∞-Lie theory

# Contents

## Idea

A hyperbolic Kac-Moody Lie algebra in the E-series

E6, E7, E8, E9, $E_{10}$, E11, …

## Properties

### As U-duality group of 1d supergravity

$E_{10}$ is conjectured (e.g. Nicolai 08) to be the U-duality group (see there) of 11-dimensional supergravity compactified to 1 dimension.

supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
$SL(2,\mathbb{R})$1$SL(2,\mathbb{Z})$ S-duality10d type IIB supergravity
SL$(2,\mathbb{R}) \times$ O(1,1)$\mathbb{Z}_2$$SL(2,\mathbb{Z}) \times \mathbb{Z}_2$9d supergravity
SU(3)$\times$ SU(2)SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$$O(2,2;\mathbb{Z})$$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$8d supergravity
Spin(10)$SL(5,\mathbb{R})$$O(3,3;\mathbb{Z})$$SL(5,\mathbb{Z})$7d supergravity
SU(5)$Spin(5,5)$$O(4,4;\mathbb{Z})$$O(5,5,\mathbb{Z})$6d supergravity
E6$E_{6(6)}$$O(5,5;\mathbb{Z})$$E_{6(6)}(\mathbb{Z})$5d supergravity
E7$E_{7(7)}$$O(6,6;\mathbb{Z})$$E_{7(7)}(\mathbb{Z})$4d supergravity
E8$E_{8(8)}$$O(7,7;\mathbb{Z})$$E_{8(8)}(\mathbb{Z})$3d supergravity
E9$E_{9(9)}$$O(8,8;\mathbb{Z})$$E_{9(9)}(\mathbb{Z})$2d supergravityE8-equivariant elliptic cohomology
E10$E_{10(10)}$$O(9,9;\mathbb{Z})$$E_{10(10)}(\mathbb{Z})$
E11$E_{11(11)}$$O(10,10;\mathbb{Z})$$E_{11(11)}(\mathbb{Z})$

## References

### General

Lecture notes include

The fact that every simply laced hyperbolic Kac-Moody algebra is a sub Lie algebra of $E_{10}$ is due to

• Sankaran Viswanath, Embeddings of hyperbolic Kac-Moody algebras into $E_{10}$ (pdf)

### Relation to supergravity

Reviews include

• Hermann Nicolai, Wonders of $E_{10}$ and $K(E_{10})$ (2008) (pdf)
Revised on May 16, 2014 10:03:50 by Urs Schreiber (89.204.135.51)