The appearance of exceptional Lie algebra, Kac-Moody algebras and Borcherds algebra in gravitational theories is a very elegant and exciting corner of current research in supergravity and string theory. I would like to discuss how exceptional Lie algebras naturally appear in the context of maximal supergravity theories and how this is connected to $E_{11}$ and Borcherds superalgebras. I will also discuss how Kac-Moody algebra occurs naturally in the context of gravitational singularities.

## 11 dimensional supergravity and exceptional Lie algebra

11 dimensional supergravity was constructed in 1978 by Cremmer, Julia and Scherk.
Nowadays it is considered as the low energy limit of M-theory.
The field content of 11-dimensional supergravity is simply given by a metric $g$ and a 3-form $A_{(3)}$.
We can construct all the (massless)
maximal supergravity theory in
$D$-dimension with $2 < D <11$, by
considering (Kaluza-Klein) reduction
of 11 dimensional supergravity on a
$(11-D)$-torus. This process produces a lot of additional fields (2-forms, 1-forms and 0-forms) coming from the reduction of the metric and the 3-form.
In general for a compactification of 11 dimensional supergravity on
a torus $T^{11-D}$ to a D-dimensional
spacetime, we produce a scalar
manifold $$
\frac{E_{11-D}}{K(E_{11-D})},$$
where $K(G)$ is the maximal
compact subalgebra of $G$. In particular, we have in dimension 5,
4 and 3

$$ 5D\rightarrow \frac{E_6}{USp(8)},
\quad 4D\rightarrow \frac{E_7}{SU(8)},
\quad 3D\rightarrow
\frac{E_8}{SO(16)}. $$

## $E_{11}$ conjecture and Borcherds algebras

We recall that $E_9=E_8^+$ is understood as the extended Dynkin diagram of $E_8$. In the same way $E_{10}=E_8^{++}$ and $E_{11}=E_8^{+++}$ are the over-extended and the very-extended Dynkin diagram of $E_8$.
There is a conjecture introduced by
Peter West in 2001 and supported by
several facts that the Kac-Moody
algebra $E_{11}$ is related to a
non-linear realization of M-theory and
that $E_{11}$ can provide an 11
dimensional origin not only of all
massless maximal supergravity theories
(including type IIB) but also of the
massive ones.

A beautiful duality was discovered by Iqbal, Neitzke and Vafa between compactifications of M-theory on tori and the second cohomology of some associated del Pezzo surfaces. Now the full cohomology of theses surfaces spans the root lattice of a Borcherds superalgebra.

Henry-Labordere Julia and Paulot have
shown that some truncations of these
Borchers algebras provide a
classification of $p$-forms coming from
tori reduction of (massive) maximal
supergravity. This classification matches the one of the $E_{11}$ conjecture of Peter West. The Borcherds description was recently proven to be systematically derived from the split real form of $E_{11}$ by Henneaux, Julia and Levie.

## Space-time singularities, Kac-Moody algebra and Cosmic billiards

A fascinating and non-speculative occurrence of $E_9$ and $E_{10}$ in a theory of gravity occurs when studying the behavior of gravity near a spacetime singularity.
Belinskii, Khalatnikov and Lifchitz (BKL) have studied in details the general solution of Einstein equations near a spacetime singularity. As one reaches the singularity, the Einstein equations admit a chaotic behavior in time. Chitre and Misner has reformulated the BKL analysis in terms of a billiard motion in a 2 dimensional hyperbolic space.

In higher dimension, the chaotic behavior disappear in spacetime dimensions greater than 10.
In particular, in 11 dimensions, there is no chaos at all. But if one add a 3-form (like the one of 11 dimensional supergravity), chaos comes back. In higher dimension one can also describe the chaotic behavior by a billiard in a higher dimensional hyperboloic space.

When a theory admits a
compactification to three dimensions
on a higher dimensional torus such
that in the reduced 3 dimensional
theory, the Lagrangian is given by
Einstein-Hilbert action and a sigma
model with target space a $G/H$ such
that $G$ is a simple Lie group and $H$
its maximal compact subgroup, the
billiard table is a Coxeter polyhedron
and the billiard group is a Coxeter
group. The table billiard can be
described by the over-extended
Kac-Moody algebra $G^{++}$ associated
with the group $G$.
In particular $$ \text{The billiard
associated with eleven supergravity is
} E_8^{++}=E_{10}.$$

One can formulate the billiard dynamics as a motion in the Cartan subalgebra of the Kac-Moody algebra.

determinantof the matrix, etc. Lie theory originated in questions about mathematics and only gradually got taken up by various kinds of physicists overcoming fear of the "Gruppenpest". At first only the classical Lie groups were thought important, but in theoretical areas like string theory the exceptional ones have now caught on. Affine Kac-Moody algebras were at first just an abstract curiosity but then turned out to fit needs of conformal field theory. $\endgroup$