E10, billiards, and M-theory

Eleven-dimensional supergravity on tori "T^k" has exceptional non-compact symmetries, and discrete subgroups of them - uniformly written as "E_{k(k)} (Z)" - are expected to be exact symmetries (U-duality groups) of the full quantum theory, namely M-theory.

For "k" smaller than 6, the exceptional symmetry can really be written as a classical symmetry. For example, for "k=5", the symmetry "E_{5(5)}(Z)" is nothing else than "SO(5,5,Z)" and this U-duality group may be seen as the T-duality group of the (1,1) little string theory - which is the BFSS-like matrix model for M-theory on "T^5" and the stringy UV completion of 5+1-dimensional Yang-Mills theory. The T-duality group of the (1,1) little string theory - which is the decoupled limit of type IIB string theory on "N" NS5-branes for the coupling at infinity going to zero - is inherited from the "big" string theory: it's the same SO(5,5,Z).

For "k=6,7,8", the U-duality group is the honest exceptional Lie group, and no proper geometric interpretation is known. For "k=9", the "E_k" group does not exist as a finite-dimensional algebra. But in a proper sense, "E_9" is nothing else than the affine "E_8". This infinite-dimensional construction is relevant for M-theory on "T^9" which has two large dimensions.

One can go on to even more speculative realms - M-theory on "T^{10}". There, naively, the relevant algebra is "E_{10}" which is even more infinite-dimensional than "E_{9}". It is a hyperbolic algebra - the signature of its Cartan subalgebra is 9+1 and the roots appear both in the space-like region (the imaginary roots) as well as the time-like region (the real roots, analogous to the roots of the compact Lie groups), and most of them have infinite degeneracy. Incidentally, the next natural step, "E_{11}", is so horribly infinite-dimensional that almost nothing is known about its properties.

One of the proposals to illuminate the meaning of the exceptional groups, recently studied by Damour and Nicolai in

• hep-th/0504153,

is that the classical motion of a particle along null geodesics in the E_{10} group manifold describes the classical evolution of bosonic fields in M-theory (i.e. the UV-completed 11-dimensional supergravity, including higher order terms). This kind of idea was first obtained by the investigation of the evolution of toroidally compactified M-theoretical universes when the circles are very short - we have also played with things like that years ago. In this context, one sees similar behavior as in chaos theory - billiards, quantum billiards, and so forth.

Even if this picture describes the classical bosonic sector of M-theory, the obvious question is how to quantize it and how to incorporate the fermions?

At any rate, I am among those who believe that a proper generalized geometric understanding of the existence of exceptional symmetries in maximally supersymmetric flat backgrounds of M-theory is one of the most natural paths to a more universal definition of string/M-theory. This is why the picture of Nicolai et al. - much like the mysterious duality and other directions based on properties of the exceptional groups - should be looked at.