Wednesday, May 04, 2005

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
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.


  1. I think you will like this quote?

    When we rise above the pool table, and think of sound as a analogy of the balls hitting each other, this classical discription paves the way for more "illustrous views" of what you are talking about.

    But if people do not want to embrace this ideology, then you will have the most inappropriate resistance to retaining this theoretical position?:)No refutations, and wide spread debunking status?

    Why will people who reject string theory care? Will they like to stay misinformed or not understand the ideology and place it to factions of, "for and against?"

    Gia and his hammered plate is a extension of this view. Amazing correlations, in classical defintions, to a more resounding view of reality. Good for you and all others in string theory who lead the way.

  2. Hi Lubos,

    After reading your post on cosmological billiards a few days ago, I was inspired to post some comments on the state of E11 in reply, which can be found on my blog.

    Keep up the good work.

    Best wishes,