The Quantum Structure of Space and Time (293 pages)

Mathematical Structures 155

e

3k + 2

3k + 3

3k + 1

the Dg and A8 x A1 decompositions. Remarkably, for the bosonic sector of these

theories, El0 yields exactly the same information as local supersymmetry, namely

0 unique (bosonic) actions with the correct (Chern-Simons) couplings;

0 incompatibility of D = 11 supergravity with a cosmological constant;

0 self-duality of the 5-form field strength in IIB supergravity;

0 information about the higher order R4, R7,... corrections to M theory.

This may indicate, perhaps surprisingly, that (maximal local) supersymmetry may

not play such a prominent role at the most fundamental level. Indeed, in a theory,

where space and time are ‘emergent’, the distinction between bosons and fermions
must also be regarded an ‘emergent’ phenomenon, and not as a feature of the theory
at its most basic level 5.

Ag module Tensor

[k00000100] E,l...,k bi bz b3

[k00100000] Eal..,ak bl...b6

[kl000000l] E,,...akbl...b81b9

Towards higher levels: Understanding the (exponentially growing) spectrum

of higher level representations and their correct physical interpretation is the key
problem; on the mathematical side, this is the (still unsolved) problem of finding
a manageable realization for indefinite Kac-Moody algebras. An analysis of the

higher level representations w.r.t. Ag has revealed the existence of a distinguished

series of representations, the so-called gradient representations, which we tentatively
associate with (non-local functionals of) the higher order spatial gradients, corre-
sponding to the differential operators a,,... a,, acting on the D = 11 supergravity
fields.

(All tensors in the above table are symmetric in the lower indices al,... , ak.) We

thus face the following questions:

Can the complete evolution of D = 11 supergravity (or some M theoretic exten-

sion thereof) be mapped to a null geodesic motion on the infinite-dimensional

coset space Elo/K(Elo)?

Can the expansion in spatial gradients be understood in terms of a ‘zero tension

limit’, where space-time is regarded as some kind of ‘elastic medium’? And is

the level expansion (or an expansion in the height of the roots) the correct

mathematical framework for studying this limit?

Does El0 thereby provide a new Lie algebraic mechanism for the emergence of

space 6, and can the initial singularity be described in these terms as the place

5This point of view might receive further support if, contrary to widespread expectations, no

6Whereas time could emerge ‘operationally’, as it is expected to in canonical approaches based

evidence for supersymmetry is found at the Large Hadron Collider.

on a Wheeler-DeWitt equation.