156 The Quantum Structure of Space and Time
where space ‘de-emerges’?Even if some variant of the gradient hypothesis turns out to be correct, there remains
the question how to interpret the remaining (M theoretic?) degrees of freedom. E.g.,
up to level! = 28, there are already 4400 752 653 representations of Ag, out of which
only 28 qualify as gradient representations!
Fermions: The above considerations can be extended to the fermionic sector, and
it can be shown that K(El0) indeed plays the role of a generalised ‘R symmetry’
[6]. Because K(E1o) is not a Kac-Moody group, many of the standard tools are
not available. double-valued) rep-
resentations which cannot be obtained from (or lifted to) representations of Elo.
Remarkably, it has been shown very recently [6, 71 that the gravitino field of D = 11
supergravity (at a fixed spatial point) can be promoted to a bona fide, albeit un-
faithful, spinorial representation of K(El0). This result strengthens the evidence
for the correspondence proposed in [2], and for the existence of a map between the
time evolution of the bosonic and fermionic fields of D = 11 supergravity and the
dynamics of a massless spinning particle on Elo/K(Elo). However, the existence
(and explicit construction) of a faithful spinorial representation, which might also
accommodate spatially dependent fermionic degrees of freedom, remains an open
problem.
This applies in particular to fermionic (i. e.
For further references and details on the results reported in this comment see
[2, 3, 61. The potential relevance of El0 was first recognized in [8]; an alterna-
tive proposal based on Ell has been developed in [9].
Acknowledgments: It is a great pleasure to thank T. Damour, T. Fischbacher,
M. Henneaux and A. Kleinschmidt for enjoyable collaborations and innumerable
discussions, which have shaped my understanding of the results reported here.
Bibliography
[l] V. Kac, Infinite Dimensional Lie Algebras, 3rd edition, C.U.P. (1990).
[2] T. Damour, M. Henneaux and H. Nicolai, Phys. Rev. Lett. 89 (2002) 221601,
hep-th/0207267.[3] T. Damour and H. Nicolai, hep-th/0410245.
[4] A. Kleinschmidt and H. Nicolai, hep-th/0407101.
[5] T. Damour, prepared comment in this volume.
[6] T. Damour, A. Kleinschmidt and H. Nicolai, hep-th/0512163.
[7] S. de Buyl, M. Henneaux and L. Paulot, hep-th/0512292.
[8] B. Julia, in: Lectures in Applied Mathematics, Vol. 21 (1985), AMS-SIAM, p. 335;
preprint LPTENS 80/16.