Fermionic Kac-Moody billiards and supergravity

Abstract

We study the "fermionic billiards", i.e. the chaotic dynamics of the gravitino, that arise in the near-spacelike-singularity limit of elevendimensional supergravity and of its dimensional truncations (notably fourdimensional simple supergravity). By exploiting the gravity-coset correspondence, we show that the billiard dynamics of the gravitino is described by a 'spin extension' of the Weyl group of the hyperbolic Kac-Moody algebra E 10 . This spin extension is a discrete subgroup of (a spin covering of) the maximal compact subgroup K(E 10 ) of E 10 that is generated by ten (simple-root-related) idempotent elements of order 8. The 'super-billiard' that combines the bosonic and fermionic billiards is found to have a remarkably simple structure, which exhibits a striking analogy with a polarized photon propagating in the tendimensional Lorentzian Weyl chamber of E P 10 a=1 β a collapses at each spatial point). This limit shall also be referred to as the 'BKL-limit' in the following. Let us first recall that the 1 See Appendix A for our conventions in the Cremmer-Julia-Scherk action [20]. 2 The proper time T is related to t by dT = N (t, x)dt at each spatial point x [14].

Spin covers of maximal compact subgroups of Kac–Moody groups and spin-extended Weyl groups

“…We can now derive a generalization of [9 X i X j D´ X j X i ; if ¹i; j º 2 E.…/; X j X i ; if ¹i; j º … E.…/;…”

On higher spin realizations of K(E 10)

“…Even for the affine subgroup E 9 and its compact subgroup K(E 9 ) this is a non-trivial problem [14]. First results for K(E 10 ) have been obtained in [1,[15][16][17][18][19][20], where it was shown how to write the supersymmetry parameter and the gravitino field as consistent albeit unfaithful representations of the Lie algebra of K(E 10 ). 1 The qualifier 'unfaithful' here signals that -although the Lie algebra K(E 10 ) is infinite-dimensional -the representation spaces are finite-dimensional.…”

“…First results for K(E 10 ) have been obtained in [1,[15][16][17][18][19][20], where it was shown how to write the supersymmetry parameter and the gravitino field as consistent albeit unfaithful representations of the Lie algebra of K(E 10 ). 1 The qualifier 'unfaithful' here signals that -although the Lie algebra K(E 10 ) is infinite-dimensional -the representation spaces are finite-dimensional. The corresponding transformation rules have been worked out for maximal supergravity (and also for half-maximal supergravity [21]); they have reappeared in recent studies of fermions in the context of generalised geometry, see e.g.…”

Reconstruction of the equation of state for the cyclic universes in homogeneous and isotropic cosmology

“…More recently, f-essence is dealt with a Grassmann-valued quantity in Ref. [132] or an operator, i.e., q-number in Ref. [133].…”