Kac-Moody Symmetries of Ten-dimensional Non-maximal Supergravity Theories

Abstract

A description of the bosonic sector of ten-dimensional N = 1 supergravity as a non-linear realisation is given. We show that if a suitable extension of this theory were invariant under a Kac-Moody algebra, then this algebra would have to contain a rank eleven Kac-Moody algebra, that can be identified to be a particular real form of very-extended D 8 . We also describe the extension of N = 1 supergravity coupled to an abelian vector gauge field as a non-linear realisation, and find the Kac-Moody algebra governing the symmetries of this theory to be very-extended B 8 . Finally, we discuss the related points for the N = 1 supergravity coupled to an arbitrary number of abelian vector gauge fields.

E10andSO(9, 9) invariant supergravity

“…This formula is easily read off from the list of highest weight vectors (see appendix B), and it is also easy to check that α D 0 is a real root. This root also appeared in [27] in the context of very-extended symmetries of type I theories as a subsector of type II. We can thus adjoin this root to the nine (common) simple roots of so(9, 9) called α i (i = 1, .…”

“…More specifically, the elements DE 10 form a subset of the NSNS sector of E 10 , corresponding to the even levels in the D 9 decomposition of E 10 . We note that the analogous embedding of DE 11 into E 11 was recently established in [27] also by invoking the embedding of type I into type IIA supergravity, but the level decomposition via their common D 10 has so far not been studied in any detail.…”

“…A non-linear realization of massive IIA supergravity [26], and the embedding of the bosonic sector of type I into that of type II supergravity [27] were already investigated in the context of an earlier, and conceptually different, proposal concerning the realization of hidden Kac-Moody symmetries in M-theory [28,29,30,31] (see also [32,33,34]). According to that proposal, D = 11 supergravity admits an M-theoretic extension possessing an even larger symmetry containing a hidden E 11 , which is also supposed to accommodate type IIB supergravity [34].…”

Towards a classification of branes in theories with eight supercharges

“…Indeed, it was shown in [22] that what one is actually counting in both cases are the longest weights of each representation of the fields. The theories with sixteen supercharges also admit a Kac-Moody description [41], which was used in [42] to determine the full spectrum of the theory. The real form of the algebra is in this case not maximally non-compact, and thus the weights of the representations of the fields are not necessarily real.…”

Pure type I supergravity and DE 10

“…K(DE 10 ) refers to the (formally) maximal compact subgroup of DE 10 which plays the role of a generalised spatial Lorentz group. In the context of the E 11 approach to Kac-Moody symmetries [21,22], the bosonic sectors of D = 10 type I theories (also with abelian vector fields) have been investigated in [23] and the equations of motion were derived from a DE 11 analysis in the pure type I case. The non-maximal pure D = 5, N = 2 supergravity has been studied from a Kac-Moody perspective in [24].…”