Towards a Background Independent Quantum Gravity in Eight Dimensions

Abstract

We start a program of background independent quantum gravity in eight dimensions. We first consider canonical gravity a la "Kaluza-Klein" in D = d + 1 dimensions. We show that our canonical gravity approach can be applied to the case of self-dual gravity in four dimensions. Further, by using our previously proposed classical action of Ashtekar self-dual gravity formalism in eight dimensions, we proceed to develop the canonical approach in eight dimensions. Our construction considers different SO(8) symmetry breakings. In particular, the breaking SO(8) = S 7 R × S 7 L × G 2 plays an important role in our discussion.

Alternative self-dual gravity in eight dimensions

“…But the curvature tensor R a 1 a 2 µ 1 µ 2 contains only the two indices a 1 and a 2 . This is one of the reason why in Refs [6]- [9] an Ashtekar formalism with an octonionic structure was proposed. Although this an interesting route here we shall follow an alternative approach.…”

“…Higher dimensional Ashtekar formalism has shown to be a promising proposal [1][2][3][4][5] (see also Ref. [6][7][8][9][10]). The main physical reason for this is that it allows a possible connection with string theory [11]; an alternative approach for quantum gravity.…”

Towards a Canonical Gravity in Two Time and Two Space Dimensions

“…The general believe is that self-dual gravity makes sense only in four dimensions since in this case the dual of a two form (the curvature) is again a two form. However, there are a number of evidences that self-dual gravity can also be implemented in eight dimensions [5]- [8]. It turns out that even in four dimensions self-dual gravity does not determine the signature of the 'space-time'.…”

Oriented matroid theory and loop quantum gravity in (2+2) and eight dimensions

“…This seems to indicate that, from a quantum gravity perspective, 4-dimensions is an exceptional signature. However, in references [4][5][6][7] it is shown that also makes sense to consider self-dual gravity in 8-dimensions, but one should define * R AB in terms of the η-symbol. (see Refs.…”

“…Here, Ê i (a) is an operator associated with the E (a) i part of the general vielbein on a M D -manifold (see Ref. [7] and references therein)…”