New Canonical Variables for d = 11 Supergravity

A set of new canonical variables for d = 11 supergravity is proposed which renders the supersymmetry variations and the supersymmetry constraint polynomial. The construction is based on the SO(1, 2) × SO(16) invariant reformulation of d = 11 supergravity given in [4], and has some similarities with Ashtekar’s reformulation of Einstein’s theory. The new bosonic variables fuse the gravitational degrees of freedom with those of the three-index photon AMNP in accordance with the hidden symmetries of the dimensionally reduced theory. Although E8 is not a symmetry of the theory, the bosonic sector exhibits a remarkable E8 structure, hinting at the existence of a novel type of “exceptional geometry”. Recent advances in string theory (see e.g. [1]) have lent renewed support to the long held belief that d = 11 supergravity [2] has a fundamental role to play in the unification of fundamental interactions. In this letter, we present an unconventional formulation of this theory, developing further the results of refs. [3, 4] where new versions of d = 11 supergravity with local SO(1, 3) × SU(8) and SO(1, 2) × SO(16) tangent space symmetries, respectively, were presented. In both versions the supersymmetry variations were shown to acquire a polynomial form from which the corresponding formulas for the maximal supergravities in four and three dimensions can be read off directly and without the need for complicated duality redefinitions. Our reformulation can thus be regarded as a step towards the complete fusion of the bosonic degrees of freedom of d = 11 supergravity (i.e. the elfbein and the antisymmetric tensor AMNP ) in a way which is in harmony with the hidden symmetries of the dimensionally reduced theories [5, 6]1. The results are very suggestive of a novel kind of “exceptional geometry” for d = 11 supergravity (or some bigger theory containing it) that would be intimately tied to the special properties of the exceptional groups, and would be characterized by relations such as (1)–(4) below, which have no analog in ordinary Riemannian geometry. The hamiltonian formulation of our results reveals surprising similarities with Ashtekar’s reformulation of Einstein’s theory [8] (for a conventional hamiltonian treatment of d = 11 supergravity, cf. [9]2). More specifically, the “248-bein” to be introduced below is the analog of the inverse densitized dreibein (or “triad”) in [8]. Furthermore, in terms of the canonical variables proposed here the supersymmetry constraints become polynomial; the polynomiality of the remaining canonical constraints is then implied by supersymmetry and the polynomiality of the canonical brackets. Unfortunately, not all sectors of the theory are as simple as one might have wished, and a further simplification will very likely require a better understanding of the exceptional structures alluded to above, as well as the further extension of the results of [3, 4] to incorporate the even larger (infinite dimensional) symmetries arising in the dimensional reductions of d = 11 supergravity to two and one dimensions, respectively. Let us first recall the main results, conventions and notation of [4] (further details will be provided in a forthcoming thesis [11]). To derive the new version from the original formulation of d = 11 supergravity, one first breaks the original tangent space symmetry SO(1,10) to its subgroup SO(1, 2)× SO(8) through a partial choice of gauge for the elfbein, and subsequently enlarges it again to SO(1, 2) × SO(16) by introducing new gauge degrees of freedom. This symmetry enhancement requires suitable redefinitions of the bosonic and fermionic fields, or, more succinctly, their combination into tensors w.r.t. the new tangent space symmetry. The basic strategy underlying this construction goes back to [5], but the crucial difference is that the dependence on all eleven coordinates is retained here. The construction thus requires a 3+8 split of the d = 11 coordinates and indices, implying a similar split for all tensors of the theory. Accordingly, In a different context, the fusion of gravitational and matter (Yang Mills) degrees of freedom was also attempted in [7]. The Chern-Simons part of the d = 11 action was recently considered in [10].