The canonical structure of maximally extended supergravity in three dimensions

Abstract

The algebra of local and rigid symmetries of N = 16 supergravity in three dimensions is discussed in detail. The conserved charges associated with the rigid E8 symmetry are constructed and shown to be physical observables in that they weakly commute with all constraints. The phase-space variables, which render the constraints polynomial, are identified; they are related to the variables used in the SO(1,2) × SO(16)-invariant reformation of d = 11 supergravity. This indicates that Ashtekar's approach to canonical gravity can be generalized to higher dimensions if suitable matter degrees of freedom are added to gravity

New canonical variables for d=11 supergravity

“…The brackets between the internal components P A m and Q IJ m all vanish (this is not true for the d = 3 components). In the dimensional reduction to three dimensions, where ∂ m ≡ 0 and Q IJ m = P A m ≡ 0, these brackets coincide with the brackets predicted by the σ-model formulation [13,15] (modulo different normalizations).…”

“…where, of course, Γ I AȦ are the standard SO(16) Γ-matrices and Γ IJ AB ≡ (Γ [I Γ J] ) AB , etc. The identities (2) were already given in [13]; the minus sign in (1) reflects the fact that we are dealing with the non-compact form E 8(+8) . In addition we will need to make use of the more general relations…”

“…The correctness of the ensuing brackets can be tested in alternative ways, for instance by inspection of the composite connections and their field content in terms of the original d = 11 fields. Furthermore, in the dimensional reduction to d = 3, the canonical brackets must match with those derived in [13].…”

“…which in the reduction to three dimensions are just the canonical momenta associated with the scalar fields, cf. [13]. We emphasize that bosonic brackets may vanish only up to bilinear fermionic contributions which we have neglected here; this may necessitate redefinitions by fermionic bilinears, such as e.g.…”

Loop quantum gravity: an outside view

“…In fact, one of the authors (H.N.) was first enticed into learning about Ashtekar's variables when realising that they simplify the calculation of the constraint algebra of supergravity considerably [75,76]. This is because the local supersymmetry constraint (i.e.…”