Holst-MacDowell-Mansouri action for (extended) supergravity with boundaries and super Chern-Simons theory

Abstract

In this article, the Cartan geometric approach toward (extended) supergravity in the presence of boundaries will be discussed. In particular, based on new developments in this field, we will derive the Holst variant of the MacDowell-Mansouri action for $$ \mathcal{N} $$ N = 1 and $$ \mathcal{N} $$ N = 2 pure AdS supergravity in D = 4 for arbitrary Barbero-Immirzi parameters. This action turns out to play a crucial role in context of boundaries in the framework of supergravity if one imposes supersymmetry invariance at the boundary. For the $$ \mathcal{N} $$ N = 2 case, it follows that this amounts to the introduction of a θ-topological term to the Yang-Mills sector which explicitly depends on the Barbero-Immirzi parameter. This shows the close connection between this parameter and the θ-ambiguity of gauge theory.We will also discuss the chiral limit of the theory, which turns out to possess some very special properties such as the manifest invariance of the resulting action under an enlarged gauge symmetry. Moreover, we will show that demanding supersymmetry invariance at the boundary yields a unique boundary term corresponding to a super Chern-Simons theory with OSp($$ \mathcal{N} $$ N |2) gauge group. In this context, we will also derive boundary conditions that couple boundary and bulk degrees of freedom and show equivalence to the results found in the D’Auria-Fré approach in context of the non-chiral theory. These results provide a step towards of quantum description of supersymmetric black holes in the framework of loop quantum gravity.

Towards black hole entropy in chiral loop quantum supergravity

“…In this section, let us briefly review the Cartan geometric description of pure AdS Holst-supergravity with N -extended supersymmetry with N = 1, 2. For more details, we refer to [15][16][17] (see also [21,22] using standard variables). Pure AdS (Holst-)supergravity can be described in terms of a super Cartan geometry modeled on the super Klein geometry (OSp(N |4), Spin + (1, 3) × SO(N )) with super Cartan connection…”

“…The present work is not the first that is considering black hole entropy in supergravity from a loop quantum gravity perspective. We are using variables that were first proposed in [14] and whose geometric meaning was recently clarified in [15][16][17]. A super Chern-Simons theory as a source of entropy was first considered in [6].…”

“…Let us explain the setup and strategy. The quantum theory is obtained from a canonical formulation of N = 1, D = 4 supergravity in terms of a supersymmetric generalization [14][15][16][17] of the (chiral) Ashtekar connection A + which can be obtained from a Holst modification of the McDowell Mansouri action [15,16]. The structure group in this formulation is OSp(1|2).…”

$$ \mathcal{N} $$ = 2 extended MacDowell-Mansouri supergravity

“…From this perspective, the group G puts together the structure group H and fiber sections, unifying the H-principal bundle and the associated fiber bundle with sections in f. For a more rigorous exposition of these subjects see e.g. [18,19].…”

“…Replacing (5.2) in the torsion-like conditions(5.15) this yields,α ± D Ω e a − α ∓ he a = 0 , (5.18)which, in the non-degenerate case α 2 + = α 2 − , requires h = 0, D Ω e a = 0, and spacetime to be of constant curvature, R ab (w) + (α 2 + − α 2 − )e a e b = 0 . (5 19).…”