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} = 1 and mathcal{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} = 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} |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.
Highlights
We will 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
Boundaries in string theory have recently been explored in [43] where a specific brane configuration in the framework of type IIB superstring theory has been considered consisting of a stack of D3 branes on two sides of a NS5 brane where the worldvolume theory on the D3 branes corresponds to a maximally supersymmetrc YangMills theory with U(n) gauge group
There, a systematic approach for D = 4 pure supergravity theories both with and without a cosmological constant has been developed, by studying the most general class of possible boundary terms that are compatible with the symmetry of the bulk Lagrangian
Summary
In order to see that the super electric field E defines the canonical conjugate of the super Ashtekar connection, let us go back to (3.45) and vary the action with respect to A+ In this way, it follows i δSbulk(A) = κ (3.47). As in the N = 1 case, due to the splitting of the full action into a bulk and boundary term, one needs to derive a matching condition relating bulk and boundary degrees of freedom at the boundary This is equivalent to requiring consistency with the equation of motion of the full theory , i.e. δS = δSbulk + δSbdy = 0. That is, the pullback of the curvature components F (A)IJ , F (A)αr and Fcorresponding to the OSp(2|4) super Cartan connection A to the boundary are constrained to vanish at the boundary in accordance with the boundary condition as derived in [82] in context of the non-chiral theory
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