Tensor calculus for supergravity on a manifold with boundary

Abstract

Using the simple setting of 3D N = 1 supergravity, we show how the tensor calculus of supergravity can be extended to manifolds with boundary. We present an extension of the standard F -density formula which yields supersymmetric bulk-plus-boundary actions. To construct additional separately supersymmetric boundary actions, we decompose bulk supergravity and bulk matter multiplets into co-dimension one submultiplets. As an illustration we obtain the supersymmetric extension of the York-Gibbons-Hawking extrinsic curvature boundary term. We emphasize that our construction does not require any boundary conditions on off-shell fields. This gives a significant improvement over the existing orbifold supergravity tensor calculus.

N=1 and N=2 pure supergravities on a manifold with boundary

“…At the full supergravity level, boundary contributions were considered from several authors, using different approaches, and in particular in [24][25][26][27][28][29][30][31][32][33][34]. While in [30][31][32][33] boundary conditions on the fields are imposed, in [24][25][26][27][28][29] it is pointed out that the supergravity action should be invariant under local supersymmetry without imposing Dirichlet boundary conditions on the fields, in contrast to the Gibbons-Hawking prescription [3].…”

“…While in [30][31][32][33] boundary conditions on the fields are imposed, in [24][25][26][27][28][29] it is pointed out that the supergravity action should be invariant under local supersymmetry without imposing Dirichlet boundary conditions on the fields, in contrast to the Gibbons-Hawking prescription [3]. The explicit construction of an AdS supergravity theory with a boundary and with no boundary conditions on the fields was achieved in reference [24][25][26][27][28][29] using superconformal tensor calculus, in the particular case of N = 1, D = 3 (off-shell) supergravity. Within that approach, it was shown in particular that N = 1, D = 3 pure supergravity, including its appropriate boundary term, actually reproduces not only the Gibbons-Hawking-York boundary term, but also the counterterm which regularizes the total action, in the language of holographic renormalization.…”

“…In the present paper we work out the construction of N = 1 and N = 2, D = 4 simple supergravities with negative cosmological constant and a non trivial boundary, thus generalizing to four dimensional extended supergravity the results of [17][18][19][20][21] and [24][25][26][27][28][29]. To deal with this problem we take an approach different from that of reference [24][25][26][27][28][29], namely we introduce in a geometric way appropriate boundary terms to the lagrangian in such a way that the action, including the boundary contributions, be invariant under supersymmetry transformations.…”

“…To deal with this problem we take an approach different from that of reference [24][25][26][27][28][29], namely we introduce in a geometric way appropriate boundary terms to the lagrangian in such a way that the action, including the boundary contributions, be invariant under supersymmetry transformations. In a sense our approach extends to superspace the geometric approach of [17][18][19][20][21].…”

On the supersymmetry invariance of flat supergravity with boundary

“…The inclusion of boundary terms in supergravity has been studied by diverse authors in [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. In particular, in [26,27,30,31] it was pointed out that, unlike the Gibbons-Hawking prescription [2], the supersymmetry invariance of a supergravity action should be satisfied without imposing Dirichlet boundary conditions. Interestingly, for the N = 1 three-dimensional supergravity, it was proven that the boundary term reproduces not only the Gibbons-Hawking-York boundary term but also the counterterm allowing to regularize the action [31].…”