Minimal AdS-Lorentz supergravity in three-dimensions

The $\mathcal{N}=1$ AdS-Lorentz superalgebra is studied and its relationship to semigroup expansion developed. Using this mathematical tool, the invariant tensors and Casimir operators are found. In terms of these invariants, a three-dimensionnal Chern--Simons supergravity action with AdS-Lorentz symmetry is constructed. The Killing spinors for a BTZ black-hole like solution of the theory are discussed.


Introduction
Since the original works of Deser, Jackiw and 't Hooft [1,2], three dimensional gravity has attracted attention. Despite having no propagating degrees of freedom, the BTZ black-hole solution [3,4] and the quantization of the theory by Witten [5] are its highly non-trivial trademarks. These features seem to be rooted in the fact that the Einstein-Hilbert (EH) Lagrangian with cosmological constant can be written (up to a boundary term) as a Chern-Simons (CS) three-form. Therefore, three-dimensional gravity corresponds to an off-shell quasi-invariant gauge theory (for AdS, dS or Poincaré depending on the cosmological constant). The locally supersymmetric extension of Einstein gravity in three dimensions was carried out by Deser and Kay in Ref. [6]. Regarding the CS formulation, three-dimensional supergravity arises very naturally in the case of negative [7] and vanishing [8,9] cosmological constant. However, there is still the possibility of having other families of supergravity theories containing gauge groups larger than the supersymmetric AdS or Poincaré groups [10][11][12]. This is particularly interesting because symmetries enhancements usually invokes new generators in the Lie algebra. Subsequently, this requires the inclusion of extra gauge fields in the gauge potential, giving rise to non-minimal couplings of "matter" fields with geometry in such a way that gauge invariance is preserved.
The purpose of this work is to analyze the construction of three-dimensional CS supergravity theories whose symmetry groups are obtained by an S-expansion of the N = 1 supersym-* Corresponding author. metric AdS algebra osp(1|2) ⊗ sp(2). The S-expansion method [13,14] is a powerful tool in order for obtaining new Lie algebras starting from a given one. Moreover, it provides the associated invariant tensors of the expanded algebra in a simple way. Since the invariant tensor is an essential ingredient in the construction of gauge theories and in particular of CS (super)gravities, it is a welcomed feature.
The application of S-expansion methods in the context of supergravity was first introduced in [15] and subsequently in [16] as an attempt to describe the low energy regime of M-Theory. More recently, a wide range of theories of S-expanded (super)gravities have been studied in different contexts, and with different motivations (see for instance [17][18][19][20] and references therein). Also, in Ref.
[21], three-dimensional gravity is constructed using the semisimple extension of the Poincaré algebra [22,23] as a gauge symmetry. The Lie algebra behind this symmetry can be obtained as a S-expansion of AdS algebra so(d − 1, 2). This article is organized as follows: In section 2, the supersymmetric extension of the three-dimensional AdS-Lorentz algebra is written. In section 3, we review the general properties of the S-expansion method. Also, it is explicitly shown that three-dimensional AdS-Lorentz superalgebra corresponds to a Sexpansion of the AdS superalgebra. The components of the invariant tensor are worked out. In section 4 we extend the notion of S-expansion to Casimir operators and the invariant operators associated to the expanded superalgebra are constructed. Section 5 is devoted to the analysis of three-dimensional AdS-Lorentz CS supergravity. Field equations and symmetry transformations are worked out. In section 6 we compute stationary solutions and its Killing spinors equation are found. Finally, section 7 concludes this paper with some remarks and future developments.

AdS-Lorentz superalgebra
In Ref.
[23] the semi-simple extension of the Poincaré algebra iso(d − 1, 1), generated by Lorentz rotations { J ab } and translations {P a }, has been carried out by the inclusion of a second-rank tensor generator {Z ab }. Interestingly, this Lie algebra enhancement is isomorphic to the direct sum of the AdS and Lorentz algebra so(d − 1, 2) ⊕ so(d − 1, 1) in any dimension. More recently, it has been shown in Refs. [21,24] that the so called AdS-Lorentz algebra can be obtained as a S-expansion of the AdS algebra and its Inönü-Wigner contraction leads to the Maxwell algebra. The supersymmetric extension in four-dimensions has been also considered in Refs. [25,26]. Remarkably, both algebras, pure bosonic and supersymmetric, are semi-simple in contrast to the (super) Poincaré algebras.
In this work we are interested in the N = 1 AdS-Lorentz superalgebra in three-dimensions, which is defined by the following commutation relations: where J a denote the generators of the Lorentz subalgebra so(2, 1), P a the translations, Z a are a new set of non-abelian generators, and Q α the supercharges. Lorentz indices a, b, ... = 0, 1, 2 are raised and lowered with the Minkowski metric η ab , abc is the three-dimensional Levi-Civita symbol. Greek indices α, β... = 0, 1 are raised and lowered by the charge conjugation matrix C, and a denote the 2 × 2 gamma matrices representation (see Appendix A for spinor conventions). In the following section, we show that the (super)AdS-Lorentz algebra (2.1) can be derived as an application of the S-expansion procedure.

Abelian semigroup expansion
The Lie algebra expansion procedure was introduced for the first time in Ref. [27], and subsequently studied in Refs. [28,29]. In this expansion method, we must consider the Maurer-Cartan forms on the group manifold. Some of the group parameters are rescaled by a factor λ, and the Maurer-Cartan forms are expanded as a power series in λ. The series is finally truncated in such a way that the closure of the expanded algebra is assured.
In Refs. [13,14,30] a natural outgrowth of the power series expansion method was proposed. The idea is to start with a Lie algebra g and to combine it with the binary product structure of an abelian semigroup S in order to define a new Lie algebra. This new algebra is known in general as a S-expanded algebra. In fact, from [13, Theorem 3.1], it is possible to prove that the direct product S × g retains Lie algebra structure (see also [27,28,31]). The most relevant cases are provided when subalgebras of S × g can be systematically extracted. For instance, any Lie algebra can be written as a direct sum of subspaces g = p∈I V p , where I is a set of indices. The subspace structure of the algebra can be analyzed defining a mapping i : I × I → 2 I such that the Lie algebra g can be written as V p , V q ⊂ r∈i (p,q) V r . Now, whenever the semigroup S admits a decomposition S = p∈I S p , satisfying the resonant condition S p ·S q ⊂ r∈i (p,q) S r , then it follows that S R = p∈I S p × V p is a subalgebra of S × g [13,Theorem 4.2]. The procedure is practical because the subspace structure is arbitrary, but we use it in order to codify our physicist's intuition on the meaning of the symmetry (e.g. a subspace corresponds to Lorentz transformations, another to AdS boosts, etc.). Thus, using the S-expansion it is possible to find bigger symmetries in a simple way, and to do this preserving some valuable structure from a physical point of view. Without it, constructing bigger symmetries requires long and careful work regarding the closure of Jacobi's identity (or the self-consistency of d 2 = 0 when working with Maurer-Cartan forms).
The S-expansion procedure has already been used in different contexts with different motivations. For instance, the so called B m -algebras [17] (also known as generalized Poincaré algebras), were constructed from the AdS-algebra and a particular semigroup 1 denoted by S (N) α=0 . Moreover, in Ref.
Another interesting application is in the context of nonrelativistic algebras. Recently, in Ref. [35] it was shown that it is possible to obtain the non-relativistic versions of both generalized Poincaré algebras and generalized AdS-Lorentz algebras. These were called generalized Galilean type I and type II, denoted by GB n and GL n respectively. It seems likely that new non-relativistic CS gravity theories may be constructed following a similar procedure as the one presented in Ref. [36]. Its symmetries would correspond to deformations of the symmetries of the Newton-Cartan formulation of Newtonian gravity. This problem will be addressed in the near future.

S-expansion and the AdS superalgebra
In this section we construct the three-dimensional AdS-Lorentz superalgebra as a S-expansion of the AdS superalgebra g = osp ( 2| 1) ⊗ sp (2), given by the commutation relations ˜J a ,J b = abcJ c , P a ,Q α = − 1 2 aQ α , P a ,P b = abcJ c , ˜J a ,Q α = − 1 2 aQ α , ˜J a ,P b = abcP c , Q α , Q β = ( a C) αβ ˜J a +P a .
(3.1) Let us start by choosing the following subspace decomposition where V 0 = Span ˜J a , V 1 = Span Q α and V 2 = Span P a . This decomposition obeys the following structure At this point it is convenient to apply the S-expansion resonance theorem using (3.3) and a specific semigroup S M . A similar treat-1 This semigroup is endowed with the multiplication rule λ α ·λ β = λ α+β when α + β ≤ N + 1; and λ α ·λ β = λ N+1 otherwise.