Exploring new boundary conditions for $$\mathcal {N}=(1,1)$$ extended higher-spin $$AdS_3$$ supergravity

Abstract

In this paper, we present a candidate for $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) extended higher-spin $$AdS_3$$ A d S 3 supergravity with the most general boundary conditions discussed by Grumiller and Riegler recently. We show that the asymptotic symmetry algebra consists of two copies of the $$\mathfrak {osp}(3|2)_k$$ osp ( 3 | 2 ) k affine algebra in the presence of the most general boundary conditions. Furthermore, we impose some certain restrictions on gauge fields on the most general boundary conditions and that leads us to the supersymmetric extension of the Brown–Henneaux boundary conditions. We eventually see that the asymptotic symmetry algebra reduces to two copies of the $$\mathcal {SW}(\frac{3}{2},2)$$ SW ( 3 2 , 2 ) algebra for $$\mathcal {N}=(1,1)$$ N = ( 1 , 1 ) extended higher-spin supergravity.

$\mathcal{N} = (2,2)$ extended $\mathfrak{sl}(3|2)$ Chern-Simons $AdS_3$ supergravity with new boundaries

“…It is convenient to get started with standard basis for sl(2|1) Lie superalgebra. We have denoted the bosonic generators by L i (i = ±1, 0), J and the fermionic ones by G M r (r = ± 1 2 , M = ±), whose commutations relations read…”

“…Note that η AB is the supertrace matrix and f AB C 's are the structure constants of the related algebra with (A, B = 0, ±1, ± 1 2 ), i.e, η ip = 0 and f ij i+j = (i − j). Lastly, by repeating the same analysis for the barred sector also, the asymptotic symmetry algebra of N = (2, 2) supergravity for the loosest set of boundary conditions is given by two copies of the affine sl(2|1) k algebra.…”

“…The simplest extension of Grumiller and Riegler's procedure for the most general N = (1, 1), and N = (2, 2) extended higher spin supergravity is introduced by Valcarcel [43] where the asymptotic symmetry algebra for the loosest set of boundary conditions for (extended) supergravity has been obtained. The most general N = (1, 1) extended AdS 3 higher spin supergravity theory has been similarly presented, including further [1].…”

“…

We present the first example of N = (2, 2) formulation for the extended higherspin AdS 3 supergravity with the most general boundary conditions as an extension of the N = (1, 1) work, discovered recently by us [1]. Using the method proposed by Grumiller and Riegler, we construct a consistent class of the most general boundary conditions to extend it.

…”

“…Note that η AB is the supertrace matrix and f AB C 's are the structure constants of the related algebra with (A, B = 0, ±1, ± 1 2 , 0, ±1, ± 1 2 , ± 3 2 ), i.e, η ip = 0 and f ij i+j = (i − j). Since the barred sector is completely analogous, the same results are obtained.…”