Exploring new Boundary Conditions for $\mathcal{N}=(1,1)$ Extended Higher Spin $AdS_3$ Supergravity

In this paper, we present a candidate for $\mathcal{N}=(1,1)$ extended higher spin $AdS_3$ 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 $\mathfrak{osp}(3|2)_k$ affine algebra in the presence of the most general boundary conditions. Furthermore, we impose some certain restrictions to 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 $\mathcal{SW}(\frac{3}{2},2)$ algebra for $\mathcal{N}=(1,1)$ extended higher spin supergravity.


I. INTRODUCTION
The most common asymptotic symmetries of AdS 3 gravity with a negative cosmological constant in 3D are known as two copies of the Virasoro algebras and this has been written first by Brown and Henneaux in their seminal paper 1 . So this work is well known as both a pionner of AdS 3 /CF T 2 correspondance 2 and also a realization of Holographic Principle 3 . Clearly, one of the biggest breakthroughs in theoretical physics in the past few decades is undoubtedly the discovery of the AdS 3 /CF T 2 correspondence describing an equivalence between the Einstein gravity and a large N gauge field theory.
The pure Einstein gravity in this context is simply a Chern -Simons gauge theory, that is, it is rewritten as a gauge field theory, in such a way that the structure simplify substantially. This recalls us that there is no local propagating degrees of freedom in the theory, and hence no graviton in threedimensions. Therefore, this gauge theory is said purely topological and only global effects are of physical relevence. Finally, one must emphasize here that the dynamics of the theory is controlled entirely by the boundary conditions, because its dynamical content is far from insignificant due to the existence of boundary conditions. This fact was first discovered by Achucarro and Townsend 4 , and subsequently developed by Witten 5 . The things found here is that the gravity action in three -dimensions and equations of motions are in the same class with a Chern -Simons theory for an suitable gauge group. Under an convenient choice of boundary conditions, there is actually an infinite number of degrees of freedom living on the boundary. These boundary conditions are required, but these conditions are not unique in the selections. In fact, the dynamical properties of the theory are also highly sensitive to the selection of these boundary conditions. Thus, the situations of asymptotic symmetries as the remnant global gauge symmetries occur.
In the case of AdS 3 gravity above with a negative cosmological constant in 3D, the most famous of these boundary conditions is worked in the same paper 1 . These boundary conditions also contain BT Z a) Electronic mail: ozert@itu.edu.tr b) Electronic mail: aytulfiliz@itu.edu.tr black holes 6,7 . Besides, the Chern -Simons higher spin theories are purely bosonic theories 8,9 as versions of Vasiliev higher spin theories 10,11 with higher spin fields of integer spin, they are based on the sl(N, R) algebras and higher spin algebras hs(λ) respectively. The Chern -Simons higher spin theories can also realize the W N algebras as the asymptotic symmetries of the two dimensional CFT's [12][13][14][15] . The validity of these results can be extended to supergravity theory 1,16 , as well as higher spin theory 8,9 . Beyond that a supersymmetric generalization of these bosonic theories can be achieved by considering Chern -Simons theories based on superalgebras such as sl(N |N − 1), see e.g 14,16-20 , or osp(N |N − 1) 21 which can be obtained by truncating out all the odd spin generators and one copy of the fermionic operators in sl(N |N −1).
Our motivation is to construct a candidate solution for the most general N = (1, 1) extended higher spin supergravity theory in AdS 3 . Our theory falls under the same metric class as the recently constructed most general AdS 3 boundary condition by Grumiller and Riegler 22 . This method recently has also been applied to flat-space 23 and chiral higher spin gravity 24 which showed a new class of boundary conditions for higher spin theories in AdS 3 . This is an alternative to the non -chiral Drinfeld -Sokolov type boundary conditions. In particular we first focus on the simplest example, N = (1, 1) Chern -Simons theory based on the superalgebra osp(2|1). It is clear that the related asymptotic symmetry algebra is two copies of the osp(2|1) k affine algebra. Then one can extend this study to the N = (1, 1) Chern -Simons theory based on the superalgebra osp(3|2) and the symmetry algebras are given by two copies of the osp(3|2) k affine algebra. Furthemore, we also impose certain restrictions to the gauge fields on the most general boundary conditions and that lead us to the supersymmetric extensions of the Brown -Henneaux boundary conditons. From these restrictions we see that the asymptotic symmetry algebras reduce to two copies of the SW( 3 2 , 2) algebra for the most general N = (1, 1) extended higher spin supergravity theory in AdS 3 . So, one can think that this method provides a good labratory for investigating the rich asymptotic structure of extended supergravity.
The outline of the paper is as follows. In the next section, we give briefly a fundamental formulation of N = (1, 1) supergravity in the perspective of osp(2|1) ⊕ osp(2|1) Chern -Simons gauge theory for both affine and superconformal boundaries respectively in three-dimensions. In section 3, we carry out our calculations to extend the theory to osp(3|2) ⊕ osp(3|2) higher spin Chern -Simons supergravity in the presence of both affine and superconformal boundries, in where we showed explicitly principal embedding of osp(2|1) ⊕ osp(2|1) and also demonstrated how asymptotic symmetry and higher spin Ward identities arise from the bulk equations of motion coupled to spin s, (s = 3 2 , 2, 2, 5 2 ) currents. Finally, the last section is devoted to the case of classical two copies of the osp(3|2) k affine algebra on the affine boundary and SW( 3 2 , 2) symmetry algebra on the superconformal boundary as asymptotic symmetry algebras, and also the chemical potentials related to source fields appearing through the temporal components of the connection are obtained. The final section contains our summary and conclusion.

II. SUPERGRAVITY IN THREE -DIMENSIONS, A REVIEW:
In this section we review the Chern -Simons formalism for higher spin supergravity. In particular, we use this formalism to study supergravity in the osp(2|1) superalgebra basis under the same metric class as the recently constructed most general AdS 3 boundary condition by Grumiller and Riegler 22 .

A. Connection to Chern -Simons Theory
The three -dimensional Einstein -Hilbert action for N = (1, 1) supergravity with negative cosmological constant is classically equivalent to the Chern -Simons action, as it was first proposed by Achucarro and Townsend in 4 and developed by Witten in 5 . One can start by defining 1 -forms (Γ,Γ) taking values in the gauge group's osp(2|1) superalgebra, and also the supertrace str is taken over the superalgebra generators. The Chern -Simons action can be written in the form, Here k = ℓ 4Gstr(L0L0) = c 6str(L0L0) is the level of the Chern -Simons theory depending on the AdS radius l and the Newton's constant G with the related central charge c of the superconformal field theory. Nevertheless, supertrace shows a metric on the osp(2|1) superalgebra. If L i , (i = ±1, 0) and G p , (p = ± 1 2 ) are the generators of osp(2|1) superalgebra. We have expressed osp(2|1) superalgebra such that The equations of motion for the Chern -Simons gauge theory give the flatness condition F =F = 0 where is the same as the Einstein's equation. Γ andΓ are related to the metric g µν through the veilbein e = ℓ 2 (Γ −Γ) One can choose a radial gauge of the form with state-independent group element as 22 which manifests all the osp(2|1) charges and chemical potentials and also the choice of b is irrelevant in the case of asymptotic symmetry, as long as δb = 0. Moreover, a (t, φ) andā (t, φ) in the radial gauge are the osp(2|1) superalgebra valued fields, which are independent from the radial coordinate, ρ as

1) Supergravity for Affine Boundary
Affine case is given by reviewing asymptotically AdS 3 boundary conditions for a osp(2|1) ⊕ osp(2|1) Chern -Simons theory, and how to determine the asymptotic symmetry algebra using the method described in 22 . Thus the most general solution of the Einstein's equation that is asymptotically AdS 3 , as a generalization of Fefferman -Graham method is given by with a flat boundary metric ij + e ρ g (1) ij + g We need to choose the most general boundary conditions for N = (1, 1) supergravity such that they maintain this metric form. In the following we only focus on the Γ -sector. Therefore, one can propose to write the components of the osp(2|1) superalgebra valued connection in the form, where (α i , β p )'s are some scaling parameters to be determined later and we have five functions: three bosonic L i and two fermionic G p . They are usually called charges and also the time component of the connection a (t, ϕ) Here, the time component have in total five independent functions (µ i , ν p ). They are usually called chemical potentials. But, they are not allowed to wary The flat connection conditions (4) for fixed chemical potentials impose the following additional conditions as the temporal evolution of the five independent source fields (L i , G p ) as After the temporal evolution of the source fields, one can start to compute the gauge transformations for asymptotic symmetry algebra by considering all transformations that preserve the boundary conditions with the gauge parameter in the osp(2|1) superalgebra Here, the gauge parameter have in total five arbitrary functions : bosonic ǫ i and fermionic ζ p on the boundary. The condition (17) impose that transformations on the gauge are given by Since the chemical potentials (µ m , ν p ) are fixed, they obey the following transformations As a final step, one now has to determine the canonical boundary charge Q[λ] that generates the transformations (19)- (21). Therefore, the corresponding variation of the boundary charge Q[λ] [25][26][27][28] , in order to show the asymptotic symmetry algebra, is given by The canonical boundary charge Q[λ] can be integrated which reads We now prefer to work in complex coordinates for affine boundary, z(z) ≡ ϕ ± i t ℓ . After both the infinitesimal transformations and the canonical boundary charge have been determined, one can yields the Poisson bracket algebra by using the methods 29 with for any phase space functional ̥ : where α +1 = − 2π k for the convention in the literature and η ij is the bilinear form in the fundamental representation of osp(2|1) superalgebra. One can also expand L(z) and G(z) charges into Fourier modes L i (z) = 1 2π n L i n z −n−1 , and G p (z) = 1 2π r G p r z −r− 1 2 , and also replacing i{·, ·} P B → [·, ·]. A mode algebra can then be defined as: {G p r , G q s } = −2L p+q r+s + ksκ pq δ r+s,0 .
Besides, this mode algebra in these space is equivalent to operator product algebra, where z 12 = z 1 − z 2 , or in the more compact form, Here, 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). After repeating the same algebra forΓ -sector, one can say that the asymptotic symmetry algebra for the most general boundary conditions of N = (1, 1) supergravity is two copies of the affine osp(2|1) k algebra as in Ref. 30 .
on the boundary conditions with the osp(2|1) superalgebra valued connection (10), one can get the superconformal boundary conditions as the supersymmetric extension of the Brown -Henneaux boundary conditions proposed in 1 for AdS 3 supergravity. Therefore we have the supersymmetric connection as, where µ ≡ µ +1 , ν ≡ ν + 1 2 can be interpreted as the independent chemical potentials. This means that we assume the chemical potential to be fixed at infinity, i.e. δµ = 0. The functions µ 0 , µ −1 and ν − 1 2 are fixed by the flatness condition (4) as For the fixed chemical potentials µ and ν, the time evolution of canonical boundary charges L and G can be written as where α −1 and β − 1 2 are some scaling parameters to be determined later. Now, we are in a position to work the superconformal asymptotic symmetry algebra under the Drinfeld -Sokolov reduction. This reduction imply that the only independent parameters ǫ ≡ ǫ +1 , and ζ ≡ ζ + 1 2 . One can start to compute the gauge transformations for asymptotic symmetry algebra by considering all transformations (17) that preserve the boundary conditions. with the gauge parameter in the osp(2|1) superalgebra (18) as The condition (17) impose that transformations on the gauge with α −1 = 6 c and β − 1 2 = − 3 c are given by The variation of canonical boundary charge Q[λ] (25) can be integrated which reads This leads to operator product expansions in the complex coordinates by using (27) After repeating the same algebra forΓ -sector, one can say that the asymptotic symmetry algebra for a set of boundary conditions of N = (1, 1) supergravity is two copies of the super -Virasoro algebra with central charce c = 6k.

A. For Affine Boundary
In this section, an extension of the N = (1, 1) higher spin supergravity theory constructed here as osp(3|2) ⊕ osp(3|2) is defined by the osp(2|1) ⊕ osp(2|1) algebra that is present as a sub-superalgebra as the ordinary case of osp(N |N − 1) gauge algebra. If L i (i = ±1, 0), G p (p = ± 1 2 ), A i (i = ±1, 0), and S p (p = ± 1 2 , ± 3 2 )are the generators of osp(3|2) superalgebra, we have expressed osp(3|2) superalgebra such that The super Jacobi identities give us the nontrivial relations for some constants σ ′ i s, (i = 1, 2, . . . , 12) appearing on the RHS of eq.(52) − (57) as, For the corresponding affine algebra, the resulting relations are as σ 3 = 2, σ 7 = −1, σ 8 = 0, σ 9 = 1. We are now ready to formulate most general boundary conditions for asymptotically AdS 3 spacetimes: where (α i , γ i , β p , δ p )'s are some scaling parameters to be determined later and we have twelve functions: six bosonic L i , A i and six fermionic (G p , S p ) as the charges and also here, the time component have in total twelve independent functions (µ i , χ i , f p , ν p ) as the chemical potentials. The flat connection conditions (4) for fixed chemical potentials impose the following additional conditions as the temporal evolution of the twelve independent source fields, After the temporal evolution of the source fields, one can start to compute the gauge transformations for asymptotic symmetry algebra by considering all transformations (17) that preserve the boundary conditions, with the gauge parameter λ in the osp(3|2) superalgebra as, Here, the gauge parameter have in total twelve arbitrary functions :six bosonic (ǫ i , κ i ) and six fermionic (ζ p , ̺ p ) on the boundary. The condition (17) impose that transformations on the gauge are given by Since the chemical potentials (µ i , χ i , f p , ν p ) are fixed, they obey the following transformations Here, 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). After repeating the same algebra for Γ -sector, one can say that the asymptotic symmetry algebra for the most general boundary conditions of N = (1, 1) supergravity is two copies of the affine osp(3|2) k algebra.

V. ACKNOWLEDGMENTS
This work was supparted by Istanbul Technical University Scientific Research Projects Department(ITU BAP, project number:40199).