Eur. Phys. J. C (2020) 80:1072https://doi.org/10.1140/epjc/s10052-020-08613-4

Regular Article - Theoretical Physics

Exploring new boundary conditions forN = (1, 1) extendedhigher-spin AdS3 supergravity

H. T. Özera , Aytül Filizb

Faculty of Science and Letters, Physics Department, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey

Received: 28 July 2020 / Accepted: 28 October 2020 / Published online: 20 November 2020© The Author(s) 2020

Abstract In this paper, we present a candidate for N =(1, 1) extended higher-spin AdS3 supergravity with the mostgeneral boundary conditions discussed by Grumiller andRiegler recently. We show that the asymptotic symmetryalgebra consists of two copies of the osp(3|2)k affine alge-bra in the presence of the most general boundary conditions.Furthermore, we impose some certain restrictions on gaugefields on the most general boundary conditions and that leadsus to the supersymmetric extension of the Brown–Henneauxboundary conditions. We eventually see that the asymptoticsymmetry algebra reduces to two copies of the SW( 3

2 , 2)

algebra for N = (1, 1) extended higher-spin supergravity.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . 12 Supergravity in three-dimensions, a review . . . . . 3

2.1 Connection to Chern–Simons theory . . . . . . 32.2 osp(1|2)⊕osp(1|2) Chern–Simons N = (1, 1)

supergravity for affine boundary . . . . . . . . 32.3 osp(1|2)⊕osp(1|2) Chern–Simons N = (1, 1)

supergravity for superconformal boundary . . . 53 N = (1, 1) osp(3|2) ⊕ osp(3|2) higher-spin Chern–

Simons supergravity . . . . . . . . . . . . . . . . . 63.1 For affine boundary . . . . . . . . . . . . . . . 63.2 For superconformal boundary . . . . . . . . . . 9

4 Other extended (super)gravity checks in the mostgeneral boundary conditions . . . . . . . . . . . . . 114.1 Avery–Poojary–Suryanarayana for sl(2, R) ⊕ sl(2) . 114.2 Avery–Poojary–Suryanarayana N = 1 super-

gravity for osp(1|2) ⊕ sl(2) . . . . . . . . . . . 124.3 The on-shell action and the variational principle 124.4 osp(1|2)-case . . . . . . . . . . . . . . . . . . 134.5 osp(3|2)-case . . . . . . . . . . . . . . . . . . 13

a e-mail: ozert@itu.edu.tr (corresponding author)b e-mail: aytulfiliz@itu.edu.tr

5 Summary and comments . . . . . . . . . . . . . . . 13References . . . . . . . . . . . . . . . . . . . . . . . . 14

1 Introduction

The most common asymptotic symmetries of AdS3 gravitywith a negative cosmological constant in 3D are known astwo copies of the Virasoro algebras and this has been writtenfirst by Brown and Henneaux in their seminal paper [1]. Sothis work is well known as both a pioneer of AdS3/CFT2 cor-respondence [2,3] and also a realization of the HolographicPrinciple [4]. One of the biggest breakthroughs in theoreticalphysics in the past few decades is undoubtedly the discoveryof the AdS/CFT correspondence describing an equivalencebetween 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 fieldtheory, in such a way that the structure simplifies substan-tially. This recalls us that there are no local propagatingdegrees of freedom in the theory, and hence no gravitonin three-dimensions. Therefore, this gauge theory is saidpurely topological and only global effects are of physical rel-evance. Finally, one must emphasize here that the dynamicsof the theory is controlled entirely by the boundary condi-tions, because its dynamical content is far from insignificantdue to the existence of boundary conditions. This fact wasfirst discovered by Achucarro and Townsend [5], and sub-sequently developed by Witten [6]. The things found hereis that the gravity action in three-dimensions and equationsof motions are in the same class with a Chern–Simons the-ory for a suitable gauge group. Under a convenient choice ofboundary conditions, there is an infinite number of degreesof freedom living on the boundary. These boundary condi-tions are required, but these conditions are not unique in theselections. The dynamical properties of the theory are alsohighly sensitive to the selection of these boundary conditions.

123

1072 Page 2 of 15 Eur. Phys. J. C (2020) 80 :1072

Thus, the situations of asymptotic symmetries as the residualglobal gauge symmetries emerge.

In the case of AdS3 gravity above with a negative cosmo-logical constant in 3D, the most famous of these boundaryconditions is worked in the same paper [1]. These bound-ary conditions also contain BT Z black holes [7,8]. Besides,the Chern–Simons higher-spin theories are purely bosonictheories [9,10] as versions of Vasiliev higher-spin theories[11,12] with higher-spin fields of integer spin, they are basedon the sl(N , R) algebras and higher-spin algebras hs(λ)

respectively. The Chern–Simons higher-spin theories canalso be resulted with a realization of the classical WN asymp-totic symmetry algebras as in the related two-dimensionalCFT’s [13–18]. The validity of these results can be extendedto supergravity theory [1,19,20], as well as a higher-spintheory [9,10]. Beyond that a supersymmetric generaliza-tion of these bosonic theories can be achieved by consider-ing Chern–Simons theories based on superalgebras such assl(N |N − 1), see e.g [19–24], or osp(N |N − 1) [25] whichcan be obtained by truncating out all the odd spin generatorsand one copy of the fermionic operators in sl(N |N − 1).

The critical role of boundary conditions in field theoriesand in particular in gravity theories has been well understoodso far. In three dimensions, as previously mentioned, asymp-totic AdS3 boundary conditions by Brown and Henneaux[1] provided an important precursor to the AdS3/CFT2. Fora long time, these boundary conditions have been changed(see, e.g. [26,38,39,41–44]) and generalized (see, e.g. [45–52]) many times. Our motivation is to construct a candidatesolution for the most general N = (1, 1) extended higher-spin supergravity theory in AdS3. Our theory falls under thesame metric class as the recently constructed most generalAdS3 boundary condition by Grumiller and Riegler [26].They showed that all charges and chemical potentials thatappear in the Chern–Simons formulation can also be seen inthe metric formulation. It is also mentioned that the possibil-ity to obtain asymptotic symmetries that are the affine versionof the gauge algebras of the Chern–Simons theory is not a sur-prise given the relation between Chern–Simons theories andWess–Zumino–Witten models [27]. If one examines the cor-respondence between 2+1 dimensional pure Chern–Simonsgauge theories and 1+1 dimensional Wess–Zumino–Wittenconformal field theories, it can be seen that this is the core ofthe relationship between these theories. This method recentlyhas also been applied to flat-space [28] and chiral higher-spingravity [29] which showed a new class of boundary condi-tions for higher-spin theories in AdS3. The simplest exten-sion of the Grumiller and Riegler’s analysis for the most gen-eral N = (1, 1) and N = (2, 2) extended higher spin super-gravity theory in AdS3 is the Valcarcel’s paper [30] wherethe asymptotic symmetry algebra for the loosest set of bound-ary conditions for (extended) supergravity has been obtained.This is an alternative to the non-chiral Drinfeld–Sokolov type

boundary conditions. In particular, we first focus on the sim-plest example, N = (1, 1) Chern–Simons theory based onthe superalgebra osp(1|2). The related asymptotic symme-try algebra is two copies of the osp(1|2)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 thesymmetry algebras are given by two copies of the osp(3|2)kaffine algebra. Furthermore, we also impose certain restric-tions on the gauge fields on the most general boundary condi-tions and that leads to the supersymmetric extensions of theBrown–Henneaux boundary conditions. From these restric-tions, we see that the asymptotic symmetry algebras reduceto two copies of the SW( 3

2 , 2) algebra for the most gen-eral N = (1, 1) extended higher-spin supergravity theory inAdS3. It will be also very interesting to show another class ofboundary conditions that appeared in the literature (see, e.g.[39,41–43]) for (super)gravity case, whose higher-spin gen-eralization is less clear than the Grumiller–Riegler boundaryconditions. Therefore, one can think that this method pro-vides a good laboratory for investigating the rich asymptoticstructure of extended supergravity.

The outline of the paper is as follows. In the next sec-tion, we give briefly a fundamental formulation of N =(1, 1) supergravity in the perspective of osp(1|2)⊕ osp(1|2)

Chern–Simons gauge theory for both affine and supercon-formal boundaries respectively in three-dimensions. In Sect.3, we carry out our calculations to extend the theory toosp(3|2) ⊕ osp(3|2) higher-spin Chern–Simons supergrav-ity in the presence of both affine and superconformal bound-aries, in where we showed explicitly principal embeddingof osp(1|2) ⊕ osp(1|2) and also demonstrated how asymp-totic symmetry and higher-spin Ward identities arise from thebulk equations of motion coupled to spin s, (s = 3

2 , 2, 2, 52 )

currents. This section is devoted to the case of classical twocopies of the osp(3|2)k affine algebra on the affine bound-ary and SW( 3

2 , 2) symmetry algebra on the superconfor-mal boundary as asymptotic symmetry algebras, and also thechemical potentials related to source fields appearing throughthe temporal components of the connection are obtained. InSect. 4, we present two most general boundary conditionsfor pure bosonic gravity and N = 1 extended supergrav-ity. That is, The Avery–Poojary–Suryanarayana gravity forsl(2, R)⊕ sl(2) and osp(1|2)⊕ sl(2) respectively with somefurther checks. On the other hand, it is shown that the Chern–Simons action which compatible with our boundary condi-tions leads to a finite action and a well-defined variationalprinciple for the higher-spin fields. The final section containsour summary and conclusion.

123

Eur. Phys. J. C (2020) 80 :1072 Page 3 of 15 1072

2 Supergravity in three-dimensions, a review

In this section, we review the Chern–Simons formalism forhigher-spin supergravity. In particular, we use this formalismto study supergravity in theosp(1|2) superalgebra basis underthe same metric class as the recently constructed most generalAdS3 boundary condition by Grumiller and Riegler [26].

2.1 Connection to Chern–Simons theory

The three-dimensional Einstein–Hilbert action for N =(1, 1) supergravity, which is defined on any manifold M,with a negative cosmological constant is classically equiv-alent to the Chern–Simons action, as it was first proposedby Achucarro and Townsend in [5] and developed by Wittenin [6]. One can start by defining 1-forms (�, �) taking val-ues in the gauge group’s osp(1|2) superalgebra, and also thesupertrace str is taken over the superalgebra generators. TheChern–Simons action can be written in the form,

S = SCS [�] − SCS[�

](2.1)

where

SCS [�] = k

4π

∫

M〈� ∧ d� + 2

3� ∧ � ∧ �〉. (2.2)

Here k = �4Gstr(L0L0)

= c6str(L0L0)

is the level of the Chern–Simons theory depending on the AdS radius l and Newton’sconstant G with the related central charge c of the supercon-formal field theory. If Li , (i = ±1, 0) and Gp, (p = ± 1

2 ) arethe generators of osp(1|2) superalgebra. We have expressedosp(1|2) superalgebra such that

[Li ,L j

] = (i − j)Li+ j ,[Li ,Gp

] =(i

2− p

)Gi+p,

{Gp,Gq

} = −2Lp+q . (2.3)

The equations of motion for the Chern–Simons gauge theorygive the flatness condition F = F = 0 where

F = d� + � ∧ � = 0 (2.4)

is the same as Einstein’s equation. � and � are related to themetric gμν through the veilbein e = �

2 (� − �)

gμν = 1

2str(eμeν). (2.5)

One can choose a radial gauge of the form

� = b−1a (t, φ) b + b−1db,

� = ba (t, φ) b−1 + bdb−1

(2.6)

with state-independent group element as [26]

b(ρ) = eL−1eρL0 (2.7)

which manifests all the osp(1|2) charges and chemical poten-tials and also the choice of b is irrelevant in the case of asymp-totic symmetry, as long as δb = 0. Moreover, a (t, φ) anda (t, φ) in the radial gauge are the osp(1|2) superalgebra val-ued fields, which are independent from the radial coordinate,ρ as

a (t, ϕ) = at (t, ϕ) dt + aϕ (t, ϕ) dϕ. (2.8)

2.2 osp(1|2) ⊕ osp(1|2) Chern–Simons N = (1, 1)

supergravity for affine boundary

The affine case is given by reviewing asymptotically AdS3

boundary conditions for a osp(1|2) ⊕ osp(1|2) Chern–Simons theory, and how to determine the asymptotic symme-try algebra using the method described in [26]. Thus the mostgeneral solution of Einstein’s equation that is asymptoticallyAdS3, as a generalization of Fefferman–Graham method isgiven by with a flat boundary metric

ds2 = dρ2 + 2[eρN (0)

i + N (1)i + e−ρN (2)

i + O(e−2ρ

)]dρdxi

+[e2ρg(0)

i j + eρg(1)i j + g(2)

i j + O(e−ρ

)]dxidx j . (2.9)

We need to choose the most general boundary conditionsfor N = (1, 1) supergravity such that they maintain thismetric form. In the following, we only focus on the �-sector.Therefore, one can propose to write the components of theosp(1|2) superalgebra valued connection in the form,

aϕ (t, ϕ) =+1∑

i=−1

αiLi (t, ϕ)Li ++1/2∑

p=−1/2

βpG p (t, ϕ)Gp

(2.10)

where(αi , βp

)’s are some scaling parameters to be deter-

mined later and we have five functions: three bosonic Li andtwo fermionic G p. They are usually called charges and alsothe time component of the connection a (t, ϕ)

at (t, ϕ) =+1∑

i=−1

μi (t, ϕ)Li ++1/2∑

p=−1/2

ν p (t, ϕ)Gp. (2.11)

Here, the time component has in total of five indepen-dent functions (μi , ν p). They are usually called chemicalpotentials. But, they are not allowed to vary

δaϕ =+1∑

i=−1

αiδLiLi ++1/2∑

p=−1/2

βpδG pGp, (2.12)

δat = 0. (2.13)

123

1072 Page 4 of 15 Eur. Phys. J. C (2020) 80 :1072

The flat connection conditions (2.4) for fixed chemical poten-tials impose the following additional conditions as the tem-poral evolution of the five independent source fields (Li ,G p)

as

α±1∂tL±1 = ∂ϕμ±1 ∓ α0L0μ±1 ± α±1L±1μ0

−2β± 12G± 1

2 ν± 12 , (2.14)

1

2α0∂tL0 = 1

2∂ϕμ0 + α+1L+1μ−1 − α−1L−1μ+1

−β+ 12G+ 1

2 ν− 12 − β− 1

2G− 1

2 ν+ 12 , (2.15)

β± 12∂tG± 1

2 = ∂ϕν± 12 ± α±1L±ν∓ 1

2 ∓ 1

2α0L0ν± 1

2

∓β∓ 12μ±G∓ 1

2 ± 1

2β± 1

2μ0G± 1

2 . (2.16)

After the temporal evolution of the source fields, one canstart to compute the gauge transformations for asymptoticsymmetry algebra by considering all transformations

δλ� = dλ + [�, λ] (2.17)

that preserve the boundary conditions with the gauge param-eter in the osp(1|2) superalgebra

λ = b−1

⎡

⎣+1∑

i=−1

εi (t, ϕ)Li ++1/2∑

p=−1/2

ζ p (t, ϕ)Gp

⎤

⎦ b.

(2.18)

Here, the gauge parameter has in total of five arbitrary func-tions : bosonic εi and fermionic ζ p on the boundary. Thecondition (2.17) impose that transformations on the gaugeare given by

α±1δλL±1 = ∂ϕε±1 ∓ α0L0ε±1 ± α±1L±1ε0

−2β± 12G± 1

2 ζ± 12 , (2.19)

1

2α0δλL0 = 1

2∂ϕε0 + α+1L+1ε−1 − α−1L−1ε+1

−β+ 12G+ 1

2 ζ− 12 − β− 1

2G− 1

2 ζ+ 12 , (2.20)

β± 12δλG± 1

2 = ∂ϕζ± 12 ± α±1L±ζ∓ 1

2 ∓ 1

2α0L0ζ± 1

2

∓β∓ 12ε±G∓ 1

2 ± 1

2β± 1

2ε0G± 1

2 . (2.21)

With analogical reasoning, they obey the following transfor-mations

α±1δλμ±1 = ∂tε

±1 ∓ α0μ0ε±1 ± α±1μ

±1ε0

−2β± 12ν± 1

2 ζ± 12 , (2.22)

1

2α0δλμ

0 = 1

2∂tε

0 + α+1μ+1ε−1 − α−1μ

−1ε+1

−β+ 12ν+ 1

2 ζ− 12 − β− 1

2ν− 1

2 ζ+ 12 , (2.23)

β± 12δλν

± 12 = ∂tζ

± 12 ± α±1μ

±ζ∓ 12 ∓ 1

2α0μ

0ζ± 12

∓β∓ 12ε±ν∓ 1

2 ± 1

2β± 1

2ε0ν± 1

2 . (2.24)

As a final step, one now has to determine the canonical bound-ary charge Q[λ] that generates the transformations (2.19)-(2.21). Therefore, the corresponding variation of the bound-ary charge Q[λ] [31–34], to show the asymptotic symmetryalgebra, is given by

δλQ = k

2π

∫dϕ str

(λδ�ϕ

). (2.25)

The canonical boundary chargeQ[λ] can be integrated whichreads

Q[λ] =∫

dϕ[L0ε0 + L+1ε−1 + L−1ε+1

+G+ 12 ζ− 1

2 + G− 12 ζ+ 1

2

]. (2.26)

We now prefer to work in complex coordinates for the affineboundary, z(z) ≡ ϕ ± i t

�. After both the infinitesimal trans-

formations and the canonical boundary charge have beendetermined, one can yield the Poisson bracket algebra byusing the methods [35] with

δλ� = {�,Q[λ]} (2.27)

for any phase space functional � :{Li (z1) ,L j (z2)

}

PB= (i − j)Li+ j (z2) δ (z1 − z2)

− 1

α+1ηi j∂ϕδ (z1 − z2) , (2.28)

{Li (z1) ,G p (z2)

}

PB=

(i

2− p

)Gi+p (z2) δ (z1 − z2) ,

(2.29){G p (z1) ,Gq (z2)

}PB

= −2Lp+q (z2) δ (z1 − z2)

− 1

α+1ηpq∂ϕδ (z1 − z2) (2.30)

where α+1 = − 2πk for the convention in the literature and

ηi j is the bilinear form in the fundamental representation ofosp(1|2) superalgebra. One can also expand L(z) and G(z)charges into Fourier modesLi (z) = 1

2π

∑n L

inz

−n−1, and

G p(z) = 12π

∑r G

pr z−r− 1

2 , and also replacing i{·, ·}PB →[·, ·]. A mode algebra can then be defined as:[Lim,L j

n

]= (i − j)Li+ j

m+n + knηi jδm+n,0, (2.31)

[Lim,Gp

r

]=

(i

2− p

)Gi+pm+r , (2.32)

{Gpr ,Gqs

} = −2Lp+qr+s + ksκ pqδr+s,0. (2.33)

123

Eur. Phys. J. C (2020) 80 :1072 Page 5 of 15 1072

Besides, this mode algebra in this space is equivalent to oper-ator product algebra,

Li (z1)L j (z2) ∼k2ηi j

z212

+ (i − j)

z12Li+ j (z2) , (2.34)

Li (z1)G p (z2) ∼( i

2 − p)

z12Gi+p (z2) , (2.35)

G p (z1)Gq (z2) ∼k2ηpq

z212

+ 2

z12Lp+q (z2) (2.36)

where z12 = z1 − z2, or in the more compact form,

JA (z1) JB (z2) ∼

k2ηAB

z212

+ fABCJC (z2)

z12. (2.37)

Here, ηAB is the supertrace matrix and fABC ’s are the structureconstants of the related algebra with (A, B = 0,±1,± 1

2 ),

i.e, ηi p = 0 and fi ji+ j = (i − j). After repeating the same

algebra for �-sector, one can say that the asymptotic sym-metry algebra for the most general boundary conditions ofN = (1, 1) supergravity is two copies of the affine osp(1|2)kalgebra as in Ref. [30].

2.3 osp(1|2) ⊕ osp(1|2) Chern–Simons N = (1, 1)

supergravity for superconformal boundary

Under the following restrictions as the Drinfeld–Sokolovhighest weight gauge condition,

L0 = G+ 12 = 0,L−1 = L,G− 1

2 = G, α+1L+1 = 1 (2.38)

on the boundary conditions with the osp(1|2) superalgebravalued connection (2.10), one can get the superconformalboundary conditions as the supersymmetric extension of theBrown–Henneaux boundary conditions proposed in [1] forAdS3 supergravity. Therefore we have the supersymmetricconnection as,

aϕ = L1 + α−1LL−1 + β− 12GG− 1

2, (2.39)

at = μL1 +0∑

i=−1

μiLi + νG 12

+ ν− 12 G− 1

2(2.40)

where μ ≡ μ+1 andν ≡ ν+ 12 can be interpreted as an inde-

pendent chemical potentials. This means that we assumethe chemical potential to be fixed at infinity, i.e. δμ = 0. The

functions μ0, μ−1 and ν− 12 are fixed by the flatness condition

(2.4) as

μ0 = −μ′, (2.41)

μ−1 = 1

2μ′′ + α−1Lμ + β− 1

2Gν, (2.42)

ν− 12 = −ν′ + β− 1

2Gν. (2.43)

For the fixed chemical potentials μ and ν, the time evolutionof canonical boundary charges L and G can be written as

∂tL = − μ′′′

2α−1

+2Lμ′ + L′μ +β− 1

2

α−1

(3Gν′ + G′ν

), (2.44)

∂tG = − ν′′

α−1− α−1

β− 12

Lν + 3

2μ′G + μG′ (2.45)

where α−1 and β− 12

are some scaling parameters to be deter-mined later. Now, we are in a position to work the super-conformal asymptotic symmetry algebra under the Drinfeld–Sokolov reduction. This reduction implies that the only inde-

pendent parameters ε ≡ ε+1, and ζ ≡ ζ+ 12 . One can start

to compute the gauge transformations for asymptotic sym-metry algebra by considering all transformations (2.17) thatpreserve the boundary conditions. with the gauge parameterin the osp(1|2) superalgebra (2.18) as

λ = b−1[εL1 − ε′L0 +

(1

2ε′′ + α−1Lε + β− 1

2Gζ

)L−1

+ζG+ 12

+(β− 1

2Gε − ζ ′)G− 1

2

]b. (2.46)

The flat connection condition (2.17) with α−1 = 6c and

β− 12

= − 3c becomes:

δλL = c

12ε′′′ + 2Lε′ + L′ε − 1

2

(3Gζ ′ + G′ζ

), (2.47)

δλG = c

3ζ ′′ + 2Lζ + 3

2ε′G + εG′. (2.48)

The variation of canonical boundary charge Q[λ] (2.25) canbe integrated which reads

Q[λ] =∫

dϕ [Lε + Gζ ] . (2.49)

This leads to operator product expansions in the complexcoordinates by using (2.27)

L (z1)L (z2) ∼c2

z412

+ 2Lz2

12

+ L′

z12, (2.50)

L (z1)G (z2) ∼32Gz2

12

+ G′

z12, G (z1)G (z2) ∼

2c3

z312

+ 2Lz12

.

(2.51)

After repeating the same algebra for �-sector, one can saythat the asymptotic symmetry algebra for a set of boundaryconditions of N = (1, 1) supergravity is two copies of thesuper-Virasoro algebra with central charce c = 6k.

123

1072 Page 6 of 15 Eur. Phys. J. C (2020) 80 :1072

3 N = (1, 1) osp(3|2) ⊕ osp(3|2) higher-spinChern–Simons supergravity

3.1 For affine boundary

In this section, we will construct an extension of theN = (1, 1) higher-spin supergravity theory. To this end,osp(3|2) algebra is defined by using an higher-spin exten-sion of osp(1|2) algebra which can be presented as a sub-superalgebra as the ordinary case of osp(N |M) gauge alge-bra. If Li (i = ±1, 0), Gp (p = ± 1

2 ), Ai (i = ±1, 0), andSp (p = ± 1

2 ,± 32 )are the generators of osp(3|2) superalge-

bra, we have expressed osp(3|2) superalgebra such that

[Li ,L j

] = (i − j)Li+ j ,[Li ,Gp

] =(i

2− p

)Gi+p, (3.1)

{Gp,Gq

} = σ1Ap+q + σ2Lp+q ,

{Gp,Sq

} =(

3p

2− q

2

)(σ3Ap+q + σ4Lp+q

), (3.2)

[Li ,A j

] =(i

2− j

)Ai+ j ,

[Li ,Sp

] =(

3i

2− p

)Si+p, (3.3)

[Ai ,A j

] = (i − j)(σ5Li+ j + σ6Ai+ j

),

[Ai ,Gp

] = σ7Si+p + σ8

(i

2− p

)Gi+p, (3.4)

[Ai ,Sp

] = σ9

(3i

2− p

)Si+p + σ10

(3i2 − 2i p + p2 − 9

4

)Gi+p,

(3.5){Sp,Sq

} =(

3p2 − 4pq + 3q2 − 9

2

)(σ11Ap+q + σ12Lp+q

). (3.6)

The super Jacobi identities give us the nontrivial relationsfor some constants σ ′

i s, (i = 1, 2, . . . , 12) appearing on theRHS of Eqs. (3.1)–(3.6) as,

σ1 = 0, σ2 = −σ3σ7, σ4 = −σ3σ8,

σ5 = 1

2(σ8 − 2σ9) (σ8 + σ9) , σ6 = 1

2(5σ9 − σ8) ,

σ10 = − (σ8 − σ9)2

4σ7, σ11 = −σ3 (σ8 − σ9)

4σ7,

σ12 = −σ3σ9 (σ9 − σ8)

4σ7. (3.7)

For the corresponding algebra, the resulting relations are asσ3 = 2, σ7 = −1, σ8 = 0, σ9 = 1. We are now readyto formulate most general boundary conditions for asymp-totically AdS3 spacetimes:

aϕ = αiLiLi + γiAiAi + βpG pGp + τpS pSp, (3.8)

at = μiLi + χ iAi + f pGp + ν pSp (3.9)

where(αi , γi , βp, τp

)’s are some scaling parameters to be

determined later and we have twelve functions: six bosonic(Li ,Ai

)and six fermionic (G p,S p) as the charges and also

here, the time component has in total of twelve independentfunctions (μi , χ i , f p, ν p) as the chemical potentials.

The flat connection conditions (2.4) for fixed chemical poten-tials impose the following additional conditions as the tem-poral evolution of the twelve independent source fields,(Li ,Ai ,G p,S p) as

α±1∂tL±1 = ∂ϕμ±1 ± α±1μ0L±1

∓α0μ±1L0 + 2β± 1

2G± 1

2 ν± 12

∓A±1γ±1χ0 ± A0γ0χ

±1

+3τ∓ 12f± 3

2 S∓ 12 − 2τ± 1

2f± 1

2 S± 12

+3τ± 32f∓ 1

2 S± 32 , (3.10)

1

2α0∂tL0 = 1

2∂ϕμ0 + α+1μ

−1L1 − α−1μ+1L−1

+β+ 12G+ 1

2 ν− 12 + β− 1

2G− 1

2 ν+ 12

−A+1γ+1χ−1 + A−1γ−1χ

+1

+9

2τ− 3

2f+ 3

2 S− 32 − 1

2τ− 1

2f+ 1

2 S− 12

−1

2τ+ 1

2f− 1

2 S+ 12 + 9

2τ+ 3

2f− 3

2 S+ 32 , (3.11)

g± 12∂tG± 1

2 = ∂ϕν± 12 ∓ 1

2α0ν

± 12 L0 ± α±1ν

∓ 12 L±1

∓β∓ 12G∓ 1

2 μ±1 ± 1

2β± 1

2G± 1

2 μ0

+3

2A∓1γ∓1f

± 32 − 1

2A0γ0f

± 12

+1

2A±1γ±1f

∓ 12

−3

2τ± 3

2S± 3

2 χ∓1 + 1

2τ± 1

2S± 1

2 χ0

−1

2τ∓ 1

2S∓ 1

2 χ±1, (3.12)

1

2γ0∂tA0 = 1

2∂ϕχ0 + A+1γ+1μ

−1 − A−1γ−1μ+1

+5

2A+1γ+1χ

−1

−5

2A−1γ−1χ

+1 − β− 12f+ 1

2 G− 12

+β+ 12f− 1

2 G+ 12 − 9

2τ− 3

2f+ 3

2 S− 32

+1

2τ− 1

2f+ 1

2 S− 12

+1

2τ+ 1

2f− 1

2 S+ 12 − 9

2τ+ 3

2f− 3

2 S+ 32

−τ+ 12ν− 1

2 S+ 12 + τ− 1

2ν+ 1

2 S− 12 + α+1χ

−1L+1

−α−1χ+1L−1, (3.13)

γ±1∂tA±1 = ∂ϕχ+1 ± A±1γ±1μ0 ∓ A0γ0μ

±1

±5

2A±1γ±1χ

0 ∓ 5

2A0γ0χ

±1 ∓ 3β∓ 12f± 3

2 G− 12

±β± 12f± 1

2 G± 12 − 3τ∓ 1

2f± 3

2 S∓ 12 + 2τ± 1

2f± 1

2 S± 12

−3τ± 32f∓ 1

2 S± 32 − 3τ± 3

2ν∓ 1

2 S± 32

+ τ± 12ν± 1

2 S± 12 ± α±1χ

0L±1 ∓ α0χ±1L0, (3.14)

123

Eur. Phys. J. C (2020) 80 :1072 Page 7 of 15 1072

τ± 12∂tS± 1

2 = ∂ϕ f± 1

2 − A±1γ±1ν∓ 1

2

−A0γ0ν± 1

2 ∓ 3A∓1γ∓1f± 3

2

∓1

2A0γ0f

± 12 ± 2A±1γ±1f

∓ 12

∓3α∓1f± 3

2 L∓1 − 1

2α0f

± 12 L0 ± 2α±1f

∓ 12 L±1

+β± 12G± 1

2 χ0 + β∓ 12G∓ 1

2 χ±1

±3τ± 32μ∓1S± 3

2 ± 1

2τ± 1

2μ0S± 1

2 ∓ 2τ∓ 12μ±1S∓ 1

2

±τ± 32S± 3

2 χ∓1 ± 1

2τ± 1

2S± 1

2 χ0

∓2τ∓ 12S∓ 1

2 χ±1, (3.15)

τ± 32∂tS± 3

2 = ∂ϕ f± 3

2 − A±1γ±1ν± 1

2

∓3

2A0γ0f

± 32 ± A±1γ±1f

± 12

∓3

2α0f

± 32 L0 + α±1f

± 12 L±1

+β± 12G± 1

2 χ±1 ± 3

2τ± 3

2μ0S± 3

2 − τ± 12μ±1S± 1

2

±3

2τ± 3

2S± 3

2 χ0 ∓ τ± 12S± 1

2 χ±1. (3.16)

After the temporal evolution of the source fields, one can startto compute the gauge transformations for asymptotic sym-metry algebra by considering all transformations (2.17) thatpreserve the boundary conditions, with the gauge parameterλ in the osp(3|2) superalgebra as,

λ = b−1[εiLi + κ iAi + ζ pGp + �pSp

]b. (3.17)

Here, the gauge parameter has in total of twelve arbitraryfunctions :six bosonic (εi , κ i ) and six fermionic (ζ p, �p) onthe boundary. The condition (2.17) impose that transforma-tions on the gauge are given by

α±1δλL±1 = ∂ϕε±1 ± α±1ε0L±1

∓α0ε±1L0 + 2β± 1

2G± 1

2 �± 12

∓A±1γ±1κ0 ± A0γ0κ

±1

+3τ∓ 12ζ± 3

2 S∓ 12 − 2τ± 1

2ζ± 1

2 S± 12

+3τ± 32ζ∓ 1

2 S± 32 , (3.18)

1

2α0δλL0 = 1

2∂ϕε0 + α+1ε

−1L1 − α−1ε+1L−1

+β+ 12G+ 1

2 �− 12 + β− 1

2G− 1

2 �+ 12

−A+1γ+1κ−1 + A−1γ−1κ

+1

+9

2τ− 3

2ζ+ 3

2 S− 32 − 1

2τ− 1

2ζ+ 1

2 S− 12

−1

2τ+ 1

2ζ− 1

2 S+ 12 + 9

2τ+ 3

2ζ− 3

2 S+ 32 , (3.19)

g± 12δλG± 1

2 = ∂ϕ�± 12 ∓ 1

2α0�

± 12 L0 ± α±1�

∓ 12 L±1

∓β∓ 12G∓ 1

2 ε±1 ± 1

2β± 1

2G± 1

2 ε0

+3

2A∓1γ∓1ζ

± 32 − 1

2A0γ0ζ

± 12

+1

2A±1γ±1ζ

∓ 12

−3

2τ± 3

2S± 3

2 κ∓1 + 1

2τ± 1

2S± 1

2 κ0

−1

2τ∓ 1

2S∓ 1

2 κ±1, (3.20)

1

2γ0δλA0 = 1

2∂ϕκ0 + A+1γ+1ε

−1 − A−1γ−1ε+1

+5

2A+1γ+1κ

−1

−5

2A−1γ−1κ

+1 − β− 12ζ+ 1

2 G− 12

+β+ 12ζ− 1

2 G+ 12 − 9

2τ− 3

2ζ+ 3

2 S− 32

+1

2τ− 1

2ζ+ 1

2 S− 12

+1

2τ+ 1

2ζ− 1

2 S+ 12 − 9

2τ+ 3

2ζ− 3

2 S+ 32

−τ+ 12�− 1

2 S+ 12 + τ− 1

2�+ 1

2 S− 12

+α+1κ−1L+1 − α−1κ

+1L−1, (3.21)

γ±1δλA±1 = ∂ϕκ+1 ± A±1γ±1ε0 ∓ A0γ0ε

±1

±5

2A±1γ±1κ

0 ∓ 5

2A0γ0κ

±1 ∓ 3β∓ 12ζ± 3

2 G− 12

±β± 12ζ± 1

2 G± 12 − 3τ∓ 1

2ζ± 3

2 S∓ 12 + 2τ± 1

2ζ± 1

2 S± 12

−3τ± 32ζ∓ 1

2 S± 32 − 3τ± 3

2�∓ 1

2 S± 32

+τ± 12�± 1

2 S± 12 ± α±1κ

0L±1 ∓ α0κ±1L0, (3.22)

τ± 12δλS± 1

2 = ∂ϕζ± 12 − A±1γ±1�

∓ 12

−A0γ0�± 1

2 ∓ 3A∓1γ∓1ζ± 3

2

∓1

2A0γ0ζ

± 12 ± 2A±1γ±1ζ

∓ 12

∓3α∓1ζ± 3

2 L∓1 − 1

2α0ζ

± 12 L0 ± 2α±1ζ

∓ 12 L±1

+β± 12G± 1

2 κ0 + β∓ 12G∓ 1

2 κ±1

±3τ± 32ε∓1S± 3

2 ± 1

2τ± 1

2ε0S± 1

2 ∓ 2τ∓ 12ε±1S∓ 1

2

±τ± 32S± 3

2 κ∓1 ± 1

2τ± 1

2S± 1

2 κ0

∓2τ∓ 12S∓ 1

2 κ±1, (3.23)

τ± 32δλS± 3

2 = ∂ϕζ± 32 − A±1γ±1�

± 12 ∓ 3

2A0γ0ζ

± 32

±A±1γ±1ζ± 1

2 ∓ 3

2α0ζ

± 32 L0 + α±1ζ

± 12 L±1

+β± 12G± 1

2 κ±1 ± 3

2τ± 3

2ε0S± 3

2

−τ± 12ε±1S± 1

2 ± 3

2τ± 3

2S± 3

2 κ0

∓τ± 12S± 1

2 κ±1. (3.24)

123

1072 Page 8 of 15 Eur. Phys. J. C (2020) 80 :1072

With analogical reasoning, they obey the following transfor-mations

α±1δλμ±1 = ∂tε

±1 ± α±1ε0μ±1

∓α0ε±1μ0 + 2β± 1

2f± 1

2 �± 12

∓χ±1γ±1κ0 ± χ0γ0κ

±1

+3τ∓ 12ζ± 3

2 ν∓ 12 − 2τ± 1

2ζ± 1

2 ν± 12

+3τ± 32ζ∓ 1

2 ν± 32 , (3.25)

1

2α0δλμ

0 = 1

2∂tε

0 + α+1ε−1μ1 − α−1ε

+1μ−1

+β+ 12f+ 1

2 �− 12 + β− 1

2f− 1

2 �+ 12

−χ+1γ+1κ−1 + χ−1γ−1κ

+1 + 9

2τ− 3

2ζ+ 3

2 ν− 32

−1

2τ− 1

2ζ+ 1

2 ν− 12

−1

2τ+ 1

2ζ− 1

2 ν+ 12 + 9

2τ+ 3

2ζ− 3

2 ν+ 32 , (3.26)

g± 12δλf

± 12 = ∂t�

± 12 ∓ 1

2α0�

± 12 μ0

±α±1�∓ 1

2 μ±1 ∓ β∓ 12f∓ 1

2 ε±1 ± 1

2β± 1

2f± 1

2 ε0

+3

2χ∓1γ∓1ζ

± 32 − 1

2χ0γ0ζ

± 12

+1

2χ±1γ±1ζ

∓ 12

−3

2τ± 3

2ν± 3

2 κ∓1 + 1

2τ± 1

2ν± 1

2 κ0

−1

2τ∓ 1

2ν∓ 1

2 κ±1, (3.27)

1

2γ0δλχ

0 = 1

2∂tκ

0 + χ+1γ+1ε−1 − χ−1γ−1ε

+1

+5

2χ+1γ+1κ

−1

−5

2χ−1γ−1κ

+1 − β− 12ζ+ 1

2 f− 12

+β+ 12ζ− 1

2 f+ 12 − 9

2τ− 3

2ζ+ 3

2 ν− 32

+1

2τ− 1

2ζ+ 1

2 ν− 12 + 1

2τ+ 1

2ζ− 1

2 ν+ 12

−9

2τ+ 3

2ζ− 3

2 ν+ 32

−τ+ 12�− 1

2 ν+ 12 + τ− 1

2�+ 1

2 ν− 12

+α+1κ−1μ+1 − α−1κ

+1μ−1, (3.28)

γ±1δλχ±1 = ∂tκ

+1 ± χ±1γ±1ε0 ∓ χ0γ0ε

±1

±5

2χ±1γ±1κ

0 ∓ 5

2χ0γ0κ

±1 ∓ 3β∓ 12ζ± 3

2 f− 12

±β± 12ζ± 1

2 f± 12 − 3τ∓ 1

2ζ± 3

2 ν∓ 12 + 2τ± 1

2ζ± 1

2 ν± 12

−3τ± 32ζ∓ 1

2 ν± 32 − 3τ± 3

2�∓ 1

2 ν± 32

+τ± 12�± 1

2 ν± 12 ± α±1κ

0μ±1 ∓ α0κ±1μ0, (3.29)

τ± 12δλν

± 12 = ∂tζ

± 12 − χ±1γ±1�

∓ 12 − χ0γ0�

± 12

∓3χ∓1γ∓1ζ± 3

2 ∓ 1

2χ0γ0ζ

± 12 ± 2χ±1γ±1ζ

∓ 12

∓3α∓1ζ± 3

2 μ∓1 − 1

2α0ζ

± 12 μ0 ± 2α±1ζ

∓ 12 μ±1

+β± 12f± 1

2 κ0 + β∓ 12f∓ 1

2 κ±1

±3τ± 32ε∓1ν± 3

2 ± 1

2τ± 1

2ε0ν± 1

2 ∓ 2τ∓ 12ε±1ν∓ 1

2

±τ± 32ν± 3

2 κ∓1 ± 1

2τ± 1

2ν± 1

2 κ0

∓2τ∓ 12ν∓ 1

2 κ±1, (3.30)

τ± 32δλν

± 32 = ∂tζ

± 32 − χ±1γ±1�

± 12 ∓ 3

2χ0γ0ζ

± 32

±χ±1γ±1ζ± 1

2 ∓ 3

2α0ζ

± 32 μ0 + α±1ζ

± 12 μ±1

+β± 12f± 1

2 κ±1 ± 3

2τ± 3

2ε0ν± 3

2 − τ± 12ε±1ν± 1

2

±3

2τ± 3

2ν± 3

2 κ0 ∓ τ± 12ν± 1

2 κ±1. (3.31)

As in the osp(1|2) ⊕ osp(1|2) case one can now determinethe canonical boundary chargesQ[λ] that generates the trans-formations (3.18)–(3.24). Therefore, the corresponding vari-ation of the boundary charge Q[λ] [31–34], to show theasymptotic symmetry algebra, is given by (2.25). The canon-ical boundary charge Q[λ] can be integrated which reads

Q[λ] =∫

dϕ[Liε−i + Aiκ−i + G pζ−p + S p�−p

].

(3.32)

Having determined both the infinitesimal transformationsand the canonical boundary charges as the generators of theasymptotic symmetry algebra, one can also get the Poissonbracket algebra by using the methods [35] with (2.27) againfor any phase space functional �, the operator product alge-bra can then be defined as,

Li (z1)L j (z2) ∼k2 ηi j

z212

+ (i − j)

z12Li+ j , (3.33)

Li (z1)G p (z2) ∼( i

2 − p)

z12Gi+p, (3.34)

G p (z1)Gq (z2) ∼k2 ηpq

z212

+ 1

z12

(σ1Ap+q + σ2Lp+q) , (3.35)

G p (z1)Sq (z2) ∼(

3p2 − q

2

)

z12

(σ3Ap+q + σ4Lp+q) , (3.36)

Li (z1)A j (z2) ∼( i

2 − j)

z12Ai+ j ,

Li (z1)S j (z2) ∼( 3i

2 − j)

z12S i+ j , (3.37)

123

Eur. Phys. J. C (2020) 80 :1072 Page 9 of 15 1072

Ai (z1)A j (z2) ∼k2 ηi j

z212

+ (i − j)

z12

(σ5Li+ j + σ6Ai+ j

),

(3.38)

Ai (z1)G p (z2) ∼ 1

z12

(σ7Si+p + σ8

(i

2− p

)Gi+p

),

(3.39)

Ai (z1)S p (z2) ∼ 1

z12

(σ9

(3i

2− p

)Si+p + σ10

(3i2 − 2i p

+p2 − 9

4

)Gi+p

), (3.40)

S p (z1)Sq (z2) ∼k2 ηpq

z212

+ 1

z12

((3p2 − 4pq + 3q2

−9

2

)(σ11Ap+q + σ12Lp+q

))

(3.41)

where z12 = z1 − z2, or in the more compact form,

JA (z1) JB (z2) ∼

k2ηAB

z212

+ fABCJC (z2)

z12. (3.42)

Here, ηAB is the supertrace matrix and fABC ’s are thestructure constants of the related algebra with (A, B =0,±1,± 1

2 , 0,±1,± 12 ,± 3

2 ), i.e, ηi p = 0 and fi ji+ j =

(i − j). After repeating the same algebra for �-sector, onecan say that the asymptotic symmetry algebra for the mostgeneral boundary conditions of N = (1, 1) supergravity istwo copies of the affine osp(3|2)k algebra.

3.2 For superconformal boundary

Under the following restrictions as the Drinfeld–Sokolovhighest weight gauge condition,

L0 = A0 = A+1 = G+ 12 = S+ 1

2 = S+ 32 = 0,

L−1 = L,A−1 = A,G− 12 = G,S− 3

2 = S, α+1L+1 = 1

(3.43)

on the boundary conditions with the osp(3|2) superalgebravalued connection (2.10), one can get the superconformalboundary conditions as the supersymmetric extension of theBrown–Henneaux boundary conditions proposed in [1,20]for AdS3 supergravity. Therefore we have the supersymmet-ric connection as,

aϕ = L1 + α−1LL−1 + γ−1AL−1

+β− 12GG− 1

2+ τ− 3

2SS− 3

2, (3.44)

at = μL1 + χA1 + fG+ 12

+νS+ 32

+0∑

i=−1

μiLi +0∑

i=−1

χ iAi

+f− 12 G− 1

2+

12∑

p=− 32

ν pSp (3.45)

where μ ≡ μ+1, χ ≡ χ+1, ν ≡ ν+ 12 , and f ≡ f+ 3

2 can beinterpreted as the independent chemical potentials. Onecan choose for the corresponding asymptotic symmetry alge-bra, the resulting relations as σ3 = 2i√

5, σ7 = i

√5, σ8 =

0, σ9 = 1,appearing on the RHS of Eqs. (3.1)–(3.6) inthe super Jacobi identities of the osp(3|2) superalgebra. Thefunctions, except the chemical potentials are fixed by theflatness condition (2.4). For the fixed chemical potentials, thetime evolution of canonical boundary charges can be writtenas,

∂tL = μ′′′

2α−1+ μL′ + 2Lμ′ + γ−1χA′

α−1

+2Aγ−1χ′

α−1−

3β− 12Gf ′

α−1

−β− 1

2fG′

α−1−

9δ− 32νS ′

10α−1−

3δ− 32Sν′

2α−1, (3.46)

∂tG = 3iγ−1νA′

2√

5β− 12

+ 2iAγ−1ν′

√5β− 1

2

− f ′′

β− 12

− α−1fLβ− 1

2

+ μG′ + 3Gμ′

2+

3iδ− 32Sχ

2√

5β− 12

, (3.47)

∂tA = μA′ − 5

2iχA′ −

9Aβ− 12Gν

√5

+ 2Aμ′ − 5iAχ ′ + χ ′′′

2γ−1

+3iδ− 3

2fS

√5γ−1

−3iβ− 1

2νG′′

2√

5γ−1

−4iβ− 1

2G′ν′

√5γ−1

−3iβ− 1

2Gν′′

√5γ−1

−9iα−1β− 1

2GνL

2√

5γ−1+

9iδ− 32νS ′

10γ−1

+3iδ− 3

2Sν′

2γ−1+ α−1χL′

γ−1+ 2α−1Lχ ′

γ−1, (3.48)

∂tS = iγ−1νA′′

2δ− 32

+ 5iγ−1A′ν′

3δ− 32

− i√

5γ−1fA′

3δ− 32

+5iAγ−1ν′′

3δ− 32

− 4i√

5Aγ−1f ′

3δ− 32

−3√

5Aβ− 12γ−1Gχ

δ− 32

+3iα−1Aγ−1νLδ− 3

2

− ν(4)

6δ− 32

−i√

5β− 12χG′′

6δ− 32

−7β2

− 12GνG′

2δ− 32

−2i

√5β− 1

2G′χ ′

3δ− 32

−i√

5β− 12Gχ ′′

δ− 32

−3i

√5α−1β− 1

2GχL

2δ− 32

+ μS ′ − iχS ′ + 5Sμ′

2− 5

2iSχ ′

−α−1νL′′

2δ− 32

− 3α2−1νL2

2δ− 32

−5α−1ν′L′

3δ− 32

− 5α−1Lν′′

3δ− 32

(3.49)

123

1072 Page 10 of 15 Eur. Phys. J. C (2020) 80 :1072

where α−1, γ−1, β− 12

and τ− 32

are some scaling parametersto be determined later. Now, we are in a position to workthe superconformal asymptotic symmetry algebra under theDrinfeld–Sokolov reduction. This reduction implies that the

only independent parameters ε ≡ ε+1, κ ≡ κ+1, ζ ≡ ζ+ 12

and � ≡ �+ 32 . One can start to compute the gauge trans-

formations for asymptotic symmetry algebra by consideringall transformations (2.17) that preserve the boundary condi-tions with the gauge parameter λ in the osp(3|2) superalgebra(3.1)–(3.6) as

λ = b−1[εL+1 − ε′L0 +

(Aγ−1κ − β− 1

2ζG − 9

10δ− 3

2ρS

+α−1Lε + ε′′

2

)L−1

+ ζG+ 12

+(

3iγ−1�A2√

5− ζ ′ + β− 1

2Gε

)G− 1

2

+ κA+1 +(

−κ ′ +3iβ− 1

2�G

√5

)

A0

+(

−5

2iAγ−1κ + Aγ−1ε −

3iβ− 12ρG′

2√

5

−1

2i√

5β− 12G�′ + κ ′′

2+ 9

10iδ− 3

2ρS + α−1κL

)A−1

+ �S 32

− �′S 12

+(

−3

2iAγ−1ρ

+1

2i√

5β− 12Gκ + �′′

2+ 3

2α−1�L

)S− 1

2

+(

1

2iγ−1�A′ − 1

3i√

5Aγ−1ζ

+7

6iAγ−1�

′ + β2− 1

2− G2� − 1

6i√

5β− 12κG′

− 1

2i√

5β− 12Gκ ′ − �(3)

6− iδ− 3

2κS

+δ− 32Sε − 1

2α−1�L′ − 7

6α−1L�′

)S− 3

2

]b. (3.50a)

The condition (2.17) impose that transformations on thegauge with α−1 = γ−1 = 6

c , β− 12

= − 3c and τ− 3

2= − 10

care given by

δλL = cε′′′

12+ εL′ + 2Lε′

+κA′ + 2Aκ ′ + ζG′

2+ 3Gζ ′

2+ 3ρS ′

2+ 5Sρ′

2, (3.51)

δλG = −3iρA′√

5− 4iAρ′

√5

+cζ ′′

3+ εG′ + 3Gε′

2+ i

√5κS + 2ζL, (3.52)

δλA = −5

2iκA′ + εA′ − 5iAκ ′ + 2Aε′

+27AGρ√5c

+ 27iGρL2√

5c+ cκ ′′′

12

+3iρG′′

4√

5+ 2iG′ρ′

√5

+ 3iGρ′′

2√

5− 3

2iρS ′ − i

√5ζS

−5

2iSρ′ + κL′ + 2Lκ ′, (3.53)

δλS = − 3

10iρA′′ + iζA′

√5

− iA′ρ′ + 4iAζ ′√

5− iAρ′′ − 27AGκ√

5c

−54iAρL5c

+ 63GρG′

20c− 27iGκL

2√

5c+ cρ(4)

60

+27ρL2

5c− iκG′′

4√

5− iG′κ ′

√5

− 3iGκ ′′

2√

5− iκS ′ + εS ′

−5

2iSκ ′ + 5Sε′

2+ 3ρL′′

10+ ρ′L′ + Lρ′′. (3.54)

The variation of canonical boundary charge Q[λ] (2.25) canbe integrated which reads

Q[λ] =∫

dϕ[Lε + Aκ + Gζ + S�

]. (3.55)

This leads to operator product expansions in the complexcoordinates by using (2.27)

L (z1)L (z2) ∼c2

z412

+ 2Lz2

12

+ L′

z12, (3.56a)

L (z1)G (z2) ∼32Gz2

12

+ G′

z12, G (z1)G (z2) ∼

2c3

z312

+ 2Lz12

,

(3.56b)

L (z1)A (z2) ∼ 2Az2

12

+ A′

z12, L (z1)S (z2) ∼

52Sz2

12

+ S ′

z12,

(3.56c)

G (z1)A (z2) ∼ − i√

5Az12

, G (z1)S (z2) ∼4i√

5A

z212

−i√5A′

z12,

(3.56d)

A (z1)A (z2) ∼c2

z412

+ 2L − 5iAz2

12

+ L − 5i2 A

z12, (3.56e)

A (z1)S (z2) ∼ −3i√

5G

z312

−iG′√

5+ 5iS

2

z212

− 1

z12

(27AG√

5c

+27iGL2√

5c+ iS ′ + iG′′

4√

5

), (3.56f)

S (z1)S (z2) ∼2c5

z512

+ 2(L − iA)

z312

+ L′ − iA′

z212

+ 1

z12

(27L2

5c

−54iAL5c

+ 63GG′

20c− 3iA′′

10+ 3L′′

10

). (3.56g)

After repeating the same algebra for �-sector, one can saythat the asymptotic symmetry algebra for a set of boundaryconditions of N = (1, 1) supergravity is two copies of theSW( 3

2 , 2) algebra with central charce c = 6k. Finally, onecan also take into account normal ordering (quantum) effectsof this algebra as in Ref’s. [36,37].

123

Eur. Phys. J. C (2020) 80 :1072 Page 11 of 15 1072

4 Other extended (super)gravity checks in the mostgeneral boundary conditions

In this section, it will be very interesting to show anotherclass of boundary conditions that appeared in the litera-ture (see, e.g. [39,41–43]) for gravity case, whose higher-spin generalization is less clear than the Grumiller–Rieglerboundary conditions. Because of the possibility to obtainasymptotic symmetries that are the affine version of thegauge algebras of the Chern–Simons theory is not a sur-prise given the relation between Chern–Simons theories andWess–Zumino–Witten models [27]. Therefore, we presenttwo most general boundary conditions for pure bosonicgravity and N = 1 extended supergravity. That is, TheAvery–Poojary–Suryanarayana gravity for sl(2, R) ⊕ sl(2)

and osp(1|2)⊕sl(2) respectively. In order to check, we com-pare the most general boundary conditions used so far withthe Avery–Poojary–Suryanarayana boundary conditions forsl(2, R) ⊕ sl(2) pure gravity in Sect. 4.1 and N = 1 super-gravity for osp(1|2) ⊕ sl(2) in Sect. 4.2 respectively afterwe have worked with the Brown–Henneaux boundary con-ditions for N = 1 supergravity for osp(1|2) ⊕ sl(2) in Sect.3.2. We also mention the checks that our boundary conditionspassed and discuss conservation of the charges and consis-tency of the variational principleN = (1, 1) supergravity forosp(1|2) in Sect. 4.4 and osp(3|2) in Sect. 4.5 respectively.

4.1 Avery–Poojary–Suryanarayana for sl(2, R) ⊕ sl(2)

The Avery–Poojary–Suryanarayana boundary conditions pro-posed in [43] are a generalization of the Brown–Henneauxboundary conditions and that can be obtained by choos-ing for �-sector the Brown–Henneaux boundary conditionswhile leaving all charges to vary for the �-sector. Con-sequently, these lead to an interesting set of asymptoticsymmetries in the form of sl(2, R) ⊕ sl(2)k . To this end,we recapitulate Avery–Poojary–Suryanarayana gravity forsl(2, R) ⊕ sl(2) from the most general Grumiller–Rieglergravity for sl(2, R)⊕sl(2, R) as in [26]. Therefore, the mostgeneral connections for Grumiller–Riegler boundary condi-tions is given by

� − sector : aϕ = αiLiLi , at = μiLi , (4.1)

� − sector : aϕ = −αi LiLi , at = μiLi . (4.2)

Besides, in the Avery–Poojary–Suryanarayana gravity theboundary conditions is given by

� − sector : aϕ = L1 − κL−1,

at = L1 − κL−1, (4.3)

� − sector : aϕ = L−1 − κL1 + f iLi ,

at = −L−1 + κL1 + f iLi . (4.4)

In this gravity language, this amounts to the following restric-tions on the charge and the chemical potential:

� − sector : L0 = 0, L−1 = L = − κ

α−1,

L1 = 1

α1,

μ0 = 0, μ1 = μ = 1, μ−1 = κ, (4.5)

� − sector : L0 = − f 0

α0, L−1 = − 1

α−1

(1 + f−1) ,

L1 = 1

α1

(κ − f 1) ,

μ0 = f 0, μ1 = μ = κ + f 1,

μ−1 = f−1 − 1, and La = T a . (4.6)

Next we want to find two sets of gauge transformations thatpreserve the boundary conditions. First, we define the gaugeparameter λ for �-sector as:

aϕ = L1 − α−1LL−1, λ = εiLi , (4.7)

where α−1 = 6c . If Li , (i = ±1, 0) are the generators of

sl(2, R) algebra such that[Li ,L j

] = (i − j)Li+ j that is,this is the Brown–Henneaux boundary condition that producethe following asymptotic symmetry algebra :

L (z1)L (z2) ∼c2

z412

+ 2Lz2

12

+ ∂Lz12

(4.8)

with central charge c = 6k. Second, we define the gaugeparameter λ for �-sector as:

aϕ = L−1 − α−1LL1 + βiT iTi0, λ = ηiLi + σ iTi0.

(4.9)

where α−1 = 6c , β−1 = β1 = − 2

k , and β0 = 4k . If Tan, (a =

±1, 0) are the generators of sl(2) current algebra. We haveexpressed sl(2, R) ⊕ sl(2) algebra such that

[Ln,Lm] = (n − m)Ln+m,[Ln,T

am

] = −mTan+m,[Tan,T

bm

]= (a − b)Ta+b

n+m .

(4.10)

that is, this is an affine boundary condition that produce thefollowing sl(2, R) ⊕ sl(2)k asymptotic symmetry algebra :

L (z1)L (z2) ∼c2

z412

+ 2Lz2

12

+ L′

z12, (4.11)

L (z1) T a (z2) ∼ T a

z212

+ T a′

z12,

T a (z1) T b (z2) ∼k2ηab

z212

+ (a − b)

z12T a+b (z2) . (4.12)

123

1072 Page 12 of 15 Eur. Phys. J. C (2020) 80 :1072

with central charge c = 6k and performing a Sugawara shift[26],

L → L−1 + 1

k

(T 0T 0 − T +T −)

. (4.13)

4.2 Avery–Poojary–Suryanarayana N = 1 supergravity forosp(1|2) ⊕ sl(2)

Inspired by previous calculations, in this section, we cal-culate Avery–Poojary–Suryanarayana N = 1 gravity forosp(1|2) ⊕ sl(2) from the most general Grumiller–RieglerN = (1, 1) supergravity for osp(1|2)⊕osp(1|2). Therefore,the most general connections for Grumiller–Riegler bound-ary conditions is given by

� − sector: aϕ = αiLiLi + βpG pGp, (4.14)

at = μiLi + ν pGp, (4.15)

� − sector: aϕ = −(αi LiLi + βpG pGp

), (4.16)

at = μiLi + ν pGp. (4.17)

Besides, in the Avery–Poojary–Suryanarayana gravity theboundary conditions is given by

� − sector: aϕ = L1 − κL−1 − ωG− 12, (4.18)

at = L1 − κL−1 − ωG− 12, (4.19)

� − sector: aϕ = L−1 − κL1 − ωG 12

+ f iLi + gpGp,

(4.20)

at = −L−1 + κL1 + ωG 12

+ f iLi + gpGp.

(4.21)

In this gravity language, this amounts to the following restric-tions on the charges and the chemical potentials:

� − sector : L0 = 0, L−1 = L = − κ

α−1, L1 = 1

α1,

G− 12 = − ω

β− 12

, G 12 = 0,

μ0 = 0, μ1 = μ = 1, μ−1 = κ,

ν− 12 = −ω, ν

12 = 0, (4.22)

� − sector : L0 = − f 0

α0,

L−1 = − 1

α−1

(1 + f−1

),

L1 = 1

α1

(κ − f 1

),

G− 12 = g− 1

2

β− 12

, G 12 = ω − g

12

β 12

,

μ0 = f 0, μ1 = μ = κ + f 1, (4.23)

μ−1 = f−1 − 1, ν− 12 = g− 1

2 ,

ν12 = ω + g

12 , and La = T a . (4.24)

Next, we want to find two sets of gauge transformations thatpreserve the boundary conditions. First, we define the gaugeparameter λ for �-sector as:

aϕ = L1 − α−1LL−1 − β− 12GG− 1

2, λ = εiLi + ζ pGp.

(4.25)

This is the Brown–Henneaux boundary condition for the �−sector that produces one copy of the asymptotic symmetryalgebra as in Sect. 2.3. Second, we define the gauge parameterλ for �-sector as:

aϕ = L−1 − α−1LL1 − β− 12GG 1

2+ γiT iTi0,

λ = εiLi + ζ pGp + σ iTi0. (4.26)

where α−1 = 6c , β− 1

2= − 3

c , γ−1 = γ1 = − 2k , and γ0 = 4

k .

IfTan, (a = ±1, 0) are the generators of sl(2) current algebra.We have expressed osp(1|2) ⊕ sl(2) superalgebra such that

[Ln,Lm] = (n − m)Ln+m,[Ln,Gp

] =(n

2− p

)Gn+p,

{Gp,Gq

} = −2Lp+q ,[Ln,T

am

] = −mTan+m,[Tan,T

bm

]= (a − b)Ta+b

n+m . (4.27)

that is, this is an affine boundary condition that produces thefollowing osp(1|2) ⊕ sl(2)k asymptotic symmetry algebra :

L (z1)L (z2) ∼c2

z412

+ 2Lz2

12

+ L′

z12,

L (z1)G (z2) ∼32Gz2

12

+ G′

z12,

G (z1)G (z2) ∼2c3

z312

+ 2Lz12

,

L (z1) T a (z2) ∼ T a

z212

+ T a′

z12,

T a (z1) T b (z2) ∼k2ηab

z212

+ (a − b)

z12T a+b. (4.28)

with central charge c = 6k, and the same Sugawara shift(4.13).

4.3 The on-shell action and the variational principle

It is well known that on manifold M with boundary ∂Mthe Chern–Simons action (2.2) in general is neither differen-tiable nor gauge invariant. Therefore, it is important to showthat the boundary conditions admit a well-defined variationalprinciple. To see this, take a closer look at the variation ofthe Chern–Simons action,

δSCS[�] = k

2π

∫

M〈δ� ∧ F〉 + k

4π

∫

∂M〈δ� ∧ �〉,

123

Eur. Phys. J. C (2020) 80 :1072 Page 13 of 15 1072

(4.29)

where, due that on-shell F = 0, the on-shell bulk part iszero and we are left with only a boundary term :

δSCS[�] = k

4π

∫

∂Mdtdϕ 〈�ϕ ∧ δ�t − �t ∧ δ�ϕ〉. (4.30)

Therefore, it would be good to have an appropriate way ofextending the possible consistent boundary conditions. Aconsistent way is to add a boundary term SB [�] to the Chern–Simons action. So, the total action will then be of the form :

Stot = SCS + k

4π

∫

∂Mdtdϕ 〈�t ∧ �ϕ〉. (4.31)

4.4 osp(1|2)-case

Then, considering a solution of the form (4.14), for ourboundary conditions is very easy to see that the radial depen-dent group element plays no role in this analysis, in factplugging (4.14) into (4.31) we have :

δStot[�] = − k

2π

∫

∂Mdtdϕ 〈aϕ δat 〉 (4.32)

= k

2π

∫

∂Mdtdϕ 〈Liδμ−i + G pδν−p〉

= 0 (4.33)

because of δat = 0. A similar calculation should be madefor the barred sector. This solution ensures for that α1 =− 2π

k , α0 = 4πk , α−1 = − 2π

k , β 12

= πk and β− 1

2= −π

k in(2.10).

4.5 osp(3|2)-case

After the calculation of osp(1|2)-case, it could be an interest-ing straightforward exercise to establish a well-defined vari-ational principle also for N = (1, 1) extended higher-spinsupergravity with two copies of the osp(3|2)k affine algebraas follows. To this end, considering a solution of the form(3.8), plugging these into (4.31) we have :

δStot[�] = k

2π

∫

∂Mdtdϕ 〈Liδμ−i

+ Aiδχ−i + G pδf−p + S pδν−p〉= 0 (4.34)

because of δat = 0, under the re-definitions of Li = 3Li +5Ai and Ai = 5Li + 3Ai , A similar calculation should bemade for the barred sector. This solution finally ensures forthat α1 = − 2π

k , α0 = 4πk , α−1 = − 2π

k , β 12

= π3k , β− 1

2=

− π3k , γ1 = − 2γ

k , α0 = 4πk , γ−1 = − 2π

k , τ 32

= π9k , τ 1

2=

π3k , τ− 1

2= − π

3k and τ− 32

= π9k in(3.8).

5 Summary and comments

In this work, a relation between AdS3 and osp(1|2) ⊕osp(1|2) Chern–Simons theory was first reviewed. TheChern–Simons formulation of AdS3 allows for a straight-forward generalization to a higher-spin theory as in thebosonic cases. The higher-spin gauge fields have no prop-agating degrees of freedom, but we noted that there canbe a large class of interesting non-trivial solutions. Specifi-cally, AdS3 in the presence of a tower of higher-spin fieldsup to spin 5

2 is obtained by enlarging osp(1|2) ⊕ osp(1|2)

to osp(3|2) ⊕ osp(3|2) for the most general N = (1, 1)

extended higher-spin supergravity theory in AdS3. Finally,classical two copies of the osp(3|2)k affine algebra on theaffine boundary and two copies ofSW( 3

2 , 2) symmetry alge-bra on the superconformal boundary as asymptotic symmetryalgebras, and also the chemical potentials related to sourcefields appearing through the temporal components of the con-nection are obtained. Finally, higher-spin generalization ofother class of boundary conditions, in particular for the Avery-Poojary - Suryanarayana gravity, that appeared in the litera-ture for gravity case is also shown and checked. On the otherhand, it is shown that the Chern–Simons action which com-patible with our boundary conditions leads to a finite actionand a well-defined variational principle for the higher-spinfields. Therefore, one can think that this method provides agood laboratory for investigating the rich asymptotic struc-ture of extended supergravity.

It could be rewarding to cast our results in the metric for-mulation, since this may help in lifting our boundary condi-tions to higher dimensions where Chern–Simons formulationexists. Such generalizations are interesting in itself and havethe potential for novel applications in holography. Finally,in our discussion of the most general boundary conditionsof(super)gravity, we have barely scratched the boundary.Therefore, there could be numerous open questions that callfor further investigations. For instance, which other boundaryconditions can be obtained from a similar starting point? Orhow to explain the puzzling result that the related geometriesappear to have an entropy?

In conclusion, we consider it gratifying that the asymp-totically AdS story of the most general boundary conditionsinitiated recently by Grumiller and Riegler to inspire new andsurprising developments even in the simple cases of three-dimensional gravity.

Our results presented in this paper can be extended in var-ious ways. One possible extension is by enlarging sl(2|1) ⊕sl(2|1) to sl(3|2) ⊕ sl(3|2) supergravity for the most gen-eral N = (2, 2) extended higher-spin supergravity theory inAdS3. In this context, one can finally emphasise here that therather comprehensive analysis of the sl(3|2) ⊕ sl(3|2) the-ory for the Brown–Henneaux boundary conditions has been

123

1072 Page 14 of 15 Eur. Phys. J. C (2020) 80 :1072

already given in Ref. [19]. The details of this possible exten-sion will be examined in our forthcoming paper.

Acknowledgements This work was supparted by Istanbul TechnicalUniversity Scientific Research Projects Department (ITU BAP, projectnumber:40199).

Data Availability Statement This manuscript has associated data in adata repository. [Authors’ comment: This is a theoretical study and noexternal data associated with the manuscript.]

Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.

References

1. J.D. Brown, M. Henneaux, Central charges in the canonical realiza-tion of asymptotic symmetries: an example from three-dimensionalgravity. Commun. Math. Phys. 104, 207 (1986)

2. J.M. Maldacena, The Large N limit of superconformal field the-ories and supergravity. Int. J. Theor. Phys. 38, 1113 (1999).arXiv:hep-th/9711200

3. J.M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998)4. L. Susskind, The World as a hologram. J. Math. Phys. 36, 6377

(1995). arXiv:hep-th/94090895. A. Achucarro, P.K. Townsend, A Chern–Simons action for three-

dimensional anti-de Sitter supergravity theories. Phys. Lett. B 180,89 (1986)

6. E. Witten, (2+1)-Dimensional gravity as an exactly soluble system.Nucl. Phys. B 311, 46 (1988)

7. M. Banados, C. Teitelboim, J. Zanelli, The Black hole inthree-dimensional space-time. Phys. Rev. Lett. 69, 1849 (1992).arXiv:hep-th/9204099

8. M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli, Geom-etry of the (2+1) black hole. Phys. Rev. D 48, 1506 (1993).arXiv:gr-qc/9302012, Erratum: [Phys. Rev. D 88, 069902 (2013)]

9. E. Bergshoeff, M.P. Blencowe, K.S. Stelle, Area preserving diffeo-morphisms and higher-spin algebra. Commun. Math. Phys. 128,213 (1990)

10. M.P. Blencowe, A consistent interacting massless higher-spin fieldtheory in D = (2+1). Class. Quantum Gravity 6, 443 (1989)

11. M. A. Vasiliev, Higher-spin gauge theories: Star product and AdSspace, In Shifman, M.A. (ed.): The Many Faces of the Superworld,533–610 arXiv:hep-th/9910096

12. M.A. Vasiliev, Higher-spin symmetries, star product and relativisticequations in AdS space, arXiv:hep-th/0002183

13. M. Henneaux, S.J. Rey, Nonlinear Win f ini t y as asymptotic symme-try of three-dimensional higher-spin anti-de Sitter gravity. JHEP1012, 007 (2010). arXiv:1008.4579 [hep-th]

14. A. Campoleoni, S. Fredenhagen, S. Pfenninger, S. Theisen,Asymptotic symmetries of three-dimensional gravity coupled to

higher-spin fields. JHEP 1011, 007 (2010). arXiv:1008.4744 [hep-th]

15. A. Campoleoni, S. Fredenhagen, S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories. JHEP1109, 113 (2011). arXiv:1107.0290 [hep-th]

16. A. Campoleoni, D. Francia, C. Heissenberg, Asymptotic symme-tries and charges at null infinity: from low to high spins. EPJ WebConf. 191, 06011 (2018). arXiv:1808.01542 [hep-th]

17. H.S. Tan, Aspects of three-dimensional spin-4 gravity. JHEP 02,035 (2012). arXiv:1111.2834 [hep-th]

18. H.T. Özer, A. Filiz, On the explicit asymptotic W5 symmetry of3D Chern–Simons higher-spin AdS3 gravity. J. Math. Phys. 59(8),083504 (2018). arXiv:1707.09514

19. H.S. Tan, Exploring three-dimensional higher-spin supergravitybased on sl(n, n − 1) Chern–Simons theories. JHEP 1211, 063(2012). arXiv:1208.2277 [hep-th]

20. M. Banados, K. Bautier, O. Coussaert, M. Henneaux, M. Ortiz,Anti-de Sitter / CFT correspondence in three-dimensional super-gravity. Phys. Rev. D 58, 085020 (1998). arXiv:hep-th/9805165

21. C. Candu, M.R. Gaberdiel, Supersymmetric holography on AdS3.JHEP 1309, 071 (2013). arXiv:1203.1939 [hep-th]

22. M. Henneaux, G.L. Gomez, J. Park, S.J. Rey, Super-W(infinity)asymptotic symmetry of higher-spin AdS3 supergravity. JHEP1206, 037 (2012). arXiv:1203.5152 [hep-th]

23. C. Peng, Dualities from higher-spin supergravity. JHEP 1303, 054(2013). arXiv:1211.6748 [hep-th]

24. K. Hanaki, C. Peng, Symmetries of holographic super-minimalmodels. JHEP 1308, 030 (2013). arXiv:1203.5768 [hep-th]

25. B. Chen, J. Long, Y.N. Wang, Conical defects, black holesand higher-spin (super-)symmetry. JHEP 1306, 025 (2013).arXiv:1303.0109 [hep-th]

26. D. Grumiller, M. Riegler, Most general AdS3 boundary conditions.JHEP 1610, 023 (2016). arXiv:1608.01308 [hep-th]

27. S. Elitzur, G.W. Moore, A. Schwimmer, N. Seiberg, Remarks on thecanonical quantization of the Chern–Simons–Witten theory. Nucl.Phys. B 326, 108 (1989)

28. D. Grumiller, W. Merbis, M. Riegler, Most general flat spaceboundary conditions in three-dimensional Einstein gravity. Class.Quantum Gravity 34(18), 184001 (2017). arXiv:1704.07419 [hep-th]

29. C. Krishnan, A. Raju, Chiral higher-spin gravity. Phys. Rev. D95(12), 126004 (2017). arXiv:1703.01769 [hep-th]

30. C. E. Valcarcel, New boundary conditions for (extended)AdS3 supergravity, Class. Quantum Gravity 36, 065002 (2019).arXiv:1812.02799 [hep-th]

31. M. Banados, Three-dimensional quantum geometry and blackholes, AIP Conf. Proc. 484(1), 147 (1999). arXiv:hep-th/9901148

32. S. Carlip, Conformal field theory, (2+1)-dimensional gravity, andthe BTZ black hole. Class. Quantum Gravity 22, R85 (2005).arXiv:gr-qc/0503022

33. M. Banados, Global charges in Chern–Simons field theoryand the (2+1) black hole. Phys. Rev. D 52, 5816 (1996).arXiv:hep-th/9405171

34. M. Banados, T. Brotz, M.E. Ortiz, Boundary dynamics and thestatistical mechanics of the (2+1)-dimensional black hole. Nucl.Phys. B 545, 340 (1999). arXiv:hep-th/9802076

35. M. Blagojevic, Gravitation and Gauge Symmetries (IOP publish-ing, Bristol, 2002)

36. J.M. Figueroa-O’Farrill, S. Schrans, Extended superconformalalgebras. Phys. Lett. B 257, 69 (1991)

37. J.M. Figueroa-O’Farrill, A Note on the extended superconformalalgebras associated with manifolds of exceptional holonomy. Phys.Lett. B 392, 77 (1997). arXiv:hep-th/9609113

38. H. Afshar, S. Detournay, D. Grumiller, B. Oblak, Near-horizongeometry and warped conformal symmetry. JHEP 1603, 187(2016). arXiv:1512.08233 [hep-th]

123

Eur. Phys. J. C (2020) 80 :1072 Page 15 of 15 1072

39. G. Compere, W. Song, A. Strominger, New Boundary Conditionsfor AdS3, JHEP 1305, 152 (2013) arXiv:1303.2662 [hep-th]

40. L. Donnay, G. Giribet, H.A. Gonzalez, M. Pino, Supertransla-tions and superrotations at the black hole horizon. Phys. Rev. Lett.116(9), 091101 (2016). arXiv:1511.08687 [hep-th]

41. H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez,D. Tempo, R. Troncoso, Soft Heisenberg hair on black holesin three dimensions. Phys. Rev. D 93(10), 101503 (2016).arXiv:1603.04824 [hep-th]

42. C. Troessaert, Enhanced asymptotic symmetry algebra of AdS3.JHEP 1308, 044 (2013). arXiv:1303.3296 [hep-th]

43. S.G. Avery, R.R. Poojary, N.V. Suryanarayana, An sl(2,R)current algebra from AdS3 gravity. JHEP 1401, 144 (2014).arXiv:1304.4252 [hep-th]

44. A. Perez, D. Tempo, R. Troncoso, Boundary conditions for GeneralRelativity on AdS3 and the KdV hierarchy, JHEP 1606, 103 (2016)arXiv:1605.04490 [hep-th]

45. M. Henneaux, C. Martinez, R. Troncoso, J. Zanelli, Black holesand asymptotics of 2+1 gravity coupled to a scalar field. Phys. Rev.D 65, 104007 (2002). arXiv:hep-th/0201170

46. M. Henneaux, C. Martinez, R. Troncoso, J. Zanelli, Asymptoti-cally anti-de Sitter spacetimes and scalar fields with a logarithmicbranch. Phys. Rev. D 70, 044034 (2004). arXiv:hep-th/0404236

47. D. Grumiller, N. Johansson, Consistent boundary conditions forcosmological topologically massive gravity at the chiral point. Int.J. Mod. Phys. D 17, 2367 (2009). arXiv:0808.2575 [hep-th]

48. M. Henneaux, C. Martinez, R. Troncoso, Asymptotically anti-deSitter spacetimes in topologically massive gravity. Phys. Rev. D79, 081502 (2009). arXiv:0901.2874 [hep-th]

49. J. Oliva, D. Tempo, R. Troncoso, Three-dimensional black holes,gravitational solitons, kinks and wormholes for BHT massive grav-ity. JHEP 0907, 011 (2009). arXiv:0905.1545 [hep-th]

50. G. Barnich, P.H. Lambert, Einstein–Yang–Mills theory: asymptoticsymmetries. Phys. Rev. D 88, 103006 (2013). arXiv:1310.2698[hep-th]

51. C. Bunster, A. Perez, Superselection rule for the cosmologicalconstant in three-dimensional spacetime, Phys. Rev. D 91, no. 2,024029 (2015) arXiv:1412.1492 [hep-th]

52. A. Perez, M. Riquelme, D. Tempo, R. Troncoso, Asymptotic struc-ture of the Einstein-Maxwell theory on AdS3. JHEP 1602, 015(2016). arXiv:1512.01576 [hep-th]

123