On asymptotic symmetries of 3d extended supergravities

We study asymptotic symmetry algebras for classes of three dimensional supergravities with and without cosmological constant. In the first part we generalise some of the non-dirichlet boundary conditions of AdS3 gravity to extended supergravity theories, and compute their asymptotic symmetries. In particular, we show that the boundary conditions proposed to holographically describe the chiral induced gravity and Liouville gravity do admit extension to the supergravity contexts with appropriate superalgebras as their asymptotic symmetry algebras. In the second part we consider generalisation of the 3d BMS computation to extended supergravities without cosmological constant, and show that their asymptotic symmetry algebras provide examples of nonlinear extended superalgebras containing the BMS3 algebra.


Introduction
In the standard statement of the AdS/CFT correspondence as a duality between a gravitational theory in AdS d+1 spacetime and a CF T d , the definition of the bulk theory requires specifying boundary conditions for all its fields near the boundary of the AdS d+1 space. The most extensively used boundary conditions for the bulk metric are the ones first proposed by Brown and Henneaux (BH) [1] -which consist of holding the boundary metric fixed and specifying certain fall-off conditions of the metric components away from the boundary. Then the asymptotic symmetry algebra of BH is obtained by studying the asymptotic Killing vector fields and the algebra of the corresponding charges. In particular, for the AdS 3 gravity the BH boundary conditions give rise to the 2-dimensional conformal algebra -two commuting copies of the Virasoro algebra -as the asymptotic symmetry algebra with the central charge c = 3ℓ/2G. Their results have since been generalised to include

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AdS 3 supergravites [2][3][4] where people have uncovered 2-dimensional superconformal agebras as the corresponding asymptotic symmetry algebras. The results thereof enabled successful embedding of BH boundary conditions into string theory context with applications in AdS 3 /CF T 2 . The 2-dimensional superconformal algebras with extended supersymmetry are generically nonlinear. The computations of [2][3][4] thus have also provided holographic realizations of such nonlinear superconformal algebras as asymptotic symmetry algebras of AdS 3 supergravities.
However, BH boundary conditions are not the only admissible ones for the AdS 3 gravity. In recent times it has been shown (see for instance, [5][6][7][8][9][10][11][13][14][15] etc.) that non-dirichlet boundary conditions for the metric also lead to consistent sets of boundary conditions of AdS 3 gravity which admit appropriately defined asymptotic symmetry algebras that are infinite dimensional.
The non-dirichlet boundary conditions of AdS 3 gravity would involve bulk configurations which have some (if not all) components of the boundary metric fluctuating. Two types of fluctuating boundary metrics have been considered so far in the literature. Working with co-ordinates (x + , x − ) for the 2-dimensional boundary these boundary metrics can be written as: • the light-cone gauge: • the conformal gauge: For the class of boundary metrics in the light-cone gauge in AdS 3 gravity the following boundary conditions [9,11] (which we refer to as CIG boundary conditions) were proposed for the metric g rr = l 2 r 2 + O(r −4 ), g r+ = O(r −1 ), g r− = O(r −3 ), where x + , x − are again the boundary coordinates and r is the radial coordinate with the asymptotic boundary at r −1 = 0. After imposing the conditions coming from variational principle that allowed the function F (x + , x − ) to fluctuate, it was shown in [9] that the asymptotic symmetry algebra consists of one copy of Virasoro and one copy of sl(2, R) k current algebra with level k given by l/4G. Further imposing the condition that the Ricci scalar of the boundary metric vanishes leads one to the boundary conditions considered earlier by Compere et al. in [7], where the boundary metrics were restricted to be ds 2 = −dx + dx − + P (x + )(dx + ) 2 . Then the asymptotic symmetry algebra was found to be a u(1) k ⋉ Viarasoro, with the u(1) k current algebra at level k = c/6.
For the class of boundary metrics in the conformal gauge in AdS 3 gravity the boundary conditions (such as those proposed by Troessaert [8] -which we refer to as Liouville JHEP02(2019)168 boundary conditions) take the following form: g rr = l 2 r 2 + O(r −4 ), g r+ = O(r −1 ), g r− = O(r −1 ), Troessaert in [8] further required that G(x + , x − ) satisfies ∂ + ∂ − log G = 0, coming from the additional condition that the boundary metric has zero scalar curvature. 1 Let us briefly motivate these classes of boundary conditions. Consider a 2-dimensional CFT. Starting from its Lagrangian one can construct a diffeomorphism invariant theory by simply coupling the CFT to an arbitrary background metric minimally. Such diffeomorphic theory will also exhibit Weyl symmetry classically. This Weyl invariance in general may not survive quantisation. An example being the string worldsheet theory where demanding vanishing Weyl anomaly restricts the matter sector of the theory -this then allows one to gauge fix the 2d worldsheet metric completely. When the Weyl anomaly survives the metric cannot be completely gauged away and there will be one component left dynamical in the metric. The problem of quantising such a gravitational theory was first addressed by Ployakov [16,17]. The effective theory one obtains by integrating out the original 2d CFT matter content is non-local in terms of the metric and is termed as an induced gravity theory. However, the residual diffeomorphism invariance can be used to bring the metric into either of the two standard forms, namely: the light-cone gauge ds 2 = −dx + dx − + F (x + , x − )(dx + ) 2 , and the conformal gauge ds 2 = −e φ(x + , x − ) dx + dx − , in terms of which the induced gravity action becomes local. Therefore, to describe such local induced gravity theories holographically one is naturally led to the boundary conditions alluded to above.
Polyakov studied the 2d induced gravity theory in the light-cone gauge, namely, the Chiral induced gravity (CIG), and uncovered a hidden sl(2, R) current algebra symmetry. This feature is recovered holographically in [9,11] using the CIG boundary conditions for the AdS 3 gravity.
Soon after Polaykov's work [16,17] on CIG it was generalised to include supersymmetry in the 2d induced gravity theory (see [18,19]). To describe these chiral induced supergravity theories also holographically one needs to generalise the results of [9,11] to appropriate AdS 3 supergravities. Further to be able to embed the CIG boundary conditions of [9] into some string theory context one needs to generalise them to supergravity contexts first. One of the aims of this paper is to provide supersymmetric generalisation of [9].
As mentioned above one can work in the conformal gauge for the 2d induced gravity and this leads to the Liouville theory. To model such theory holographically one has to let the conformal factor of the boundary metric to fluctuate. In [8] Troessaert provided the first example of such boundary conditions as mentioned above. However, [8] imposed an additional condition (as it was done for the light-cone gauge in [7]) that the boundary metric has vanishing scalar curvature -making the conformal factor to satisfy the free field equation

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-and not the more general Liouville equation. Just as [9] considers relaxation of the Ricci flatness of the boundary metric of [7], one can relax the boundary conditions of [8] so that the boundary metric has non-zero (constant) scalar curvature. It is then easy (see also [10]) to show that the bulk equations of motion match those of the appropriate Liouville theory. In the appendix B to this paper we study the asymptotic symmetry algebra of this case too.
In this paper we embed the boundary conditions of [8] also into supergravity contexts and we find that generically the asymptotic symmetry algebra generalises to nonlinear superalgebra extending the cases of [4] and [8].
In the second part of the paper we change tracks to investigate R 1,2 supergravity generalisations of well known flat space boundary conditions first introduced by Bondi, van der Burg and Metzner in [20] and Sachs in [21] at null infinity. Recently there has been renewed interest in analysing aspects of asymptotic symmetry of flat-space at null infinity [22][23][24][25][26] relating them to Weinberg's soft theorems which are interpreted as Ward identities associated with these symmetries. There have also been works [27][28][29][30] conjecturing how the infinite charges associated with such asymptotic symmetries could be used to better understand if not resolve the black hole information paradox. Attempts have also been made to uncover a 2d CFT on the celestial sphere at null infinity with a stress-tensor which gives rise to Virasoro Ward identities when inserted in the 4d quantum-gravity S-matrix [31,32]. Here gravity in 3d flat-space provides a simpler set up to study some if the properties associated with asymptotic symmetries as there are no bulk degrees of freedom. The 3d version of the BMS symmetry was first explored in [33] and subsequently in [35][36][37]. There it was shown that an appropriate l → ∞ takes BH symmetries (V ir ⊕ V ir) of AdS 3 gravity go over to the BM S 3 symmetries of R 1,2 gravity.
The super-extensions of the BM S 3 algebra for N = 1 case was analysed in [38], and the N = 2, 4 cases have been worked out in [39,40]. The extended super-Virasoro algebras are genereically non-linear and are realised as asymptotic symmetries of AdS 3 supergravities [4]. 2 However, so far there has been no example of super-extension of BM S 3 algebra that has such non-linearities. Motivated to find non-linear super-BM S 3 algebras, in this paper we study many ways of taking the flat space limit (l → ∞) of the super-algebras uncovered in [4] for arbitrary N . We show that among the many possible flat-space limits of asymptotic super-algebras in [4], at least one allows for a flat space asymptotic algebra to retain non-linearities, thus providing the first example of a super-symmetric non-linear BM S 3 algebras. This paper is organized as follows: in section 2 we extend the chiral boundary conditions of [9,11] to the super-symmetric setting -first to the simplest minimal supergravities and then to extended super-gravities. In section 3 we embed the results of [8] to extended super-gravities. In section 4 we take a suitable flat limit of the results in [4] and obtain BM S 3 embedded in extended asymptotic super-algebra. In appendix B we study a generalization of the conformal boundary conditions in [8] with non-vanishing boundary curvature. This will further be analysed in the Chern-Simons formulation of 3d gravity. We conclude with discussion of our results in section 6.

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2 Holographic chiral induced supergravities Motivated to describe the 2-dimensional Chiral Induced Gravity of Polyakov holographically, in the second order formulation of AdS 3 gravity the following boundary conditions [9,11] were proposed for the metric where x + , x − are the boundary coordinates and r is the radial coordinate with the asymptotic boundary at r −1 = 0. After imposing the conditions coming from variational principle that allowed the function F (x + , x − ) to fluctuate, it was shown in [9] that the asymptotic symmetry algebra consists of one copy of Virasoro and one copy of sl(2, R) current algebra with level k given by l/4G. Those boundary conditions were translated to the first order Chern-Simons (CS) formulation of AdS 3 gravity in [12].
In this section we generalise the CIG boundary conditions of [9,11] to supersymmetric contexts.

N = (1, 1) supergravity in AdS 3
We first begin with a simple set-up where we generalise the chiral boundary conditions of [9]. We will work in the Chern-Simons (CS) formulation as the calculations are simpler in this formulation of supergravity in AdS 3 . The graded Lie algebra of interest here is osp(1|2) which contains in it the bosonic sl(2, R). The commutation relations are as follows: where σ a s label the space-time sl(2, R) symmetry algebra, while the R ± are the fermionic generators. The gauge-invariant, bi-linear, non-degenerate metric on the algebra is: 3) The N = (1, 1) supergravity action can be written as a difference to two Chern-Simons actions [42,43], where the gauge algebra for the two CS terms is osp(1|2). 3 The equation of motion imposes flatness condition on the two gauge fields valued in the adjoint of osp(1|2). In order to obtain the required generalization, we first notice the form of the gauge fields corresponding to the chiral boundary conditions written down in [9,12]. We take them to be the following gauge fields A &Ã as given below: can a priori depend on both the boundary coordinates x ± , but flatness of the gauge field forces them to be of a specific form. Note that the r dependence only enters via finite gauge transformation b. It is easy to check that these also yield the same chiral boundary conditions as in [9] on the metric but the metric will no longer be in Fefferman-Graham gauge. Therefore these would also correspond to the chiral boundary conditions studied therein.
Here we note that the fluctuating field at the boundary comes from f (+) i.e. coefficient of σ + dx + component ofã + in (2.5). 4 The components ofã − play the role of sources, i.e. functions that need to be specified like a chemical potential. We further notice that all theã + components are a priori turned on and are determined in terms of f (+) . On the other hand, theÃ − component with leading r dependence is fixed to be −1, while only the sub-leading component, σ − ofã − is allowed to have coordinate dependence.
Therefore taking a cue from the above observation, we propose the following fall-off conditions for the supersymmetric case: Here the dx − component of the gauge field 1-formã is that of a super-gauge field corresponding to Dirichlet boundary condition as given in [44,45]. All functions above are a priori functions of both the boundary coordinates. The equations of motion imply flatness:

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For the left gauge field a this implies that the functions are independent of the x − coordinate. i.e. ∂ − a = 0.
While for the right gauge fieldã components, equation of motion allows one to express the dx + components ofã in terms of dx − components: whereÃ is treated as an algebra valued 1-form on the boundary; the first subscript labels the sl(2, R) algebra component while the second labels the space-time component of the 1-form. The remaining relations imposed by equations of motion are differential equations relating the dx + components and the dx − components ofã. These are interpreted as Ward identities for the boundary theory: Here, conventionally (according to Brown-Henneaux analysis) theÃ ++ ,ψ + , are sources i.e. chemical potentials coupling to conserved currents labelled byL,ψ − respectively. But our boundary conditions would require that the currentsL,ψ − play the role of sources. We will later choose these sources such that global AdS 3 , corresponding toL = − 1 4 andψ = 0, is part of the space of bulk solutions.
The boundary terms required to make the set of solutions with fixedL andψ variationally well defined are: whereL (0) ,ψ (0) are some desired fixed functions. The variation of the total action therefore reads: Here, choosing the fluctuations δÃ ++ and δψ + to vanish would impose Dirichlet type boundary condition, whereas allowing for their fluctuations demands settingL =L 0 and ψ − =ψ 0 to satisfy the variational principal. We choose the later case as this implies fields fluctuating on the boundary of AdS 3 as we seek. In [7] new boundary conditions were discovered by allowing the boundary metric to fluctuate in the one of the light-cone directions ds 2 = − 1 2 dx + dx − + g ++ (x + )(dx + ) 2 . This basically translates to lettingÃ ++ be a function of x + alone. The authors in [7] discover that such boundary conditions lead to a different asymptotic algebra of a U(1) current times a Virasoro.
One may try and generalise the boundary conditions of Compère et al [7] by choosing only x + dependence forÃ ++ for arbitrary values of the sourcesL 0 andψ (0)− ; but on solving JHEP02(2019)168 the above Ward identities, one sees thatψ + cannot just be a function of x + . Therefore allowingψ + to depend on x − leads to generalisation of the boundary conditions of [7] to the super-gravity case. As we will see this case can be thought of as a special case of more general analysis of the next subsection 2.2.

Charges and symmetry algebra
We first solve the equation of motion (2.10) for a particular value ofL = − 1 4 andψ − = 0. This choice ofL allows for global AdS 3 to be one of the allowed solutions. This implies that the boundary fieldsÃ ++ andψ + take the following form: The residual gauge transformations that leaveã form-invariant are: (2.14) One can solve for the residual gauge transformation parameters: The left gauge field components are independent of x − and the corresponding residual gauge transformations are parametrised by: where all the parameters are determined in terms of ζ − (x + ) and ε − (x + ). We observe that the arbitrary functions specifying the space of gauge fields a andã and the space of residual gauge transformations are specified by functions of x + alone. Therefore the x + dependence of the functions will be suppressed here onwards. The variation of the above parameters under the residual gauge transformations for the right sector are: Similarly those for the left sector are: The charges corresponding to these transformation (which can be computed using the formalism of [46,47]) are given by [4]: The above charge can be integrated and is finite. The charges for the two gauge fields decouple: This charge is the generator of canonical transformations on the space of solutions parametrized by set of functions F via the Poisson bracket.
Therefore the Poisson bracket algebra for the right sector has to be: While the fermionic Poisson brackets are: where α = 2π k . Rescaling the above currents to:

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and expanding them in the Fourier modes in x + yields the following commutators: and anti-commutators: This yields the familiar affine sl(2, R) current algebra at level k = c/6 with two additional fermionic current parametrized by χ,χ. Here a = 0 is the NS sector and a = 1/2 is the Ramond sector. 5 Of these only the NS sector admits a maximal finite dimensional subsuperalgebra. These fermionic currents form a semi-direct sum with the sl(2, R) current algebra elements with their anti-commutators yielding the latter.
Similarly the left sector yields the familiar Brown-Henneaux result [44]. We first redefine the currents by suitable scaling: After these redefinitions one gets the following Poisson algebra: The Fourier modes for the above algebra satisfy the following Dirac brackets: Here unlike the right sector a = 0 implies Ramond and a = 1/2 implies the NS boundary condition for the fermions. This is the Virasoro algebra with an affine super-current parametrised by ψ + .

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2.2 Generalization to extended AdS 3 supergravity Next, we will generalise the analysis of the last subsection to extended supergravity setting with negative cosmological constant. Since the number for gauge field components would now increase to include the ones corresponding to the internal bosonic directions, it would be interesting to see whether the chiral boundary conditions proposed admit a unique nontrivial generalisation. That is, we would seek boundary fall-off conditions for the gauge field components such that all of them admit a fluctuating mode at the AdS 3 boundary with the solutions elucidated in the pure gravity case [9] being a subset. The N = (4, 4) case which would be of interest for realising these boundary conditions in a string theoretic setting would therefore be a special case of this analysis. One first begins by classifying the superalgebras possible in AdS 3 [4,48]. Let G denote the graded Lie algebra, such that G = G 0 ⊕ G 1 , where G 0 denotes the even part whereas G 1 denotes the odd part. The even part, G 0 must contain a direct sum of sl(2, R) and an internal symmetry algebra denoted byG. The fermions must transform in the 2-dimensional spinor representation of sl(2, R). The dimension of the internal algebraG is denoted by D while the fermions transform under a representation ρ (dim ρ = d) ofG which is real but not necessarily unitary. This is possible in a set of seven cases, whose list can be found for example in [4]. 6 The gauge field is then written as a super gauge field valued in the adjoint of any of the allowed (graded Lie) superalgebra: The gauge field as written above, separates as a sum of sl(2, R),G and fermionic 1-forms. The parameters A a and B a commute, 7 while ψ ±α are anti-commuting Grasmann parameters. The gauge field Γ is a G valued 1-form. The supergravity action is then given as the difference of two Chern-Simons action at level k written for two such gauge fields Γ andΓ. (2.31) We will concern ourselves with the case where both Γ andΓ are valued in the same G. The cases where this is not so leads to chiral action and can be regarded as one of the ways to generate chiral asymptotic symmetries.

Boundary conditions
Now, we would like to impose boundary conditions on the gauge fields-just as in the higher-spin case [12], which generalize the chiral boundary conditions on pure AdS 3 of [9].
Here we would allow one of the super-gauge fields-Γ, to obey the boundary conditions of [4] i.e. consistent with Brown-Henneaux, while proposing new boundary conditions on

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Γ. The BH type boundary conditions on Γ are already analysed in [4] and so we skip the details here. The chiral boundary conditions onΓ generalised to extended supergravity will be analysed in detail below. The fall-off conditions in terms of the gauge fields are: where b = e σ 0 ln(r/ℓ) , refer to appendix A.1 for more details. The flatness of the super-connectionΓ requires that the functions therein satisfy the following set of differential equations: These are the Ward identities expected to be satisfied by the induced supergravity theory on the boundary. Generalising the minimal case of the previous subsection we choose boundary conditions such that global AdS 3 withL = − 1 4 andB = 0 =ψ is a part of the space of bulk solutions. The boundary term to be added so as to make the right sector with fixedL,B andψ variationally well-defined is given by: This results in the following desired variation of the total action: Here, one has an option of choosing δ() functions to vanish at the AdS asymptote, implying a Brown-Henneaux type boundary condition whereÃ ++ ,B a+ ,ψ +α act as chemical potentials. Or, alternatively, treatL (0) ,B (0)a ,ψ (0)−α as chemical potentials allowing A ++ ,B a+ ,ψ +α to fluctuate -thus describing a theory of induced gravity on the boundary. In our present case, we choose the later by fixingL (0) = −1/4,B (0)a = 0 =ψ (0)−α . Thus the variational principle is satisfied for configurations withL = − 1 4 andB a− = 0 =ψ −α which describes global AdS 3 .

Charges and symmetry algebra
Just as in the previous subsection, one needs to find the space of gauge transformations that maintains the above form of the gauge fields, thus inducing transformations on the functionsÃ a+ ,B a+ ,ψ +α , L, B a , ψ +α which parametrize the space of solutions. Once this is achieved, one can define asymptotic conserved charge associated with the change induced by such residual gauge transformations on the space of solutions.
The left sector. The analysis of the left sector is exactly as in [4] which we skip giving the details of here. One basically gets the super-Virasoro with quadratic nonlinearities as the asymptotic algebra for the modes of parameters labelling the left sector gauge filed Γ.
with a = 0 being Ramond and a = 1/2 being NS boundary conditions on the fermions. This is the nonlinear super-conformal algebra. The central extension is k = c/6, and is the same for all the seven cases listed in the table of [4] mentioned previously. This algebra, although a supersymmetric extension of the Virasoro algebra, is not a graded Lie algebra in the sense that the right hand sides of the fermionic (Rarita-Schwinger) anti-commutators contain quadratic nonlinearities in currents for the internal symmetry directions.
The right sector. The analysis of the right sector is similar to the one covered in the N = (1, 1) case whose details can be found in the appendix A.
with the reversed identification of a = 0 corresponding to NS and a = 1/2 corresponding to Ramond boundary conditions on the fermions. This is the affine Kač-Moody super-algebra.
Here, it is evident that the central extension to the sl(2, R)-current sub-algebra spanned by (f, g,ḡ) is k = c/6. The quadratic nonlinearities that occur in the super-Virasoro are not present here. Also, as in the N = (1, 1) case the right & left sector with NS boundary conditions on the fermions give a maximal global subalgebra.
Thus demanding that one considers all types of fields (Ã,B,ψ) have fluctuating components on the boundary of asymptotic AdS 3 , we have constructed a unique generalisation of the boundary condition studied in [9] to extended supergravity in asymptotically AdS 3 spaces. In doing so we uncovered the expected super-Virasoro algebra with quadratic nonlinearities for the left sector and a Kač-Moody super-current algebra at level k = c/6 for the right sector. Here, we have demanded as before that the global AdS 3 remains in the space of allowed solutions.
There exists a consistent truncation of this superalgebra with g andḡ set to zero. This would correspond to a supergravity generalisation of Compère et al [7]. In that case the residual gauge transformations could be appropriately chosen so that they do not turn on g andḡ. 8 It then corresponds to a super-extension of the u(1) Kač-Moody algebra of [7].

Holographic induced super-Liouville theory
To describe the 2-dimensional Induced Gravity in the conformal gauge holographically one would start with boundary conditions of AdS 3 gravity that allow the conformal factor of the boundary metric to fluctuate. Such boundary conditions proposed by Troessaert [8] JHEP02(2019)168 look like: , yielding the boundary metric to have zero curvature. Therefore, the boundary conditions proposed by [8] fix the boundary metric to be flat up to a conformal factor; and further demanding that the boundary metric has vanishing Ricci curvature. 9 We now turn to generalising boundary conditions proposed by [8] to extended supergravity in AdS 3 . For this we again use the CS formulation of AdS 3 gravity. The notations and conventions below are again taken from Henneaux et al [4] and are summarised in appendix A.
As the boundary conditions in this case are non-chiral the analyses for the two gauge fields are identical; hence we will give the details of only one of the gauge fields -Ã. One begins with an ansatz, The equation of motion, ∂ +ã− − ∂ −ã+ + [ã + ,ã − ] = 0 is readily satisfied. The above form of the gauge field ansatz doesn't need extra boundary terms added to the Chern-Simons action to make it variationally consistent. This is made apparent due to the fact that the gauge fieldã as an 1-form only has a dx − component along the boundary while the fluctuation of the action yields, We now look for the space of gauge transformations which keep the above 1-formã forminvariant.
Solving these constraints onΛ, we get The variations these gauge transformations induce on the functions parametrizing the 1formã are: Associated to the above fluctuations are infinitesimal variations of a well defined asymptotic charge.
The next step is to be able to write the above change in the charge such that it is a total variation δQ so that δ can be taken out of the integral and (the charge) can be integrated from a suitable point (vacuum) on the solution space to any arbitrary point. Therefore, it is important to recognise field independent parameters parametrising the gauge transformations and accordingly functions of the fields parametrising the space of solutions on which a phase space can be defined via the Poisson brackets induced by the integrated charge. The above expression for / δQ can be simplified if one redefines ψ ±α =Ψ ±α e ±Φ ,ǫ ±α = ε ±α e ±Φ ,ξ + =Ξ + eΦ.
Therefore the total integrated charge is The redefined gauge transformation parameters in (3.9) are therefore to be considered field independent. Also, one sees that it isΦ ′ and notΦ which is appropriate for defining the phase space structure on the space of solutions. The variations of the redefined fields in terms of the field independent gauge parameters are: This leads to the following Poisson brackets amongst the solution space variables The first Poisson bracket amongκ s can be easily recognised as that of the Witt algebra (Virasoro without the central extension). As was done in [8] the phase-space variableΦ ′ can be used to generate the central term in the Witt algebra ofκ by redefininĝ κ =κ + α cΦ ′′ . (3.14)

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After further rescalingκ → k 2π one can show the central term in the Virasoro ofκ to be 2α c (α c +1)k/(2π); which for α c = − 1 2 ± 1 √ 2 yields the central extension of the Virasoro found in Brown-Henneaux analysis. The redefined chargesκ can then be shown to be associated with the gauge transformation parameters which yield the Brown-Henneaux symmetries in the Chern-Simons formulation of AdS 3 gravity. One can mode expand the above Poisson brackets and find the relevant algebra. Depending on integer or half-integer moding of the fermionic charges the algebra will either fall into Ramond or the NS sector respectively.
The above algebra is obtained at the large-c limit and one would expect a finite c completion of this algebra. Also, the studying the representation theory of the above algebra can tell us which theories could exhibit such symmetries. We leave these questions for future investigations. 10 4 Flat limit of extended AdS 3 supergravity In this section we deviate from the main theme of the paper so far to suggest a possible flat limit of extended AdS 3 supergravity studied by Henneaux et al [4]. In [4] a thorough asymptotic symmetry analysis of all allowed but arbitrary AdS 3 supergravities was carried out. The boundary conditions on these AdS 3 supergravities are the most general ones which are consistent with the boundary conditions proposed by Brown and Henneaux for AdS 3 gravity. The boundary metric in [4] is that of 2d flat Minkowski. The asymptotic symmetries in the case of such generic AdS 3 supergravities was found to be a super extension of the Virasoro algebra with quadratic non-linearities. 11 In this section we answer the question whether similar boundary conditions exist in flat 3d Minkowski space and what could be their asymptotic symmetry algebra. We answer this by taking a suitable flat space limit: l → ∞; of the analysis in [4]. Since the analysis in [4] maintains the boundary metric to be of 2d Minkowski, the l → ∞ limit would correspond limits of such supergravities with a flat boundary metric in AdS.
Such limits were first considered by [37] and subsequently investigated in [38][39][40][41]. Here the specific AdS 3 supergravities were considered which did not yield any non-linearities in their asymptotic symmetry algebra. In this section we show that certain flat limits exists which allow us to retain the non-linearities found in [4]. We begin by first analysing different l → ∞ limits of the graded algebras considered in [4]. In principle one can then take any of these limits on the solutions of AdS 3 supergravity in [4] and therefore proceed analogously to determine boundary conditions, compute asymptotic charges and determine the asymptotic symmetry algebra [38][39][40][41]. We however choose one out these many possible limits which enable us to retain the quadratic non-linearities even in the flat space.
The sl(2, R) algebra we use here differ from that in [4] as L ± = ±σ ∓ and L 0 = σ 0 . Therefore the gauge fields of [4] after having their r-dependence stripped off take the form (4.1)

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We then perform the finite gauge transformation given by Barnich et al. [37,38]: Note the asymmetric manner in which the two gauge fields are being treated under the finite gauge transformation. One can proceed in an even handed manner but this doesn't yield the known metric in the l → ∞ limit at I + in BMS gauge. 12 After the gauge transformations the gauge fields take the form The solution to the flat space equation of motion would be given by the sum A = A + + A − in the limit l → ∞ after replacing x ± = u l ± φ.

l → ∞ limit of the algebra
We redefine (such as in [49]) the spacetime algebra elements as follows: Thus yielding where P i are translations that commute among themselves only in l → ∞. We further choose to define f (l)S ±α = (R ±α +R ±α ),f (l)S ±α = (R ±α −R ±α ), here the functions f (l),f (l), g(l) &ḡ(l) are as of yet undefined. The commutators independent of these functions are

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While the ones which do depend on them are and This is the term which is indeed responsible for the quadratic terms in the r.h.s. in the asymptotic symmetry algebra of [4]. The internal bosonic symmetries scale like Now we would like to specify the functions f (l),f (l), g(l),ḡ(l) as powers in l such that a consistent l → ∞ limit exists for the algebra i.e. the r.h.s. of the above algebra should not diverge as l → ∞.
Different limits as l → ∞. We start by observing that (4.10) demands ff ≥ l and 2 nd eq. of (4.12) demandsḡ 2 ≥ g. We will categorize them as follows with (4.5) and (4.7) holding in all cases below: Case I: f =f .
The algebra studied in [39][40][41] fall in this last category. Notice in the all above cases the anti-commutators never produce a J i since both f &f scale with l. This will not be true in the next case.

Charges and symmetry algebras
We now proceed to express A = A + + A − for the different cases mentioned above.
where M = κ +κ, N = l(κ −κ) and x ± = u l ± φ. We have furter supressed the coordinate dependencies for simplicity. From here on one would have to make a consistent choice of functions {f,f , g,ḡ} as specified in the last subsection.
We will workout the limits in the last case above, namely, Case II: f =ḡ = l &f = g = 1. In order that the terms in (4.18) do not blow-up in the l → ∞ limit we redefine recalling that κ, B + a and Q +α only depended on x + whileκ, B − a andQ −α on x − , and given that x ± = u l ± φ we get The gauge field (4.18) after taking the limit l → ∞ looks like The r-dependence can be further gauged away by working with a = e −rP 0 A e rP 0 + P 0 dr = P 0 (M du + N dφ) + P 1 du + J 0 M dφ + J 1 dφ where the l → ∞ limit implies The infinitesimal gauge transformation that keep the above gauge field form-invariant is The corresponding fluctuations are In order to express the difference of two CS theories-each valued in sl(2, R); as one CS theory valued in sl(2, R) × sl(2, R), the (super) trace (Killing metric) in one must differ from the other in sign. Therefore the non-zero Killing metric components are: The 1 l in the super trace exactly cancels the l in front of the action integral. The symplectic structure is then given by Integrating on the space of fluctuations we get (4.28)

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where we have used function which only depends on φ, defined as; Evaluating the Poisson brackets one finds:

Now we would like to shift
This is a shift by Sugawara tensor as it leaves invariant the Poisson brackets between M and N (0) , and between N (0) and N (0) . The Poisson brackets then become (4.32) The modes of which satisfy the following algebra in terms of Dirac brackets 13 13 Here we have absorbed the factors of

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The nonlinearity on the r.h.s of the anti-commutator of fermionic charges is proportional to α which is the same as the one that occurs in the super-conformal algebra obtained in [4]. In (4.33) m, n in the fermionic charges Q andQ being integers correspond to the Ramond sector, while m, n being half-integers correspond to Neveau-Schwarz sector. The NS sector admits the full global sub-algebra spanned by }, while the Ramond sector only contains sub-algebra spanned by {M 0 , N 0 , B a 0 ,B a 0 }. Using the second last anti-commutation relation above one can clearly see for the Neveau-Schwarz case that the Minkowski vacuum corresponds to M 0 = − 1 16G with any other excitation having higher M 0 . This is exactly the bound obtained in [40] after having scaling M n such that the first commutation relation above matches the one in [40]. Therefore Minkowski vacuum clearly is a ground state for such boundary conditions.
As the above algebra has been obtained in the large c limit, one could investigate how they look like at finite c. Further, studying the representation theory of this algebra can tell us which theories can admit such symmetries. We leave these questions for future investigations. 14

Conclusion
In this paper we have surveyed, generalised and studied supersymmetric extensions of the boundary conditions of AdS 3 and R 1,2 supergravities and their asymptotic symmetry algebras.
In the first part of this paper we studied the generalisation of the non-dirichlet type boundary conditions introduced in [8,9] to AdS 3 (extended) supergravity contexts. The extension of [9] reveals the existence of Kaĉ-Moody current of the relevant extended superalgebra concerned which is linear in its generators. This is not the case for the super-Virasoro algebra found in extended supergravity in [4]. In studying the chiral case we do uncover the Ward identities of the boundary chiral induced extended super-gravities which occur in [18,19]. The supersymmetric boundary conditions studied here should be considered essential for exploring how such non-Dirichlet boundary conditions may occur in string theoretic settings.
In the second part we generalised asymptotically flat boundary conditions of R 1,2 gravity to, again (extended) supergravity theories. In the latter case we found superalgebras

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containing BM S 3 that are nonlinear. These are the analogs of the nonlinear superalgebras of 2d CFTs studied long time ago [50][51][52][53][54][55][56][57][58][59][60]. The nonlinear extensions of Virasoro algebra found some applications in the AdS/CFT context before (see for instance [61]). It is conceivable that the nonlinear BMS 3 superalgebras of the kind uncovered here will find suitable applications. A systematic classification of superalgebras containing BM S 3 is still under progress. The R 1,2 supergravity calculation above predicts/suggests the existence of such nonlinear superalgebras whose large-k limits are the ones found here. It will be interesting to uncover such nonlinear superalgebras on the lines of [50][51][52][53][54][55][56][57][58][59][60]. There are other types of boundary conditions for R 1,2 gravity such as [62,63] and it will be interesting to generalise such boundary conditions also to supergravity contexts.
In section B in the appendices we imposed conformal boundary conditions such that the conformal factor of the boundary metric obeys the Liouville equation ∂ + ∂ − log F = 2χF , which in general allows for non-vanishing boundary curvature. We uncovered the asymptotic symmetry algebra consisting of two copies of Virasoro corresponding to BH boundary conditions and two more copies of Virasoro with c = −3ℓ/3G corresponding to the stress-tensor modes of the Liouville field F on the boundary. More recently people have uncovered many interesting generalisations of the non-dirichlet boundary conditions considered in the text. For example the ones in [14] give the KdV equation as the bulk equation of motion. The Liouville case presented here is in similar spirit. It should be possible to extend these boundary conditions to supersymmetric contexts too.

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• The sl(2, R) generators are denoted as before by (σ 0 , σ ± ) The Killing form on sl(2, R) is: • The generators ofG which commute with σ a s are: 15 • The fermionic generators are denoted by R ±α , where ± denotes the spinor indices with respect to the sl(2, R) and α (Greek indices) denotes the vector index in the representation ρ ofG.
Since the underlining algebra is now promoted to a graded Lie algebra, its generators satisfy the generalized Jacobi identity. The three-fermion Jacobi identity thus yields an identity for the matrices in the representation ρ of the internal algebraG: The super-traces defined above are consistent, invariant and non-degenerate with respect to the super-algebra defined above and would be used in defining the action and the charges.

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The action. The super Chern-Simons action is defined as: The above integration is over a three manifold M = D × R, where D has a topology of a disk. The level k of the Chern-Simons action is related to the Newton's constant G in three dimension and the AdS length ℓ through k = ℓ/(4G). The product of two fermions differs by a factor of i from the standard Grasmann product ((ab) * = b * a * ).This basically requires one the multiply a factor of −i where ever η αβ occurs, and where ever d−1 2Cρ (λ a ) αβ occurs while evaluating anti-commutator between fermionic generators in the calculations below. 16 In the Chern-Simons formulation of (super-)gravity, the metric (and other fields) which occur in Einstein-Hilbert (Hilbert-Palatini) action are a derived concept. The equations of motion for the Chern-Simons action can for example be satisfied by gauge field configurations which may yield a non-singular metric. There fore one has to make sure that such configurations are not considered in the analysis. The supergravity action for the above super-algebra can be written in full detail yielding the action in the Hilbert-Palatini form: The square brackets denote the two-component sl(2, R) spinor representations. The spin covariant operators D andD are: From the form of the above action it is quite evident that the analysis done in the Hilbert-Palatini formulation of supergravity would be quite cumbersome if not difficult. Further, 16 This basically so because the product of two real Grasmann fields is imaginary. This is equivalent to using instead of the one stated in the commutation relations of the extended super-algebra.

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it was found that computation of the asymptotic charge associated with gauge transformations which vary the super-gauge field at the AdS asymptote 17 via the prescription of Barnich et al [46] for the above form of the action is too difficult. The same prescription of computing asymptotic charges in the Chern-Simons formalism yields a know expression for asymptotic charge in Chern-Simons theory. Therefore we proceed as before with the analysis in the Chern-Simons prescription.

A.1 Boundary conditions
The fall-off conditions in terms of the gauge fields are: Here the dx − component of the gauge fieldã one form is that of a super-gauge field corresponding to Dirichlet boundary condition as given in [4]. All functions above are a priori functions of both the boundary coordinates. The equation of motion-as mentioned earlier, is implied by the flatness condition imposed on the two gauge fields. For the right gauge field this implies that the functions are independent of the x − co-ordinate. i.e. ∂ − a = 0.
For the left gauge field we would like to use the equations of motion to solve for thẽ a + components. This gives theã + components in terms ofÃ ++ ,B a+ ,ψ +α+ and theã − components:Ã Provided they satisfy the following set of differential equations:

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These are the Ward identities expected to be satisfied by the induced gravity theory on the boundary. We will later choose the() functions such that global AdS 3 is a part of the moduli space of bulk solutions.i.e.L = −1 4 andB = 0 =ψ. In the following analysis we will consider the sources i.e. the bared functions as constants along the boundary directions. There is no need to assume this, and we have done so only for simplicity in the expressions for change in the moduli space parameters. Either ways, demanding that the bared functions-L,B,ψ, be treated as sources which determine aspects of the theory requires adding of specific boundary term to the bulk action. As explained previously, this is done so that the required set of bulk solutions obey the variational principle.
The boundary term to be added is given by: This implies the following desired variation of the total action: 15) In our present case, we would be choosing the later by fixingL = −1/4,B 0a = 0 = (ψ 0 ) −α . Thus the variational principle is satisfied for configurations withL = −1 4 andB a− = 0 = ψ −α− which describes global AdS 3 .

A.2 Charges and symmetries
Just as in the previous sections, one needs to find the space of gauge transformations that maintains the above form of the gauge fields, thus inducing transformations on the functionsÃ a+ ,B a+ ,ψ +α+ , L, B a , ψ +α+ which parametrize the space of solutions. Once this is achieved, one can define asymptotic conserved charge associated with the change induced by such residual gauge transformations on the space of solutions. For the boundary conditions to be well defined, this asymptotic charge must be finite and be integrable on the space of solutions.
Right sector. The analysis of the left sector i.e. on the gauge field Γ is exactly the one done in [4]. Here we analyze the right sector. For the choice ofL = − 1 4 ,B = 0 =ψ the eom can be solved and the solutions can be parametrized as below: We would now seek the residual gauge tranformation parameters that would keep the above form of the gauge fieldΓ form invariant. The residual gauge transformations are generated byΛ = ξ a σ a + b a T a + ǫ +α R +α + ǫ −α R −α with the constraint that δã − = 0:

JHEP02(2019)168 B Holographic Liouville theory
In this appendix we begin with generalising the boundary conditions of [8] so as to let the boundary metric have a non-vanishing curvature. The motivation is to provide a holographic description of the Liouville equation (instead of the free-field equation as in [8]). The boundary conditions in [8] for the metric components look like: where F (x + , x − ) satisfies ∂ + ∂ − log F = 0, yielding the boundary metric to have zero curvature. In contrast we impose on F the generic Liouville equation: ∂ − ∂ + log F = 2χF . Here x + , x − are treated to be the boundary coordinates and r is the radial coordinate with the asymptotic boundary at r −1 = 0. One can write a general nonlinear solution of AdS 3 gravity in Fefferman-Graham coordinates [64] as: Therefore, the full set of nonlinear solutions consistent with our boundary conditions is obtained when −− =κ(x + , x − ), db , where in the last line g cd cd inverse. Imposing the equations of motion R µν − 1 2 R g µν − 1 l 2 g µν = 0 one finds that these equations are satisfied for µ, ν = +, −. Then the remaining three equations coming from (µ, ν) = (r, r), (r, +), (r, −) impose the following relations: In general the equations can be solved for κ andκ in terms of F as follows: We now have to specialise to some subset of solutions such that we have Liouville equation satisfied by F . For this observe that when ∂ − κ = ∂ +κ = 0 we have σ = χ F for some constant χ. Then the ward identity σ = 1 2 ∂ + ∂ − log F reads:

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which is the famous Liouville's equation. So if we add boundary terms such that we keep σ = χ F then it follows that ∂ − κ = ∂ +κ = 0. For this, it is useful to note that the boundary (holographic) stress tensor T ij for the class of metrics we have is proportional to Taking the trace with respect to the boundary metric gives Therefore the constraint σ = χ F simply translates into demanding g µν (0) (g µν ) = −20 χ. The variation of the action along the solution space is So we add the boundary term: such that the total variation of the action is Now we could choose either δF = 0 (Dirichlet) or σ = χ F (Neumann). Choosing the latter gives rise to the Liouville equation as we desire.

B.1 Classical solutions and asymptotic symmetries
It is well known that the general solution of the Liouville equation ∂ + ∂ − log F = 2χ F is given by The χ = 0 case was considered by [8]. We now proceed to obtain the asymptotic symmetries for the above boundary conditions. The residual diffeomorphisms that the leave the metric in the above form are: the subleading functions in r are all determined from the boundary values of components. For convenience of calculation, let us introduce a field Φ = log(χF ). The equation for Φ then is: (B.14)

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The first order variation of the above differential equation satisfies: therefore δΦ satisfies the same equation as ξ r . Reading off δΦ from the general solution of F and labelling δf = gf ′ and δf =gf ′ , the expression for ξ r reads: We now use the covariant prescription prescribed in [46] to compute the asymptotic conserved charges assiciated with the above diffeomorphisms. The infinitesimal change in the asymptotic charge under such diffeomorphisms is given by: The above charge is required to be integrable on the space of solutions. It can be shown to be so upto terms which vanish due to the integrand being a total derivative in the angular co-ordinate φ.
The total integrated charge can be written again (after similarly throwing away total derivatives in φ): The factors multiplying g andg can be recognized as the stress-tensor modes of the Liouville theory. One can proceed to construct the classical Poisson brackets by demanding that the above charge gives rise to the fluctuations of the metric components F , κ andκ produced by the residual diffeomorphisms (B.13). The change in the parameters under such boundary condition preserving gauge transformations are:

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The variation in F can be cast in terms off andf as: Therefore the space of classical solutions allowed by the proposed boundary condition (3.1) are parametrized by the functions (f ,f, κ,κ), where as the diffeomorphisms that would keep the metric under Lie derivative in this form are parametrized by (g,g, ξ + (0) , ξ − (0) ). Redefining functions as:f the Poisson algebra reads: f (x + ′ , κ(x + )) = 0. (B.23) Similarly for the left sector, which commutes with the right sector. This shows that central charge associated with the Virasoros of the Liouville theory to be negative of the central charge of the Virasoros obtained from the Brown-Henneaux boundary conditions. The space of bulk geometries allowed by the Brown-Henneaux boundary conditions is contained in the space of solutions allowed by the above boundary condition. If one begins with a generic 3d asymptotically locally AdS 3 metric in the Fefferman and Graham gauge then the residual diffeomorphisms are the ones which would generate the Diff×Weyl 18 for the boundary metric. Restricting the boundary metric to have the form as the one in [9] restricts the residual diffeomorphisms further to have an algebra chiral-Diff×Witt to the leading order. Sub-leading order corrections to the residual diffeomorphisms further restrict it to an sl(2, R)×Virasoro. Similarly, if on the other hand one imposed the above boundary conditions then the Diff×Weyl reduces to two copies of left-right Virasoro with opposite central charges.
The analyses here were done in the second order formalism of gravity using the Einstein-Hilbert action. For the sake of completeness next we look at the same problem from the first order formulation of AdS 3 gravity.

B.2 Liouville boundary conditions in CS formulation
For the case of AdS 3 the gauge algebra of the CS theories is sl(2, R). Here k = ℓ/4G. We first find the gauge fields which yield the metric proposed in (3.1). As before, one mods out the radial r dependence with a finite gauge transformation, The corresponding analysis in the second order formulation made use of the Fefferman-Graham (FG) gauge for the metric. One may impose this gauge on the above gauge fields by demanding that the metric corresponding to them be in the FG gauge. This is not strictly necessary but this has a benefit of reducing the number of solution space parameters by those ones which do not contribute to the asymptotic charge. Imposing the FG gauge on the metric translates to the following condition on the gauge field components: The above expression for charge turns out to be the same as (B.18). Since the expressions for the fluctuations and the charges are the same in both the formalisms, we get the same asymptotic symmetry algebra as expected.
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