A note on the $\mathcal{N}=2$ super-$\mathcal{W}_3$ holographic dictionary

This is a long-overdue companion paper to arXiv:1512.00073. We study the relation between $sl(3|2)$ Chern-Simons supergravity on AdS$_3$ and two-dimensional CFT's with $\mathcal{N}=2$ super-$\mathcal{W}_3$ symmetry. Specifically, we carry out a complete analysis of asymptotic symmetries in a basis that makes the superconformal structure transparent, allowing us to establish the precise dictionary between currents and transformation parameters in the bulk and their boundary counterparts. We also discuss the incorporation of sources and display in full detail the corresponding holographic Ward identities. By imposing suitable hermiticity conditions on the CFT currents, we identify the superalgebra $su(2,1|1,1)$ as the appropriate real form of $sl(3|2)$ in Lorentzian signature. We take the opportunity to review some of the properties of the $\mathcal{N}=2$ super-$\mathcal{W}_3$ conformal algebra, including its multiplet structure, OPE's and spectral flow invariance, correcting some minor typos present in the literature.


Introduction
It is a well-known fact that higher spin algebras obtained from the Hamiltonian reduction of Kac-Moody current algebras can be realized in terms of flat connections in Drinfeld-Sokolov form [2]. This fits the general scheme of higher spin dualities involving three-dimensional Chern-Simons theory and two-dimensional CFT's with W-symmetry, where some recent and notable examples are the relation between sl(N ) Chern-Simons theory and CFT's with W N symmetry [3,4], and, more generally, the duality between hs[λ] Chern-Simons theory and CFT's with W ∞ [λ] symmetry [5,6]. See [7] for a review on this subject. One appeal of these higher spin dualities is that they serve as "simple" examples of holography: the abundance of symmetry opens an opportunity to build solvable and tractable instances of the AdS/CFT correspondence. Higher spin dualities also allow us to introduce non-linear and non-geometrical features classically, via the Chern-Simons formulation of the gravitational side of the correspondence. These are features we expect to arise in string theory dualities, but are generically difficult to quantify.
Our focus in this note is on supersymmetric examples of higher spin holography, with the specific goal of building a detailed dictionary between flat connections in Chern-Simons theory and the corresponding currents and sources in the dual supersymmetric W-algebra. As a complement to [1], we study what it is perhaps the simplest three-dimensional higher spin supergravity, namely, Chern-Simons theory based on two copies of the sl(3|2) gauge algebra. 1 As we prove in detail below, this theory is holographically dual to a CFT with N = 2 super-W 3 symmetry, in the semiclassical (large c) limit. It is important to mention that the relation between sl(3|2) connections and CFT's with W (3|2) symmetry was uncovered long ago in the context of Hamiltonian reduction of current algebras [9]. More recent studies from an AdS/CFT perspective can be found in [10][11][12][13][14]. However, the explicit form of the holographic dictionary between the basic Chern-Simons and CFT variables for this case has not been fully laid out so far. The original reference [9] included a map for the currents (although in a rather cumbersome basis), but it did not provide the corresponding map between transformation parameters, implying in practice that the relation between bulk and boundary sources is missing. By analyzing the CFT Ward identities, the authors in [14] produced a dictionary for both currents and sources, albeit only in the bosonic sector. In what follows we fill these gaps. We perform a complete analysis of the asymptotic symmetries of the sl(3|2) Chern-Simons theory in a basis that makes the supersymmetries transparent and allows to make contact with the modern literature. In addition to the dictionary for the currents, we establish the holographic relation for the infinitesimal transformation parameters as well as for the sources, which is a key ingredient in the thermodynamic analysis of black hole solutions [1]. This enables us to reproduce the holographic Ward identities in full detail.

The N = super-W 3 algebra
We begin by reviewing some basic aspects of the N = 2 super-W 3 algebra, often refered to as W (3|2) . Far from being a comprehensive survey, this section is intended as a summary of the essential features that are pertinent for our purposes. In particular, we review the N = 2 multiplet structure, the OPE's and the spectral flow invariance of the W (3|2) algebra. We also comment on the semiclassical limit needed to make contact with the holographic description and on the hermiticity properties of the generators on the cylinder. Important issues such as the commutator algebra and unitarity and BPS bounds are purposefully omitted (see [1,[18][19][20]). The material below closely follows references [18,21].

General structure and OPE's
Let us briefly recall the structure of the N = 2 super-Virasoro algebra. Besides the stress tensor T , this algebra contains a weight-1 U (1) current J and two weight-3/2 fermionic currents G + and G − with U (1) charges +1 and −1, respectively. In standard conventions, their OPE's are given by For notational convenience, we denote these fields collectively by W (1) = {J, G + , G − , T }. According to [21], the W-algebras that are relevant for the N = 2 version [22] of minimal model holography [23] contain, in addition to the super-Virasoro generators W (1) , an infinite number of higher spin multiplets W (s) , one for every integer spin s ≥ 2. Each of these multiplets compromises four Virasoro primaries, W (s) = W , 2 whose OPE's with the generators of the superconformal algebra are 2 As usual, starting from the superconformal primary W (s) 0 , the other fields in the multiplet are generated by acting with G ± as W (s) (2.14) In particular, the conformal dimensions ∆ and U (1) charges q of the different fields in the spin-s multiplet W (s) are given in As explained in [21,24], the Jacobi identities determine the full structure of the N = 2 super-W algebras up to two free parameters: the central charge c and the self-coupling c 22,2 of the spin-2 primary W (2) 0 . Indeed, the singular part of the OPE of fields in the W (2) multiplet has the schematic form Once a normalization for the currents is chosen, the parameters n 2 and c 22,3 are fixed in terms of c and c 22,2 , which are physically meaningful. The same is true for the structure constants appearing in the OPE's of all the other higher spin multiplets. Quite interestingly, it possible to consistently decouple the multiplets W (s) with s ≥ 3 and truncate the full higher spin algebra to one containing just the super-Virasoro currents W (1) and the spin-2 multiplet W (2) . The resulting algebra is precisely the W (3|2) algebra we want to study, and from now on we focus exclusively on this case. As discussed in [21], the truncation happens when the parameter c 22,3 introduced above is zero, implying that the self-coupling c 22,2 is no longer independent but a particular function of the central charge. Fixing n 2 = c/2 to comply with the standard normalization for the spin-2 current W (2) 0 , the decoupling occurs for Here we have introduced the constant κ so as to make contact with the notation in Romans' paper [18]. As pointed out there, the sign ambiguity corresponds to the freedom of simultaneously flipping the sign of all fields in the W (2) multiplet. Notice that κ is real only for −6 < c < 1 or 3 2 < c < 15. In particular, it is purely imaginary as c → ∞, which makes the representations of the algebra non-unitarity [18].
In order to spell out the explicit form of the N = 2 super-W 3 algebra we adopt the notation W (2) = {V, U + , U − , W } for the currents in the spin-2 multiplet. According to table 1, V has conformal dimension 2 and U (1) charge zero, U ± have weight 5/2 and U (1) charge ±1, and W has conformal dimension 3 and U (1) charge zero. The OPE's among these fields are given in [18] and read 3 3 Notice that W (z) and U ± (z) do not have the standard CFT normalization for a weight-s current, namely, Instead, we have adopted the same normalization as in [18] for ease of comparison. In particular, the currents U ± and W are related to their canonically-normalized counterparts by U ± = √ 5U ± canon and W = 15 2 Wcanon.
± are built out of primary and quasi-primary composite operators, the precise form of which is fixed by the Jacobi identities [18]. We have reproduced them all in appendix A.1. For example, the operator appearing in the V V OPE is Of course, normal ordering is assumed. The full W (3|2) OPE algebra is then given by equations (2.1)-(2.6), (2.7)-(2.14) with s = 2, and (2.17)-(2.23).
It is important to mention that we have detected some minor typos in the G ± U ∓ , G ± W and V U + OPE's in reference [18], which have been corrected in the expressions above. More importantly, however, [18] has typos in the definitions of the composite operators C [1] and Φ [7/2] ± entering in the V W and V U ± OPE's. 4 These are highlighted in (A.5) and (A.15). The correct expressions can be found by checking that all the Jacobi identities are in fact satisfied, a task that we have performed exhaustively using the Mathematica package 5 introduced in [25]. These corrections are also required by the spectral flow invariance of the algebra, as we explain next.

Spectral flow
It is a well-known fact that the N = 2 superconformal algebra enjoys a continuous class of automorphisms called spectral flow. Although usually expressed in terms of modes, we find it more convenient for our purposes to write this symmetry directly in terms of the currents. It is easy to check that the transformations where η is a continuous parameter, leave the OPE's (2.7)-(2.14) invariant. In particular, for η ∈ Z + 1 2 this operation interpolates between the NS sector and the Ramond sector, whereas for η ∈ Z it maps each sector onto itself.
As pointed out in [18], the extension of spectral flow to the W (3|2) case is achieved by letting the spin-2 multiplet currents transform as appearing in the V U ± OPE (cf. (A.15)).
The spectral flow automorphism of the W (3|2) algebra will be an important guiding principle in the upcoming discussion of the holographic dictionary. For the time being, we anticipate that some of the bulk fields turn out to be spectral flow-invariant, so their CFT duals must posses the same property. Here we construct two such quantities. Using (2.25) and (2.26), it is easy to see that (normal ordering is implicit) leading to two invariant quasi-primary operators of dimensions 2 and 3, respectively. As we will see below, these particular combinations of fields appear naturally from the bulk perspective. Coincidentally, it follows from the first relation in (2.28) that A [2] ′ = A [2] , resulting in the invariance of the V V OPE. The fact that the composite A [2] does not change under spectral flow makes this particular check fairly simple. Proving the invariance of the remaning OPE's, however, is more involved.

Semiclassical limit
In order make contact with the holographic description in the next section we need to extract the semiclassical limit of the OPE relations (2.1)-(2.6), (2.7)-(2.14) and (2.17)-(2.23), which in the bulk Chern-Simons theory translate into classical Poisson brackets. This involves taking a "large-c" and "large-current" limit, procedure that is more subtle than a naive expansion in 1/c. As outlined in [26], the proper way to proceed is to rescale all the CFT currents as J i (z) = cJ i (z) (2.29) and then expand for c → ∞ while keepingJ i fixed. One finds that the leading term in thẽ JJ OPE's is of order 1/c, that is, Notice that by construction the OPE's on the right hand side do not depend explicitly on the central charge. We can now express everything back in terms of the original currents J i and write (2.31) This defines the semiclassical limit of the algebra. Following the above procedure, we find that the semiclassical version of the N = 2 super-W 3 algebra is identical to its quantum progenitor (2.1)-(2.6), (2.7)-(2.14) and (2.17)-(2.23) with the proviso that we use the semiclassical limit of the composites ± , as opposed to their full quantum expressions (A.2)-(A. 19). For example, the V V OPE is still given by The full list of semiclassical composite fields is written in the next section. As we will see, they are in perfect agreement with the corresponding expressions obtained from the bulk analysis.

From the plane to the cylinder
In the final portion of this section we collect some useful relations regarding the hermiticity and spectral flow properties of the CFT currents cast on the Euclidean cylinder versus the complex plane. Since bulk observables are naturally defined on the AdS 3 cylinder, this will ease the comparison with the dual Chern-Simons description, especially when discussing the continuation of the higher spin theory from Euclidean to Lorentzian signature and identifying the correct real form of the sl(3|2) gauge algebra. Recall that in radial quantization a real quasi-primary field of dimensions (h,h) satisfies the hermiticity condition Similarly, for a pair of charge conjugate fields this is One can check that in the conventions adopted here the W (3|2) algebra is consistent with these conditions when imposed on the genetators {J, T, κV, κW } and {G ± , κU ± }. Importantly, as required by the OPE structure, the currents in the spin-3 multiplet must always be accompanied by the coupling κ, which is imaginary in the semiclassical limit [1]. The labels in (2.34) and (2.35) emphasize the fact that these relations are valid on the complex plane, where the OPE's are defined. The transition from the plane to the cylinder is achieved through the conformal transformation z → e ζz , where ζ is a bookkeeping device that allows us to accommodate different conventions relating the real cylinder coordinates Then, taking into account the conformal weights of the fields, the hermiticity conditions (2.34) and (2.35) become, respectively, Ultimately, the different phases can be understood by recalling that in the Euclidean formalism the effect of complex conjugation on the time direction t = −it E must be compensated in the definition of Hermitian conjugate by taking t E → −t E . 6 Lastly, we spell out the spectral flow transformations for the CFT currents on the cylinder. Upon implementing the conformal map z → e ζz , one readily finds that relations (2.25) and (2.26) become and Of course, the operators appearing in (2.28) remain spectral flow-invariant. This version of the transformations is better suited for comparison with the analogous relations appearing in the bulk Chern-Simons theory. 6 In terms of the mode expansions (settingh = 0 for simplicity) both Hermiticity conditions imply that Φ † n = Φ−n.

The N = 2 super-W holographic dictionary
Having reviewed some of the properties of the N = 2 super-W 3 algebra, we now move on to study its realization in terms of higher spin fields on AdS 3 . We first provide a short summary of Chern-Simons supergravity theory, touching only on those points that are relevant to the construction of the holographic dictionary. Subsequently, a detailed derivation of the asymptotic symmetry algebra is given for the sl(3|2) case, followed by a discussion of sources and the corresponding holographic Ward identities, which are relevant for the study of higher spin black hole solutions. The correct choice of real form in Lorentzian signature is also discussed. Our conventions for sl(3|2) follow [1] and are reproduced in appendix B for completeness. Since this superalgebra has dimension 24 and involves 5 × 5 matrices, we have found it necessary to use a mathematical software such as Maple and Mathematica to perform most of the calculations.

Higher spin supergravity on AdS 3
In its simplest version, the action for three-dimensional higher spin gravity with negative cosmological constant is 7 where A andĀ are two independent connections valued in a real Lie algebra g and Tr denotes the trace in some representation of choice. As appropriate to AdS 3 , the topology of the spacetime M is assumed to be that of a solid cylinder, with coordinates (ρ, t, φ) such that −∞ < t < ∞, φ ∼ φ + 2π and the boundary is located at ρ → ∞. The corresponding equations of motion read implying that both connections are flat. Local symmetries include diffeomorphisims and gauge transformations although the former can be seen as a particular case of the latter when on-shell. The gravitational sector of the theory is associated with an sl(2; R) subalgebra, whose generators L i satisfy 8 (3.4) 7 We refer the reader to the extensive literature for a more detailed review; see for example [4,7,27,28] and references therein. 8 The constant kcs appearing in the action is related to Newton's constant G3 and the AdS3 radius l by . The Chern-Simons level of the sl(2) gravitational theory is k = l 4G3 .
The spectrum of higher spin fields then depends on the precise way in which this subalgebra is embedded in g; different embeddings give rise to different field contents in the bulk. A supersymmetric extension can be obtained by considering instead a Lie superalgebra and replacing the trace Tr by the supertrace sTr. In this case one must specify the embedding of osp(1|2) ⊃ sl(2) in the gauge superalgebra. Two-dimensional CFT's are usually discussed in Euclidean signature, so is convenient to also formulate the supergravity theory in this language. Our conventions follow [29,30] and are such that after the Wick rotation t = −it E , the light-cone directions x ± = t ± φ become complex coordinates x + → z and x − → −z, subject to the periodicity conditions z ∼ z + 2π andz ∼z +2π. Depending on the type of solutions one is interested in, one can generalize this condition and let (z,z) parametrize any Riemann surface. The cylinder is the topology most fitting for the analysis of asymptotic symmetries with AdS 3 boundary conditions, whereas the torus (z ∼ z + 2π ∼ z + 2πτ ) is appropriate for the discussion of black hole solutions [1,13,14,[27][28][29][31][32][33][34][35][36][37][38][39]. In this paper, will be concerned exclusively with the former. Notice that in these conventions the map between the cylinder and the plane is z → e iz .
As it turns out, when continuing to imaginary time, the algebra g needs to be complexified and the two connections A andĀ are no longer independent. Rather, they are related bȳ a condition that ensures the reality of the action and of all other physical observables. Recall, however, that a complex algebra can have several real forms, so special care must be taken in order to reconstruct the appropriate Lorentzian theory. We will come back to this point below. From now on we focus on the unbarred sector only. Boundary conditions are a crucial ingredient in the context of the AdS/CFT correspondence. As shown in [3], using the gauge symmetries (3.3) of the Chern-Simons theory, we can eliminate the radial dependence of the connection and write for some suitable choice of group element b(ρ). This allows us to work only with the twodimensional boundary connection a(z,z), whose components satisfy the equation As further argued in [3][4][5][6], by a series of gauge transformations, any asymptotically AdS connection can be brought to the so-called highest weight gauge, or Drinfeld-Sokolov form, where Q ∈ g is a matrix satisfying [L −1 , Q] = 0. The holomorphicity of a z follows from the flatness condition (3.7). These represent source-free solutions. In particular, the connection corresponding to pure AdS 3 has Q = 1 4 L −1 . Other source-free configurations include boundary gravitons (higher spin generalizations of Brown-Henneaux states [40]) and conical defect solutions [41]. The deformation of AdS boundary conditions by the incorporation of sources will be discussed in section 3.3.

Dictionary part I: currents and asymptotic symmetries
The asymptotic symmetries of the higher spin theory with AdS boundary conditions are defined as those residual gauge transformations (3.3) that preserve the form of the Drinfeld-Sokolov connection (3.8). Concretely, one looks for gauge parameters Λ such that Since Λ must be holomorphic in order for the transformation to be compatible with az = 0, we can expand it as where [L 1 , λ] = 0, i.e. a lowest-weight condition. The dots represent higher-weight terms that are fixed algebraically in terms of Q, λ and their derivatives by the condition (3.9). Naturally, Q is allowed to change under the asymptotic symmetry transformations, this being precisely the algebra one is interested in uncovering. According to the AdS/CFT correspondence, the different components of Q(z) and λ(z) are then identified, respectively, with the currents J s (z) and parameters ǫ s (z) generating the same symmetry algebra in a dual CFT. We will now go through some of the details of this analysis in the case of Chern-Simons supergravity based on the superalgebra g = sl(3|2) and show that the resulting structure is given by the semiclassical limit of the N = 2 super-W 3 algebra reviewed in section 2.

Spectrum and operator content
The first step in the construction of the holographic dictionary is to derive the spectrum of fields in the bulk and verify that it matches the operator content of the dual CFT. To this purpose, we look at the decomposition of the adjoint representation of sl(3|2) into irreducible representations of osp(1|2). Concretely, we focus our attention on the principal embedding of osp(1|2) in sl(3|2), since this is the case that makes contact with the W (3|2) algebra. The decomposition reads (see e.g. [12]) where R j denotes a spin-j representation of the osp(1|2) superalgebra. In turn, these can be written as with D j being a spin-j representation of sl (2). Thus, the bulk theory contains fields of spin 9 (1, 2, 2, 3) and 3 2 , 3 2 , 5 2 , 5 2 , which under the holographic dictionary map to the scaling weights of the dual CFT operators. Of course, this coincides with the operator content (J, T, V, W ) and (G + , G − , U + , U − ) of the W (3|2) algebra.
Following the notation of appendix B, we label the sl(3|2) generators by (J, L i , A i , W m ) and (H r , G r , T s , S s ). These correspond to sl(2) multiplets of spin (0, 1, 1, 2) and 1 2 , 1 2 , 3 2 , 3 2 , respectively. Then, according to the decomposition (3.11), the Drinfeld-Sokolov connection (3.8) takes the explicit form (3.13) Similarly, the gauge transformation parameter (3.10) becomes while the asymptotic symmetry condition (3.9) reads It is important to emphasize that the currents Q G (z), Q H (z), Q S (z) and Q T (z), as well as the parameters λ G (z), λ H (z), λ S (z) and λ T (z) are Grassmann variables since they are associated with odd elements of the superalgebra. Expression (3.15) encodes 24 equations, 16 of which (the lower-weight components) determine the coefficients in front of the higher-weight generators in (3.14), with the remaining 8 (the highest-weight components) allowing us to solve for the variations δQ(z) in terms of the fields Q(z), the parameters λ(z) and their derivatives. For brevity, we omit the solution to the former. 10 We shall write the solution to the latter in a more convenient basis momentarily.

Field redefinitions
Were we to directly transform the field variations δQ(z) obtained from (3.15) into semiclassical OPE's we would find the symmetry algebra in a quite awkward form that obscures the superconformal structure discussed in section 2. This is actually a rather generic feature of the asymptotic symmetry computations in Chern-Simons theory, present even in much simpler setups. One such example involves the bosonic theory based on the sl(3) algebra with diagonally-embedded sl (2), which results in the so-called W (2) 3 algebra [4,42]. In this case one finds that the naive bulk stress tensor (analogous to Q L (z) in (3.13)) requires a Sugawara shift by the spin-1 current (analogous to Q J (z) in (3.13)) squared. A simultaneous redefinition of the naive infinitesimal U (1) parameter (analogous to λ J in (3.14)) is necessary in order for the U (1) current to have the appropriate conformal dimension. In the present context we expect that even more involved modifications are needed because, in addition to a U (1) current, there is a second bulk spin-2 field. Our goal in the reminder of this subsection is to provide the precise combinations of bulk fields Q(z) and gauge transformation parameters λ(z) such that the asymptotic symmetries of the sl(3|2) Chern-Simons theory with AdS 3 boundary conditions take the form dictated by the semiclassical limit of the N = 2 super-W 3 algebra discussed in section 2. Not surprisingly, this turns out to be a laborious task, but there are a few guiding principles we can use to our advantage.
The starting point to derive necessary field redefinitions is to recognize that the stress tensor and central charge in the dual CFT are given by This bit of the holographic dictionary can be derived in multiple ways, e.g. by putting the Chern-Simons theory on a solid torus with modular parameter τ and studying the variation of the action under τ → τ + δτ [29,30]. Consequently, the redefinitions of fields should be such that the resulting combinations transform as primaries under this stress tensor. In the present case we find Here we see explicitly the Sugawara shift by the U (1) current Q J (z). An additional clue comes from the observation that when bulk and CFT quantities are properly aligned, the connection (3.13) and gauge parameter (3.14) should satisfy where ǫ(z) parameterizes infinitesimal conformal transformations, J s (z) denotes a current of weight s , ǫ s (z) is the associated infinitesimal parameter, and the sum runs over all the spins present in the spectrum (minus the stress tensor itself, which is singled out). This relation has been shown to be valid in the bosonic theory based on the sl(N ) algebra [30], even in non-principal embeddings where U (1) currents are involved. We will verify that it remains true for sl(3|2) as well. Using (3.17) we get from where we infer that the parameter of conformal transformations is One also learns from this calculation that the U (1) parameter λ J (z) does in fact need to be modified as a result of the Sugawara shift in the stress tensor (3.17), although we have not written it explicitly here since additional changes are required due to the presence of the other fields and parameters. The final guiding principle in the construction of the holographic dictionary is the spectral flow automorphism (2.25) and (2.26) of the W (3|2) algebra. As expected, this piece of information is properly encoded in the symmetries of the Chern-Simons theory, being implemented by a gauge transformation associated with the U (1) generator J ∈ sl(3|2). Indeed, it is easy to see that a (finite) transformation with parameter Λ(z) = λ J (z)J induces the change 11 which for λ J (z) ∼ ηz resembles the second equation in (2.38). This naturally leads to the identification of Q J (z) with the CFT current J(z). Moreover, since the sl(2) multiplets L i , A i and W m in sl(3|2) commute with J, the corresponding fields in the Drinfeld-Sokolov connection (3.13) are inert under this transformation, that is, As a consequence, the map between these charges and their dual CFT variables can only involve spectral flow-invariant combinations such as (2.28). As for the fermions one finds that Looking at (2.39), spin and charge assignments then clearly imply that Q H (z) ∼ G + (z), . This way, the spectral flow invariance of the dual CFT severely restricts the form that the bulk/boundary map can take. Taking all of these insights into account, and after a detailed look at the transformation rules for the different charges, we are led to the following redefinitions of bulk fields: (3.24) Accordingly, the gauge parameters must be redefined as: Not coincidentally, this corresponds to the c → ∞ limit of (2.16). The transformation parameters ǫ s (z) associated to the symmetries generated by each conserved current are identified as in table 2. Relations (3.24) and (3.25) are one of the main results of this paper. They constitute the first piece of the holographic dictionary, establishing the map between bulk currents and symmetry parameters in the sl(3|2) Chern-Simons theory and their boundary CFT counterparts, and completing the partial analysis in [9,10,14]. Of course, the ultimate check of this result is the agreement between the OPE algebra for the redefined currents and the semiclassical limit of the N = 2 super-W 3 algebra reviewed in section 2.

Variations and semiclassical OPE's
In order to exhibit the transformation rules of the redefined fields under the asymptotic symmetries, it is convenient to introduce the operator Its interpretation is as follows: M s" s,s ′ (λ; φ) gives the contribution of a field φ of spin-s ′′ to the variation of a spin-s ′ primary under the symmetry generated by a spin-s primary with associated infinitesimal parameter λ. The reason this particular operator simplifies the task of writing down the field variations is that the structure constants appearing in the OPE's of Virasoro primaries are constrained by the sl(2, R) covariance of the algebra; the coefficients appearing in (3.27) are then related to Clebsch-Gordan coefficients.
Using the above notation, and in terms of the redefined charges and parameters (3.24) and (3.25), the asymptotic symmetry variations coming from (3.15) are found to be

28)
δT = ∂ηJ + ǫ∂T + 2∂ǫT + c 12 ∂ 3 ǫ + γ∂V + 2∂γV + 2χ∂W + 3∂χW   ± are given by It is reassuring to verify that expressions (3.34)-(3.51), which emerge entirely from a bulk analysis, correspond precisely to the semiclassical limit, taken as explained in section 2.3, of the full quantum composites (A.2)-(A. 19) that appear in the W (3|2) OPE algebra. 12 At this point it also becomes clear that the fields defined through (3.24) have conformal dimensions ∆ and U (1) charges q as given in table 3, in agreement with the N = 2 multiplet structure described in section 2. These two facts are a non-trivial test for the validity of our results. The final step in identifying the asymptotic symmetry algebra is to convert the variations (3.28)-(3.33) into semiclassical OPE's using Noether's theorem. To this purpose, following the assignments exhibited in table 2, we define the total current Then, after mapping the boundary cylinder to the complex plane via z → e iz , the transformations can be rewritten as expression from which the OPE algebra can be read. For example, setting all the transformation parameters but γ to zero, the variation δV in (3.30) becomes c 12 ∂ 3 γ(w) + 2∂γ(w)A [2] (w) + γ(w)∂A [2] We have checked that, quite satisfactorily, the full OPE algebra derived from the asymptotic symmetry variations (3.28)-(3.33) is given precisely by the semiclassical limit of the N = 2 super-W 3 algebra written in (2.1)-(2.6), (2.7)-(2.14) and (2.17)- (2.23). This shows that the identifications (3.24) and (3.25) between bulk and CFT quantities are indeed correct.

Dictionary part II: sources and Ward identities
Having successfully aligned bulk and boundary currents J s and infinitesimal parameters ǫ s , we now turn to the study of sources (or chemical potentials). These play a central role in the discussion of black hole solutions and their thermodynamics [1,13,14,[27][28][29][31][32][33][34][35][36][37][38][39]. We closely follow the work of [30], which offered a detailed account of sources in the context of the AdS 3 /CFT 2 correspondence. As before, the treatment of the two connections A andĀ is completely analogous, so we focus on the unbarred sector for concreteness.
Consider deforming a two-dimensional CFT with W-symmetry by coupling the set of (would-be) conserved higher spin currents J s (z,z) to some external fields µ s (z,z). Restricting to chiral deformations, one natural possibility is to perturb the CFT action by Another is to write the deformed Hamiltonian In either case, the W-symmetry is still realized at the level of the partition function provided that one transforms the sources µ s (z,z) accordingly [30]. This results in the existence of Ward identities for the one-point functions of the currents in the presence of sources. It is common knowledge that a holographic description of these deformations requires generalizing the AdS boundary conditions (3.8), such that the bulk Chern-Simons fields now include the deformation parameters in their asymptotics. As argued in [30], the ensuing structure is best described in terms of a "Drinfeld-Sokolov pair", consisting of one component of the connection a(z,z) carrying the bulk currents as highest-weights, and a conjugate component carrying the corresponding sources as lowest-weights. Since all source-free solutions satisfy az = 0, sources should certainly be included in this component of the connection. However, in their presence, the question arises of whether the currents should be incorporated in a z or in a z + az = a z . This leads to two natural choices of boundary conditions for the gauge fields, namely, holomorphic boundary conditions, given by the Drinfeld-Sokolov pair a z = L 1 + Q(z,z) , 2az = ν(z,z) + · · · , (3.58) and canonical boundary conditions, implemented by a z + az = L 1 + Q(z,z) , 2az = ν(z,z) + · · · . (3.59) The matrices Q and ν are such that [L −1 , Q] = 0 and [L 1 , ν] = 0 (highest and lowest-weight, respectively), and the dots represent higher-weight terms that are fixed algebraically by the flatness condition (3.7). Of course, the source-free solution (3.8) is recovered for ν = 0. From the field theory point of view, it was shown in [30,31,42] that holomorphic boundary conditions correspond precisely to deformations (3.56) of the CFT action, whereas the the canonical choice maps to deformations (3.57) of the Hamiltonian. In either case the connection a(z,z) is no longer holomorphic, and the bulk equations of motion become the CFT's Ward identities. Below we will exemplify in detail the incorporation of sources in the holomorphic case, where they are more symmetrical. Then we will point out the changes needed to accomplish this in the canonical case.

Action deformations and holomorphic boundary conditions
Most of the analysis of asymptotic symmetries in the previous section consisted in finding the correct combinations of bulk currents in Q such that the transformation rules took the form dictated by the W (3|2) algebra. By the same token, the analysis of deformed boundary conditions boils down to finding the precise combination of bulk sources in ν such that the equations of motion for the Drinfeld-Sokolov pair of connections reproduce the CFT's Ward identities. Happily, in the case of holomorphic boundary conditions (3.58), all of the necessary algebra can be recycled from the asymptotic symmetry calculations by noticing that the flatness equation which must also be satisfied when bulk and boundary quantities are properly aligned [30], follows from (3.18) by the same replacements. These considerations allow us to directly state the second piece of the holographic dictionary, namely, the one relating the chemical potentials in the sl(3|2) Chern-Simons theory with the sources coupling to the conserved currents in a CFT displaying W (3|2) symmetry. Writing a lowest-weight ansatz similar to (3.14), we find that the correct combination of bulk fields coupling to the CFT currents is 13 (cf. (3.25)) (3.63) 13 We omit the dependence in (z,z) for simplicity where κ = ±5i/2 as before. Expression (3.13) for a z , as well as the map (3.24) between bulk and boundary charges, still apply in the deformed theory, albeit with an additional antiholomorphic dependence. Importantly, with these redefinitions, the Drinfeld-Sokolov connection automatically verifies relation (3.61) for the principally-embedded osp(1|2) ⊂ sl(3|2) spectrum, leading to the pairing of the different CFT sources and currents shown in table 4. Finally, the solution to the highest-weight components of the equation of motion, that is, the Current Source solution for∂Q(z,z), follows directly from the transformation rules (3.28)-(3.33), yieldinḡ ; D [3] . (3.69) The composite fields ± and the operators M s" s,s ′ (λ; φ) are the same as in section 3.2. These are the holographic Ward identities corresponding to the deformation (3.56) of the CFT action.

Hamiltonian deformations and canonical boundary conditions
For the implementation of canonical boundary conditions it is convenient to switch to the real coordinates φ = (z +z) /2 and t E = i (z −z) /2, in terms of which the Drinfeld-Sokolov pair (3.59) becomes (3.70) Bulk charges and sources are then included as in (3.13) and (3.62), but with a z → a φ and az → a φ − ia t E . Since the holographic dictionary for the charges is derived from the sourcefree solutions, where a φ = a z , we still have (3.24). Moreover, the Drinfeld-Sokolov pair now satisfies the relation [30] −k cs sTr a φ (a φ − ia t E ) = c 12 The remaining Ward identities are derived in a similar fashion. We close this section by commenting that when putting the theory at finite temperature, i.e. on a torus, one also needs to specify how the thermal sources scale with the temperature; only then is the partition function (and consequently the free energy) well-defined [30]. Moreover, in this context it is redundant to include the source for the stress tensor in the connection a φ − ia t E , since it can be incorporated as the modular parameter of the torus. From (3.63), we see that setting µ 2 = 0 amounts to fixing ν A = − 3 5 ν L , so that only the source for the combination of generators L 1 − 3 5 A 1 is turned on. These issues are discussed in our companion paper [1], where the thermodynamics of black hole solutions in the sl(3|2) theory with canonical boundary conditions were studied.

Lorentzian connections and su(2, 1|1, 1)
The preceding construction of the holographic dictionary was carried out in the Euclidean formalism, where the Chern-Simons connection a(z,z) is valued in the complex superalgebra sl(3|2; C). When continuing back to Lorentzian signature the question arises of which real form appropriately describes the CFT structure that we have uncovered. According to [43], the candidate superalgebras are (3.73) As we will now see, the real form su(2, 1|1, 1) is singled out after imposing the standard hermiticity conditions on the generators of the N = 2 super-W 3 algebra and applying the holographic map. To make the analysis more transparent we will focus on source-free solutions (3.8); having understood the reality properties of the charges those of the sources follow directly. All throughout section 3 we have complied with the conventions of [29,30], which implemented the continuation of the Lorentzian higher spin theory to Euclidean signature by taking t + φ → z and t − φ → −z. In particular, the definition (3.16) of the stress tensor and the condition (3.18) between bulk and boundary quantities, both of which where crucial in the derivation of the holographic dictionary, pend on this choice. This means that our conventions are such that the N = 2 super-W 3 generators on the cylinder satisfy as follows from (2.36) and (2.37) with ζ = i. Thus, in light of the map (3.24), the different components in the Drinfeld-Sokolov connection must obey the following reality conditions: (3.75) In order to figure out the appropriate real form of sl(3|2) that is compatible with (3.75) we seek for combinations of currents that are real when continued back to Lorentzian signature via z → x + . Then, the correct superalgebra will be the one spanned by those generators accompanying these charges in a z → a + . With this in mind, a more suggestive way of writing the Drinfeld-Sokolov connection (3.13) is

(3.77)
It is easy to check using (3.75) that all these combinations of charges satisfy Q(z) = Q(z), implying that they are real if restricted to real arguments. In other words, Q(x + ) = Q(x + ). This way we learn that the CFT structure of the Euclidean theory dictates that a general Lorentzian connection on the cylinder takes values in the real form of the sl(3|2) algebra generated by As discussed in appendix B, this corresponds to the superalgebra su(2, 1|1, 1). 14

Conclusion
The purpose of this note was to provide the explicit form of the holographic dictionary between sl(3|2) Chern-Simons supergravity on AdS 3 and two-dimensional CFT's with N = 2 super-W 3 symmetry that was used in [1]. The main entries of the dictionary are the identification of bulk and boundary currents given in (3.24), with their corresponding transformation parameters (3.25), and the identification of the Chern-Simons and CFT sources in (3.63). This allowed us to display in full detail the holographic Ward identities (3.64)-(3.69). The other entry is the identification of su(2, 1|1, 1) as the correct real form of sl(3|2) in Lorentzian signature. Along the way, we also corrected some typos in the original reference [18] on the W (3|2) algebra. These corrections were verified by three independent methods: i) fulfillment of the Jacobi identities for the OPE's, ii) spectral flow invariance of the OPE algebra, and iii) agreement with the asymptotic symmetries of Chern-Simons theory. It is worth highlighting the role of the spectral flow automorphism in building the holographic dictionary. Our discussion in section 3.2.2 applies broadly to any gauge algebra g that contains a U (1) generator that would lead to spectral flow symmetry in the boundary W-algebra. In particular, following the argument around (3.21), it should be straightforward to identify the components of Q(z) that are spectral flow invariant in a highest-weight gauge (3.8) for the connection. This, combined with (3.18), leads to a clear and simple basis in which to setup the dictionary with CFT variables.
Naturally, some of our results can be extended to the analysis of Lorentzian solutions in the sl(N |N − 1) theory, where the same reasoning shows that the correct real form of the bulk gauge algebra that is consistent with the structure of N = 2 supersymmetric higher spin symmetries is su(p, N − p|q, N − 1 − q) and not sl(N |N − 1; R) as naively expected. Perhaps the easiest way to see this is to note that su(p, m−p|q, n−q) is the only real form of sl(m|n; C) that possesses a compact Abelian generator in the bosonic subalgebra [43]. As shown [1], this property is crucial for compatibility with R-charge quantization and the existence of Killing spinors with non-trivial angular dependence. Ultimately, this is tied to the fact that we have performed a Drinfeld-Sokolov reduction based on an sl(2|1) embedding instead of an osp(2|2) 14 Another way to derive this result, purely from the bulk perspective, is to start from az = L1 + Q(z) and make the change of coordinates z ′ = ζz. Then, an additional gauge transformation with parameter Λ = ζ L 0 is necessary in order to bring the new connection a z ′ = ζ −1 az back to Drinfeld-Sokolov form, , it follows that Q ′ ∈ sl(3|2; R) ⇔ Q ∈ su(2, 1|1, 1). embedding as required to reproduce the N = 2 CFT structure (see e.g. [10,12]). To our knowledge, the issue of identifying the appropriate real form has not been discussed in the literature so far. It is, however, an important ingredient in the holographic dictionary if one is to match bulk and boundary results correctly.
An important asset of the dualities addressed in this paper is that they exploit the topological formulation of the Chern-Simons theory in order to set up and perform tractable calculations that are quite challenging in CFT's with W-algebra using solely field-theoretical techniques. The analysis here is one very modest example in the context of supersymmetric dualities. In recent years this asset has been applied to Wilson lines in Chern-Simons theory as one efficient approach to evaluate W-conformal blocks in the CFT [44][45][46]. In our case it would be interesting to evaluate a supersymmetric Wilson line and include quantum corrections following the approach of [47][48][49][50][51][52]. This would allow us to improve the semiclassical limit discussed in section 2.3, and attempt to study the duality in a healthier regime where the representations of the W-algebra are unitary.

Acknowledgments
We are extremely grateful to our dear friend Juan Ignacio Jottar, without whom this work would not have been possible. In fact, most of the heavy lifting was done by Juan in 2015 while working on our companion paper [1]. We will always value and cherish our discussions and collaborations.
On behalf of Juan and ourselves we are especially grateful to Max Bañados and Jan de Boer for useful discussions. J.I.J. would also like to thank Matthias Gaberdiel ± appearing in the W (3|2) algebra together with their transformation properties under the spectral flow automorphism.

A.1 Definition
In what follows, all combinations of fields enclosed by square brackets are quasi-primary. Naturally, normal-ordering is assumed. As in [18] we define The composite operators appearing in the W (3|2) algebra then read , , , (A.8) , We have highlighted in red the coefficients in C [1] and Φ [7/2] ± that need to be modified with respect to [18] so that the Jacobi identities are satisfied and the algebra is spectral flowinvariant.

A.2 Spectral flow
In order to derive the spectral flow rules for the composite operators, we need to recall the definition of normal-ordering as the regular part of the OPE in the coincidence limit. Following the conventions of [18,25] such that the OPE between two holomorphic operators A and B is written as with h ≤ h A + h B , the normal-ordering prescription is Products of more than two fields are defined recursively, grouping them as These definitions, together with the OPE algebra, allow us to compute the spectral flow of any composite starting from the transformation rules for the fundamental fields. Take as an example the composite :G + G − :(w), entering in the definition of C [3] and D [3] . Using (2.25), we first compute the OPE between the spectral flowed operators, which in the above notation becomes Since the z dependence on the right hand side can only involve powers of z − w, we expand Relabeling the indices and swapping the sums we find The new OPE can be read directly from this expression. In particular, the regular part is which, after extracting from (2.6), reads It also follows from the singular part that showing the invariance of this particular OPE. The procedure is similar in all other cases, always relying on the OPE algebra and the expansion (A.23) for different values of η. This way, starting from the definitions (A.2)-(A.19) of the quantum composites and using the transformation rules (2.25) and (2.26) for the fundamental fields, we find that the spectral flowed-version of these operators is Collecting poles of same order we find that spectral flow invariance demands that C [4]′ (w) + ∂C [3]′ (w) = C [4] (w) + ∂C [3] (w) + 2η w ∂A [2] (w) , C [3]′ (w) = C [3] (w) + 4η 3w A [2] (w) , C [1]′ (w) = C [1] (w) + cη 36w .

B.1 Definition and (anti-)commutation relations
The superalgebra sl(m|n; C) consists of all complex (m + n) × (m + n) supermatrices of the form The complex dimension of the superalgebra is (m + n) 2 − 1. Elements with B = 0 and C = 0 are called even or bosonic, while those with A = 0 and D = 0 are termed odd or fermionic. The even subalgebra is sl(m; C) ⊕ sl(n; C) ⊕ C. In what follows we deal specifically with m = 3 and n = 2. We comment on the real form of interest below.
In the principal embedding of sl(2|1) in sl(3|2) [12,13], the even-graded sector of the superalgebra is decomposed into the sl(2) generators, L i , one spin 1 multiplet, A i , one spin 2 multiplet, W m , and a spin 0 element, J. By spin we mean the sl(2) spin, S.

B.2 Matrix representation
For convenience, we have chosen to work in a representation where all matrices are real and satisfy Notice that K has the correct eigenvalues. Therefore, these particular combinations of generators, with the above pre-factors included, form a basis for the real superalgebra su(2, 1|1, 1).