Supersymmetric localization in AdS 5 and the protected chiral algebra

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Introduction
Any four-dimensional N = 2 superconformal field theory (SCFT) admits a subsector of correlation functions that exhibits the structure of a two-dimensional chiral algebra [1]. This is in particular the case for N = 4 super Yang-Mills (SYM) theories. The associated chiral algebra is labelled by the gauge algebra g and is independent of the complexified gauge coupling. It encodes an infinite amount of information about a very rich protected subsector of the SYM theory. In this paper we start addressing the question of finding a holographic description of this protected chiral algebra for g = su(N ), in the large N limit. Answering this question would provide us with a new solvable model of holography. Rather than a mere toy example, this would be an intricate yet tractable model carved out naturally from the standard holographic duality.

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While in the general N = 2 case the protected chiral algebra has no residual supersymmetry, the chiral algebra associated to an N = 4 SCFT contains the small N = 4 superconformal algebra (SCA) as a subalgebra. Conjecturally [1], the chiral algebra for N = 4 SYM theory with gauge algebra g is a novel N = 4 super chiral algebra, strongly generated by a finite number of currents. The super chiral algebra generators descend from the generators of the one-half BPS chiral ring of the SYM theory, and are thus in one-to-one correspondence with the Casimir invariants of g. For example, for g = su(N ), the super chiral algebra is conjectured to have N −1 generators, 1 of holomorphic dimension h = 1, 3 2 , . . . N 2 , in correspondence with the familiar single-trace one-half BPS operators of the SYM theory, namely Tr X 2h in the symmetric traceless representation of the so(6) Rsymmetry. As we will review in detail below, only an su(2) F subalgebra of so (6) is visible in the chiral algebra, where it is in fact enhanced to the affine Kac-Moody algebra su(2) F that is part of the small N = 4 SCA. The super chiral algebra generator of dimension h transforms in the spin h representation of su(2) F . This is a BPS condition -the generators with h > 1 are the highest-weight states of short representations of the N = 4 subalgebra. 2 The central charge of the chiral algebra is given by c 2d = −3 dim g = −3(N 2 − 1). It is not known whether the chiral algebra for fixed N > 2 admits a deformation 3 to general values of the central charge. In fact, for this special value of c 2d one finds several null relations that might be essential to ensure associativity of the operator algebra.
Let us now consider the large N holographic description. As familiar, N = 4 SYM theory is dual to IIB string theory on AdS 5 × S 5 , with the 1/N expansion on the field theory side corresponding to the genus expansion on the string theory side. It would be extremely interesting to construct a "topological" string theory whose genus expansion reproduces the 1/N expansion of the N = 4 SYM chiral algebra. Here we will address the simpler question of finding a holographic description for the leading large N limit of the chiral algebra, in terms of a classical field theory in the bulk. There are two ways we can imagine to proceed: attempting to construct the bulk theory by bottom-up guesswork; or deriving it from the top-down as a subsector of AdS 5 × S 5 string field theory.
From the bottom-up perspective, the natural conjecture is that the bulk theory is a Chern-Simons field theory in AdS 3 , with gauge algebra a suitable infinite-dimensional supersymmetric higher-spin algebra. Such a duality would mimic several examples of "higherspin holography" that have been studied in recent years in the context of the AdS 3 /CFT 2 correspondence. A duality has been proposed in [4] 4 between higher-spin Vasiliev theory 1 We are counting the psu(1, 1|2) primaries; the chiral algebra is generated by the psu(1, 1|2) primaries and their supersymmetry descendants. 2 Other examples of N = 4 super chiral algebra have been considered in the literature, see e.g., [2,3], where the generators are taken to be su(2)F singlets and are thus highest-weight states of long representations of the N = 4 SCA. Long generators give a qualitatively different behavior: the higher-spin currents encountered in the CFT duals of Vasiliev theories are not protected and are in fact expected to be lifted when one moves away from the tensionless point. 3 For N = 2, the chiral algebra coincides with the small N = 4 SCA, which is of course consistent for any central charge. 4 This proposal, reviewed in [5], has passed several tests [6,7] and has been extended in various directions, both in a purely bosonic context [8,9] and in models with supersymmetry, see for instance [10][11][12][13][14][15][16].

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in AdS 3 [17] and a suitable 't Hooft limit of W N minimal models, i.e. the coset CFTs su(N ) k ⊗ su(N ) 1 /su(N ) k+1 . In this example, the chiral algebra that controls the large N limit is the W ∞ [µ] algebra, where the parameter µ is identified with the 't Hooft coupling N/(N + k), kept fixed as N → ∞. The bulk dual description involves Chern-Simons theory with gauge algebra the infinite-dimensional Lie algebra hs [µ]. The (non-linear) W ∞ [µ] algebra arises at the asymptotic symmetry of this Chern-Simons theory [18][19][20]. We find it likely that our example will work along similar lines, but we have not yet been able to identify the correct supersymmetric higher-spin algebra. An obvious feature of the sought after higher-spin algebra is that it must contain psu(1, 1|2) as a subalgebra. Indeed the asymptotic symmetry of AdS 3 Chern-Simons theory with algebra psu(1, 1|2) is the small N = 4 SCA (see for instance [21,22] and references therein), which, as reviewed above, is a consistent truncation of the full super W-algebra. Higher-spin holography with small N = 4 superconformal symmetry has also been studied in [23,24]. In our case, the construction of the complete higher-spin algebra is made more challenging by the absence of an obvious deformation parameter analogous to the 't Hooft coupling of the W N minimal models 5 -as we have remarked, the chiral algebra for su(N ) SYM theory might be isolated, stuck at a specific value of the central charge.
The top-down approach is conceptually straightforward. The dual bulk theory must be a subsector of IIB supergravity on AdS 5 × S 5 . Indeed, the generators of the chiral algebra descend from the single-trace one-half BPS operators of N = 4 SYM, which are dual to the infinite tower of Kaluza-Klein (KK) supergravity modes on S 5 . In principle, our task is clear. In the boundary SYM theory, the 2d chiral subsector is carved out by passing to the cohomology of either one of two nilpotent supercharges [1]. The bulk supergravity admits analogous nilpotent supercharges. We then expect to find the bulk dual to the large N limit of the chiral algebra by localization of the supergravity theory with respect to either supercharge. In practice however, this program is difficult to implement rigorously. The technique of supersymmetric localization requires an off-shell formalism, but we are not aware of such a formalism for AdS 5 × S 5 supergravity, or even for its consistent truncation to N = 8 AdS 5 supergravity.
In this paper, we give a proof of concept that this localization program works as expected, producing an AdS 3 Chern-Simons theory out of AdS 5 supergravity. We consider the simplest truncation of the supergravity theory for which a convenient off-shell formalism is readily available: the theory of an N = 4 vector multiplet in AdS 5 , covariant under an su(2, 2|2) subalgebra of the full psu(2, 2|4) superalgebra. We obtain this model by a straightforward analytic continuation of the analogous model on S 5 [25]. When viewed as part of the N = 8 supergravity multiplet, the N = 4 vector multiplet transforms in the adjoint representation of su(2) F (the centralizer of the embedding su(2, 2|2) ⊂ psu(2, 2|4)), but it is no more difficult to consider a general simple Lie algebra g F . We show by explicit calculation that supersymmetric localization with respect to the relevant supercharge yields Chern-Simons theory in AdS 3 , with gauge algebra g F , and level k related to the Yang-Mills 5 Recall that the usual 't Hooft coupling g 2 Y M N is not visible in the chiral algebra, which describes a protected subsector of observables of the SYM theory.

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coupling. As is well-known, its dual boundary theory is the affine Kac-Moody algebra g F at level k. Apart from confirming the general picture that we have outlined, we believe that the details of our calculations are interesting in their own right, and may find a broader range of applications. Localization computations involving non-compact AdS backgrounds have been considered in the literature, see for instance [26][27][28][29][30][31][32][33][34] and more recently [35,36]. It is worth pointing out that the Killing spinor used in our localization computation satisfies somewhat unusual algebraic properties compared to those usually assumed in past work. This is a consequence of the fact that our choice of supercharge mimics the (somewhat unusual) cohomological construction on the field theory side.
Localization of the full maximally supersymmetric AdS 5 supergravity would be technically challenging, but it seems very plausible (by supersymmetrizing the above result) that it would yield AdS 3 Chern-Simons theory with gauge algebra psu(1, 1|2), whose boundary dual is the small N = 4 superconformal algebra. Inclusion of the KK modes is however much harder, and at present the quest for the full holographic dual seems best pursued by bottom-up guesswork of the higher-spin superalgebra.
The rest of the paper is organized as follows. In section 2 we review the construction and main features of the chiral algebra associated to an N = 2 SCFT. Section 3 contains our main result, the localization of the N = 4 super Yang-Mills action in AdS 5 to bosonic Chern-Simons theory in AdS 3 . In section 4 we collect some useful facts and offer some speculations for the construction of the full holographic dual of the N = 4 SYM chiral algebra. We conclude in section 5 with a brief discussion. An appendix contains conventions and technical material.

Review of the chiral algebra construction
In an effort to make this paper self-contained, we briefly review in this section the construction of the two-dimensional chiral algebra associated to a four-dimensional N = 2 superconformal field theory [1]. Our main focus is on N = 4 SYM theory, but the calculations of section 3 will be relevant for any N = 2 SCFT that admits a supergravity dual and enjoys a global symmetry. To this end, we review in section 2.2 the special properties of chiral algebras associated to SCFTs with additional (super)symmetries.

Cohomological construction
The spacetime signature and the reality properties of operators are largely inessential in the following. We thus consider the complexified theory on flat C 4 . The complexified superconformal algebra is sl(4|2), with maximal bosonic subalgebra sl(4) ⊕ sl(2) R ⊕ C r . The first term corresponds to the action of the complexified conformal algebra on C 4 , while the other terms constitute the complexification of the R-symmetry of the theory.

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The first two summands comprise the complexified two-dimensional conformal algebra acting on the fixed plane, while C ⊥ corresponds to complexified rotations in the two directions orthogonal to the plane. We use the notation L n , n = −1, 0, 1 for the generators of the first summand, and L n for the second summand, while the generator of C ⊥ will be denoted M ⊥ . It is natural to adopt coordinates ζ,ζ on the selected plane C 2 , with sl(2) acting on ζ via Möbius transformations, and sl(2) acting similarly onζ. The R-symmetry of the superconformal theory allows us to define a suitable diagonal subalgebra given explicitly by where R, R ± denote the generators of sl(2) R with commutators The relevance of the twisted subalgebra sl(2) stems from the following crucial fact. There exist two linear combinations Q 1 , Q 2 of the supercharges of sl(4|2), inequivalent under similarity transformations, that enjoy the following properties: for suitable odd generators F 1,n , F 2,n of sl(4|2) , (2.5) where r denotes the generator of C r . In other words, the supercharges Q 1 , Q 2 are nilpotent, are invariant under the action of the holomorphic factor sl(2) of the conformal algebra of the plane, and are such that the twisted antiholomorphic factor sl(2) is both a Q 1 -and a Q 2 -commutator. Explicit expressions for Q 1 , Q 2 in a convenient basis are found in [1], where it is also shown that The chiral algebra associated to the four-dimensional superconformal field theory is then defined by considering cohomology classes of operators with respect to Q i (i = 1 or 2), i.e. the set of operators (anti)commuting with Q i modded out by addition of arbitrary Q i -commutators.
Let O be a local operator of the four-dimensional theory such that its insertion at the origin O(0) defines a non-trivial Q i cohomology class (i = 1 or 2), i.e. [Q i , O(0)} = 0, but O(0) is not itself a Q i -commutator. It follows that O(0) necessarily commutes with L 0 and r + M ⊥ . In terms of the four dimensional quantum numbers of O, this amounts to  .7) are not only necessary but also sufficient conditions for O(0) to define a non-trivial Q i cohomology class. Furthermore, in that case Q 1 and Q 2 define the same cohomology. We refer the reader to [1] for an explanation of these points.
Suppose O(0) defines a non-trivial Q i -cohomology class. We cannot translate this operator away from the origin along the directions orthogonal to the (ζ,ζ) plane without losing Q i -closure, since Q i is not invariant under translations in those directions. We can, however, construct the following twisted translated operator which is still annihilated by Q i . This object can also be written as aζ-dependent linear combination of the R-symmetry components of the multiplet to which the Schur operator belongs, Crucially, thanks to the fact that the generators sl(2) are Q i -exact, the antiholomorphic dependence ofÕ(ζ,ζ) is trivial in cohomology,

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This quantity is generically a half-integer, but as a consequence of four-dimensional sl(2) R selection rules, the OPE of any two cohomology classes χ[O 1 ], χ[O 2 ] is single-valued in the ζ-plane. 6

Affine enhancement of symmetries
The stress tensor of a four-dimensional N = 2 theory sits in a supersymmetry multiplet of typeĈ 0(0,0) in the notation of [37]. The same multiplet contains the sl(2) R symmetry current of the theory, J αβ . Its Lorents and R-symmetry highest-weight component is a Schur operator and determines an element of the chiral algebra with holomorphic dimension two, (2.14) This object is identified with the stress tensor of the chiral algebra. 7 The meromorphic TT OPE is determined by the OPE of R-symmetry currents in four dimensions, and has the expected form with a two-dimensional central charge where c 4d is one of the two conformal anomaly coefficients of the four-dimensional theory [1]. Unitarity in four dimensions requires c 4d > 0, yielding a non-unitary chiral algebra in two dimensions. If the four-dimensional theory is invariant under a continuous flavor symmetry group G F , its spectrum contains a conserved current in the adjoint of the flavor symmetry algebra g F . The latter is contained in a supersymmetry multiplet of typeB 1 , which also includes an sl(2) R triplet of scalars M (IJ) in the adjoint of g F with ∆ = 2. The R-symmetry highest weight component of M IJ is a Schur operator, yielding an element of the chiral algebra with holomorphic dimension one, (2.16) The JJ OPE reveals that this object can be identified with an affine current in two dimensions, satisfying a Kac-Moody algebra based on the Lie algebra g F with level where k 4d is an anomaly coefficient entering the four-dimensional OPE of two flavor currents [1]. The cohomological construction of the previous section can also be performed in theories with N = 3 or N = 4 superconformal symmetry. The spectrum of such a theory, expressed in N = 2 language, contains additional conserved spin 3/2 supersymmetry currents. The latter are contained in supermultiplets of type D 1 2 (0,0) ,D 1 2 (0,0) , which also include 6 This implies that the NS sector is the relevant sector of the chiral algebra from the point of view of the parent four-dimensional SCFT. 7 We refer the reader to [1] for a careful discussion of the relative normalization of T, J 11 ++ in a standard set of conventions for four-dimensional and two-dimensional operators, and similarly for other pairs of four-dimensional and two-dimensional operators discussed below.

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an sl(2) R triplet of spin 1/2 operators Ψ (IJ) α ,Ψ (IJ) α with ∆ = 5/2. The highest-weight components of Ψ,Ψ are Schur operators and yield elements of the chiral algebra with holomorphic dimension 3/2, The operators G,G are supersymmetry currents in two-dimensions. Both Ψ,Ψ and G, G carry implicitly flavor symmetry indices associated to the commutant of the N = 2 Rsymmetry sl(2) R ⊕ C r inside the larger R-symmetry group of the N = 3 or N = 4 theory. Focusing on the case of an N = 4 theory, the larger (complexified) R-symmetry group is sl(4) R and the commutant of sl(2) R ⊕ C r is sl(2) F . The relevant branching rule is where we denoted the fundamental representation of sl(4) R by its Dynkin indices, sl(2) representations by their half-integral spin, and the subscript is the C r charge. Fundamental indices of sl(2) F will be denotedÎ,Ĵ = 1, 2. It follows that the chiral algebra always contains the two-dimensional small N = 4 chiral algebra [38]. The latter is generated by the stress tensor T, two supersymmetry currents GÎ ,GÎ with holomorphic dimension 3/2 in the fundamental of sl(2) F , and an sl(2) F current J (ÎĴ) . The Virasoro modes L 0,±1 , the supercurrent modes GÎ ±1/2 ,GÎ ±1/2 and the modes JÎĴ 0 of the affine current generate a global psl(2|2) symmetry.

Localization argument
Given a four-dimensional N = 2 theory admitting a holographic dual, it is natural to ask what is the bulk analog of the field-theoretic cohomological construction that we have just reviewed. In this section we address this problem in a simplified model. The superconformal algebra on the field theory side is realized on the gravity side as the algebra of superisometries of the background. In particular, the background admits suitable Killing spinors that can be identified with the linear combinations Q 1 , Q 2 of section 2. In light of the cohomological construction on the field theory side, we expect the following picture on the gravity side. If we only switch on sources dual to twisted translated Schur operators on the field theory side, the partition function on the gravity side should be subject to supersymmetric localization and should define an effective dynamics localized on an AdS 3 slice of the original AdS 5 spacetime. The boundary of the AdS 3 slice is identified to the preferred (ζ,ζ) plane singled out by the cohomological construction on the field theory side. Implementing this program rigorously appears challenging in any realistic holographic duality, e.g., in the canonical duality between large N N = 4 SYM theory and IIB string theory on AdS 5 × S 5 . We are not aware of the requisite off-shell formalism for IIB supergravity on AdS 5 ×S 5 , or even for its consistent truncation to N = 8 gauged supergravity on AdS 5 . We can, however, address explicitly a simplified version of the problem, along the following lines.

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Consider an N = 2 SCFT with a flavor symmetry algebra g F . Our main target is N = 4 SYM theory, for which g F = su(2) F (the centralizer of the 4d N = 2 superconformal algebra su(2, 2|2) inside the N = 4 superconformal algebra psu(2, 2|4)), but we may as well keep g F general. According to the standard AdS/CFT dictionary, on the gravity side we find massless gauge fields with gauge algebra g F , which must belong to an N = 4 vector multiplet (half-maximal susy). The vector multiplet is part of the spectrum of a suitable half-maximal supergravity in five dimensions admitting an AdS 5 vacuum. We will consider the truncation of the full supergravity to the N = 4 supersymmetric five-dimensional gauge theory with gauge algebra g F on a non-dynamical AdS 5 background. This setup can be explicitly analyzed using available localization techniques.
We should point out from the outset that the restriction to the vector multiplet is not a bona fide consistent truncation of the full equations of motion of five-dimensional supergravity. 8 It is however guaranteed to be a "twisted" consistent truncation, i.e., to hold in Q-cohomology. Indeed, the corresponding sector of the chiral algebra is just the affine Kac-Moody algebra g F , which is clearly a closed subalgebra.

Summary of the localization results
As the details of our calculations are somewhat technical, we begin with a summary of the main results. Our goal is to show that the five-dimensional super Yang-Mills action defined on AdS 5 localizes to an effective action defined on an AdS 3 slice inside AdS 5 , and determine this effective action. The relevant AdS 3 slice is specified as follows. We can write the Euclidean AdS 5 background in Poincaré coordinates as where R is the AdS 5 radius, z is the AdS 5 radial coordinate, ζ,ζ are complex coordinates on a selected plane on the boundary, ρ, ϕ are polar coordinates along the two other directions on the boundary. The coordinates ζ,ζ are identified with those used in section 2 in the discussion of the chiral algebra. In particular, the plane selected by the cohomological construction is the plane spanned by ζ,ζ. With this notation, the relevant AdS 3 slice of AdS 5 is the one located at ρ = 0 and spanned by ζ,ζ, z, Let us remind the reader that the bosonic field content of maximal super Yang-Mills theory in five dimensions consists of a gauge connection A and five real adjoint scalars, JHEP08(2018)098 denoted here φ 6 , φ 7 , φ 8 , φ 9 , φ 0 . (Our terminology is related to the ten-dimensional origin of these fields, described in the following subsection.) The realization of off-shell supersymmetry used in the localization computation induces a split of the five scalars into (φ 6 , φ 7 ) and (φ 8 , φ 9 , φ 0 ).
After these preliminaries, we can exhibit the value of the localized super Yang-Mills action. It can be written as the sum of two decoupled contributions, (3.5) Here g 2 YM denotes the Yang-Mills coupling of the five-dimensional super Yang-Mills theory, and the symbol tr stands for the trace in a reference representation of the gauge algebra (the fundamental for gauge algebra su(N )). The scalars φ 6 , φ 7 are implicitly evaluated at ρ = 0, i.e. on the AdS 3 slice of AdS 5 . The object A is an emergent complex gauge connection living on the AdS 3 slice of AdS 5 . Its expression in terms of the fields of the original Yang-Mills theory reads

8)
The symbols A ζ , Aζ, A z denote the components of the pullback of the original Yang-Mills connection A from AdS 5 to the AdS 3 slice. The scalars φ 8 , φ 9 , φ 0 are implicitly evaluated on the AdS 3 slice. The quadratic action S free for φ 6,7 is expected to be completely decoupled from the rest of the dynamics on the AdS 3 slice, even if suitable supersymmetric insertions are considered in the path integral. As a result, φ 6,7 are expected to provide only an inconsequential field-independent Gaussian factor in the computation of correlators, and can be effectively ignored. The emergent gauge field A, on the other hand, has dynamics specified by the Chern-Simons action S CS , which according to the classic results of [39][40][41] defines a WZWN theory on the boundary of AdS 3 based on the group G F . This provides a realization of the two-dimensional affine current algebra of the Lie algebra g F , as expected from the cohomological construction on the field theory side.

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The outline of the rest of this section is as follows. The derivation of the above results is described in subsections 3.2, 3.3, 3.4. Further comments about our results are collected in subsection 3.5, in which we also test our findings against predictions from the chiral algebra based on the case of N = 4 super Yang-Mills with gauge algebra su(N ).

Lagrangian with off-shell supersymmetry
We consider maximally supersymmetric Yang-Mills theory in five dimensions on a Euclidean AdS 5 background. Following [25,42] this theory can be constructed in two steps. Firstly, the flat-space 10d maximally supersymmetric Yang-Mills theory with signature (1,9) is formally dimensionally reduced on a five-torus with signature (1,4). Secondly, the external flat metric is replaced with the curved Euclidean AdS 5 metric, minimal coupling to gravity is introduced, as well as extra non-minimal couplings needed for supersymmetry. In order to set up our notation, we review here the field content, Lagrangian, and off-shell supersymmetry variations, following closely [25]. Maximally supersymmetric Yang-Mills theory with off-shell supersymmetry has also been studied in [43] in a variety of backgrounds, including AdS 5 .
Curved 5d spacetime indices are denoted µ, ν = 1, . . . 5. We adopt Poincaré coordinates for the background AdS 5 metric, where a hat is used to denote flat 5d spacetime indices. All dynamical bosonic fields of the 5d theory originate from the 10d gauge connection A M , M = 0, 1, . . . , 9, where 0 denotes the time direction. Upon dimensional reduction we obtain the 5d gauge connection A µ , µ = 1, . . . , 5, as well as five real scalars φ I ≡ A I , I = 6, . . . , 9, 0 in the adjoint representation of the gauge group. The index I is a vector index of the R-symmetry so(4, 1) R . The latter, however, is explicitly broken to so(2) R ⊕ so(2, 1) R by the way off-shell supersymmetry is realized below. Correspondingly, it is useful to introduce the notation We use anti-Hermitian generators for the gauge algebra and the 10d field strength reads Its components after dimensional reduction are given by In order to close supersymmetry off-shell we also need to introduce seven real auxiliary scalars K m , m = 1, . . . , 7 in the adjoint representation of the gauge group. Their vector so (7) index is raised and lowered with the flat invariant δ mn .

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All fermionic degrees of freedom are encoded in a 16-component Grassmann-odd 10d Majorana-Weyl gaugino Ψ α , α = 1, . . . , 16, in the adjoint representation of the gauge group. The chiral blocks of 10d gamma matrices are denoted Γ M αβ ,Γ M αβ , and we also use the notation (3.15) Weyl indices are henceforth suppressed. After dimensional reduction and coupling to the curved AdS 5 background, the 10d covariant derivative of the gaugino D M Ψ gives rise in five dimensions to where ω µλτ is the spin connection associated to the background vielbein (3.11).
The off-shell supersymmetric Lagrangian reads where tr denotes the trace in a reference representation (the fundamental for gauge algebra su(N )), and we defined Note that we adopted the customary compact notation in which the spacetime indices µ, ν are curved and thus raised with the metric (3.10), while the indices I, J are flat and raised with the so(4, 1) R metric η IJ = diag(1, 1, 1, 1, −1). In a similar way we have where eλ µ denotes the inverse of the 5d vielbein (3.11). All spinor bilinears in (3.17) and in the following are Majorana bilinears. Further details about our spinor conventions are collected in appendix A. The Lagrangian (3.17) is invariant up to total derivatives under the off-shell supersymmetry transformations

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In these expressions is a Grassmann-even 16-component Majorana-Way spinor with the same chirality as Ψ. It satisfies the AdS 5 Killing spinor equation Note that in this equationΓ µ =Γλeλ µ , ∇ µ = ∂ µ + 1 4 ω µλτ Γλτ . Let us also stress that the compact notation Γ M N F M N in (3.23) is subject to remarks similar to those around (3.19) and (3.20) above. We have also introduced a set ν m , m = 1, . . . , 7 of auxiliary Grassmanneven spinors with the same chirality as , determined up to an so(7) rotation by the algebraic relations The so (7) index m on ν m is raised with δ mn . The square of the supersymmetry transformations (3.21)-(3.24) can be written as combinations of the bosonic symmetries of the theory without using the equations of motion. More precisely, one has where we utilized the spinor bilinears The 5d vector v µ is a Killing vector for the AdS 5 background metric, ∇ (µ v ν) = 0. Note that δ 2 K m contains an so (7) rotation, which is a symmetry of the Lagrangian. All our formulae can be obtained as an analytic continuation of the formulae given in [25] for the case of the five-sphere. More precisely, the radius r of S 5 is related to the radius R of Euclidean AdS 5 as R = ir .
Note, however, that the coordinate system utilized in [25] is different from the one adopted here, and would correspond in the case of Euclidean AdS 5 to the disk model of hyperbolic space, rather than the half-space model.

Identification of the relevant supercharge
Our first task in the implementation of the localization argument is the identification of the Killing spinor corresponding to the relevant supercharge on the field theory side. As JHEP08(2018)098 reviewed in section 2, for a unitary theory Q 1 and Q 2 define the same cohomology classes on the field theory side. From the point of view of localization it is most convenient to consider (3.34) The holomorphic sl (2) factor on the fixed plane is Q-closed, and the twisted antiholomorphic factor sl(2) is Q-exact. Note, however, that Q is not nilpotent, but rather satisfies On the gravity side, if we select the Killing spinor corresponding to Q , the associated Killing vector v contains the spacetime action M ⊥ , consisting of rotations in the directions orthogonal to the fixed plane. As a result, we localize on the fixed point set of M ⊥ , consisting of the fixed plane itself.
In order to identify the Killing spinor corresponding to Q we have to analyze the space of solutions to the Killing spinor equation (3.25) with Λ given in (3.18). We refer the reader to section A.2 in the appendices for a thorough discussion and for the explicit expression for the Killing spinor . Let us summarize here some of its key properties. To this end, it is convenient to use complex coordinates ζ,ζ in the x 1 x 2 plane and polar coordinates ρ, ϕ in the x 3 x 4 plane, while the field-dependent gauge parameter that enters the square of the supersymmetry transformations (3.27)-(3.31) is given by Our Killing spinor induces no so(2, 1) R R-symmetry rotation, but yields a non-zero so(2) R rotation, Recall that the Lie derivative of a spinor in the direction of a Killing vector k µ is given by Using this expression one can check that our Killing spinor is invariant under the action of the Killing vectors associated to the holomorphic conformal generators in the (ζ,ζ) plane. More precisely, if we consider the Killing vectors This corresponds to the fact that our supercharge commutes with the holomorphic conformal generators in the (ζ,ζ) plane. Furthermore, we expect the anti-holomorphic generators to be exact. This expectation is confirmed by checking that each of the Killing vectors can be written in the form Γ µ for a suitable Killing spinor .
Once the suitable Killing spinor is identified, we are left with the task of finding the associated auxiliary spinors ν m satisfying (3.26). We refer the reader to section A.4 in the appendix for more details on this point.

BPS locus and classical action
The localization argument ensures that in the computation of Q-closed observables the path integral localizes to the BPS locus 9 In particular this implies that δ 2 annihilates all fields on the BPS locus. Making use of the expression for δ 2 recorded in the previous section one can verify that, for our choice of supercharge, this implies where we introduced a new 3d curved spacetime indexμ = ζ,ζ, z and v I φ I is given in (3.38). Let us point out that, in all the above equations, the covariant derivative acts on spacetime scalars and therefore contains the gauge field but no spacetime connection. Once the constraints coming from δ 2 = 0 are implemented, one can show that the 16 equations δΨ = 0 are all solved by determining the seven auxiliary scalars K m as a functional of all other bosonic fields. In summary, BPS locus: (3.50) 9 The fact that only field configurations invariant under the action of Q can contribute to the path integral can be argued on very general grounds, independently of the choice of localizing terms. See, for instance, section 3.1 of [44]. We restrict our attention to the bosonic part of the BPS locus because we are interested in extracting the classical action. The integral over the fermionic component of the BPS locus is best thought of as part of the computation of the one-loop determinant, which lies beyond the scope of this work.

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We refrain from recording here the explicit expressions for the auxiliary scalars K m in terms of A µ , φ I , which are lengthy and not particularly illuminating.
As a next step in the localization we evaluate the classical Lagrangian (3.17) on the BPS locus (3.50). A straightforward but tedious computation shows that the entire bosonic Lagrangian, including the appropriate volume form, collapses on the BPS locus to a sum of total derivatives. More precisely, one finds where the quantities Y are suitable functionals of the gauge field and scalars whose explicit expressions are not recorded for the sake of brevity. On the l.h.s. the notation d 4 xdz is a shorthand for the five form dx 1 ∧ dx 2 ∧ dx 3 ∧ dx 4 ∧ dz, and by a similar token we have omitted wedge products on the r.h.s. In checking (3.51) it is essential to take into account the factors coming from the expression of the AdS 5 volume form in the (ζ,ζ, ρ, ϕ, z) The classical action on the BPS locus is given by the integral of (3.51) over the factorized domain Let us discuss the possible boundary contributions. Of course, since all fields are periodic in the angular variable ϕ no boundary term can be generated by integrating ∂ ϕ Y ϕ . We assume that all fields fall off sufficiently rapidly at infinity in all directions orthogonal to the radial coordinate z of AdS 5 . As a result, we get no contributions from ∂ ζ Y ζ + ∂ζYζ, while ∂ ρ Y ρ contributes exclusively via the lower limit of integration ρ = 0. The asymptotic behavior of fields in the z direction is more subtle and is related to the implementation of the AdS/CFT prescription for the computation of correlators. The goal of our localization computation is the identification of an effective 3d bulk theory that could be then used to compute correlators of twisted-translated Schur operators according to the standard prescription. For the purpose of identifying the 3d theory we do not need to consider boundary terms coming from the z direction.
In conclusion, the relevant classical action on the BPS locus can be written as where we anticipated that Y ρ is actually independent of ϕ and we performed the ϕ integration. The fact that all fields are evaluated at ρ = 0 shows manifestly the expected localization of the dynamics on the ζζ plane which is fixed under the action of the Killing vector (3.37) associated to our Killing spinor. Let us now record the expression of the integrand in (3.54) in a convenient way. To this end, it is useful to trade the scalar fields φ A , A = 8, 9, 0 of the original Yang-Mills theory with the components of a one-form Φμ living on the AdS 3 slice of AdS 5 identified JHEP08(2018)098 by ρ = 0 and parametrized by ζ,ζ, z. This twist is achieved by means of a suitable object V Aμ built from bilinears of the Killing spinor in the following way. To begin with, let us define the 5d three-vector where µ 1 , µ 2 , µ 3 are curved 5d indices. Our choice of spinor breaks 5d covariance by selecting the plane spanned by ρ, ϕ. It is thus natural to consider the components X Aμρϕ withμ = ζ,ζ, z. The sought-for intertwiner V Aμ is then constructed as where the prefactor ρ has been introduced to guarantee finiteness of the limit. The relation between the scalars φ A and the twisted one-form Φ µ is then where the normalization factor has been chosen for later convenience, and all fields are implicitly evaluated at ρ = 0. More explicitly, in our conventions the components of Φμ are given by We can finally present the explicit expression for the classical action (3.54). It can be written as the sum of a non-topological and a topological term, Let us remind the reader that all quantities are implicitly evaluated at ρ = 0. In the second line we have adopted a differential form notation suppressing wedge products and, by slight abuse of notation, A, F denote the restriction of the 5d gauge connection and field strength the AdS 3 slice spanned by coordinates ζ,ζ, z. The symbol d A denotes the exterior gauge-covariant derivative

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Let us point out that the appearance of the topological term tr Φ 3 in (3.63) is a consequence of the cubic term ABC φ A [φ B , φ C ] in the scalar potential of the original Yang-Mills Lagrangian (3.17).
It is useful to construct the quantity which transforms as a connection since Φ is an adjoint-valued one-form. Thanks to the identity the topological part of the action S top can be written compactly as a Chern-Simons action, (3.67) The minus sign in front of the Chern-Simons term is introduced because, in our conventions, the pairing tr(ab) is negative definite, since we are using antihermitian generators. For instance, for gauge algebra su(n), 10 we adopt the standard normalization with tr denoting the trace in the fundamental representation, where a, b = 1, . . . , n 2 − 1 are adjoint indices of su(n). As a result, we may also write (3.69) The localization technique can be applied with arbitrary insertions of Q-closed functionals of the fields. The functionals should be well-defined on the AdS 3 slice and have a vanishing supersymmetry variation (3.21)-(3.22) on that slice. It is not hard to check that functionals built from A ζ , Aζ and A z (and independent of φ 6,7 ) satisfy these requirements, 11 in agreement with the conclusion that φ 6,7 can be consistently decoupled. 10 This 5d/3d gauge algebra should not be confused with the gauge algebra su(N ) of 4d N = 4 SYM. The case n = 2 will be relevant below, where we make contact to the chiral algebra dual to 4d su(N ) N = 4 SYM by specializing the 5d gauge algebra to be su(2)F ⊃ su(4)R, the R-symmetry algebra of SYM. 11 It also appears that these are the only admissible functionals. At first sight, the linear combinations seem admissible, since they have a vanishing supersymmetry variation for ρ → 0. However, assuming that the Cartesian components A3, A4 of Aµ are smooth near ρ = 0, the quantity Aρ = cos ϕ A3 + sin ϕ A4 does not admit a unique limit for ρ → 0, but rather depends on the ϕ angle with which we approach ρ = 0, while the quantity Aϕ = ρ(− sin ϕ A3 + cos ϕ A4) vanishes for ρ → 0. These combinations must therefore be discarded.

Remarks
Our implementation of the localization recipe is different from the one usually applied to supersymmetric theories on Euclidean compact manifolds. In the latter case it is customary to supplement the classical Lagrangian by an explicit Q-exact localizing term, S tot = S + t QV . The functional V and the reality conditions on the fields are chosen in such a way as to guarantee that, as t → ∞, the path integral converges and localizes on a suitable real slice of (a subspace of) the BPS locus. Different choices for V and for the reality conditions yield different localization schemes. The computation of the previous section shows that, in any localization scheme, the classical action must reduce to (3.61). This conclusion only relies on the form of BPS locus (3.50) without choosing a specific real slice in the space of field configurations. For example, the BPS locus allows for a non-zero profile for the scalars φ i , but they enter the classical action via the quadratic, algebraic action S non-top only, and therefore decouple from the dynamics after localization.
We are ready to go back to our main physical goal -the determination of the bulk holographic dual to the 2d chiral algebra. The bulk action (3.61) must be supplemented with suitable boundary conditions for the fields A, φ 6,7 in order to implement the holographic recipe for the computation of correlators in the boundary theory. We should also contemplate the possibility of additional boundary terms to the AdS 3 action. The boundary conditions and boundary terms for the theory on the AdS 3 slice could be derived via localization of an appropriate set of boundary conditions and boundary terms in the original super Yang-Mills theory defined in AdS 5 . These 5d data are constrained by the requirement of compatibility with the action of the supercharge Q selected for localization. 12 For the problem at hand, we can follow a simpler route without making reference to the parent 5d bulk theory. Since we have already argued that the scalars φ 6,7 play no relevant role, we focus on A only.
To begin with, we observe that supercharge Q induces an asymmetry in the treatment of holomorphic and antiholomorphic components A ζ , Aζ. This is most easily detected by looking at the expressions (3.58)-(3.60) for the components of the twisted one-form Φ. Inspection of (3.58)-(3.60) reveals a hierarchy of the three components with respect to the radial coordinate of AdS 3 . In particular, if we prescribe the boundary conditions φ 8,9,0 ∼ z 2 as z → 0 in order to get a finite Φ ζ , then Φζ and Φ z necessarily vanish at the boundary. We can regard A ζ , Aζ, A z as the supersymmetrizations of Φ ζ , Φζ, Φ z , and argue that the asymmetric pattern for the holomorphic and antiholomorphic components persists. This mechanism is the bulk dual of the emergence of a purely meromorphic dynamic on the field theory side, once we consider cohomology classes of the supercharges Q 1 , Q 2 .
From this observation we can deduce the correct boundary terms that must be added to the action. As explained for instance in [41,[46][47][48], meromorphic boundary conditions are selected by supplementing the Chern-Simons action with the boundary term

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where s is a constant to be fixed momentarily and, by slight abuse of notation, we used µ, ν to denote two dimensional curved indices on ∂AdS 3 . The combined variation of the bulk action (3.67) and the boundary action (3.71) with respect to the gauge field takes the form where The bulk term in the variation imposes that the connection be flat. The currents J ζ , Jζ entering the boundary terms of the variation are identified with the currents of the boundary CFT. Because of our choice of supercharge we know that the antiholomorphic component Jζ of the boundary current should be zero. Thus, we must select s = 1. This implies that k 2d = k, which is negative in our case (recall (3.69)), in agreement with field theory expectations. Our choice differs from the one in [48], where s = sgn k is advocated on the basis of the following argument. The boundary action (3.71) contributes to the boundary stress tensor, which in complex components reads As explained in [48], with these conventions a positive coefficient of A a ζ A b ζ in T bdy ζζ corresponds to a positive definite contribution to the boundary energy in a semi-classical picture, leading to the prescription s = sgn k and k 2d = |k|. In other terms, in the standard case unitarity of the boundary CFT is enforced by hand. In our case, we must let supersymmetry dictate the correct boundary conditions, and we naturally land on a non-unitary chiral algebra.
We can now specialize (3.67) to the case in which the boundary theory is 4d N = 4 super Yang-Mills with gauge algebra su(N ). Regarded as a 4d N = 2 theory, this theory has a flavor symmetry su(2) F , the commutant of su(2) R × u(1) r inside su(4) R . The gravity dual of N = 4 super Yang-Mills is maximal gauged supergravity with gauge group su(4) R . The corresponding gauge coupling function, evaluated at the origin of the scalar potential corresponding to the AdS 5 vacuum, is [49] at leading order in 1/N . We have to consider the branching su(4) R → su(2) R × u(1) r × su(2) F and restrict to the su(2) F factor. One can easily check that this does not affect the normalization of the Yang-Mills kinetic term, so g YM = g su(2) F = g su(4) R . As a result, in this case the Chern-Simons level k in (3.67) is

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We can compare this result with the level of the affine current algebra of the su(2) F current JÎĴ in the chiral algebra dual to N = 4 super Yang-Mills with gauge algebra su(N ). One finds [1] which agrees with the Chern-Simons level k (3.67) in the large N limit. Finally, let us briefly comment about quantum corrections to the classical result (3.61). While it may be of some technical interest to compute the one-loop determinant factor associated to fluctuations of super Yang-Mills fields transversely to the localization locus, the physical relevance of such a calculation is a priori unclear. What would be physically relevant is a calculation of quantum fluctuations in a fully consistent holographic theory, e.g. in IIB string field theory on AdS 5 × S 5 , but this is clearly beyond the scope of this work -indeed even the classical problem seems prohibitively hard in the complete theory. Given the agreement of (3.61) with the expected large N result, we may speculate that the only effect of quantum fluctuations is an O(1) shift of the Chern-Simons level.

Towards the complete holographic dual
In this section we propose a strategy to determine the complete holographic dual for the chiral algebra of N = 4 SYM theory. A straightforward task is the identification of the linearized bulk modes that correspond to non-trivial boundary operators in Q-cohomology.
Their non-linear interactions are however extremely complicated, and extending the localization procedure to the full theory is presently beyond our technical abilities. Encouraged by the emergence of a Chern-Simons action in the simplified model discussed above, and drawing inspiration from minimal model holography, we will outline a bottom-up construction of the dual theory. We will argue that it is a Chern-Simons theory with gauge algebra a suitable higher-spin Lie superalgebra, defined implicitly by the large N OPE coefficients of N = 4 SYM.
We begin in section 4.1 with a review of the super chiral algebra conjecture of [1]. We give a simple argument in favor of the conjecture in the large N limit. The generators 13 of the chiral algebra are the single-trace Schur operators, which are in 1-1 correspondence with KK sugra modes obeying the Schur condition. In section 4.2 we give the details of this correspondence. Following the blueprint of minimal model holography, we propose in section 4.3 that the sought after holographic dual is AdS 3 Chern-Simons theory with gauge algebra given by the wedge algebra of the large N super W-algebra. While it is unclear whether such a wedge algebra exists for finite N , we outline its construction at infinite N .

The N = 4 SYM chiral algebra at large N
As we have reviewed in section 2.2, the chiral algebra always admits the small N = 4 superconformal algebra (SCA) as a subalgebra. The global part of the small N = 4 SCA is psu (1, 1|2), and we henceforth organize the operator content in terms of psu(1, 1|2) 13 A generator of a chiral algebra is an operator that does not appear in the non-singular OPE of other operators. In the mathematics literature, what we call a generator is usually referred to as a strong generator.
14 A fully explicit statement of the super W-algebra conjecture for N ≥ 3 is then the following: (i) as a vector space, the chiral algebra is the linear span of derivatives of J (n) , G (n) ,G (n) , T (n) (n = 0, . . . , N − 2) and their conformally ordered products; (ii) the operators J (n) , G (n) ,G (n) , T (n) (n = 0, . . . , N − 2) cannot be written as conformally ordered products of derivatives of other operators. 15 In fact, for c 2d = −9, the stress tensor T is not an independent generator, but is rather identified with the Sugawara stress tensor built from the affine current JÎĴ .

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(The ellipsis in (4.2) represents additional multiplets that are not relevant for our discussion, since they do not contain Schur operators.) The degeneracies in (4.2) are accounted for by the dimensions of the su(2) F representations. The fact that the operators J (n) are supergenerators of the chiral algebra is easily established. Indeed, they arise from 4d 1/2 BPS operators, which are absolutely protected against quantum corrections; 16 as they correspond to generators of the 1/2 BPS chiral ring it is clear that they cannot appear in the non-singular OPE of other operators. The hard part of the conjecture of [1] is showing that these are all the supergenerators. The chiral algebra is specified by a non-trivial BRST procedure, which in physical terms amounts to selecting operators obeying the Schur shortening condition in the interacting theory.
The conjecture, however, can be proved at infinite N . All Schur operators are in particular 1/16 BPS operators of N = 4 SYM, i.e., operators in the cohomology of a single Poincaré supercharge. This cohomology was studied in [50], where it was proved that at infinite N it is obtained by taking arbitrary products of 1/16 BPS single-trace operators, 17 which were further shown to be in 1-1 correspondence with single 1/16 BPS gravitons in the dual AdS 5 × S 5 supergravity. Schur operators are in the simultaneous cohomology of two Poincaré supercharges of opposite chirality, say Q 1 − andQ 2− in the conventions of [1]. Specializing the results of [50] to this double cohomology, we find that at infinite N it is given by arbitrary products 18 of the single-trace Schur operators corresponding to J (n) , G (n) ,G (n) , T (n) , n ∈ Z + . This shows that these operators comprise the full set of generators for the chiral algebra at infinite N .

Single-trace Schur operators of N = 4 SYM and supergravity
Having established the super chiral algebra conjecture of [1] for infinite N , we now proceed to give more details on the single-trace Schur operators in (4.2) and to map them to the Kaluza-Klein modes in type IIB supergravity on AdS 5 × S 5 [51]. Our conclusions are summarized in table 4.2.
All Kaluza-Klein modes in the compactification of type IIB supergravity on AdS 5 × S 5 are dual to operators that are organized in 1/2-BPS short N = 4 multiplets of type [0,n+2,0](0,0) . In Lagrangian language, the N = 4 superprimaries of these multiplets can be written as where A 1 , . . . , A n+2 are vector indices of so(6) R , X A are the real scalars of N = 4 super Yang-Mills, and curly brackets denote the traceless symmetric part. 16 Recall that 1/2 BPS multiplets cannot recombine into long multiplets [37]. 17 The product operation relevant here is the commutative product induced by the ordinary OPE of the 4d theory, which is non-singular for 1/16 BPS operators. 18 In this statement, the product operation is again the commutative product induced by the standard OPE of the 4d theory. The twisted translation prescription of [1] deforms this commutative algebra into the chiral algebra that we are interested in. There is an easy general argument that the generators of the chiral algebra are a subset of the generators of the commutative algebra. This is just what we need (the other inclusion is obvious in our case).

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Schur operator tr (Q n+2 ) tr (λ + Q n+1 ) tr (λ+ Q n+1 ) tr (λ +λ+ Q n ) ∆ n + 2 n + 5 2 n + 5 2 n + 3 chiral algebra operator KK mass m 2 R 2 = n 2 − 4 |m|R = n + 1 2 |m|R = n + 1 2 |m|R = n 2 + 2n Table 1. Families of Schur operators of N = 4 super Yang-Mills theory. For each family we give the N = 2 supermultiplet in the notation of [37], the schematic form of the superprimary in the multiplet, and the schematic form of the Schur operator. The quantum numbers ∆, h, J F are the 4d scaling dimension of the Schur operator, the 2d holomorphic dimension of the chiral algebra element, and the half-integer sl(2) F spin of both, respectively. The scalar fields Q, Z,Z are defined in (4.4), while λ,λ denote the gaugini in N = 2 language, which are a subset of all gaugini in N = 4 language. The KK modes are given in the notation of [51]. All families are labelled in such a way that the range of n is n = 0, 1, 2, . . . .
We have already reviewed the decomposition of an N = 4 multiplet of type B into superconformal multiplets of N = 2, see (4.2). In order to elucidate the connection between the branching rule (4.2) and the Lagrangian presentation (4.3), it is convenient to reorganize the scalars X A schematically as where a = 1, . . . , 4, I = 1, 2 is a fundamental index of su(2) R ,Ĵ = 1, 2 is a fundamental index of su(2) F , and σ a IĴ are chiral blocks of so(4) gamma matrices. The scalar Z is the complex scalar in the N = 2 vector multiplet, while Q IĴ are the scalars in the N = 2 hypermultiplet. We can now easily identify a Lagrangian realization of the N = 2 superconformal primary for each of the multiplets on the r.h.s. of (4.2). Let us list them together with their R-and F -isospins and r charges, Each of these N = 2 supermultiplets yields a Schur operator. Let us discuss them in turn and relate them to the associated Kaluza-Klein mode in the spectrum of type IIB supergravity on AdS 5 × S 5 .
Multiplets of typeB 1 2 (n+2) . In this case the Schur operator is directly the su(2) R highest-weight component of the superconformal primary listed in (4.3). It follows that JHEP08(2018)098 the operator in the chiral algebra is simply (n + 2) , J F = 1 2 (n + 2) , n = 0, 1, . . . , (4.5) where we summarized its 2d quantum numbers. The gravity duals of the N = 4 chiral primaries in (4.3) are given by the Kaluza-Klein modes named π I 1 in table III of [51]. It follows that the gravity duals of the operators (4.5) are given by the subset of the modes π I 1 corresponding to the J R = 1 2 (n + 2), J F = 1 2 (n + 2) representation of su(2) R ⊕ su(2) F inside the [0, n + 2, 0] of so(6) R . The masses of the π I 1 Kaluza-Klein tower are m 2 R 2 = (n − 2)(n + 2) , n = 0, 1, . . . The case n = 0 deserves special attention. The superconformal primary ofB 1 is the moment map for the su(2) F flavor symmetry. The associated operator in the chiral algebra J (0)ÎĴ ≡ JÎĴ is the affine su(2) F current of the small N = 4 subalgebra. The dual scalar mode in supergravity has a negative mass-squared that saturates the Breitenlohner-Friedmann bound [52].
Multiplets of types D 1 2 (n+1)(0,0) andD 1 2 (n+1)(0,0) . In this case the Schur operator is a component of a super-descendant of the scalar operator listed in (4.2). More precisely, for D 1 2 (n+1)(0,0) we need to act withQ Iα , obtaining a right-handed spinor operator of the schematic form where we recorded explicitly the part coming from the action ofQ Iα on Z, which yields the N = 2 gauginoλ Iα , but we omitted additional terms arising from the action ofQ Iα on the N = 2 hypermultiplet scalars. The quantum numbers of the operator in (4.7) are and its su(4) R orbit is the one of the N = 4 superdescendant of (4.3) in the [0, n + 1, 1] representation of su(4) R . The Schur operator is the highest-weight component of Ψ (n) and the associated chiral operator is then Completely analogous considerations hold for typeD 1 2 (n+1)(0,0) multiplets. The analog of Ψ (n) , denotedΨ (n) , is built using the supercharge Q I α and thus contains a λ I α insertion. It has the same quantum numbers as Ψ (n) , except r = − 1 2 , andG (n) is the associated operator in the chiral algebra.

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The minus sign is relative to the positive masses of the excited Kaluza-Klein modes in the tower of the 5d gravitino.
Multiplets of typeĈ 1 2 n(0,0) . In this case the Schur operator is a component of the operator obtained acting with one Q and oneQ on the superprimary in (4.2). Schematically, we have . . Q InĴn ) + . . . , n = 0, 1, . . . , (4.11) where we omitted several other terms for the sake of brevity. The quantum numbers of this 4d operator are For n = 0 the multipletĈ 0(0,0) contains the 4d stress tensor and the Schur operator is a component of the su(2) R symmetry current. The operator T (0) ≡ T in the chiral algebra is the 2d stress tensor. On the gravity side, we find the massless vectors associated to the Killing vectors of S 5 . The families discussed above have a natural Z 2 grading corresponding to even modes n = 0, 2, . . . , and odd modes n = 1, 3, . . . . The series for even n constitutes a consistent truncation of the chiral algebra. For n even, J (n) and T (n) have integer holomorphic dimension, while G (n) andG (n) have half-integer holomorphic dimension. These assignments obey the standard spin/statistics connection. On the other hand, for n odd the situation is reversed, and the spin/statistics connection is violated. There is of course no contradiction -this is the generic case for chiral algebras associated to N = 2 SCFTs.

Comments on the full higher-spin algebra
Motivated by the emergence of an AdS 3 Chern-Simons theory in the localization computation of section 3, we believe that the bulk dual of the full chiral algebra is a higher-spin AdS 3 Chern-Simons theory. This expectation is in line with known examples of minimal JHEP08(2018)098 model holography (see [5] for a review). From this perspective, we are left with the task of determining the correct higher-spin Lie superalgebra in which the Chern-Simons gauge connection takes values.
Before proceeding, it is useful to review the well-understood case in which the bulk theory is AdS 3 Chern-Simons with gauge algebra sl(n) ⊕ sl(n). This bulk theory describes gravity coupled to massless higher spin fields. In order to identify the states associated to the physical graviton it is necessary to specify an embedding of sl(2) in sl(n). As explained in [19], the bulk theory must be supplemented by suitable boundary conditions in order to guarantee an asymptotically AdS 3 geometry. The interplay between the sl(2) embedding and these boundary conditions determines the asymptotic symmetry algebra of the bulk theory, which is furnished by two copies (left-moving and right-moving) of the same classical infinite-dimensional Poisson algebra. Interestingly, this physical construction based on the asymptotic symmetry algebra is equivalent to the classical Drinfel'd-Sokolov (DS) Hamiltonian reduction of sl(n) associated to the prescribed embedding sl(2) ⊂ sl(n). In the case of the principal embedding, the outcome of the DS reduction is the classical W n algebra, whose quantization yields the quantum W n algebra.
If the DS reduction provides the natural way to get the boundary W-algebra from the bulk Lie algebra, the notion of wedge algebra, explored in [53] in great generality, proves extremely useful for proceeding in the opposite direction. Let the generators of the Walgebra be denoted as W s (ζ), where s labels the integer holomorphic dimension of the generator. Let W s , ∈ Z be the modes in the Laurent expansion of W s (ζ). The vacuum preserving modes are W s , | | < s , (4.15) and preserve both the left and right sl(2) invariant vacuum. Our goal is to define a finitedimensional Lie algebra generated by the vacuum preserving modes (4.15). A naïve truncation of the commutators of the original W-algebra fails in general, due to the non-linear terms that may appear on r.h.s. of the commutators of the vacuum preserving modes. The crucial observation is that, if the W-algebra can be defined for arbitrary values of the central charge c and satisfies additional non-degeneracy assumptions listed in [53], 19 then all non-linear terms on the r.h.s.s of commutators of vacuum preserving modes are suppressed in the limit c → ∞. Furthermore, central terms do not contribute if we restrict to vacuum preserving modes. It follows that the algebra becomes linear and, since associativity of the parent W-algebra holds for any c, we are guaranteed to obtain a bona fide finite-dimensional Lie algebra satisfying all Jacobi identities. An essential property of the wedge algebra construction is that, if the starting point is a W-algebra W DS (g) obtained by DS reduction of a finite-dimensional Lie algebra g, then the wedge algebra of W DS (g) reproduces g itself. In particular, the wedge algebra of W n is sl(n). Even though we have reviewed a purely bosonic example, the extension of these considerations to graded Lie JHEP08(2018)098 algebras does not pose any essential difficulty. In particular, the arguments above can be straightforwardly applied to W-superalgebras and Lie superalgebras in the NS sector, which is the sector relevant from the four-dimensional point of view. In our problem, the role of W n is played by the chiral algebra of N = 4 SYM with gauge algebra su(N ). The existence of this chiral algebra is guaranteed if its central charge is tuned to the value determined by the cohomological construction, c 2d = −3(N 2 − 1). It is not clear, however, if for N ≥ 3 this chiral algebra can be deformed to arbitrary c. As a result, we cannot guarantee the existence of a wedge algebra, which would be the natural candidate for the sought after Lie algebra in the bulk.
If we consider the case of infinite N , however, we can infer the existence of a wedge algebra, which is an ordinary (linear) Lie algebra, albeit infinite dimensional. The argument relies on large N factorization, and goes as follows. We have established in section 4.1 that, in the large N limit, the supergenerators of the chiral algebra are in 1-to-1 correspondence with single trace 1/2 BPS operators of N = 4 SYM theory. Thanks to the protection ensured by supersymmetry, their correlators can be computed in the free field theory limit. We normalize the fundamental adjoint scalars of N = 4 SYM in such a way that their contraction yields schematically where x, y, z, w are fundamental indices of su(N ), we suppressed all spacetime and Rsymmetry dependence, and we restricted to the leading term at large N . With the aid of the standard double-line notation, it is elementary to show the following schematic scalings, : trX k 1 trX k 2 : : trX k 1 trX k 2 : ∼ g 2(k 1 +k 2 ) YM N k 1 +k 2 = λ k 1 +k 2 , trX k 1 trX k 2 : trX k 3 trX k 4 : ∼ g k 1 +k 2 +k 3 +k 4 YM N 1 2 (k 1 +k 2 +k 3 +k 4 )−2 = λ 1 2 (k 1 +k 2 +k 3 +k 4 ) N −2 , (4.17) where : : denotes normal ordering and λ = g 2 YM N is the 't Hooft coupling. 20 If we modify the normalization of the single trace operators, setting the previous relations may then be written in the simpler form

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These relations constrain the N dependence of the OPE coefficients in the OPE of two O k operators. Very schematically, we may then write where we have only kept track of the N dependence. Furthermore we have only focused on potential singular terms in the OPE, and in particular we supposed (k 3 , k 4 ) = (k 1 , k 2 ). 21 As we can see, if double trace operators enter the singular part of the OPE of two single trace operators, the corresponding OPE coefficient is suppressed by a power of N −2 . It is not hard to convince oneself that this pattern persists for all multi-trace operators: if a trace-m operator enters the singular part of the OPE, it appears with a power N 2−2m . This argument implies that all non-linear terms in the chiral algebra must be suppressed at large N . As a result, the obstruction to the consistency of the wedge algebra generated by the vacuum preserving modes is removed, and we obtain a well-defined, infinite-dimensional Lie (super)algebra. This is our candidate for the higher-spin Lie (super)algebra in the bulk. All the necessary information for determining the structure constants of this Lie algebra is contained in the OPE of single-trace operators in the 1/2 BPS chiral ring of N = 4 SYM. It would be desirable, however, to have a more direct construction of this higherspin algebra, along the lines of [5] in the context of minimal model holography. Such investigation is left for future work. Let us list here the expected vacuum preserving modes of the operators in (4.1) that generate the wedge Lie algebra, suppressing su(2) F indices for simplicity: = 0, ±1, · · · ± (1 + 1 2 n), = ± 1 2 , ± 3 2 , · · · ± (1 + 1 2 n) .

Discussion
In this work we have addressed the problem of determining the holographic dual of the protected chiral algebra of N = 4 SYM theory with gauge algebra su(N ) in the large N limit. The resulting picture is the following. The cohomological construction on the field theory side is mirrored by supersymmetric localization in the bulk. By virtue of this localization, type IIB supergravity on AdS 5 × S 5 reduces to a Chern-Simons theory defined on an AdS 3 slice of the AdS 5 space. The gauge algebra of the Chern Simons theory is an infinite-dimensional supersymmetric higher spin Lie algebra, whose structure can a priori be extracted from the coefficients in the OPE of the single-trace 1/2 BPS generators of the chiral ring of N = 4 SYM theory.

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Although we were not able to provide a proof for all aspects of the above picture, we have collected several pieces of evidence in favor of it. To begin with, we have implemented the localization program explicitly in a simplified setup, illustrating how an AdS 3 Chern-Simons theory emerges non-trivially from a five-dimensional gauge theory on AdS 5 . Secondly, we have established the super chiral algebra conjecture of [1] in the case of infinite N , providing the correspondence between supergenerators of the chiral algebra, single-trace 1/2 BPS generators of the chiral ring of N = 4 SYM, and Kaluza-Klein modes of type IIB supergravity on AdS 5 × S 5 . Finally, we have identified a natural candidate for the higherspin algebra in which the Chern-Simons connection takes values. It is the wedge algebra of the chiral algebra, i.e. the (infinite-dimensional) Lie algebra generated by the vacuum preserving modes of the generators of the chiral algebra. We furnished an argument based on large N factorization for the existence of this wedge algebra, and we have connected its structure to the OPEs of single-trace 1/2 BPS scalar operators of N = 4 SYM theory.
It is interesting to contrast this four-dimensional setup to the six-dimensional case in which the superconformal field theory is the (2, 0) theory of type A N −1 . As established in [54], the protected chiral algebra in this case coincides with W N . The latter is defined for arbitrary values of the central charge c and admits sl(N ) as its wedge algebra. The gravity dual of a W N chiral algebra is thus an AdS 3 Chern-Simons theory with gauge algebra sl(N ). These facts are well-known in the context of minimal model holography [5], and the large N limit is also well understood. This problem is thus simpler from a bottom-up point of view. From a top-down perspective, however, this case is considerably more complicated. In the large N limit we can access the holographic dual of the (2, 0) theory of type A N −1 via eleven-dimensional supergravity on AdS 7 ×S 4 . In contrast to the case studied in this paper, it is not possible to single out a simplified setup without dynamical gravity to perform the localization computation. As a result, a direct check of the emergence of the claimed Chern-Simons theory would require a full-fledged localization computation in supergravity.
The cohomological construction of the protected chiral algebra in 4d SCFTs also has a counterpart for 3d SCFTs [55,56]: the protected sector gives rise to a one-dimensional topological algebra. The construction requires at least N = 4 in three dimensions, and is in particular applicable to the maximally supersymmetric case N = 8. In the latter situation the holographic dual can be accessed via eleven-dimensional supergravity on AdS 4 × S 7 . In analogy to the case discussed in this work, it is possible to single out a simplified model, involving the dynamics of a vector multiplet in AdS 4 , in order to perform the localization computation. The outcome of the localization procedure is expected to live on an AdS 2 slice of AdS 4 , and it would be interesting to show this explicitly.
A general formalism for defining twisted supergravity theories has been recently introduced in [57], with the motivation to discuss twisted versions of the AdS/CFT correspondence. It would be extremely interesting to apply this formalism to our setup.

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Rees for useful conversations. We would like to thank the referee for useful comments and suggestions.
Let us combine these objects to obtain a convenient representation of the chiral 16 × 16 blocks Γ M ,Γ M of gamma matrices in ten dimensions, where M = 1, . . . , 9, 0 and the flat metric is η M N = diag(+ 9 , −). Let us set We

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As a final remark, recall that we adopt an off-shell supersymmetry formalism that realizes manifestly only a subalgebra so(2) R × so(2, 1) R of the full R-symmetry group so(4, 1) R , associated to the split I = (i, A), i = 6, 7, A = 8, 9, 0. It is therefore convenient to specify further the representation of gamma matrices ρ I by requiring ρ 8 ρ 9 ρ 0 = I 2 0 0 −I 2 . (A.8) An explicit realization of all the gamma matrices considered in this section in terms of Pauli matrices can be found in subsection A.3 below.
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