Boundary Conditions and Partition Functions in Higher Spin AdS 3 /CFT 2
BBoundary Conditions and Partition Functions in Higher SpinAdS /CFT Jan de Boer and Juan I. Jottar
Institute for Theoretical Physics, University of Amsterdam,Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
[email protected], [email protected]
Abstract
We discuss alternative definitions of the semiclassical partition function in two-dimensionalCFTs with higher spin symmetry, in the presence of sources for the higher spin currents.Theories of this type can often be described via Hamiltonian reduction of current alge-bras, and a holographic description in terms of three-dimensional Chern-Simons theorywith generalized AdS boundary conditions becomes available. By studying the CFT Wardidentities in the presence of non-trivial sources, we determine the appropriate choice ofboundary terms and boundary conditions in Chern-Simons theory for the various types ofpartition functions considered. In particular, we compare the Chern-Simons descriptionof deformations of the field theory Hamiltonian versus those encoding deformations of theCFT action. Our analysis clarifies various issues and confusions that have permeated theliterature on this subject. a r X i v : . [ h e p - t h ] J u l ontents W theory in Hamiltonian form . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Adding central extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Symmetries of the action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Ward identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 U (1) example 43B Useful W formulae 44C Non-chiral stress tensor deformations 46D Tr (cid:2) a z (cid:3) and the OPE 49 N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52D.2 N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54D.3 N = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 The study of higher spin theories in anti-de Sitter (AdS) space has been recently revitalized,partly because they provide an example of holographic duality in which the field theory isessentially non-interacting, and one often has good analytic control over both local and non-local observables. At least in principle, this feature allows for a very precise holographicdictionary to be established and tested: roughly speaking, the higher spin symmetries emergein a regime in which one can compute reliably in both the bulk and the boundary sides of thecorrespondence. A very interesting example of these dualities is the conjecture [1] of Klebanovand Polyakov relating three-dimensional critical O ( N ) vector models and the Fradkin-Vasilievhigher spin theories in AdS [2, 3, 4], for which robust evidence has been provided recently(see [5] and references therein).Another setup where both sides of the duality are amenable to study is that of AdS /CFT :here the boundary theories correspond to two-dimensional CFTs with extended current alge-bras, and the gauge sector of the three-dimensional bulk gravitational theory can be formulatedas a Chern-Simons gauge theory. Indeed, starting with the proposal of Gaberdiel and Gopaku-mar [6, 7] relating the three-dimensional interacting higher spin theories [8, 9] to a family ofminimal model coset CFTs with W -symmetry, several results have been obtained that showagreement between quantities computed in CFT and from the bulk duals. These include thespectrum [11, 12, 13, 14, 15], partition functions [16, 17, 18], scalar correlators [19, 20], andentanglement entropies [21, 22, 23], to name a few. While the full realization of the duality alsoinvolves matter fields in the bulk, which couple to operators other than conserved currents,the pure higher spin sector of the correspondence already provides an interesting arena whereuniversal aspects of the duality can be explored.In the present article we will focus on the sector of the latter dualities describing the CFT’sconserved currents, where the corresponding symmetries emerge via Hamiltonian reduction ofcurrent algebras and admit a simple holographic description in terms of two copies of Chern-Simons theory. Our main goal will be to clarify the interpretation of different boundaryconditions in Chern-Simons theory from the point of view of the dual CFT, in the presence ofsources for the conserved currents furnishing the extended (possibly higher spin) symmetries. See [10] for a comprehensive review of W -symmetry in CFT.
2n particular, we will argue that certain boundary conditions correspond to a deformation ofthe CFT Hamiltonian, while others correspond to deformations of the CFT action.More precisely, given a CFT with Hamiltonian H CFT and action S CFT , we can distinguishat least four types of deformations depending on whether they are chiral or non-chiral andwhether they are defined as modifications of S CFT or H CFT : S = S CFT + (cid:90) d z (cid:88) s µ s W s (1.1) S = S CFT + (cid:90) d z (cid:88) s µ s W s + (cid:90) d z (cid:88) s ¯ µ s W s + · · · (1.2) H = H CFT + (cid:73) dσ (cid:88) s µ s W s (1.3) H = H CFT + (cid:73) dσ (cid:88) s µ s W s + (cid:73) dσ (cid:88) s ¯ µ s W s . (1.4)Here W s and W s are a set of currents of weight ( s,
0) and (0 , s ), respectively, obeying appro-priate Poisson or Dirac bracket chiral algebras which will typically be non-linear extensions ofthe Virasoro algebra, and σ denotes a compact coordinate on the cylinder. The deformationparameters µ s and ¯ µ s can be thought of as chemical potentials or background gauge fields:provided they transform suitably, the partition functions defined from the above Hamiltoni-ans/actions will be invariant under the symmetry algebra furnished by the currents. The dotsin (1.2) denote the fact that, in the presence of deformations of both chiralities, the corre-sponding action requires terms to all orders in the chemical potentials in order to realize thesymmetry. On the other hand, as we will discuss in detail in due course, at the level of theHamiltonian the linear couplings suffice, even when both chiralities are present, because W s and W s Poisson-commute.The program that we will follow can be summarized quite simply. The fact that thepartition functions associated with the various types of deformations above enjoy a symmetrywill as usual result in Ward identities for the one-point functions of the currents in the presenceof sources. The precise form of these Ward identities will depend on the particular type ofdeformation under consideration, but in all cases one can encode them as a flatness conditionon suitable 2 d gauge connections in “Drinfeld-Sokolov form” [24]. If we now regard the CFT asbeing defined on the boundary of a 3 d manifold, these 2 d gauge connections become boundaryconditions for 3 d Chern-Simons gauge fields, with the flatness conditions enforced by the Chern-Simons equations of motion. From a practical point of view, the advantage of this formulationis that one can now use Chern-Simons theory to derive a number of universal results for theboundary theories quite efficiently, including thermodynamic quantities such as entropy andfree energy, and even non-local observables such as entanglement and R´enyi entropies which3re usually quite difficult to obtain using solely CFT techniques. For example, formulae forthe thermal entropy in the presence of higher spin chemical potentials written entirely in termsof the Chern-Simons connections were derived in [25, 26], and two proposals for higher spinentanglement entropy in terms of Wilson lines in Chern-Simons theory were put forward in[21] and [22].It is worth mentioning that the logic behind the holographic formulation of the currentsector of these theories predates the advent of the AdS/CFT correspondence, and can be seenas a special case of the usual connection between Chern-Simons theory and Wess-Zumino-Witten (WZW) models. In fact, the different types of deformations we discuss as well as theirassociated symmetries were studied more than two decades ago in the context of gauging of W -algebras and the so-called W -gravity. Similarly, the connection between deformations ofthe Hamiltonian and chiral deformations of the CFT action was discussed in [27] from a field-theoretical perspective. Our main goal will be to derive the implications of these results for theWard identities and their connection to Chern-Simons theory, in the hope that these consid-erations will help to bridge the gap between the existing literature and the recent discussionsin the context of higher spin AdS /CFT .Importantly, in order to derive the Chern-Simons formulation one does not use holographyor the existence of a holographic dual of the starting CFT. Our analysis, however, is onlyvalid at the classical level (which in the dual CFT corresponds to a limit where c → ∞ , with c the central charge), and uses no properties of the CFT except that it possesses particularsymmetries. It is only when studying subleading corrections to various quantities that onewould need to have a more detailed knowledge of the matter content of the field theory, whichin the bulk corresponds to specific couplings of matter fields to Chern-Simons theory. In thelatter situation the details of the full-fledged holographic correspondence become important.While the problem at hand may appear to be of a fairly technical nature, it is conceivablethat the techniques developed in the context of the higher spin AdS /CFT duality may findan application to realistic systems. In fact, Hamiltonians of the form (1.3) feature prominentlyin the study of the dynamics of one-dimensional integrable condensed matter systems followinga quantum quench, where they are referred to as a “generalized Gibbs ensemble” or GGE (seee.g. [28] and references therein). Similarly, the large- N limit of certain coset CFTs proposedto describe strange metals in one spatial dimension has been related to higher spin theories onAdS [29]. Furthermore, even though most of the results that we will derive are strictly speakingapplicable in the large central charge regime, one may hope that some of the conclusions andlessons from the holographic analysis will retain their validity in other corners of parameterspace, which would make these results appealing to a wider community. In fact, some of thepredictions for entanglement entropy in the presence of sources derived in [21] using a novel4olographic proposal have been recently argued to apply beyond the large central charge limitfrom a purely CFT perspective [30, 23], with the first perturbative correction in the higherspin sources being moreover universal.The rest of the article is organized as follows. In section 2 we consider Hamiltonian defor-mations of the CFT and rewrite the canonical partition function as a path integral in first orderform, and exploit this representation to derive the Ward identities obeyed by the one-pointfunction of currents in the presence of sources. Although we employ a free boson realizationto perform the calculation, we will find that the resulting Ward identities take a generic form,independent of the specific realization and particular symmetry algebra. Moreover, we willdiscover that these Ward identities have a slightly different structure from the ones usuallydiscussed in the literature. In section 3 we consider deformations of the CFT action instead,and exhibit the form of the corresponding Ward identities. In section 4 we determine theprecise “Drinfeld-Sokolov pair” that allows to rewrite the Ward identities for Hamiltonian andaction deformations as the flatness condition on 2 d connections. Using holography, we thenextend them into 3 d flat connections with suitable boundary conditions, and use the associ-ated variational principle to derive expressions for the free energy and entropy, for example.We also revisit and discuss a few results that have generated some confusion in the recentliterature, and point out a useful relation obeyed by flat connections in Drinfeld-Sokolov form.We conclude in section 5. Useful formulas and examples that complement the discussion arecollected in the appendices. The basic object of interest is the canonical torus partition function Z can [ τ, α s , ¯ α s ] = Tr H exp 2 πi (cid:34) τ (cid:16) L − c (cid:17) − ¯ τ (cid:16) ¯ L − c (cid:17) + (cid:88) s (cid:16) α s W (0) s − ¯ α s W (0) s (cid:17)(cid:35) (2.1)where the trace is assumed to be taken over the Hilbert space H of the CFT, W ( s )0 and W ( s )0 denote the zero modes of conserved currents of weight ( s,
0) and (0 , s ), respectively, and α s ,¯ α s the corresponding sources. In our conventions the torus has volume Vol( T ) = 4 π Im( τ )with τ = τ + iβ/ (2 π ) , where β is the inverse temperature. The sum over s runs over theparticular spectrum of operators present in the theory, which depends on the symmetry algebrain question. Before proceeding further it is convenient to clarify our terminology: in agreementwith common usage in the literature, we will refer to the CFT operators of conformal dimension In the holographic realization that we will study in section 4, the spectrum is fixed by the choice of gauge πiτ (cid:16) L − c (cid:17) − πi ¯ τ (cid:16) ¯ L − c (cid:17) = − βH + 2 πiτ J , (2.2)where H = L + ¯ L − c is the Hamiltonian and J = L − ¯ L the angular momentum, with L , ¯ L the Virasoro generators on the cylinder. Defining the chemical potentialsΩ ≡ iτ β , µ s ≡ iα s β , ¯ µ s ≡ − i ¯ α s β (2.3)we see that the partition function describes a theory with density operatorˆ ρ = e − βH µ Z can [ β, Ω , µ s , ¯ µ s ] , (2.4)where the deformed Hamiltonian H µ is given by H µ ≡ H − π Ω J − π (cid:88) s (cid:16) µ s W (0) s + ¯ µ s W (0) s (cid:17) . (2.5) We will now assume the theory possesses a Lagrangian representation. Denoting the set offields collectively by φ , and their (Euclidean) conjugate momenta by P , the partition functioncan be written in a path integral representation as Z can [ β, Ω , µ s , ¯ µ s ] = (cid:90) D φ D P e ˜ I ( E ) ( P,φ ) , (2.6)where the Hamiltonian form of the action is˜ I ( E ) ( P, φ ) = (cid:90) β dt E (cid:90) π dσ (cid:34) − P ˙ φ − H + Ω J + (cid:88) s (cid:0) µ s W s + ¯ µ s W s (cid:1)(cid:35) (2.7)with ˙ φ = ∂ t E φ and (cid:73) dσ H = H , (cid:73) dσ π J = J , (cid:73) dσ π W s = W ( s )0 , (cid:73) dσ π W s = W ( s )0 . (2.8) algebra g ⊕ g for the bulk Chern-Simons theory, plus a choice of embedding of the sl (2 , R ) factor correspondingto the gravitational (spin-2) degrees of freedom into g . As usual, one thinks of the torus as a cylinder of finite length with the ends identified up to a twist. µ = iβ − α ,namely the chemical potentials, that enter in the action. This is the usual result in finite-temperature field theory, and can be established by carefully discretizing the operator trace (see[31, 32] for example). Secondly, in the reasoning above the potential Ω for angular momentumwas treated in the same footing as the other deformations. We can instead “geometrize” thispotential by introducing a twist in the boundary conditions. Doing so the partition functionbecomes Z can [ β, Ω , µ s , ¯ µ s ] = (cid:90) D φ D P e I ( E ) ( P,φ ) (2.9)with I ( E ) ( P, φ ) = (cid:90) T d z (cid:34) − P ˙ φ − H + (cid:88) s (cid:16) µ s W s ( P, φ ) + ¯ µ s W s ( P, φ ) (cid:17)(cid:35) (2.10)where d z is the standard measure on the Euclidean plane (we are assuming a flat torus) andthe path integral is performed with boundary conditions φ ( z ) = φ ( z + 2 π ) = φ ( z + 2 πτ ) . (2.11)Notice that while we have been working with constant µ s , ¯ µ s up to now, we are free to make µ s and ¯ µ s time- and space-dependent in this path integral representation of Z can , as long aswe specialize to constant µ s , ¯ µ s when we want to compute Z can .A point that will be crucial for the considerations to follow is that in general the currents W s corresponding to higher spin operators are at least cubic in momenta. Therefore, if wetransition to a Lagrangian path integral description by integrating out the momenta (i.e.Legendre-transforming) we find that the resulting action is non-linear in the sources, and infact it will generically involve mixing between the two chiral sectors. What this means is thatthe canonical partition function is in general quite different from a simple second order versionof the path integral with linear couplings, which we denote by Z Lag,naive : Z Lag,naive [ β, Ω , µ s , ¯ µ s ] = (cid:90) D φ e − S ( φ ) e − (cid:82) T d z (cid:80) s ( µ s W s ( φ )+¯ µ s W s ( φ ) ) (2.12)where S is the Lagrangian action in the absence of deformations. Fortunately, as we willdiscuss in detail in the rest of this section, for the purpose of deriving the Ward identitiesobeyed by the partition function Z can it will suffice to stay within the first order form of theaction, where the deformations appear only linearly and the two chiral sectors do not mix.It is important to emphasize that the action deformed by linear couplings which entersthe path integral (2.12) is not invariant under the higher spin symmetries furnished by thecurrents when both chiral sectors are deformed simultaneously, even if one allows the sourcesto transform. When both chiralities are present an invariant action involves corrections to all7rders in the sources [33, 34, 35], and we will return to this point in section 3.3. In general,this means that the naive partition function Z Lag,naive does not obey the usual Ward identitieswhen both µ s and ¯ µ s are switched on. The fact that Z can and Z Lag,naive are different objectsis in fact true even for deformations involving “lower spin” currents (relevant operators), andhas important consequences for modular invariance, for example. To illustrate this point, inappendix A we review an example involving U (1) currents in a free compact boson realization.Our next goal is to derive the Ward identities obeyed by the canonical partition function Z can . For the sake of concreteness, we will often resort to a theory with W symmetry deformedby sources for the stress tensor and weight-3 currents as our basic example. Even though we willuse a simple boson realization to derive these identities, we will find that the result is completelyfixed by the symmetry algebra and does not rely on details of the explicit realization. By thesame token, our conclusions will be general enough to later allow us to make a connectionwith flat connections in three dimensions and to find the appropriate boundary conditionsthese should obey in order to reproduce the canonical computations (c.f. section 4). We willfirst work in Lorentzian signature, where the discussion of symmetries, conserved charges andWard identities is more transparent. When discussing the Lorentzian theory on the cylinderwe will often refer to the chemical potentials µ , ¯ µ as the sources. On the other hand, oncewe transition to the finite temperature theory defined on the torus we will reserve the termsources to denote the α = − iβµ , ¯ α = iβ ¯ µ . W theory in Hamiltonian form Free field realizations of the W -current algebras were originally discussed in [36, 37]. Here wewill follow the Hamiltonian approach employed in [38], which will prove very advantageous.Consider then a theory of n real bosons X i ( i = 1 , . . . , n ) on the cylinder with coordinates( t, σ ) (where σ (cid:39) σ + 2 π ). We denote the canonical momentum conjugate to X i by P i , withequal-time Poisson brackets (cid:110) P i ( σ, t ) , X j ( σ (cid:48) , t ) (cid:111) = δ ji δ ( σ − σ (cid:48) ) , (2.13)and raise and lower Latin indices with the flat metric δ ij . Define nowΠ i ± = 1 √ (cid:0) P i ± ∂ σ X i (cid:1) , (2.14)which satisfy (cid:110) Π i ± ( σ, t ) , Π j ∓ (cid:0) σ (cid:48) , t (cid:1)(cid:111) = 0 (2.15) (cid:110) Π i ± ( σ, t ) , Π j ± (cid:0) σ (cid:48) , t (cid:1)(cid:111) = ∓ δ ij ∂ σ δ ( σ − σ (cid:48) ) . (2.16) If so desired, it is possible to introduce a non-trivial metric on the target space [33, 34]. W ( s ) ± = 1 s d i ...i s Π i ± . . . Π i s ± , (2.17)with s = 2 , , . . . N , where the d i ...i s are constant symmetric tensors of rank s . The basicPoisson brackets (2.15)-(2.16) imply that these generators fulfill two decoupled copies of the W N algebra (with no central extensions) provided the coefficients d i ...i s satisfy certain algebraicrelations that guarantee the closure of the algebra [34]. For example, defining T ± = W (2) ± and W ± = W (3) ± , in the W case one finds (cid:110) T ± ( σ ) , T ± ( σ (cid:48) ) (cid:111) = ∓ (cid:104) T ± ( σ ) ∂ σ δ (cid:0) σ − σ (cid:48) (cid:1) + δ (cid:0) σ − σ (cid:48) (cid:1) ∂ σ T ± ( σ ) (cid:105) (2.18) (cid:110) T ± ( σ ) , W ± ( σ (cid:48) ) (cid:111) = ∓ (cid:104) W ± ( σ ) ∂ σ δ (cid:0) σ − σ (cid:48) (cid:1) + 2 δ (cid:0) σ − σ (cid:48) (cid:1) ∂ σ W ± ( σ ) (cid:105) (2.19) (cid:110) W ± ( σ ) , W ± ( σ (cid:48) ) (cid:111) = ∓ κ (cid:104) T ± ( σ ) ∂ σ δ (cid:0) σ − σ (cid:48) (cid:1) + δ ( σ − σ (cid:48) ) T ± ( σ ) ∂ σ T ± ( σ ) (cid:105) (2.20)provided [33, 39] d ij = δ ij , d ( ijk d km ) n = κ δ ( ij δ m ) n . (2.21)Below we will discuss how to generalize this construction to allow for a semiclassical centralcharge c , in terms of which κ = − /c . The condition on d ijk guarantees that the spin-4 termin the r.h.s. of the { W, W } bracket is proportional to T , closing the algebra of the stresstensor T and the dimension-3 current W , albeit non-linearly. We stress that, since we areusing Poisson brackets and working at the semiclassical level, we have considered the productof currents such as T without worrying about operator ordering issues.Before integrating over momenta, the partition function for the deformed theory involvesthe first-order action I = (cid:90) dσdt (cid:20) P i ˙ X i − (cid:0) P i P i + ∂ σ X i ∂ σ X i (cid:1) − µ +2 T + − µ − T − − µ +3 W + − µ − W − (cid:21) (2.22)whose symmetries we want to study. The dot notation indicates time derivatives as usual. Aconvenient feature of the first order formalism is that the W ( s )+ and W ( s ) − generators Poisson-commute, so the separation of left- and right-movers is exact. To avoid unnecessary clutter wewill often work exclusively with the + sector and drop the subindex to simplify the notation,i.e. we use T ≡ T + , W ≡ W + and so forth when there is no risk for confusion. Naturally, allthe conclusions apply to the other chiral sector as well.The key point we want to stress is that integrating out the momenta one obtains thesecond order form of the action, which is non-linear in the sources and mixes left- and right-movers in a non-trivial way. A related observation is that, in the absence of deformations Since we are working with equal-time Poisson brackets, in order to simplify the notation we will oftensuppress the explicit time dependence of the currents and other quantities. µ ± = µ ± = 0) the equation of motion for P i implies P i = ∂ t X i , so that Π i ± = ∂ ± X i inthe undeformed theory. The undeformed currents are then schematically of the form W ( s ) ± ∼ ( ∂ ± X ) s and obviously chiral. On the other hand, when the chemical potentials are switchedon the P i acquire explicit dependence on them to all orders, and so do the currents themselves.For the purpose of studying the symmetries of the partition function and the associated Wardidentities it will be very advantageous to stay within the first order formulation, because thesources enter linearly and the chiral sectors remain factorized. We will now extend the Hamiltonian analysis of [38] to include classical central extensions.This can be achieved by adding improvement terms to the generators, often times called“background charges” in the literature, along the lines of [37, 33, 39]: T = 12 δ ij Π i Π j + a i ∂ σ Π i (2.23) W = 13 d ijk Π i Π j Π k + e ij ∂ σ Π i Π j + f i ∂ σ Π i , (2.24)where the a i , e ij and f i are constant coefficients. With these additions, the W Poisson algebrabecomes (cid:110) T ( σ ) , T ( σ (cid:48) ) (cid:111) = − (cid:104) T ( σ ) ∂ σ δ (cid:0) σ − σ (cid:48) (cid:1) + δ (cid:0) σ − σ (cid:48) (cid:1) ∂ σ T ( σ ) + c ∂ σ δ (cid:0) σ − σ (cid:48) (cid:1)(cid:105) (2.25) (cid:110) T ( σ ) , W ( σ (cid:48) ) (cid:111) = − (cid:104) W ( σ ) ∂ σ δ (cid:0) σ − σ (cid:48) (cid:1) + 2 δ (cid:0) σ − σ (cid:48) (cid:1) ∂ σ W ( σ ) (cid:105) (2.26) (cid:110) W ( σ ) , W ( σ (cid:48) ) (cid:111) = 64 c (cid:104) T ( σ ) ∂ σ δ ( σ − σ (cid:48) ) + δ ( σ − σ (cid:48) ) T ( σ ) ∂ σ T ( σ ) (cid:105) + 3 ∂ σ δ ( σ − σ (cid:48) ) ∂ σ T ( σ ) + 5 ∂ σ δ ( σ − σ (cid:48) ) ∂ σ T ( σ ) (2.27)+ 23 δ ( σ − σ (cid:48) ) ∂ σ T ( σ ) + 103 ∂ σ δ ( σ − σ (cid:48) ) T ( σ ) + c ∂ σ δ ( σ − σ (cid:48) )provided the various coefficients satisfy (B.3)-(B.13) (in particular a i a i = − c , where c isthe semiclassical central charge), and similarly in the other chiral sector. A feature thatdistinguishes the non-linear Poisson algebras such as (2.25)-(2.27) from their linear counterpartsis that, upon normal-ordering the products of currents, the Jacobi identities (associativity)will imply that the structure constants in the quantum version of the algebra acquire O (1 /c )corrections (see e.g. [39]). It is in this sense that the non-linear Poisson bracket algebra is a“large- c ” version of the full quantum algebra.With the Poisson algebra at our disposal, we can compute the transformation of the currents10nder the various symmetries. Defining the integrated spin-2 and spin-3 charges Q (2) = (cid:90) dσ (cid:48) (cid:15) (cid:0) σ (cid:48) (cid:1) T (cid:0) σ (cid:48) (cid:1) (2.28) Q (3) = (cid:90) dσ (cid:48) χ (cid:0) σ (cid:48) (cid:1) W (cid:0) σ (cid:48) (cid:1) , (2.29)under an infinitesimal spin-2 transformation one finds δ (cid:15) T = (cid:110) Q (2) , T (cid:111) = (cid:15) ∂ σ T + 2 T ∂ σ (cid:15) + c ∂ σ (cid:15) (2.30) δ (cid:15) W = (cid:110) Q (2) , W (cid:111) = (cid:15) ∂ σ W + 3 W ∂ σ (cid:15) (2.31)(with similar expressions in the other chiral sector) and we recognize the transformation ofthe stress tensor and a weight-3 primary operator under diffeomorphisms of the form x + → x + + (cid:15) ( σ ). Similarly, under the spin-3 symmetry one finds δ χ T = (cid:110) Q (3) , T (cid:111) = 2 χ∂ σ W + 3 W ∂ σ χ (2.32) δ χ W = (cid:110) Q (3) , W (cid:111) = − (cid:34) c (cid:16) χ T ∂ σ T + T ∂ σ χ (cid:17) + c ∂ σ χ + 13 (cid:16) χ ∂ σ T + 9 ∂ σ χ ∂ σ T + 15 ∂ σ χ ∂ σ T + 10 T ∂ σ χ (cid:17)(cid:35) . (2.33) Let us now discuss the symmetries of the action. To this end it is useful to think of the sourcesas gauge fields, i.e. Lagrange multipliers imposing constraints that generate the W algebraor any other symmetry in question. We emphasize however that the sources are backgroundfields which are not being integrated over in the path integral. We will denote the currentsgenerating the symmetry of interest by a vector (cid:126)J with components J α , and the correspondingLagrange multipliers by a vector (cid:126)µ with components µ α . In our W example we will have (cid:126)J = { T + , T − , W + , W − } and (cid:126)µ = { µ +2 , µ − , µ +3 , µ − } . The action we consider is then of thegeneric form I = (cid:90) dσdt (cid:16) P i ˙ X i − H − µ α J α (cid:17) (2.34)where H denotes the undeformed Hamiltonian. We will study the symmetries of the associatedpartition function using the improved generators, i.e. when the algebra acquires semiclassicalcentral extensions: (cid:110) J α ( σ ) , J β ( σ (cid:48) ) (cid:111) = (cid:90) dx f γαβ ( σ, σ (cid:48) , x ) J γ ( x ) + c αβ ( σ, σ (cid:48) ) , (2.35)11here as before we have suppressed the explicit time dependence of the currents for the sakeof notational simplicity. The functions c αβ are proportional to the semiclassical central charge c , but do not depend on the phase space variables.Before moving forward, we can take two steps that will simplify the task of finding the Wardidentities obeyed by the currents in the presence of sources. First, we note that the undeformedHamiltonian is simply H = T + + T − . It is then possible to eliminate H from the actionby shifting the spin-2 chemical potentials as µ ± → ν ± −
1, while keeping the higher spinchemical potentials the same. Consequently, for practical purposes we will define a new vector (cid:126)ν with components { ν +2 , ν − , µ +3 , µ − , . . . } and drop H . Even though the shift in the spin-2deformation could be thought of as a “gauge choice”, the undeformed theory has generically anon-zero Hamiltonian H , so we must remember to translate our results back to the µ α at theend if we are to interpret the sources strictly as deformations of the original theory. Secondly,we will define for convenience an auxiliary action I c which includes an “identity gauge field” ν c which can be thought of as coupling to an extra Abelian generator [40]: I c = (cid:90) dσdt (cid:16) P i ˙ X i − ν α J α − ν c · (cid:17) . (2.36)The role of this additional Lagrange multiplier, which is purely a bookkeeping device, is tocancel contributions to the variation of the action coming from central extensions. Naturally,at the end of the day we will set ν c = 0 in order to obtain the Ward identities obeyed by theoriginal partition function.We are now in position to discuss the symmetries of the action in the presence of defor-mations. It is straightforward to check that under an infinitesimal transformation of the fieldsand sources of the form δP i ( σ ) = (cid:90) dσ (cid:48) (cid:15) α ( σ (cid:48) ) (cid:110) J α ( σ (cid:48) ) , P i ( σ ) (cid:111) (2.37) δX i ( σ ) = (cid:90) dσ (cid:48) (cid:15) α ( σ (cid:48) ) (cid:110) J α ( σ (cid:48) ) , X i ( σ ) (cid:111) (2.38) δν α ( σ ) = ˙ (cid:15) α ( σ ) − (cid:90) dσ (cid:48) dx ν β ( x ) (cid:15) γ ( σ (cid:48) ) f αγβ ( σ (cid:48) , x, σ ) (2.39) δν c ( σ ) = ˙ ζ ( σ ) − ν β ( σ ) (cid:90) dσ (cid:48) (cid:15) γ ( σ (cid:48) ) c γβ ( σ (cid:48) , σ ) , (2.40)the auxiliary action (2.36) changes by a boundary term: δI c = (cid:90) dtdσ ∂ t (cid:0) P i δX i − (cid:15) α J α − ζ (cid:1) . (2.41) More precisely, in the presence of improvement terms we have H = (cid:0) P i P i + ∂ σ X i ∂ σ X i (cid:1) = T + + T − − a + j ∂ σ Π j + − a − j ∂ σ Π j − , but the total σ -derivatives do not contribute to the Hamiltonian (cid:82) dσ H . W example. Using (B.15)-(B.20) we canobtain the explicit transformation of the sources from (2.39) (with (cid:15) α = { (cid:15), χ, . . . } ) δν = ∂ t (cid:15) − ν ∂ σ (cid:15) + (cid:15)∂ σ ν − c T (cid:0) χ∂ σ µ − µ ∂ σ χ (cid:1) + 23 µ ∂ σ χ − χ∂ σ µ − ∂ σ µ ∂ σ χ + ∂ σ χ∂ σ µ (2.42) δµ = ∂ t χ − ν ∂ σ χ + 2 χ∂ σ ν + (cid:15)∂ σ µ − µ ∂ σ (cid:15) (2.43)where we used the shorthand ν ≡ ν +2 and µ ≡ µ +3 , with similar expressions for the sourcesin the other chiral sector. Shifting back to the original spin-2 source µ = ν − δµ = ∂ − (cid:15) − µ ∂ σ (cid:15) + (cid:15)∂ σ µ − c T (cid:0) χ∂ σ µ − µ ∂ σ χ (cid:1) + 23 µ ∂ σ χ − χ∂ σ µ − ∂ σ µ ∂ σ χ + ∂ σ χ∂ σ µ (2.44) δµ = ∂ − χ − µ ∂ σ χ + 2 χ∂ σ µ + (cid:15)∂ σ µ − µ ∂ σ (cid:15) . (2.45)Notice the appearance of the chiral derivative defined as ∂ − = ∂ t − ∂ σ ( ∂ + = ∂ t + ∂ σ ). We thensee that the only effect of the undeformed Hamiltonian H is to turn the time derivatives in(2.39) into chiral derivatives.It is worth emphasizing that the theory and in particular the partition function are definedat fixed values of the sources. The fact that one needs to transform the µ α in order to realize thesymmetry, therefore moving in the space of theories, shows that generically these deformationswill explicitly break the original conformal as well as higher spin and Lorentz symmetries. Having derived the transformation of the sources, the basic result (2.41) showing the invarianceof the action under the combined transformation of background sources and fundamental fieldswill imply a Ward identity for the currents. From the point of view of the path integral,changing the fields X i and momenta P i is a just a change of integration variables. Hence, thesymmetry (2.37)-(2.40) implies (cid:28)(cid:90) dσdt (cid:18) δI c δν α δν α + δI c δν c δν c (cid:19)(cid:29) (cid:39) , (2.46) The theory naively has new higher spin and Lorentz symmetries which one obtains by (i) performing ahigher spin transformation that puts all sources equal to zero, (ii) performing a higher spin transformation inthe undeformed theory and (iii) performing the inverse higher spin transformation that puts all sources back totheir original value. As we will discuss in section 5, it is not entirely clear whether this is a proper symmetry ofthe deformed theory. (cid:39) denotes equivalence up to surface terms (the integral of total time derivatives). Setting ν c = 0 in order to recover the Ward identity obeyed by the original partition function we obtain (cid:90) dσdt (cid:18) − J α δν α + (cid:90) dσ (cid:48) ν β ( σ ) (cid:15) γ ( σ (cid:48) ) c γβ ( σ (cid:48) , σ ) (cid:19) (cid:39) , (2.47)where the J α are interpreted as the one-point function of the currents in the presence of externalsources. Plugging the explicit form (2.39) of the variations δν α and integrating by parts wefind the identity ∂ t J α ( σ ) + (cid:90) dσ (cid:48) dx ν β ( x ) f γαβ ( σ, x, σ (cid:48) ) J γ ( σ (cid:48) ) + (cid:90) dσ (cid:48) ν β ( σ (cid:48) ) c αβ ( σ, σ (cid:48) ) = 0 . (2.48)Note that defining the extended Hamiltonian H ν ≡ (cid:90) dσ ν α ( σ ) J α ( σ ) , (2.49)the above Ward identity takes a very compact form: ∂ t J α ( σ ) = (cid:110) H ν , J α ( σ ) (cid:111) . (2.50)In other words, in Hamiltonian language the Ward identities are just the equations of motionof the currents, with the time evolution generated by the deformed Hamiltonian H ν .We emphasize that the source vector ν α in (2.48) and (2.50) contains the shifted spin-2deformation ν = µ + 1 that allowed us to absorb the undeformed Hamiltonian H . In thespecific W example, using (B.15)-(B.20) it is easy to see that (2.48) yields, after shifting backto µ , ∂ − T = µ ∂ σ T + 2 T ∂ σ µ + c ∂ σ µ + 3 W ∂ σ µ + 2 µ ∂ σ W (2.51) ∂ − W = µ ∂ σ W + 3 W ∂ σ µ − c (cid:0) T ∂ σ µ + µ T ∂ σ T (cid:1) − T ∂ σ µ − ∂ σ T ∂ σ µ − ∂ σ T ∂ σ µ − µ ∂ σ T − c ∂ σ µ , (2.52)where ∂ − = ∂ t − ∂ σ . Just as before, shifting back to µ produced an extra term that combinedwith the time derivatives in (2.48) to turn them into chiral derivatives. This was to be expected,because ∂ − T = 0 and ∂ − W = 0 are the Ward identities in the free theory (i.e. when µ = µ = 0). More generally, if in a slight abuse of notation we let J α , f γαβ and c αβ denote thecurrents, structure constants and central extensions on a single chiral copy of the algebra, ourresults for the Ward identity in terms of the original sources µ β becomes ∂ − J α ( σ ) + (cid:90) dσ (cid:48) dx µ β ( x ) f γαβ ( σ, x, σ (cid:48) ) J γ ( σ (cid:48) ) + (cid:90) dσ (cid:48) µ β ( σ (cid:48) ) c αβ ( σ, σ (cid:48) ) = 0 , (2.53)14ith a similar expression in the other chiral sector ( ∂ + ¯ J α ( σ ) + . . . = 0).Even though the Ward identities were derived using an explicit realization in terms ofscalars, it is clear from (2.48) and (2.53) that the end result is completely fixed by the symmetryalgebra and therefore independent of the particular realization we have chosen. In other words,(2.51)-(2.52) are the semiclassical (large- c ) Ward identities associated to the canonical partitionfunction in any theory with W symmetry, in the presence of sources. It is also clear from thederivation that (2.53) extends to any other closed symmetry algebra.It is somewhat peculiar that the right-hand side of the Ward identities involves σ -derivativesas opposed to x + -derivatives, which to our knowledge has not been emphasized in the literaturebefore. As we have seen this is an automatic consequence of our canonical treatment, withthe Hamiltonian as the starting point. In section 4.1 we will show that these Ward identitiescan be written as the flatness condition on sl ( N, R ) ⊕ sl ( N, R ) gauge fields with appropriateboundary conditions. By now we have established the structure of the Ward identities corresponding to a deformationof the CFT Hamiltonian by higher spin currents. Our next task is to consider a differentpartition function obtained by deforming the CFT action. Many of the technical aspects inthe analysis below are analogous to the canonical case discussed in depth in the previoussection, so in what follows we will omit unessential details for the sake of brevity.
We will begin by studying the symmetries of the partition function and the associated Wardidentities in the presence of chiral deformations. To this end we will again resort to the freeboson realization, and consider an action of the form S = (cid:90) d x (cid:18) ∂ + X i ∂ − X i − λ α G α (cid:19) (3.1)where the vector G = {L , W} contains the currents L = 12 ∂ + X i ∂ + X i , W = 13 d ijk ∂ + X i ∂ + X j ∂ + X k (3.2)and λ = { λ , λ } the corresponding sources. Following the Noether procedure, it was estab-lished long ago that this linear coupling is in fact enough for the action with chiral deformations As the notation indicates, these chiral currents are different from their canonical counterparts (2.17), andonly agree with them in the absence of sources.
15o enjoy a gauge invariance [33, 34], akin to a chiral half of (2.37)-(2.39). A particularly trans-parent way of understanding this result, which also allows to make direct contact with thecalculations in section 2, is to realize that the above action is amenable to study in a Hamil-tonian formalism with the light-cone direction x − thought of as “time” [41, 34, 42], and wherethe undeformed Hamiltonian is identically zero, H = 0 .The key observation in [42] is that, after taking into account the presence of the secondclass constraint (1 / ∂ + X i − P i = 0 , the basic Dirac bracket { , } D reads (cid:110) ∂ + X i ( x + , x − ) , ∂ + X j ( y + , x − ) (cid:111) D = δ ij ∂ + δ (cid:0) x + − y + (cid:1) . (3.3)Given the form of this bracket and the currents (3.2) (compare with the basic canonical bracket(2.16) and currents (2.17)), from the reasoning in the previous section it is clear that theholomorphic currents enjoy the Dirac bracket algebra (cid:110) L ( x + ) , L ( y + ) (cid:111) D = − (cid:104) L ( x + ) ∂ + δ (cid:0) x + − y + (cid:1) + δ (cid:0) x + − y + (cid:1) ∂ + L ( x + ) (cid:105) (3.4) (cid:110) L ( x + ) , W ( y + ) (cid:111) D = − (cid:104) W ( x + ) ∂ + δ (cid:0) x + − y + (cid:1) + 2 δ (cid:0) x + − y + (cid:1) ∂ + W ( x + ) (cid:105) (3.5) (cid:110) W ( x + ) , W ( y + ) (cid:111) D = − κ (cid:104) L ( x + ) ∂ + δ (cid:0) x + − y + (cid:1) + δ ( x + − y + ) L ( x + ) ∂ + L ( x + ) (cid:105) (3.6)provided d ( ijk d km ) n = κ δ ( ij δ m ) n as before. Classical central extensions can be incorporatedexactly as in the canonical analysis by adding improvement terms to the generators, which willnow involve terms of higher order in chiral derivatives, e.g. L = (1 / ∂ + X i ∂ + X i + a i ∂ X i (compare with (2.23)). It is then immediate that the improved generators fulfill one copy ofthe centrally extended algebra (2.25), (2.26), (2.27), with spatial derivatives ∂ σ replaced bychiral derivatives ∂ + , provided the coefficients of the improvement terms obey the constraints(B.3)-(B.13). In simple terms, all the calculations performed in the canonical formulation carryover to the chiral deformation case provided one replaces Π i + → ∂ + X i and ∂ σ → ∂ + .Parameterizing the extended Dirac brackets of the currents as (in order to simplify thenotation we omit the explicit x − dependence below) (cid:110) G α ( x + ) , G β ( y + ) (cid:111) D = (cid:90) dz + f γαβ ( x + , y + , z + ) G γ ( z + ) + c αβ ( x + , y + ) , (3.7)it was shown in [42] that under the infinitesimal transformation δX i = (cid:88) n ≥ ( − n − ∂ n − (cid:32) (cid:15) α ∂G α ∂ (cid:0) ∂ n + X i (cid:1) (cid:33) (3.8) δλ α = ∂ − (cid:15) α − (cid:90) dy + dz + λ β ( y + , x − ) (cid:15) γ ( z + , x − ) f αγβ ( z + , y + , x + ) (3.9)16he action (3.1) transforms as δS (cid:39) − (cid:90) dx − dx + dy + λ β ( x + , x − ) (cid:15) γ ( y + , x − ) c γβ ( y + , x + ) , (3.10)where as before (cid:39) denotes equivalence up to surface terms. Repeating the manipulations thatlead to (2.48), in this case we find the Ward identities ∂ − G α ( x + ) + (cid:90) dz + dy + λ β ( y + ) f γαβ ( x + , y + , z + ) G γ ( z + ) + (cid:90) dz + λ β ( z + ) c αβ ( x + , z + ) = 0 . (3.11)Using the fact that the structure constants f γαβ and central extensions c αβ now involve ∂ + derivatives (as opposed to ∂ σ derivatives), in the W example we find ∂ − L = µ ∂ + L + 2 L ∂ + µ + c ∂ µ + 3 W ∂ + µ + 2 µ ∂ + W (3.12) ∂ − W = µ ∂ + W + 3 W ∂ + µ − c (cid:0) L ∂ + µ + µ L ∂ + L (cid:1) − L ∂ µ − ∂ + L ∂ µ − ∂ L ∂ + µ − µ ∂ L − c ∂ µ . (3.13)Just as for the algebra itself, the Ward identities associated with a chiral deformation ofthe field theory action have the same form as those associated with a chiral deformation ofthe Hamiltonian, but with spatial derivatives ∂ σ replaced by light-cone derivatives ∂ + . Asshown in e.g. [43], these Ward identities also follow by computing the one point of L and W in the presence of the insertion e (cid:82) d z λ α G α by expanding the exponential and using the OPEof the holomorphic currents. In this sense, (3.12)-(3.13) could be said to be the “usual” Wardidentities. As discussed in section 4 and appendix D, these Ward identities (in fact two chiralcopies of them) can be rewritten as the flatness condition on sl ( N ) ⊕ sl ( N ) gauge fields withappropriate boundary conditions. Consider adding to the Euclidean free boson action a chiral stress tensor deformation and achiral deformation by a weight- s current: S = (cid:90) d z (cid:18) ∂ z X i ∂ ¯ z X i − µ L − µ s W s (cid:19) (3.14)where L = 12 ∂ z X i ∂ z X i , W s = 1 s d i ...i s ∂ z X i . . . ∂ z X i s . (3.15) Just as in the case of a Hamiltonian deformation, one can alternatively introduce an extra Abelian generatorwhose transformation is such that the modified action is invariant. s , in whichcase the closure of the Dirac bracket algebra (obtained interpreting ¯ z as time) requires [34] d i ( i ...i s d ij ... ) j s = κ s − δ ( i i . . . δ j s − ) j s . (3.16)Allowing µ and µ s to have spacetime dependence, the above action is invariant under thefollowing infinitesimal transformation of fields and sources: δX i = (cid:15) δ L δ ( ∂ z X i ) + (cid:15) s δ W s δ ( ∂ z X i ) (3.17)= (cid:15) ∂ z X i + (cid:15) s d ii ...i s ∂ z X i . . . ∂ z X i s (3.18) δµ = ∂ ¯ z (cid:15) − µ ∂ z (cid:15) + (cid:15) ∂ z µ + κ L s − (cid:0) (cid:15) s ∂ z µ s − µ s ∂ z (cid:15) s (cid:1) (3.19) δµ s = ∂ ¯ z (cid:15) s + ( s − (cid:15) s ∂ z µ − µ ∂ z (cid:15) s − ( s − µ s ∂ z (cid:15) + (cid:15) ∂ z µ s (3.20)with associated Ward identities ∂ ¯ z L = µ ∂ z L + 2 L ∂ z µ + ( s − µ s ∂ z W s + s W s ∂ z µ s (3.21) ∂ ¯ z W s = µ ∂ z W s + s W s ∂ z µ + κ (cid:18) L s − ∂ z µ s + s − µ s L s − ∂ z L (cid:19) . (3.22)In order to discuss a thermal partition function, we now take µ and µ s to be constantchemical potentials and put the theory on a torus with modular parameter τ , with 2 π Im( τ ) = β as before. In the canonical formulation of section 2, the modular parameter τ of the toruscouples by definition to the Virasoro zero modes. We would now like to understand what arethe quantities that couple to τ and ¯ τ in the presence of chiral deformations of the action. Thisis an important question, as these couplings define for example the quantity that is conjugateto the inverse temperature β , namely the energy of the system. The original torus has metricand identifications given by ds = dzd ¯ z , with z (cid:39) z + 2 π (cid:39) z + 2 πτ , (3.23)and volume Vol( T ) = 4 π Im( τ ) . Since the periodicity of the coordinates depends on τ ,care must be exercised when taking variations with respect to the modular parameter. Aconvenient way of dealing with this problem consists in passing first to coordinates ( w, ¯ w ) offixed periodicity, e.g. [44] z = 1 − iτ w + 1 + iτ w , ¯ z = 1 − i ¯ τ w + 1 + i ¯ τ w , (3.24)which implies w (cid:39) w + 2 π (cid:39) w + 2 πi . (3.25)18ne then takes variations of the action in the ( w, ¯ w ) coordinates, and transforms back to ( z, ¯ z )at the end. In this way one obtains for example δ τ, ¯ τ (cid:0) ∂ z X i (cid:1) = i δτ ∂ z X i + δ ¯ τ ∂ ¯ z X i τ ) (3.26) δ τ, ¯ τ (cid:0) ∂ ¯ z X i (cid:1) = − i δτ ∂ z X i + δ ¯ τ ∂ ¯ z X i τ ) . (3.27)Denoting the free (undeformed) boson action by S and taking the variation as indicatedyields the expected result δS = (cid:90) d z i Im( τ ) (cid:0) L δτ − L δ ¯ τ (cid:1) (3.28)with L as in (3.15) and L = 12 ∂ ¯ z X i ∂ ¯ z X i . (3.29)Extending the above computation to include the effects of the chiral deformations requiressome caution, as we first have to define what exactly are the independent thermodynamicvariables that we are going to use. It might be tempting to use µ s and ¯ µ s , besides τ, ¯ τ ,as independent variables, but we will find it more natural and convenient to use Im( τ ) µ s ,Im(¯ τ )¯ µ s , τ and ¯ τ as independent variables. We make this choice because (i) a similar choicewas made in eq. (2.3), (ii) when taking µ s and W s constant the integral (cid:82) d z µ s W s reducesto 4 π Im( τ ) µ s W s , and (iii) this is also the standard procedure in thermal field theory in thepresence of chemical potentials [32]. An additional independent reason supporting this choiceof thermal sources, motivated from holographic considerations, will be given in section 4.4. Inthe present context this means that we must take the τ -variation of the action with δ (cid:0) Im( τ ) µ (cid:1) = 0 . (3.30)This illustrates a subtle yet crucial point: in the presence of deformations by conserved currents,the precise definition of the sources affects the definition of the energy and other thermodynamicquantities of interest.Taking into account the contribution of the chiral deformations and performing the varia-tion of (3.14) as described one obtains δS = (cid:90) d z i Im( τ ) (cid:16) L + 2 µ L + sµ s W s (cid:17) δτ − (cid:90) d z i Im( τ ) (cid:16) L − µ ∂ z X i ∂ ¯ z X i − µ s d i ...i s ∂ z X i . . . ∂ z X i s − ∂ ¯ z X i s (cid:17) δ ¯ τ . (3.31) Notice one keeps the invariant measure d z Im( τ ) = d w fixed in this variation.
19e would now like to rewrite this variation entirely in terms of the generators themselves.To this end we can use the reparametrization freedom of the path integral and consider a(non-local) field redefinition such that δ (cid:0) ∂ z X i (cid:1) = γ ∂ z X i + γ s d ii ...i s ∂ z X i . . . ∂ z X i s . (3.32)The variation of the free action will then cancel the offending terms in the second line of (3.31)provided we set γ = − µ i Im( τ ) δ ¯ τ and γ s = − µ s i Im( τ ) δ ¯ τ . (3.33)Taking into account the new terms generated by the variation of the ( µ L + µ s W s ) piece, thefinal result for the combined variation of the complex structure plus field redefinition is δS = (cid:90) d z i Im( τ ) (cid:0) Eδτ − Eδ ¯ τ (cid:1) (3.34)where we defined the “energies” E and E as E = L + 2 µ L + sµ s W s , E = L − µ L − κµ s L s − . (3.35)The above simple-minded calculation glossed over many details: it did not take centralterms into account, it applies to a single higher spin deformation only, and the field redefini-tion we have performed is non-local. A rigorous calculation should involve e.g. treating thekernel of the derivative operator (in particular zero modes) carefully. Barring these technicalcomplications, the naive calculation exemplifies some facts that should remain true once thesesubtleties are taken into account. In particular, it shows that even for a chiral deformation thenotion of energy on the opposite chiral sector is modified. In fact, the generalization of (3.35)was obtained in [25] using Chern-Simons theory. We now see that the mixing of chiralitieshas a very simple origin in field theory: it arises due to the mixing of left- and right-moversin (3.26) and (3.27). We will return to this result and its interpretation in section 4.3 andappendix D.
Having studied chiral deformations of the CFT action, a natural question is whether one cansimultaneously turn on sources for both left- and right-moving chiral algebras in such a waythat the Ward identities consist of two copies of (3.11) (with ∂ + and ∂ − interchanged). As we The calculation performed in [25] moreover involved non-chiral deformations, but reduces to the aboveresults once the chemical potentials in the barred sector are switched off. Z Lag,naive [ β, µ s , ¯ µ s ] = (cid:90) D φ e − S ( φ ) e − (cid:82) T d z (cid:80) s ( µ s W s ( φ )+¯ µ s W s ( φ ) ) (3.36)would not lead to the desired Ward identities. A simple way to appreciate the problemsassociated with this definition is to notice that in order to derive the desired Ward identitiesone would need to assume that the chiral sectors are decoupled, whereas in practice the OPEbetween e.g. W s and W s involves contact terms. In terms of free bosons X i , these contactterms arise for example from ∂X i ( z, ¯ z ) ¯ ∂X j ( w, ¯ w ) ∼ δ ij δ (2) ( z − w, ¯ z − ¯ w ) . Though contactterms are perhaps often associated to quantum effects, it is straightforward to see that here W s transforms non-trivially under a higher spin transformation generated by W s already atthe classical level, thereby spoiling the derivation of the Ward identities.In certain cases one can indeed write down a partition function whose symmetries result intwo copies of the chiral Ward identities, at the expense of introducing auxiliary fields [35]. Asanticipated, integrating out the auxiliary fields results in an action involving infinitely manyhigher order terms in µ s and ¯ µ s , which would be the Lagrangian version of the theory with well-separated left- and right-movers. Even though we will not discuss the auxiliary field formalismin detail, in appendix C we review an example involving non-chiral stress tensor deformationsthat illustrates various general features of the construction. We emphasize that the difficultiesassociated with non-chiral deformations do not arise in the holographic formulation usingChern-Simons theory. In particular, it was already shown in [43] that two copies of the chiralWard identities arise as the flatness condition on sl ( N ) ⊕ sl ( N ) gauge fields with appropriateboundary conditions (c.f. section 4.3). The difficult only emerges when one tries to associatea deformed CFT path integral to the bulk theory with these boundary conditions.In our considerations above, the non-decoupling of the chiral sectors in the path integral(3.36) was easy to see because it involved classical field variations only. One could howevercontemplate other definitions of the path integral, for example using conformal perturbationtheory, where contact terms play no role since one regularizes the integrated correlators byexcising small disks around each operator insertion. At first sight, this prescription then leadsto a factorization of the chiral and non-chiral deformations, since they only interact throughcontact terms in correlation functions of the form (cid:104)WW(cid:105) , and therefore also to the correctseparate Ward identities. It is somewhat puzzling that conformal perturbation theory naivelyyields an answer which differs from that obtained using classical field variations, especially sincethe disagreement is already there at the classical level and has nothing to do with quantumissues. One possibility is that the treatment using conformal perturbation theory becomessubtle when going to higher orders, since one needs to separate the chiral and anti-chiral We thank Per Kraus for bringing this scenario to our attention.
The semiclassical analysis in sections 2 and 3 culminating in the Ward identities (2.53) and(3.11), respectively, was purely field-theoretical and did not presume the existence of a holo-graphic description. In particular, for any theory with W symmetry we showed that thesymmetries of the canonical partition function in the presence of sources result in equations(2.51)-(2.52) for the one-point function of the stress tensor and the dimension 3 operator, inthe semiclassical limit. For chiral deformations of the field theory action, the analogous results(3.12)-(3.13) hold. Solely a consequence of symmetries, these results are generally valid inthe large central charge limit, and in particular independent of the existence of a holographicrealization.As we have mentioned, the W N algebras are an example of symmetries that emerge via theso-called Hamiltonian or Drinfeld-Sokolov [24] reduction of current algebras, and such theoriescan be described in terms of Chern-Simons theories on a three-dimensional manifold withboundary (see the recent [6, 45, 46, 47, 48, 49, 43, 50, 18, 17], and [51, 52, 53, 54, 55, 56, 57]for earlier work). The pure Chern-Simons sector is in fact a consistent truncation of thefull interacting Prokushkin-Vasiliev theory [8, 9], where the matter sector decouples. Whenthe connections are valued in sl ( N, R ) ⊕ sl ( N, R ) the dual CFT possesses W N symmetry[46, 45]. Replacing the gauge algebra by two copies of the infinite-dimensional hs[ λ ] algebra,the resulting theory enjoys W ∞ [ λ ] symmetry [47, 49]. For a succinct overview of the basicfacts concerning the formulation of three-dimensional gravitational theories in Chern-Simonslanguage, as applied to higher spin AdS /CFT , we refer the reader to [25, 21]. Full detailscan be found in the comprehensive reviews [48, 7, 58].We do not need the full machinery referred to in the previous paragraph to describe theconnection to Chern-Simons theory, however. Consider Chern-Simons theory, augmented witha suitable boundary term and boundary conditions, on a three-manifold with boundary. TheChern-Simons gauge fields will depend in some particular way on the sources µ s and ¯ µ s , andalso on the expectation values (EVs) (cid:104)W s (cid:105) µ and (cid:104)W s (cid:105) µ in the deformed CFT. If the variationof the Chern-Simons action with boundary term and boundary conditions takes the schematicform δS ∼ (cid:90) M Tr [ δA ∧ F ] + (cid:90) ∂M d x (EVs) δ (sources) (4.1)22nd F = 0 restricted to the boundary agrees with the Ward identities of the dual deformedCFT, then the on-shell value of the Chern-Simons action plus boundary term yields a functionalwhich will automatically solve the Ward identities. Interestingly, we obtain a solution for eachchoice of three-manifold M with the same boundary ∂M . What this analysis does not tellus is which M to pick, whether to sum over all possible M , and whether all solutions of theWard identities can be obtained in this way. In the remainder, we will assume the latter to betrue, and in order to select a three-manifold we will pick the dominant saddle point suggestedby AdS/CFT in the case where the sources are turned off. We expect this to remain thedominant saddle for sufficiently small values of the sources, but an analysis of exactly whichsaddle dominates for which values of the sources is beyond the scope of the present paper.Note that (4.1) requires one to identify precisely what sources one chooses, and differentchoices of sources or thermodynamic variables will correspond to different boundary terms andboundary conditions, as recently discussed in [25]. A class of boundary conditions was studiedin [59, 60, 61, 62, 63, 64, 65] that lead to the so-called “canonical thermodynamics”, consistentwith canonical definitions of conserved charges and thermodynamics in gravitational theories,with a perturbative application of Wald-like formulae for the entropy and energy [66], and withthe thermal limit of entanglement entropy calculations in higher spin theories [21, 22]. A featureof the canonical approach is that, once sources for the currents are switched on, quantitiessuch as the energy and higher spin charges, for example, acquire an explicit dependence onthe chemical potentials and differ from their undeformed counterparts. On the other hand,alternative “holomorphic” boundary conditions were employed in [43, 50, 17, 26] which yieldedresults consistent with various independent CFT calculations [18, 20, 19]. The question thatconcerns us here is what is the precise interpretation of these boundary conditions in terms ofthe dual field theory.The picture we want to put forward is that while canonical boundary conditions are as-sociated with deformations of the CFT Hamiltonian, of the type studied in section 2, theholomorphic ones are related to deformations of the CFT action as described in section 3. We will begin our discussion from the perspective of holography by deriving the boundaryconditions that realize the canonical structure discussed in section 2. Taking the W case asour guiding example, we then ask what are the boundary conditions in Chern-Simons theorythat are consistent with the symmetry transformations (2.30)-(2.33) of the currents, and the See [27] for a detailed discussion of the relation between chiral deformations of the action and Hamiltonianin CFT. ρ and works with the reducedor “two-dimensional” connection a defined through A = b − ( ρ ) a ( t, σ ) b ( ρ ) + b − ( ρ ) db ( ρ ) . (4.2)To obtain the right Ward identities, one needs to choose the asymptotic boundary conditionsto be of Drinfeld-Sokolov form, i.e. a σ = L + Q (4.3)where Q a highest weight matrix ([ Q, L − ] = 0) whose entries contain the stress tensor andthe higher spin currents. For example, in the spin-3 case we write the spatial component ofthe sl (3 , R ) connection as a σ = L + Tk L − − W k W − (4.4)where k = (cid:96) G . By definition, the asymptotic symmetry algebra is generated by the gaugetransformations that respect these boundary conditions, and it corresponds to the (infinite)global symmetries of the dual CFT. Perfoming an infinitesimal gauge transformation withparameter λ as δa = dλ + [ a, λ ], one finds that (4.4) is preserved if λ takes the form [46] λ = (cid:88) i = − (cid:15) i L i + (cid:88) m = − χ m W m (4.5)with the parameters fixed in terms of (cid:15) ≡ (cid:15) and χ ≡ χ by (B.23)-(B.28). Under suchtransformations, the change in the currents is precisely given by (2.30)-(2.33).The remaining question is how to incorporate the sources µ , µ in the connection. Theguiding principle is that the asymptotic equations of motion, namely the flatness conditionon the reduced connection a ( t, σ ) , should reproduce the Ward identities (2.51)-(2.52). Havingfixed the form of a σ , the complete sl (3 , R ) flat connection is found to be a σ = L + Tk L − − W k W − (4.6) a t = a σ + µ L + µ W − ∂ σ µ L − ∂ σ µ W + (cid:18) ∂ σ µ + 2 Tk µ (cid:19) W + (cid:18) ∂ σ µ + 2 Wk µ + Tk µ (cid:19) L − + (cid:18) − ∂ σ µ − k T ∂ σ µ − k µ ∂ σ T (cid:19) W − (4.7)+ (cid:18) ∂ σ µ + 23 k T ∂ σ µ + 712 k ∂ σ T ∂ σ µ + (cid:18) T k + 16 k ∂ σ T (cid:19) µ − k µ W (cid:19) W − . We adopt the same convention as [46] for the sl (3 , R ) generators L , L , L − and W j ( j = − , − , . . . , π . a σ = L + Q (4.8) a t − a σ = M + . . . (4.9)where as before Q is linear in the currents and satisfies [ Q, L − ] = 0, M is a matrix linear inthe sources which satisfies [ M, L ] = 0 , and the dots stand for higher weight terms completelyfixed by the equations of motion once a suitable normalization of the sources is chosen (see[25, 63] and appendix D for details). In particular, in the above example we have M = µ L + µ W . (4.10)As a further consistency check, acting on (4.7) with the gauge parameter (4.5) (subject to(B.23)-(B.28)) one easily verifies that the change in the lowest weights of the connection is δ ( a t − a σ ) = δµ L + δµ W + (higher weights) , (4.11)with δµ and δµ given precisely by (2.44)-(2.45). We have then shown that the equations ofmotion of Chern-Simons theory with boundary conditions (4.8)-(4.9) (and a Dirichlet varia-tional principle for the sources) agree with the Ward identities we obtained from the canonicalpartition function in field theory. To make sure that the partition functions also agree, all thatremains is to find an appropriate boundary term which is compatible with the Dirichlet bound-ary conditions on the sources, and which will guarantee that the charges are indeed coupledin the right way to the currents. We will turn back to these boundary terms momentarily.In order to facilitate comparison with the recent literature, we note that the above boundaryconditions written in light-cone coordinates x ± = t ± σ read a + − a − = L + Q (4.12)2 a − = M + . . . (4.13)Recalling that the a − component is zero for undeformed solutions (such as pure AdS), wesee that incorporating the sources in a − we can readily interpret them as deformations of theoriginal theory. In the N = 3 case, these boundary conditions have been recently advocated in [62, 65] from a purelybulk perspective. Here we have arrived at them following a different route, using as guiding principle the(1 + 1)-dimensional field theory Ward identities in the presence of Hamiltonian deformations. M reads I CS = k cs π (cid:90) M Tr (cid:104) CS ( A ) − CS ( ¯ A ) (cid:105) (4.14)where [11] k cs = k L L ] (4.15)in order to match with the normalization of the Einstein-Hilbert action in the pure gravity case.With this normalization, the central charge in the dual CFT is given by c = 12 k cs Tr [ L L ] .The total action will be of the form I = I CS + I B , (4.16)where I B is the required boundary term. In terms of the ρ -independent connections a , ¯ a , thevariation of the bulk action I CS , evaluated on-shell, is easily seen to be δI CS | os = − k cs π (cid:90) ∂M Tr (cid:104) a ∧ δa − ¯ a ∧ δ ¯ a (cid:105) (4.17)= − k cs π (cid:90) ∂M d x Tr (cid:104) a + δa − − a − δa + − ¯ a + δ ¯ a − + ¯ a − δ ¯ a + (cid:105) , (4.18)where d x ≡ (1 / dx − ∧ dx + = dt dσ . The necessary boundary term is I B = − k cs π (cid:90) ∂M d x Tr (cid:104) ( a + − a − − L ) a − (cid:105) − k cs π (cid:90) ∂M d x Tr (cid:104) (¯ a − − ¯ a + + 2 L − ) ¯ a + (cid:105) , (4.19)and the variation of the full action I , evaluated on-shell, is then δI | os = − k cs π (cid:90) ∂M d x Tr (cid:104) ( a + − a − − L ) δ (2 a − ) + (¯ a − − ¯ a + + L − ) δ (2¯ a + ) (cid:105) . (4.20)This confirms that the boundary term above is well-suited to the Dirichlet problem (fixedsources). We will now describe the boundary conditions in the Euclidean formulation of Chern-Simonstheory, and derive general expressions for the free energy and entropy in the dual theory. Inorder to introduce temperature, the Euclidean time direction is compactified and the topologyof the three-dimensional manifold M becomes that of a solid torus. Complex coordinates ( z, ¯ z ) All traces in this section are taken in the fundamental representation. x + → z , x − → − ¯ z , withidentifications z (cid:39) z + 2 π (cid:39) z + 2 πτ , where τ is the modular parameter of the boundary two-torus. In the semiclassical limit (large temperature and central charges), the CFT partitionfunction is obtained from the saddle point approximation of the Euclidean on-shell action:ln Z = − I ( E ) os = − (cid:16) I ( E ) CS + I ( E ) B (cid:17)(cid:12)(cid:12)(cid:12) os , (4.21)where I ( E ) CS = ik cs π (cid:90) M Tr (cid:104) CS ( A ) − CS ( ¯ A ) (cid:105) (4.22)and I ( E ) B denotes the Euclidean continuation of the boundary term (4.19), I ( E ) B = − k cs π (cid:90) ∂M d z Tr (cid:104) ( a z + a ¯ z − L ) a ¯ z (cid:105) − k cs π (cid:90) ∂M d z Tr (cid:104) (¯ a ¯ z + ¯ a z − L − ) ¯ a z (cid:105) . (4.23)Mirroring the field theory discussion in section 3.2, when computing the variation of theChern-Simons action one should acknowledge that the modular parameter of the torus isvarying. As before, a convenient way of dealing with this fact is to compute the variation incoordinates with fixed-periodicity (where τ appears in the connection itself), and change backto the z coordinates at the end. Following the steps detailed in [25], in the present case wefind that the variation of the full action, evaluated on-shell, is given by δI ( E ) os = − πik cs (cid:90) ∂M d z π Im( τ ) Tr (cid:20)
12 ( a z + a ¯ z ) δτ + ( a z + a ¯ z − L ) δ ((¯ τ − τ ) a ¯ z ) −
12 (¯ a z + ¯ a ¯ z ) δ ¯ τ + (¯ a z + ¯ a ¯ z − L − ) δ ((¯ τ − τ )¯ a z ) (cid:21) . (4.24)First, we notice that the quantities conjugate to τ and ¯ τ , namely the left- and right-movingenergies T and T , are given by T = − k cs (cid:104) ( a z + a ¯ z ) (cid:105) , T = − k cs (cid:104) (¯ a z + ¯ a ¯ z ) (cid:105) . (4.25)In particular one notices that the mixing of chiralities encountered in 3.2 from the field theoryperspective, and in [25] from the Chern-Simons perspective, does not arise when using canonicalboundary conditions. Secondly, we see that the quantities coupling to the higher spin currentsare (¯ τ − τ ) a ¯ z and (¯ τ − τ ) ¯ a z , so the Euclidean version of the boundary conditions (4.12)-(4.13)is a z + a ¯ z = L + Q ¯ a z + ¯ a ¯ z = L − − Q (4.26)(¯ τ − τ ) a ¯ z = M + . . . (¯ τ − τ ) ¯ a z = M + . . . (4.27)with the difference that the matrix M does not contain the spin-2 source anymore, becausethe latter has been incorporated as the modular parameter of the torus. Equation (4.37)27akes manifest the fact that the sources get rescaled by the temperature when transitioningto the Euclidean formalism. In other words, the matrix elements of a ¯ z and ¯ a z contain thechemical potentials µ (i.e. the deformation parameters in the Lorentzian description), whilethe matrices M and M contain the actual sources α (cid:39) Im( τ ) µ . This agrees with our fieldtheory discussion in section 2.1 (c.f. (2.3)).As explained in [25], for the theory determined by choosing the principal embedding of sl (2) into sl ( N ), resulting in W N as the asymptotic symmetry algebra, the normalization ofthe currents and sources can be chosen such that − k cs (¯ τ − τ ) Tr (cid:2) Qa ¯ z (cid:3) = N (cid:88) s =3 α s W s (4.28) − k cs (¯ τ − τ ) Tr (cid:2) L a ¯ z (cid:3) = N (cid:88) s =3 ( s − α s W s (4.29)Similar expressions hold in the other chiral sector. To adapt the above formulae to non-principalembeddings one simply replaces s by the conformal weight of the operator, with the sumrunning over the appropriate spectrum. The above formulae rely solely on the lowest/highestweight structure of the solutions, and therefore are valid even for non-constant connections.See appendix D for a general derivation.So far we have been discussing the variation of the on-shell value of the Chern-Simonsaction, for which the choice of three-manifold was irrelevant. To find the actual value of theChern-Simons action, we however need to pick a three-manifold M . In the absence of sourcesthe dominant saddle point in the high-temperature regime is the one where the Euclideantime-circle is smoothly contractible in the interior. We will therefore pick this particular three-manifold M , as we expect this to still be the dominant saddle point for sufficiently small valuesof the sources. In this particular case, where moreover the connections are constant, we canexplicitly evaluate the on-shell action and therefore the partition function Z and the free energy F as − βF = ln Z = − I ( E ) (cid:12)(cid:12) os , obtainingln Z can = − πik cs Tr (cid:20)
12 ( a z + a ¯ z ) τ + (¯ τ − τ ) L a ¯ z −
12 (¯ a z + ¯ a ¯ z ) ¯ τ + (¯ τ − τ ) L − ¯ a z (cid:21) . (4.30) A slightly different definition of the sources was employed in [63], as lowest weights in τ a z + ¯ τ a ¯ z . One noteshowever that τ a z + ¯ τ a ¯ z = (¯ τ − τ ) a ¯ z + τ ( a z + a ¯ z ), and since a z + a ¯ z is a highest weight matrix, this impliesthat the lowest weights in τ a z + ¯ τ a ¯ z and (¯ τ − τ ) a ¯ z are in fact the same. By the same token, in the Lorentziantheory the sources can be said to be the lowest weights in a t − a σ or equivalently in a t , because a σ is a highestweight matrix. The difference between these two approaches amounts simply to a shift in the definition of thespin-2 source. See [59] for a discussion of the subtleties associated with the evaluation of the bulk piece.
28s usual, the free energy is a function of the temperature and chemical potentials. A standardLegendre transform produces the entropy, a function of the charges. The term implementingthe Legendre transformation can be read off from (4.24), and the thermal entropy is then S can = ln Z can − πik cs Tr (cid:20)
12 ( a z + a ¯ z ) τ −
12 (¯ a z + ¯ a ¯ z ) ¯ τ + ( a z + a ¯ z − L ) (¯ τ − τ ) a ¯ z + (¯ a z + ¯ a ¯ z − L − ) (¯ τ − τ ) ¯ a z (cid:21) (4.31)which after using (4.30) yields S can = − iπk cs Tr (cid:104) ( a z + a ¯ z ) ( τ a z + ¯ τ a ¯ z ) − (¯ a z + ¯ a ¯ z ) ( τ ¯ a z + ¯ τ ¯ a ¯ z ) (cid:105) . (4.32)This formula for the entropy was first derived in [25]. In the particular case of the W theory( N = 3) in the principal embedding, it agrees with a result derived in the metric formulation[61], as well as the perturbative application of Wald’s entropy formula [66]. We emphasizehowever that equations (4.30) and (4.32) are valid for any N , and any choice of embedding.Moreover, they are valid for the hs[ λ ] theory as well, provided the trace is interpreted ac-cordingly (see section 4 of [63] for a complete discussion of this case). The above form of theentropy has been also recovered as the thermal limit of entanglement entropy proposals forhigher spin theories [21, 22].It is important to mention that the charges and their conjugate sources have to be related ina particular way for the first law of thermodynamics to hold. In the Chern-Simons formulationthis requirement has been encoded in an elegant way in terms of holonomies of the connection[43]. In a few words, one demands that the connection has trivial holonomy around the thermalcycle of the boundary torus, that becomes contractible in the bulk. This is the Chern-Simonsanalogue of the familiar statement for Euclidean black holes that the thermal circle should besmoothly contractible. Using these holonomy conditions, it was also shown in [25] that theabove formula for the entropy can be written very compactly as S can = 2 πk cs Tr (cid:104)(cid:0) λ − λ (cid:1) L (cid:105) , (4.33)where λ and λ are diagonal matrices containing the eigenvalues of the component of theconnection along the non-contractible cycle of the boundary torus, i.e. λ ≡ Eigen ( a z + a ¯ z ) , λ ≡ Eigen (¯ a z + ¯ a ¯ z ) . (4.34)Given the boundary conditions (4.26)-(4.37), it is evident from (4.33) that the entropy isa function of the charges. The particular combination of zero modes implied by (4.33) canbe then viewed as the generalization of the Cardy formula for higher spin theories. As a29ide remark we note that in the principally-embedded sl ( N, R ) ⊕ sl ( N, R ) theory, the aboveexpression for the entropy can be also written in a representation-independent way as [21] S can = 2 πk cs (cid:68) (cid:126)λ − (cid:126)λ , (cid:126)ρ (cid:69) (4.35)where (cid:126)λ , (cid:126)λ are the weight vectors dual to λ and λ (which belong to the Cartan subalgebra), (cid:126)ρ denotes the Weyl vector of sl ( N ) (which is dual to L ), and the brackets denote the usualinner product induced by the Killing form. The holomorphic partition functions we discussed in this paper correspond to deformations ofthe Lagrangian instead of the Hamiltonian. The analysis proceeds exactly as for the canonicalcase, the main difference being that the boundary conditions become a z = L + Q ¯ a ¯ z = L − − Q (4.36)(¯ τ − τ ) a ¯ z = M + . . . (¯ τ − τ ) ¯ a z = M + . . . (4.37)instead of (4.26), and similarly in Lorentzian signature (see [25]) where they result in two copiesof the Ward identities (3.11) [43]. Accordingly, instead of (4.23) the appropriate boundary termnow reads I ( E ) B = − k cs π (cid:90) ∂M d z Tr (cid:104) ( a z − L ) a ¯ z (cid:105) − k cs π (cid:90) ∂M d z Tr (cid:104) (¯ a ¯ z − L − ) ¯ a z (cid:105) . (4.38)As discussed in [25], the corresponding free energy is − βF holo = ln Z holo = − πik cs Tr (cid:20) τ (cid:18) a z a z a ¯ z − ¯ a z (cid:19) − ¯ τ (cid:18) ¯ a z a ¯ z ¯ a z − a z (cid:19) + (¯ τ − τ ) ( L a ¯ z + L − ¯ a z ) (cid:21) . (4.39)There are several marked differences with the canonical case. For example, under variationsof the complex structure one finds [25] δ τ, ¯ τ ln Z holo = − πi (cid:90) d z π Im( τ ) (cid:0) Eδτ − Eδ ¯ τ (cid:1) (4.40)with E = − k cs (cid:104) a z + 2 a z a ¯ z − ¯ a z (cid:105) , E = − k cs (cid:104) ¯ a z + 2¯ a ¯ z ¯ a z − a z (cid:105) . (4.41)We then see that the operator that couples to τ is now much more complicated, and involvesa mixture of left and right movers. The content of (4.41) is that the energy of the system is30anifestly modified by the higher spin sources, and in particular it does no longer correspondto the zero modes of the stress tensor as defined in the undeformed theory. We presenteda qualitative field theory explanation of this mixture in section 3.2, and it would be veryinteresting to derive the form of this operator directly from the path integral. A hint as to howthis might come about comes from a property of Drinfeld-Sokolov connections that we discussbelow.In order to characterize the operators E and E , we note that the Drinfeld-Sokolov formof the connection with holomorphic boundary conditions is easily seen to imply (c.f. appendixD) − k cs (cid:2) a z (cid:3) = L (4.42) − k cs Tr (cid:2) a z a ¯ z (cid:3) = (cid:88) s ≥ sµ s W s , (4.43)with L the stress tensor, and similarly in the other chiral sector. The only quantity left tocharacterize is then − k cs Tr (cid:2) a z (cid:3) . In appendix D we point out the useful relation − k cs Tr (cid:2) a z (cid:3) = Res z → w (cid:104) ( z − w )∆ L N ( z )∆ L N ( w ) (cid:105) + ∂ ( P N ) , (4.44)where ∆ L N ≡ (cid:80) Ns =3 µ s W s is the deformation operator, and provide the explicit form of P N for N = 2 , ,
4. We notice however that P N does not contribute under the integral sign in(4.40). In other words, the contribution of − k cs Tr (cid:2) a z (cid:3) to the energies is given precisely bythe second order pole in the OPE of the Lagrangian deformation with itself. We emphasizethat (4.44) holds for arbitrary spacetime-dependent sources, and it therefore applies beyondthe thermodynamic analysis. We leave a further study of this curious relation to future work.In order to obtain the entropy we need to perform an appropriate Legendre transform ofthe free energy, which in this case reads S holo = ln Z holo − πik cs Tr (cid:20) (¯ τ − τ ) ( a z − L ) a ¯ z + τ (cid:18) a z a z a ¯ z − ¯ a z (cid:19) − (¯ τ − τ ) ( − ¯ a ¯ z + L − ) ¯ a z − ¯ τ (cid:18) ¯ a z a ¯ z ¯ a z − a z (cid:19)(cid:21) (4.45)and evaluates to the same expression (4.32) for the entropy as in the canonical case, namely S holo = − πik cs Tr (cid:104) ( a z + a ¯ z ) ( τ a z + ¯ τ a ¯ z ) − (¯ a z + ¯ a ¯ z ) ( τ ¯ a z + ¯ τ ¯ a ¯ z ) (cid:105) . (4.46) Note that we have not included a spin-2 source µ in the connection, as we would have done if usingcoordinates with fixed periodicity, i.e. for a square torus. The full general expressions containing µ can befound in appendix D.
31n particular, this result can be written in the same form as 4.33. However, despite the apparentsimilarity, there is an important difference once again: whereas in the canonical case ( a z + a ¯ z )depends on the charges only and the eigenvalues λ and ¯ λ immediately yield an expression forthe entropy as a function of the charges, in the holomorphic case a z + a ¯ z depends on both thecharges and chemical potentials in a complicated way. Hence, in order to find an expression forthe entropy, in the latter case one needs to explicitly solve the monodromy conditions whichallow to express the sources in terms of the charges. In the context of holography, the first discussion of boundary conditions in the presence ofhigher spin sources and the associated thermodynamics was given in the original work [43] ofGutperle and Kraus on higher spin black holes. For the bulk theory based on sl (3 , R ) ⊕ sl (3 , R )with principally embedded sl (2 , R ), for example, the boundary conditions advocated thereinagree with our holomorphic boundary conditions, and resulted in two copies of the Wardidentities (3.12)-(3.13). For chiral deformations of this sort, we have given a particular partition function in CFTwhich corresponds to a deformation of the action by a linear coupling, which indeed reproducesa single chiral copy of these Ward identities. We have also pointed out that when sources forcurrents of both chiralities are present, the corresponding partition function in CFT thatreproduces the Ward identities involves terms to all orders in the sources, including terms thatmix both chiralities. This can be understood, for example, using the auxiliary field formalismintroduced in [35], which unfortunately needs to be formulated on a case by case basis.By considering thermodynamics of Chern-Simons theory on a solid torus, an entropy wasfound in [43] whose precise form was determined by the first law of thermodynamics basedon a definition of the sources ( α, ¯ α ) that involved rescaling the chemical potentials ( µ, ¯ µ ) bythe modular parameter τ of the boundary torus torus, e.g. α = ¯ τ µ and ¯ α = τ ¯ µ . It was thenshown in [25] that the entropy formula obtained in [43] can be written quite generally as S G-K = − iπk cs Tr (cid:104) a z ( τ a z + ¯ τ a ¯ z ) − ¯ a ¯ z ( τ ¯ a z + ¯ τ ¯ a ¯ z ) (cid:105) , (4.47)or, equivalently, S G-K = 2 πk cs Tr (cid:2)(cid:0) λ z − λ z (cid:1) L (cid:3) , (4.48) The structure of a general Drinfeld-Sokolov connection obeying these boundary conditions is described inappendix D, and detailed examples are provided for the theory based on the sl ( N, R ) ⊕ sl ( N, R ) algebra for N = 2 , , λ z and λ z are diagonal matrices containing the eigenvalues of the a z and ¯ a ¯ z componentsof the connection.It may appear strange that although the Ward identities take the same “holomorphic” formin [43] as we obtained from deformations of the action, the entropy (4.46) one obtains fromthe latter formulation is clearly distinct from (4.47). The explanation of this discrepancy liesin the different choices of sources α , ¯ α in the thermal case: even with identically looking Wardidentities (in terms of the chemical potentials µ , ¯ µ ), different choices of sources α , ¯ α can giverise to different notions of free energy and entropy. Moreover, the modular parameter τ of thetorus does not enter the Ward identities directly and always needs a separate treatment.These results illustrate a rather subtle point that was alluded to from a field theory per-spective in section 3.2, namely that the precise definition of the sources in the thermal theoryaffects the notion of energy and other thermodynamic quantities. The bottom line is thatan unambiguous definition of the thermal partition function requires to specify not only theboundary conditions on the plane/cylinder, but also the precise scaling of the sources withthe complex structure of the torus. In this light it should come as no surprise that the sameflat connection can yield two different results (4.32) and (4.47) for the entropy, depending onprecisely how the thermal sources are related to the chemical potentials and the temperature.We would like to mention, in passing, one more argument in favor of the definition α (cid:39) Im( τ ) µ for the sources and the resulting canonical form (4.46) of the holomorphic entropy. Inthe N = 2 theory, corresponding to pure gravity, there is an independent holographic notionof entropy in the dual CFT in terms of the thermal entropy of black hole solutions. Thelatter can be of course computed by the Bekenstein-Hawking area law or any other standardmethod. As shown in [63], it is (4.46) and not (4.47) that coincides with the area of the blackhole horizon in the N = 2 theory. Moreover, the canonical entropy was derived in [63] byadapting the Wald formalism to Chern-Simons theory, and recently rederived in [67] usingthese techniques. These results indicate that the definition α (cid:39) Im( τ ) µ and consequently theentropy (4.46) are preferred from a bulk perspective. This is reassuring, because it impliesthat both the canonical boundary conditions studied in sections 4.1-4.2 and the holomorphicboundary conditions studied in [25] and reviewed in section 4.3 yield the same expression forthe thermal entropy, consistent with the idea that there should be a single notion of thermalentropy in a bulk theory containing gravity.Finally, to see how the Gutperle-Kraus result fits in our general framework, we would liketo present a computation in deformed 2 d CFT which reproduces the appropriate free energyand entropy. To this end, we first recall that one can find the free energy by e.g. varying the33ntropy to read off the sources and charges δS G-K = − πik cs Tr (cid:34) τ δ (cid:18) a z (cid:19) − ¯ τ δ (cid:18) ¯ a z (cid:19) + ¯ τ a ¯ z δ ( a z − L ) + τ ¯ a z δ ( − ¯ a ¯ z + L − ) (cid:35) , (4.49)and the free energy is then given by the corresponding Legendre transform [25] − βF G-K = ln Z G-K = − πik cs Tr (cid:20) τ (cid:18) a z (cid:19) − ¯ τ (cid:18) ¯ a z (cid:19) + (¯ τ L a ¯ z − τ L − ¯ a z ) (cid:21) . (4.50)One can derive from the results in [25] that this free energy follows by computing the partitionfunction on a square torus of the following deformed CFT S = S CFT + c (cid:18) iτ (cid:19) (cid:90) d z T CFT + c ¯ τ (cid:90) d z (cid:88) s µ s W s + c.c. (4.51)with some numerical constants c , c which we did determine explicitly. Let us reemphasizethat this theory lives on a square torus of fixed periodicities and that the dependence on τ isonly through the explicit appearance in the action. It remains to be seen whether deformationsof the type (4.51) have any particularly nice intrinsic properties, or whether they were merelystumbled upon by accident as a by-product of the definitions in [43]. Even though different boundary conditions in Chern-Simons theory describe different partitionfunctions in the two-dimensional boundary theory, it was already pointed out in [25] that fieldredefinitions exist which allow to map between different Drinfeld-Sokolov pairs. This suggeststhat redefinitions of the sources might be possible which allow to relate partition functionscorresponding to, say, a chiral deformation of the Hamiltonian and a chiral deformation of theaction. We will now discuss to what extent this is indeed possible. It will in general turn out tobe relatively easy to find redefinitions of the charges in such a way that the entropies transforminto each other, but difficult to find redefinitions of the sources to map the free energies intoeach other.We will first use chiral stress tensor deformations as an example. This is, for the theorydefined on a torus T with modular parameter τ , we would like to relate a Hamiltoniandeformation of the form Z can [ τ, α ] = Tr H (cid:104) q L − c ¯ q ¯ L − c exp (2 πiα L ) (cid:105) (4.52)with q = e πiτ , and an action deformation of the form Z holo [ τ, λ ] = (cid:90) D φ e − S ( φ ) e − i (cid:82) T d z π Im( τ ) λ L (4.53)34ith L the left-moving stress tensor.Given a Drinfeld-Sokolov pair, for constant sources the flatness of the gauge connectionimplies that the conjugate components of the gauge field commute. In the particular case of astress tensor deformation, that can be described by sl (2 , R ) connections, the two componentsare actually proportional to each other. Denoting the connection describing the Hamiltoniandeformation by a and that describing an action deformation by b , from our discussion above inthe canonical case plus the corresponding results for the holomorphic case (see [25]) it followsthat a ¯ z = α ¯ τ − τ ( a z + a ¯ z ) and b ¯ z = λ ¯ τ − τ b z . (4.54)The precise proportionality coefficient is fixed by relations such as (4.28)-(4.29). The idea is tonow relate the two sets of gauge fields on the torus to each other through gauge transformations.Recall now that the only gauge-invariant information carried by the connection is containedin the holonomy around cycles, with a z + a ¯ z being the component of the connection along thenon-contractible cycle of the boundary torus, and τ a z + ¯ τ a ¯ z the component along the thermalcycle, which becomes contractible in the bulk (and similarly for b ). Hence, the two sets ofgauge fields are gauge-equivalent if their spectrum matches (up to conjugation):spec (cid:0) a z + a ¯ z (cid:1) ∼ spec (cid:0) b z + b ¯ z (cid:1) (4.55)spec (cid:0) τ a z + ¯ τ a ¯ z (cid:1) ∼ spec (cid:0) τ b z + ¯ τ b ¯ z (cid:1) . (4.56)Using the on-shell relations (4.54) these conditions becomespec (cid:0) a z + a ¯ z (cid:1) ∼ (cid:18) λ ¯ τ − τ (cid:19) spec ( b z ) (4.57)( τ + α ) spec (cid:0) a z + a ¯ z (cid:1) ∼ (cid:18) τ + ¯ τ λ ¯ τ − τ (cid:19) spec (cid:0) b z (cid:1) , (4.58)implying 1 + iλ τ ) = (cid:18) − iα τ ) (cid:19) − . (4.59)This is precisely the same relation obtained in [27] using field theory techniques, reproducedhere with very simple manipulations in terms of flat connections in Chern-Simons theory.Why did this work? The reason is that (4.55) implies that the gauge fields transform undera global gauge transformation, i.e.( a z + a ¯ z ) = U − ( b z + b ¯ z ) U (4.60)( τ a z + ¯ τ a ¯ z ) = U − ( τ b z + ¯ τ b ¯ z ) U , (4.61)35nd therefore any quantity which consists of the trace of the products of gauge fields will beleft invariant under this transformation. The entropy is of this form in general, and that is whytransformations of this type can be used to find charge redefinitions which leave the entropyinvariant. For the free energy, however, the situation is more complicated. The on-shell valueof the Chern-Simons action is left invariant under the global gauge transformations (4.60),but the boundary terms are not, because these contains terms like Tr [ L a ¯ z ] which are notinvariant under (4.60), given that L is kept fixed. Then why did the computation for chiralstress-tensor deformations work? It did because it so happens that the boundary terms vanishfor such deformations.The general lesson is therefore that although we can relate in a fairly straightforward waythe different entropies to each other with redefinitions of the charges, the free energies donot share this property. Although this does not say that there can not exist redefinitions ofthe sources which relate the free energies, Chern-Simons theory does not appear to provide anatural candidate, except in the case of stress-tensor deformations.Let us now comment on an application involving non-chiral deformations. As we havementioned, using Chern-Simons theory the results (4.33) and (4.48) were derived in [25], thefirst corresponding to canonical boundary conditions, and the second to holomorphic boundaryconditions, a particular choice of stress tensor coupling to τ and a particular scaling α = ¯ τ µ and ¯ α = τ ¯ µ of the sources with the modular parameter. On the other hand, the componentsof the connection carrying the charges in either case are a z + a ¯ z = L + ˜ Q and a z = L + Q ,with similar expressions in the barred sector. Since both Q and ˜ Q are highest weight matriceswhich are linear in the corresponding charges, it follows that the matrix a z + a ¯ z in the canonicaldescription has the same form as function of the tilded charges that a z has as function of theuntilded charges in the holomorphic description. From (4.33) and (4.48) it is then immediatethat the functional form of the canonical entropy, as a function of the canonical charges, isexactly the same as the functional form of the Gutperle-Kraus entropy as a function of theholomorphic charges. This agreement was first noticed in [63], and while establishing it from afield theory perspective would be presumably quite involved, it emerges in a very transparentway when using the holographic description in terms of Chern-Simons theory.To be a bit more explicit about the above map at the level of free energies, consider thecanonical free energy (4.30) and the Gutperle-Kraus free energy (4.50). It is easy to see thatif we start with the former, and make the following change of variable (restricting to the chiralsector for simplicity) (¯ τ − τ ) a ¯ z → ¯ τ b ¯ z , a z + a ¯ z → b z (4.62)we get precisely the Gutperle-Kraus free energy in terms of b z , b ¯ z . In addition, if ( a z +36 ¯ z , a ¯ z ) was a Drinfeld-Sokolov pair, then so is ( b z , b ¯ z ), and the trivial monodromy around thecontractible cycle is preserved because τ a z + ¯ τ a ¯ z = τ b z + ¯ τ b ¯ z . The field redefinition (4.62) thenrealizes a map between Hamiltonian deformations, dual to canonical boundary conditions, anddeformations of the type (4.51), which are dual to Gutperle-Kraus boundary conditions.Given that the Gutperle-Kraus boundary conditions are dual to action deformations, whilethe canonical boundary conditions are dual to Hamiltonian deformations, it might seem sur-prising that a detailed agreement was found between the free energies computed from the bulktheory with Gutperle-Kraus boundary conditions [17] and a CFT calculation that involvedHamiltonian deformations by zero modes [18]. From the map (4.62) and the above discussionit is clear that the functional form of the canonical free energy, as a function of the canonicalsources α (cid:39) Im( τ ) µ , is exactly the same as that of the Gutperle-Kraus free energy as a func-tion of the sources α G-K = ¯ τ µ . This explains why the two calculations seemingly agreed, eventhough they involve two a priori different partition functions. We will further comment on theimplications of these findings in section 5.
Recall that in 2 d CFT modular transformations can be understood as a change of coordinatesfollowed by a scale transformation, which are symmetries of the deformed action when thecurrents and sources transform appropriately [27]. In our example involving stress tensordeformations, this implies in particular that λ above transforms covariantly under modulartransformations; from (4.59) it then follows that the canonical source α transforms in acomplicated way. Another way to understand this fact is to notice that the rescaling amountsto a gauge transformation, and that the combined effect of the change of coordinates andgauge transformation preserves the Drinfeld-Sokolov form of the pair ( b z , b ¯ z ) , but not that of( a z + a ¯ z , a ¯ z ). An additional compensating transformation would be necessary to put the gaugefield a back into the appropriate Drinfeld-Sokolov form, which explains why the canonicalsource α transforms in a complicated way under modular transformations. Whether suchtransformations exist when higher sources are turned on is not clear.It is instructive to describe the Chern-Simons perspective on modular transformations, anissue that was recently investigated in [68]. Here we will provide a succinct derivation that willonce more make it clear why modular transformations are simple for deformed Lagrangiansand complicated for deformed Hamiltonians. We will only consider chiral deformations in whatfollows, but the results can be generalized to the non-chiral case in a straightforward way.As we have discussed, there are different possible three-manifolds we can use to evaluatethe Chern-Simons action. If the boundary is a two-torus, there is an entire SL (2 , Z ) family37f three-manifolds we can choose, each yielding a different answer for the on-shell value of theaction. To write this answer explicitly, we rewrite (4.39) for one chiral sector asln Z holo = − πik cs Tr (cid:20) ( τ a z + ¯ τ a ¯ z )( a z + a ¯ z ) + (¯ τ − τ ) (cid:0) (2 L − a z ) a ¯ z (cid:1)(cid:21) (4.63)where the first term is the contribution from the on-shell value of the Chern-Simons action, andthe second term is the contribution from the boundary term. For a different three-manifold,labeled by an SL (2 , Z ) matrix R = (cid:32) α βγ δ (cid:33) (4.64)the partition function becomes ln Z holo [ R ] = − πik cs Tr (cid:20)(cid:16) α ( τ a z + ¯ τ a ¯ z ) + β ( a z + a ¯ z ) (cid:17)(cid:16) γ ( τ a z + ¯ τ a ¯ z ) + δ ( a z + a ¯ z ) (cid:17) + (¯ τ − τ ) (cid:16) (2 L − a z ) a ¯ z (cid:17)(cid:21) . (4.65)In the first term, we recognize the product of the monodromies of the gauge field along thenew a-cycle and b-cycle of the boundary two-torus.Suppose that in the above me make the substitution a z → ( γτ + δ ) − U b z U − , a ¯ z → ( γ ¯ τ + δ ) − U b ¯ z U − (4.66)with U = exp (cid:104) ln ( γτ + δ ) L (cid:105) . (4.67)This substitution preserves the Drinfeld-Sokolov form of the gauge field, i.e. if ( a z , a ¯ z ) isa Drinfeld-Sokolov pair then so is ( b z , b ¯ z ). Morever, by direct calculation, we observe thatafter this substitution the partition function takes the original form (4.39) with τ replaced by( ατ + β ) / ( γτ + δ ). Thus, to summarize, we have shown thatln Z holo [ R ] (cid:2) τ ; ( γτ + δ ) − U b z U − , ( γ ¯ τ + δ ) − U b ¯ z U − (cid:3) = ln Z holo [ ] (cid:20) ατ + βγτ + δ ; b z , b ¯ z (cid:21) , (4.68)where Z holo [ ] on the r.h.s. denotes the partition function in the original manifold, labeledby R = . This is the Chern-Simons version of modular invariance, and we see that (4.66)provides the transformation rules for the sources, in agreement with what one gets directlyfrom the deformed action. As we have emphasized the boundary term is the same for any choice of three-manifold, but the on-shellvalue of the Chern-Simons action depends on how the two-torus is filled, namely the choice of contractible andnon-contractible cycles in the bulk.
Starting from two-dimensional CFTs with a (possibly higher spin) current symmetry algebra,we have reviewed different types of deformations that are possible once sources are switchedon. While some of these can be understood as deformations of the CFT Hamiltonian, othersare defined as changes directly at the level of the action. Associated with each of these theoriesthere is a notion of partition function that is a function of the background sources, and whoseassociated Ward identities we have studied. Using the Ward identities as the guiding prin-ciple, we have argued that these different theories map to different boundary conditions in aholographic realization in terms of Chern-Simons theory on a three-dimensional manifold withboundary. The issue of boundary conditions in the higher spin AdS /CFT correspondencehas proven to be particularly subtle, and it is therefore worth summarizing how our analysisfits with the recent literature.In the holographic context, a first set of boundary conditions in the presence of higherspin sources was proposed in [43, 50], with the flatness condition on the connection resultingin Ward identities of the form (3.11). We have argued that these boundary conditions mostnaturally correspond to a deformation of the CFT action of the form (3.1) in the chiral case,and to an action involving infinitely many higher order terms in the sources in the non-chiralcase. The latter can be rewritten linearly in the sources at the expense of introducing auxiliaryfields, but this formulation has to be constructed on a case by case basis.For the finite temperature version of the holomorphic theory on the torus, two definitionsof the thermal higher spin sources have been proposed. The first alternative was put forwardin [43, 50] and identifies the sources schematically as α = ¯ τ µ and ¯ α = τ ¯ µ , where µ and ¯ µ arethe chemical potentials. This choice implies in particular that the expression for the energyis the same as in the absence of sources, and leads to an entropy of the form (4.47). Thisidentification of the sources is not what one gets from deformations of the form (3.1), butmaps instead to a peculiar deformation of the form (4.51), with the theory defined on a squaretorus. An alternative definition was studied in [25], which consists in defining the thermalsources as α = − iβµ , ¯ α = iβ ¯ µ with β = 2 π Im( τ ) the inverse temperature. This case preciselydescribes deformed actions of the form (3.1) and the expression for the energy is explicitlymodified with respect to the undeformed theory, a fact that we have rederived from a field39heory perspective in section 3.2, and one is led in particular to the formula (4.46) for theentropy [25].A different set of “canonical” boundary conditions in the presence of sources was proposedin [64, 63, 62, 65] from a bulk perspective, with the flatness condition on the connection re-sulting in Ward identities of the form (2.53). We have shown that these boundary conditionscorrespond to deformations of the CFT Hamiltonian of the form (2.5). In our discussion ofthe finite temperature version of this theory on the torus, we have exploited the holographicdescription in terms of Chern-Simons theory to provide expressions for the stress tensor (4.25),free energy (4.30), and entropy (4.32), which are written entirely in terms of the gauge con-nections and are valid in any embedding. It is satisfying to note that, provided the thermalsources are always identified as α = − iβµ , Chern-Simons theory yields the same functionalfor the entropy in theories corresponding to Hamiltonian and Lagrangian deformations (c.f.(4.32) and (4.46)), consistent with the expectation that there should exist an unambiguousfunctional that computes the thermal entropy in a bulk theory containing gravity.It has been proposed [62, 65] that the solutions constructed in [43, 50] that realize theholomorphic W boundary conditions are in fact W (2)3 boundary conditions in disguise. Thisconclusion was arrived at by interpreting the solutions of [43, 50] in light of a canonical Drinfeld-Sokolov pair of the form (4.8)-(4.9). In the original proposal, however, these solutions areinterpreted instead in terms of a holomorphic Drinfeld-Sokolov pair of the form a z = L + Q , a ¯ z = M + . . . . For chiral deformations, we have shown that the latter choice realizes a canonicalstructure where one of the light-cone directions is chosen as the “time” coordinate [41], c.f. theDirac bracket algebra (3.7) obtained by acknowledging the presence of a second class constraint P i = (1 / ∂ + X i . It is conceivable that a canonical structure based on a null coordinate couldbe at odds with a well-posed Cauchy problem in the bulk when sources of both chiralities areswitched on, and this issue deserves further scrutiny. On the other hand, we have argued thatwell-defined partition functions exist in CFT whose Ward identities are indeed those obtainedin [43, 50], and we expect them to have a dual description in the bulk. Consequently, ourpoint of view is that the W boundary conditions proposed in [43, 50] do indeed give rise to W symmetry, and that no conflict arises when they are interpreted in a light-cone frameworkas in [41] (or a suitable generalization thereof in the non-chiral case).To add to this, we emphasize than just providing a solution of the Chern-Simons fieldequations, i.e. a flat connection, is not sufficient; we also need to specify an a priori choiceof boundary conditions, boundary terms, and identification of sources and dual expectationvalues, and different choices can provide different interpretations for the same flat connections.For one choice the flat connections in [43, 50] describe a solution with W boundary conditions,and for another choice they describe a solution with W (2)3 boundary conditions. Both are valid40ut inequivalent points of view.Regarding the matching between bulk and boundary computations, it might appear assomewhat surprising that a chiral half of the partition function (free energy) derived in [17]from the bulk theory with Gutperle-Kraus boundary conditions, which as we have seen herecorrespond to a linear deformation of the CFT action, has been matched by a CFT calculationinvolving a chiral deformation of the Hamiltonian by zero modes [18]. To clarify this issue, insection 4.5 we have shown that the functional form of the partition function (as a functionof the sources) and of the entropy (as a function of the charges) is the same with canonicalor Gutperle-Kraus boundary conditions, even though different definitions of the sources andcharges themselves are been used in one version of the theory or the other. As we have discussedin depth, the detailed matching between charges and sources in the bulk and boundary, namelythe holographic dictionary, will however change depending on what precise version of the theorywe want to describe. As a consequence, one should in principle expect observables such ascorrelators, which are generically not fixed by symmetry or otherwise, to be sensible to thesechoices. These subtle differences have indeed been noticed in calculations of thermal correlatorsof scalar primaries in CFTs with higher spin symmetry [20], and we expect our analysis toshed light on these issues as well.It is perhaps worthwhile to briefly discuss the validity and interpretation of the irrelevantdeformations that we considered. A priori, theories deformed by irrelevant deformations areill-defined. In the present case we are deforming by conserved currents, which might improvethe situation. Let us first think what happens when we expand the theories as a powerseries in terms of the sources, with each term being an integrated correlation function. Thesecorrelation functions are singular when points coincide and some regularization has to beemployed. In standard conformal perturbation theory, one cuts out small disks around thepoints and subtracts all singularities that arise when shrinking the disks to zero size. Weexpect this procedure to yield finite, well-defined answers, in particular since the conservedcurrents cannot develop anomalous dimensions. Therefore, the theories we consider maywell have well-defined perturbative expansions in µ s and ¯ µ s . These perturbative expansionspresumably have zero radius of convergence, and it is an interesting questions whether onecan directly define the deformed theories non-perturbatively e.g. by choosing suitable complexcontours.There are two other arguments that these deformed theories make sense. First, one the One might worry that contact terms produce divergences containing new operators which would need tobe added to the theory to make it consistent. For example, the OPE of two spin-three currents contains T ,the square of the energy-momentum tensor, which does not appear in the deformed theory. We do not see anyneed, at least classically, to add such deformations to the theory. µ s = 0 , mapping the de-formed theory to the original, undeformed theory. The latter is clearly well-defined, and soshould the former? Perhaps, except that it is not clear that the required higher spin transfor-mations act in a reasonable way, they could for example map normalizable field configurationsinto non-normalizable field configurations. Moreover, on a torus one cannot get rid of thezero modes of the µ s and ¯ µ s in this way and the argument no longer applies. A second argu-ment that these deformed theories are well-defined is that we can use Chern-Simons theory tocompute their partition functions, and the result is a non-pathological function of µ s and ¯ µ s .Clearly, more work is required before we can make a definite statement about the existence ofCFT’s deformed by irrelevant deformations of conserved currents.We have by no means exhausted the possible deformations of 2 d conformal field theories,nor have we exhausted the possible list of boundary conditions in Chern-Simons theory. Itwould be interesting to examine whether other interesting boundary conditions exist and ifso what their 2 d CFT interpretation is. Similarly, one could extend our considerations toencompass the non-AdS (non-CFT) higher spin dualities studied in [69, 70]. As discussedin section 4.6 and appendix A, the different types of partition functions we have studiedmoreover differ in their modular transformations properties. We have not found a change ofvariable which directly connects deformations of the action to deformations of the Hamiltonian,however, and in particular we have not been able to determine the behavior of the latterunder modular transformations. It is possible that in order to find such a change of variableadditional operators need to be included, such as normal-ordered products of higher spinfields and their derivatives, and it would be interesting to explore whether such more generaldeformed theories still admit dual Chern-Simons descriptions. These interesting questions willbe discussed elsewhere.
Acknowledgments
It is a pleasure to thank Per Kraus and Daniel Robbins for enlightening discussions and com-ments on a draft of this paper. We are also grateful to Marco Baggio, Max Ba˜nados, AlejandraCastro, Geoffrey Comp`ere, Matthias Gaberdiel, Daniel Grumiller, Diego Hofman, RomualdJanik, Rob Leigh, Wei Li, Eric Perlmutter, Wei Song, Hai-Siong Tan and Erik Verlinde forhelpful conversations. J.I.J. is supported by funding from the European Research Council, ERCGrant agreement no. 268088-EMERGRAV. This work is part of the research programme ofthe Foundation for Fundamental Research on Matter (FOM), which is part of the NetherlandsOrganization for Scientific Research (NWO). 42 A U (1) example Here we will briefly review a non-higher spin example from [71], involving deformations by U (1) currents in a compact boson realization. The canonical partition function with sourcesfor left- and right-momenta is Z can [ τ, α L , α R ] = 1( q ¯ q ) / Tr (cid:104) q L ¯ q ¯ L e πiα L p L e − πiα R p R (cid:105) (A.1)where q = exp(2 πiτ ) and p L = (cid:73) dσ π (cid:0) ∂ σ X − i∂ t E X (cid:1) , (A.2) p R = (cid:73) dσ π (cid:0) ∂ σ X + i∂ t E X (cid:1) . (A.3)It is tempting to conclude that the path integral representation of this partition function is Z Lag,naive = (cid:90) D X e − S + (cid:82) T d σ √ gA i ∂ i X , (A.4)where S is the free action S = 14 π (cid:90) T d σ √ g g ij ∂ i X∂ j X = 14 π (cid:90) T d σ (cid:104) ( ∂ t E X ) + ( ∂ σ X ) (cid:105) , (A.5)(we consider a flat torus with ds ( T ) = dzd ¯ z = dt E + dσ ) and the background gauge field,whose components are the chemical potentials, given by A z = − i α R π Im( τ ) = µ R , A ¯ z = i α L π Im( τ ) = µ L . (A.6)In particular, since modular transformations correspond to a change of coordinates followed bya Weyl rescaling, which are symmetries of the deformed action, Z Lag,naive is modular invariant: Z Lag,naive (cid:20) aτ + bcτ + d , α L cτ + d , α R c ¯ τ + d (cid:21) = Z Lag,naive [ τ, α L , α R ] . (A.7)On the other hand, following the standard steps to discretize the operator trace, the pathintegral representation of Z can is found to be Z can = (cid:90) D P D X e (cid:82) T d σ [ − π ( P ˙ X − ( P ) + ( ∂ σ X ) ) + A tE P + A σ ∂ σ X ] (A.8)with P the momentum conjugate to X . Integrating out P one concludes [71] Z can [ τ, α L , α R ] = e − π ( αL + αR ) τ ) Z Lag,naive [ τ, α L , α R ] , (A.9)43hich in particular implies (c.f. (A.7)) Z can (cid:20) aτ + bcτ + d , α L cτ + d , α R c ¯ τ + d (cid:21) = e πiccτ + d α L e − πicc ¯ τ + d α R Z can [ τ, α L , α R ] . (A.10)Therefore, the canonical partition function in the presence of sources is not modular invariant,but rather modular covariant. The bottom line is that, even in simple examples such asa deformation of the Hamiltonian by constant U (1) chemical potentials, it is important toacknowledge that the proper representation of the canonical partition function involves thepath integral in first order form, and to exercise care when Legendre-transforming to pass tothe Lagrangian version of the theory. B Useful W formulae As explained in the main text, the improved W generators in the bosonic realization are T = 12 δ ij Π i Π j + a i ∂ σ Π i (B.1) W = 13 d ijk Π i Π j Π k + e ij ∂ σ Π i Π j + f i ∂ σ Π i . (B.2)The improved T T bracket takes the usual form (2.25) provided a i a i = − c , (B.3)where c denotes the classical central charge. Similarly, the form of the T W bracket requires a i f i = 0 , f i = a j e ji , a i e ij = a i e ji , e ( ij ) = d kij a k . (B.4)Finally, the improved W W bracket (2.27) requires (2.21) to be satisfied with κ = − c , (B.5)and f i f i = − c
36 (B.6) a i = e ji f j (B.7) e ij f j = 13 e ji f j (B.8) d kij ( e k(cid:96) − e (cid:96)k ) − d k(cid:96) ( j e ki ) = 32 c δ ij a (cid:96) (B.9) − d kij f k + e ki e kj = 53 δ ij (B.10) e k [ i e kj ] = 0 (B.11) e ( ik e kj ) = δ ij (B.12)6 d kij f k + e ik e kj + 64 c a i a j = − δ ij (B.13)44n addition to (B.3) and (B.4). It is worth pointing out that there is some degree of redun-dancy in these constraints; if so desired, one could choose a minimal set that contains all theinformation. We emphasize that the above conditions were derived semiclassically, at the levelof Poisson brackets, and therefore ignoring operator ordering issues. The resulting expressionscan be viewed as the “large- c ” version of the full constraints obtained from the quantum W algebra, derived in [39]. An immediate consequence of the conditions on the various coefficientsis that at least two scalars are needed in order to support an arbitrary semiclassical centralcharge.When the above conditions are satisfied, the improved generators satisfy the Poisson algebra (cid:8) J α ( σ ) , J β ( σ (cid:48) ) (cid:9) = (cid:90) dx f γαβ ( σ, σ (cid:48) , x ) J γ ( x ) + c αβ ( σ, σ (cid:48) ) , (B.14)with f TT T ( σ, σ (cid:48) , x ) = − δ ( σ − x ) ∂ x δ (cid:0) x − σ (cid:48) (cid:1) + δ (cid:0) x − σ (cid:48) (cid:1) ∂ x δ ( σ − x ) (B.15) f WT W ( σ, σ (cid:48) , x ) = − δ ( σ − x ) ∂ x δ (cid:0) x − σ (cid:48) (cid:1) + 2 δ (cid:0) x − σ (cid:48) (cid:1) ∂ x δ ( σ − x ) (B.16) f WW T ( σ, σ (cid:48) , x ) = δ (cid:0) σ (cid:48) − x (cid:1) ∂ x δ ( x − σ ) − δ ( x − σ ) ∂ x δ (cid:0) σ (cid:48) − x (cid:1) (B.17) f TW W ( σ, σ (cid:48) , x ) = 2 κ T ( x ) (cid:2) − δ ( σ − x ) ∂ x δ (cid:0) x − σ (cid:48) (cid:1) + δ (cid:0) x − σ (cid:48) (cid:1) ∂ x δ ( σ − x ) (cid:3) − ∂ x δ ( σ − x ) δ ( x − σ (cid:48) ) + ∂ x δ ( σ − x ) ∂ x δ ( x − σ (cid:48) ) − ∂ x δ ( σ − x ) ∂ x δ ( x − σ (cid:48) ) + 23 δ ( σ − x ) ∂ x δ ( x − σ (cid:48) ) (B.18)and c T T ( σ, σ (cid:48) ) = − c ∂ σ δ ( σ − σ (cid:48) ) (B.19) c W W ( σ, σ (cid:48) ) = c ∂ σ δ ( σ − σ (cid:48) ) . (B.20)Writing the boundary Chern-Simons connection in highest weight gauge, a σ = L + Tk L − − W k W − , (B.21)it was found in [46] that the gauge transformations δa = dλ + [ a, λ ] that respect the Drinfeld-Sokolov form of a σ are generated by an infinitesimal parameter λ = (cid:88) i = − (cid:15) i L i + (cid:88) m = − χ m W m (B.22) We follow the conventions of [46] up to a rescaling of the currents by a factor of 2 π . (cid:15) = − ∂ σ (cid:15) (B.23) (cid:15) − = 12 ∂ σ (cid:15) + Tk (cid:15) + 2
Wk χ (B.24) χ = − ∂ σ χ (B.25) χ = 12 ∂ σ χ + 2 k χT (B.26) χ − = − ∂ σ χ − k T ∂ σ χ − k χ∂ σ T (B.27) χ − = 124 ∂ σ χ + 23 k T ∂ σ χ + 712 k ∂ σ T ∂ σ χ + 16 k χ∂ σ T + 1 k χT − (cid:15) k W (B.28)where (cid:15) ≡ (cid:15) and χ ≡ χ . Under such transformations, the change in the charges is preciselygiven by (2.30)-(2.33). C Non-chiral stress tensor deformations
In certain cases it is possible to write down a partition function whose symmetries give rise totwo decoupled copies of Ward identities of the type (3.12)-(3.13), at the expense of introducingauxiliary fields [35] (see [72] for a review). The auxiliary field formalism is non-universal andhas to be constructed on a case-by-case basis, but we can illustrate many of its importantfeatures by considering a simple example involving stress tensor deformations. Consider thenthe action for the scalar field theory with both left- and right-moving stress tensor deformations S aux = 2 (cid:90) d x (cid:18) − ∂ + X i ∂ − X i − Π i + Π i − + Π i + ∂ − X i + Π i − ∂ + X i − µ −− T ++ − µ ++ T −− (cid:19) (C.1)where Π i ± denote the auxiliary fields and T ±± ≡
12 Π i ± Π i ± . (C.2)For the sake of simplicity, we have omitted improvement terms that would generate classicalcentral extensions. When µ ±± = 0 , S aux yields the free boson action upon integrating out theauxiliary fields. When deformations are present, the action is invariant under the infinitesimaltransformation δX i = p − Π i + + p + Π i − (C.3) δµ ±± = ∂ ± p ± + p ± ∂ ∓ µ ±± − µ ±± ∂ ∓ p ± (C.4) δ Π i ± = ∂ ± (cid:0) p ∓ Π i ± (cid:1) . (C.5)46ince the path integral contains an integration over X and Π, this symmetry yields the Wardidentity (cid:90) d x (cid:28) δS aux δµ ++ δµ ++ + δS aux δµ −− δµ −− (cid:29) = 0 . (C.6)Plugging the explicit variation (C.4) of the sources we obtain ∂ − T ++ = µ −− ∂ + T ++ + 2 T ++ ∂ + µ −− (C.7) ∂ + T −− = µ ++ ∂ − T −− + 2 T −− ∂ − µ ++ (C.8)which are the familiar holomorphic Ward identities (in the absence of central extensions).In the context of holography, these Ward identities (including central extensions) and theirsupersymmetric extension were derived in [73] using the Chern-Simons formulation of three-dimensional anti-de Sitter gravity.One notices that the equation of motion for the auxiliary fields isΠ i ± = ∂ ± X i − µ ±± Π i ∓ , (C.9)which can be solved to give Π i ± = ∂ ± X i − µ ±± ∂ ∓ X i − µ −− µ ++ . (C.10)From (C.2) we see that the stress tensor obeying the holomorphic Ward identities is not merely ∼ ( ∂ ± X ) , but rather T ±± = 12 (cid:18) ∂ ± X i − µ ±± ∂ ∓ X i − µ −− µ ++ (cid:19) . (C.11)The fact that the naive free-field expressions for the currents are modified in a source-dependentway in the presence of non-chiral deformations is a general feature of the construction.Another general feature we have emphasized in the body of the paper is that the process ofintegrating out the auxiliary fields results in a second order action which contains correctionsto all orders in the sources. To illustrate this point we can replace (C.10) back into the action,obtaining a flat space theory with LagrangianLag ≡ − µ −− µ ++ ) (cid:20) (1 + µ −− µ ++ ) ∂ + X i ∂ − X i − µ ++ ∂ − X i ∂ − X i − µ −− ∂ + X i ∂ + X i (cid:21) . (C.12)The spin-2 symmetries are of course still present: under the transformations (note the infinites-imal parameters k ± below are different from the p ± above) δX i = k + ∂ + X i + k − ∂ − X i (C.13) δµ ++ = ∂ + (cid:0) k − + µ ++ k + (cid:1) + (cid:0) k − + µ ++ k + (cid:1) ∂ − µ ++ − µ ++ ∂ − (cid:0) k − + µ ++ k + (cid:1) (C.14) δµ −− = ∂ − (cid:0) k + + µ −− k − (cid:1) + (cid:0) k + + µ −− k − (cid:1) ∂ + µ −− − µ −− ∂ + (cid:0) k + + µ −− k − (cid:1) (C.15)47he second order action changes as δS = (cid:90) d x (cid:104) ∂ + (cid:0) k + Lag (cid:1) + ∂ − (cid:0) k − Lag (cid:1)(cid:105) . (C.16)The fact that the action is non-linear in the sources should come as no surprise if werecall that the gauging of spin-2 deformations is equivalent to putting the theory on a curvedbackground metric. Indeed, the second order action involving the Lagrangian (C.12) can bewritten covariantly as S = 12 (cid:90) d x √− gg µν ∂ µ X i ∂ ν X i (C.17)with metric [72] g µν = Ω(1 − µ −− µ ++ ) (cid:32) µ ++ µ −− µ ++ µ −− µ ++ µ −− (cid:33) . (C.18)As emphasized in [72], (C.18) does not correspond to partial gauge fixing: it is a generalparameterization of a two-dimensional metric. In our conventions √− g = Ω is the conformalmode of the metric, which as usual drops from the action because of Weyl invariance. Itis straightforward to verify that the transformations of the covariant fields induced by thetransformation (C.13)-(C.15) of the sources are simply δX i = £ k X i (C.19) δg µν = £ k g µν − ( ∇ ρ k ρ ) g µν . (C.20)In other words, the symmetry transformations (C.13)-(C.15) are a combination of diffeomor-phism generated by k µ plus a Weyl rescaling generated by − ( ∇ ρ k ρ ) .Note that the components of the covariant stress tensorˆ T µν = 12 (cid:16) ∂ µ X i ∂ ν X i − g µν ∂ α X i ∂ α X i (cid:17) (C.21)are given by ˆ T ++ = T ++ + µ T −− (C.22)ˆ T −− = T −− + µ −− T ++ (C.23)ˆ T + − = µ ++ T −− + µ −− T ++ . (C.24)This illustrates yet another subtle point: the definition of the stress tensor depends on whatthe sources are, namely what is kept fixed in the variation. While the covariant stress tensorcouples to the metric g µν , the currents T ±± satisfying the usual Ward identities (C.7)-(C.8)couple instead to the sources µ ±± . Notice that our parameterization of the metric differs slightly from that in [72]. Tr (cid:2) a z (cid:3) and the OPE In what follows we will exemplify various relations satisfied by flat connections in Drinfeld-Sokolov form. A 2 d Drinfeld-Sokolov connection consists of a component a J that contains aset of currents as highest weights, and a conjugate component a µ whose lowest weights arelinear in the corresponding sources. The various relations we discuss below rely exclusively onthis lowest/highest weight structure, and therefore apply to any choice of boundary conditions.However, for the sake of concreteness we will exemplify them for holomorphic boundary condi-tions, where the currents sit in a z and the sources in a ¯ z and ( z, ¯ z ) denote complex coordinates.We will moreover work with the theory defined by the principal sl (2) embedding into sl ( N ) ,but the expressions adapt straightforwardly to other embeddings as well (see [25] for example).In the principal embedding, the sl ( N ) generators organize into N − sl (2)spin s − s = 2 , . . . , N ), spanned by generators W ( s ) j with j = − s + 1 , . . . , s −
1. In particular,the sl (2) generators L j ( j = − , ,
1) correspond to the spin one multiplet W (2) j = L j . Thestructure of the general Drinfeld-Sokolov connection is then a z = L + T ( z, ¯ z ) k L − + N (cid:88) s =3 α s J s ( z, ¯ z ) W ( s ) − s +1 (D.1) a ¯ z = µ ( z, ¯ z ) L + N (cid:88) s =3 β s µ s ( z, ¯ z ) W ( s ) s − + (higher weights) . (D.2)Here k ≡ c/ α s and β s are normalization constants which will be fixed as indicated below,and the higher weight terms in a ¯ z are completely determined by solving the flatness conditions.The latter contain N − J s in the presence of sources µ s .In order to derive the symmetries associated to the above connection, one notices that themost general gauge transformation δa = d Λ+[ a, Λ] that preserves the form of a z contains N − (cid:15) , . . . , (cid:15) N . Moreover, given that the flatness condition F z ¯ z = 0 and the condition δa z = 0 are essentially the same equation (save for two componentsthat yield the Ward identities in the former case and the transformation of the currents inthe latter), it is not hard to see that the matrix parameter Λ that generates such a gaugetransformation is obtained from a ¯ z by simply replacing µ s → (cid:15) s for s = 2 , . . . , N :Λ = a ¯ z | µ s → (cid:15) s . (D.3)Under this transformation the stress tensor and higher spin currents J s will transform, so that δa z = δTk L − + N (cid:88) s =3 α s δJ s ( z, ¯ z ) W ( s ) − s +1 . (D.4)49omparing these transformations with Noether’s theorem δ λ O ( w ) = Res z → w (cid:104) λ ( z ) J ( z ) O ( w ) (cid:105) (D.5)one reads off the semiclassical (large- c ) OPEs of the W N currents. Their normalization α s canbe then determined (up to a sign) by fixing the normalization of the OPEs to be J s ( z ) J s ( w ) ∼ c/s ( z − w ) s + . . . (D.6)Having determined the normalization of the currents in this way, the normalization β s of thesources is fixed by demanding − k cs Tr (cid:2) ( a z − L ) a ¯ z (cid:3) = µ ( z, ¯ z ) T ( z, ¯ z ) + N (cid:88) s =3 µ s ( z, ¯ z ) J s ( z, ¯ z ) (D.7) − k cs Tr (cid:2) L a ¯ z (cid:3) = µ ( z, ¯ z ) T ( z, ¯ z ) + c ∂ µ ( z, ¯ z ) + N (cid:88) s =3 ( s − µ s ( z, ¯ z ) J s ( z, ¯ z ) (D.8)where ∂ ≡ ∂ z ( ¯ ∂ ≡ ∂ ¯ z ), all traces are taken in the fundamental representation, and k cs = k L L ] = c L L ] (D.9)is the Chern-Simons level. The trace relations (D.7)-(D.8) follow from properties of the sl ( N )algebra and the flatness condition on the Drinfeld-Sokolov pair, and are valid for arbitraryspacetime-dependent sources as we now show.Without loss of generality, for the purpose of proving (D.7)-(D.8) we choose the normal-ization of the generators in the principal embedding such that[ L m , L n ] = ( m − n ) L m + n (D.10) (cid:104) L m , W ( s ) n (cid:105) = ( m ( s − − n ) W ( s ) m + n (D.11)and the Cartan-Killing form on sl ( N, R ) is thenTr (cid:104) W ( s ) m W ( r ) n (cid:105) = t ( s ) m δ r,s δ m, − n (D.12)where the explicit form of the coefficients t ( s ) m can be found in e.g. [11]. Since highest-weightgenerators have non-vanishing trace only against lowest-weight generators in the same multi-plet, it is immediate from (D.1)-(D.2) that − k cs Tr (cid:2) ( a z − L ) a ¯ z (cid:3) = − Tr (cid:2) L − L (cid:3) L L ] µ T ( z, ¯ z ) − k cs N (cid:88) s =3 α s β s µ s ( z, ¯ z ) J s ( z, ¯ z ) t ( s ) s − . (D.13)50ur normalization implies Tr [ L − L ] = − L L ], so the last equation will be precisely(D.7) provided we choose β s = − k cs α s t ( s ) s − = − L L ] kα s t ( s ) s − . (D.14)Since we are always free to normalize the sources in this way, this proves (D.7).In order to prove (D.8) we will first obtain the useful intermediate results − k cs Tr [ L a ¯ z ] = k ∂µ (D.15) k cs Tr (cid:104)(cid:2) ( a z − L ) , L (cid:3) a ¯ z (cid:105) = µ ( z, ¯ z ) T ( z, ¯ z ) + N (cid:88) s =3 ( s − µ s ( z, ¯ z ) J s ( z, ¯ z ) . (D.16)To this end, consider the flatness condition F z ¯ z = ∂a ¯ z − ¯ ∂a z + [ a z , a ¯ z ] = 0 and its trace against L − : 0 = Tr [ L − F z ¯ z ] = ∂ Tr [ L − a ¯ z ] + Tr (cid:2) [ L − , a z ] a ¯ z (cid:3) , (D.17)where we used the cyclicity of the trace and ¯ ∂ Tr [ L − a z ] = 0, which follows from Tr [ L − a z ] =Tr [ L − L ] = constant. Noticing Tr [ L − a ¯ z ] = Tr [ L − L ] µ and also [ L − , a z ] = [ L − , L ] = − L , which follows from (D.1) and (D.11), (D.17) becomes0 = Tr [ L − L ] ∂µ − L a ¯ z ] ⇒ Tr [ L a ¯ z ] = − Tr [ L L ] ∂µ . (D.18)Multiplying this last equation by − k cs we obtain (D.15).In order to derive (D.16), let us define the matrix Q = a z − L . It follows that[ Q, L ] = T ( z, ¯ z ) k [ L − , L ] + N (cid:88) s =3 α s J s ( z, ¯ z ) (cid:104) W ( s ) − s +1 , L (cid:105) = − T ( z, ¯ z ) k L − − N (cid:88) s =3 ( s − α s J s ( z, ¯ z ) W ( s ) − s +1 (D.19)and therefore k cs Tr (cid:104) [ Q, L ] a ¯ z (cid:105) = − Tr [ L − L ]2Tr [ L L ] µ ( z, ¯ z ) T ( z, ¯ z ) − k cs N (cid:88) s =3 ( s − α s β s µ s ( z, ¯ z ) J s ( z, ¯ z ) t ( s ) s − = µ ( z, ¯ z ) T ( z, ¯ z ) + N (cid:88) s =3 ( s − µ s ( z, ¯ z ) J s ( z, ¯ z ) (D.20)where in the last equality we used Tr [ L − L ] = − L L ] and the normalization (D.14).51ith these results in hand we can now compute − k cs Tr [ L a ¯ z ] = − k cs Tr (cid:2) [ L , L ] a ¯ z (cid:3) = − k cs Tr (cid:2) [ a z − Q, L ] a ¯ z (cid:3) = − k cs Tr (cid:2) [ a ¯ z , a z ] L − [ Q, L ] a ¯ z (cid:3) = − k cs Tr (cid:2)(cid:0) ∂a ¯ z − ¯ ∂a z (cid:1) L − [ Q, L ] a ¯ z (cid:3) = − k cs ∂ Tr (cid:2) a ¯ z L (cid:3) + k cs Tr (cid:104) [ Q, L ] a ¯ z (cid:105) (D.21)where as before we used the flatness condition and the cyclicity of the trace. Using (D.15) and(D.16), equation (D.21) becomes precisely (D.8), completing the proof. As it should be clearfrom the above derivations, it is a straightforward matter to extend these general results tonon-principal embeddings.Let us continue with our discussion of symmetries. The transformation δµ s of the sourcescan be read off from the lowest weights of δa ¯ z under the same allowed gauge transformationwith parameter (D.3) we employed above. We also note that, by construction, the Wardidentities in the presence of sources are obtained from the variation of the currents by simplyreplacing the infinitesimal parameters (cid:15) s by the sources µ s , i.e. ∂J s = δJ s | (cid:15) s → µ s . (D.22)Define now the quantity∆ L N ( z, ¯ z ) = µ ( z, ¯ z ) T ( z, ¯ z ) + N (cid:88) s =3 µ s ( z, ¯ z ) J s ( z, ¯ z ) (D.23)which is the deformation of the CFT Lagrangian in the chiral case. With the above normal-ization one finds the curious relation − k cs Tr (cid:2) a z (cid:3) = Res z → w (cid:104) ( z − w )∆ L N ( z )∆ L N ( w ) (cid:105) + ∂ ( P N ) , (D.24)where P N will be determined below for N = 2 , , − k cs Tr (cid:2) a z (cid:3) is the coefficient of the second order pole in the ∆ L N ( z )∆ L N ( w )OPE. D.1 N = 2 We employ the usual two-dimensional representation of sl (2 , R ) in terms of matrices L = 12 (cid:32) − (cid:33) , L = (cid:32) (cid:33) , L − = (cid:32) −
10 0 (cid:33) . (D.25)52he Drinfeld-Sokolov connection is a z = L + 1 k T ( z, ¯ z ) L − (D.26) a ¯ z = µ ( z, ¯ z ) L − ∂µ L + (cid:18) k T µ + 12 ∂ µ (cid:19) L − (D.27)and the flatness condition amounts to the Ward identity¯ ∂T = µ ∂T + 2 T ∂µ + k ∂ µ . (D.28)The general infinitesimal gauge transformation that preserves the form of a z has parameterΛ = a ¯ z | µ → (cid:15) = (cid:15) ( z, ¯ z ) L − ∂(cid:15) L + (cid:18) k T (cid:15) + 12 ∂ (cid:15) (cid:19) L − . (D.29)Under such a transformation, the stress tensor changes as δT = (cid:15)∂T + 2 T ∂(cid:15) + k ∂ (cid:15) . (D.30)Similarly, from δa ¯ z = δµ L + (higher weights) we read off the transformation of the source δµ = ¯ ∂(cid:15) − µ ∂(cid:15) + (cid:15)∂µ . (D.31)Comparing the variation (D.30) with Noether’s theorem δ (cid:15) T ( w ) = Res z → w [ (cid:15) ( z ) T ( z ) T ( w )] weobtain the stress tensor OPE. The standard normalization requires k = c L L ] k cs = k cs (D.32)and we find T ( z ) T ( w ) ∼ c/ z − w ) + 2 T ( w )( z − w ) + ∂T ( w ) z − w (D.33)as expected.With the normalization (D.32) the Drinfeld-Sokolov flat connection satisfies − k cs Tr (cid:2) a z (cid:3) = µ (2 T ) + c µ ∂ µ − ∂ (cid:16) c µ (cid:17) . (D.34)In the N = 2 case we have ∆ L = µ T andRes z → w (cid:104) ( z − w )∆ L ( z )∆ L ( w ) (cid:105) = µ (2 T ) + c µ ∂ µ . (D.35)Therefore, − k cs Tr (cid:2) a z (cid:3) is indeed of the form (D.24) with P = − c µ .53 .2 N = 3 Our convention for the sl (3 , R ) generators in the principal embedding is L = − , L = , L − = − ,W = 2 , W = − , W − = 8 ,W − = 2 − , W = 23 − . The Drinfeld-Sokolov connection is of the form a z = L + 1 k T ( z, ¯ z ) L − + 1 k β W ( z, ¯ z ) W − (D.36) a ¯ z = µ ( z, ¯ z ) L − β µ ( z, ¯ z ) W + (cid:88) j = − f j ( z, ¯ z ) L j + (cid:88) m = − g m ( z, ¯ z ) W m . (D.37)Solving the flatness condition yields f = − ∂µ (D.38) f − = 1 k T µ + 2 k µ W + 12 ∂ µ (D.39) g = β ∂µ (D.40) g = − β (cid:18) k µ T + 14 ∂ µ (cid:19) (D.41) g − = β k (cid:18) µ ∂T + 5 T ∂µ + k ∂ µ (cid:19) (D.42) g − = − β k (cid:18) k µ T − β µ W + 7 ∂T ∂µ + 2 µ ∂ T + 8 T ∂ µ + k ∂ µ (cid:19) (D.43)plus two additional constraints that correspond to the Ward identities ∂T = µ ∂T + 2 T ∂µ + 2 µ ∂W + 3 W ∂µ + k ∂ µ (D.44) ∂W = µ ∂W + 3 W ∂µ − β µ (cid:18) ∂ T + 16 k T ∂T (cid:19) − β ∂µ (cid:18) ∂ T + 32 k T (cid:19) − β ∂ µ ∂T − β T ∂ µ − β k∂ µ . (D.45)54n agreement with (D.3), the generator Λ of a general infinitesimal gauge transformationthat preserves the form of a z is obtained by replacing µ → (cid:15) and µ → χ in (D.37):Λ = a ¯ z | µ → (cid:15), µ → χ . (D.46)Under such gauge transformations, the currents transform as δT = (cid:15)∂T + 2 T ∂(cid:15) + k ∂ (cid:15) + 2 χ∂W + 3 W ∂χ . (D.47) δW = (cid:15)∂W + 3 W ∂(cid:15) − β χ (cid:18) ∂ T + 16 k T ∂T (cid:19) − β ∂χ (cid:18) ∂ T + 32 k T (cid:19) − β ∂ χ∂T − β T ∂ χ − β k∂ χ . (D.48)Similarly, from δa ¯ z = δµ L − β δµ W + higher weights (D.49)we find the transformation of the sources δµ = ¯ ∂(cid:15) + (cid:15)∂µ − µ ∂(cid:15) − β k χ (cid:0) k∂ µ + 16 T ∂µ (cid:1) + β k ∂χ (cid:0) T µ + 3 k∂ µ (cid:1) − β ∂ χ∂µ + β ∂ χ µ (D.50) δµ = ¯ ∂χ + 2 χ∂µ − µ ∂χ + (cid:15)∂µ − µ ∂(cid:15) . (D.51)Comparing the variations (D.47)-(D.48) with Noether’s theorem (D.5) we can read off thelarge- c W OPEs. The standard normalization (D.6) requires k = c k cs Tr [ L L ] = 4 k cs , β = −
85 (D.52)and we obtain T ( z ) W ( w ) ∼ W ( w )( z − w ) + ∂W ( w ) z − w (D.53) W ( z ) W ( w ) ∼ c/ z − w ) + 2 T ( w )( z − w ) + ∂T ( w )( z − w ) + 110 3 ∂ T ( w ) + c T ( w )( z − w ) + 115 ∂ T ( w ) + c T ( w ) ∂T ( w ) z − w (D.54)with T T as in (D.33).With the normalization (D.52) we find that the flat connection in Drinfeld-Sokolov formsatisfies − k cs Tr (cid:2) a z (cid:3) = 2 µ T + 6 µ µ W + 110 µ (cid:18) ∂ T + 64 c T (cid:19) + µ ∂ (cid:16) c ∂µ (cid:17) + µ ∂ (cid:16) T ∂µ + c ∂ µ (cid:17) − ∂ (cid:18) µ T + c µ − c
180 ( ∂µ ) + c µ ∂ µ (cid:19) . (D.55)55n the other hand, in the N = 3 case we have ∆ L = µ T + µ W andRes z → w (cid:104) ( z − w )∆ L ( z )∆ L ( w ) (cid:105) = 2 µ T + 6 µ µ W + 110 µ (cid:18) ∂ T + 64 c T (cid:19) + µ ∂ (cid:16) c ∂µ (cid:17) + µ ∂ (cid:16) T ∂µ + c ∂ µ (cid:17) . (D.56)Hence, Tr (cid:2) a z (cid:3) verifies (D.24) with P = − µ T − c µ + c
180 ( ∂µ ) − c µ ∂ µ . (D.57) D.3 N = 4 We employ the matrix realization of the sl (4 , R ) generators given in [74]. The Drinfeld-Sokolovconnection is of the form a z = L + 1 k T ( z, ¯ z ) L − + 1 k β W ( z, ¯ z ) W − + 1 k γ U ( z, ¯ z ) U − (D.58) a ¯ z = µ ( z, ¯ z ) L − β µ ( z, ¯ z ) W + 5 γ µ ( z, ¯ z ) U (D.59)+ (cid:88) j = − f j ( z, ¯ z ) L j + (cid:88) m = − g m ( z, ¯ z ) W m + (cid:88) n = − h n ( z, ¯ z ) U n (D.60)where the constants k , β and γ will be fixed by demanding the OPEs to have the standardnormalization. The flatness conditions yields f = − ∂µ (D.61) f − = 1 k ( T µ + 2 µ W + 3 µ U ) + 12 ∂ µ (D.62) g = 5 β ∂µ (D.63) g = − β k (cid:18) µ T + γβ µ W + k ∂ µ (cid:19) (D.64) g − = 5 β k (cid:18) µ ∂T + 5 T ∂µ + 4 γβ ( µ ∂W + 2 W ∂µ ) + k ∂ µ (cid:19) (D.65) g − = 1 βk µ W − β k (cid:20) µ (cid:18) ∂ T + 3 k T − Uγ (cid:19) + 74 ∂µ ∂T + 2 T ∂ µ + k ∂ µ + γβ µ (cid:18) ∂ W + 485 k T W (cid:19) + 3 γβ ∂µ ∂W + 13 γ β W ∂ µ (cid:21) (D.66)56 = − γ ∂µ (D.67) h = 5 γ (cid:18) ∂ µ + 6 k µ T (cid:19) (D.68) h = − γ k (cid:18) µ ∂T + 2 T ∂µ + k ∂ µ (cid:19) (D.69) h − = − k µ W + 5 γ k (cid:20) µ (cid:18) ∂ T + 6 k T + 6 γ U (cid:19) + 112 ∂T ∂µ + 7 T ∂ µ + k ∂ µ (cid:21) (D.70) h − = 16 k ( µ ∂W + 4 W ∂µ ) − γ k (cid:20) µ (cid:18) ∂ T + 9 k T ∂T + 3 γ ∂U (cid:19) + ∂µ (cid:18) ∂ T + 11 k T + 9 γ U (cid:19) + 2512 ∂ µ ∂T + 53 T ∂ µ + k ∂ µ (cid:21) (D.71) h − = 136 k (cid:20) γ µ U − µ (cid:18) ∂ W + 30 k T W (cid:19) − ∂W ∂µ − W ∂ µ + µ (cid:18) γ k T ∂ T + 3 γk ( ∂T ) + γ ∂ T + 22 k T U + 10 γk T + ∂ U − γkβ W (cid:19) + ∂µ (cid:18) γ k T ∂T + 17 γ ∂ T + 4 ∂U (cid:19) + ∂ µ (cid:18) γ ∂ T + 68 γ k T + 5 U (cid:19) + 5 γ (cid:18) ∂T ∂ µ + 59 T ∂ µ + k ∂ µ (cid:19)(cid:21) (D.72)plus the Ward identities ∂T = µ ∂T + 2 T ∂µ + 2 µ ∂W + 3 W ∂µ + 3 µ ∂U + 4 U ∂µ + k ∂ µ (D.73) ∂W = µ ∂W + 3 W ∂µ − β µ (cid:18) k T ∂T + 172 ∂ T − γ ∂U (cid:19) − β ∂µ (cid:18) ∂ T + 29 k T − γ U (cid:19) − β ∂T ∂ µ − β T ∂ µ − kβ ∂ µ − γk µ (cid:18) W ∂T + 179
T ∂W + 5 k ∂ W (cid:19) − γ ∂µ (cid:18) k T W + 5 ∂ W (cid:19) − γ ∂W ∂ µ − γ W ∂ µ (D.74)57 U = µ ∂U + 4 U ∂µ − γ k µ (cid:18) W ∂T + 18
T ∂W + k ∂ W (cid:19) − γ ∂µ (cid:18) ∂ W + 523 k T W (cid:19) − γ ∂W ∂ µ − γ W ∂ µ + γ k µ (cid:20) ∂T (cid:18) ∂ T + 4 k T + 73 γ U (cid:19) + 1336 T ∂ T + k ∂ T + 73 γ T ∂U + k γ ∂ U − β W ∂W (cid:21) + γ k ∂µ (cid:18) T ∂ T + 29572 ( ∂T ) + 5 k ∂ T + 14 γ T U + 8 k T + 5 k γ ∂ U − β W (cid:19) + γ ∂ µ (cid:18) k T ∂T + 7162 ∂ T + 14 γ ∂U (cid:19) + γ ∂ µ (cid:18) ∂ T + 4927 k T + 1 γ U (cid:19) + γ (cid:18) ∂T ∂ µ + 14 T ∂ µ + k ∂ µ (cid:19) (D.75)In agreement with (D.3), the generator Λ of the most general infinitesimal gauge transfor-mation that preserves the form of a z is obtained asΛ = a ¯ z | µ → (cid:15), µ → χ, µ → ξ . (D.76)Under such gauge transformations, the transformation of the currents is obtained from theright hand side of the Ward identities (D.73)-(D.75) by replacing µ → (cid:15) , µ → χ and µ → ξ (c.f. (D.22)). Comparing these variations with Noether’s theorem (D.5) we read off the large- c W OPEs. The standard normalization (D.6) requires k = c k cs Tr [ L L ] = 10 k cs , β = − , γ = 2735 . (D.77)In addition to (D.33) and (D.53) we then obtain the following OPEs T ( z ) U ( w ) ∼ U ( w )( z − w ) + ∂U ( w ) z − w (D.78) W ( z ) W ( w ) ∼ c/ z − w ) + 2 T ( w )( z − w ) + ∂T ( w )( z − w ) + 110 3 ∂ T ( w ) + c T ( w ) − γ U ( w )( z − w ) + 115 ∂ T ( w ) + c T ( w ) ∂T ( w ) − γ ∂U ( w ) z − w (D.79) W ( z ) U ( w ) − γ ∼ W ( w )( z − w ) + 79 ∂W ( w )( z − w ) + 16 ∂ W ( w ) + c T ( w ) W ( w )( z − w ) + 136 ∂ W ( w ) + c W ( w ) ∂T ( w ) + c T ( w ) ∂W ( w ) z − w (D.80)58 ( z ) U ( w ) ∼ c/ z − w ) + 2 T ( w )( z − w ) + ∂T ( w )( z − w ) + 310 ∂ T ( w ) + c T ( w ) + γ U ( w )( z − w ) + 15 ∂ T ( w ) + c T ( w ) ∂T ( w ) + γ ∂U ( w )( z − w ) + 184 ∂ T ( w ) + γ ∂ U ( w )( z − w ) (D.81)+ 1 c W ( w ) + γ T ( w ) U ( w ) + ( ∂T ( w )) + T ( w ) ∂ T ( w ) + c T ( w ) ( z − w ) + 1 c c γ ∂ U ( w ) + γ ( U ( w ) ∂T ( w ) + T ( w ) ∂U ( w )) + W ( w ) ∂W ( w ) z − w + 17 c c ∂ T ( w ) + ∂T ( w ) ∂ T ( w ) + T ( w ) ∂ T ( w ) + c T ( w ) ∂T ( w ) z − w We have left explicit factors of γ in the OPEs in order to have the freedom to choose the overallsign in the normalization of U (c.f. (D.77)).With the normalization (D.52) the flat connection in Drinfeld-Sokolov form satisfies − k cs Tr (cid:2) a z (cid:3) = 2 µ T + 6 µ µ W + 8 µ µ U + µ (cid:18) c T + 310 ∂ T − γ U (cid:19) − γ µ µ (cid:18) ∂ W + 48 c T W (cid:19) + µ (cid:18) c T + 8835 c T ∂ T + 5928 c ( ∂T ) + 365 cγ T U + 22514 c W + 184 ∂ T + 328 γ ∂ U (cid:19) + µ (cid:16) c ∂ µ (cid:17) + µ (cid:18) c ∂ µ + ∂T ∂µ + T ∂ µ − γ ∂W ∂µ − γ W ∂ µ (cid:19) + µ (cid:18) T ∂ µ + 115 ∂ T ∂µ + 320 ∂ T ∂ µ + 425 c T ∂T ∂µ + 215 c T ∂ µ + 16 ∂T ∂ µ + 2770 γ ∂ ( U ∂µ ) − γ ∂W ∂µ − γ W ∂ µ + c ∂ µ (cid:19) (D.82)+ ∂ ( P )where the quantity P in the last line is defined as P = − c µ − T µ − µ (cid:18) ∂ T + 84 c T + 10 γU (cid:19) + 7 γ W µ µ − µ (cid:18) c ∂ µ + 160 ∂T ∂µ + 120 T ∂ µ (cid:19) + 130 T ( ∂µ ) − c µ ∂ µ + c
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