Higher Spin Gravities and Presymplectic AKSZ Models
aa r X i v : . [ h e p - t h ] F e b Higher Spin Gravities and Presymplectic AKSZ Models
Alexey
Sharapov a & Evgeny Skvortsov ∗ b,ca Physics Faculty, Tomsk State University,Lenin ave. 36, Tomsk 634050, Russia b Service de Physique de l’Univers, Champs et Gravitation,Universit´e de Mons, 20 place du Parc, 7000 Mons, Belgium c Lebedev Institute of Physics,Leninsky ave. 53, 119991 Moscow, Russia
Abstract
As a step towards quantization of Higher Spin Gravities we construct the presymplecticAKSZ sigma-model for 4 d Higher Spin Gravity which is AdS/CFT dual of Chern–Simonsvector models. It is shown that the presymplectic structure leads to the correct quantumcommutator of higher spin fields and to the correct algebra of the global higher spinsymmetry currents. The presymplectic AKSZ model is proved to be unique, it dependson two coupling constants in accordance with the AdS/CFT duality, and it passes somesimple checks of interactions. ∗ Research Associate of the Fund for Scientific Research – FNRS, Belgium ontents d HSGRA 257 Higher spin waves and currents 378 Final comments and discussion 39A Hochschild, cyclic, and Lie algebra cohomology 44B Cohomology of Weyl algebras and their smash products 50C Cohomology of Grassmann algebras 54D Presymplectic structures in 4 d HSGRA 57E Integrability of zero-form equations 62Bibliography 63
Introduction
Quantization of Higher Spin Gravities (HSGRA) is an important open problem together withthe problem of constructing more viable HSGRA. Concrete quantum checks of HSGRA requireconcrete models. Topological theories in 3 d , which are higher spin extensions of Chern–Simonsformulation of 3 d gravity [1–4] or 3 d conformal gravity [5–7], are well-behaved perturbativelydue to the lack of any propagating degrees of freedom, but their non-perturbative definitionrequires further study, see e.g. [8]. Amplitudes of 4 d conformal HSGRA [9–11] were studied in[12–14]. The most elaborate checks have been performed for Chiral HSGRA [15–19], which wasshown to be one-loop finite [18–20]. In addition, there is a number of interesting computations ofvacuum one-loop corrections [21–30]. Finally, the one-loop correction to the four-point functionin the bulk dual of the free vector model can be reduced to a CFT computation and seems tobe consistent with the vacuum one-loop results [31]. The only class of perturbatively local HSGRAs with propagating massless fields and anaction seems to be given by Chiral HSGRA and its various truncations. This was proved forflat space in [17] and it is likely to be the case for
AdS as well. However, Chiral Theory doesnot have a covariant action at the moment and this hampers the study of quantum corrections.Furthermore, the chiral theory constitutes [34] only a subsector of the HSGRA that is dual toChern–Simons Matter theories [35–40]. The latter HSGRA is known to feature non-localitiesbeyond what is admissible by the field theory methods [41–45].Nevertheless, the bulk dual of vector models [46, 47] can be approached from the oppositeend: abandoning perturbative locality one can get a very explicit description of this HSGRAat the level of formal algebraic structures; hence the name “Formal HSGRA”. In practice,this means that one can construct an L ∞ -algebra that extends a given higher spin algebra.Equivalently, one can think of the corresponding Q -manifold, where Q — an odd, nilpotent Q = 0 vector field — can be expanded as Q = f ABC w B w C ∂/∂w A + ... with f ABC being thestructure constants of a higher spin algebra. The same idea can be encoded in the formaldynamical system [48] E A ≡ d w A − Q A ( w ) = 0 , ( ∗ )which is a sigma-model with the Q -manifold as a target and the space-time manifold as a base.It is rather remarkable that the Q ’s of HSGRAs remain non-trivial despite non-localities andnon-local redefinitions hidden in the very definition of Q . With the proviso that the equations In this regard it is worth mentioning the collective dipole approach, see e.g. [32, 33], which is bound toreproduce the correct holographic correlation functions and all the other observables of vector models. However,its relation to the field theory approach is to be clarified [33]. Q with certain prescriptions thatwould lead to a reasonable field theory or the formal approach itself will allow one to definethe theory as to be able to compute various physical observables.The classical equations of motion ( ∗ ) are non-Lagrangian as they stand. There is no theoremprohibiting the action principle for HSGRA extended with suitable auxiliary fields; and yetno one has succeeded in formulating it. This prevents applying the standard quantizationmethods (canonical or path-integral) to this class of field-theoretical models. Therefore, itseems reasonable to try other approaches to the quantization problem that are not so closelyrelated to the Lagrangian or Hamiltonian form of classical dynamics. Let us stress that allholographic HSGRAs are Lagrangian theories, more or less by definition. The actions, howevernon-local they may be, can be reconstructed [42] from the CFT correlators. At the free levelsuch actions reduce to the Fronsdal action [49] for an appropriate set of fields.A possible alternative to the conventional quantization methods has been proposed long agoin [50]. It is based on the concept of a Lagrange anchor, which can be regarded as a covariantcounterpart of (degenerate) Poisson brackets. In the non-degenerate case, the Lagrange anchoris just inverse to the integrating factor in the inverse problem of the calculus of variations.Degeneracy manifests itself in the fact that not all classical degrees of freedom may fluctuatein quantum theory; some of them remain purely classical. Nonetheless, it is possible to definea full-fledged path integral for transition amplitudes and quantum averages. With the helpof Lagrange anchor every (non-)Lagrangian field theory in d dimensions can equivalently bereformulated as a topological Lagrangian theory in d +1 dimensions. Applying then the standardBV quantization to the latter induces a quantization of the former (non-)Lagrangian theory.It should be noted that the extension of a (non-)Lagrangian theory on the boundary to atopological Lagrangian theory in the bulk depends crucially on the choice of Lagrange anchorand different choices may result in different quantizations of one and the same classical theory.In the HSGRA context, the quantization method above was first considered in [51]. However,the Lagrange anchor behind the topological model of [51] does not reproduce the standardpropagators for free higher spin fields in Fronsdal’s theory. Therefore, the relation betweenthe two theories is unclear beyond the classical equations of motion. In [52], the canonicalLagrange anchor was constructed for the subsystem of ( ∗ ) that describes a free scalar field.Contrary to the Lagrange anchor of the work [51], it involves an infinite number of spacetimederivatives, and it seems unlikely that one can remove them all by the inclusion of higher3pin fields and interactions. As shown in [53], the Lagrange anchors for higher spin fieldsdescribed by the Bargmann–Wigner equations also contain higher derivatives, the number ofwhich increases with spin. In the two most recent papers [54], [55] a similar quantizationmethod was implemented to reproduce propagators for free higher spin fields.In the present paper, we consider another approach to quantizing (non-)Lagrangian theo-ries. It employs the concept of a covariant presymplectic structure introduced in [56], [57]. Thenotion of presymplectic structure is, in a sense, dual to that of Lagrange anchor; both coincidefor Lagrangian theories, being non-degenerate and inverse to each other. The degeneracy ofa presymplectic structure reduces the algebra of physical observables admitting quantization.More precisely, the kernel distribution of the presymplectic structure must annihilate quantiz-able observables. Hence, the bigger the kernel, the smaller the algebra of quantum observables.Since all the gauge symmetry generators of the classical equations of motion belong to thekernel distribution, the quantizable observables are automatically gauge-invariant. As withthe Lagrange anchor, classical equations of motion do not specify a compatible presymplecticstructure uniquely, and different choices may lead to different algebras of quantum observables.While the Lagrange anchor aims at the path-integral quantization of (non-)Lagrangian dynam-ics, the concept of presymplectic structure is more adapted to the deformation quantization.As was proved in [58], each covariant presymplectic structure defines and is defined bysome Lagrangian. The corresponding Euler–Lagrange equations, however, are weaker thanthe original ones whereby admitting more solutions. That is why we call such Lagrangians‘weak’. One may regard a weak Lagrangian as a solution to the inverse problem of the calculusof variations where the integrating factor is not necessarily invertible. Finding a covariantpresymplectic structure is thus fully equivalent to constructing a weak Lagrangian.We will show that for 4 d HSGRA the corresponding weak Lagrangian has the form of anAKSZ sigma-model [61], or more precisely, its presymplectic counterpart [62], [63]. For freehigher spin fields such a weak Lagrangian was proposed in [64]. There is a number of immediateadvantages of the presymplectic approach. Among these are ( i ) minimality: one does not haveto introduce any auxiliary fields on top of what are already present in Eq. ( ∗ ); ( ii ) backgroundindependence and gauge invariance: we do not have to pick any particular vacuum, like AdS and the gauge symmetry is fully taken into account to every order in the weak curvatureexpansion; ( iii ) relation to the canonical quantization and to the Lagrangian formulation thatHSGRAs must have in principle; ( iv ) a complete classification and an explicit description Historically, the importance of the AKSZ construction for the development of action principles for HSGRAwas pointed out in [51], see also [59, 60], which have been an important reference frame for our work eventhough we deviate significantly from these works by considering the presymplectic case.
4f admissible presymplectic structures in terms of the Chevalley–Eilenberg cohomology of theunderlying higher spin algebra, which can be reduced to a much simpler Hochschild cohomologyand, finally, computed with the help of the techniques of [65].The main results of this paper can be summarized as follows: • we prove the existence of the weak action – presymplectic AKSZ sigma-model – for 4 d formal HSGRA and construct the first few terms in the weak-field expansion. Higher ordercorrections are proved to be unobstructed and can be found with the help of generaltechniques from [66–69]. The action depends on one additional coupling constant inaccordance with conjectured duality to Chern–Simons vector models [35]; • In arbitrary dimension d the presymplectic AKSZ action for HSGRA begins with S = Z (cid:10) V ( ω , ..., ω , C ) ⋆ ( d ω − ω ⋆ ω ) i + O ( C ) , where ω and C are a connection and a matter field valued in a given higher spin algebra, V ( ω , . . . , ω , C ) is a cocycle of this higher spin algebra and h−i is the invariant trace; • for free fields we show that the corresponding presymplectic structure gives the correctquantization for fields with s ≥ AdS it turns to be a genuine action, rather than a weak action; • it is shown that the presymplectic structure obtained from the Fradkin–Vasiliev part [70]of the cubic action agrees with the one resulting from the presymplectic AKSZ model,as well as the cubic vertex does. Therefore, the perturbative analysis performed upto the cubic level over the anti-de Sitter background is consistent with the backgroundindependent considerations of this paper. We also clarify the origin of the action.The paper is organized as follows. In the first sections we, to a large extent, review all therequired material: formal dynamical systems are defined in Sec. 2; the details specific toHSGRA in four dimensions are introduced in Sec. 3; we discuss presymplectic AKSZ sigma-models in Sec. 4 with an example of gravity elaborated on. The covariant phase space ofpresymplectic models is discussed in Sec. 5. The central section, to which a skilled readercan immediately scroll down, is Sec. 6, where we present the presymplectic AKSZ action for This action contains a number of non-abelian cubic interactions that are consistent at the cubic level withthe gauged higher spin algebra. However, it is not a complete action of the HSGRA up to the cubic level. Thecomplete cubic action was found in [71] and with one more parameter available only in 4 d in [34]. d HSGRA and discuss its properties and relations to other results in the literature. Sec. 7elaborates on the general properties of the presymplectic AKSZ models over the maximallysymmetric higher spin backgrounds. The main statements of Sec. 6 are supported by technicalAppendices A - D, where all necessary cohomology groups are computed.
The formal dynamical systems we are going to discuss are defined in terms of Q -manifolds, seee.g. [72–74]. By a Q -manifold we understand a Z -graded manifold endowed with an integrablevector field Q of grade one. Since Q is odd, the integrability condition[ Q, Q ] = 2 Q = 0 (2.1)is a non-trivial restriction on Q . In mathematics, such vector fields Q are usually called homo-logical or Q -structure [61, 75]. Let us present some typical constructions of Q -manifolds thatwe will need later for reference. Example 2.1.
A classical example of a Q -manifold is the ‘shifted’ tangent bundle T [1] M ofan ordinary manifold M ; here, the tangent space’s coordinates θ µ are assigned the grade one,while the local coordinates x µ on the base manifold M have grade zero. Let M denote the totalspace of T [1] M considered as a graded manifold. Then the algebra of ‘smooth functions’ on M is clearly isomorphic to the exterior algebra of differential forms Λ • ( M ). The isomorphismis established by the relation C ∞ ( M ) ∋ f ( x, θ ) ⇐⇒ f ( x, dx ) ∈ Λ • ( M ) . (2.2)Upon this identification the de Rham differential on Λ • ( M ) passes to the canonical homologicalvector field d = θ µ ∂∂x µ (2.3)on M . The cohomology of the operator d : C ∞ ( M ) → C ∞ ( M ) is obviously isomorphic to thede Rham cohomology of M . Example 2.2.
Let G = L n ∈ Z G n be a graded Lie algebra with a homogeneous basis { e A } andthe commutation relations [ e A , e B ] = f CAB e C . Then one can endow the vector space G [1], viewed Here prefer the term grade to a more conventional degree ; the latter is reserved to the “degree of a differentialform”. The grade of a homogeneous element a is denoted by | a | ∈ Z , e.g. | Q | = 1.
6s a Z -graded manifold, with the quadratic homological vector field Q = −
12 ( − | e A | ( | e B |− ξ A ξ B f CAB ∂∂ξ C . (2.4)Here ξ A are global coordinates on G [1] relative to the basis { e A } . By definition, | ξ A | = −| e A | +1.It is easy to see that the condition Q = 0 is exactly equivalent to the Jacobi identity in G .Endowed with the action of Q , the space of smooth functions C ∞ ( G [1]) becomes a cochaincomplex computing the cohomology of the Lie algebra G with trivial coefficients, the Chevalley–Eilenberg (CE) complex.Extending C ∞ ( G [1]) to the algebra T ( G [1]) of smooth tensor fields on G [1] wherein Q actsthrough the operator of Lie derivative L Q = i Q d − di Q , we obtain a more general CE complexwith coefficients in the tensor powers of adjoint and coadjoint representations of the Lie algebra G . For instance, considering the action of Q in the space of smooth vector fields on G [1] yieldsthe standard CE complex with coefficients in the adjoint representation. The cohomologyof this complex controls both the deformation of the homological vector field (2.4) and theunderlying Lie algebra G . For our present purposes, the most interesting is the algebra ofexterior differentials forms Λ • ( G [1]). This corresponds to the CE complex with coefficients inthe symmetrized tensor powers S • G ∗ of the coadjoint module of G , see Sec. 6. Example 2.3.
Given a Q -manifold ( M, Q ), consider the shifted tangent bundle T [ n ] M . Theoperator of Lie derivative L Q allows one to extend the homological vector Q from M to thetotal space of T [ n ] M considered as a Z -graded manifold. Let us denote the latter by M . If x i are local coordinates on M and v i are linear coordinates in the tangent spaces T x M relative tothe natural frame { ∂/∂x i } , then the extended vector field on M is given by Q = Q i ∂∂x i + ( − n v j ∂Q i ∂x j ∂∂v i . (2.5)By definition, | v i | = | x i | + n . We will refer to the Q -manifold ( M , Q ) as the first (tangent)prolongation of ( M, Q ). One can obviously iterate this construction producing bigger and bigger Q -manifolds. Moreover, the shifted tangent bundle may well be replaced by an arbitrary tensorbundle of M . The resulting Q -manifolds are particular examples of Q -vector bundles [76], [77].In order to define a formal dynamical system we need a pair of Q -manifolds: the source and the target . As a source manifold M we always take the total space of the shifted tangentbundle T [1] M of a space-time manifold M equipped with the canonical homological vector field(2.3). Let N denote a target manifold with homological vector field Q . We will treat N in7he sense of formal differential geometry [78], identifying ‘smooth functions’ on N with formalpower series in globally defined coordinates w A : w A w B = ( − | w A || w B | w B w A . In particular, the homological vector field Q is given by the series Q = ∞ X n =0 w A n · · · w A Q AA ··· A n ∂∂w A (2.6)for some structure constants Q AA ··· A n . Besides, two more assumptions about the structure ofthe target Q -manifold N will be made.1. There are no coordinates of negative grade among { w A } , so that the manifold N isactually N -graded.2. The expansion (2.6) starts with quadratic terms in coordinates, that is, Q A = Q AA = 0.Such homological vector fields Q are called minimal .Notice that the structure constants Q AA A of a minimal homological vector field (2.6) satisfythe Jacobi identity of a graded Lie algebra as in Example 2.2.Given the pair of Q -manifolds above, we identify the classical fields with the smooth maps w : T [1] M → N of degree zero. In terms of local coordinates, each map w is given bya set of relations w A = w A ( x, θ ), where w A ( x, θ ) are smooth functions of x ’s and θ ’s with | w A ( x, θ ) | = | w A | . The true field configurations are, by definition, those relating the homologicalvector fields, i.e., w ∗ (d) = Q . Upon identification (2.2), the last condition takes the form of asystem of differential equations, namely, d w A = ∞ X n =2 Q AA ··· A n w A n ∧ · · · ∧ w A . (2.7)The l.h.s. is given here by the differentials of the forms w A ∈ Λ • ( M ), while the r.h.s. involvesexterior products of the same forms. Equations (2.7) are thus adopted as field equations for acollection of form fields on M . In what follows we will systematically omit the wedge productsign and write Eq. (2.7) simply as E A ≡ d w A − Q A ( w ) = 0 , (2.8) Q A ( w ) being exterior polynomials in the w ’s. Applying now the de Rham differential to (2.8),one can readily see that the field equations contain no hidden integrability conditions whenever Q = 0. Besides general covariance, system (2.8) enjoys the gauge symmetry transformations δ ε w A = dε A + ε B ∂ B Q A , δ ε E A = ( − | w A | + | w B | ε C (cid:0) ∂ B ∂ C Q A (cid:1) E B , (2.9)8here the infinitesimal gauge parameters ε A are differential forms of appropriate degrees. Actu-ally, the general coordinate transformations of the form fields w A are specifications of (2.9). If ξ is a vector field generating a one-parameter group of diffeomorphisms of M , then by Cartan’sformula δ ξ w A = L ξ w A = di ξ w A + i ξ d w A ≈ d ( i ξ w A ) + i ξ Q A ( w ) . (2.10)Hereinafter, the sign ≈ means “equal when the equations of motion hold”, i.e., on-shell. There-fore we can set ε A = i ξ w A to reproduce the action of an infinitesimal diffeomorphism ξ on thesolution space to the field equations (2.8). Remark . The gauge symmetry (2.9) is known to be strong enough to gauge away all localdegrees of freedom for any finite collection of fields { w A } provided that dim M >
1. The lastfact can be seen as follows. Notice that each form field w A of degree > ε A such that deg w A = deg ε A + 1. In a topologically trivial situation, thecorresponding gauge transformation δ ε w A = dε A + . . . allows one to remove all physical modesof the field w A subject to the first-order differential equation d w A = . . . This suggests that allphysical degrees of freedom are accommodated in the zero-form fields, let us enumerate them w a . Without a detailed analysis it is clear that the solution space of the equations d w a = . . . is parameterized by constants c a whose number is equal to the number of zero-forms w a ’s. Therefore if we are interested in non-topological field theories, i.e., models with propagatingdegrees of freedom, then we have to consider infinite multiplets of zero-form fields.
Remark . It is known that any system of partial differential equations can be brought intothe form (2.7) at the expense of introducing an infinite number of auxiliary fields [79]. In thecontext of HSGRA this is known as an unfolded representation [48, 80]. Of course, care mustbe exercised in treating (2.7) as a system of partial differential equations whenever an infinitenumber of fields and interaction vertices are involved into the game. That is why we refer to(2.7) as a formal dynamical system. In this paper, we leave aside all subtle analytical issuesrelated to the field equations (2.7) focusing upon their formal consistency.
Remark . Notice that the system remains consistent if we omit all the vertices in the r.h.s.of (2.7) except the quadratic ones. The resulting system is defined solely in terms of theunderlying Lie algebra G . Although the truncated equations of motion are still non-linear, onemay think of them as describing free field dynamics; the ‘genuine’ interaction vertices startform the cubic order. From this perspective switching on a consistent interaction amounts to In general, the higher-degree forms w A may also add up to the physical sector a finite number of globaldegrees of freedom associated with the de Rham cohomology of the space-time manifold M . L as discussed in Example 2.2. This leads to a refined version of the N¨oether procedure (seee.g. [81]) and may be regarded as a main technical advantage of the formal dynamical systemapproach. We will illustrate this point in the next section.
The gauge theory of interacting higher spin fields delivers the major class of formal dynamicalsystems with propagating degrees of freedom. Being general covariant and involving a masslessfield of spin two, they are usually referred to as Higher Spin Gravities (HSGRA). As explained inthe previous section, the structure of the corresponding equations of motion is largely controlledby the underlying graded Lie algebra G .Below we specify the algebra G for the case of 4 d HSGRA.(1) G is concentrated in degrees zero and one, so that G = G ⊕ G as a vector space and[ G , G ] ⊂ G , [ G , G ] ⊂ G , [ G , G ] = 0 . This allows us to regard the odd commutative ideal G as a module over the even subal-gebra G .(2) The G -module G is given by the adjoint representation of G . Therefore G ≃ G asvector spaces.(3) G = gl n ( A ), that is, G is the Lie algebra of n × n -matrices with entries in an associativealgebra A .(4) A = A ⊗ A , where the complex associative algebra A is given by the smash product A = A ⋊ Z of the first Weyl algebra A and the cyclic group Z generated by aninvolutive automorphism of A . More precisely, A is the unital C -algebra on the threegenerators y , y , and κ obeying the relations[ y , y ] = 2 i , { κ, y α } = 0 , κ = 1 . (3.1)Hereinafter, α = 1 , y and y . Adding the generator κ makes the outer automorphism y α → − y α of A into an inner automorphism of A .10n what follows we will denote the generators of the left and right tensor factors in A = A ⊗ A by ( y α , κ ) and (¯ y ˙ α , ¯ κ ), so that the general element of A can be written as a = f ( y, ¯ y ) + g ( y, ¯ y ) κ + h ( y, ¯ y )¯ κ + v ( y, ¯ y ) κ ¯ κ , (3.2) f , g , h , and v being some ordered complex polynomials in y ’s and ¯ y ’s. The complex conjugationof the ground field C extends to the semi-linear anti-involution of A that takes y α and κ to( y α ) ∗ = ¯ y ˙ α and κ ∗ = ¯ κ . Remark . The Lie algebra G contains sp ( R ) ≃ so (3 ,
2) as a real subalgebra. The latter isspanned by the unit n × n -matrix 11 multiplied by real quadratic polynomials in y ’s and ¯ y ’s; inso doing, the complex conjugate polynomials { y α , y β } and { ¯ y ˙ α , ¯ y ˙ β } generate the complexifiedLorentz subalgebra so (3 , y α ¯ y ˙ β correspond to the AdS transvections. The algebra so (3 , d anti-de Sitter space, suggests that the empty AdS may appear as a natural vacuum solution of 4 d HSGRA.
Remark . The full associative algebra generated by y ’s and ¯ y ’s is clearly isomorphic tothe second Weyl algebra A = A ⊗ A . Let hs ⊂ A denote the subalgebra of all elementscommuting with κ ¯ κ . It is spanned by the even polynomials f ( y, ¯ y ) = f ( − y, − ¯ y ) and is calledthe higher spin algebra [82, 83], hence the notation. Since the element κ ¯ κ generates a Z subgroup in the Klein four-group Z × Z = { , κ, ¯ κ, κ ¯ κ } , one can also characterize hs as a Z -invariant subalgebra of A . Extending hs with κ and ¯ κ , we get the smash product algebra hs ⋊ ( Z × Z ) called usually the extended higher spin algebra . Remark . For technical reasons explained in Appendix A, we will consider the Lie algebraof ‘big matrices’. Formally, it is defined as the projective limit gl ( A ) = lim → gl n ( A ) associatedwith the natural embedding gl n ⊂ gl n +1 (an n × n -matrix is augmented by zeros). The resultis the Lie algebra G = gl ( A ) of infinite matrices with only finitely many entries different fromzero. The algebra can also be augmented without harm by the matrices proportional to theunit matrix 11.Given the graded Lie algebra G = G ⊕ G above, we can define 4 d HSGRA as a formaldynamical system with a one-form field ω and a zero-form field C , both taking values in gl ( A ).Eqs. (2.7) assume now the form d ω = ω ⋆ ω + V ( ω , ω , C ) + V ( ω , ω , C , C ) + · · · , (3.3a) dC = ω ⋆ C − C ⋆ ω + V ( ω , C , C ) + V ( ω , C , C , C ) + · · · . (3.3b)11ere ⋆ combines the wedge product of differential forms with the matrix product in gl ( A ). Tolook for the dynamical equations for interacting massless fields of all spins in this form wasfirst proposed in [48]. Geometrically, the target space of fields C and ω is given by an infinite-dimensional Q -manifold coordinatized by variables of degree zero and one. In the smoothsetting such Q -manifolds are known to be equivalent to Lie algebroids [84]. This allows one toconsider the field equations (3.3) as originating from a (formal) Lie algebroid over the targetspace of fields C . The natural vacuum solution C = 0 corresponds then to a singular point ofthe Lie algebroid with isotropy Lie algebra G = gl ( A ).Omitting the interaction vertices V k , we are left with the system describing a free HSGRA.Indeed, in that case one can view Eq. (3.3a) as the zero-curvature condition for the connectionone-form ω associated with the gauge algebra gl ( A ). In topologically trivial situation, one cansolve the equation in a purely gauge form as ω = dg ⋆ g − , where g is a zero-form with values in GL ( A ). Then Eq.(3.3b) identifies C as a covariantly constant section with values in the adjointrepresentation of the gauge group GL ( A ). Again, one can write the general solution for C as C = g⋆C ⋆g − , where C is an arbitrary element of the algebra gl ( A ). We thus see that, modulogauge invariance, the solutions to the free equations form a linear space isomorphic to gl ( A ).When endowed with an invariant Hermitian inner product, the space can be identified with theHilbert space of one-particle states of 4 d HSGRA. More precisely, the usual interpretation interms of particles arises from decomposition of the adjoint representation of GL ( A ) into thedirect sum of irreducible unitary representations of the anti-de-Sitter group SO (3 , ⊂ GL ( A ).In a slightly different language this is the content of the Flato–Fronsdal theorem [85] statingthat the tensor product of free massless 3 d scalar and fermion with themselves decomposes intoa direct sum of massless fields with all spins. As was shown by Dirac [86] many years beforethat, the free massless 3 d scalar and fermion, as representations of so (3 , A acts naturally. Together with the Flato–Fronsdal theorem, this equips the higher spin multiplet with the action of A (understoodas a Lie algebra). The appearance of Z is a bit harder to explain. The group Z realizesan isomorphism that allows one to treat elements from the tensor product | v i| w i (states) aselements | v ih w | from the higher spin algebra (operators). As a result one can embed the statesinto the higher spin algebra. This automorphism is realized by κ , ¯ κ . The smash product algebra A , which extends the higher spin algebra hs with κ , ¯ κ , is useful: it is the deformation of A thatallows one to reconstruct all vertices V k in (3.3), see below and [69] for explicit formulas. Let us choose creation and annihilation operators, [ a α , a † β ] = δ αβ , instead of y α and ¯ y ˙ α . Then, sp ( R )is realized by the bilinears in a and a † . The scalar/fermion states | v i correspond to the span of f ( a † ) | i foreven/odd f ( a † ) that act on the vacuum | i . emark . It should be noted that the spectrum of fields generated by the graded Lie algebra G above is superfluous, containing more fields than actually needed to describe 4 d HSGRA.The true physical degrees of freedom are accommodated in the graded subalgebra G ′ ⊂ G ,where G ′ = gl ( hs )Π, Π = (1 + κ ¯ κ ) / G ′ = gl ( hs ) K , K = ( κ + ¯ κ ) / K = Π, see Remark3.2. The corresponding field configurations are given by ω = ω ( y, ¯ y )Π and C = C ( y, ¯ y ) K such that ω ( − y, − ¯ y ) = ω ( y, ¯ y ) and C ( − y, − ¯ y ) = C ( y, ¯ y ). The use of the ‘extended’ algebra G offers considerable technical advantages over G ′ . In particular, it makes possible applying theK¨unneth formula to the tensor product A = A ⊗ A . The redundant fields do not interfere thedynamics of the physical ones and can easily be excluded at the end of all calculations. Thisfinal projection onto the subspace of physical fields will always be implied in the sequel.
Remark . One can also deduce the spectrum of 4 d HSGRA over
AdS background by theconventional field-theoretic analysis. For this end, put ω = ◦ ω
11, where the one-form field ◦ ω = − i { y α , y β } w αβ − i { ¯ y ˙ α , ¯ y ˙ β } w ˙ α ˙ β − i y α y ˙ β h α ˙ β (3.4)takes values in the subalgebra so (3 , ⊂ gl ( A ), as discussed in Remark 3.1. The forms w αβ and w ˙ α ˙ β are then naturally identified with the components of a unique spin-connection compatiblewith the vierbein h α ˙ β of AdS , see Example 4.4 below for λ = 1. Upon this identificationall the structure relations of AdS geometry are compactly encoded by the Maurer–Cartanequation d ◦ ω = ◦ ω ⋆ ◦ ω . On substituting (3.4) into (3.3b) and restricting to the physical sector,one gets an infinite number of relativistic wave equations on the component spin-tensor fieldsaccommodated in C = ∞ X n,m =0 C α ··· α n ˙ β ··· ˙ β m ( x ) y α · · · y α n ¯ y ˙ β · · · ¯ y ˙ β m K . (3.5)A closer inspection of these equations shows that they are equivalent to the Bargmann–Wignerequations for the (matrix-valued) massless fields of all integer spins, see e.g. [48, 87].Although C = 0 looks like a natural vacuum solution of 4 d HSGRA, suppose that Eqs.(3.3) admit a constant solution of the form C = ( µ + νκ + ¯ ν ¯ κ + λκ ¯ κ )11 , (3.6)where µ, ν, λ are complex parameters. This would provide the most general Lorentz- and gl -invariant family of vacuums. On substituting this hypothetical solution, Eq. (3.3a) takes theform d ω = ω ◦ ω , where by ◦ we denoted the bilinear operator determining the r.h.s. of theequation. Since d = 0, the ◦ -commutator must obey the Jacobi identity, i.e., [[ ω , ω ] , ω ] = 0.A simple way to satisfy the last condition is to require the ◦ -product to be associative. Then13he r.h.s. of Eq. (3.3a) gives rise to a multi-parameter deformation of the associative productin A = A ⊗ A . The existence and non-triviality of such a deformation (and hence, interaction)depend on the properties of the algebra itself. As explained in Appendix B, either A factor in A admits a non-trivial deformation. This is obtained by alteration of Heisenberg’s commutationrelation in (3.1). Now it reads [ y , y ] = 2 i (1 + νκ ) , (3.7) ν being a complex parameter. The resulting one-parameter family of algebras A ( ν ) is knownunder the name of deformed oscillator algebra . Implicitly, it was first introduced by E. Wignerin his 1950 paper [88] on foundations of quantum mechanics. See [89–91] for subsequent dis-cussions. Clearly, the commutation relation (3.7) and the similar relation for ¯ y ’s correspond tothe vacuum solution (3.6) with µ = λ = 0. The general element of A ( ν, ¯ ν ) = A ( ν ) ⊗ A (¯ ν ) isstill given by an ordered polynomial in y ’s, ¯ y ’s, κ , and ¯ κ . There are explicit, albeit somewhatcumbersome, formulas for the product of such polynomials [92–97]. Choosing, for example,the symmetric (or Weyl) ordering for y ’s and ¯ y ’s, while keeping the generators κ and ¯ κ in therightmost position, one can write the following expansion for the ◦ -product of two polynomials a ( y, ¯ y ) and b ( y, ¯ y ): a ◦ b = a ∗ b + X n + m> φ nm ( a, b )( νκ ) n (¯ ν ¯ κ ) m . (3.8)Here ∗ stands for the usual Weyl–Moyal product in A and the collection of bilinear operators { φ nm } defines a two-parameter deformation of the full algebra A ‘in the directions of κ and¯ κ ’. A nice integral representation for the first-order deformations φ and φ can be found inAppendix B.The surprising thing is that knowledge of the interaction vertices for a particular vacuumsolution (3.6) permits reconstruction of the r.h.s. of Eqs. (3.3a, 3.3b) for arbitrary C ! As wasshown in [69], it is possible to express all the V ’s through compositions of the bilinear maps φ nm entering the expansion (3.8). For example, V ( ω , ω , C ) ij = X n + m =1 ν n ¯ ν m φ nm ( ω ii , ω i i ) ⋆ C i j , V ( ω , ω , C , C ) ij = X n + m =2 ν n ¯ ν m φ nm ( ω ii , ω i i ) ⋆ C i i ⋆ C i j + X n + m = k + l =1 ν n + k ¯ ν m + l φ nm ( φ kl ( ω ii , ω i i ) , C i i ) ⋆ C i j . (3.9)Here we wrote down explicitly the gl -indices i ’s and j ’s. As is seen, they are all contracted inchain. Of course, all the vertices are defined modulo field redefinitions. The modulus of the In other words, the algebra A ( ν, ¯ ν ) satisfies the Poincar´e–Birkhoff–Witt condition. ν can obviously be absorbed by rescaling C . Therefore, setting ν = e iθ weare left with the only free parameter θ , which allows one to interpolate between the HSGRAs oftype A ( θ = 0) and type B ( θ = π/
2) [47]. In the context of AdS/CFT correspondence, thesetwo theories should be dual [46, 47, 98], respectively, to the free boson and fermion vectormodels on the boundary of AdS . For general θ , this family of HSGRAs is expected to be dualto Chern–Simons matter theories [35].Historically, explicit expressions for the first two vertices V and V were found in the works[48], [99]. In the subsequent paper [100], a systematic method was developed for generating allthe formal interaction vertices in 4 d HSGRA. A direct relation of the interaction problem withthat of deformations of extended higher spin algebras was established in our papers [68, 69];this solves the problem of formal HSGRA for any given higher spin algebra and allows oneto construct classical integrable systems out of any family of associative algebras. There is anumber of formal HSGRA models available in the literature [60, 69, 100–107], which includeoriginal models, variations and different realizations.
Like the previous two sections this one is mostly expository. In order to formulate a class offield-theoretic models in the title we need to equip the target space N with one more geometricstructure discussed below.First of all, we note that the N -grading on the target space N can be conveniently describedby means of the Euler vector field N = X A | w A | w A ∂∂w A . (4.1)By our assumption about target spaces all | w A | ≥
0. We say that T ∈ T ( N ) is a homogeneoustensor field of grade n if L N T = nT , where L N denotes the Lie derivative along N . In particular,the commutation relations [ N, N ] = 0 and [
N, Q ] = Q mean that | N | = 0 and | Q | = 1. Thehighest grade of the coordinates { w A } is called the degree of N and is denoted by deg N . Forinstance, the shifted tangent bundle T [1] M provides an example of an N -graded Q -manifold ofdegree one.A presymplectic structure of grade n is, by definition, a closed two-form Ω on N such that L N Ω = n Ω . (4.2) to an extent to which formal dynamical systems represent the actual field theories behind them N , Ω) is called a presymplectic manifold . If the matrix (Ω AB ) of the two-form Ωhappens to be non-degenerate in some (and hence any) coordinate system, then one speaks of asymplectic structure and a symplectic manifold. In that case the inverse tensor Π = Ω − definesa Poisson structure on N of grade ( − n ). Denote by ker Ω the space of all vector fields V on N such that i V Ω = 0. Since d Ω = 0, the vector fields of ker Ω span an integrable distribution on M . We will refer to it as the kernel distribution of Ω.A vector field X (a function H ) is said to be Hamiltonian if there exists a function H (avector field X) such that i X Ω = dH . (4.3)The function H is called the Hamiltonian of the vector field X = X H . It follows immediatelyfrom the definition that ( i ) | X | + | Ω | = | H | ; ( ii ) the presymplectic structure is invariant underthe action of Hamiltonian vector fields, i.e., L X Ω = 0; ( iii ) given a Hamiltonian H , Eq. (4.3)defines X up to adding a vector field from ker Ω; ( iv ) each Hamiltonian is invariant under thekernel distribution, i.e., L V H = 0 for all V ∈ ker Ω.An important fact about the geometry of presymplectic manifolds (graded or not) is thatthe Hamiltonians of Hamiltonian vector fields form a graded Poisson algebra w.r.t. point-wisemultiplication and the Poisson bracket { H, F } = ( − | H | L X H F = ( − | H | i X H i X F Ω = − ( − ( | H | + | Ω | )( | F | + | Ω | ) { F, H } . (4.4)One can easily verify that the bracket is well-defined and satisfies all the required properties:bi-linearity, graded anti-symmetry, the Leibniz rule, and the Jacobi identity. In the case ofsymplectic manifolds any function is a Hamiltonian and { H, F } = Π( dH, dF ). Proposition 4.1 ([72]) . For any N -graded presymplectic manifold ( N , Ω) the following hold.1. If | Ω | = n > , then Ω = d Θ , where Θ = n i N Ω .
2. If V is a vector field of degree m > − n such that L V Ω = 0 , then i V Ω = dH , where H = − ( − m nn + m i V Θ .3. If Ω is non-degenerate, then deg N ≤ | Ω | ≤ N . The first two statements follow immediately from Cartan’s homotopy formula L N = i N d + di N . To prove the last statement, write the presymplectic form in terms of local coordinates asΩ = dw A Ω AB ( w ) dw B , whence | Ω | = | w A | + | Ω AB | + | w B | . It follows from the definition that Ω AB = Ω BA ( − | w A || w B | + | Ω | ( | w A | + | w B | )+1 .
16n case deg N > | Ω | some of the differentials dw A do not enter Ω and the two-form is necessarilydegenerate. If now | Ω | > N , then | Ω AB | > AB ) is again non-invertible being composed of non-invertible elements.The one-form Θ of item (1) is called a presymplectic potential for Ω. It is clear that | Ω | = | Θ | and the equation Ω = d Θ (4.5)defines Θ up to an exact one-form d Φ, e.g. one can always take Θ = n i N Ω.In the following we are interested in presymplectic Q -manifolds ( N , Ω , Q ) that satisfy theadditional condition L Q Ω = 0. According to item (2) of the proposition this means that thehomological vector field Q is Hamiltonian, i.e., i Q Ω = d H , (4.6) H being the Hamiltonian. Since ( L Q ) = L Q = 0, one may also say that Ω is a cocycle of thedifferential L Q : Λ ( N ) → Λ ( N ) increasing the grade of a two-form by one unit. As one mightsuspect, only non-trivial Q -cocycles will be of interest to us below.Consider now the N -graded presymplectic Q -manifold ( N , Q, Ω) as the target space of aformal dynamical system (2.8), whose source manifold M fulfills the only condition dim M = | Ω | + 1. Given these data, the AKSZ Lagrangian for the form fields w A reads L = Θ A ( w ) d w A − H ( w ) , (4.7)where Θ A ( w ) and H ( w ) are exterior polynomials in w ’s defined by Eqs. (4.5) and (4.6). Bydefinition, L is a form of top degree on M . With the help of (4.6) one can readily bring thevariation of L into the form δ L = δ w A Ω AB ( d w B − Q B ) − d (Θ A δ w A ) . (4.8)As usual the first term defines the Euler–Lagrange (EL) equations E A ≡ Ω AB (cid:0) d w B − Q B ( w ) (cid:1) = 0 , (4.9) To avoid confusion we note that the grade is defined by the Euler vector field (4.1); it counts the totaldegree of w ’s irrespective of whether they appear as variables w A or differentials dw A . In other words, the Eulervector field does not take the exterior differential d into account. We emphasize that w A ≡ w A ( x, dx ) are fields, while w A are just target space coordinates. Correspondingly, d w A is a space-time form of degree | w A | + 1 and should not be confused with the 1-form dw A of grade | w A | onthe target space. The grade (as opposite to degree) of all fields d w A is 1. Since d w A and dw A have differentgrade/degree, the position of various variables is important in the formulas below. AB ) is invertible and theEL equations (4.9) are fully equivalent to the field equations (2.8). In this case, (Ω AB ) playsthe role of the so-called integrating multiplier in the inverse problem of variational calculus.Moreover, passing to global Darboux coordinates on ( N , Ω) considerably simplifies the ‘kineticterm’ in (4.7), bringing the Lagrangian into the form L = 12 Ω AB w A d w B − H ( w ) , Ω AB ∈ R . (4.10) Remark . The Lagrangians of the form (4.10) are called AKSZ (sigma-)models after Alexan-drov, Kontsevich, Schwartz, and Zaboronsky [61], who proposed them in the mid-1990s. A gooddeal of topological field theories can be formulated or re-formulated within the AKSZ approachfor an appropriate quartet ( M, N , Q, Ω), see [72, 108–110] for reviews. There is also a deep rela-tionship between the AKSZ construction of topological field theories and the Batalin–Vilkovisky(BV) formalism of gauge theories [111, Ch. 17]. This manifests itself in a simple and elegantform of the BV master action associated to the gauge invariant Lagrangian (4.10). In order toconstruct the corresponding BV action one simply allows the fields w : T [1] M → N to be themaps of arbitrary Z -degree and this yields automatically the right spectrum of auxiliary (anti-)fields of the BV formalism. When evaluated on such promoted form fields w and integratedover M , the Lagrangian (4.10) gives the desired BV master action. Remark . Contrary to the symplectic case, the general presymplectic AKSZ models havereceived much less attention in the literature. Here we should mention the three papers [62, 63,112], where some analysis of the models has been undertaken in different contexts. In the firstpaper, the authors discuss the presymplectic AKSZ models from the perspective of frame-likeformulations known for many gauge theories, including higher spin fields. The second paperintroduces the AKSZ formalism into the geometric theory of PDEs through the notion ofan intrinsic Lagrangian . The most recent paper [112] focuses on the presymplectic AKSZformulation for Einstein’s gravity. We also review the last model in Example 4.4 below.Turning back to the general case, we need to examine the equality E A = Ω AB E B (4.11)relating Eqs. (2.8) and (4.9). Rather than discuss this in full generality, let us only highlightsome key points. First of all, any solution to the equations E B = 0 obviously satisfies E A = 0whatever the matrix (Ω AB ). Furthermore, if the matrix is degenerate it seems reasonable todeclare the EL equations to be ‘weaker’ than the original ones. This, however, is not always18he case. The crux of the matter is hidden integrability conditions that must be allowed for.In order to make them explicit, suppose that all null-vectors of the matrix (cid:0) Ω AB ( w ) (cid:1) comefrom the kernel distribution of the presymplectic structure Ω. Let us further assume that thedistribution ker Ω is spanned by a set of vector fields K a on N . The integrability of ker Ωimplies the commutation relations [ K a , K b ] = f cab K c (4.12)for some structure functions f ’s. Besides, it follows from the identity L Q i X a Ω = 0 that thekernel distribution is Q -invariant, i.e., [ Q, K a ] = U ba K b (4.13)for some functions U ’s. Now Eq. (4.9) says that the forms E A determining the l.h.s. of Eq.(2.8) constitute a null-vector of the matrix (cid:0) Ω AB ( w ) (cid:1) . Under the assumptions above, thisamounts to the equality d w A − Q A ( w ) = λ a K Aa ( w ) . (4.14)Here λ a are new form fields of appropriate degrees and identification (2.2) is implied. Clearly,the last equations are completely equivalent to the EL equations (4.9). Although the newfields λ a enter the equations in a pure algebraic way, their dynamics are not entirely arbitrary.Indeed, applying the de Rham differential to both sides of (4.14) and using Rels. (4.12) and(4.13), we find ( dλ a + U ab λ b + f abc λ b λ c ) K a = 0 . (4.15)If all the vector fields K a , being linearly independent, are contained in the r.h.s. of Eq. (4.14),then the last condition amounts to dλ a + U ab λ b + f abc λ b λ c = 0 . (4.16)By construction, Eqs. (4.14) and (4.16) are compatible to each other and define a new formaldynamical system extending (2.8). The question now is whether the extended system is dy-namically equivalent to the original one. If the total number of λ ’s is finite, then the numberof physical degrees of freedom they can bring in is finite as well, see Remark 2.4. In casedim M >
1, these are global modes associated to the boundary conditions and/or topology of M . Hence, for genuine field theories the formal dynamical systems in question are essentiallyequivalent to each other (i.e., equivalent up to a finite ambiguity). Another typical situationis when M ≃ R d , d >
1, and all the form fields λ a are of strictly positive degree. Being puregauge and having no topological modes, the fields λ a can safely be set to zero. In this gauge,the equivalence of the formal dynamical systems at hand is again obvious.19he analysis of the general case is complicated by two points. First, it may happen that every generating set { K a } for ker Ω consists of linearly dependent vectors. This implies theexistence of left null-vectors Z α for the rectangular matrix { K Aa } , so that Z aα K Aa = 0. If thesystem of null-vectors { Z α } is complete, then we can still divide the l.h.s. of Eq. (4.15) by the K a ’s at the expense of introducing new form fields ξ α and adding the term ξ α Z aα to r.h.s. ofEq. (4.16). The verification of formal integrability gives further equations for ξ α and so on. The second point concerns the case where not all the null-vectors K a are actually present inEq. (4.14). Here, besides the differential equations, one can find some algebraic constraints on λ a associated with those K a ’s that dropped out of the r.h.s. of Eq. (4.14) on account of degree.(Both the points are exemplified below.) Under reasonable assumptions one may repeat thisconstruction once and again. The result is a formal dynamical system equivalent to the ELequations (4.9). All the above comments on equivalence to the original equations E A = 0 arestill valid for the fully extended system. We are going to detail this construction elsewhere.The last comment concerns gauge symmetries. One should realize that not every gaugetransformation (2.9) of the equations E A = 0 can be promoted to a symmetry of the correspond-ing AKSZ model (4.7), unless Ω is non-degenerate. Some necessary and sufficient conditionfor this to happen are discussed in [62]. In particular, the general coordinate transformations(2.10) may no longer be part of (2.9). On the other hand, the null-vectors of the presymplec-tic structure Ω give rise to extra gauge symmetries of the form δ ε w A = ε a K Aa provided that | K Aa | ≤ | w A | . Example 4.4 (Pure gravity) . In order to construct a first-order formulation of 4 d gravity withnegative cosmological constant, we can start with the Lie algebra G = so (3 , Q GR = ω ac ω cb ∂∂ω ab + ω ab e b ∂∂e a + λe a e b ∂∂ω ab (4.17)on the graded manifold G [1] with global coordinates e a and ω ab . Here the degree-one coordinates ω ab = − ω ba correspond to the generators of the Lorentz subalgebra so (3 , ⊂ so (3 , e a are associated with the AdS transvections. As usual we raise and lower the Lorentz indices a, b, ... = 0 , , , η ab . Finally, the parameter λ < A relevant Q -invariant presymplectic form Ω GR The problem we have to deal with here is quite similar to that of reducible gauge symmetries, see e.g. [111,Ch. 10]. From the algebraic standpoint, each extension defines and is defined by a certain free resolution of the C ∞ ( N )-module ker Ω. As λ → so (3 ,
2) contracts to the Poincar´e algebra iso (3 , so (4 ,
1) taking λ > G [1] reads Ω GR = ǫ abcd e a de b dω cd . (4.18)Notice that it does not depend on λ . The 2-form Ω GR , being obviously closed, turns out to bedegenerate. The kernel distribution is concentrated in degrees one and two. It is spanned bythe family of null-vectors K C = C ab,cd e a e b ∂∂ω cd , K H = H abc,d e a e b e c ∂∂e d , (4.19)where the constant parameters C ab,cd and H abc,d form traceless Lorentz tensors.In accordance with the general philosophy, the target space coordinates e a and ω ab arepromoted to the one-form fields e a = e aµ dx µ and ω ab = ω abµ dx µ associated, respectively, withthe vierbein and the spin-connection on M . Applying now the general formulas of Proposition4.1, we can write the following AKSZ Lagrangian: L GR = 12 ǫ abcd e c e d (cid:16) d ω ab − ω an ω nb − λ e a e b (cid:17) . (4.20)Varying it w.r.t. the spin connection and vierbein, we learn that ǫ abcd e c D e d = 0 , (4.21a) ǫ abcd e b ( R cd − λ e c e d ) = 0 . (4.21b)Here D e a = d e a − ω ab e b is the Lorentz-covariant differential of the vierbein and R ab = d ω ab − ω ac ω cb is the curvature tensor of D . As the presymplectic form (4.18) has no null-vectors ofdegree 1 ‘in the direction of e ’, the first equation (4.21a) is actually equivalent to D e a = 0 . (4.22)It says that the spin connection ω ab is torsion free and can be uniquely expressed via e a and d e a . As to the second equation (4.21b), being Lorentz and general covariant and dependingon second derivatives of the vierbein through ω ( e , d e ), it may be nothing else but the vacuumEinstein’s equation with the cosmological constant. Thus, we are lead to conclude that thefirst-order formulation (4.20) is fully equivalent to that relied on the Einstein–Hilbert action.But wait a minute, not so fast! What about the kernel of the presymplectic form?If the presymplectic form (4.18) were non-degenerate, we would get just the structure rela-tions of anti-de Sitter geometry, D e a = 0 , R ab = λ e a e b , (4.23)21ncoded in the homological vector field (4.17). Clearly, this leaves no room for local degrees offreedom. The explicit expression for the kernel distribution (4.19) suggests that Eq. (4.21b) isactually equivalent to the following one: R ab − λ e a e b = C ab,cd e c e d , (4.24)where C ab,cd is now a collection of zero-form fields. Considered as a Lorentz tensor, C ab,cd isanti-symmetric in the first and second pairs of indices and has zero trace. Checking the formalintegrability of the full system (4.22), (4.24), as explained above, we get both algebraic anddifferential constraints on C ’s. The former are given by e a e b e c C ab,cd = 0 and force the tensor C ab,cd to have the symmetry of ‘window’ Young diagram. Then Eq. (4.24) identifies C ab,cd asthe Weyl tensor. As for the differential constraints, they have the standard form dC ab,cd = . . . ,where the r.h.s. involves new zero-form fields C ab,cd,e with the symmetry of a two-row Youngdiagram. These new fields owe their existence to the over-completeness of the null-vector system(4.19): as C ab,cd,e cannot have three anti-symmetric indices, the shift C ab,cd → C ab,cd + C ab,cd,n e n does not affect K C . This gives the null-vectors Z ’s of the second generation and so on. Theabove extension procedure never stops generating an infinite number of zero-form fields C ab,cd, ··· .This results in a formal dynamical system of the Einstein gravity (with infinitely many zero-forms, as promised for a field theory), whose explicit form is not known, but can, in principle,be obtained with the help of [79]. Example 4.5 (Gravity + scalar field) . The above AKSZ formulation of pure gravity can easilybe upgraded to include interaction with a scalar field. To illustrate the idea of how one canextend the construction based on an algebra with some of its modules we consider the simplestfinite-dimensional module. Specifically, we extend the underlying Lie algebra so (3 ,
2) to theinhomogeneous anti-de Sitter algebra iso (3 ,
2) and prescribe the abelian ideal of the latterdegree one. This results in the graded Lie algebra G = G ⊕ G , where G = so (3 ,
2) and G ≃ R , (as vector spaces). The associated homological vector field on G [1] takes the form Q = Q GR − Q S , Q S = π a e a ∂∂ϕ + ( π b ω ba + λϕe a ) ∂∂π a . (4.25)Here ( ϕ, π a ) are new coordinates of grade zero associated with the ideal G . In order to be Q -invariant the presymplectic form (4.18) is extended toΩ = Ω GR + d Θ S , Θ S = ǫ abcd e a e b e c π d dϕ . (4.26) This can also be regarded as a trivial extension of so (3 ,
2) by its fundamental representation. K C = C ab e a ∂∂π b , ˜ K H = H ab e a e b ∂∂ϕ . (4.27)Here the constant parameters H ab and C ab form, respectively, anti-symmetric and tracelessLorentz tensors. The corresponding AKSZ Lagrangian now reads L = L GR + ǫ abcd e a e b e c (cid:16) π d d ϕ + (cid:0) π a π a + λ ϕ (cid:1) e d (cid:17) . (4.28)Eliminating the auxiliary fields π a with the help of their equations of motion, one obtains thestandard second-order formulation for the scalar ϕ coupled to Einstein’s gravity with cosmo-logical constant. The mass of ϕ , however, is below the unitarity bound unless λ = 0. Bydefinition, the space G corresponds to the vector representation of so (3 , so (3 ,
2) associated with the totally-symmetric tracelesstensors of definite rank and get other discrete values of masses below unitarity. In order tolet the mass be arbitrary one has to start with an infinite-dimensional so (3 , A glance at the AKSZ Lagrangian (4.7) is enough to observe its similarity with the least actionprinciple in Hamiltonian mechanics and this is more than just an analogy. Actually, the target-space presymplectic form Ω induces a presymplectic structure on the covariant phase space offields, which then endows the algebra of physical observables with a Poisson bracket. We willnot dwell here on the covariant Hamiltonian formalism in field theory referring the reader tothe papers [56, 57, 64, 114] for general discussions and examples. The basic idea is to treat theboundary term in (4.8) as defining a ‘functional one-form’ on the space of fields:ˆΘ = Z Σ Θ A ( w ) δ w A . (5.1)Here Σ ⊂ M is an arbitrary Cauchy surface of initial data for the equations of motion. Treatingnow the variation symbol δ as an exterior differential on the functional space of fields andapplying it to (5.1), we get the two-formˆΩ = δ ˆΘ = 12 Z Σ δ w A Ω AB ( w ) δ w B , (5.2)which is just a functional counterpart of the presymplectic form Ω on N . By construction, theform ˆΩ defines a presymplectic structure on the functional space of all field configurations and,23hrough restriction, on every subspace therein. Let Sol( E A ) and Sol( E A ) denote the subspacesof all solutions to Eqs. (2.8) and (4.9). In view of Rel. (4.11) we have the natural embeddingSol( E A ) ⊂ Sol( E A ). Either space is then identified with the covariant phase space of thecorresponding field theory. It is not hard to see that the restriction of the presymplectic form(5.2) onto Sol( E A ) (and hence, on Sol( E A )) does not depend on the choice of a Cauchy surfaceΣ; this justifies the adjective “covariant” in the name.In general, the induced presymplectic structures on the solution spaces above are degenerateeven if the original presymplectic form (5.2) is not. This is due to the gauge symmetries of theequations of motion. Let us check, for example, that the infinitesimal gauge transformations(2.9) belong to the kernel of the form ˆΩ restricted to Sol( E A ). We have i δ ε w ˆΩ = Z Σ δ ε w A Ω AB ( w ) δ w B = Z Σ (cid:0) dε A + ε C ∂ C Q A ( w ) (cid:1) Ω AB ( w ) δ w B = Z Σ ( − | w A | ε A d (cid:0) Ω AB δ w B (cid:1) + ε C ∂ C Q A Ω AB δ w B ≈ − Z Σ ε A ( L Q Ω) AB δ w B = 0 . (5.3)Our convention here is that δd = − dδ, δ w A w B = ( − ( | w A | +1) | w B | w B δ w A , δ w A d w B = ( − ( | w A | +1)( | w B | +1) d w B δ w A . It is fit to and motivated by the concept of variational bi-complex, see e.g. [115–117].Following the general recipe, the space of Hamiltonian function(al)s ˆ H = R Σ H ( w ), w ∈ Sol( E A ), is then endowed with the covariant Poisson bracket (4.4), which can be viewed as aprecursor for the canonical quantization of the theory.Another important remark is that the correspondence between the presymplectic structureΩ on the target space N and its functional counterpart ˆΩ restricted to either solution spaceis far from being one-to-one. Indeed, shifting the integrand in (5.2) by an on-shell exact form χ AB δ w A δ w B ≈ d Ψ( w ) does not affect the induced presymplectic structure (Stokes theorem).In the case of Sol( E A ) this point can further be refined. The equations of motion w ∗ (d) = Q tell us that the action of the de Rham differential is on-shell equivalent to the action ofthe homological vector field. As a result Q -exact presymplectic forms L Q Ψ on the targetspace N pass to the d -exact forms on the covariant phase space Sol( E A ). This motivatesthe following definition: two presymplectic forms Ω and Ω ′ on N are said to be equivalent if Ω − Ω ′ = L Q Ψ for some two-form Ψ; correspondingly, the presymplectic structures of theform Ω = L Q Ψ are considered trivial. Equivalent presymplectic structures on N give rise to24he same presymplectic structure on the covariant phase space Sol( E A ). In other words, thespace of non-trivial presymplectic structures for the formal dynamical system (2.8) is identifiedwith the cohomology group H dim M − ( L Q , Λ ( N )). It might be well to point out that a non-trivial presymplectic structure on Sol( E A ) may become trivial upon restriction to the subspaceSol( E A ). The last point is best exemplified by Einstein’s gravity with cosmological constant. Example 5.1.
Proceeding with Example 4.4, we note that the presymplectic structure (4.18)is Q -exact whenever λ = 0:Ω GR = L Q GR Ψ , Ψ = 14 λ ǫ abcd dω ab dω cd . (5.4)This is no surprise as the Lie algebra G underlying the homological vector field (4.17) is simple.It is easy to realize that Q -invariant presymplectic structures Ω of grade 3 correspond toone-cocycles of G with coefficients in S G ∗ , the symmetrized tensor square of the coadjointrepresentation. But by Whitehead’s first lemma the first cohomology of any simple Lie algebravanishes. The Q -exactness of the form (5.4) by no means implies that the induced presymplecticstructure on the solution space of Einstein’s equations (4.21) is trivial. It is true, however, thatthe further restriction of ˆΩ to the subspace of locally (anti-)de Sitter geometries describedby Eq. (4.23) does lead to the trivial presymplectic structure. This is in contrast with thecase λ = 0 when G degenerates to the Poincar´e algebra iso (3 , Q -cocycle. (The Poincar´e algebra beingnot simple, the first Whitehead’s lemma is not applicable anymore.) The moduli spaces of flatgeometries are finite-dimensional and depend on the topology of the space-time manifold M ,see e.g. [118] for a physicist-oriented discussion. Using ˆΩ, one can equip them with presumablynon-trivial symplectic structures. d HSGRA
In this section, we put 4 d HSGRA into the framework of presymplectic AKSZ models andthereby provide its ‘weakly’ Lagrangian description. All we need is a presymplectic structure Ωcompatible with the homological vector field Q defined by the right-hand sides of the field equa-tions (3.3). We also expect the corresponding Euler–Lagrange equations (4.9) to be essentiallyequivalent to the original equations of 4 d HSGRA and this rules out some trivial choices likeΩ = 0. Once an appropriate presymplectic 2-form Ω is found, the corresponding presymplecticpotential Θ = i N Ω defines immediately the ‘kinetic’ term of an AKSZ Lagrangian (4.10) inquestion as well as the Hamiltonian H = i Q Θ.25he form of the equations of motion (3.3) suggests to look for a compatible presymplecticstructure as an expansion in powers of C ’s. In order to make contact with the general notationof Sec. 4 it is convenient to endow the target space N = G [1] of fields ω and C with the globalcoordinate system w A = ( ω a , C a ) w.r.t. some basis e A = ( e − a , e a ) in the graded Lie algebra G = G − ⊕ G underlying the free theory. To control formal powers series C a we also introducethe vector field N C = C a ∂∂C a , (6.1)which prescribes degree one to C ’s and degree zero to ω ’s. Now the r.h.s. of Eqs. (3.3) comefrom a homological vector field on N of the form Q = Q + Q + · · · , [ N C , Q n ] = nQ n , (6.2)where Q n = C a · · · C a n Q aa ··· a n bc ω b ω c ∂∂ω a + C a · · · C a n Q aa a ··· a n b ω b ∂∂C a . (6.3)The leading term Q , being a homological vector field by itself, determines the graded Liealgebra G as explained in Example 2.2. A similar expansion for the thought-for presymplecticstructure can be written asΩ = Ω m + Ω m +1 + · · · , L N C Ω n = n Ω n , (6.4)Ω n = C a · · · C a n Ω (0) a ··· a n abc ω a dω b dω c + C a · · · C a n − Ω (1) a ··· a n abc ω a ω b dω c dC a n + C a · · · C a n − Ω (2) a ··· a n abc ω a ω b ω c dC a n − dC a n − , (6.5)where we suppose that Ω m = 0. The defining conditions of a presymplectic structure d Ω = 0 , L Q Ω = 0 , (6.6)impose an infinite set of linear relations on the structure constants Ω (0) , Ω (1) , and Ω (2) . Inparticular, on substituting expansions (6.2) and (6.4) into the condition of Q -invariance (6.6),we obtain the chain of equations L Q Ω m = 0 , (6.7) L Q Ω n = B n (Ω m , . . . , Ω n − ) , B n ≡ m + n X k =1 L Q k Ω n − k , n = m + 1 , m + 2 , . . . (6.8)Since ( L Q ) = 0, we are lead to the standard problem of homological perturbation theory. Thefirst equation (6.7) tells us that the leading term of the expansion (6.4) is a Q -cocycle. As26xplained in Sec. 5, we are interested in non-trivial Q -cocycles, which then induce non-zeropresymplectic structures on the solution space to the field eqiations. Given such a non-trivialcocycle Ω m , the remaining equations (6.8) are solved one after the other provided that no coho-mological obstacles arise. Arguing by induction, one can see that the r.h.s. of the n -th equation, B n , is Q -closed whenever all the previous equations for Ω , . . . , Ω n − are satisfied. Thereforethe 2-form B n defines a class of Q -cohomology, whose vanishing provides the necessary andsufficient condition for solvability of the n -th equation (6.8). Notice that Eq. (6.8) defines Ω n ,if it exists, up to adding to it an arbitrary Q -cocycle Ψ n (perhaps trivial). If the equivalenceclass Ω n + Ψ n contains a d -closed representative, then one can take it to extend the sought-forsolution one step further.On the other hand, the cohomology of the coboundary operator L Q coincides with that ofthe graded Lie algebra G = gl ( A ) as discussed in Example 2.2. In particular, Ω m is determinedby an element of H • ( G , S G ∗ ) and B n defines a cohomology class of H • ( G , S G ∗ ), where thesubscripts refer to grades. The existence of a non-trivial presymplectic structure thus implies H • ( G , S G ∗ ) = 0, while the property that H • ( G , S G ∗ ) = 0 ensures extendibility of the leadingterm Ω m to all higher orders in C ’s. A detailed analysis of the relevant cohomology is presentedin Appendix D.A short summary is that there is no non-trivial solutions to (6.7) for m = 0; and hence,expansion (6.4) necessarily starts with terms involving at least one C or dC . The solutions with m > C = 0,which covers all maximally (higher spin) symmetric backgrounds. The remaining possibilitiesgive a 2-parameter family of presymplectic structures and Lagrangians. This is in line withChern–Simons matter theories, which we discuss below. Equations recap.
While the general theorems of Appendix D guarantee that the presym-plectic structure to be discussed below is unobstructed, we would like to work out a few termsexplicitly as they are the most important ones. To this end let us write down Eq. (3.3) up tothe first order in C as d ω = ω ⋆ ω + V ν ( ω , ω , C ) + O ( C ) , (6.9a) d C = ω ⋆ C − C ⋆ ω + O ( C ) . (6.9b)The simplest vertex has the factorized form [69, 119], see [48] for the explicit expression, V ν ( ω , ω , C ) = Φ ν ( ω , ω ) ⋆ C , (6.10)27here Φ ν represents the first deformation, cf. (3.8), of the extended higher spin algebra A : Φ ν ≡ Φ ν ( ω , ω ) = νφ ( ω , ω ) κ + ¯ νφ ( ω , ω )¯ κ . (6.11)For manipulations below, it is convenient to rewrite (6.9) in a more compact form as R = Φ ν ⋆ C mod C , D C = 0 mod C . (6.12)Here D = d − [ ω , − ] ⋆ , D = [ R , − ] ⋆ , R = d ω − ω ⋆ ω , (6.13)and we do not write the gl -indices of fields explicitly; it is assumed that all the matrix indicesare contracted in chain as in (3.9). The main property of the space-time 2-form Φ ν ( ω , ω ),which provides the formal integrability of the differential equations (6.12) modulo C , is D Φ ν ( ω , ω ) ≈ C . (6.14)The formulas above will suffice for all formal manipulations. In order to make contact with thefield theory approach we have to adjust the cubic vertex (6.10). We will only need the factthat there exists a better representative V can ν , one may call it ‘canonical’, such that V can ν ( ◦ ω , ◦ ω , C ) = νH αα ∂ α ∂ β C ( y, ¯ y = 0) + ¯ νH ˙ α ˙ α ∂ ˙ α ∂ ˙ β C ( y = 0 , ¯ y ) . (6.15)Here ◦ ω is the AdS connection (3.4), H αβ ≡ h α ˙ γ ∧ h β ˙ γ , idem for H ˙ α ˙ β , and we projected thevertex onto the physical sector C = C ( y, ¯ y ) K , see Remark 3.4. The consistency of V can ν implies D V can ν ( ω , ω , C ) = 0 mod C . (6.16)More generally, one can perform a field redefinition in equations (6.9) that maps the vertex(6.10) to another representative, e.g. V can ν , which always obeys (6.16). Thanks to the product structure A = A⊗A the deformation is built from a single two-cocycle φ ( − , − ) of theWeyl algebra A ; by definition, a ⋆ φ ( b, c ) − φ ( a ⋆ b, c ) + φ ( a, b ⋆ c ) − φ ( a, b ) ⋆ ˜ c = 0, where ˜ c ( y ) = c ( − y ). Then ondecomposable elements a ( y, ¯ y ) = a ( y ) ⊗ ¯ a (¯ y ) ∈ A , we have φ ( a , b ) = φ ( a, b ) ⊗ ¯ a⋆ ¯ b and φ ( a , b ) = a⋆b ⊗ φ (¯ a, ¯ b ). As it stands the vertex (6.10) does not lead to the desired field equations even at the free level [48]. Whathappens is that the free equations mix fields of different spins. Nevertheless, one can diagonalize the equations byperforming a linear change of variables; in so doing, the non-commutativity of covariant derivatives, [ ∇ a , ∇ b ] ∼ λ ,plays a crucial role. This example reveals a general fact that a formally integrable Q , i.e., Q = 0, has to beadjusted as to make contact with field theory. resymplectic structure and AKSZ to LO. With the conventions and notation abovethe leading order term of the thought-for family (6.4) of presymplectic structures on N = G [1]is given by Ω = Ω (0) + Ω (1) = d Θ , Θ = h C ⋆ Φ µ ( ω, ω ) ⋆ dω i . (6.17)Here µ is an arbitrary complex parameter and the brackets h−i stand for the matrix trace and the trace in the ⋆ -product algebra A . It is important that µ differs from ν in the vertex (6.10).For the case of HSGRA in d > φ ’s in (6.10) does not take place and thereis a clear distinction between the cocycle participating in Θ and the cocycle that deforms theequations, which we will make a few remarks about at the end.As is clear from the explicit form of Θ , one can instead chooseΘ = hV µ ( ω, ω, C ) ⋆ dω i (6.18)for any representative V µ in (6.9) as long as µ = ν . The last inequality ensures non-trivialityof the presymplectic structure to the leading order. Explicitly,Ω (0) = hV µ ( dω, ω, C ) ⋆ dω i − hV µ ( ω, dω, C ) ⋆ dω i , (6.19a)Ω (1) = hV µ ( ω, ω, dC ) ⋆ dω i . (6.19b)Notice that the component Ω (0) vanishes on the maximally symmetric background C = 0. Withthe help of (6.15) we find for free fields in AdS : Ω = µH αα h ∂ α ∂ α dC ( y, ¯ y = 0) ⋆ dω i + ¯ µH ˙ α ˙ α h ∂ ˙ α ∂ ˙ α C ( y = 0 , ¯ y ) ⋆ dω i == X s =1 µ (2 s − H αα dC α (2 s ) dω α (2 s − + ¯ µ (2 s − H ˙ α ˙ α dC ˙ α (2 s ) dω ˙ α (2 s − . (6.20)This is an admissible presymplectic structure [64]. For free fields, the parameter ν in (6.15) doesnot play any role and can be eliminated by rotating (anti-)holomorphic Weyl tensors ( C ˙ α (2 s ) ) C α (2 s ) by the phase of ν . It is also clear that for µ = ν the presymplectic structure (6.20), beingproportional to the r.h.s. of (6.15), is trivial.It is especially interesting to see how the presymplectic AKSZ action look like to the leadingorder. The equations to this order d ω = ω ⋆ ω , d C = ω ⋆ C − C ⋆ ω , (6.21)describe free fields C propagating over a maximally symmetric higher spin background ω .When linearized over AdS the C -equation reduces to a collection of the Bargmann–Wigner To save letters we denote all symmetric (or to be symmetrized) indices by the same letter. ∇ α ˙ β C ˙ α (2 s −
1) ˙ β = 0, ∇ β ˙ α C α (2 s − β = 0 for fields of all spins. It is clear that equations(6.21) are non-Lagrangian. For d = 3, the first one could be obtained from the Chern–Simonsaction; the second one, however, requires a two-form Lagrange multiplier, say B , to write theLagrangian L = h B ⋆ ( d C − ω ⋆ C + C ⋆ ω ) i . In four dimensions, even the first equation doesnot come from an action. Therefore, the main function of Θ is to provide a presymplectictreatment of (6.21). The corresponding presymplectic AKSZ Lagrangian reads L = (cid:10) V µ ( ω , ω , C ) ⋆ R i = (cid:10) V µ ( ω , ω , C ) ⋆ ( d ω − ω ⋆ ω ) i (6.22)It acquires a particularly nice form for the canonical vertex V can µ . Surprisingly, in that caseit describes the right propagating degrees of freedom on AdS [120]. (In general, one mightexpect to obtain only a Lagrangian whose EL equations are weaker than the ones we need).The presymplectic structure (6.20) or, equivalently, the action (6.22) can be checked againstthe well-known Fradkin–Vasiliev action [70] and its precursor [121]. First, we need to decomposethe curvature R = d ω − ω ⋆ ω as follows: R ( y, ¯ y ) = X m,n im ! n ! R α ( m ) , ˙ α ( n ) y α · · · y α ¯ y ˙ α · · · ¯ y ˙ α = R − + R + R + , (6.23)where R − ( R + ) takes the m < n ( m > n ) part of the sum and R corresponds to the termswith m = n . Note that the sum is over even m + n since we consider bosonic fields only. Thetwo-form R contains the (higher spin) torsions. As with the conventional gravity, one cansolve the zero-torsion equation R = 0 (6.24)for some auxiliary fields in a purely algebraic way. With account of this equation, we can writethe following integral for the Pontryagin topological invariant: S top = Z h R ⋆ R i = Z h R + ⋆ R + i + Z h R − ⋆ R − i . (6.25)Here h−i combines the matrix trace and the trace on the Weyl algebra A . The Fradkin–Vasilievaction now reads S FV = Z h R ⋆ σ ( R ) i = Z h R + ⋆ R + i − Z h R − ⋆ R − i , (6.26)where σ = sgn( N y − N ¯ y ) and N y = y ν ∂ ν , N ¯ y = ¯ y ˙ ν ∂ ˙ ν are the number operators for y and ¯ y . Thepurpose of σ -map is to project out the torsion and to flip the sign of R − terms. For example,30he usual Pontryagin’s invariant and the MacDowell–Mansouri action [122] have the form S top = Z R αα R αα + R ˙ α ˙ α R ˙ α ˙ α , S MM = i Z R αα R αα − R ˙ α ˙ α R ˙ α ˙ α . (6.27)As is well-known in the case of Yang–Mills theory and gravity one can add up the action andthe topological invariant to get S = τ + Z h R + ⋆ R + i + τ − Z h R − ⋆ R − i , (6.28)where τ ± ∈ C are combinations of the coupling constant g and the theta-angle θ , τ ± =( a/g ± iθ/b ) (here, a , b are some numerical constants). This introduces one more coupling θ ,which is invisible in the equations of motion. This trick makes the action reminiscent that of3 d (higher spin) gravity, which is a non-degenerate sum of two Chern–Simons actions. Withthe help of τ the SL (2 , Z )-action on the space of 3 d conformal field theories [123] extends tothe higher spin fields in bulk [124, 125]. Any non-degenerate combination ( τ − = τ + ) givesan action which is consistent with the required gauge symmetries up to the cubic order since(6.26) has this property [70]. The cubic part of the action can be evaluated on-shell to − S = τ + Z (cid:10) V can µ =1 , ¯ µ =0 ( h, h, C ) ⋆ ( ω ⋆ ω ) + (cid:11) + τ − Z (cid:10) V can µ =0 , ¯ µ =1 ( h, h, C ) ⋆ ( ω ⋆ ω ) − (cid:11) , (6.29)where we indicated that V can consists of two terms and only one of them contributes to thefirst (second) part of the action. (Note that (6.15) involves the AdS vierbein h α ˙ α rather thanthe full SO (3 , ◦ ω .)It is worth noting that each action of the family (6.28) fixes the relative coefficient betweencubic vertices V s ,s ,s for various spins s , , that contribute to the action, which is usuallynot the case for cubic interactions (any linear combination of consistent cubic vertices V s ,s ,s is consistent again). This rigidity/uniqueness of the action is due to taking the higher spinalgebra into account. Moreover, the relative normalization of free kinetic terms is also fixedin the free limit. The latter point is subtle: one could rescale all ω α ( n ) , ˙ α ( m ) , n + m = 2 s − a s and change this normalization, but then one would haveto change the star-product accordingly. Therefore, the relative normalization makes sense as We put i in front of MacDowell–Mansouri action assuming the usual reality conditions, ( R αα ) ∗ = R ˙ α ˙ α ,but will ignore all such factors below. Note also that the RR -action does not contain the Maxwell/Yang–Millsaction. Nevertheless, it can be added by hand as C αα C αα − C ˙ α ˙ α C ˙ α ˙ α . The Pontryagin term does include the s = 1 component. It would be interesting to check if the normalization via a , b that can be read off from the spin-two part ofthe action is consistent for higher spin fields as well. R and for the gauge symmetries δ ω = d ξ − [ ω , ξ ] ⋆ .After this detour into the Fradkin–Vasiliev action and Pontryagin invariant, we can extractthe presymplectic structure and compare it with (6.20). The free part of the action (6.28)differs from the usual first order (AKSZ-type) actions by a total derivative. Varying the action(6.28) and extracting the boundary term, we find the following presymplectic potential:ˆΘ = 2 τ + Z Σ h R + ⋆ δ ω i + 2 τ − Z Σ h R − ⋆ δ ω i . (6.30)On restricting to the solution space, we can replace the curvature R with its on-shell value(6.15) and get an equivalent presymplectic potential Θ on the target space that yieldsΩ = d Θ = 2 τ + H αα h ∂ α ∂ α dC ( y, ¯ y = 0) ⋆ dω i + 2 τ − H ˙ α ˙ α h ∂ ˙ α ∂ ˙ α C ( y = 0 , ¯ y ) ⋆ dω i . (6.31)This is exactly (6.20) with µ ’s replaced by 2 τ ’s. Furthermore, (6.31) is equivalent to thepresymplectic form Ω = i ( h β ˙ α dω α ( s − , ˙ α ( s − dω α ( s − β, ˙ α ( s − − c.c.) , (6.32)which comes from the free first-order action of Ref. [121]. It coincides in form with the presym-plectic structure of the linearized gravity, cf. (4.18), and is one of the admissible presymplecticstructures found in [64].As another small test of interactions, the presymplectic AKSZ Lagrangian (6.21) does re-produce the correct cubic on-shell vertex (6.29) upon appropriate identification between τ and µ . However, since beyond the free level it is only a weak action (unless otherwise shown) wedo not suggest to literally compare the actions in the literature, e.g. [34, 42, 71] with (6.21).Presymplectic AKSZ Lagrangian (6.21) can also clarify the very form of the original action(6.26) or its refinement (6.28), i.e., the fact that it is very close to a topological one. Freeparameters τ ± of (6.28) get expressed in terms of a single coupling constant by requiring theaction to be Hermitian and parity invariant. On relaxing these conditions (or at least theparity in view of the importance of the θ -term) we get a two-parameter family of actions, whichis closely related to the fact that there are two independent deformations of the equations ofmotion and of the extended higher spin algebra, cf. (6.11). Therefore, the two-term structureof action (6.28) and the appearance of ‘almost’ the trace is explained by (6.21) without anyreference to AdS or to cubic approximation. E. S. is grateful to N. Boulanger for asking this question many years ago.
AdS ; it does reproduce the correct presymplectic structure, which leads to thecanonical quantization of free higher spin fields; it does reproduce the presymplectic structurecoming from cubic action (6.28) together with the normalization that is fixed by the higher spinsymmetry. The last point should not be too surprising in view of the fact that both (6.22) and(6.28) are fixed by the higher spin symmetry. However, (6.28) is fixed up to the cubic termsover AdS , while (6.22) is fixed up to C -terms, which is much more powerful. Nevertheless, itprovides an additional check of the structure of interactions that goes beyond the classification[126–128] of cubic vertices V s ,s ,s within the Noether procedure.Let us make a few comments on the presymplectic structure of higher spin fields and itsrelevance for quantization. In the simplest, non-gauge, case of the scalar field C ( x ), Ω directlycorresponds to the canonical commutation relations:ˆΩ = Z Σ ǫ abcd dx b dx c dx d δ C ∂ a δ C ⇐⇒ [ C ( x ) , ˙ C ( x ′ )] = iδ ( x − x ′ ) , (6.33)with Σ ⊂ R , being a Cauchy surface x = const. In the case of gauge fields, the situation ismore complicated. For example, for s = 1 we haveˆΩ = Z Σ ǫ abcd dx b dx c dx d δ F an δ A n , (6.34)which allows one to work out the commutators of physical observables. In particular, for thecomponents of magnetic field B i and electric field E j we get[ B i ( x ) , B j ( x ′ )] = 0 , [ B i ( x ) , E j ( x ′ )] = iǫ ijk ∂ k δ ( x − x ′ ) , [ E i ( x ) , E j ( x ′ )] = 0 . (6.35)Similar commutation relations can be found for the curvatures of higher spin fields. Indeed, let φ α ( s ) , ˙ α ( s ) be the traceless part of the Fronsdal field, which can be associated with the totallysymmetric part of the higher spin vierbein e α ( s − , ˙ α ( s − . The expressions for the Weyl tensorsare very simple in the spinorial language: C α (2 s ) = ∇ α ˙ α · · · ∇ α ˙ α φ α ( s ) , ˙ α ( s ) , C ˙ α (2 s ) = ∇ α ˙ α · · · ∇ α ˙ α φ α ( s ) , ˙ α ( s ) . (6.36)Upon 3 + 1 split we get out of C α (2 s ) , C ˙ α (2 s ) a pair of rank-2 s spin-tensors, which can be mappedto higher spin electric E j ··· j s and magnetic B i ··· i s fields; both of the fields are symmetric andtraceless SO (3)-tensors. The only non-trivial commutator implied by ˆΩ in Minkowski space-time has the form[ B i ··· i s ( x ) , E j ··· j s ( x ′ )] = iǫ i j k ∂ k ∂ i ∂ j ...∂ i s ∂ j s δ ( x − x ′ ) + . . . , (6.37)33here the dots complete the r.h.s. to a traceless, symmetric and transverse bi-tensor. Notethat (6.32) and (6.31) are equivalent in AdS , but not in flat space-time.The correct quantization of free higher spin fields is guaranteed by the fact that Ω co-incides with the one obtained from the standard actions, all of which are equivalent to theFronsdal action. In particular, the frame-like actions are known [121] to reduce to the Fronsdalaction. Note that the presymplectic structure should be used to compute the commutator ofobservables. The complete classification of observables in 4 d HSGRA was obtained in [65] withthe important results found first in [129, 130]. For the free theory one set of gauge invariantobservables is given by the components of C , e.g. by the electric and magnetic fields (6.35) andby the higher spin generalizations thereof. Another set of observables, currents of the globalhigher spin symmetry, is considered in Sec. 7. Presymplectic structure and AKSZ to NLO.
For the formal manipulations below, wewill use the presymplectic potential Θ in the form (6.17). Applying the homological perturba-tion theory (6.7, 6.8), one can extend (6.17) to higher orders in C ’s to make it invariant w.r.t.the full homological vector field (6.2) underlying 4 d HSGRA. In particular, the sub-leadingterm of expansion (6.4, 6.5) has the following structure:Ω = Ω (0) + Ω (1) + Ω (2) . (6.38)By definition, the coefficients of the two-form Ω (0) vanish modulo C ; this allows us to ignorethe contribution of Ω (0) in the first-order approximation (6.12) to the field equations. As forthe rest two terms, we findΩ (1) + Ω (2) = d Θ , Θ = h Λ µν ⋆ dC i , (6.39)where Λ µν ( ω, ω, ω, C ) = Φ µ ( ω, ω ) ⋆ Ψ ν ( ω, C ) + Ψ µ ( ω, C ) ⋆ Φ ν ( ω, ω ) , (6.40)Ψ ν ( ω, C ) = Φ ν ( ω, C ) − Φ ν ( C, ω ) . (6.41)One can check that D Ψ ν ( ω , C ) ≈ [ C , Φ ν ( ω , ω )] ⋆ mod C . (6.42)All this allows us to reconstruct the corresponding AKSZ Lagrangian up to the second orderin the zero-form field C . A straightforward calculation gives L = (cid:10) C ⋆ Φ µ ( ω , ω ) ⋆ R − Λ µν ( ω , ω , ω , C ) ⋆ D C − C ⋆ Φ µ ( ω , ω ) ⋆ Φ ν ( ω , ω ) ⋆ C (cid:11) + O ( C ) (6.43)34et us check directly that the Lagrangian above leads to the equations of the form (4.9). Wefind δ L = h δ C ⋆ Φ µ ⋆ R + C ⋆ δ Φ µ ⋆ R + C ⋆ Φ µ ⋆ Dδ ω − δ Λ µν ⋆ D C − Λ µν ⋆ Dδ C − δ C ⋆ Φ µ ⋆ Φ ν ⋆ C − C ⋆ Φ µ ⋆ Φ ν ⋆ δ C i + O ( C ) ≈ d h C ⋆ Φ µ ⋆ δ ω i + d h Λ µν ⋆ δ C i− h ( D Λ µν − [ C , Φ µ ⋆ Φ ν ] ⋆ ) ⋆ δ C i + O ( C ) . (6.44)It remains to observe that the last term vanishes due to the identity D Λ µν ≈ [ C , Φ µ ⋆ Φ ν ] ⋆ mod C (6.45)that follows immediately form (6.14) and (6.42).We see that the EL equations for the Lagrangian (6.44) are satisfied by all solutions to theequations (6.12) of 4 d HSGRA. As expected, the total derivatives in (6.44) exactly reproducethe presymplectic potential Θ = Θ + Θ + . . . defined by Rels. (6.17, 6.39), cf. (4.8). We claimthat the corresponding on-shell presymplectic structureˆΩ = δ ˆΘ , ˆΘ = Z Σ Θ , where Θ = h C ⋆ Φ µ ( ω , ω ) ⋆ δ ω i + h Λ µν ( ω , ω , ω , C ) ⋆ δ C i + O ( C ) (6.46)and ω , C obey equations (6.12), becomes trivial modulo C whenever µ = ν . Indeed, Θ ≈ h Φ ν ⋆ C ⋆ δ ω i + h D Ψ ν ⋆ δ ω i + h Λ νν ⋆ δ C i + O ( C ) ≃ h R ⋆ δ ω i + h Ψ ν ⋆ Dδ ω i + h Λ νν ⋆ δ C i + O ( C ) ≃ h R ⋆ δ ω i − h Ψ ν ⋆ δ R i + h Λ νν ⋆ δ C i + O ( C )= d h ω ⋆ δ ω i + δ (cid:10) ω ⋆ d ω − ω ⋆ ω ⋆ ω (cid:11) − h Ψ ν ⋆ δ R i + h Λ νν ⋆ δ C i + O ( C ) . (6.47)Denoting J = (cid:10) ω ⋆ ω ⋆ ω + Φ ν ( ω , ω ) ⋆ C ⋆ ω (cid:11) + O ( C ) , (6.48)35e proceed as Θ ≃ δ J − h Ψ ν ⋆ Φ ν ⋆ δ C i + h Λ νν ⋆ δ C i + O ( C )= δ J + h (Φ ν Ψ ν − Ψ ν Φ ν ) ⋆ δ C i + O ( C )= δ J + h [ δ C , Φ ν ] ⋆ ⋆ Ψ ν + Ψ ν ⋆ [ δ C , Φ ν ] ⋆ i + O ( C ) ≈ δ J + h D Ψ ν ( ω , δ C ) ⋆ Ψ ν ( ω , C ) + Ψ ν ( ω , C ) ⋆ D Ψ ν ( ω , δ C ) i + O ( C ) ≈ δ (cid:0) J + h [ C , Φ ν ( ω , ω )] ⋆ ⋆ Ψ ν ( ω , C ) i (cid:1) − d h Ψ ν ( ω , C ) ⋆ Ψ ν ( ω , δ C ) i + O ( C ) . (6.49)Hence, Θ degenerates into the sum of on-shell d - and δ -exact terms, so that ˆΩ = 0 modulo C . The current (6.48) is conserved up to the first order in C , that is, d J ≈ C ); itsconservation, however, inevitably breaks down at the next order, see [65].The general conclusion is that for µ = ν , the presymplectic structure (6.46) is equivalent tothat of the form (6.4) with m >
2. In other words, even for µ = ν the presymplectic structureremains non-trivial, but starts at NNLO. However, this does not seem to be satisfactory fromthe physical point of view as long as we want to reproduce the correct quantization of freehigher spin fields on maximally symmetry backgrounds, C = 0, e.g. on AdS . Therefore, wekeep µ = ν .Another important comment is that the scalar field of the higher spin multiplet is somewhatseparated from s > of the form (6.18) and perform the corresponding redefinitionsof Θ (6.39), which is a problem closely related to the issue of (non-)locality of HSGRA. Wehope that many useful statements can be proved first at the formal level, while the non-localityproblem awaits its satisfactory resolution.Lastly, the presymplectic structure is unobstructed, see Appendix D for the proof. There-fore, it can be extended to any order and the general expression can be written with the helpof the techniques introduced in [69] for equations. The presymplectic structure and AKSZaction depend on two coupling constants, as required by the AdS/CFT duality with Chern–Simons matter theories (one should set µ = ν = ˜ N − e iθ and ¯ ν = ν ∗ , ¯ µ = − µ ∗ , where θ = π λ ,˜ N = 2 N sin πλπλ are expressed in terms of the number of fields N and t’Hooft coupling λ = N/k for Chern–Simons level k ). 36 Higher spin waves and currents
As has been mentioned above, the most symmetric vacuum in HSGRA corresponds to C = 0.In this case, the highly non-linear equations of motion (3.3) are greatly simplified taking theform of zero-curvature condition for the gauge connection ω associated with the Lie algebra gl ( A ). In other words, the solutions to d ◦ ω = ◦ ω ⋆ ◦ ω (7.1)describe the most symmetric geometric backgrounds against which the free higher spin fieldscan consistently propagate. A particular solution to these equations is provided by the anti-deSitter space, in which case ◦ ω ∈ so (3 , ⊂ gl ( A ). We let ˜ C and ˜ ω denote the small fluctuationsof fields about the vacuum. Then the linearized equations of motion (3.3) read ◦ D ˜ ω = Φ ν ( ◦ ω , ◦ ω ) ⋆ ˜ C , ◦ D ˜ C = 0 . (7.2)Here we introduced the background covariant differential ◦ D = d − [ ◦ ω , − ] ⋆ . By definition, ◦ D = 0.The equations are invariant under the gauge transformations δ ε ˜ ω = ◦ D ε , δ ε ˜ C = 0 , (7.3) ε being an infinitesimal gauge parameter. Besides, system (7.2) enjoys the global symmetriesof the form δ ξ ˜ ω = [ ξ , ˜ ω ] ⋆ − Ψ ν ( ◦ ω , ξ ) ⋆ ˜ C , δ ξ ˜ C = [ ξ , ˜ C ] ⋆ , (7.4)where the infinitesimal parameter ξ obeys the condition ◦ D ξ = 0. Notice the term involvingΨ ν , which survives even for the anti-de Sitter background. In that case the κ -independent partof ξ accommodates the whole set of Killing’s tensors of AdS . Noteworthy also is the fact thatthe global symmetries form an associative algebra, Mat( A ), w.r.t. the ⋆ -product of ξ ’s, andnot just the Lie algebra gl ( A ) w.r.t. their ⋆ -commutators.A nice property of (7.1, 7.2) is that this is a fully consistent system on its own. It requiresonly the Hochschild cocycle Φ ν and does not suffer from non-localities. It describes propaga-tion of higher spin fields on backgrounds more complicated than just the anti-de Sitter space,providing thus a way to overcome the Aragone–Deser no-go [131]. In d = 3 there is no r.h.s.in (7.2) since higher spin fields do not have propagating degrees of freedom. For d >
3, theHochschild cocycle is required. The simplest solutions beyond empty
AdS d could be topologicalblack holes [132] and higher spin generalizations thereof along the 3 d lines, see e.g. [133]. We are grateful to Per Sundell for this suggestion. C = 0 and ω = ◦ ω gives the following quadraticLagrangian: L ◦ = h ˜ C ⋆ Φ µ ( ◦ ω , ◦ ω ) ⋆ ◦ D ˜ ω − Λ µν ( ◦ ω , ◦ ω , ◦ ω , ˜ C ) ⋆ ◦ D ˜ C − ˜ C ⋆ Φ µ ( ◦ ω , ◦ ω ) ⋆ Φ ν ( ◦ ω , ◦ ω ) ⋆ ˜ C i . The Lagrangian is obviously invariant under the gauge transformations (7.3), but not underthe action of global symmetries (7.4) as one can easily check. Nonetheless, we can still usethe underlying presymplectic structure to assign conserved currents to the global symmetrytransformations. The linearized presymplectic form Ω ◦ = d Θ ◦ on the target space induces thaton the configuration space of fields ˜ C and ˜ ω . The latter is given byˆΩ ◦ = Z Σ Ω ◦ , Ω ◦ = δ Θ ◦ , Θ ◦ = h ˜ C ⋆ Φ µ ( ◦ ω , ◦ ω ) ⋆ δ ˜ ω i + h Λ µν ( ◦ ω , ◦ ω , ◦ ω , ˜ C ) ⋆ δ ˜ C i . (7.5)In the case of AdS background, this presymplectic structure and the Lagrangian L ◦ were firstfound in [64].Let X ξ denote the variational vector field defined by the r.h.s. of equations (7.4). Weleave it to the reader to check that the vector field X ξ , being tangent to the solution space, isHamiltonian relative to ˆΩ ◦ , i.e., i X ξ Ω ◦ ≃ δ J ξ , (7.6)where J ξ = (cid:10) ˜ C ⋆ Φ µ ( ◦ ω , ◦ ω ) ⋆ ([ ξ , ˜ ω ] − Ψ ν ( ◦ ω , ξ ) ⋆ ˜ C ) − Λ µν ( ◦ ω , ◦ ω , ◦ ω , ˜ C ) ⋆ [ ξ , ˜ C ] (cid:11) + h ˜ C ⋆ Λ µν ( ◦ ω , ◦ ω , ◦ ω , ξ ) ⋆ ˜ C i . (7.7)By construction, the Hamiltonian J ξ defines a conserved current for the higher-spin waveequations (7.2). The last fact is easy to verify directly: d J ξ ≈ h ˜ C ⋆ Φ µ ⋆ ([ ξ , Φ ν ⋆ ˜ C ] ⋆ − [ ξ , Φ ν ] ⋆ ˜ C ) i− h [ ˜ C , Φ µ ⋆ Φ ν ] ⋆ ⋆ [ ξ , ˜ C ] ⋆ i − ˜ C ⋆ [ ξ , Φ µ ⋆ Φ ν ] ⋆ ⋆ ˜ C i = 0 . (7.8) Note that to this order it is easy to write the Lagrangian in a more suitable form as L ◦ = hV can µ ( ◦ ω , ◦ ω , ˜ C ) ⋆ ◦ D ˜ ω − ˜Λ µν ( ◦ ω , ◦ ω , ◦ ω , ˜ C ) ⋆ ◦ D ˜ C − V can µ ( ◦ ω , ◦ ω , ˜ C ) ⋆ V can ν ( ◦ ω , ◦ ω , ˜ C ) i , see footnote 17. One just needs to perform the field redefinitions that lead from the factorized vertex to V can ,inducing a certain change of Λ µν into ˜Λ µν . It is the Λ-term that gives the presymplectic structure for the freescalar field. µ = ν .Notice that the conserved currents J ξ are not invariant under the gauge transformations(7.3) in the sense that δ ε J ξ ≈ d h ˜ C ⋆ Φ µ ( ◦ ω , ◦ ω ) ⋆ [ ξ , ε ] ⋆ i 6 = 0 . (7.9)The violation of gauge invariance occurs due to the explicit dependence of ˜ ω and cannot beremoved by adding d -exact terms to the currents J ξ . Of course, this non-invariance does notaffect the integrated conserved charges. It is significant that the currents (7.7) have non-zeroprojections onto the physical sector if one takes ξ ∈ gl ( hs ), see Remark 3.4.Gauge invariant and non-invariant conserved currents for the free higher spin fields on the AdS background have been intensively studied in the literature, see [134–136] and referencestherein. In [136], an infinite family of gauge non-invariant currents of the form ˜ ω × ˜ ω havebeen explicitly constructed. Our gauge non-invariant currents (7.7) have a different structure,schematically ˜ C × ˜ ω + ˜ C × ˜ C . The gauge invariant currents were completely classified in Ref.[134, 135]. Being gauge invariant, they may only depend on the zero-form field as ˜ C × ˜ C . Itgoes without saying that none of the currents above can be extended to or come from the non-linear theory. Furthermore, a straightforward cohomological analysis exposed in the Appendicesshows that the aforementioned currents ˜ ω × ˜ ω and ˜ C × ˜ C cannot be even extended from AdS to the general higher spin background (7.1).The Poisson bracket associated with the presymplectic structure (7.5) makes the conservedcurrents (7.7) into the Lie algebra: { J ξ , J ξ ′ } = i X ξ i X ξ ′ Ω ◦ ≃ δ ξ J ξ ′ ≃ J [ ξ , ξ ′ ] ⋆ . (7.10)In general, the currents J ξ being quadratic in fields, one may expect a non-trivial centralcharge to appear in the r.h.s. of (7.10) upon quantization. The vanishing of the second coho-mology group H ( gl ( A )) (see [137, Sec. B.4]), however, precludes such a possibility, so that thecommutation relations (7.10) have to survive quantization as they are. We have constructed a presymplectic AKSZ sigma-model for 4 d HSGRA at the formal level, thatis, we showed that it does exist and depends on the right number of coupling constants. We alsoworked out the first two orders explicitly and proved that the higher orders are unobstructed.To a great surprise the free action turns out to be a genuine action for higher spin fields rather39han just a weak action. The action leads to the right quantum commutators for variousobservables: the gauge-invariant field strengths and the currents of the global higher spinsymmetry (leftover of the gauge symmetry at the free level). The AKSZ action reproducessome of the cubic vertices, while the rest should come from higher orders in the C -expansion.Among the future developments of interest we can mention: ( i ) to find a compact form forthe presymplectic AKSZ action of HSGRA in arbitrary dimension, which should be a variationof the techniques from [66–69]; ( ii ) guided by the examples of lower spin theories, to developa path-integral quantization of HSGRA by means of presymplectic AKSZ models; ( iii ) to seeif concrete quantum checks of HSGRA can be done already at the formal level, i.e., withoutpaying attention to the non-locality of the models. Quantization I.
The main purpose of the approach advocated in this paper is to shed somelight on the quantization of HSGRA. The basic ingredient of any quantization method is aPoisson structure on the space of physical observables of a classical theory. This is, for example,a starting point of the deformation quantization technique and the canonical quantization. Inother approaches, the Poisson brackets manifest themselves and can be recovered through thesemi-classical limit of equal-time commutation relations. Each Lagrangian field theory enjoysa canonical presymplectic structure that induces a non-degenerate Poisson bracket on gauge-invariant functionals of fields.It is particularly remarkable that one can define and classify all suitable presymplecticstructures independently of Lagrangians; the only input one needs for that are classical equa-tions of motion. In this paper, we solved this classification problem for 4 d HSGRA. Applyingcohomological analysis, we found that all physically relevant presymplectic structures form atwo-parameter family that can be read off or encoded in an AKSZ-type sigma-model (6.43).Moreover, we were able to find the explicit expressions (6.46) for (some representatives of)these presymplectic structures up to the second-order in C . As an immediate application ofthese presymplectic structures, we computed the Poisson brackets of the conserved currents(7.7) in free theory. We also argued that the classical commutation relations of the currents(7.10) should survive quantization.The deformation quantization of finite-dimensional presymplectic manifolds has been con-sidered in the works [138, 139]. Appropriate adaptation of this method to the context of fieldtheory (see e.g. [140] and references therein) will hopefully work for the model at hand. Asan alternative, one can try to apply the path-integral quantization to the presymplectic AKSZmodel of Sec. 6. It is well to bear in mind that the weak Lagrangian (6.43) may not be aLagrangian in the ordinary sense and its relevance to the path-integral quantization of HSGRA40alls for further investigation. Quantization II.
More abstractly one could check what are the possible counterterms in aHSGRA, which does not require actual quantization. The beauty of the counterterm argumentis that one might be able to prove renormalizability or finiteness even without having to quantizeanything. However, the naive argument – the more symmetries, the less counterterms – doesnot seem to apply without further fuss: there exists infinitely many invariants of the higherspin symmetry that can serve as potential counterterms [65]. Nevertheless, additional physicalrestrictions might potentially rule most of them out. Therefore, the direct quantization ofHSGRA is an important open problem too.
Vacuum corrections.
The on-shell AKSZ action is proportional to the corresponding Hamil-tonian; the latter is an example of on-shell observables which were classified in [65]. The on-shellaction starts with ω C -type terms and appears to vanish on any maximally symmetric back-ground ( C = 0).However, the value of the classical action on AdS vacuum should not be zero. Moreover, itis known to contain a rather peculiar numerical factor (cid:0) log(4) − ζ (3) π (cid:1) for the ∆ = 1 boundaryconditions [46, 47], the free energy of the free scalar field on the three-sphere. Some contributionmight come from the boundary terms (a higher spin analog of the Gibbons–Hawking term) thatthe action should be supplemented with. All possible on-shell non-trivial three-forms can befound in [65] and again there is no candidate among ω -type observables.It should be borne in mind that the on-shell action is actually ill-defined and requiresregularization. Formally, it is the indeterminate form 0 · ∞ , where 0 corresponds to the on-shell value of the integrand, while ∞ comes from integration over an infinite volume. Toevaluate the integral properly, one could choose a family of solutions approaching the AdS vacuum; in so doing, each C of the family must satisfy an appropriate fall-off condition toensure convergence. The limit of the integral as C → AdS vacuum. Unfortunately, not muchexact solutions to HSGRA are available in the literature to implement the above regularizationprocedure. It seems reasonable to use a family of solutions that preserves as much symmetriesas possible, e.g. as in [129].Note that in the ordinary (non-higher-spin) cases the on-shell value is proportional to thevolume of (Euclidian) AdS d , which is divergent and needs to be regularized. One can also regardthe space-time integral as the volume of the quotient SO ( d + 1 , /SO ( d, g µν with all higher spin fields and theon-shell value of the action should not be thought of as the volume of AdS d , rather it shouldbe related to the (regularized) volume of the higher spin group. How degenerate the presymplectic AKSZ action is?
In general, as is discussed in Sec.4, presymplectic actions cannot reproduce all of the equations of motion. Nevertheless, insome cases, e.g. gravity, the hidden integrability conditions allow one to get equations that arecompletely equivalent to the one we started with. In order to have such a miracle one needs theright balance between fields and equations (e.g. in gravity the torsion constrain has the samenumber of components as spin-connection ω a,b and the Einstein equations can, in principle, bereproduced from the variation of vielbein e a , which is what happens). However, in any HSGRAin d > C and ω are one- and two-forms valued in ahigher spin algebra. This argument is not precise enough since we do not have to reproduce allof the equations, only the ones that lead to the differential equations for the dynamical fields(for example, torsion-like constraints can be imposed by hand).Nevertheless, the following observation can save the day: variation with respect to ω can,in principle, reproduce all of the C -equations, (C.5). Suppose this is true. Then we can show,see Appendix E, that the ω -equations resulting from the integrability of the C -equations haveto be exactly (3.3a) up to a two-form B d ω = ω ⋆ ω + V ( ω , ω , C ) + O ( C ) + B , (8.1)where B belongs to the center of the higher spin algebra, i.e. it is B = 1 · B and B is anon-trivial two-form belonging to the de Rham cohomology, if any. Higher dimensions.
It is instructive to have a look at the presymplectic AKSZ model forthe d -dimensional formal HSGRA. As argued in [65], the basis of the Hochschild cohomologyof the (extended) higher spin algebra is given by a two-cocycle φ and a ( d − ψ . Thetwo-cocycle is the one that drives the deformation of the free equations. The ( d − d = 4 both of these, rather different innature, cocycles happen to be two-cocycles. Algebraically the doubling of two-cocycles is due tothe fact that the extended higher spin algebra A = A ⊗ A is the tensor square of A = A ⋊ Z . We are grateful to Ergin Sezgin for a useful discussion on the ambiguities of the equations, see also [59] forthe discussion of interactions’ ambiguities related to de Rham cohomology. ψ we can write the following natural expression for the presymplecticpotential on fields: Θ = h ψ ( ω , . . . , ω ) ⋆ C ⋆ δ ω i . (8.2)It is also plausible that there exists a field redefinition that brings this Θ into the form Θ = hV ( ω , . . . , ω , C ) ⋆ δ ω i (8.3)such that for the anti-de Sitter vacuum ◦ ω = h a P a + w a,b L ab we have Θ = ǫ v ··· v d − u u h v · · · h v d − C u a ( s − ,u b ( s − δ ˜ ω a ( s − ,b ( s − . (8.4)Here, h a is a vielbein, w a,b is a spin-connection, P a and L ab are the generators of transvectionsand Lorentz transformations, and ˜ ω is the fluctuation about ◦ ω . This gives the canonicalpresymplectic form Ω ◦ = ǫ v ··· v d − u u h v · · · h v d − δ ˜ C u a ( s − ,u b ( s − δ ˜ ω a ( s − ,b ( s − (8.5)for free fields ˜ ω , ˜ C on C = 0 background. The non-linear presymplectic AKSZ action shouldhave then the form S = Z hV ( ω , . . . , ω , C ) ⋆ ( d ω − ω ⋆ ω ) i + O ( C ) . (8.6)We are going to detail this construction elsewhere, but it is tempting to compare it with theEinstein–Hilbert action in the frame-like formulation S [ e, ω ] = Z ǫ v ··· v d − u u e v · · · e v d − ( dω u ,u − ω u c ω c,u ) . (8.7)The first factor is nothing else but the Chevalley–Eilenberg cocycle of the Poincar´e algebra.In 3 d , one should not be surprised that to leading order the only presymplectic structure isthe one of Chern–Simons theory and the presymplectic AKSZ action is just the Chern–Simonsaction. It would be interesting to explore the 3 d case further.As a step towards the complete AKSZ sigma-model it is worth noting that the non-linearequations of any HSGRA are integrable in the sense of being equivalent to the following ‘freesystem’ [69]: d ˆ ω = ˆ ω ∗ ˆ ω , d ˆ C = ˆ ω ∗ ˆ C − ˆ C ∗ ˆ ω . (8.8)43ere ∗ is the product in the deformed (extended) higher spin algebra A ~ , a ∗ b = a ⋆ b + ~ φ ( a, b ) + · · · , and the fields ˆ ω , ˆ C take values in A ~ . The deformed algebra features the same structure ofHochschild cocycles, e.g. it has 2-cocycle ϕ ( a, b ) = ∂ ~ ( a ∗ b ), it also has an invariant trace h−i and should have an appropriate cocycle b V ( ˆ ω , . . . , ˆ ω , ˆ C ). Now, one can write a complete actionas S = Z h b V ( ˆ ω , . . . , ˆ ω , ˆ C ) ∗ ( d ˆ ω − ˆ ω ∗ ˆ ω ) i , (8.9)which may be a good starting point for exploring the quantum properties of HSGRA in thefuture. Acknowledgments
We are grateful to Nicolas Boulanger, Maxim Grigoriev, Ergin Sezgin and Per Sundell for usefuldiscussions and correspondence. E. S. is grateful to the Erwin Schr¨odinger Institute in Viennafor hospitality during the program “Higher Structures and Field Theory” while this work was inprogress. The work of A. Sh. was supported by the Ministry of Science and Higher Educationof the Russian Federation, Project No. 0721-2020-0033. The work of E. S. was supported bythe Russian Science Foundation grant 18-72-10123 in association with the Lebedev PhysicalInstitute. Each author blames the faults that remain on the other. A Hochschild, cyclic, and Lie algebra cohomology
In this appendix, we collect some basic definitions and constructions related to the cohomologyof associative and Lie algebras. For a more coherent exposition of the material we refer thereader to [142], [143], [144]. A word about notation: all unadorned tensor products ⊗ andHom’s are taken over k , a ground field of characteristic zero. We systematically follow theKoszul sign convention. As in the main text, the grade of a homogeneous element a is denotedby | a | . Many formulas below are considerably simplified if one uses the shifted grade ¯ a = | a | − HH • ( A, M ) of a graded associative k -algebra A with coef-ficients in a graded A -bimodule M is the cohomology of the Hochschild cochain complex We are grateful to Karapet Mkrtchyan for suggesting this peaceful resolution. • ( A, M ) composed by the vector spaces C p = Hom( A ⊗ p , M ) , A ⊗ p = A ⊗ · · · ⊗ A | {z } p and the homomorphisms ∂ : C p → C p +1 defined by( ∂f )( a , . . . , a p +1 ) = ( − (¯ a +1)( ¯ f +1) a f ( a , . . . , a p +1 ) − ( − ¯ a + ··· +¯ a p f ( a , . . . , a p ) a p +1 (A.1)+ p X k =1 ( − ¯ a + ··· +¯ a k f ( a , . . . , a k a k +1 , . . . , a p +1 ) . In the special case M = A ∗ it is convenient to identify the spaces C p ( A, A ∗ ) with Hom( A ⊗ ( p +1) , k ).Then the formula for the Hochschild differential takes the form( ∂g )( a , a , . . . , a p +1 ) = p X k =0 ( − ¯ a + ··· +¯ a k g ( a , a , . . . , a k a k +1 , . . . , a p +1 )+ ( − (¯ a +1)(¯ a + ··· +¯ a p +1 ) g ( a , . . . , a p , a p +1 a ) , (A.2)where, by definition, g ( a , . . . , a p − , a p ) = ( − | a p | f ( a , . . . , a p − )( a p ).As was first observed by A. Connes, the complex C • ( A, A ∗ ) contains a subcomplex C • cyc ( A )of cyclic cochains , i.e., cochains g ∈ Hom( A ⊗ ( p +1) , k ) satisfying the additional condition g ( a , a , . . . , a p ) = ( − ¯ a (¯ a + ··· +¯ a p ) g ( a , . . . , a p , a ) . (A.3)The cohomology of the complex C • cyc ( A ) is called the cyclic cohomology of A and the corre-sponding cohomology groups are denoted by HC • ( A ). Upon restricting to cyclic cochains, onecan bring the differential (A.2) into a more familiar form ( ∂g )( a , a , . . . , a p +1 ) = p X k =0 ( − ¯ a + ··· +¯ a k g ( a , a , . . . , a k a k +1 , . . . , a p +1 )+ ( − ¯ a p +1 (¯ a + ··· +¯ a p +1) g ( a p +1 a , a , . . . , a p ) . (A.4)Considering, for example, the ground field k as a one-dimensional algebra over itself one readilyconcludes that C n cyc ( k ) ≃ k and C n +1cyc ( k ) = 0. Hence, HC n cyc ( k ) ≃ k and HC n +1cyc ( k ) = 0.It follows from the definition that the cyclic cohomology groups are contravarinat functorsof the algebra, so that any algebra homomorphism h : A → B induces a homomorphism h ∗ : HC p ( B ) → HC p ( A ) in cohomology. Notice that the signs in either form obey the
Koszul sign rule if one shifts the grade of all a ’s by −
1, sothat the dot product acquires degree 1. Upon this interpretation the dot product and cyclic permutation of a ’sgo first and the map g after. A and B with that of their tensor product A ⊗ B and is defined as follows.Let M and N be bimodules over algebras A and B , respectively. Then for any f ∈ C q ( A, M )and g ∈ C p ( B, N ) we put ( f ⊔ g )( a ⊗ b , . . . , a q + p ⊗ b q + p )= ( − ǫ f ( a , . . . , a q ) a q +1 · · · a q + p ⊗ b · · · b q g ( b q +1 , . . . , b q + p ) , (A.5)where ( − ǫ is the Koszul sign resulting from permutations of a ’s, b ’s, and g . By definition, f ⊔ g ∈ C q + p ( A ⊗ B, M ⊗ N ). The cup product (A.5) is differentiated by the Hochschildcoboundary operator (A.1) thereby inducing a product in cohomology: ⊔ : HH q ( A, M ) ⊗ HH p ( B, N ) → HH q + p ( A ⊗ B, M ⊗ N ) . (A.6)If all the cohomology groups HH p ( B, N ∗ ) turn out to be finite-dimensional, then (A.6) definesa natural isomorphism HH n ( A ⊗ B, M ⊗ N ) ≃ M p + q = n HH p ( A, M ) ⊗ HH q ( B, N ) (A.7)for any A -bimodule M . This is just the dual version of the K¨unneth formula for Hochschildhomology, see [145, Ch. X, Thm. 7.4].Unlike the Hochschild cohomology, the cup product ⊔ : HC q ( A ) ⊗ HC p ( B ) → HH q + p ( A ⊗ B ) (A.8)for cyclic cohomology groups cannot be defined at the level of complexes; one has to multiplycyclic cocycles. The explicit formula for this cup product is somewhat cumbersome and we donot present it here. The reader can found it in many places, e.g. [142, Sec. 4.4.10], [146, II.1].The cyclic analog of the isomorphism (A.7) is given now by the exact sequence0 → HC • ( A ) O HC • ( k ) HC • ( B ) ⊔ → HC • ( A ⊗ B ) → Tor HC • ( k ) ( HC • ( A ) , HC • ( B )) → HC p ( B ) are finite-dimensional, see [147, Thm. 1]. (Thecommutative algebra HC • ( k ) and its left/right action on cyclic cohomology are defined below.) Not to be confused with Gerstenhaber’s ∪ -product on HH • ( A, A ).
46s is seen, the ⊔ -product homomorphism from the tensor product of HC • ( k )-modules, beinginjective, is not generally surjective, yet it becomes an isomorphism whenever either of the HC • ( k )-modules is torsion free.By way of illustration let us take B to be the matrix algebra Mat n ( k ) viewed as a bimoduleover itself. Then A ⊗ B = Mat n ( A ) and M ⊗ N = Mat n ( M ). Since the algebra Mat n ( k ) is separable [142, Sec. 1.2.12], HH • (Mat n ( k ) , Mat n ( k )) ≃ HH (Mat n ( k ) , Mat n ( k )) ≃ k , (A.10)where the group HH (Mat n ( k ) , Mat n ( k )), being isomorphic to the centre of Mat n ( k ), is gener-ated by the unit matrix 11. By the K¨unneth formula (A.7), HH p ( A, M ) ≃ HH p (Mat n ( A ) , Mat n ( M )) . (A.11)At the level of cochains the isomorphism is induced by the so-called cotrace map : cotr( f ) = f ⊔ f ∈ C p ( A, M ). As Rel. (A.5) suggests,cotr( f )( a ⊗ m , . . . , a p ⊗ m p ) = f ( a , . . . , a p ) ⊗ m · · · m p (A.12)for a i ⊗ m i ∈ A ⊗ Mat n ( k ).In the case of cyclic p -cochains (A.3) the map (A.12) takes the formcotr( g )( a ⊗ m , . . . , a p ⊗ m p ) = g ( a , . . . , a p )tr( m · · · m p ) (A.13)and gives rise to the isomorphism HC p ( A ) ≃ HC p (Mat n ( A )) (A.14)of cyclic cohomology groups.As was mentioned above HC n ( k ) ≃ k . Let σ denote the basis 2-cocyle for HC ( k ) obey-ing the normalization condition σ (1 , ,
1) = 1. Note that for each k -algebra A , there is thenatural isomorphism A ⊗ k ≃ A . Using this isomorphism and the 2-cocycle σ , we can define ahomomorphism S : HC p ( A ) → HC p +2 ( A ) (A.15)by setting Sf = σ ⊔ f = f ⊔ σ , ∀ f ∈ HC p ( A ) . (A.16)The homomorphism S of degree 2 is called the periodicity map . For example, applying S to a1-cocycle φ , one obtains( Sφ )( a , a , a , a ) = ( − | a | +( | a | + | a | )( | a | + | a | ) φ ( a a a , a ) + ( − | a | φ ( a a a , a ) .
47t is instructive to verify the cyclic property (A.3) of the resulting 3-cocycle Sφ . If we take A = k , then (A.15) makes HC • ( k ) into an associative commutative algebra; in fact HC • ( k ) ≃ k [ S ]. This allows one to regard each k -vector space HC • ( A ) as a bimodule over the k -algebra HC • ( k ).Let Der( A ) denote the space of all derivations of the graded algebra A . By definition,homogeneous elements of Der( A ) are homomorphism D : A → A obeying the graded Leibnizrule D ( ab ) = ( Da ) b + ( − | a || D | a ( Db ) . The derivations are known to form a graded Lie algebra w.r.t. the commutator. Furthermore,each derivation D gives rise to a cochain transformation L D : C p cyc ( A ) → C p cyc ( A ) defined by( L D g )( a , a , . . . , a p ) = p X k =0 ( − ¯ D (¯ a + ··· +¯ a k − ) g ( a , . . . , Da k , . . . , a p ) . (A.17)As usual, this induces a homomorphism L ∗ D : HC p ( A ) → HC p ( A ) in cohomology. The inducedhomomorphism is known to be trivial for inner derivations. A similar action of derivationscan be defined for Hochschild cohomology as well.Each derivation D of an algebra A trivially extends to the derivation ˆ D of the tensor product A ⊗ B by setting ˆ D ( a ⊗ b ) = Da ⊗ b . This extension appears to be compatible with the cupproduct of cyclic cocycles in the most natural way: L ˆ D ( f ⊔ g ) = L D f ⊔ g . (A.18)As a result, each D ∈ Der( A ) generates a homomorphism L ∗ ˆ D : HC p ( A ⊗ B ) → HC p ( A ⊗ B )in cohomology.The fact that the cyclic complex C • cyc ( A ) is a subcomplex of the Hochschild complex C • ( A, A ∗ ) gives rise to the long exact sequence in cohomology · · · / / HH p ( A, A ∗ ) B / / HC p − ( A ) S / / HC p +1 ( A ) I / / HH p +1 ( A, A ∗ ) / / · · · . (A.19)The sequence involves the periodicity map (A.15) and is known as Connes’ Periodicity Exact Se-quences. In many interesting cases it reduces the problem of computation of cyclic cohomologyto that of Hochschild cohomology. The map I is induced by the inclusion C • cyc ( A ) → C • ( A, A ∗ ),while the definition of B is more complicated, see [143].Among important applications of cyclic cohomology is computation of the cohomology ofthe Lie algebra gl ( A ) of ‘big matrices’. By definition, the algebra gl ( A ) consists of infinitematrices with only finitely many entries different from zero. Formally, it is defined through the48nductive limit gl ( A ) = lim → gl n ( A ) corresponding to the natural inclusions gl n ( A ) ⊂ gl n +1 ( A )(an n × n -matrix is augmented by zeros). A precise relationship between the cohomology ofthe Lie algebra of matrices and cyclic cohomology is established by the Tsygan–Loday–Quillentheorem [148], [149].In order to formulate this theorem precisely we need some more terminology. Recall that theChevalley–Eilenberg cochain complex of a graded Lie algebra L = L L n consists of the sequenceof groups C p ( L ) = Hom(Λ p L, k ) endowed with a coboundary operator δ : C p ( L ) → C p +1 ( L ).By definition, c ( a , . . . , a k , a k +1 , . . . , a p ) = ( − ¯ a k ¯ a k +1 c ( a , . . . , a k +1 , a k , . . . , a p ) (A.20)and ( δc )( a , . . . , a p +1 ) = X ≤ k
The image of the map (A.24) lies in the indecomposable part of the algebra H • ( gl n ( A )) and induces an isomorphism HC p − ( A ) ≃ Indec H p ( gl n ( A )) for all n ≥ p . As an exterior algebra, H • ( gl ( A )) is freely generated by the graded vector space HC •− ( A ) . In other words, the cohomology group H p ( gl n ( A )) does not depend on the size of matricesprovided it is large enough. B Cohomology of Weyl algebras and their smash prod-ucts
The polynomial Weyl algebra A n over C is a unital algebra on 2 n generators q i and p j subjectto Heisenberg’s commutation relations[ q i , q j ] = 0 , [ p i , p j ] = 0 , [ q i , p j ] = δ ij . (B.1) There is no need to antisymmetrise all p + 1 arguments due to cyclicity of g and tr.
50t is known to be a simple Noetherian domain with a k -basis consisting of the ordered monomialsin q ’s and p ’s, see e.g. [150].The Hochschild cohomology groups of Weyl algebras are known for various coefficients. Forinstance, applying the Koszul resolution (see e.g. [151], [152]) yields HH • ( A n , A n ) ≃ HH ( A n , A n ) ≃ C (B.2)and HH p ( A n , M ) ≃ HH n − p ( A n , M ∗ ) , HH p ( A n , M ) = 0 ∀ p > n (B.3)for any bimodule M . Among other things, the isomorphisms (B.2) mean that all Weyl alge-bras are rigid, have only inner derivations, and their center is generated by the unit element.Combining (B.2) and (B.3), one also obtains HH • ( A n , A ∗ n ) ≃ HH n ( A n , A ∗ n ) ≃ C . (B.4)An explicit formula for a non-trivial 2 n -cocycle τ n generating the group HH n ( A n , A ∗ n ) wasfound in the 2005 paper [153] by Feigin, Felder, and Shoikhet. It was derived as a consequenceof Shoikhet’s proof [154] of Tsygan’s formality conjecture. It should be noted that fifteen yearsearlier Vasiliev had found an explicit expression for τ in the context of 4 d HSGRA [48]. Inorder to present the cocycle τ explicitly it is convenient to identify the elements of A withthe polynomials a ( q, p ) in (commuting) indeterminates q and p endowed with the Weyl–Moyalstar-product a ⋆ b = m exp α ( a ⊗ b ) , (B.5)where α = 12 (cid:18) ∂∂p ⊗ ∂∂q − ∂∂q ⊗ ∂∂p (cid:19) ∈ End( A ⊗ A ) (B.6)and m ( a ⊗ b ) = ab . We also introduce the maps α , α , α : α ( a ⊗ a ⊗ a ) = 12 (cid:18) ∂a ∂p ⊗ ∂a ∂q ⊗ a − ∂a ∂q ⊗ ∂a ∂p ⊗ a (cid:19) ∈ End( A ⊗ A ⊗ A ) (B.7)and similarly for α and α . Finally, we define the homomorphism µ : A ⊗ A ⊗ A → C by µ ( a ⊗ a ⊗ a ) = a (0) a (0) a (0) . (B.8)Here a (0) is the constant term of the polynomial a ( q, p ). Now the expression for the 2-cocyclereads τ ( a , a , a ) = µ ◦ F ( α , α , α )( a ⊗ a ⊗ a ) , (B.9)51here the operator F ( α , α , α ) ∈ End( A ⊗ A ⊗ A ) is determined by the following entireanalytic function of three variables: F ( x, y, z ) = ( z − y ) e z + y − x + ( y − x ) e y + x − z + ( x − z ) e x + z − y ( x − z )( z − y )( y − x ) . (B.10)Clearly, the action of the operator F is well defined on polynomials. Unlike the Hochschild2-cocycles of Refs. [153] and [48], the cocycle (B.9) enjoys cyclic invariance, τ ( a , a , a ) = τ ( a , a , a ) , (B.11)thereby generating the cyclic cohomology group HC ( A ) ≃ C .In the context of HSGRA, Weyl algebras usually appear in smash products with finitegroups of their automorphisms. Recall that, given an associative k -algebra A and a finitegroup G ⊂ Aut( A ), the skew group algebra A ⋊ G (aka smash product algebra ) is defined to bethe k -vector space A ⊗ k [ G ] endowed with the product( a ⊗ g )( a ⊗ g ) = a a g ⊗ g g . (B.12)Here a g denotes the action of g ∈ G on a ∈ A .Notice that the automorphism group of A n contains a subgroup Sp n ( C ) acting by lineartransformations on the 2 n -dimensional complex space V spanned by the generators q ’s and p ’s. If G is a finite subgroup of Sp n ( k ), then g | G | = e for any g ∈ G and the action of g is diagonalizable in V . Denote by 2 µ g the multiplicity of the eigenvalue 1 of the operator g : V → V . Notice that µ g = µ hgh − and the set of all element g ∈ G with 2 µ g = p is invariantunder conjugation. The next theorem is due to Alev, Farinati, Lambre, and Solotar [151] (seealso [152]). Theorem B.1.
Let n p ( G ) denote the number of conjugacy classes of elements g ∈ G with µ g = p , then dim HH n − p ( A n ⋊ G, A n ⋊ G ) = dim HH p (cid:0) A n ⋊ G, ( A n ⋊ G ) ∗ (cid:1) = n p ( G ) . As an example, consider the most simple skew group algebra A = A ⋊ Z . Here the group Z = { e, κ } acts on A by the involution q κ = − q , p κ = − p . (B.13)Since 2 µ e = 2 and 2 µ κ = 0, all non-trivial groups of Hochschild cohomology mentioned in thetheorem above are HH ( A , A ) ≃ HH ( A , A ∗ ) ≃ C ≃ HH ( A , A ∗ ) ≃ HH ( A , A ) . (B.14)52n particular, we see that the algebra A admits a unique non-trivial deformation.Since HH n ( A , A ∗ ) = 0 for n > HC k − ( A ) = 0 , HC ( A ) ≃ C , HC k ( A ) ≃ C , k = 1 , , , . . . , (B.15)see [137, Appendix B.4] for details. Furthermore, the groups HC • ( A ) form a free HC • ( C )-module generated (via S ) by the pair of elements φ ∈ HC ( A ) and φ ∈ HC ( A ). The firstone is the trace φ = Tr : A → C defined by the projection onto the one-dimensional subspace C (1 ⊗ κ ) ⊂ A . We can normalize it by setting Tr(1 ⊗ κ ) = 1. The trace is known to give rise to anon-degenerate inner product on A defined by ( a, b ) = Tr( ab ). This inner product allows one toidentify the A -bimodules A and A ∗ . An explicit expression for a 2-cocycle representing the class φ is given in [137, Eq. (4.15)]. We refer to φ and φ as primary classes of cyclic cohomology.All the other classes in (B.15) are obtained from these two by successive application of theperiodicity operator: S n φ ∈ HC n ( A ), S n φ ∈ HC n +2 ( A ).As discussed in Sec. 3, the skew group algebra A = A ⋊ Z is a building block for theextended higher spin algebra A underlying 4 d HSGRA. The latter is given by the tensor square A = A ⊗ A = A ⋊ ( Z × Z ) . (B.16)One may also regard it as the smash product of the Weyl algebra A and the Klein four-group Z × Z acting on A by symplectic reflections. By the K¨unneth formula (A.7) for Hochschildcohomology HH n ( A , A ) = M q + p = n HH q ( A , A ) ⊗ HH p ( A , A ) , whence HH ( A , A ) ≃ C , HH ( A , A ) ≃ C , HH ( A , A ) ≃ C , (B.17)and the other groups vanish. Again, the standard interpretations of the second and thirdgroups of Hochschild cohomology suggest that the algebra A admits a two-parameter familyof formal deformations. A representative cocycle generating the group HH ( A , A ) defines anon-degenerate trace on A , which implies the isomorphism A ≃ A ∗ . Hence, HH ( A , A ∗ ) ≃ C , HH ( A , A ∗ ) ≃ C , HH ( A , A ∗ ) ≃ C . (B.18)We could also arrive at these isomorphisms by the direct application of Theorem B.1.Now the cyclic cohomology of A can be computed by means of the Connes exact sequence(A.19) or by the K¨unneth formula HC • ( A ) O HC • ( C ) HC • ( A ) ≃ HC • ( A ) . (B.19)53n either approach one finds HC ( A ) ≃ C , HC ( A ) ≃ C , HC k ( A ) ≃ C ,HC k +1 ( A ) = 0 , k = 0 , , . . . . (B.20)For detail, see [137, Appendix B.5]. As we already know HC • ( A ) is a free HC • ( C )-module ofrank two generated by the primary cohomology classes φ and φ in degrees 0 and 2. Denotingby ¯ φ and ¯ φ the same cohomology classes for the second copy of A in (B.19), we see that HC • ( A ) is a rank-four HC • ( C )-module freely generated by the cup productsΦ = φ ⊔ ¯ φ , Φ = φ ⊔ ¯ φ , ¯Φ = φ ⊔ ¯ φ , Φ = φ ⊔ ¯ φ . (B.21)These are the primary cocycles of HC • ( A ). C Cohomology of Grassmann algebras
Let Λ n denote the Grassmann algebra over C on n generators θ , . . . , θ n subject to the relations θ i θ j = − θ j θ i . As the algebra Λ n is supercommutative one concludes immediately that HH (Λ n , Λ n ) = Z (Λ n ) = Λ n and HH (Λ n , Λ n ) = Der(Λ n ) . Geometrically, one can think of Λ n as the algebra of smooth functions on a supermanifold G with odd coordinates θ ’s. Then the Lie superalgebra Der(Λ n ) of derivations of Λ n can beidentified with the algebra of smooth vector fields on G w.r.t. the supercommutator. The lastfact is a particular manifestation of the Hochschild–Kostant–Rosenberg theorem for smoothgraded-commutative algebras [142, Sec. 5.4.5]. In the case under consideration it states theisomorphism HH p (Λ n , Λ n ) ≃ Λ p (cid:0) Der(Λ n ) (cid:1) , (C.1)the r.h.s. being the space of polyvector fields on G . In terms of the odd coordinates θ i , each p -vector φ is given by φ = φ i ··· i p ( θ ) ∂ i ∧ · · · ∧ ∂ i p , ∂ i ≡ ∂∂θ i , (C.2)where the coefficients φ i ··· i p ∈ Λ n are totally symmetric in permutation of indices.The cyclic cohomology of Λ n is also well known. As was shown by Kassel [147, Prop. 2] HC p (Λ n ) ≃ HC p ( C ) ⊕ V p , (C.3)54here HC k ( C ) ≃ C , HC k +1 ( C ) = 0, and the complex dimensions of the vector spaces V p areencoded by the Poincar´e seriesch( V ) = ∞ X p =0 t p dim V p = 2 n − (1 − t ) n (1 + t )(1 − t ) n . (C.4)In particular, dim HC (Λ n ) = dim Λ n = 2 n . Actually, Λ n is a bialgebra with the standardcoproduct ∆ θ i = 1 ⊗ θ i + θ i ⊗ . This enables us to equip the complex vector space HC • (Λ n ) with the structure of a graded-commutative algebra. Indeed, composing the k -algebra homomorphism ∆ : Λ n → Λ n ⊗ Λ n withthe cup product (A.8), we get a new product ∆ ∗ ◦ ⊔ : HC • (Λ n ) ⊗ HC • (Λ n ) → HC • (Λ n ) (C.5)that makes HC • (Λ n ) into a graded associative algebra. The algebra HC • (Λ n ) contains HC • ( C )as a subalgebra and the action of the periodicity map S on HC • (Λ n ) is induced by the inclusion HC ( C ) ⊂ HC • (Λ n ). The map S generates the first summand in (C.3) and acts trivially onthe V p ’s.As was first shown in [155] cyclic cocycles representing the spaces V p admit a nice interpre-tation in terms of the polyvector fields (C.2). See [156] for subsequent discussions. Specifically,to each p -vector φ one first associates a Hochschild p -cocycle by the rule φ ( a , a , . . . , a p ) = Z a φ i ··· i p ∂ i a · · · ∂ i p a p ∈ C p (Λ n , Λ ∗ n ) . (C.6)Here a k ∈ Λ n and the integral sign stands for the Berezin integral on the Grassmann algebra.One can easily check that the property ∂φ = 0 is automatically satisfied for any φ . The cyclicitycondition φ ( a , a , . . . , a p ) = ( − ( | a | +1)( | a | + ··· + | a p | + p ) φ ( a , . . . , a n , a ) , (C.7)however, requires the p -vector φ to be divergence-free, i.e., ∂ i φ i ··· i p ( θ ) = 0 . (C.8)In order to analyse the last condition, it is convenient to identify the polyvector fields with theelements of the Berezin algebra B n on n anti-commuting generators θ i and the same number ofcommuting generators y i : θ i θ j = − θ j θ i , y i y j = y j y i , θ i y j = y j θ i . Recall that cyclic cohomology is a contravariant functor of algebra. φ i ··· i p ( θ ) ∂ i ∧ · · · ∧ ∂ i p ⇐⇒ φ ( θ, y ) = φ i ··· i p ( θ ) y i · · · y i p . (C.9)Upon this identification the operator of divergence (C.8) passes to the odd Laplace operator∆ = ∂ ∂θ i ∂y i . (C.10)Since ∆ = 0, we may say that the divergence-free polyvectors correspond to the cocycles(perhaps trivial) of the odd Laplacian: ∆ φ = 0 . The non-trivial cocycle can easily be computed with the help of the homotopy operator h = θ i y i · . Clearly, ∆ h + h ∆ = n + y i ∂∂y i − θ i ∂∂θ i . (C.11)In view of this relation there is the only (up to a multiplicative constant) non-trivial ∆-cocyle C = θ · · · θ n . (C.12)All other cocycles are of the form φ = ∆ ψ .Since ∆ is a second-order differential operator, the product of two ∆-cocyles is not a cocyclein general. Instead, the ∆-cocycles form a graded Lie algebra w.r.t. the bracket( φ , φ ) = ∆( φ φ ) − (∆ φ ) φ − ( − | φ | φ ∆ φ . (C.13)This is just the Schouten bracket on polyvector fields. It follows from the definition that thebracket is differentiated by ∆; and hence, it maps ∆-cocycles to ∆-cocycles. Taken togetherthe odd Laplacian and the bracket endow B n with the structure of Batalin–Vilkovisky algebra.One can use the 0-vector (C.12) to write non-trivial 2 m -cocycles representing the first directsummand in (C.3) in terms of the Berezin integral, namely, c m ( a , . . . , a m ) = Z Ca a · · · a m = a a · · · a m | θ =0 . (C.14)Notice that the first member of this family, c , also comes from (C.6).Let us now specify the above constructions to the case of Λ . The corresponding Berezinalgebra B is generated by θ , ¯ θ , y , and ¯ y . As a complex vector space, B is spanned by themonomials y n ¯ y m , θy n ¯ y m , ¯ θy n ¯ y m , θ ¯ θy n ¯ y m . A nm = n ¯ θy n − ¯ y m − mθy n ¯ y m − , B nm = y n ¯ y m , C = θ ¯ θ (C.15)for m, n = 0 , , , . . . . They form the following Lie superalgebra:( C, C ) = 0 , ( C, A nm ) = 0 , ( C, B nm ) = A nm , ( B nm , B kl ) = 0 , ( A nm , B kl ) = ( ln − mk ) B n + k − ,m + l − , ( A nm , A kl ) = ( ln − mk ) A n + k − ,m + l − . Thus, we are lead to conclude that the cyclic cohomology of Λ is generated by the ∆-exactpolynomials (C.15) together with the cyclic cocycles (C.14) associated with the non-trivial∆-cocycle C = θ ¯ θ . D Presymplectic structures in 4 d HSGRA
We begin with an algebraic reformulation of the problem. At the free level the homologicalvector field underlying 4 d HSGRA comes from the graded Lie algebra G described in items(1) - (4) of Sec. 3. The construction of the algebra G also admits the following geometricinterpretation. Starting from the Lie algebra G = gl ( A ) of ‘big matrices’ associated with theextended higher spin algebra A , we can define the canonical homological vector field Q = 12 ω A ω B f CAB ∂∂ω C (D.1)on the N -graded manifold N = G [1], as explained in Example 2.2. Here ω A are global coordi-nates on N associated with a basis { e A } ⊂ G wherein the commutation relations take the form[ e A , e B ] = f CAB e C . Next, following the recipe of Example 2.3, we construct the first prolongationof (D.1) to the total space of the shifted tangent bundle N = T [ − N . This is given by thehomological vector field Q = 12 ω A ω B f CAB ∂∂ω C + ω A C B f KAB ∂∂C K , (D.2) C A being linear coordinates in the tangent spaces. By definition, | C A | = 0. Since Q is quadratic,it defines and is defined by some graded Lie algebra G = G − ⊕ G , so that N = G [1]. Onecan regard G as the trivial extension of the Lie algebra G with its adjoint module put indegree − d HSGRA. One can iterate this construction to produce higher prolongations ofthe homological vector field (D.1). Of particular interest to us is the second prolongation of Q to the homological vector field Q on the total space N of the tangent bundle T [1] N = This can also be seen as a double tangent bundle of N . The construction of the homological vector field Q by Q is a particular example of a tangent prolongation Lie algebroid [157, Ch. 9]. [1] T [ − N . From the algebraic viewpoint this yields the double adjoint extension G = G − ⊕G ⊕G ⊕G of the Lie algebra G . Geometrically, we can identify the algebra of smooth functionson N = G [1] with the algebra of exterior differential forms Λ( N ), see Example 2.1. Upon suchidentification the Q -invariant differential forms on N correspond to Q -invariant functions on N and vice versa. To emphasise this correspondence we denote the linear coordinates in thetangent spaces of T [1] N by δω A and δC A . Then C ∞ ( N ) ∋ f ( ω, C, δω, δC ) ⇐⇒ f ( ω, C, dω, dC ) ∈ Λ • ( N ) . (D.3)It should be pointed out that | δω A | = 2 and | δC A | = 1, while | dω A | = 1 and | dC A | = 0.Therefore, equivalence (D.3) does not mean the equality of N -degree, if one regards the symbol δ as the ‘exterior differential’ of the coordinates ω A and C A .As an intermediate summary of our discussion we state the following isomorphisms of co-homology groups: H • ( L Q , Λ( N )) ≃ H • ( Q , C ∞ ( N )) ≃ H • ( G ) . (D.4)This reduces the classification of non-trivial presymplectic structures on the N Q -manifold( N , Q ) to the computation of certain Lie algebra cohomology groups with trivial coefficients.Notice that the form degree on the left induces an additional grading in the Lie algebra coho-mology groups H • ( G ), which will be introduced in a moment under the name of weight . Lookingfor Q -invariant presymplectic structures on N , we are thus interested in certain elements of H • ( G ) of weight two.The next step is to reinterpret the Lie algebra G – the double adjoint extension of G = gl ( A )– in terms of the underlying associative algebra A . A simple observation is that G ≃ gl ( A ⊗ Λ ),where Λ is the Grassmann algebra on two odd generators θ and ¯ θ . For our purposes, it isconvenient to prescribe them the following Z -degrees: | θ | = − , | ¯ θ | = 1 . (D.5)Since θ = 0 , ¯ θ = 0 , θ ¯ θ + ¯ θθ = 0 , (D.6)the general element of gl ( A ⊗ Λ ) has the form f = f + θf − + ¯ θf + ¯ θθ ¯ f , f i ∈ gl ( A ) . (D.7)It is easy to see that the commutation relations in gl ( A ⊗ Λ ) coincide exactly with those in G = G − ⊕ G ⊕ G ⊕ G if we set f , ¯ f ∈ G , f − ∈ G − , and f ∈ G . On passing to field theory,58he element (D.7) is promoted to the ‘superfield’ f ( θ, ¯ θ ) = ω + θ C + ¯ θδ ω + ¯ θθδ C , (D.8)which accommodates the zero- and one-form fields C and ω together with their variationaldifferentials δ C and δ ω .By Theorem A.1, H • ( gl ( A ⊗ Λ )) is a graded associative algebra freely generated by theelements of the subspace Indec H • ( gl ( A ⊗ Λ )) ≃ HC •− ( A ⊗ Λ ) . (D.9)We also know that the cyclic cohomology groups HC • ( A ) constitute a free HC • ( C )-modulegenerated by the four primary classes (B.21). This implies the K¨unneth isomorphism HC • ( A ⊗ Λ ) ≃ HC • ( A ) O HC • ( C ) HC • (Λ ) (D.10)defined by the cup product (A.8). The problem thus reduces to identifying those cohomologyclasses on the right that correspond to free presymplectic structures and their obstructions todeformation. We proceed with a closer examination of the right tensor factor in (D.10).Keeping the notation of Appendix C, we endow the Berezin algebra B with an auxiliary Z -grading by setting w ( θ ) = w ( y ) = 0 , w (¯ θ ) = − , w (¯ y ) = 1 . (D.11)We will refer to this grading as the weight , lest one confuse it with many other degrees. Besides,we introduce the differential δa = ( a, ¯ y ) = ∂a∂ ¯ θ , ∀ a ∈ B . (D.12)Clearly, w ( δ ) = 1, δ = 0, and [∆ , δ ] = 0. Taken together with the Schouten bracket (C.13)this differential makes the space B into a differential graded Lie algebra. When restricted toΛ ⊂ B , δ becomes a derivation of the Grassmann algebra Λ . By formula (A.17) it induces ahomomorphism L ∗ δ : HC p (Λ ) → HC p (Λ ) in cyclic cohomology, which then trivially extends tothe tensor product (D.10). As should be evident from (D.8) the differential (D.12) just mimicsthe action of the de Rham differential on Λ( N ). The non-trivial presymplectic structures comefrom those elements of B that are both ∆- and δ -closed and have weight one. It follows from It is well to bear in mind that the Berezin integral (C.6) implies one more differentiation by ¯ θ , so that theresulting cyclic cocycles have weight two and correspond to 2-forms on N . α n +1 = A n = n ¯ θy n − ¯ y − θy n ¯ y , β n +1 = B n = y n ¯ y . (D.13)Applying to them δ , we readily find δα n +1 = ny n − ¯ y , δβ n +1 = 0 . (D.14)Hence, only the first element α = − θ ¯ y of the α -series and all elements of the β -series (D.13)may generate presymplectic structures. Notice that β n +1 = δ (¯ θy n ¯ y ) = δ (cid:16) n + 1 A n +1 , (cid:17) . (D.15)Translated into the language of presymplectic geometry the last equality means that thedivergence-free polyvectors γ n +1 = ¯ θy n ¯ y − n + 1 θy n +1 , w ( γ n +1 ) = 0 , (D.16)generate Q -invariant potentials for the presymplectic forms Ω n +1 on N associated with β n +1 ,see Rels. (6.2, 6.5). The corresponding cyclic cocycles read γ n ( a , . . . , a n ) = n X k =1 Z a ¯ θ∂a · · · ¯ ∂a k · · · ∂a n − Z a θ∂a · · · ∂a n = − ( − | a | (cid:16) ¯ ∂ ( a ∂a · · · ∂a n ) + n X k =1 ∂ ( a ∂a · · · ¯ ∂a k · · · ∂a n ) (cid:17) θ =¯ θ =0 = n X k =0 ∂a · · · ¯ ∂a k · · · ∂a n (cid:12)(cid:12)(cid:12) θ =¯ θ =0 , (D.17) β n ( a , a , . . . , a n ) = ( L δ γ n )( a , a , . . . , a n ) = n X k =1 Z a ∂a · · · ¯ ∂a k · · · ∂a n = n X k =1 ¯ ∂∂ ( a ∂a · · · ¯ ∂a k · · · ∂a n ) = n X k =0 ¯ ∂ ( ∂a · · · ¯ ∂a k · · · ∂a n ) , (D.18) α ( a , a ) = Z a θ ¯ ∂a = ( − | a | ¯ ∂a ¯ ∂a | θ =¯ θ =0 . (D.19)This agrees with explicit computations for n = 0 , , α is a non-trivial δ -cocycle in the space of ∆-closed elements of B means that any presymplectic60tructure associated with α admits no Q -invariant presymplectic potential in distinction tothe case of β n ’s.Turning now to the left tensor factor in (D.10), we recall that the HC • ( C )-module HC • ( A )is freely generated by the four primary classes (B.21). Of these, only Φ and ¯Φ can generatepresymplectic structures on N of degree 3. In other words, all ‘free’ presymplectic structuresof 4 d HSGRA come from the two infinite series of cyclic cohomology classes ̟ n = Φ ⊔ β n , ¯ ̟ n = ¯Φ ⊔ β n . (D.20)The direct computation of the ⊔ -products shows that the resulting presymplectic structuresare of the form Ω (1) n , see Eq. (6.5). The corresponding presymplectic potentials originate fromthe classes ϑ n = Φ ⊔ γ n , ¯ ϑ n = ¯Φ ⊔ γ n . (D.21)Notice that the remaining class (D.19) gives no presymplectic structure in degree 3.We claim that all free presymplectic structures associated with the series (D.20) surviveupon switching on interaction. More precisely, by means of homological perturbation theoryof Sec. 6, they can always be deformed so as to become compatible with non-linear fieldequations (3.3a, 3.3b), whatever the interaction vertices. The existence of such a deformationis ensured by the absence of obstructing cocycles. The last fact can be seen in two equivalentways. First, one can try to deform a free presymplectic 2-form by itself. For reasons of degree,all the obstructing cohomology classes, if any, must belong to the linear span of Φ ⊔ α n and¯Φ ⊔ α n . However, Eq. (D.14) says that the corresponding 2-forms are not closed for n >
1; andhence, they cannot appear as obstructions to deformation. The remaining case n = 1 is alsoexcluded as the corresponding 2-forms do not depend on C , while the interaction vertices do.An alternative possibility is to deform the free presymplectic structure through the deformationof its presymplectic potential. Again, by degree considerations, all potential obstructions arespanned by the classes Φ ⊔ B n = L ∗ δ (Φ ⊔ B n ), so that the corresponding 1-forms turn outto be exact. The exact 1-forms represent natural ambiguity in the choice of a presymplecticpotential and can thus be disregarded. The details are left to the reader. All in all, we see thatthe classes (D.20) span the space of all presymplectic structures in 4 d HSGRA.Finally, let us note that the map I : HC ( A ) → HH ( A , A ∗ ) of the long exact sequence(A.19) is actually an isomorphism, so that each Hochschild 2-cocycle is cohomologous to acyclic one. The existence of a non-degenerate trace on A implies further isomorphisms A ≃ A ∗ and HH ( A , A ∗ ) ≃ HH ( A , A ). In view of these isomorphisms it is little wonder that thesame pair of the Hochschild 2-cocycles of HH ( A , A ) define the cubic vertices (6.10) and thepresymplectic structure (6.17) of 4 d HSGRA gravity.61
Integrability of zero-form equations
Since equations (3.3) for ω and C do not seem to be Lagrangian, it is interesting to to ask thefollowing question: to which extent do the equations of motion for C ’s control those for ω ’s?The equations in question have the form d ω α = 12 f γαβ ( C ) ω α ω β , (E.1a) d C i = V iα ( C ) ω α . (E.1b)Here the indices α and i , labelling the fields, are essentially equivalent and originate from thesame algebra gl ( A ). At this point, however, it is useful to consider them as independent. Wealso assume (and this is indeed the case for HSGRA) that the right sides of equations (E.1)define a Lie algebroid with anchor V . In other words, the vector fields V α = V iα ∂/∂C i form anintegrable distribution on a manifold coordinatized by C i :[ V α , V β ] = f γαβ V γ . (E.2)Notice that we do not assume the vector fields V α to be linearly independent in general position.Nevertheless, the Jacobi identity [[ V, V ] , V ] = 0 is presumably satisfied in the strongest form: f λαµ f µβγ + V α f λβγ + cycle ( α, β, γ ) = 0 . (E.3)Let us now examine the integrability conditions for the second equation (3.3b). Applyingthe de Rham differential d to both sides, we find (cid:18) f γαβ ω α ω β − dω γ (cid:19) V γ = 0 . (E.4)If the V α ’s are linearly independent, then the last condition implies the first equation (E.1a).In that case Eq. (E.1b) ‘knows’ about Eq. (E.1a), and we may omit the later without anyconsequences for the system. In the general case, there may be some null-vectors Z αA = Z αA ( C )spanning the kernel of the anchor V , that is, Z αA V iα = 0 . (E.5)Then Eq. (E.4) implies that d ω λ = f λαβ ω α ω β + Z λA B A (E.6)for some collection of 2-forms B A . These 2-forms should be regarded as new independentvariables describing arbitrariness in the dynamics of ω ’s whenever equations (E.1a) are omitted.62he dynamics of B ’s are not completely arbitrary. Checking the integrability condition andassuming the null-vectors Z A to be linearly independent, one can find d B A = ω α U AαB ( C ) B B (E.7)for some structure functions U ’s. In such a way we arrive at the natural extension of the originalsystem (E.1) by the 2-form fields B A subject to (E.7).When applied to 4 d HSGRA, these ideas instruct us to look for null-vectors (E.5). Toleading order in C , one gets then the equation Z ⋆ C − C ⋆ ˜ Z = 0, where ˜ Z ( y, ¯ y ) = Z ( y, − ¯ y )(we deal with the physical fields only). It is easy to see that the last equation has only constantsolutions, Z = const ·
11, which also satisfy the entire equation (E.5).
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