ABJ Triality: from Higher Spin Fields to Strings

# ABJ Triality: from Higher Spin Fields to Strings

Chi-Ming Chang, Shiraz Minwalla, Tarun Sharma, Xi Yin
Center for the Fundamental Laws of Nature, Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Dept. of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Rd, Mumbai 400005, India.
Email:  cmchang@physics.harvard.edu, minwalla@theory.tifr.res.in, tarun@theory.tifr.res.in, xiyin@fas.harvard.edu
###### Abstract:

We demonstrate that a supersymmetric and parity violating version of Vasiliev’s higher spin gauge theory in AdS admits boundary conditions that preserve or supersymmetries. In particular, we argue that the Vasiliev theory with Chan-Paton and boundary condition is holographically dual to the 2+1 dimensional ABJ theory in the limit of large and finite . In this system all bulk higher spin fields transform in the adjoint of the gauge group, whose bulk t’Hooft coupling is . Analysis of boundary conditions in Vasiliev theory allows us to determine exact relations between the parity breaking phase of Vasiliev theory and the coefficients of two and three point functions in Chern-Simons vector models at large . Our picture suggests that the supersymmetric Vasiliev theory can be obtained as a limit of type IIA string theory in AdS, and that the non-Abelian Vasiliev theory at strong bulk ’t Hooft coupling smoothly turn into a string field theory. The fundamental string is a singlet bound state of Vasiliev’s higher spin particles held together by gauge interactions. This is illustrated by the thermal partition function of free ABJ theory on a two sphere at large and even in the analytically tractable free limit. In this system the traces or strings of the low temperature phase break up into their Vasiliev particulate constituents at a deconfinement phase transition of order unity. At a higher temperature of order Vasiliev’s higher spin fields themselves break up into more elementary constituents at a deconfinement temperature, in a process described in the bulk as black hole nucleation.

preprint: TIFR/TH/12-29

## 1 Introduction and Summary

It has long been speculated that the tensionless limit of string theory is a theory of higher spin gauge fields. One of the most important explicit and nontrivial construction of interacting higher spin gauge theory is Vasiliev’s system in . It was conjectured by Klebanov and Polyakov [1], and by Sezgin and Sundell [2, 3], that the parity invariant A-type and B-type Vasiliev theories are dual to 2+1 dimensional bosonic and fermionic or vector models in the singlet sector. Substantial evidence for these conjectures has been provided by comparison of three-point functions [4, 5], and analysis of higher spin symmetries [6, 7, 8, 9].

It was noted in [10, 11] that, at large , the free and theories described above each have a family of one parameter conformal deformations, corresponding to turning on a finite Chern-Simons level for the or gauge group. It was conjectured in [11] that the bulk duals of the resultant Chern-Simons vector models is given by a one parameter family of parity violating Vasiliev theories. In the bulk description parity is broken by a nontrivial phase in function in Vasiliev’s theory that controls bulk interactions. This conjecture appeared to pass some nontrivial checks [11] but also faced some puzzling challenges [11]. In this paper we will find significant additional evidence in support of the proposal of [11] from the study of the bulk duals of supersymmetric vector Chern-Simons theories.

The duality between Vasiliev theory and 3d Chern-Simons boundary field theories does not rely on supersymmetry, and, indeed, most studies of this duality have been carried out in the non-supersymmetric context. However it is possible to construct supersymmetric analogues of the Type A and type B bosonic Vasiliev theories [12, 13, 14, 3, 15, 16]. With appropriate boundary conditions, these supersymmetric Vasiliev theories preserve all higher spin symmetries and are conjectured to be dual to free boundary supersymmetric gauge theories. In the spirit of [11] it is natural to attempt to construct bulk duals of the one parameter set of interacting supersymmetric Chern-Simons vector theories obtained by turning on a finite level for the Chern-Simons terms (recall that Chern Simons coupled gauge fields are free only in the limit ). Interacting supersymmetric Chern-Simons theories differ from their free counterparts in three ways. First, as emphasized above, their Chern-Simons level is taken to be finite. According to the conjecture of [11] this is accounted for by turning on the appropriate phase in Vasiliev’s equations. Second the Lagrangian includes potential terms of the schematic form and Yukawa terms of the schematic form , where and are fundamental and antifundamental scalars and fermions in the field theory. These terms may be regarded as double and triple trace deformations of the field theory; as is well known, the effect of such terms on the dual bulk theory may be accounted for by an appropriate modification of boundary conditions [17]. Lastly, supersymmetric field theories with and supersymmetry necessarily have two gauge groups with matter in the bifundamental. Such theories may be obtained by from theories with a single Chern-Simons coupled gauge group at level and fundamental matter by gauging a global symmetry with Chern-Simons level . In the dual bulk theory this gauging is implemented by a modification of the boundary conditions of the bulk vector gauge field [18].

These elements together suggest that it should be possible to find one parameter families of Vasiliev theories that preserve some supersymmetry upon turning on the parity violating bulk phase, if and only if one also modifies the boundary conditions of all bulk scalars, fermions and sometimes gauge fields in a coordinated way. In this paper we find that this is indeed the case. We are able to formulate one parameter families of parity violating Vasiliev theory (enhanced with Chan-Paton factors, see below) that preserve or supersymmetries depending on boundary conditions. In every case we identify conjectured dual Chern-Simons vector models dual to our bulk constructions.111A similar analysis of the breaking of higher spin symmetry by boundary conditions allows us to demonstrate that all deformations of type A or type B Vasiliev theories break all higher spin symmetries other than the conformal symmetry. We are also able to use this analysis to determine the functional form of the double trace part of higher spin currents that contain a scalar field.

The identification of parity violating Vasiliev theory with prescribed boundary conditions as the dual of Chern-Simons vector models pass a number of highly nontrivial checks. By considering of boundary conditions alone, we will be able to determine the exact relation between the parity breaking phase of Vasiliev theory, and two and three point function coefficients of Chern-Simons vector models at large . These imply non-perturbative relations among purely field theoretic quantities that are previously unknown (and presumably possible to prove by generalizing the computation of correlators in Chern-Simons-scalar vector model of [19] using Schwinger-Dyson 222See [11] for these equations in the Chern-Simons fermion model. equations to the supersymmetric theories). The results also agree with the relation between and Chern-Simons ’t Hooft coupling determined in [11] by explicit perturbative computations at one-loop and two-loop order.

From a physical viewpoint, the most interesting Vasiliev theory presented in this paper is the theory. It was already suggested in [11] that a supersymmetric version of the parity breaking Vasiliev theory in should be dual to the vector model limit of the ABJ theory, that is, a Chern-Simons-matter theory in the limit of large but finite . Since the ABJ theory is also dual to type IIA string theory in with flat -field, it was speculated that the Vasiliev theory must therefore be a limit of this string theory. The concrete supersymmetric Vasiliev system presented in this paper allows us to turn the suggestion of [11] into a precise conjecture for a duality between three distinct theories that are autonomously well defined atleast in particular limits.

The Vasiliev theory, conjectured below to be dual to theory has many elements absent in more familiar bosonic Vasiliev systems. First theory is ‘supersymmetric’ in the bulk. This means that all fields of the theory are functions of fermionic variables which obey Clifford algebra commutation relations (all bulk fields are also functions of the physical spacetime variables () as well as Vasiliev’s twistor variables , , , , as in bosonic Vasiliev theory). Next the star product used in the bulk equations is the usual Vasiliev star product times matrix multiplication in an auxiliary space. The physical effect of this maneuver is to endow the bulk theory with a gauge symmetry under which all bulk fields transform in the adjoint. Finally, for the reasons described above, interactions of the theory are also modified by a bulk phase, and bulk scalars, fermions and gauge fields obey nontrivial boundary conditions that depend on this phase.

The triality between ABJ theory, type IIA string theory on , and supersymmetric parity breaking Vasiliev theory may qualitatively be understood in the following manner. The propagating degrees of freedom of ABJ theory consist of bifundamental fields that we denote by and antibifundamental fields that we will call . A basis for the gauge singlet operators of the theory is given by the traces . As is well known from the study of ABJ duality, these single trace operators are dual to single string states. The basic ‘partons’ (the and fields) out of which this trace is composed are held together in this string state by the ‘glue’ of and gauge interactions.

Let us now study the limit . In this limit the glue that joins type fields to type fields (provided by the gauge group ) is significantly weaker than the glue that joins fields to fields (this glue is supplied by interactions). In this limit the trace effectively breaks up into weakly interacting particles , . These particles, which transform in the adjoint of , are the dual to the adjoint fields of the dual Vasiliev theory. Indeed the spectrum of operators of field theory operators of the form precisely matches the spectrum of bulk fields of the dual Vasiliev system.

If our picture is correct, the fields of Vasiliev’s theory must bind together to make up fundamental IIA strings as is increased. We now describe a qualitative way in which this might happen. The bulk Vasiliev theory has gauge coupling , It follows that the bulk ’t Hooft coupling is . In the limit , the bulk Vasiliev theory is effectively weakly coupled. As increases, a class of multi-particle states of higher spin fields acquire large binding energies due to interactions, and are mapped to the single closed string states in type IIA string theory. Roughly speaking, the fundamental string of string theory is simply the flux tube string of the non abelian bulk Vasiliev theory.

Note that although we claim a family of supersymmetric Vasiliev theory with Chan-Paton factors and certain prescribed boundary conditions is equivalent to string theory on , we are not suggesting that Vasiliev’s equations are the same as the corresponding limit of closed string field equations. Not all single closed string states are mapped to single higher spin particles; infact the only closed strings that are mapped to Vasiliev’s particles are those dual to the operators of the form . Closed string field theory is the weakly interacting theory of the ‘glueball’ bound states of the Vasiliev fields; it is not a weakly interacting description of Vasiliev’s fields themselves.

We have asserted above that the glue between and partons is significantly weaker than the glue between and partons in the limit . This claim may be made quantitatively precise in a calculation in the ABJ theory with taken to be an arbitrary parameter. The computation in question is the partition function of free ABJ field theories on a sphere in the t’Hooft large and limit. We use the fact that the path integral that computes this partition function, even in the limit , is not completely free [20]. This path integral includes the effects of strong interactions between matter and the Polyakov line of and gauge fields. This computation of the partition function is a straightforward application of the techniques described in [20], but yields an interesting result (see Section 7, and see [21, 11] for related earlier work in the context of models with fundamental matter). We find that the theory undergoes two phase transitions as a function of temperature. At low temperature the theory is in a confined phase. This phase may be thought of as a gas of traces of the form , or, roughly, closed strings. Upon raising the temperature the field theory undergoes a first order phase transition at a temperature of order unity. Above the phase transition temperature, group deconfines while the group continues to completely confine333Throughout this paper we assume without loss of generality that . (we make this statement precise below.) The intermediate temperature phase has an effective description in terms of the partition function of a gauge theory whose effective matter degrees of freedom are simply the set of adjoint ‘mesons’ of the form . These adjoint degrees of freedom are deconfined. In other words the traces of the low temperature phase (dual to fundamental strings of ABJ theory) split up into a free gas of smaller - but not yet indivisible units, i.e. the fields of Vasiliev’s theory. Upon further raising the temperature, the theory undergoes yet another phase transition, this time of third order. This transition occurs at a temperature of order and is associated with the complete ‘deconfinement’ of the gauge group . At temperatures much higher than the second phase transition temperature, the system may be thought of as a plasma of the bifundamental and anti-bifundamental letters and . In other words the basic units, , of the intermediate temperature phase, split up into their basic building blocks in the high temperature phase. This extreme high temperature phase is presumably dual to a black hole in the bulk theory. 444In the very high temperature limit, this phase has recently been studied in closely related supersymmetric Chern Simons theories even away from the free limit [22] (generalizing earlier computations in nonsupersymmetric theories in [11].In the special case the intermediate phase never exists; the system directly transits from the string to the black hole phase. The fact that the deconfinement temperature is much smaller than the deconfinement temperature demonstrates that the glue between and type partons is much weaker than than between and type partons. Our computations also strongly suggests that the string dual to ABJ theory has a new finite temperature phase - one composed of a gas of Vasiliev’s particles - even at finite values of .

Let us note a curious aspect of the conjectured duality between Vasiliev’s theory and ABJ theory. The gauge groups and appear on an even footing in the ABJ field theory. In the bulk Vasiliev description, however, the two gauge groups play a very different role. The gauge group is manifest as a gauge symmetry in the bulk. However symmetry is not manifest in the bulk (just as the symmetry is not manifest in the bulk dual of Yang Mills); the dynamics of this gauge group that leads to the emergence of the background spacetime for Vasiliev theory. The deconfinement transition for is simply a deconfinement transition of the adjoint bulk degrees of freedom, while the deconfinement transition for is associated with the very different process of ‘black hole formation’. If our proposal for the dual description is correct, the gauged Vasiliev theory must enjoy an symmetry, which, from the bulk viewpoint is a sort of level – rank duality. Of course even a precise statement for the claim of such a level rank duality only makes sense if Vasiliev theory is well defined ‘quantum mechanically’ (i.e. away from small ) at least in the large limit.

We have noted above that Vasiliev’s theory should not be identified with closed string field theory. There may, however, be a sense in which it might be thought of as an open string field theory. We use the fact that there is an alternative way to engineer Chern-Simons vector models using string theory [23], that is by adding D6-branes wrapped on inside the , which preserves supersymmetry and amounts to adding fundamental hypermultiplets of the Chern-Simons gauge group. In the “minimal radius” limit where we send to zero, with flat -field flux , the geometry is entirely supported by the D6-branes [24].555We thank Daniel Jafferis for making this important suggestion and O. Aharony for related discussions. This type IIA open+closed string theory is dual to Chern-Simons vector model with hypermultiplet flavors. The duality suggests that the open+closed string field theory of the D6-branes reduces to precisely a supersymmetric Vasiliev theory in the minimal radius limit. Note that unlike the ABJ triality, here the open string fields on the D6-branes and the nonabelian higher spin gauge fields in Vasiliev’s system both carry Chan-Paton factors, and we expect one-to-one correspondence between single open string states and single higher spin particle states.

## 2 Vasiliev’s higher spin gauge theory in AdS4 and its supersymmetric extension

The Vasiliev systems that we that we study in this paper are defined by a set of bulk equations of motion together with boundary conditions on the bulk fields. In this section we review the structure of the bulk equations. We turn to the consideration of boundary conditions in the next section.

In this section we first present a detailed review of bulk equations of the ‘standard’ Vasiliev theory. We then describe nonabelian and supersymmetric extensions of these equations. Throughout this paper we work with the so-called non-minimal version of Vasiliev’s equations, which describe the interactions of a field of each non-negative integer spin in . Under the AdS/CFT correspondence non-minimal Vasiliev equations are conjectured to be dual to gauged Chern-Simons-matter boundary theories.666 The non minimal equations admit a consistent truncation to the so-called minimal version of Vasiliev’s equations; this truncation projects out the gauge fields for odd spins and are conjectured to supply the dual to Chern-Simons boundary theories.

There are exactly two ‘standard’ non-minimal Vasiliev theories that preserve parity symmetry. These are the type A/B theories, which are conjectured to be dual to bosonic/fermionic vector models, restricted to the -singlet sector. Parity invariant Vasiliev theories are particular examples of a larger class of generically parity violating Vasiliev theories. These theories appear to be labeled by a real even function of one real variable. In subsection 2.1 we present a review of these theories. It was conjectured in [11] that a class of these parity violating theories are dual to Chern-Simons vector models.

In subsection 2.2 we then present a straightforward nonabelian extension of Vasiliev’s system, by introducing Chan-Paton factors into Vasiliev’s star product. The result of this extension is to promote the bulk gauge field to a gauge field; all other bulk fields transform in the adjoint of . The local gauge transformation parameter of Vasiliev’s theory is also promoted to a local matrix field that transforms in the adjoint of . The nature of the boundary CFT dual to the non abelian Vasiliev theory depends on boundary conditions. With ‘standard’ magnetic type boundary conditions for all gauge fields (that set prescribed values for the field strengths restricted to the boundary) the dual boundary CFT is obtained simply by coupling copies of (otherwise non interacting) matter multiplets to the same boundary Chern-Simons gauge field. The boundary theory has a ‘flavour’ global symmetry that acts on the identical matter multiplets.

In subsection 2.3 we then introduce the so called -extended supersymmetric Vasiliev theory (generalizing the special cases studied earlier in [12, 13, 14, 3, 15]). The main idea is to enhance Vasiliev’s fields to functions of fermionic fields (; we assume to be even) which obey a Clifford algebra777We emphasize that should not be confused with the number of globally conserved supercharges (equivalently is the number of supercharges in the superconformal algebra of the dual three-dimensional CFT). characterizes only the local structure of Vasiliev’s equations of motion. on the other hand depends on the choice of boundary condition for bulk fields of spin , and . As we will see is for parity violating Vasiliev theories, as expected from the dual CFT (, or course, can be arbitrarily large ). . This extension promotes the usual Vasiliev’s fields to dimensional matrices (or operators) that act on the dimensional representation of the Clifford algebra. The local Vasiliev gauge transformations are also promoted to functions of , and so matrices or operators888The bulk equations of motion the extended supersymmetric Vasiliev theory is identical to the theory extended by Chan Paton factors. However, the language of extended supersymmetric Vasiliev theory is more convenient when the boundary conditions of the problem break part of this symmetry, as will be the case later in this paper. . Half of the resultant fields (and gauge transformations) are fermionic; the other half are bosonic.

### 2.1 The standard parity violating bosonic Vasiliev theory

In this section we present the ‘standard’ non minimal Vasiliev equations, allowing, however, for parity violation.

#### 2.1.1 Coordinates

In Euclidean space the fields of Vasiliev’s theory are functions of a collection of bosonic variables . () are an arbitrary set of coordinates on the four dimensional spacetime manifold. and are spinors under while and are spinors under a separate . As we will see below, Vasiliev’s equations enjoy invariance under local (in spacetime) rotations of , , and . This local rotational invariance, which, as we will see below is closely related to the tangent space symmetry of the first order formulation of general relativity, is only a small part of the much larger gauge symmetry of Vasiliev’s theory.

#### 2.1.2 Star Product

Vasiliev’s equations are formulated in terms of a star product. This is just the usual local product in coordinate space; whereas in auxiliary space it is given by

 f(Y,Z)∗g(Y,Z) (2.1) =f(Y,Z)exp[ϵαβ(←∂yα+←∂zα)(→∂yβ−→∂zβ)+ϵ˙α˙β(←∂y˙α+←∂z˙α)(→∂y˙β−→∂z˙β)]g(Y,Z) =∫d2ud2vd2¯ud2¯veuαvα+¯u˙α¯v˙αf(y+u,¯y+¯u,z+u,¯z+¯u)g(y+v,¯y+¯v,z−v,¯z−¯v).

In the last line, the integral representation of the star product is defined by the contour for along in the complex plane, and along the contour . It is obvious from the first line of (2.1) that ; this fact may be used to set the normalization of the integration measure in the second line. The star product is associative but non commutative; in fact it may be shown to be isomorphic to the usual Moyal star product under an appropriate change of variables. In Appendix A.1 we describe our conventions for lowering spinor indices and present some simple identities involving the star product.

Below we will make extensive use of the so called Kleinian operators and defined as

 K=ezαyα,    ¯¯¯¯¯K=e¯z˙α¯y˙α. (2.2)

They have the property (see Appendix A.1 for a proof)

 K∗K=¯¯¯¯¯K∗¯¯¯¯¯K=1, (2.3) K∗f(y,z,¯y,¯z)∗K=f(−y,−z,¯y,¯z),   ¯¯¯¯¯K∗f(y,z,¯y,¯z)∗¯¯¯¯¯K=f(y,z,−¯y,−¯z).

#### 2.1.3 Master fields

Vasiliev’s master fields consists of an -space 1-form

 W=Wμdxμ,

a -space 1-form

 S=Sαdzα+S˙αd¯z˙α,

and a scalar , all of which depend on spacetime as well as the internal twistor coordinates which we denote collectively as . It is sometimes convenient to write and together as a 1-form on -space

 A=W+S=Wμdxμ+Sαdzα+S˙αd¯z˙α.

will be regarded as a gauge connection with respect to the -algebra.

We also define

 ^S =S−12zαdzα−12¯z˙αd¯z˙α, (2.4) ^A =W+^S=A−12zαdzα−12¯z˙αd¯z˙α=Wμdxμ+(−12zα+Sα)dzα+(−12¯z˙α+S˙α)d¯z˙α.

Let be the exterior derivative with respect to the spacetime coordinates and denote by the exterior derivative with respect to the twistor variables . We will write . We will also find it useful to define the field strength

 F=dx^A+^A⋆^A=(dxW+W∗W)+(dx^S+{W,^S}∗)+(^S∗^S). (2.5)

Note also that

 ^S∗^S=dzS+S∗S+14(ϵαβdzαdzβ+ϵ˙α˙βd¯z˙αd¯z˙β). (2.6)

#### 2.1.4 Gauge Transformations

Vasiliev’s master fields transform under a large set of gauge symmetries. We will see later that the vacuum solution partially Higgs or breaks this gauge symmetry group down to a subgroup of large gauge transformations - either the higher spin symmetry group or the conformal group depending on boundary conditions.

Infinitesimal gauge transformations are generated by an arbitrary real function . By definition under gauge transformations

 δ^A=dxϵ+^A∗ϵ−ϵ∗^A,δB=−ϵ∗B+B∗π(ϵ). (2.7)

In other words the 1-form master field transforms as a gauge connection under the star algebra while transforms as a ‘twisted’ adjoint field. The operation that appears in (2.7) is defined as follows

 π(y,z,dz,¯¯¯y,¯¯¯z,d¯¯¯z)=(−y,−z,−dz,¯¯¯y,¯¯¯z,d¯¯¯z)

Since does not involve differentials in , the action of on is equivalent to conjugation by , namely . ( acting on a 1-form in acts by conjugation by together with flipping the sign of ).

It follows from (2.7) that the field strength ( and so each of the three brackets on the RHS of the second line of (2.5)) transform in the adjoint. The same is true of .

 δF=[F,ϵ]∗,δ(B⋆K)=−ϵ∗(B∗K)+(B∗K)∗ϵ, (2.8)

When expanded in components the first line of (2.7) implies that

 δWμ=∂μϵ+Wμ∗ϵ−ϵ∗Wμ,δ^Sα=^Sα∗ϵ−ϵ∗^Sα. (2.9)

In terms of unhatted variables,

 δA=dϵ+A∗ϵ−ϵ∗A,δSα=∂ϵ∂zα+Sα∗ϵ−ϵ∗Sα. (2.10)

#### 2.1.5 Truncation

The following truncation is imposed on the master fields and gauge transformation parameter . Define

 R=K¯¯¯¯¯K.

We require

 [R,W]∗={R,S}∗=[R,B]∗=[R,ϵ]∗=0. (2.11)

More explicitly, this is the statement that , and are even functions of whereas are odd in ,

 Wμ(x,y,¯y,z,¯z)=Wμ(x,−y,−¯y,−z,−¯z),Sα(x,y,¯y,z,¯z)=−Sα(x,−y,−¯y,−z,−¯z),S˙α(x,y,¯y,z,¯z)=−S˙α(x,−y,−¯y,−z,−¯z),B(x,y,¯y,z,¯z)=B(x,−y,−¯y,−z,−¯z).ϵ(x,y,¯y,z,¯z)=ϵ(x,−y,−¯y,−z,−¯z). (2.12)

A physical reason for the imposition of this truncation is the spin statistics theorem. As the physical fields of Vasiliev’s theory are all commuting, they must also transform in the vector (rather than spinor) conjugacy class of the tangent group; the projection (2.12) ensures that this is the case. One might expect from this remark that the consistency of Vasiliev’s equations requires this truncation; we will see explicitly below that this is the case.

#### 2.1.6 Reality Conditions

It turns out that Vasiliev’s master fields admit two consistent projections that may be used to reduce their number of degrees of freedom. These two projections are a generalized reality projection (somewhat analogous to the Majorana condition for spinors) and a so called ‘minimal’ truncation (very loosely analogous to a chirality truncation for spinors). These two truncations are defined in terms of two natural operations defined on the master field; complex conjugation and an operation defined by the symbol . In this subsection we first define these two operations, and then use them to define the generalized reality projection. We will also briefly mention the minimal projection, even though we will not use the later in this paper.

Vasiliev’s fields master fields admit a straightforward complex conjugation operation, , defined by complex conjugating each of the component fields of Vasiliev theory and also the spinor variables 999As complex conjugation of interchanges left and right moving spinors, our definition of complex conjugation (the analytic continuation of the Lorentzian notion) must also have this property.

 (yα)∗=¯y˙α,    (¯y˙α)∗=yα,    (zα)∗=¯z˙α,    (¯z˙α)∗=zα. (2.13)

It is easily verified that

 (M∗N)∗=M∗∗N∗ (2.14)

where is an arbitrary form and and arbitrary form. In other words complex conjugation commutes with the star and wedge product, without reversing the order of either of these products. Note also that the complex conjugation operation squares to the identity.

We now turn to the definition of the operation ; this operation is defined by

 ι: (y,¯y,z,¯z,dz,d¯z)→(iy,i¯y,−iz,−i¯z,−idz,−id¯z), (2.15)

The signs in (2.15) are chosen101010Changing the RHS of (2.15) by an overall sign makes no difference to fields that obey (2.12) to ensure

 ι(f∗g)=ι(g)∗ι(f) (2.16)

(see (A.7) for a proof). In other words reverses the order of the product. Note however that by definition does not affect the order of wedge products of forms. As a consequence picks up an extra minus sign when acting on the product of two oneforms

 ι(C∗D)=−ι(D)∗ι(C)

(see (A.8) for a proof; the same equation is true if is a form and a form provided and are both odd; if atleast one of and is even we have no minus sign).

We now define the generalized reality projection that we will require Vasiliev’s master fields to obey throughout this paper (this projection defines the non-minimal Vasiliev theory which we study through this paper). The projection is defined by the conditions

 ι(W)∗=−W,   ι(S)∗=−S,   ι(B)∗=¯¯¯¯¯K∗B∗¯¯¯¯¯K=K∗B∗K (2.17)

The equality of the two different expressions supplied for in (2.17) follows upon using the fact commutes with (see (2.11)).

It is easily verified that (2.17) implies that

 ι(F)∗=−F (2.18)

(see (A.13) for an expansion in components) and that

 ι(B∗K)∗=B∗¯¯¯¯¯K,   ι(B∗¯¯¯¯¯K)∗=B∗K. (2.19)

(2.17) may be thought of as a combination of two separate projections. The first is the ‘standard’ reality projection (see (A.9)). The second is the ‘minimal truncation’(A.10). As discussed in Appendix 2.12, it is consistent to simultaneously impose invariance of Vasiliev’s master field under both these projections. This operation defines the minimal Vasiliev theory (dual to Chern-Simons field theories). We will not study the minimal theory in this paper.

#### 2.1.7 Equations of motion

Vasiliev’s gauge invariant equations of motion take the form

 F=dx^A+^A∗^A=f∗(B∗K)dz2+¯¯¯f∗(B∗¯¯¯¯¯K)d¯z2, (2.20) dxB+^A∗B−B∗π(^A)=0.

where is a holomorphic function of , its complex conjugate, and the corresponding -function of . Namely, is defined by replacing all products of in the Taylor series of by the corresponding star products.

Note that both sides of the first of (2.20) are gauge adjoints, while the second line of that equation transforms in the twisted adjoint. In Appendix A.4 we have demonstrated that the second equation of (2.20) may be derived from the first (assuming that is a non-degenerate function) using the Bianchi identity

 dxF+[A,F]∗=0 (2.21)

In Appendix A.4 we have also expanded Vasiliev’s equations in components to clarify their physical content. As elaborated in (A.14) and (A.15), it follows from (2.20) that the field strength is flat and that the adjoint fields , and are covariantly constant. In addition, various components of these adjoint fields commute or anticommute with each other under the star product (see Appendix A.24 for a listing). The fields and , however, fail to commute with each other; their commutation relations are given by

 [^Sα,^Sβ]∗=ϵαβf∗(B∗K)[^S˙α,^S˙β]∗=ϵ˙α˙β¯f∗(B∗¯K) (2.22)

Using various formulae presented in the Appendix (see e.g. (A.11)) it is easily verified that the Vasiliev equations, (expanded in the Appendix as (A.14) and (A.15)) map to themselves under the reality projection (2.17). The same is true of the minimal truncation projection.

#### 2.1.8 Equivalences from field redefinitions

Vasiliev’s equations are characterized by a single complex holomorphic function . In this subsection we address the following question: to what extent to different functions label different theories?

Any field redefinition that preserves the gauge and Lorentz transformation properties of all fields, but changes the form of clearly demonstrates an equivalence of the theories with the corresponding choices of . The most general field redefinitions consistent with gauge and Lorentz transformations and the form of Vasiliev’s equations are

 B→g∗(B∗K)∗KˆSz≡(−12zα+Sα)dzα→ˆSz∗h∗(B∗K),ˆS¯z≡(−12¯z˙α+¯S˙α)∗d¯z˙α→ˆS¯z∗~h∗(−B∗¯¯¯¯¯K). (2.23)

Several comments are in order. First note that the field redefinitions above obviously preserve form structure and gauge transformations properties. In particular these redefinitions preserve the fact that , and transform in the adjoint representation of the gauge group. Second the field redefinitions above are purely holomorphic (e.g. is a function only of but not of ). It is not difficult to convince oneself that this is necessary in order to preserve the holomorphic form of Vasiliev’s equations. Finally we have chosen to multiply the redefined functions and with functions from the right rather than the left. There is no lack of generality in this, however, as

 ˆSz∗h∗(B∗K)=h∗(−B∗K)∗ˆSz,    ˆSz∗¯¯¯h∗(B∗¯¯¯¯¯K)=¯¯¯h∗(B∗¯¯¯¯¯K)∗ˆSz, (2.24) ˆS¯z∗h∗(B∗K)=h∗(B∗K)∗ˆS¯z,    ˆS¯z∗¯¯¯h∗(B∗¯¯¯¯¯K)=¯¯¯h∗(−B∗¯¯¯¯¯K)∗ˆS¯z,

((2.24) follows immediately from (A.24) derived in the Appendix). Finally, we have inserted a minus sign into the argument of the function for future convenience.

The reality conditions (2.17) impose constraints on the functions , and . It is not difficult to verify that is forced to be an odd real function . is forced to be odd because the complex conjugation operation turns into . When is odd, however, the truncation (2.11) may be used to turn back into . For instance, with , the field redefinition is

 B→g1B+g3B∗K∗B∗K∗B+⋯ (2.25)

The RHS is still real because (it would not be real if were not odd).

In order to examine the constraints of (2.17) on the functions and note that

 ι(Sz∗h(B∗K)+S¯z∗~h∗(−B∗¯¯¯¯¯K))∗=¯¯¯h(B∗¯¯¯¯¯K)∗(−S¯z)+¯¯¯~h(−B∗K)∗(−Sz)=−(S¯z∗¯¯¯h(−B∗¯¯¯¯¯K)+Sz∗¯¯¯~h(B∗K)) (2.26)

(where in the last step we have used (2.24)). It follows that the redefined function obeys the reality condition (2.17) if and only if

 ~h=¯¯¯h

where is the complex conjugate of the function .

The effect of the field redefinition of is simply to permit a redefinition of the argument of the function in Vasiliev’s equations by an arbitrary odd real function. The effect of the field redefinition of may be deduced as follows. The component of Vasiliev’s - the assertion that is a flat connection (see (A.14)) - is clearly preserved by this field redefinition. The components of the equation asserts that and are covariantly constant. As and are also covariantly constant (see (A.15)) the redefinition (2.23) clearly preserves this equation as well. However the components of the equations become

 ˆSz∗h∗(B∗K)∗ˆSz∗h∗(B∗K)=f∗(B∗K)dz2, (2.27) {ˆS∗h∗(B∗K),ˆS¯z∗¯¯¯h∗(−B∗¯¯¯¯¯K)}∗=0, ˆS¯z∗¯¯¯h∗(−B∗¯¯¯¯¯K)∗ˆS¯z∗¯¯¯h∗(−B∗¯¯¯¯¯K)=¯¯¯f∗(B∗¯¯¯¯¯K)d¯z2.

Using (2.24) and the fact that commutes with (this is obvious as and commute), these equations may be recast as

 h∗(−B∗K)∗(ˆSz∗ˆSz)∗h∗(B∗K)=f∗(B∗K)dz2, (2.28) h∗(−B∗K)∗({ˆS,ˆS¯z}∗)∗¯¯¯h∗(−B∗¯¯¯¯¯K)=0, ¯¯¯h∗(B∗¯¯¯¯¯K)∗(ˆS¯z∗ˆS¯z)∗¯¯¯h∗(−B∗¯¯¯¯¯K)=¯f∗(B∗¯¯¯¯¯K)d¯z2.

(2.28) is precisely the component of the Vasiliev equation (the third equation in (A.14) ) with the replacement

 f∗(X)→h∗(−X)−1∗f∗(X)∗h∗(X)−1, (2.29)

or simply .

So we see that the theory is really defined by up to a change of variable for some odd real function and multiplication by an invertible holomorphic even function. Provided that the function admits a power series expansion about and that ,111111This condition can probably be weakend, but cannot be completely removed. For example if is an odd function, it is easy to convince oneself that it cannot be cast into the form (2.30). In this paper we will be interested in the Vasiliev duals to field theories. In the free limit, the dual Vasiliev theories to the field theory in question are given by of the form (2.30) with . It follows that, atleast in a power series in the field theory coupling, the Vasiliev duals to the corresponding field theories are defined by an that can be put in the form (2.30). in Appendix A.6 we demonstrate that we can can use these field redefinitions to put in the form

 f(X)=14+Xexp(iθ(X)) (2.30)

where is an arbitrary real even function.

Ignoring the special cases for which cannot be cast into the form (2.30), the function determines the general parity-violating Vasiliev theory.

While Vasiliev’s system is formulated in terms of a set of background independent equations, the perturbation theory is defined by expanding around the vacuum. In order to study this solution it is useful to establish some conventions. Let and () denote the usual vielbein and spin connection one-forms on any space (the index transforms under the vector representation of the tangent space ). We define the corresponding bispinor objects

 eα˙β=14eaσaα˙β,    wαβ=116wabσabαβ,    w˙α˙β=−116wab¯σab˙α˙β. (2.31)

(see Appendix A.7 for definitions of the matrices that appear in this equation.) Let and be the vielbein and spin connection of Euclidean with unit radius. It may be shown that (see Appendix A.8 for some details)

 A=W0(x|Y)≡e0(x|Y)+ω0(x|Y) (2.32) =(e0)α˙βyα¯y˙β+(ω0)αβyαyβ+(ω0)˙α˙β¯y˙α¯y˙β,   B=0.

solves Vasiliev’s equations. We refer to this solution as the vacuum (as we will see below this preserves the invariance of space).

In the sequel we will find it convenient to work with a specific choice of coordinates and a specific choice of the vielbein field. For the metric on AdS space we work in Poincaré coordinates; the metric written in Euclidean signature takes the form

 ds2=d→x2+dz2z2, (2.33)

We also define the vielbein oneform fields

 ei0=−dxiz,   e40=−dzz (2.34)

( runs over the index and ). The corresponding spin connection one form fields are given by

 wab0=dxi4z[Tr(σizσab)+Tr(¯σiz¯σab)] (2.35)

Using (2.31) we have explicitly

 ω0(x|Y)=−18dxiz(yσizy+¯y¯σiz¯y), (2.36) e0(x|Y)=−14dxμzyσμ¯y.

Here our convention for contracting spinor indices is , etc (see Appendix A.7).