Generalized BRST symmetry for arbitrary spin conformal field theory

We develop the finite field-dependent BRST (FFBRST) transformation for arbitrary spin-s conformal field theories. We discuss the novel features of the FFBRST transformation in these systems. To illustrate the results we consider the spin-1 and spin-2 conformal field theories in two examples. Within the formalism we found that FFBRST transformation connects the generating functionals of spin-1 and spin-2 conformal field theories in linear and non-linear gauges. Further, the conformal field theories in the framework of FFBRST transformation are also analysed in Batalin-Vilkovisky (BV) formulation to establish the results.


Introduction
Conformal field theories (CFT) [1] have been at the center of much attention during the last seventeen years mainly because they provide models for genuinely interacting quantum field theories, they describe two-dimensional critical phenomena, and they play a central role in string theory, at present the most promising candidate for a unifying theory of all forces. Much attention has been given to conformal field theories in higher dimensions due to their role in the AdS/CFT correspondence [2,3]. AdS 3 /CFT 2 is one of the most hot topics nowadays as it may be amenable to the integrability approach that proved very successful especially in the case of AdS 5 /CFT 4 [4]. The AdS/CFT correspondence has also been investigated for scalar fields [5][6][7], gauge fields [7], spinors [8], classical gravity [9] and type IIB string theory [10,11]. The AdS/CFT correspondence is used to calculate CFT correlators from the classical AdS theories of vector and Dirac fields and the connection between the AdS and boundary fields is properly treated via a Dirichlet boundary value problem [6].
Recently, in the framework of gauge invariant approach involving Stueckelberg fields the totally symmetric arbitrary spin-s anomalous conformal current and shadow field are studied and gauge invariant two-point vertex of the arbitrary spin anomalous shadow field is also obtained [12]. In Stueckelberg gauge frame, the two-point gauge invariant vertex becomes the standard two-point vertex of CFT. The logarithmic divergence of the BRST invariant action of arbitrary spin-s canonical shadow field turns out to be BRST invariant action of arbitrary spin-s conformal field [13]. The BRST invariant action of conformal field interprets geometrically the boundary values of massless AdS fields [13]. The study of BRST quantization which helps in proving the renormalizability of gauge theories is extremely important in the context of CFT.
Although BRST symmetry has been discussed for conformal field theory [13], the generalization of it by making the parameter field-dependent, so-called FFBRST transformation, has not yet been investigated. The FFBRST formulation, which was introduced for the first time by Joglekar and Mandal [14], has been studied considerably in various contexts [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. For example, such formulation helps in calculating a correct prescription for poles in the gauge field propagators in noncovariant gauges by connecting the covariant gauges and noncovariant gauges of the theory [15,18]. The celebrated Gribov problem [29,30] of QCD has also been addressed through FFBRST transformation in Euclidean space [20]. Further, such formulation has been investigated for YM theory explaining low-energy dynamics via Cho-Faddeev-Niemi (CFN) decomposition. So, it is worth analyzing such formulation at both classical and quantum levels for conformal field theories. This provides a motivation for the analysis of FFBRST transformation in conformal field theory in present investigation.
We further like to extend our FFBRST formulation for CFT in the framework of Batalin-Vilkovisky (BV) formalism [31][32][33][34][35] which is one of the most powerful techniques to study gauge field theories and allows us to deal with very general gauge theories, including those with open or reducible gauge symmetry algebras. The BV method provides a convenient way of analyzing the possible violations of symmetries by quantum effects [32]. It is usually used to perform the gauge-fixing in quantum field theory, but was also applied to other problems like analyzing possible deformations of the action and anomalies. The BRST-BV approach is successful for studying the manifestly Lorentz invariant formulation of string theory [36].
In this paper we generalize the FFBRST transformation for arbitrary spin-s conformal field theory by making the parameter finite and field dependent. Within the formulation, we find that the functional measure leads to a non-trivial Jacobian. This Jacobian can be exponentiated if it satisfies a certain condition. As a result the effective action gets modified. We compute the Jacobians for spin-1 and spin-2 conformal fields for particular choices of finite field-dependent parameters. We render that these calculated Jacobians play an important role in mapping of linear and nonlinear gauges. The analyzed BV formulation validates the results at quantum level. For BV formulation we extend the configuration space by introducing antifield corresponding to each field with opposite statistics. With such introduction of antifield the consequent extended action satisfies the mathematically rich quantum master equation.
The paper is presented in the following manner. In Section 2, we generalize the BRST transformation for arbitrary spin-s conformal field theory. We illustrate this generalization by two examples of spin-1 and spin-2 conformal fields in Section 3. We extend this formulation in the BV framework in Section 4. At the end we summarize the results.

Constructing FFBRST transformation for arbitrary spin-s conformal field theory
In this section we construct the FFBRST transformation for conformal field theory following the method advocated in [14]. Let us begin with the effective action for arbitrary spin-s conformal field theory defined by 1 [13], where   (4) where the [ϕ a 1 a 2 .....a s (x, κ)] is an infinitesimal but the fielddependent parameter. The FFBRST transformation (denoted by δ f ) then can be obtained by integrating the above transformation from κ = 0 to κ = 1, as follows: is the finite field-dependent parameter and f [φ] is given by This FFBRST transformation leaves effective action of a conformal field theories invariant. However, the functional measure changes non-trivially under such finite transformation. Now we compute the Jacobian of the path integral measure defined generically by (Dϕ a 1 a 2 .....a s ) for an arbitrary finite field- The Jacobian J (κ) of the path integral measure is thus obtained as a functional of fields. So we exponentiate it by defining a local functional S 1 [ϕ a 1 a 2 .....a s ] in the following manner: Preserving the quantitative (physical) changes of the functional integral in conformal field theory leads to the following condition [14] Dϕ The local functional S 1 [ϕ a 1 a 2 .....a s ] satisfies the following initial boundary condition The infinitesimal change in Jacobian, J (κ), given in (10), has the explicit expression in terms of as follows where, for bosonic fields, + sign is used and − for fermionic fields.
Therefore, performing FFBRST transformation changes the exponential action of the generating functional given in conformal field theory as following: Dϕ a 1 a 2 .....a s e i S tot −→ Dϕ a 1 a 2 .....a s e i(S tot +S 1 ) , (12) where S tot is the most general effective action for CFT given in (1). To illustrate these results we would like to consider specific examples in the next sections.

BRST invariant conformal fields
In this section, we consider the two examples of BRST symmetric conformal field theory. We study the construction and implementation of FFBRST transformation on these theories explicitly.

Spin-1 conformal field
The BRST invariant action for spin-1 conformal field (a particular form of (1)) in linear gauge is given by where field-strength Here φ a , b, c and c are spin-1 conformal field, Nakanishi-Lautrup field, ghost field and antighost field, respectively. In terms of gauge-fixing fermion the above action can be described by where transformations of the fields are The generating functional for spin-1 conformal field theory corresponding to (13) is defined by However, the BRST invariant action for spin-1 conformal field in non-linear (quadratic) gauge is given by which remains invariant under the same set of BRST transformations given in (15). Following the method given in Section 2, we construct the FFBRST transformation as follows: where [ϕ a 1 ] is an arbitrary finite field-dependent BRST parameter.
Now, we construct a particular [ϕ a 1 ] to calculate the Jacobian for path integral measure whose infinitesimal version is evaluated as follows Now we calculate the change in Jacobian with respect to continuous parameter κ as follows (20) where we have utilized the relation (11).
To exponentiate the Jacobian we propose the following local functional where ξ 1 and ξ 2 are κ-dependent arbitrary constant parameters.
Eqs. (20) and (21) together with (10) yield the following linear differential equations: The exact solutions of the above equations satisfying the boundary With these identifications the expression of local functional becomes This is evident from above expression that at κ = 0 the functional S 1 vanishes. However, at κ = 1 this takes the following form: So, according to (12), after performing the FFBRST transformation on generating functional the effective action (13) modifies by Therefore, we observe that the FFBRST transformation on generating functional of spin-1 conformal theory in linear gauge changes the effective action from linear gauge to quadratic gauge within functional integral. Here we note that the FFBRST transformation amounts the precise change on the BRST exact part of the effective action. We construct the finite parameter in such a manner that Jacobian of the path integral measure amounts change in the BRST-exact part of the effective action.

Spin-2 conformal field
The classical action for spin-2 conformal field theory (a particular form of (1)) is given by where R ab is expressed by The gauge-fixing and ghost action is given together by So, the complete action is given by which is invariant under following BRST transformation: where δλ is infinitesimal, anticommuting parameter. The FFBRST transformation is constructed by To construct the finite field-dependent parameter [ϕ a 1 a 2 ] we choose the following infinitesimal parameter: The change in Jacobian under FFBRST transformation is calculated by Keeping the forms of effective action in linear and quadratic gauges in mind we make an ansatz for S 1 in this case as follows The essential condition (10) together with (34) and (35) yields the following differential equations for ξ i : The exact solutions of these differential equations satisfying boundary condition (ξ i (κ = 0) = 0) are given by With this solutions the expression of S 1 reduces to which vanishes for κ = 0. However, for κ = 1 this reduces to Now, after performing the FFBRST transformation the extended action for spin-2 conformal field, as mentioned in (12), is calculated by Here we observe that the final action obtained in (40) has nonlinear gauge. Therefore, we observed that the FFBRST transformation relates the generating functionals corresponding to linear and non-linear gauges for spin-2 conformal field also.

Conformal field theory in BV formulation
In this section, we extend the formulation using BV technique. For this purpose, we need to introduce the antifields (ϕ a 1 a 2 .....a s ) corresponding to fields having opposite statistics in the configuration space. With the introduction of such antifields, the arbitrary extended quantum action, W [ϕ a 1 a 2 .....a s , ϕ a 1 a 2 .....a s ], satisfies a certain rich mathematical relation, the so-called quantum master equation [32], which is given by ϕ a 1 a 2 .....a s ,ϕ a 1 a 2 .....a s ] = 0, where A ≡ (a 1 a 2 .....a s ). Therefore, the extended quantum action W with different gauge-fixing fermion are solutions of the quantum master equation. We would like to show that FFBRST transformation with appropriate choice of finite field-dependent parameter relates different solutions of quantum master equation.

Spin-1 conformal field
In terms of field and antifields, the generating functional for the spin-1 conformal field theory in linear gauge is defined by where φ a L and c L are antifields corresponding to the φ a and c fields respectively with opposite statistics. The above generating functional can further be recast compactly as  ≡ φ a , b, c, c).
It is well-known that the antifields for a gauge theory can explicitly be computed from the gauge-fixed fermion. For the conformal theory in linear gauge the antifields are computed for the With these identifications of antifields the extended quantum action in (42) coincides with the total effective action (13). However, for the non-linear gauge the gauge-fixing fermion is given by The antifields for the above gauge-fixing fermion are estimated by: Likewise the linear gauge case, the generating functional for the spin-1 conformal theory in non-linear gauge can be written in compact form as where W NL [ϕ a 1 , ϕ a 1 NL ] is an extended quantum action (another solution of the quantum master equation) corresponding to nonlinear gauge. Now we construct the infinitesimal field/antifield dependent parameter as follows 2 From this infinitesimal parameter the finite field/antifield dependent parameter can be calculated using the relation (6). The FFBRST transformation with such field/antifield dependent parameter leads to the following Jacobian in the integrand of functional integral which switches the generating functional of spin-1 conformal theory from one gauge to anther. Therefore, we establish the connection of the different solutions (W L and W NL ) of the quantum master equation at quantum level through FFBRST transformation with appropriately constructed finite field-dependent parameter.

Spin-2 conformal field
Introducing the antifields corresponding to fields, the generating functional for the spin-2 conformal field theory in linear gauge is defined by 2 We note in this case that the antifields depend on fields as these are expressed in terms of gauge-fixing fermion. Therefore this field/antifield dependent parameter actually depends on field only [37].
where ϕ a 1 a 2 are antifields corresponding to the ϕ a 1 a 2 (≡ φ ab , φ a , φ, b a , b, c a , c, c a , c) fields generically with opposite statistics. This can further be written in compact notation as where W L [ϕ a 1 a 2 , ϕ a 1 a 2 L ] is the extended quantum action for spin-2 conformal theory in linear gauge.
In the same fashion, we define the generating functional for the spin-2 conformal theory for non-linear gauge in compact form as where W NL [ϕ a 1 a 2 , ϕ a 1 a 2 NL ] is the extended quantum action corresponding to non-linear gauge.
We construct the infinitesimal field/antifield dependent parameter for this case as follows: The finite field/antifield dependent parameter can be evaluated from relation (6). The FFBRST transformation with such field/antifield dependent parameter leads to the following Jacobian in the integrand of functional integral which transforms the generating functional of spin-2 conformal theory from linear gauge to non-linear. Hence, the connection of the different solutions (W L and W NL ) of the quantum master equation for spin-2 is established through FFBRST transformation with properly constructed parameter. In fact, any two solutions of quantum master equation are connected through FFBRST transformation with different finite parameter.

Conclusions
In this paper we have developed the FFBRST transformation for arbitrary spin-s conformal field theory. We construct the FFBRST transformation by making the transformation parameter finite and field-dependent. The parameter is made finite and field-dependent by making all the fields first (a continuous constant parameter) κ-dependent and then define an infinitesimal field-dependent BRST transformation. After that we integrate the parameter of infinitesimal field-dependent BRST transformation in the limiting values of κ which yields the finite field-dependent BRST parameter.
The novelty of the FFBRST transformation is that it leads to a local Jacobian for path integral measure and this Jacobian amounts a change in the BRST exact part of the effective action. Here we note that analogous to ordinary (non-conformal) quantum field theories the resulting Jacobians in the case of conformal field theories are still local in nature. This assures the consistency of generalized BRST formulation for CFTs also. For illustration purpose, we have considered the spin-1 and spin-2 conformal theories. For such theories we have explicitly constructed the specific finite fielddependent parameters. Furthermore, we have found that the Jacobians corresponding to such parameters switch the theories from one gauge to another (namely, linear to non-linear gauges). Furthermore, we have established the theory at quantum level by analyzing it through BV formulation. In BV formulation we have demonstrated that the finite field dependent BRST transformation connects the different solutions of quantum master equation for both spin-1 and spin-2 conformal theories. Thus our formulation will be helpful in estimating the observables of the conformal theory in different gauges.