ABJM theory in Batalin-Vilkovisky formulation

We analyze the quantum ABJM theory on ${\cal N} = 1$ superspace in different gauges. We study the Batalin-Vilkovisky (BV) formulation for this model. By developing field/antifield dependent BRST transformation we establish connection between the two different solutions of the quantum master equation within the BV formulation.


I. INTRODUCTION
The Aharony-Bergman-Jafferis-Maldacena (ABJM) theory is a conformal field theory in three dimensional spacetime. The ABJM theory with gauge group U (N ) × U (N ) is represented by N M 2-branes and has been constructed recently [1,2]. More precisely, it is shown that N = 6 supersymmetric Chern-Simons quiver gauge theory with bifundamental matter enjoying SO(4) flavor symmetry is dual to M -theory compactified on AdS 4 × S 7 /Z k , and describes the low energy dynamics of a stack of M 2-branes probing an orbifold singularity. This theory only has N = 6 supersymmetry but it is expected to be enhanced to the full N = 8 supersymmetry [3]. The M2-brane branes ending on M9-branes and gravitational waves have also been studied [4].
It may be noted that as the ABJM theory has gauge symmetry, it cannot be quantized without getting rid of these unphysical degrees of freedom. This can be done by fixing a gauge. The gauge fixing condition can be incorporated at a quantum level by adding ghost and gauge fixing terms to the original classical Lagrangian. It is known that for a gauge theory the new effective Lagrangian constructed as the sum of the original classical Lagrangian with the gauge fixing and the ghost terms, is invariant under a new set of transformations called the BRST transformations [5,6]. BRST symmetry has also been studied in non-linear gauges [7,8].
In this work we discuss the ABJM theory from the perspective of gauge theory by discussing different gauge conditions. We investigate the different effective actions corresponding to the different gauge choices. We establish the BRST symmetry for the theory using two Grassmann parameters. Furthermore, the general BV quantization of the model has been analyzed. We generalize the BRST symmetry of the model by making the parameters field/antifield dependent. We compute the resulting Jacobian coming from the functional measure of the general generating functional. We find that for a particular choice of field/antifield dependent parameters, (equations 36 and 37) the different gauges of ABJM theory can be connected. This result will be helpful to interrelate computations of physical quantities of the ABJM theory in linear and non-linear gauges.
The paper is presented in following way. In Sec. II, we analyze the classical ABJM theory in N = 1 superspace from the gauge symmetric point of view. Sec. III is devoted to describe the quantum analysis by studying different gauge conditions. The BV formalism is developed for ABJM theory in section IV, which widens the quantization scheme. In Sec. V, we developed a mapping between different solutions of extended quantum action using the techniques of field/antifield dependent BRST symmetry. The results are summarized in the last section.
where k is an integer playing the role of a coupling constant. ω a andω a have following expression: , The D a represents the super-derivative defined as and ′ | ′ means that the quantity is evaluated at θ a = 0. In component form the gauge connections Γ a and Γ a are expressed as The explicit expression for the Lagrangian density of the matter fields is given by where Now, the classical Lagrangian density for ABJM theory with the gauge group U (N ) × U (N ) on N = 1 superspace is given by, which remains covariant under the following gauge transformations: with the local parameters ξ andξ. The super-covariant derivatives ∇ a and∇ a are defined by

III. GAUGE CONDITIONS AND BRST SYMMETRY
In this section, we investigate the quantum action for ABJM theory in linear and non-linear gauges. The nilpotency of BRST symmetry is also demonstrated for this theory.

A. Linear gauge
Being gauge invariant, the non-Abelian Chern-Simons theory on N = 1 superspace contains some redundant degrees of freedom. To quantize the theory correctly we need to choose a gauge. The covariant (Lorentz-type) gauge fixing conditions for ABJM theory are These gauge fixing conditions can be incorporated in the theory at the quantum level by adding the following gauge fixing term to the original Lagrangian density, where b 1 andb 1 are the Nakanishi-Lautrup auxiliary fields. The Faddeev-Popov ghost terms corresponding to the above gauge fixing term is constructed as Now, we define the full quantum action for ABJM theory in Lorentz-type gauge by writing the gaugefixing and the ghost terms collectively with classical action The BRST transformations, which leaves the above effective action invariant, are written by where Λ andΛ are the infinitesimal anticommuting parameters of transformation.

B. Non-linear gauge
We start this subsection by demonstrating the ABJM theory in non-linear gauge as follows The above Lagrangian density can be obtained by performing the following shift in the Nakanishi-Lautrup auxiliary fields The BRST transformation under which the effective action in non-linear gauge remains invariant is given by The effective action is also found invariant under the another set of BRST symmetry where roles of ghost and anti-ghost fields are interchanged, called as anti-BRST transformation and given by The above BRST and anti-BRST transformations are nilpotent as well as absolutely anticommuting, i.e.
The gauge-fixing and ghost terms of the ABJM model in non-linear gauge can be expressed in terms of BRST and anti-BRST exact terms as follows In the next section we analyze the theory in BV formulation.

IV. ABJM THEORY IN BV FORMULATION
To establish the theory in BV formulation we need to introduce antifields corresponding to fields with opposite statistics. In terms of fields/antifields, the generating functional for the ABJM theory in Lorentz-type gauge is, where W L is the extended quantum action and integration dv refers to d 3 x. The gauge-fixed fermion for ABJM theory in Lorentz gauge is defined by, With the help of this gauge-fixed fermion we compute the antifields for the Lorentz gauge as following: However, the generating functional for ABJM in the non-linear gauge in terms of fields/antifields is given by, We evaluate the expression for the gauge-fixing fermion for the non-linear gauge as following: The antifields in this case are identified as, We note the difference between the two extended quantum actions as follows, The extended quantum actions, W Ψ [Φ, Φ ⋆ ] ≡ (W N L , W L ), satisfies certain rich mathematical relation so-called quantum master equation, which is given by Here we note that the extended quantum actions W N L and W L are two different possible solutions of the quantum master equation.
In the next section, our goal would be to establish a map between the two generating functionals corresponding to the above extended actions using the technique of field/antifield dependent BRST transformations.

V. A MAPPING BETWEEN SOLUTIONS OF QUANTUM MASTER EQUATION
We first analyze the field/antifield dependent BRST transformation which is characterized by the field/antifield dependent BRST parameter. To achieve the goal, we define the usual BRST transformation for the generic fields Φ α (x) andΦ α (x) written compactly as where R α (x)(s b Φ α (x)) andR α (x)(s bΦα (x)) are the Slavnov variations of the field Φ α (x) andΦ α (x) satisfying δ b R α (x) = δ bRα (x) = 0. Here the infinitesimal transformation parameters Λ andΛ are the Grassmann parameters and don't depend on any field/antifield. Now, we present the field/ antifield dependent BRST transformation as follows where the Grassmann parameters Λ[Φ, Φ ⋆ ] andΛ[Φ,Φ ⋆ ] depend on the field/antifield explicitly. The field/antifield dependent BRST transformation for the ABJM theory is constructed by making the transformation parameter of (14) and (17) field/antifield dependent. Though being symmetry of the extended action such field/antifield dependent transformation is not nilpotent any more. We notice that under such transformation the path integral measure of generating functional changes non-trivially. We compute the the change in the generating functional as follows, Furthermore, the Jacobian matrix appearing above for the field/antifield dependent BRST transformation is given by Utilizing (32) and the nilpotency of the BRST transformation (i.e. s 2 b = 0) we obtain the following relation [27] Because of the anticommuting nature of Λ[Φ, Φ ⋆ ] the determinant simplifies to Plugging this value of determinant in the relation (31) we get Now we can use the antifield expressions (23), (26) and the linear and non-linear BRST transformations (14), (17) to complete the computation. There are eight terms in the parentheses, let us calculate some of them. Firstly, we calculate The second term leads to However, the third term is computed as, Now, utilizing the Slavnov variation of (17) we have, Putting the values of (42) back in (41) gives The fourth term is calculated by,c Putting together (39), (40), (43) and (44) we obtain the following expression Following a similar computation we have for Therefore, it is easy to see from the equations (27), (35),(38),(45) and (46) that Hence we have shown that under field/antifield dependent BRST transformation with the appropriate choice of parameters (36) and (37), the different solutions of the quantum master equation can be related.

VI. CONCLUSION
In this paper we have established the ABJM theory at quantum level by investigating it in the BV formulation on N = 1 superspace. For this purpose, we have extended the configuration space by introducing the antifields corresponding to the fields of ABJM model. Further, we have calculated the exact values of antifields by choosing the suitable gauge-fixing fermion. We have mainly discussed the Lorentz-type and Curci-Ferrari type gauges from the BRST quantization perspectives. The quantum master equation for the ABJM theory, having different possible solutions, is also established. Furthermore, we have generalized the BRST symmetry of the theory by developing the field/antifield dependent parameters. Here we need two parameters of transformation rather than one. We have also successfully demonstrated how a particular choice of the transformation parameters can relate two different generating functionals in the Lorentz-type and the Curci-Ferrari type gauges.
Our analysis on BV formulation of ABJM theory will provide a convenient way to study the possible violations of the symmetries of the action by quantum effects. Such analysis may also be useful in calculating the S-matrix of the theory because we have already computed the definite values of antifields. The master equation discussed above is more fundamental than the Zinn-Justin equation which guarantees the renormalizability of the ABJM theory, since the master equation relies on the fundamental action rather than the quantum effective action. The present investigation is a step towards the study of the deformations of the action and anomalies.