aa r X i v : . [ h e p - t h ] S e p The quantum description of BF model in superspace
Manoj Kumar Dwivedi ∗ Department of Physics, Institute of Science,Banaras Hindu University,Varanasi-221005, India.
We consider the BRST symmetric four dimensional BF theory, a topological theory, containingantysymmetric tensor fields in Landau gauge and extend the BRST symmetry by introducing ashift symmetry to it. Within this formulation, the antighost fields corresponding to shift symmetrycoincide with antifields of standard field/antifield formulation. Further, we provide a superspacedescription for the BF model possessing extended BRST and extended anti-BRST transformations.
I. INTRODUCTION
Topological gauge field theories (TGFT) which came from mathematics have some peculiar features.The examples of two distinct class of TGFT are topological Yang-Mills theory and Chern-Simons (CS)theory, which are some times classified as Witten-type and of Schwarz-type respectively [1]. Except thesetwo types, there are another Schwarz-type TGFT called topological BF theory, which is an extension ofCS theory [2]. The difference between CS theory and BF model is that action of previous theory existonly in odd-dimensions while later one can be defined on manifolds of any dimensions.In string theory and non-linear sigma model, four dimensional antisymmetric (or BF) model [3] wereintroduced some years ago. This model is interested due to its topological nature [1] and their connectionwith lower dimensional quantum gravity, for example three space-time dimensional Einstein-Hilbert withor without using cosmological constant can be naturally formulated in terms of BF-models[4, 5]. Couplingof an antisymmetric tensor field with the field strength tensor of Yang-Mills is describe by these models[6]. Quantization of BF model in Landau gauge has been studied in Ref. [6]. Topological BF theory inLandau gauge has a common feature of a large class of topological models [7, 8].On the other hand, the Batalin-Vilkovisky (BV) approach, also known as field/antifield formulation,[9–12] is one of the most powerful quantization algorithms presently available. BV formulation deals withvery general gauge theories, including those with open or reducible gauge symmetry algebras. The BVmethod also address the possible violations of symmetries of the action by quantum effects. The BVformulation (independently introduced by Zinn-Justin [13]) extends the BRST approach [14]. In fact,the BRST symmetry [15, 16] is a very important symmetry for gauge theories [17]. Beside the covariantdescription to perform the gauge-fixing in quantum field theory, BV formulation was also applies to otherproblems like analysing possible deformations of the action and anomalies.A superspace description for various gauge theories in BV formulation has been studied extensively[18–22]. They have shown that the extended BRST and extended anti-BRST invariant actions of thesetheories (including some shift symmetry) in BV formulation yield naturally the proper identification ofthe antifields through equations of motion. The shift symmetry is important and gets relevance, forexample, in inflation particularly in supergravity [23] as well as in Standard Model [24]. In usual BVformulation, these antifields can be calculated from the expression of gauge-fixing fermion. We extendedBRST formulation and superspace description of the topological gauge (BF) model is still unstudied andwe try to discuss these here.In the present work, we try to generalize the superspace formulation of BV action for BF model.Particularly, we first consider BRST invariant BF model in Landau gauge and extend the BRST symmetry ∗ Electronic address: [email protected] of the theory by including shift symmetry. By doing so, we find that the antighosts of shift symmetry getidentified as antifields of standard BV formulation naturally. Further, we discuss a superspace formulationof extended BRST invariant BF model. Here we see that one additional Grassmann coordinate is requiredif action admits only extended BRST symmetry. However, for both extended BRST and extended anti-BRST invariant BF model two additional Grassmann coordinates are required.This paper is framed as follows. In section II, we discuss the BRST invariant BF model. In sectionIII, we study the extended BRST transformation of the model. Further, we describe extended BRSTinvariant action in superspace in section IV. The extended anti-BRST symmetry is discussed in sectionV. The superspace formulation of extended BRST and anti-BRST invariant action is given in section VI.The last section is reserved for concluding remarks.
II. BRST INVARIANT BF MODEL
In this section, we discuss the preliminaries of BF model with its BRST invariance. In this view, theBF model in flat (3 + 1) space-time dimensions is given by the following gauge invariant Lagrangiandensity [6]: L = − ǫ µνρσ F aµν B aρσ , (1)where B aρσ and F aµν are two-form field and field-strength tensor for vector field respectively. In order toremove discrepancy due to gauge symmetry, the gauge fixing and ghost terms are given by L gf + gh = b a ∂ µ A aµ + ¯ C a ∂ µ ( D µ C ) a + h aν ( ∂ µ B aµν ) + ω a ∂ µ ξ aµ + h aµ ( ∂ µ e a )+ ω a λ a + ( ∂ µ ¯ ξ aµ ) λ a − ( ∂ µ ¯ φ a )[( D µ φ ) a + f abc c b ξ cµ ] − ( ∂ µ ¯ ξ aν )[( D µ ξ ν ) a − ( D ν ξ µ ) a + f abc B bµν C c ]+ 12 f abc ǫ µνρσ ( ∂ µ ¯ ξ aν )( ∂ ρ ¯ ξ bσ ) φ c , (2)where fields ( C a , ξ aµ ), ( ¯ C a , ¯ ξ aµ ) and ( b a , h aµ ) are the ghosts, antighosts and the multipliers fields respectively,while the fields φ a , ¯ φ a and ω a are taken into account to remove further degeneracy due to the existenceof zero modes in the transformations.The effective Lagrangian density of BF model, L = L + L gf + gh , possesses following BRST symmetry: sA aµ = − ( D µ C ) a , sC a = 12 f abc C b C c , sξ aµ = ( D µ φ ) a + f abc C b ξ cµ ,sB aµν = − ( D µ ξ ν − D ν ξ µ ) a − f abc B bµν C c + f abc ǫ µνρσ ( ∂ ρ ¯ ξ bσ ) φ c sφ a = f abc C b φ c , s ¯ ξ aµ = h aµ , s ¯ C a = b a , s ¯ φ a = ω a , se a = λ a ,s ( h aµ , b a , ω a , λ a ) = 0 . (3)The gauge-fixing and ghost terms of the effective Lagrangian density is BRST exact and, hence, can bewritten in terms of BRST variation of gauge-fixing fermion,Ψ = (cid:0) ¯ C a ∂ µ A aµ + ¯ ξ aµ ∂ ν B aµν + ¯ φ a ∂ µ ξ aµ − e a ω a − e a ∂ µ ¯ ξ aµ (cid:1) , (4)as follows L gf + gh = s Ψ . (5)In the next section, we would like to study the extended BRST symmetry for the model which incorporatesshift symmetry together with original BRST symmetry. III. EXTENDED BRST INVARIANT LAGRANGIAN DENSITY
The advantage of studying the extended BRST transformations for BF model in BV formulation isthat antifields get identification naturally. We begin with shifting all the fields from their original valueas follows, B aµν −→ B aµν − ˜ B aµν , A aµ −→ A aµ − ˜ A aµ , C a −→ C a − ˜ C a , ¯ C a −→ ¯ C a − ˜¯ C a , b a −→ b a − ˜ b a , ξ aµ −→ ξ aµ − ˜ ξ aµ , ¯ ξ aµ −→ ¯ ξ aµ − ˜¯ ξ aµ , φ a −→ φ a − ˜ φ a , ¯ φ a −→ ¯ φ a − ˜¯ φ a , h aµ −→ h aµ − ˜ h aµ ,e a −→ e a − ˜ e a , ω a −→ ω a − ˜ ω a , λ a −→ λ a − ˜ λ a . (6)The effective Lagrangian density of BF model also get shifted under such shifting of fields respectively.This is given by ˜ L = L ( A aµ − ˜ A aµ , C a − ˜ C a , ¯ C a − ˜¯ C a , b a − ˜ b a , ξ aµ − ˜ ξ aµ , ¯ ξ aµ − ˜¯ ξ aµ ,φ a − ˜ φ a , ¯ φ a − ˜¯ φ a , h aµ − ˜ h aµ , e a − ˜ e a , ω a − ˜ ω a , λ a − ˜ λ a ) . (7)The shifted Lagrangian density is invariant under BRST transformation together with a shift symme-try transformation, jointly known as extended BRST transformation. The extended BRST symmetrytransformations under which Lagrangian density of BF model is invariant are written by sA aµ = ψ aµ , s ˜ A aµ = ψ aµ − ( D µ − ˜ D µ )( C − ˜ C ) a , sC a = ǫ a , s ˜ C a = ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c ) ,s ˜¯ C a = ¯ ǫ a − ( b − ˜ b ) a , sb a = χ a , s ˜ b a = χ a , sφ a = M a , s ˜ φ a = M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c ) ,s ¯ φ a = ¯ M a , s ˜¯ φ a = M a − ( ω a − ˜ ω a ) , se a = N a , s ˜ e a = N a − ( λ a − ˜ λ a ) , sξ aµ = L aµ , s ¯ C a = ¯ ǫ a ,s ˜ ξ aµ = L aµ − [( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ )] , s ¯ ξ aµ = ¯ L aµ , s ˜¯ ξ aµ = ¯ L aµ − ( h aµ − ˜ h aµ ) , (8)where ψ aµ , ǫ a , ¯ ǫ a , χ a , M a , ¯ M a , N a , L aµ , and ¯ L aµ are the ghost fields corresponding to shift symmetry for A aµ , C a , ¯ C a , b a , φ a , ¯ φ a , e a , ξ aµ and ¯ ξ aµ respectively. The nilpotency of extended BRST symmetry (8) leadsto the BRST transformation for the following ghost fields: sψ aµ = 0 , sǫ a = 0 , s ¯ ǫ a = 0 , sχ a = 0 , sM a = 0 ,s ¯ M a = 0 , sN a = 0 , sL aµ = 0 , s ¯ L aµ = 0 . (9)In order to make the theory ghost free, we need further antighosts A ⋆aµ , C ⋆a , ¯ C ⋆a , b ⋆a , ξ ⋆aµ , ¯ ξ ⋆aµ , φ ⋆a , ¯ φ ⋆a , and e ⋆a to be introduced corresponding to the ghost fields ψ aµ , ǫ a , ¯ ǫ a , χ a , M a , ¯ M a , N a , L aµ , and ¯ L aµ respectively.The BRST transformations of these antighosts are constructed as follows sA ⋆aµ = − ζ aµ , sC ⋆a = − σ a , s ¯ C ⋆a = − ¯ σ a , sb ⋆a = − ̟ a , sφ ⋆a = − υ a ,s ¯ φ ⋆a = − ¯ υ a , se ⋆a = − τ a , sξ ⋆aµ = − κ aµ , s ¯ ξ ⋆aµ = − ¯ κ aµ , (10)where ζ aµ , σ a , ¯ σ a , ̟ a , υ a , τ a , κ aµ , and ¯ κ aµ are the Nakanishi-Lautrup type auxiliary fields corresponding toshifted fields ˜ A aµ , ˜ C a , ˜¯ C a , ˜ b a , ˜ φ a , ˜¯ φ a , ˜ e a , ˜ ξ aµ , and ˜¯ ξ aµ having following BRST transformations: sζ aµ = 0 , sσ a = 0 , s ¯ σ a = 0 , s̟ a = 0 , sυ a = 0 ,s ¯ υ a = 0 , sτ a = 0 , sκ aµ = 0 , s ¯ κ aµ = 0 . (11)We can recover our original BF model by fixing the shift symmetry in such a way such that effect of all thetilde fields will vanish. We achieve this by adding following gauge-fixed term to the shifted Lagrangiandensity (7):˜ L gf + gh = − ζ aµ ˜ A aµ − A a⋆µ [ ψ aµ − ( D µ − ˜ D µ )( C − ˜ C ) a ] − ¯ σ a ˜ C a + ¯ C a⋆ [ ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c )] − σ a ˜¯ C a + C a⋆ [¯ ǫ a − ( b a − ˜ b a )] − υ a ˜ φ a − φ ⋆a h M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c ) i − ¯ υ a ˜¯ φ a − ¯ φ ⋆a ( ¯ M a − ω a + ˜ ω a ) − τ a ˜ e a − e ⋆ [ N a − ( λ a − ˜ λ a )] − ̟ a ˜ b a − b ⋆a χ a − κ µa ˜ ξ aµ − ¯ κ µa ˜¯ ξ aµ + ξ ⋆µa (cid:16) L aµ − h ( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ ) i(cid:17) + ¯ ξ ⋆µa h ¯ L aµ − ( h aµ − ˜ h aµ ) i . (12)One can easily check that this gauge-fixing Lagrangian density ˜ L gf + gh also admits the extended BRSTinvariance. Integrating the auxiliary fields of the above expression, we obtain˜ L gf + gh = − A a⋆µ [ ψ aµ − ( D µ C ) a ] + ¯ C a⋆ [ ǫ a − f abc C b C c ]+ C a⋆ [¯ ǫ a − b a ] − φ ⋆a ( M a − f abc C b φ c ) − ¯ φ ⋆a ( ¯ M a − ω a ) − e ⋆a [ N a − λ a ] − b ⋆a χ a + ξ ⋆µa (cid:0) L aµ − [( D µ φ ) a + f abc C b ξ cµ ] (cid:1) + ¯ ξ ⋆µa [ ¯ L aµ − h aµ ] . (13)The gauge-fixing and ghost terms of the Lagrangian density are BRST exact and can be expressed interms of a general gauge-fixing fermion Ψ as s Ψ = sA aµ δ Ψ δA aµ + sC a δ Ψ δC a + s ¯ C a δ Ψ δ ¯ C a + sb a δ Ψ δb a + sξ aµ δ Ψ δξ aµ + s ¯ ξ aµ δ Ψ δ ¯ ξ aµ + sφ a δ Ψ δφ a + s ¯ φ a δ Ψ δ ¯ φ a + se a δ Ψ δe a , = − δ Ψ δA aµ ψ aµ + δ Ψ δC a ǫ a + δ Ψ δ ¯ C a ¯ ǫ a − δ Ψ δb a χ a − δ Ψ δξ aµ L aµ − δ Ψ δ ¯ ξ aµ ¯ L aµ − δ Ψ δφ a M a − δ Ψ δ ¯ φ a ¯ M a − δ Ψ δe a N a . (14)After integrating out the auxiliary fields which set the tilde fields to zero, we have the complete effectiveaction for BF model in landau gauge possessing extended BRST symmetry as L eff = L + L gf + gh + ˜ L gf + gh , = L + (cid:18) − A ⋆aµ − δ Ψ δA µa (cid:19) ψ µa + (cid:18) ¯ C ⋆a + δ Ψ δC a (cid:19) ǫ a + (cid:18) C ⋆a + δ Ψ δ ¯ C a (cid:19) ¯ ǫ a − (cid:18) b ⋆a + δ Ψ δb a (cid:19) χ a + (cid:18) ξ ⋆aµ + δ Ψ δξ aµ (cid:19) L µa + ¯ ξ ⋆aµ + δ Ψ δ ¯ ξ aµ ! ¯ L µa − (cid:18) φ ⋆a + δ Ψ δφ a (cid:19) M a − (cid:18) ¯ φ ⋆a + δ Ψ δ ¯ φ a (cid:19) ¯ M a + (cid:18) − e ⋆a − δ Ψ δe a (cid:19) N a + A a⋆µ [( D µ C a ) − ¯ C a⋆ f abc C b ξ µc ] + C ⋆a b a + ξ ⋆µa [( D µ φ ) a + f abc C b ξ cµ + φ ⋆a f abc C b ξ cµ ] . (15)Integrating out the ghost fields associated with shift symmetry, we obtain A a⋆µ = − δ Ψ δA µa , ¯ C ⋆a = − δ Ψ δC a , C ⋆a = − δ Ψ δ ¯ C a , b ⋆a = − δ Ψ δb a ,ξ ⋆aµ = − δ Ψ δξ ⋆aµ , ¯ ξ ⋆aµ = − δ Ψ δ ¯ ξ ⋆aµ , φ ⋆a = − δ Ψ δφ a , ¯ φ ⋆a = − δ Ψ δ ¯ φ a , e ⋆a = − δ Ψ δe a . (16)For a particular choice of gauge-fixing fermion Ψ given in (4), anti-ghost fields get following identifications: A a⋆µ = ∂ µ ¯ C a , ¯ C a⋆ = 0 , C a⋆ = − ∂ µ A µa , b a⋆ = 0 , ξ ⋆aµ = ∂ µ ¯ φ a , ¯ ξ ⋆aµ = − ∂ ν B aµν − ∂ µ e a , φ ⋆a = 0 , ¯ φ ⋆a = − ∂ µ ξ aµ , e ⋆a = ω a + ∂ µ ¯ ξ aµ . (17)It is obvious to see that with these anti-ghost fields, the expression (15) changes to the original Lagrangiandensity of the BF model in Landau gauge. IV. EXTENDED BRST INVARIANT SUPERSPACE DESCRIPTION
In this section, the Lagrangian density of BF model which is invariant under the extended BRSTtransformations only is described in a superspace ( x µ , θ ), where θ is a Grassmann coordinate and x µ isthe four dimanesional spect-time coordinates. In order to give superspace description for the extendedBRST invariant theory, we first define superfields of the form: A aµ ( x, θ ) = A aµ + θψ aµ , ˜ A aµ ( x, θ ) = ˜ A aµ + θ [ ψ µ − ( D µ − ˜ D µ )( C − ˜ C )] a ,χ a ( x, θ ) = C a + θǫ a , ˜ χ a ( x, θ ) = ˜ C a + θ [ ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c )] , ¯ χ a ( x, θ ) = ¯ C a + θ ¯ ǫ a , ˜¯ χ a ( x, θ ) = ˜¯ C a + θ [¯ ǫ a − ( b − ˜ b ) a ] ,b a ( x, θ ) = b a + θχ a , ˜ b a ( x, θ ) = ˜ b a + θχ a , ξ aµ ( x, θ ) = ξ aµ + θL aµ , ˜ ξ aµ ( x, θ ) = ˜ ξ aµ + θ [ L aµ − [( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ )]] , ¯ ξ aµ ( x, θ ) = ¯ ξ aµ + θ ¯ L aµ , ˜¯ ξ aµ ( x, θ ) = ˜¯ ξ aµ + θ h ¯ L aµ − ( h aµ − ˜ h aµ ) i , φ a ( x, θ ) = φ a + θM a , ˜ φ a ( x, θ ) = ˜ φ a + θ [ M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c )] , ¯ φ a ( x, θ ) = ¯ φ a + θ ¯ M a , ˜¯ φ a ( x, θ ) = ˜¯ φ a + θ [ ¯ M a − ( ω a − ˜ ω a )] , e a ( x, θ ) = e a + θN a , ˜ e a ( x, θ ) = ˜ e a + θ [ N a − ( λ a − ˜ λ a )] . (18)The super-antifields in superspace are defined as follows˜ A ⋆aµ ( x, θ ) = A ⋆aµ − θζ aµ , ˜ χ ⋆a ( x, θ ) = C ⋆a − θσ a , ˜¯ χ ⋆a ( x, θ ) = ¯ C ⋆a − θ ¯ σ a , ˜ b ⋆a ( x, θ ) = b ⋆a − θ̟ a , ˜ ξ ⋆aµ ( x, θ ) = ξ ⋆aµ − θκ aµ , ˜¯ ξ ⋆aµ ( x, θ ) = ¯ ξ ⋆aµ − θ ¯ κ aµ , ˜ φ ⋆a ( x, θ ) = φ ⋆a − θυ a , ˜¯ φ ⋆a ( x, θ ) = ¯ φ ⋆a − θ ¯ υ a , ˜ e ⋆a ( x, θ ) = e ⋆a − θτ a . (19)From the above expressions of superfields and super-antifields, we calculate δ ( ˜ A a⋆µ ˜ A aµ ) δθ = − A a⋆µ [ ψ aµ − ( D µ − ˜ D µ )( C − ˜ C ) a ] − ζ aµ ˜ A aµ ,δ (˜¯ χ a⋆ ˜ χ a ) δθ = ¯ C a⋆ [ ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c )] − ¯ σ a ˜ C a ,δ ( ˜¯ χ a ˜ χ a⋆ ) δθ = − σ a ˜¯ C a + C a⋆ [¯ ǫ a − ( b a − ˜ b a )] ,δ (˜ b a⋆ ˜ b a ) δθ = − b a⋆ χ a − ̟ a ˜ b a ,δ ( ˜ ξ a⋆µ ˜ ξ aµ ) δθ = ξ µa⋆ [ L aµ − [( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ )]] − κ µa ˜ ξ aµ ,δ ( ˜¯ ξ a⋆µ ˜¯ ξ aµ ) δθ = ¯ ξ a⋆µ [ L µa − h µa + ˜ h µa ] − ¯ κ µa ˜¯ ξ aµ ,δ ( ˜ φ a⋆ ˜ φ a ) δθ = − ˜ φ a υ a − φ a⋆ [ M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c )] ,δ ( ˜¯ φ a⋆ ˜¯ φ a ) δθ = − ˜¯ φ a ¯ υ a − ¯ φ a⋆ [ ¯ M a − ω a + ˜ ω a ] ,δ (˜ e a⋆ ˜ e a ) δθ = − e ⋆ [ N a − ( λ a − ˜ λ a )] − ˜ e a τ a . (20)Adding all the equations of (20) side by side, we get δδθ ( ˜ A a⋆µ ˜ A aµ + ˜¯ χ a⋆ ˜ χ a + ˜¯ χ a ˜ χ a⋆ + ˜ b a⋆ ˜ b a + ˜ ξ a⋆µ ˜ ξ aµ + ˜¯ ξ a⋆µ ˜¯ ξ aµ + ˜ φ a⋆ ˜ φ a + ˜¯ φ a⋆ ˜¯ φ a + ˜ e a⋆ ˜ e a )= − ζ aµ ˜ A aµ − A a⋆µ [ ψ aµ − ( D µ − ˜ D µ )( C − ˜ C ) a ] − ¯ σ a ˜ C a + ¯ C a⋆ [ ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c )] − σ a ˜¯ C a + C a⋆ [¯ ǫ a − ( b a − ˜ b a )] − υ a ˜ φ a − φ ⋆a h M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c ) i − ¯ υ a ˜¯ φ a − ¯ φ ⋆a ( ¯ M a − ω a + ˜ ω a ) − τ a ˜ e a − e ⋆a [ N a − ( λ a − ˜ λ a )] − ̟ a ˜ b a − b ⋆a χ a − κ µa ˜ ξ aµ − ¯ κ µa ˜¯ ξ aµ + ξ ⋆µa (cid:16) L aµ − h ( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ ) i(cid:17) + ¯ ξ ⋆µa h ¯ L aµ − ( h aµ − ˜ h aµ ) i , (21)which is nothing but the gauge-fixed Lagrangian density for shift symmetry ˜ L gf + gh given in (12). Now,one can define the general super-gauge-fixing fermion in superspace as followsΦ( x, θ ) = Ψ( x ) + θ ( s Ψ) , (22)which can further be expressed asΦ( x, θ ) = Ψ( x ) + θ " − δ Ψ δA aµ ψ aµ + δ Ψ δC a ǫ a + δ Ψ δ ¯ C a ¯ ǫ a − δ Ψ δb a χ a − δ Ψ δξ aµ L aµ − δ Ψ δ ¯ ξ aµ ¯ L aµ − δ Ψ δφ a M a − δ Ψ δ ¯ φ a ¯ M a − δ Ψ δe a N a (cid:21) . (23)From this, the original gauge-fixing Lagrangian density can be defined as the left derivation of super-gauge-fixing fermion with respect to θ as h δ Φ( x,θ ) δθ i .Hence, the complete effective action for the BF model in general gauge in the superspace is now givenby L eff = L + δδθ h ˜ A a⋆µ ˜ A aµ + ˜¯ χ a⋆ ˜ χ a + ˜¯ χ a ˜ χ a⋆ + ˜ b a⋆ ˜ b a + ˜ χ a⋆µ ˜ χ aµ + ˜¯ χ a⋆µ ˜¯ χ aµ + ˜ φ a⋆ ˜ φ a + ˜¯ φ a⋆ ˜¯ φ a + ˜ e a⋆ ˜ e a + Φ] . (24)Next, we will study the extended anti-BRST symmetry for BF model. V. EXTENDED ANTI-BRST LAGRANGIAN DENSITY
In this section, we construct the extended anti-BRST transformation under which the shifted La-grangian density for BF model remains invariant as follows,¯ sA aµ = A a⋆µ + ( D µ − ˜ D µ )( ¯ C − ˜¯ C ) a , ¯ s ˜ A aµ = A a⋆µ , ¯ sC a = C a⋆ − f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ ) , ¯ s ˜ C a = C a⋆ , ¯ s ¯ C a = ¯ C a⋆ − ( b a − ˜ b a ) , ¯ s ˜¯ C a = ¯ C a⋆ , ¯ sb a = b a⋆ + χ a , ¯ s ˜ b a = b a⋆ , ¯ sξ aµ = ξ a⋆µ − h ( D µ − ˜ D µ )( φ a − ˜ φ a ) + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ ) i , ¯ s ˜ ξ aµ = ξ a⋆µ , ¯ s ¯ ξ aµ = ¯ ξ a⋆µ − h aµ + ˜ h aµ , ¯ s ˜¯ ξ aµ = ¯ ξ a⋆µ , ¯ sφ a = φ a⋆ − f abc ( C b − ˜ C b )( φ c − ˜ φ c ) , ¯ s ˜ φ a = φ a⋆ , ¯ s ¯ φ a = ¯ φ a⋆ − ω a + ˜ ω a , ¯ s ˜¯ φ a = ¯ φ a⋆ , ¯ se a = e a⋆ − ( λ a − ˜ λ a ) , ¯ s ˜ e a = e a⋆ . (25)The ghost fields associated with the shift symmetry transform under extended anti-BRST symmetry as¯ sψ aµ = ζ aµ , ¯ sǫ a = σ a , ¯ s ¯ ǫ a = ¯ σ a , ¯ sχ a = ̟ a , ¯ sL aµ = κ aµ , ¯ s ¯ L aµ = ¯ κ aµ , ¯ sM a = υ a , ¯ s ¯ M a = ¯ υ a , ¯ sN a = τ a . (26)From the nilpotency of above transformations demands that the auxiliary and antighost fields associatedwith the shift symmetry transform as¯ sζ aµ = 0 , ¯ sA a⋆µ = 0 , ¯ sσ a = 0 , ¯ sC a⋆ = 0 , ¯ s ¯ σ a = 0 , ¯ s ¯ C a⋆ = 0 , ¯ s̟ a = 0 , ¯ sb a⋆ = 0 , ¯ sκ aµ = 0 , ¯ sξ a⋆µ = 0 , ¯ s ¯ κ aµ = 0 , ¯ s ¯ ξ a⋆µ = 0 , ¯ sυ a = 0 , ¯ sφ a⋆ = 0 , ¯ s ¯ υ a = 0 , ¯ s ¯ φ a⋆ = 0 , ¯ sτ a = 0 , ¯ se a⋆ = 0 . (27)The gauge-fixing and ghost parts of the effective Lagrangian density are anti-BRST-exact also so itcan be expressed as the anti-BRST variation of this gauge-fixing fermion ( ¯Ψ). VI. EXTENDED BRST AND ANTI-BRST INVARIANT SUPERSPACE
The extended BRST and anti-BRST invariant Lagrangian density for BF model can be written insuperspace with the help of two additional Grassmannian coordinates θ and ¯ θ . Requiring the fieldstrength to vanish along unphysical directions θ and ¯ θ direction, we obtain the following superfields: A aµ ( x, θ, ¯ θ ) = A aµ ( x ) + θψ aµ + ¯ θ [ A a⋆µ + ( D µ − ˜ D µ )( ¯ C − ˜¯ C ) a ] + θ ¯ θζ aµ , ˜ A aµ ( x, θ, ¯ θ ) = ˜ A aµ ( x ) + θ [ ψ aµ − ( D µ − ˜ D µ )( C − ˜ C ) a ] + ¯ θA a⋆µ + θ ¯ θζ aµ , C a ( x, θ, ¯ θ ) = C a ( x ) + θǫ a + ¯ θ [ C a⋆ − f abc ( C b − ˜ C b )( C c − ˜ C c )] + θ ¯ θσ a , ˜ C a ( x, θ, ¯ θ ) = ˜ C a ( x ) + θ [ ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c )] + ¯ θC ⋆a + θ ¯ θσ a , ¯ C a ( x, θ, ¯ θ ) = ¯ C a ( x ) + θ ¯ ǫ a + ¯ θ [ ¯ C ⋆a − ( b − ˜ b ) a ] + θ ¯ θ ¯ σ a , ˜¯ C a ( x, θ, ¯ θ ) = ˜¯ C a ( x ) + θ [¯ ǫ a − ( b − ˜ b ) a ] + ¯ θ ¯ C ⋆a + θ ¯ θ ¯ σ a , b a ( x, θ, ¯ θ ) = b a ( x ) + θχ a + ¯ θ ( b ⋆a + χ a ) + θ ¯ θ̟ a , ˜ b a ( x, θ, ¯ θ ) = ˜ b a ( x ) + θχ a + ¯ θb ⋆a + θ ¯ θ̟ a ,ξ aµ ( x, θ, ¯ θ ) = ξ aµ ( x ) + θL aµ + ¯ θ (cid:16) ξ a⋆µ − h ( D µ − ˜ D µ )( φ a − ˜ φ a ) + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ ) i(cid:17) + θ ¯ θκ aµ , ˜ ξ aµ ( x, θ, ¯ θ ) = ˜ ξ aµ ( x ) + θ [ L aµ − ( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ )] + ¯ θξ a⋆µ + θ ¯ θκ aµ , ¯ ξ aµ ( x, θ, ¯ θ ) = ¯ ξ aµ ( x ) + θ ¯ L aµ + ¯ θ ( ¯ ξ ⋆aµ − h aµ + ˜ h aµ ) + θ ¯ θ ¯ κ aµ , ˜¯ ξ aµ ( x, θ, ¯ θ ) = ˜¯ ξ aµ ( x ) + θ ( ¯ L aµ − h aµ + ˜ h aµ ) + ¯ θ ¯ ξ ⋆aµ + θ ¯ θ ¯ κ aµ ,φ a ( x, θ, ¯ θ ) = φ a ( x ) + θM a + ¯ θ (cid:16) φ a⋆ − f abc ( C b − ˜ C b )( φ c − ˜ φ c ) (cid:17) + θ ¯ θυ a , ˜ φ a ( x, θ, ¯ θ ) = ˜ φ a ( x ) + θ ( M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c )) + ¯ θφ ⋆a + θ ¯ θυ a , ¯ φ a ( x, θ, ¯ θ ) = ¯ φ a ( x ) + θ ¯ M a + ¯ θ ( ¯ φ ⋆a − ω a + ˜ ω a ) + θ ¯ θ ¯ υ a , ˜¯ φ a ( x, θ, ¯ θ ) = ˜¯ φ a ( x ) + θ ( ¯ M a − ω a + ˜ ω a ) + ¯ θ ¯ φ ⋆a + θ ¯ θ ¯ υ a , e a ( x, θ, ¯ θ ) = e a ( x ) + θN a + ¯ θ [ e ⋆a − ( λ a − ˜ λ a )] + θ ¯ θτ a , ˜ e a ( x, θ, ¯ θ ) = ˜ e a ( x ) + θ [ N a − ( λ a − ˜ λ a )] + ¯ θe ⋆a + θ ¯ θτ a . (28)With these expressions of superfields, we can calculate − ∂∂ ¯ θ ∂∂θ (cid:16) ˜ A aµ ˜ A µa + ˜ χ a ˜¯ χ a + ˜ b a ˜ b a + ˜ ξ aµ ˜ ξ µa + ˜¯ ξ aµ ˜¯ ξ µa + ˜ φ a ˜ φ a + ˜¯ φ a ˜¯ φ a + ˜ e a ˜ e a (cid:17) = − ζ aµ ˜ A aµ − A a⋆µ [ ψ aµ − ( D µ − ˜ D µ )( C − ˜ C ) a ] − ¯ σ a ˜ C a + ¯ C a⋆ [ ǫ a − f abc ( C b − ˜ C b )( C c − ˜ C c )] − σ a ˜¯ C a + C a⋆ [¯ ǫ a − ( b a − ˜ b a )] − υ a ˜ φ a − φ ⋆a h M a − f abc ( C b − ˜ C b )( φ c − ˜ φ c ) i − ¯ υ a ˜¯ φ a − ¯ φ ⋆a ( ¯ M a − ω a + ˜ ω a ) − τ a ˜ e a − e ⋆ [ N a − ( λ a − ˜ λ a )] − ̟ a ˜ b a − b ⋆a χ a − κ µa ˜ ξ aµ − ¯ κ µa ˜¯ ξ aµ + ξ ⋆µa (cid:16) L aµ − h ( D µ − ˜ D µ )( φ − ˜ φ ) a + f abc ( C b − ˜ C b )( ξ cµ − ˜ ξ cµ ) i(cid:17) + ¯ ξ ⋆µa h ¯ L aµ − ( h aµ − ˜ h aµ ) i , (29)which is nothing but the gauge-fixed Lagrangian density for shift symmetry. Being the θ ¯ θ componentof a super field, this Lagrangian density is manifestly invariant under both the extended BRST and theanti-BRST transformations.Now, we define the general super-gauge-fixing fermion in superspace asΦ( x, θ, ¯ θ ) = Ψ( x ) + θ ( s Ψ) + ¯ θ (¯ s Ψ) + θ ¯ θ ( s ¯ s Ψ) , (30)which yields the original gauge-fixing and ghost part of the effective effective Lagrangian density upondifferentiation as follows, Tr (cid:2) ∂∂θ (cid:2) δ (¯ θ )Φ( x, θ, ¯ θ ) (cid:3)(cid:3) .Therefore, the gauge-fixed Lagrangian density corresponding to BRST and shift symmetries for BFmodel can now be given as L gf + gh + ˜ L gf + gh = − ∂∂ ¯ θ ∂∂θ (cid:16) ˜ A aµ ˜ A µa + ˜ χ a ˜¯ χ a + ˜ b a ˜ b a + ˜ ξ aµ ˜ ξ µa + ˜¯ ξ aµ ˜¯ ξ µa + ˜ φ a ˜ φ a + ˜¯ φ a ˜¯ φ a + ˜ e a ˜ e a (cid:17) + ∂∂θ (cid:2) s (¯ θ )Φ( x, θ, ¯ θ ) (cid:3) . (31)Therefore, we see that the BF model in superspace can be expressed in an elegant manner. VII. CONCLUSION
The (3 + 1) dimensional BF model is subject of great interest due to its topological nature and itssome intriguing properties. In present work, we have considered (3 + 1) dimensional BF model in Landaugauge and then we have shifted the Lagrangian to obtain the extended BRST and anti-BRST invariant(including some shift symmetry) BF model in BV formulation. The antifields corresponding to each fieldnaturally arises. Further we have provide the superfield description of BF model in superspace, where weshow that the BV action for BF model can be written in a manifestly extended BRST invariant mannerin a superspace by considering one additional Grassmann (fermionic) coordinate. However, we needtwo additional Grassmann coordinates to express both the extended BRST and extended anti-BRSTinvariant BV actions of BF model in superspace.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The author is grateful to Dr. Sudhaker Upadhyay for his suggestions in preparation to manuscript. [1] D. Birmingham, M. Blau, M. Rakowski and G. Thompson Phys. Reports 209, 209 (1991);[2] C. Lucceshi, O. Piguet and S.P. Sorella, Nucl.Phys. B 395, 325 (1993).[1] D. Birmingham, M. Blau, M. Rakowski and G. Thompson Phys. Reports 209, 209 (1991);[2] C. Lucceshi, O. Piguet and S.P. Sorella, Nucl.Phys. B 395, 325 (1993).