The quantum description of BF model in superspace

TL;DR

This paper develops a superspace formulation of the four-dimensional BRST symmetric BF topological theory, incorporating extended BRST and anti-BRST symmetries, and relates shift symmetries to antifields.

Contribution

It introduces a superspace approach for the BF model with extended BRST and anti-BRST symmetries, linking shift symmetries to antifields in a novel way.

Findings

Superspace description of the BF model with extended symmetries

Connection between shift symmetry and antifields

Enhanced understanding of topological field theories

Abstract

We consider the BRST symmetric four dimensional BF theory, a topological theory, containing antysymmetric tensor fields in Landau gauge and extend the BRST symmetry by introducing a shift symmetry to it. Within this formulation, the antighost fields corresponding to shift symmetry coincide with antifields of standard field/antifield formulation. Further, we provide a superspace description for the BF model possessing extended BRST and extended anti-BRST transformations.

Equations

$${\mathcal{L}}_0$$

$${\mathcal{L}}_{gf+gh}$$

$$s{A}_{\mu }^a=-(D_{\mu }C{)}^a,s{C}^a=\frac{1}{2}{f}^{abc}{C}^b{C}^c,s{\xi }_{\mu }^a=(D_{\mu }\varphi {)}^a+f^{abc}{C}^b{\xi }_{\mu }^c,$$

$$s{B}_{\mu \nu }^a=-(D_{\mu }{\xi }_{\nu }-D_{\nu }{\xi }_{\mu }{)}^a-f^{abc}{B}_{\mu \nu }^b{C}^c+f^{abc}{ϵ}_{\mu \nu \rho \sigma }({\partial }^{\rho }{\bar{\xi }}^{b\sigma }){\varphi }^c$$

$$s{\varphi }^a=f^{abc}{C}^b{\varphi }^c,s{\bar{\xi }}_{\mu }^a=h_{\mu }^a,s{\bar{C}}^a=b^a,s{\bar{\varphi }}^a={\omega }^a,s{e}^a={\lambda }^a,$$

$$s(h_{\mu }^a,b^a,{\omega }^a,{\lambda }^a)=0.$$

$$\Psi =\left({\bar{C}}^a{\partial }^{\mu }{A}_{\mu }^a+{\bar{\xi }}^{a\mu }{\partial }^{\nu }{B}_{\mu \nu }^a+{\bar{\varphi }}^a{\partial }^{\mu }{\xi }_{\mu }^a-e^a{\omega }^a-e^a{\partial }^{\mu }{\bar{\xi }}_{\mu }^a\right),$$

$${\mathcal{L}}_{gf+gh}=s\Psi .$$

$$B_{\mu \nu }^a⟶B_{\mu \nu }^a-{\tilde{B}}_{\mu \nu }^a,A_{\mu }^a⟶A_{\mu }^a-{\tilde{A}}_{\mu }^a,C^a⟶C^a-{\tilde{C}}^a,$$

$${\bar{C}}^a⟶{\bar{C}}^a-{\widetilde{\stackrel{ˉ}{C}}}^a,b^a⟶b^a-{\tilde{b}}^a,{\xi }_{\mu }^a⟶{\xi }_{\mu }^a-{\tilde{\xi }}_{\mu }^a,$$

$${\bar{\xi }}_{\mu }^a⟶{\bar{\xi }}_{\mu }^a-\widetilde{{\stackrel{ˉ}{\xi }}_{\mu }^a},{\varphi }^a⟶{\varphi }^a-{\tilde{\varphi }}^a,{\bar{\varphi }}^a⟶{\bar{\varphi }}^a-\widetilde{{\stackrel{ˉ}{\varphi }}^a},h_{\mu }^a⟶h_{\mu }^a-{\tilde{h}}_{\mu }^a,$$

$$e^a⟶e^a-{\tilde{e}}^a,{\omega }^a⟶{\omega }^a-{\tilde{\omega }}^a,{\lambda }^a⟶{\lambda }^a-{\tilde{\lambda }}^a.$$

$$\tilde{\mathcal{L}}$$

$$s{A}_{\mu }^a$$

$$s\widetilde{{\stackrel{ˉ}{C}}^a}$$

$$s{\bar{\varphi }}^a$$

$$s{\tilde{\xi }}_{\mu }^a$$

$$s{\psi }_{\mu }^a$$

$$s{\bar{M}}^a$$

$$s{A}_{\mu }^{\star a}$$

$$s{\bar{\varphi }}^{\star a}$$

$$s{\zeta }_{\mu }^a$$

$$s{\bar{\upsilon }}^a$$

$${\tilde{\mathcal{L}}}_{gf+gh}$$

$$s\Psi$$

$${\mathcal{L}}_{eff}$$

$$A_{\mu }^{a\star }$$

$${\xi }_{\mu }^{\star a}$$

$$A_{\mu }^{a\star }$$

$${\bar{\xi }}_{\mu }^{\star a}$$

$$A_{\mu }^a(x,\theta )$$

$${\chi }^a(x,\theta )$$

$${\bar{\chi }}^a(x,\theta )$$

$$b^a(x,\theta )$$

$${\tilde{\xi }}_{\mu }^a(x,\theta )$$

$$\widetilde{{\stackrel{ˉ}{\xi }}_{\mu }^a}(x,\theta )$$

$${\tilde{\varphi }}^a(x,\theta )$$

$$\widetilde{{\stackrel{ˉ}{\varphi }}^a}(x,\theta )$$

$${\tilde{A}}_{\mu }^{\star a}(x,\theta )$$

$${\tilde{b}}^{\star a}(x,\theta )$$

Full text

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 BF1 . Except these two types, there are another Schwarz-type TGFT called topological BF theory, which is an extension of CS theory BF2 . The difference between CS theory and BF model is that action of previous theory exist only 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 BF3 were introduced some years ago. This model is interested due to its topological nature BF1 and their connection with lower dimensional quantum gravity, for example three space-time dimensional Einstein-Hilbert with or without using cosmological constant can be naturally formulated in terms of BF-modelsBF4 ; BF5 . Coupling of an antisymmetric tensor field with the field strength tensor of Yang-Mills is describe by these models BF6 . Quantization of BF model in Landau gauge has been studied in Ref. BF6 . Topological BF theory in Landau gauge has a common feature of a large class of topological models BF7 ; BF8 .

On the other hand, the Batalin-Vilkovisky (BV) approach, also known as field/antifield formulation, BF10 ; BF11 ; BF12 ; BF13 is one of the most powerful quantization algorithms presently available. BV formulation deals with very general gauge theories, including those with open or reducible gauge symmetry algebras. The BV method also address the possible violations of symmetries of the action by quantum effects. The BV formulation (independently introduced by Zinn-Justin BF14 ) extends the BRST approach sud1 . In fact, the BRST symmetry BRS ; BRS1 is a very important symmetry for gauge theories sudup . Beside the covariant description to perform the gauge-fixing in quantum field theory, BV formulation was also applies to other problems like analysing possible deformations of the action and anomalies.

A superspace description for various gauge theories in BV formulation has been studied extensively bono ; ad ; ba ; sudp ; sudb . They have shown that the extended BRST and extended anti-BRST invariant actions of these theories (including some shift symmetry) in BV formulation yield naturally the proper identification of the antifields through equations of motion. The shift symmetry is important and gets relevance, for example, in inflation particularly in supergravity bra as well as in Standard Model he . In usual BV formulation, these antifields can be calculated from the expression of gauge-fixing fermion. We extended BRST formulation and superspace description of the topological gauge (BF) model is still unstudied and we 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 of the theory by including shift symmetry. By doing so, we find that the antighosts of shift symmetry get identified as antifields of standard BV formulation naturally. Further, we discuss a superspace formulation of extended BRST invariant BF model. Here we see that one additional Grassmann coordinate is required if 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 section III, we study the extended BRST transformation of the model. Further, we describe extended BRST invariant action in superspace in section IV. The extended anti-BRST symmetry is discussed in section V. 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, the BF model in flat $(3+1)$ space-time dimensions is given by the following gauge invariant Lagrangian density BF6 :

[TABLE]

where $B^{a}_{\rho\sigma}$ and $F^{a}_{\mu\nu}$ are two-form field and field-strength tensor for vector field respectively. In order to remove discrepancy due to gauge symmetry, the gauge fixing and ghost terms are given by

[TABLE]

where fields $(C^{a},\xi^{a}_{\mu})$, $(\bar{C}^{a},\bar{\xi}^{a}_{\mu})$ and $(b^{a},h^{a}_{\mu})$ are the ghosts, antighosts and the multipliers fields respectively, while the fields $\phi^{a},\bar{\phi}^{a}$ and $\omega^{a}$ are taken into account to remove further degeneracy due to the existence of zero modes in the transformations.

The effective Lagrangian density of BF model, ${\cal L}={\cal L}_{0}+{\cal L}_{gf+gh}$, possesses following BRST symmetry:

[TABLE]

The gauge-fixing and ghost terms of the effective Lagrangian density is BRST exact and, hence, can be written in terms of BRST variation of gauge-fixing fermion,

[TABLE]

as follows

[TABLE]

In the next section, we would like to study the extended BRST symmetry for the model which incorporates shift 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 is that antifields get identification naturally. We begin with shifting all the fields from their original value as follows,

[TABLE]

The effective Lagrangian density of BF model also get shifted under such shifting of fields respectively. This is given by

[TABLE]

The shifted Lagrangian density is invariant under BRST transformation together with a shift symmetry transformation, jointly known as extended BRST transformation. The extended BRST symmetry transformations under which Lagrangian density of BF model is invariant are written by

[TABLE]

where $\psi^{a}_{\mu},\epsilon^{a},\bar{\epsilon}^{a},\chi^{a},M^{a},\bar{M}^{a},N^{a},L^{a}_{\mu},$ and $\bar{L}^{a}_{\mu}$ are the ghost fields corresponding to shift symmetry for $A^{a}_{\mu},C^{a},\bar{C}^{a},b^{a},\phi^{a},\bar{\phi}^{a},e^{a},\xi^{a}_{\mu}$ and $\bar{\xi}^{a}_{\mu}$ respectively. The nilpotency of extended BRST symmetry (8) leads to the BRST transformation for the following ghost fields:

[TABLE]

In order to make the theory ghost free, we need further antighosts $A^{\star a}_{\mu},C^{\star a},\bar{C}^{\star a},b^{\star a},\xi^{\star a}_{\mu},\bar{\xi}^{\star a}_{\mu},\phi^{\star a},\bar{\phi}^{\star a},$ and $e^{\star a}$ to be introduced corresponding to the ghost fields $\psi^{a}_{\mu},\epsilon^{a},\bar{\epsilon}^{a},\chi^{a},M^{a},\bar{M}^{a},N^{a},L^{a}_{\mu},$ and $\bar{L}^{a}_{\mu}$ respectively. The BRST transformations of these antighosts are constructed as follows

[TABLE]

where $\zeta^{a}_{\mu},\sigma^{a},\bar{\sigma}^{a},\varpi^{a},\upsilon^{a},\tau^{a},\kappa_{\mu}^{a},$ and $\bar{\kappa}_{\mu}^{a}$ are the Nakanishi-Lautrup type auxiliary fields corresponding to shifted fields $\tilde{A}^{a}_{\mu},\tilde{C}^{a},\tilde{\bar{C}^{a}},\tilde{b}^{a},\tilde{\phi}^{a},\tilde{\bar{\phi}}^{a},\tilde{e}^{a},\tilde{\xi}^{a}_{\mu},$ and $\tilde{\bar{\xi}^{a}_{\mu}}$ having following BRST transformations:

[TABLE]

We can recover our original BF model by fixing the shift symmetry in such a way such that effect of all the tilde fields will vanish. We achieve this by adding following gauge-fixed term to the shifted Lagrangian density (7):

[TABLE]

One can easily check that this gauge-fixing Lagrangian density $\tilde{\cal L}_{gf+gh}$ also admits the extended BRST invariance. Integrating the auxiliary fields of the above expression, we obtain

[TABLE]

The gauge-fixing and ghost terms of the Lagrangian density are BRST exact and can be expressed in terms of a general gauge-fixing fermion $\Psi$ as

[TABLE]

After integrating out the auxiliary fields which set the tilde fields to zero, we have the complete effective action for BF model in landau gauge possessing extended BRST symmetry as

[TABLE]

Integrating out the ghost fields associated with shift symmetry, we obtain

[TABLE]

For a particular choice of gauge-fixing fermion $\Psi$ given in (4), anti-ghost fields get following identifications:

[TABLE]

It is obvious to see that with these anti-ghost fields, the expression (15) changes to the original Lagrangian density 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 BRST transformations only is described in a superspace $(x_{\mu},\theta)$, where $\theta$ is a Grassmann coordinate and $x_{\mu}$ is the four dimanesional spect-time coordinates. In order to give superspace description for the extended BRST invariant theory, we first define superfields of the form:

[TABLE]

The super-antifields in superspace are defined as follows

[TABLE]

From the above expressions of superfields and super-antifields, we calculate

[TABLE]

Adding all the equations of (20) side by side, we get

[TABLE]

which is nothing but the gauge-fixed Lagrangian density for shift symmetry $\tilde{{\cal L}}_{gf+gh}$ given in (12). Now, one can define the general super-gauge-fixing fermion in superspace as follows

[TABLE]

which can further be expressed as

[TABLE]

From this, the original gauge-fixing Lagrangian density can be defined as the left derivation of super-gauge-fixing fermion with respect to $\theta$ as $\left[\frac{\delta\Phi(x,\theta)}{\delta\theta}\right]$.

Hence, the complete effective action for the BF model in general gauge in the superspace is now given by

[TABLE]

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 Lagrangian density for BF model remains invariant as follows,

[TABLE]

The ghost fields associated with the shift symmetry transform under extended anti-BRST symmetry as

[TABLE]

From the nilpotency of above transformations demands that the auxiliary and antighost fields associated with the shift symmetry transform as

[TABLE]

The gauge-fixing and ghost parts of the effective Lagrangian density are anti-BRST-exact also so it can be expressed as the anti-BRST variation of this gauge-fixing fermion ($\bar{\Psi}$).

VI Extended BRST and anti-BRST invariant superspace

The extended BRST and anti-BRST invariant Lagrangian density for BF model can be written in superspace with the help of two additional Grassmannian coordinates $\theta$ and $\bar{\theta}$. Requiring the field strength to vanish along unphysical directions $\theta$ and $\bar{\theta}$ direction, we obtain the following superfields:

[TABLE]

With these expressions of superfields, we can calculate

[TABLE]

which is nothing but the gauge-fixed Lagrangian density for shift symmetry. Being the $\theta\bar{\theta}$ component of a super field, this Lagrangian density is manifestly invariant under both the extended BRST and the anti-BRST transformations.

Now, we define the general super-gauge-fixing fermion in superspace as

[TABLE]

which yields the original gauge-fixing and ghost part of the effective effective Lagrangian density upon differentiation as follows, $\mbox{Tr}\left[\frac{\partial}{\partial\theta}\left[\delta(\bar{\theta})\Phi(x,\theta,\bar{\theta})\right]\right]$.

Therefore, the gauge-fixed Lagrangian density corresponding to BRST and shift symmetries for BF model can now be given as

[TABLE]

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 its some intriguing properties. In present work, we have considered $(3+1)$ dimensional BF model in Landau gauge 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 field naturally arises. Further we have provide the superfield description of BF model in superspace, where we show that the BV action for BF model can be written in a manifestly extended BRST invariant manner in a superspace by considering one additional Grassmann (fermionic) coordinate. However, we need two additional Grassmann coordinates to express both the extended BRST and extended anti-BRST invariant 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.

Acknowledgements.

The author is grateful to Dr. Sudhaker Upadhyay for his suggestions in preparation to manuscript.