Nilpotent (Anti-)Co-BRST Symmetries and Their Consequences in a 4D Field-Theoretic System

1. Introduction

The ideas of symmetries of all varieties (i.e. discrete, continuous, spacetime, internal, supersymmetric, etc.) have been at the heart of modern developments in theoretical physics (see, e.g. [1–4] and references therein). In fact, it has been well-established that the requirement of the local continuous symmetry invariance dictates the emergence of interactions in most of the theories that are useful in the precise description of the natural phenomena. One such symmetry is the well-known gauge symmetry transformation which has been responsible for the precise theoretical description of the three out of four fundamental interactions of nature. In other words, the local gauge theories (based on the Abelian and non-Abelian gauge symmetry groups) describe the electromagnetic, weak and strong interactions of nature (see, e.g. [1]). One of the most theoretically appealing as well as mathematically elegant quantization scheme for these kinds of gauge theories is the well-known Becchi-Rouet-Stora-Tyutin (BRST) formalism [5–8] where the unitarity and quantum gauge (i,e. BRST) invariance are respected together at any arbitrary order of perturbative computations for a given physical process that is allowed by the BRST-quantized versions of the interacting gauge theories (see, e.g. [9] for details).

A couple of sacrosanct properties of the BRST formalism is the observation that, for a given local classical gauge symmetry transformation for a gauge theory, there are two nilpotent versions of the quantum gauge symmetry transformations which are called as the BRST and anti-BRST symmetry transformations for the BRST-quantized gauge theory and these nilpotent symmetry transformation operators are required to absolutely anticommute with each-other. The former property encodes the fermionic nature of the (anti-)BRST symmetry transformations under which a fermionic field of a properly BRST-quantized gauge theory transforms to its counterpart bosonic field and vice-versa. On the other hand, the latter property implies the linear independence of the BRST and anti-BRST symmetry transformations (which distinguish them from the $\mathcal{N}=2$ SUSY transformations which are also nilpotent of order two but they do not anticommute with each-other). Physically, the absolute antcommutativity property implies that the BRST and anti-BRST symmetry transformation operators have their own identity and they are important in their own right. This claim becomes quite transparent in the context of the D-dimensional (e.g. D = 2, 3, 4) BRST-quantized versions of the field-theoretic as well as the 1D toy models of Hodge theory (see, e.g. [10–14] and references therein) where the discrete and continuous symmetry transformations (and corresponding conserved charges) provide the physical realization(s) of the de Rham cohomological operators1 of differential geometry at the algebraic level.

In our present investigation, we focus on the four (3 + 1)-dimensional (4D) combined field-theoretic system of the free Abelian 3-form and 1-form gauge theories together (within the framework of BRST formalism) and discuss the dual-BRST (i.e. co-BRST) and anti-dual (i.e. anti-co-BRST) symmetry transformation operators for the coupled (but equivalent) Lagrangian densities. We establish that the total gauge-fixing terms for the Abelian 3-form and 1-form gauge fields, owing their mathematical origin to the dual-exterior (i.e. co-exterior) derivative of differential geometry (see, e.g. [15–19]), remain invariant under these nilpotent symmetry transformation operators. This observation should be contrasted against the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformation operators which leave the total kinetic terms for the Abelian 3-form and 1-form gauge fields invariant (see, e.g. [20] for details). In this context, it is worthwhile to mention that the kinetic terms for the above Abelian 3-form and 1-form gauge fields owe their mathematical origin to the exterior derivative of differential geometry [15-19].

In our present endeavor, we have been able to show that the coupled (but equivalent) Lagrangian densities ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eqs. (1),(2)] and ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ [cf. Eqs. (28),(29)] (cf. Secs. 2 and 5) respect the infinitesimal, continuous and off-shell nilpotent co-BRST (i.e. dual-BRST) and anti-co-BRST (i.e. anti-dual-BRST) symmetry transformations, respectively, because the corresponding action integrals remain invariant under them. We have derived the conserved (but non-nilpotent) Noether dual-BRST and the anti-dual-BRST charges which are found to be the generator for the above off-shell nilpotent dual-BRST and anti-dual-BRST symmetry transformations [cf. Eqs. (3),(30)]. The off-shell nilpotent versions of the co-BRST and anti-co-BRST charges have been systematically derived (cf. Secs 4 and 7) from their counterparts non-nilpotent versions of the Noether conserved charges by exploiting the theoretical proposal made in our earlier work [21]. We have discussed (i) the physicality criterion w.r.t. the conserved and nilpotent version of the co-BRST charge in our Sec. 4, and (ii) the physicality criterion w.r.t. the conserved and nilpotent version of the anti-co-BRST charge has been carried out in our Sec. 7. It has been established that the physicality criteria w.r.t. the off-shell nilpotent versions of the conserved (anti-)dual BRST charges lead to the annihilation of the physical states (existing in the total quantum Hilbert space of states) by the operator forms of the dual versions of the first-class constraints of our 4D classical field-theoretic system.

Our present investigation is urgent, essential and important on the following counts. First of all, it has been amply indicated in our earlier work (see, e.g. [22] for details) that our present 4D combined field-theoretic system is an example for Hodge theory within the framework of BRST formalism where the discrete and continuous symmetry transformations (and corresponding conserved charges) provide the physical realization(s) of the de Rham cohomological operators of differential geometry [15–19] at the algebraic level. Second, we have discussed the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST charges in our earlier work [20] where we have been able to show that the operator forms of the first-class constraints of our 4D classical gauge theory annihilate the physical states of our BRST-quantized 4D field-theoretic system at the quantum level (cf. Appendix A). In our present endeavor, we discuss the physicality criteria w.r.t. the nilpotent versions of the conserved (anti-)co-BRST charges and establish that the dual versions of the operator forms of the first-class constraints annihilate the physical states (which are identified with the harmonic states of the Hodge decomposed quantum states in the quantum Hilbert space of states). Finally, we show that the operator forms of the first-class constraints and their dual counterparts are connected with each-other by the duality symmetry transformations that are respected by the non-ghost sector of the coupled (but equivalent) Lagrangian densities for our 4D BRST-quantized field-theoretic system.

The theoretical contents of our present endeavor are organized as follows. To set up the useful notations and to explain the key mathematical symbols, in Sec. 2, we recapitulate the bare essentials of our earlier work [11] where the nilpotent co-BRST (i.e. dual-BRST) symmetry transformations have been discussed. Our Sec. 3 is devoted to the derivations of the Noether conserved co-BRST current and corresponding conserved (but non-nilpotent) co-BRST charge. The subject matter of our Sec 4 is connected with the derivation of the off-shell nilpotent conserved co-BRST charge where we also study the physicality criterion w.r.t. to it. Our Sec 5 deals with the discussion on a proper set of anti-co-BRST symmetry transformations that is respected by a coupled (but equivalent) Lagrangian density corresponding to the co-BRST invariant Lagrangian density of our Sec. 2. We derive the Noether conserved current and corresponding conserved (but non-nilpotent) anti-co-BRST charge in our Sec. 6. As far as our Sec. 7 is concerned, we derive the nilpotent version of the conserved anti-co-BRST charge and study the physicality criterion w.r.t. it in a concise manner. In our Sec 8, we devote time on the derivation of the non-trivial CF-type restrictions by requiring that both the coupled Lagrangian densities must respect the nilpotent co-BRST as well as the anti-co-BRST transformations. These CF-type restrictions are also shown to be responsible for the absolute anticommutativity of the co-BRST and anti-co-BRST transformations. Finally, in Sec. 9, we summarize our key results, mention a few novel observations and point out the future scope of our present investigation.

Our Appendix A deals with a very brief summary of our earlier work on the off-shell nilpotent (anti-)BRST symmetries of our present 4D field-theoretic system [20] and we establish that the nilpotent versions of the (anti-)BRST charges lead to the annihilation of the physical states by the operator forms of the first-class constraints on our theory when we demand the validity of the physicality criteria w.r.t. them. In our Appendix B, we very briefly mention the algebraic structure that is obeyed by the conserved (anti-)co-BRST charges and ghost charge for our 4D BRST-quantized theory.

Conventions and Notations: We adopt the convention of the left derivative w.r.t. all the fermionic fields of our theory in the computations of the EL-EoMs, canonical conjugate momenta, Noether conserved current, etc. We choose our 4D Levi-Civita tensor ${\epsilon }_{\mu \nu \sigma \rho }$ such that: ${\epsilon }_{0123}=+1=-{\epsilon }^{0123}$ and, when two of them are contrated together, they satisfy the standard relationships: ${\epsilon }_{\mu \nu \eta \kappa }{\epsilon }^{\mu \nu \eta \kappa }=-4!$, ${\epsilon }_{\mu \nu \eta \kappa }{\epsilon }^{\mu \nu \eta \rho }=-3!{\delta }_{\kappa }^{\rho },{\epsilon }_{\mu \nu \eta \kappa }{\epsilon }^{\mu \nu \sigma \rho }=-2!\left({\delta }_{\eta }^{\sigma }{\delta }_{\kappa }^{\rho }-{\delta }_{\kappa }^{\sigma }{\delta }_{\eta }^{\rho }\right)$, etc. As far as the Abelian 3-form gauge field $A_{\mu \nu \sigma }$ is concerned, we follow the convention: $(\delta A_{\mu \nu \sigma }/\delta A_{\alpha \beta \gamma })={\displaystyle \frac{1}{3!}}\left[{\delta }_{\mu }^{\alpha }({\delta }_{\nu }^{\beta }{\delta }_{\sigma }^{\gamma }-{\delta }_{\sigma }^{\beta }{\delta }_{\nu }^{\gamma })+{\delta }_{\nu }^{\alpha }({\delta }_{\sigma }^{\beta }{\delta }_{\mu }^{\gamma }-{\delta }_{\mu }^{\beta }{\delta }_{\sigma }^{\gamma })+{\delta }_{\sigma }^{\alpha }({\delta }_{\mu }^{\beta }{\delta }_{\nu }^{\gamma }-{\delta }_{\nu }^{\beta }{\delta }_{\mu }^{\gamma })\right]$, etc., for the tensorial variation/differentiation for various computational purposes. We take the background 4D flat Minkowskian metric tensor ${\eta }_{\mu \nu }$ as: ${\eta }_{\mu \nu }$ = diag $(+1,-1,-1,-1)$ so that the dot product between two non-null 4D vectors $U_{\mu }$ and $V_{\mu }$ is defined as: $U·V={\eta }_{\mu \nu }{U}^{\mu }{V}^{\nu }\equiv U_0{V}_0-U_i{V}_i$ where the Greek indices $\mu ,\nu ,\sigma ...=0,1,2,3$ denote the time and space directions and Latin indices $i,j,k...=1,2,3$ stand for the 3D space directions only. The off-shell nilpotent (i.e. $s_{\left(a\right)d}^2=0$) (anti-)dual BRST [i.e. (anti-)co-BRST] symmetry transformation operators are represented by the symbols $s_{\left(s\right)d}$ and the corresponding conserved (but non-nilpotent) Noether charges are denoted by $Q_{\left(a\right)d}$. The nilpotent versions of the conserved (anti-)co-BRST charges carry the notations $Q_{\left(A\right)D}$. The over dot (i.e. $\dot{\Phi }$) on a generic field $\Phi$ denotes the time-derivative (i.e. $\partial \Phi /\partial t$).

2. Preliminary: Co-BRST Symmetry Transformations

We begin with the BRST-invariant Lagrangian density ${\mathcal{L}}_{\left(B\right)}$ which is the sum of the non-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(NG\right)}$ and the FP-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(FP\right)}$ (see, e.g. [20,22] for details). The explicit form of the former is [20,22]

$$\begin{array}{ccc}\hfill & & {\mathcal{L}}_{\left(NG\right)}={\displaystyle \frac{1}{2}}{B}_2\left(\partial ·\varphi \right)-{\displaystyle \frac{1}{4}}{B}_2^2+{\displaystyle \frac{1}{2}}{B}_3\left(\partial ·\tilde{\varphi }\right)-{\displaystyle \frac{1}{4}}{B}_3^2+{\displaystyle \frac{1}{2}}{B}^2-B(\partial ·A)\hfill \\ & +& {\displaystyle \frac{1}{2}}{B}_1^2+B_1\left({\displaystyle \frac{1}{3!}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\mu }{A}_{\nu \sigma \rho }\right)-{\displaystyle \frac{1}{4}}{\left(B_{\mu \nu }\right)}^2+{\displaystyle \frac{1}{2}}{B}_{\mu \nu }\left[{\partial }_{\sigma }{A}^{\sigma \mu \nu }+{\displaystyle \frac{1}{2}}\left({\partial }^{\mu }{\varphi }^{\nu }-{\partial }^{\nu }{\varphi }^{\mu }\right)\right]\hfill \\ & -& {\displaystyle \frac{1}{4}}{\left({\mathcal{B}}_{\mu \nu }\right)}^2+{\displaystyle \frac{1}{2}}{\mathcal{B}}_{\mu \nu }\left[{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\sigma }{A}_{\rho }+{\displaystyle \frac{1}{2}}\left({\partial }^{\mu }{\tilde{\varphi }}^{\nu }-{\partial }^{\nu }{\tilde{\varphi }}^{\mu }\right)\right],\hfill \end{array}$$

(1)

where the fields $B,B_1,B_2,B_3,B_{\mu \nu },{\mathcal{B}}_{\mu \nu }$ are the bosonic Nakanishi-Lautrup type auxiliary fields that have been invoked to linearize the kinetic as well as the gauge-fixing terms for the fields $A_{\mu \nu \sigma },A_{\mu },{\varphi }_{\mu },{\tilde{\varphi }}_{\mu }$. To be specific, we have invoked the auxiliary fields B and ${\mathcal{B}}_{\mu \nu }$ to linearize the gauge-fixing and kinetic terms for the Abelian 1-form ($A^{\left(1\right)}=A_{\mu }d{x}^{\mu }$) gauge field $A_{\mu }$, respectively. On the other hand, the Nakanishi-Lautrup auxiliary fields $B_1$ and $B_{\mu \nu }$ have been used to linearize the kinetic and gauge-fixing terms for the Abelian 3-form field $A_{\mu \nu \sigma }$, respectively. The auxiliary fields $B_2$ and $B_3$ have been utilized to linearize the gauge-fixing terms for the Lorentz vector and axial-vector fields ${\varphi }_{\mu }$ and ${\tilde{\varphi }}_{\mu }$. We would like to clarify that the (axial-)vector fields $\left({\tilde{\varphi }}_{\mu }\right){\varphi }_{\mu }$ have been introduced in the theory due to the reducibility properties of the kinetic term for the Abelian 1-form gauge field and the gauge-fixing term for the Abelian 3-form [i.e. $A^{\left(3\right)}={\displaystyle \frac{1}{3!}}{A}_{\mu \nu \sigma }(d{x}^{\mu }\wedge d{x}^{\nu }\wedge d{x}^{\sigma })$] gauge field. The FP-ghost part of the BRST-quantized version of our 4D combined field theoretic system of the free Abelian 3-form and 1-form gauge theories is (see, e.g. [20,22] for details)

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\left(FP\right)}& =& {\displaystyle \frac{1}{2}}[\left({\partial }_{\mu }{\overline{C}}_{\nu \sigma }+{\partial }_{\nu }{\overline{C}}_{\sigma \mu }+{\partial }_{\sigma }{\overline{C}}_{\mu \nu }\right)\left({\partial }^{\mu }{C}^{\nu \sigma }\right)+\left({\partial }_{\mu }{\overline{C}}^{\mu \nu }+{\partial }^{\nu }{\overline{C}}_1\right)f_{\nu }\hfill \\ & -& \left({\partial }_{\mu }{C}^{\mu \nu }+{\partial }^{\nu }{C}_1\right){\overline{F}}_{\nu }+\left(\partial ·\overline{\beta }\right)B_4-\left(\partial ·\beta \right)B_5-B_4{B}_5-2{\overline{F}}^{\mu }{f}_{\mu }\hfill \\ & -& \left({\partial }_{\mu }{\overline{\beta }}_{\nu }-{\partial }_{\mu }{\overline{\beta }}_{\nu }\right)\left({\partial }^{\mu }{\beta }^{\nu }\right)-{\partial }_{\mu }{\overline{C}}_2{\partial }^{\mu }{C}_2]-{\partial }_{\mu }\overline{C}{\partial }^{\mu }C,\hfill \end{array}$$

(2)

where the fermionic (i.e. $C^2={\overline{C}}^2=0,C\overline{C}+\overline{C}C=0$) Lorentz scalar basic (anti-)ghost fields $\left(\overline{C}\right)C$, carrying the ghost numbers (-1)+1, are associated with the BRST-quantized version of the Abelian 1-form gauge theory (that is hidden in our combined field-theoretic system). On the other hand, the fermionic (i.e. $C_{\mu \nu }^2=0,{\overline{C}}_{\mu \nu }^2=0,C_{\mu \nu }{\overline{C}}_{\sigma \rho }+{\overline{C}}_{\sigma \rho }{C}_{\mu \nu }=0,$ etc.) basic (anti-)ghost fields $\left({\overline{C}}_{\mu \nu }\right)C_{\mu \nu }$, carrying the ghost numebrs (-1)+1, respectively, are associated with the Abelian 3-form gauge field $A_{\mu \nu \sigma }$. We have the ghost-for-ghost bosonic (i.e. ${\overline{\beta }}_{\mu }^2\ne 0,{\beta }_{\mu }^2\ne 0,{\beta }_{\mu }{\overline{\beta }}_{\nu }={\overline{\beta }}_{\nu }{\beta }_{\mu },$ etc.) Lorentz vector (anti-)ghost fields $\left({\overline{\beta }}_{\mu }\right){\beta }_{\mu }$ in our theory that carry the ghost numbers (-2)+2, respectively. Our BRST-quantized Abelian 3-from gauge theory is endowed with the fermionic (i.e. $C_2^2={\overline{C}}_2^2=0,C_2{\overline{C}}_2+{\overline{C}}_2{C}_2=0$) (anti-)ghost fields $\left({\overline{C}}_2\right)C_2$ which carry the ghost numbers (-3)+3, respectively (because $\left({\overline{C}}_2\right)C_2$ are the ghost-for-ghost-for-ghost fields in our theory). In addition, we have a pair of bosonic auxiliary fields ($B_5,B_4$) which carry the ghost numbers (-2, +2) and a pair of fermionic auxiliary fields (${\overline{F}}_{\mu },f_{\mu }$) that are endowed with the ghost numbers (-1, +1), respectively. On top of all these basonic/fremionic auxiliary and basic (anti-)ghost fields, we have the additional fermionic (i.e. $C_1^2={\overline{C}}_1^2=0,C_1{\overline{C}}_1+{\overline{C}}_1{C}_1=0$) (anti-)ghost fields $\left({\overline{C}}_1\right)C_1$ that are endowed with the ghost numbers (-1)+1, respectively. All these bosonic and fernionic (anti-)ghost fields are required in our theory to maintain the unitarity at any arbitrary order of perturbative computations for a physical process that is allowed by our 4D BRST-quantized field theoretic system of the Abelian 3-form and 1-form gauge theories.

It is very interesting to point out that the total Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eqs. (1), (2)] respects the following infinitesimal, continuous and off-shell nilpotent (i.e. $s_d^2=0$) dual-BRST (i.e. co-BRST) transformations $s_d$, namely;

$$\begin{array}{ccc}\hfill & & s_d{A}_{\mu \nu \sigma }={\epsilon }_{\mu \nu \sigma \rho }{\partial }^{\rho }\overline{C},s_d{A}_{\mu }={\displaystyle \frac{1}{2}}{\epsilon }_{\mu \nu \sigma \rho }{\partial }^{\nu }{\overline{C}}^{\sigma \rho },s_d{\overline{C}}_{\mu \nu }={\partial }_{\mu }{\overline{\beta }}_{\nu }-{\partial }_{\nu }{\overline{\beta }}_{\mu },s_d{\overline{f}}_{\mu }={\partial }_{\mu }{B}_5,\hfill \\ & & s_d{\overline{\mathcal{B}}}_{\mu \nu }={\partial }_{\mu }{\overline{F}}_{\nu }-{\partial }_{\nu }{\overline{F}}_{\mu },s_d{\overline{\beta }}_{\mu }={\partial }_{\mu }{\overline{C}}_2,s_d{C}_1=-B_3,s_d{\beta }_{\mu }=-f_{\mu },s_d{\tilde{\varphi }}_{\mu }=+{\overline{F}}_{\mu },\hfill \\ & & s_d{C}_{\mu \nu }=-{\mathcal{B}}_{\mu \nu },s_{d}C=+B_1,s_d{C}_2=B_4,s_d{\overline{C}}_1=B_5,s_d{F}_{\mu }=-{\partial }_{\mu }{B}_3,\hfill \\ & & s_d[{\overline{C}}_2,\overline{C},f_{\mu },{\overline{F}}_{\mu },{\varphi }_{\mu },B,B_1,B_2,B_3,B_4,B_5,B_{\mu \nu },{\mathcal{B}}_{\mu \nu },{\overline{B}}_{\mu \nu },]=0,\hfill \end{array}$$

(3)

because we observe that ${\mathcal{L}}_{\left(B\right)}$ transforms to a total spacetime derivative

$$\begin{array}{ccc}\hfill s_d{\mathcal{L}}_{\left(B\right)}& =& {\displaystyle \frac{1}{2}}{\partial }_{\mu }[({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\mathcal{B}}_{\nu \sigma }+{\mathcal{B}}^{\mu \nu }{\overline{F}}_{\nu }+B_4{\partial }^{\mu }{\overline{C}}_2\hfill \\ & +& B_5{f}^{\mu }+B_3{\overline{F}}^{\mu }+({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu })f_{\nu }]+{\partial }_{\mu }\left[B_1{\partial }^{\mu }\overline{C}\right],\hfill \end{array}$$

(4)

thereby rendering the action integral $S=\int d^{4}x{\mathcal{L}}_{\left(B\right)}$ invariant (i.e. $s_{d}S=0$) for all the physical fields of our theory which are supposed to vanish off as $x\to \pm \infty$ due to the Gauss divergence theorem. Thus, the infinitesimal, continuous and off-shell nilpotent co-BRST transformations (3) are the symmetry transformations for our theory.

We end this section with the following remarks. First of all, the nilpotency property (i.e. $s_d^2=0$) of the co-BRST symmetry transformation operator $s_d$ establishes its fermionic nature. As a consequence, this operator transforms (i) a bosonic field of our theory to a ferminoc field, and (ii) a fermionic field to its counterpart bosonic field. Second, the gauge-fixing terms $(\partial ·A)$ and $\left({\partial }_{\sigma }{A}^{\sigma \mu \nu }\right)$ owe their origins to the co-exterior (i.e. dual-exterior) derivative of differential geometry because we observe that: $\delta A^{\left(1\right)}=(\partial ·A)$ and $\delta A^{\left(3\right)}=-{\displaystyle \frac{1}{2!}}\left({\partial }^{\sigma }{A}_{\sigma \mu \nu }\right)(d{x}^{\mu }\wedge d{x}^{\nu })$. It is interesting to point out that the total gauge-fixing terms for the Abelian 3-form and 1-form gauge theories remain invariant under the co-BRST (i.e. dual-BRST) symmetry transformations. Hence the nomenclature (i.e. the co-BRST/dual-BRST) symmetry transformations) for the infinitesimal and continuous transformations in Equation (3), is correct. Third, we note that, it is very essential to have both the Abelian 1-form and 3-form gauge theories together so that our 4D BRST-quantized combined field-theoretic system can respect the co-BRST symmetry transformations. Finally, we would like to point out that, in addition to the off-shell nilpotent co-BRST symmetry transformations in (3), the Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ also respects the off-shell nilpotent BRST symmetry transformations (see, e.g. Appendix A for details).

3. Noether Co-BRST Conserved Current and Charge

According to Noether’s theorem, the existence of the infinitesimal continuous and off-shell nilpotent co-BRST symmetry transmissions (3) leads to the derivation of the Noether co-BRST current ($J_{\left(d\right)}^{\mu }$) as follows

$$\begin{array}{ccc}\hfill J_{\left(d\right)}^{\mu }& =& s_d{\Phi }_i{\displaystyle \frac{\partial {\mathcal{L}}_{\left(B\right)}}{\partial \left({\partial }_{\mu }{\Phi }_i\right)}}-{\displaystyle \frac{1}{2}}[({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\mathcal{B}}_{\nu \sigma }+{\mathcal{B}}^{\mu \nu }{\overline{F}}_{\nu }+B_4{\partial }^{\mu }{\overline{C}}_2\hfill \\ & +& B_5{f}^{\mu }+B_3{\overline{F}}^{\mu }+({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu })f_{\nu }]-B_1{\partial }^{\mu }\overline{C},\hfill \end{array}$$

(5)

where the generic field ${\Phi }_i$ stands for all the bosonic as well as fermionic dynamical fields of our 4D combined system of the BRST-quantized Abelian 3-from and 1-form gauge theories which is described by the perfectly (co-)BRST-invariant Lagrangian density ${\mathcal{L}}_{\left(B\right)}$. In other words, we have: ${\Phi }_i\equiv A_{\mu \nu \sigma },C_{\mu \nu },{\overline{C}}_{\mu \nu },A_{\mu },{\varphi }_{\mu },{\tilde{\varphi }}_{\mu },{\beta }_{\mu },{\overline{\beta }}_{\mu },C_1,{\overline{C}}_1,C,\overline{C}$ in the Lagrangian density ${\mathcal{L}}_{\left(B\right)}$ and the rest of the terms of the above Equation (5) are nothing but the terms on the r.h.s. of Equation (4) modulo a sign factor which is connected with the standard technique of the Noether theorem. The first term of the r.h.s. of (5) can be explicitly computed by taking into account (i) the convention of the left derivative w.r.t. the fermionic dynamical fields, (ii) the usual convention of the derivative w.r.t. the bosonic dynamical fields, and (iii) the infinitesimal, continuous and off-shell nilpotent co-BRST symmetry transformations (3) of our 4D BRST-quantized theory. The final expression for the conserved Noether co-BRST (i.e. dual-BRST) current ($J_{\left(d\right)}^{\mu }$) turns out to be the following:

$$\begin{array}{ccc}\hfill J_{\left(d\right)}^{\mu }& =& {\displaystyle \frac{1}{2}}[{\mathcal{B}}^{\mu \nu }{\overline{F}}_{\nu }-{\epsilon }^{\mu \nu \sigma \rho }B\left({\partial }_{\nu }{\overline{C}}_{\sigma \rho }\right)+({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\mathcal{B}}_{\nu \sigma }\hfill \\ & +& {\epsilon }^{\mu \nu \sigma \rho }\left({\partial }_{\nu }\overline{C}\right)B_{\sigma \rho }+B_4{\partial }^{\mu }{\overline{C}}_2-({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }){\partial }_{\nu }{\overline{C}}_2+({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu })f_{\nu }\hfill \\ & +& B_3{\overline{F}}^{\mu }+B_5{f}^{\mu }+({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu })({\partial }_{\nu }{\overline{\beta }}_{\sigma }-{\partial }_{\sigma }{\overline{\beta }}_{\nu })]+B_1{\partial }^{\mu }\overline{C}.\hfill \end{array}$$

(6)

The conservation law (i.e. ${\partial }_{\mu }{J}_{\left(d\right)}^{\mu }=0$) can be proven by taking into account the following Euler-Lagrange (EL) equations of motion (EoMs) which emerge out from the (non-)ghost sectors of the Lagrangian densities (1) and (2), namely;

$$\begin{array}{ccc}\hfill & & \square \overline{C}=0,\square {\overline{C}}_2=0,{\partial }_{\mu }\left({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }\right)={\partial }^{\nu }{B}_4,(\partial ·\overline{F})=0,\hfill \\ & & {\partial }_{\mu }\left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right)=-{\partial }^{\nu }{B}_5,{\partial }_{\mu }{\mathcal{B}}^{\mu \nu }+{\partial }^{\nu }{B}_3=0,(\partial ·f)=0,\hfill \\ & & {\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{B}_{\sigma \rho }-{\partial }^{\mu }{B}_1=0,{\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\mathcal{B}}_{\sigma \rho }+{\partial }^{\mu }B=0,\hfill \\ & & {\partial }_{\mu }\left({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }\right)=-{\displaystyle \frac{1}{2}}\left({\partial }^{\nu }{\overline{F}}^{\sigma }-{\partial }^{\sigma }{\overline{F}}^{\nu }\right),\hfill \\ & & {\partial }_{\mu }\left({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }\right)=-{\displaystyle \frac{1}{2}}\left({\partial }^{\nu }{f}^{\sigma }-{\partial }^{\sigma }{f}^{\nu }\right).\hfill \end{array}$$

(7)

The conserved Noether co-BRST current (i.e. $J_{\left(d\right)}^{\mu }$) leads to the definition of the conserved co-BRST charge $Q_d=\int d^{3}x{J}_{\left(d\right)}^0$ whose explicit expression is as follows:

$$\begin{array}{ccc}\hfill Q_d& =& \int d^{3}x\left[{\displaystyle \frac{1}{2}}\right\{{\mathcal{B}}^{0i}{\overline{F}}_i-{\epsilon }^{0ijk}B\left({\partial }_i{\overline{C}}_{jk}\right)+({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}){\mathcal{B}}_{ij}\hfill \\ & +& {\epsilon }^{0ijk}\left({\partial }_i\overline{C}\right)B_{jk}+B_4{\dot{\overline{C}}}_2-({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\partial }_i{\overline{C}}_2+({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0)f_i\hfill \\ & +& B_3{\overline{F}}^0+B_5{f}^0+({\partial }^0{C}^{ij}+{\partial }^i{C}^{jk}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)\}+B_1\dot{\overline{C}}].\hfill \end{array}$$

(8)

The above conserved charge $Q_d$ is the generator for the infinitesimal, continuous and off-shell nilpotent co-BRST symmetry transformations (3). This can be readily checked by the standard relationship between the infinitesimal continuous symmetry transformations and their generator as the Noether conserved charge. In other words, we have the following standard mathematical formula in the case of our dual-BRST transformations, namely;

$$\begin{array}{c}\hfill s_d{\Phi }_i\left(x\right)=\pm i{\left[{\Phi }_i\left(x\right),Q_d\right]}_{(\pm )},\end{array}$$

(9)

where the $(\pm )$ signs, as the subscripts on the square bracket in the above equation, correspond to the square bracket being an (anti)commutator for the generic dynamical field ${\Phi }_i\left(x\right)$ being fermionic/bosonic in nature. The ± signs, in front of the square bracket on the r.h.s., have to be chosen judiciously as far as the choice of the field ${\Phi }_i$ is concerned.

We wrap-up this section with the following decisive remarks. First of all. we observe that the total gauge-fixing terms (owing their basic mathematical origin to the co-exterior derivative) for the Abelian 3-form and 1-form gauge theories remain invariant under the nilpotent co-BRST symmetry transformations. This observation is distinctly different from the invariance of the total kinetic terms (cf. Appendix A), owing their basic origin to the exterior derivative, that remain invariant under the (anti-)BRST symmetry transformations (see, e.g. [20] for details). Second, even though the nilpotent co-BRST symmetry transformations are fermionic in nature, these transformations are not like the supersymmetric transformations which are also fermionic (cf. the Sec. 5 for more discussions). Thus, the mathematical basis for the existence of the co-BRST symmetry transformations lies in the co-exterior derivative of differential geometry. Third, the existence of the co-BRST (and corresponding anti-co-BRST) symmetry transformations are crucial for (i) the consideration of the combined 4D BRST-quantized field-theoretic system of the Abelian 3-from and 1-form gauge theories together, and (ii) the proof that our present 4D BRST-quantized field-theoretic system is an example for the Hodge theory [22]. Fourth, it is essential to point out that under the (anti-)BRST symmetry transformations (see, e.g. [20]), the BRST-quantized versions of the Abelian 3-form and 1-form gauge theories remain invariant separately and independently unlike in the case of the (anti-)co-BRST symmetry transformations where both these gauge theories are required to be present together. In other words, for the (anti-) co-BRST invariance in our 4D theory, we need to have the free Abelian 3-form and 1-form field-theoretic system together because the variation of the kinetic term for the Abelian 3-form gauge field compensates with the variation of the FP-ghost term for the BRST-quantized Abelian 1-form gauge theory. Finally, we note that the relationship, written in Equation (9), is a very general expression and it is valid for all kinds of choices for the generic field ${\Phi }_i$. Choosing the generic field ${\Phi }_i$ to be the co-BRST charge itself (i.e. ${\Phi }_i=Q_d$), we obtain the following expression from (9), namely;

$$\begin{array}{ccc}\hfill s_d{Q}_d=i\left\{Q_d,Q_d\right\}& \equiv & -{\displaystyle \frac{1}{2}}\int d^{3}x[({\partial }^0{\mathcal{B}}^{ij}+{\partial }^i{\mathcal{B}}^{jk}+{\partial }^j{\mathcal{B}}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)\hfill \\ & -& ({\partial }^0{f}^i-{\partial }^i{f}^0){\partial }_i{\overline{C}}_2]\ne 0,\hfill \end{array}$$

(10)

where we have explicitly computed the l.h.s. (i.e. $s_d{Q}_d$) by directly applying the co-BRST symmetry transformations (3) on the expression for $Q_d$ [cf. Eq. (8)]. In other words, we find that the anticommutator: $i\left\{Q_d,Q_d\right\}=2i{Q}_d^2\ne 0$ is not equal to zero [cf. the r.h.s. of Equation (10)] which implies that the Noether conserved co-BRST charge $Q_d$ is not nilpotent2 of order two (i.e. $Q_d^2\ne 0$). Hence, this Noether conserved charge is not suitable for the discussions connected with (i) the BRST cohomology (see. e.g. [23] for details), and (ii) the physicality criterion w.r.t. it (cf. Sec. 4 for more discussions).

4. Nilpotent Co-BRST Charge from Noether Charge

The central theme of this section is to derive the conserved and nilpotent (i.e. $Q_D^2=0$) version of the co-BRST charge $Q_D$ from the non-nilpotent (i.e. $Q_d^2\ne 0$) version of the Noether conserved charge $Q_d$. In this context, we shall exploit systematically the theoretical method that has been proposed in our earlier work [21]. According to the basic ideas behind our earlier work [21], the emphasis will be laid on the use of (i) the EL-EoMs that are derived from the perfectly co-BRST invariant Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$, (ii) the partial integral followed by the Gauss divergence theorem, and (ii) the infinitesimal, continuous and off-shell nilpotent (i.e. $s_d^2=0$) co-BRST symmetry transformations $s_d$ at appropriate places, to make sure that, ultimately, we obtain: $s_d{Q}_D=i\left\{Q_D,Q_D\right\}=0$ thereby leading to the derivation of the nilpotent (i.e. $Q_D^2=0$) version of the co-BRST charge $Q_D$ from the non-nilpotent (i.e. $Q_d^2\ne 0$) Noether co-BRST charge $Q_d$.

Against the backdrop of the key contents of the above paragraph, first of all, we focus on the fourth term of the Noether conserved co-BRST charge $Q_d$ [cf. Eq. (8)]. Using the partial integration followed by the Gauss divergence theorem, this term can be written as

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left({\partial }_i\overline{C}\right)B_{jk}=-{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left({\partial }_i{B}_{jk}\right)\overline{C},\end{array}$$

(11)

because we have dropped a total space derivative term. Using the EL-EoM: ${\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{B}_{\sigma \rho }={\partial }^{\mu }{B}_1$ [cf. Eq. (7)], we can recast the above integral as:

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left({\partial }_i{B}_{jk}\right)\overline{C}=-\int d^{3}x{\dot{B}}_1\overline{C}.\end{array}$$

(12)

Taking into account the last term of the expression for the Noether co-BRST charge in (8) and the above integral, we have the contribution to a part of the nilpotent (i.e. $Q_D^2=0$) version of the co-BRST charge $Q_D$ from the BRST-quantized version of the combined field-theoretic system of the Abelian 3-form and 1-form gauge theories as follows

$$\begin{array}{c}\hfill Q_D^{\left(1\right)}=\int d^{3}x\left[B_1\dot{\overline{C}}-{\dot{B}}_1\overline{C}\right],\end{array}$$

(13)

where the superscript $\left(1\right)$ on $Q_D^{\left(1\right)}$ denotes the fact that the above contribution incorporates into it the auxiliary field $B_1$. It is straightforward to note that the above charge $Q_D^{\left(1\right)}$ is co-BRST invariant because we find that: $s_d{Q}_D^{\left(1\right)}=0$ where we have directly applied the off-shell nilpotent (i.e. $s_d^2=0$) co-BRST symmetry transformations (3) on (13). Thus, the charge $Q_D^{\left(1\right)}$ will be a part of the full nilpotent version of the co-BRST charge $Q_D$. We concentrate now on the second term of the Noether co-BRST charge $Q_d$ [cf. Eq. (8)] which can be re-expressed, using the beauty of the partial integration followed by the theoretical strength of the Gauss divergence theorem, as follows

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}B\left({\partial }_i{\overline{C}}_{jk}\right)=+{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left({\partial }_{i}B\right){\overline{C}}_{jk},\end{array}$$

(14)

where we have dropped a total space derivative term because all the physical fields vanish off as $x\to \pm \infty$. We can, at this stage, use the EL-EoM: ${\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\mathcal{B}}_{\sigma \rho }=-{\partial }^{\mu }B$ [cf. Eq. (7)] which can be recast in the following form

$$\begin{array}{c}\hfill \left({\partial }^{\mu }{\mathcal{B}}^{\nu \sigma }+{\partial }^{\nu }{\mathcal{B}}^{\sigma \mu }+{\partial }^{\sigma }{\mathcal{B}}^{\mu \nu }\right)=-{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\rho }B,\end{array}$$

(15)

to obtain the following from the above integral (14), namely;

$$\begin{array}{c}\hfill +{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left({\partial }_{i}B\right){\overline{C}}_{jk}=-{\displaystyle \frac{1}{2}}\int d^{3}x\left({\partial }^0{\mathcal{B}}^{ij}+{\partial }^i{\mathcal{B}}^{j0}+{\partial }^j{\mathcal{B}}^{0i}\right){\overline{C}}_{ij}.\end{array}$$

(16)

This term would be incorporated into the nilpotent (i.e. $Q_D^2=0$) version of the co-BRST charge $Q_D$. Toward our main goal of obtaining $s_d{Q}_D=0$, we have to apply the co-BRST symmetry transformation operator $s_d$ [cf. Eq. (3)] on the r.h.s. of the above expression which, ultimately, leads to the following explicit result, namely;

$$\begin{array}{c}\hfill -{\displaystyle \frac{1}{2}}\int d^{3}x\left({\partial }^0{\mathcal{B}}^{ij}+{\partial }^i{\mathcal{B}}^{j0}+{\partial }^j{\mathcal{B}}^{0i}\right)\left({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i\right).\end{array}$$

(17)

At this juncture, we have to modify a specific term of the Noether conserved charge $Q_d$ [cf. Eq. (8)] so that, when we apply $s_d$ on a part of that term, the outcome should cancel out with our result in (17). Such a term is the last but one term: ${\displaystyle \frac{1}{2}}\int d^{3}x({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)$. This term can be modified by exploiting the simple algebraic trick as:

$$\begin{array}{ccc}\hfill & & -{\displaystyle \frac{1}{2}}\int d^{3}x({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)\hfill \\ & +& \int d^{3}x({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i).\hfill \end{array}$$

(18)

It is straightforward to note that the outcome of the application of the co-BRST transformations $s_d$ [cf. Eq. (3)] on the first term of the above equation cancels out with our result in (17). Hence, in the final expression for the nilpotent version (i.e. $Q_D^2=0$) of the co-BRST charge $Q_D$, the first term of the above Equation (18) will be present. We would like to point out that, at present stage of our exercise, we have already obtained three crucial terms of the nilpotent version of the co-BRST charge $Q_D$. These very important and useful terms are (i) the full integral in our Equation (13), (ii) the complete integral in (16), and (iii) the first integral that is present in our Equation (18).

Let us now focus on the second term of Equation (18) which can be re-expressed as: $2\int d^{3}x({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i})\left({\partial }_i{\overline{\beta }}_j\right)$. Using the partial integral followed by the Gauss divergence theorem, this term turns out to be

$$\begin{array}{c}\hfill -2\int d^{3}x{\partial }_i({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i}){\overline{\beta }}_j,\end{array}$$

(19)

where we have dropped a total space derivative term. Exploiting the theoretical strength of the appropriate3 EL-EoMs from (7), this integral can be re-written as:

$$\begin{array}{c}\hfill -\int d^{3}x({\partial }^0{f}^i-{\partial }^i{f}^0){\overline{\beta }}_i.\end{array}$$

(20)

The above integral will be present in the expression for the nilpotent version of the co-BRST charge $Q_D$. For our central goal of obtaining $s_d{Q}_D=0$, we have to apply the co-BRST symmetry transformations (3) on the above integral which leads to the following

$$\begin{array}{c}\hfill -s_d\left[\int d^{3}x({\partial }^0{f}^i-{\partial }^i{f}^0){\overline{\beta }}_i\right]=+\int d^{3}x({\partial }^0{f}^i-{\partial }^i{f}^0){\partial }_i{\overline{C}}_2,\end{array}$$

(21)

where we have taken into account (i) the fermionic natures of $s_d$ and the auxiliary vector field $f_{\mu }$, and (ii) the co-BRST symmetry transformations that have been listed in Equation (3). As per the method proposed in our earlier work [21], we have to modify an appropriate term of the non-nilpotent co-BRST charge $Q_d$ [cf. Eq. (8)] such that when $s_d$ acts on a part of that term, the outcome should cancel out with our result in (21). Such a term is: $-{\displaystyle \frac{1}{2}}\int d^{3}x({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\partial }_i{\overline{C}}_2$ which happens to be the sixth term in the expression for $Q_d$ [cf. Eq. (8)]. Using the simple algebraic trick, this integral can be re-written as follows:

$$\begin{array}{c}\hfill +\int d^{3}x({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\partial }_i{\overline{C}}_2-{\displaystyle \frac{3}{2}}\int d^{3}x({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\partial }_i{\overline{C}}_2.\end{array}$$

(22)

It is straightforward to note that the application of the co-BRST symmetry transformations: $s_d{\overline{C}}_2=0,s_d{\beta }_{\mu }=-f_{\mu }$ [cf. Eq. (3)] on the first integral of the above equation cancels out with our result in (21). Hence, the first integral of the above equation will be present in the nilpotent version of the co-BRST charge $Q_D$. We focus, at this stage, on the second integral of (22). Using the partial integration followed by the application of the Gauss divergence theorem, this integral can be re-written as

$$\begin{array}{c}\hfill +{\displaystyle \frac{3}{2}}\int d^{3}x{\partial }_i({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\overline{C}}_2,\end{array}$$

(23)

where we have dropped a total space derivative term because of obvious reason. We are in the position to use the EL-EoM: ${\partial }_{\mu }({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu })={\partial }^{\nu }{B}_4$ which leads to ${\partial }_i({\partial }^0{\beta }^i-{\partial }^i{\beta }^0)=-{\dot{B}}_4$ [cf. Eq. (7)] for the choice $\nu =0$. This specific equation is derived from the FP-ghost part of the Lagrangian density (2). Thus, ultimately, we obtain the following

$$\begin{array}{c}\hfill +{\displaystyle \frac{3}{2}}\int d^{3}x{\partial }_i({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\overline{C}}_2=-{\displaystyle \frac{3}{2}}\int d^{3}x{\dot{B}}_4{\overline{C}}_2,\end{array}$$

(24)

which turns out to be the co-BRST invariant quantity because we observe that: $s_d\left({\dot{B}}_4{\overline{C}}_2\right)=0$ [cf. Eq. (3)]. Assimilating all the useful and relevant integrals, we obtain the final expression for the nilpotent version of the co-BRST charge $Q_D$ as follows:

$$\begin{array}{ccc}\hfill Q_D& =& \int d^{3}x\left[{\displaystyle \frac{1}{2}}\right\{{\mathcal{B}}^{0i}{\overline{F}}_i+B_4{\dot{\overline{C}}}_2-3{\dot{B}}_4{\overline{C}}_2+({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}){\mathcal{B}}_{ij}+B_5{f}^0\hfill \\ & +& ({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0)f_i-({\partial }^0{C}^{ij}+{\partial }^i{C}^{jk}+{\partial }^j{C}^{0i})({\partial }_i{\overline{\beta }}_j-{\partial }_j{\overline{\beta }}_i)+B_3{\overline{F}}^0\hfill \\ & -& ({\partial }^0{\mathcal{B}}^{ij}+{\partial }^i{\mathcal{B}}^{jk}+{\partial }^j{\mathcal{B}}^{0i}){\overline{C}}_{ij}\}+\left(B_1\dot{\overline{C}}-{\dot{B}}_1{\overline{C}}_2\right)+({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\partial }_i{\overline{C}}_2\hfill \\ & -& ({\partial }^0{f}^i-{\partial }^i{f}^0){\overline{\beta }}_i].\hfill \end{array}$$

(25)

A few comments are in order now. First of all, it can be checked that $s_d{Q}_D=0$ where, for this proof, we have to apply the co-BRST symmetry transformations (3) directly on the expression for the modified version of the co-BRST charge $Q_D$ [cf. Eq. (25)]. This observation implies that we have obtained the nilpotent (i.e. $Q_D^2=0$) version of the co-BRST charge $Q_D$ from the non-nilpotent (i.e. $Q_d^2\ne 0$) version of the Noether co-BRST charge $Q_d$. Second, toward our main goal of obtaining $Q_D$ from $Q_d$, first of all, we have performed the partial integration followed by the Gauss divergence theorem on the terms of the Noether co-BRST charge $Q_d$ [cf. Eq. (8)] that contain the gauge-fixing terms for the basic gauge fields $A_{\mu }$ and $A_{\mu \nu \sigma }$. For instance, we have started off with the terms like: $-{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}B\left({\partial }_i{\overline{C}}_{jk}\right)$ and $+{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left({\partial }_{i}C\right)B_{jk}$ because we know that: $B=(\partial ·A)$ and $B_{\mu \nu }={\partial }^{\sigma }{A}_{\sigma \mu \nu }+{\displaystyle \frac{1}{2}}({\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu })$ which are nothing but the gauge-fixing terms for the Abelian 1-form and 3-form gauge fields, respectively. Third, in our derivation of $Q_D$ from $Q_d$, we have exploited (i) the appropriate EL-EoMs, and (ii) the partial integration followed by Gauss’s divergence theorem. Hence, both forms of the charges (i.e. $Q_d$ and $Q_D$) are conserved due to the basic tenets behind Noether’s theorem. Finally, it worthwhile to point out that many of the terms that are present in $Q_D$ [cf. Eq. (25)] are same as the ones that are present in $Q_d$ [cf. Eq. (8)] because such terms are trivially co-BRST invariant.

Right from the beginning [cf. Eqs. (1),(2)], we observe that the basic as well as the auxiliary (anti-)ghost fields (present in the FP-ghost part of the Lagrangian density ${\mathcal{L}}_{\left(FP\right)}$) are decoupled from the rest of the physical theory (described by the Lagrangian density ${\mathcal{L}}_{\left(NG\right)}$). As a consequence, a specific state in the quantum Hilbert space of states is the direct product [24] of the physical states (i.e. $|phys>$) and the ghost states (i.e $|ghost>$). Within the framework of BRST formalism, the physicality criterion, w.r.t. to a nilpotent and conserved charge, requires that the physical states (i.e. $|phys>$) are those that are annihilated by this conserved and nilpotent charge4. A close and careful look at the nilpotent version of the co-BRST charge $Q_D$ [cf. Eq. (25)] demonstrates that every individual term of this charge carries a ghost number equal to $-1$ which is made up of (i) the basic/auxiliary physical fields with ghost number equal to zero [cf. Eq. (1)] and the basic/auxiliary ghost field(s) [cf. Eq. (2)] with ghost number equal to $-1$, and (ii) the basic/auxiliary (anti-)ghost fields present in ${\mathcal{L}}_{\left(FP\right)}$. In the physicality criterion w.r.t. the nilpotent version of the co-BRST charge, every individual term of this charge acts on a specifically chosen quantum state. When the ghost fields of $Q_D$ act on the ghost states (i.e $|ghost>$), they always produce non-zero result. Hence, the physicality criterion (i.e. $Q_D|phys>=0$) w.r.t. the conserved and nilpotent version of the co-BRST charge $Q_D$ requires that the physical fields (carrying the ghost number equal to zero) must annihilate the true physical states (i.e. $|phys>$) which should be consistent with the Dirac quantization conditions for systems that are endowed with constraints. As far as our 4D BRST-quantized field theoretic system is concerned, the physicality criterion w.r.t. the co-BRST charge must lead to the annihilation of the physical states by the dual version of the first-class constraints of the 4D classical field-theoretic system because the physicality criterion (i.e. $Q_B|phys>=0$) w.r.t. the conserved and nilpotent version of the BRST charge leads to the annihilation of the physical states, at the quantum level, by the operator forms of purely the first-class constraints of our 4D classical field-theoretic system (see, e.g. Appendix A).

Against the backdrop of the above paragraph, we note that the physicality criterion (i.e. $Q_D|phys>=0$) w.r.t. the conserved and nilpotent version of the co-BRST charge $Q_D$, leads to the following conditions on the physical states, namely;

$$\begin{array}{ccc}\hfill & & B_1|phys>=0,{\dot{B}}_1|phys>=0,{\displaystyle \frac{1}{2}}{\mathcal{B}}_{ij}|phys>=0,\hfill \\ & & {\displaystyle \frac{1}{2}}\left({\partial }^0{\mathcal{B}}^{ij}+{\partial }^i{\mathcal{B}}^{j0}+{\partial }^j{\mathcal{B}}^{0i}\right)|phys>=0.\hfill \end{array}$$

(26)

It is worthwhile to point out a few subtle issues that have been taken care of in the above derivation from the requirement: $Q_D|phys>=0$. As mentioned earlier, we have focused on the operation of the physical fields (with ghost number equal to zero) which are associated with the basic anti-ghost fields of our 4D BRST-quantized theory. We lay emphasis on this issue because there are a couple of terms: ${\displaystyle \frac{1}{2}}\int d^{3}x({\mathcal{B}}^{0i}{\overline{F}}_i+B_3{\overline{F}}^0)$ in the expression for $Q_D$ where the auxiliary fields ${\mathcal{B}}^{0i}$ and $B_3$ carry the ghost number equal to zero. However, these fields have not contributed anything in the conditions (26) because they are not associated with the basic anti-ghost fields of our theory. Rather, they are associated with the components of the auxiliary anti-ghost fermionic vector field ${\overline{F}}_{\mu }$. A close look at the above conditions on the physical states (i.e. $|phys>$) w.r.t. the nilpotent and conserved version of the co-BRST charge $Q_D$ and the ones that have been obtained in equation (A.5) w.r.t. the nilpotent and conserved version of the BRST charge $Q_B$ (cf. Appendix A) establishes that these conditions are dual to one-another. This is due to the fact that under the following discrete duality symmetry transformations (see. e.g. [22])

$$\begin{array}{ccc}\hfill & & A_{\mu }⟶\mp {\displaystyle \frac{1}{3!}}{\epsilon }_{\mu \nu \sigma \rho }{A}^{\nu \sigma \rho },A_{\mu \nu \sigma }⟶\pm {\epsilon }_{\mu \nu \sigma \rho }{A}^{\rho },B_{\mu \nu }\to \pm {\mathcal{B}}_{\mu \nu },{\mathcal{B}}_{\mu \nu }\to \mp B_{\mu \nu },\hfill \\ & & B\to \pm B_1,B_1\to \mp B,B_2\to \pm B_3,B_3\to \mp B_2,{\varphi }_{\mu }\to \pm {\tilde{\varphi }}_{\mu },{\tilde{\varphi }}_{\mu }\to \mp {\varphi }_{\mu },\hfill \end{array}$$

(27)

the non-ghost part of the Lagrangian density ${\mathcal{L}}_{\left(NG\right)}$ [cf. Eq. (1)] remains invariant. There exists a set of duality symmetry transformations in the ghost sector, too [22]. However, for our immediate purpose, we do not need to mention them urgently here. In other words we note that the conditions (26) on the physical states (i.e. $|phys>$) w.r.t. the conserved and nilpotent co-BRST charge $Q_D$ are deeply connected with the conditions: $B|phys>=0,\dot{B}|phys>=0,{\displaystyle \frac{1}{2}}{B}_{ij}|phys>=0,{\displaystyle \frac{1}{2}}\left({\partial }^0{B}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{B}^{0i}\right)|phys>=0$ [cf. Eq. (A.5)] that emerge out from the physicality criterion (i.e. $Q_B|phys>=0$) w.r.t. the conserved and nilpotent version of the BRST charge due to the presence of the discrete duality symmetry transformations in (27) in the non-ghost sector of our theory.

5. Nilpotent Anti-Co-BRST Transformations

Corresponding to the co-BRST invariant Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eqs. (1),(2)], we have the anti-co-BRST invariant Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ which incorporates into it the non-ghost part [i.e. ${\mathcal{L}}_{\left(ng\right)}$] and the FP-ghost part [i.e. ${\mathcal{L}}_{\left(fp\right)}$]. These Lagrangian densities (i.e. ${\mathcal{L}}_{\left(ng\right)}$ and ${\mathcal{L}}_{\left(fp\right)}$) are the analogues of the non-ghost part of the Lagrangian density ${\mathcal{L}}_{\left(NG\right)}$ [cf. Eq. (1)] and FP-ghost part of the Lagrangian density ${\mathcal{L}}_{\left(FP\right)}$ [cf. Eq. (2)] of ${\mathcal{L}}_{\left(B\right)}$. First of all, let us focus on the non-ghost sector ${\mathcal{L}}_{\left(ng\right)}$ of the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ which is described by the following

$$\begin{array}{ccc}\hfill & & {\mathcal{L}}_{\left(ng\right)}={\displaystyle \frac{1}{2}}{B}^2-B(\partial ·A)+{\displaystyle \frac{1}{2}}{B}_2\left(\partial ·\varphi \right)-{\displaystyle \frac{1}{4}}{B}_2^2+{\displaystyle \frac{1}{2}}{B}_3\left(\partial ·\tilde{\varphi }\right)-{\displaystyle \frac{1}{4}}{B}_3^2\hfill \\ & +& {\displaystyle \frac{1}{2}}{B}_1^2+B_1\left({\displaystyle \frac{1}{3!}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\mu }{A}_{\nu \sigma \rho }\right)-{\displaystyle \frac{1}{4}}{\left({\overline{B}}_{\mu \nu }\right)}^2-{\displaystyle \frac{1}{2}}{\overline{B}}_{\mu \nu }\left[{\partial }_{\sigma }{A}^{\sigma \mu \nu }-{\displaystyle \frac{1}{2}}\left({\partial }^{\mu }{\varphi }^{\nu }-{\partial }^{\nu }{\varphi }^{\mu }\right)\right]\hfill \\ & -& {\displaystyle \frac{1}{4}}{\left({\overline{\mathcal{B}}}_{\mu \nu }\right)}^2-{\displaystyle \frac{1}{2}}{\overline{\mathcal{B}}}_{\mu \nu }\left[{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\sigma }{A}_{\rho }-{\displaystyle \frac{1}{2}}\left({\partial }^{\mu }{\tilde{\varphi }}^{\nu }-{\partial }^{\nu }{\tilde{\varphi }}^{\mu }\right)\right],\hfill \end{array}$$

(28)

where a couple of bosonic Nakanishi-Lautrup type auxiliary fields ${\overline{B}}_{\mu \nu }$ and ${\overline{\mathcal{B}}}_{\mu \nu }$ (with zero ghost numbers) have been invoked to linearize the gauge-fixing term for the free Abelian 3-form gauge field and kinetic term for the free Abelian 1-form field, respectively. A comparison between the Lagrangian density (1) and (28) establishes that the last four terms have a few decisive differences in the signs. The FP-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ for our 4D field-theoretic system is described by the following Lagrangian density [22]

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\left(fp\right)}& =& {\displaystyle \frac{1}{2}}[\left({\partial }_{\mu }{\overline{C}}_{\nu \sigma }+{\partial }_{\nu }{\overline{C}}_{\sigma \mu }+{\partial }_{\sigma }{\overline{C}}_{\mu \nu }\right)\left({\partial }^{\mu }{C}^{\nu \sigma }\right)-\left({\partial }_{\mu }{\overline{C}}^{\mu \nu }-{\partial }^{\nu }{\overline{C}}_1\right)F_{\nu }\hfill \\ & +& \left({\partial }_{\mu }{C}^{\mu \nu }-{\partial }^{\nu }{C}_1\right){\overline{f}}_{\nu }+\left(\partial ·\overline{\beta }\right)B_4-\left(\partial ·\beta \right)B_5-B_4{B}_5-2{\overline{f}}^{\mu }{F}_{\mu }\hfill \\ & -& \left({\partial }_{\mu }{\overline{\beta }}_{\nu }-{\partial }_{\mu }{\overline{\beta }}_{\nu }\right)\left({\partial }^{\mu }{\beta }^{\nu }\right)-{\partial }_{\mu }{\overline{C}}_2{\partial }^{\mu }{C}_2]-{\partial }_{\mu }\overline{C}{\partial }^{\mu }C,\hfill \end{array}$$

(29)

which is different from our earlier FP-ghost Lagrangian density [cf. Eq. (2)] that has been taken into account in the co-BRST invariant Lagrangian density ${\mathcal{L}}_{\left(B\right)}$. It should be pointed out that we have (i) two new fermionic (anti-)ghost [i.e. $\left({\overline{f}}_{\mu }\right)F_{\mu }$] auxiliary fields in (29) that carry the ghost numbers (-1)+1, respectively, and (ii) the second and third terms of the FP-ghost parts of the Lagrangian densities in (2) and (29) are quite different.

As per the sacrosanct requirements of the BRST formalism, it is essential that we should have the (i) off-shell nilpotent (i.e. $s_{\left(a\right)d}^2=0$), and (ii) absolutely anticommuting (i.e. $s_d{s}_{ad}+s_{ad}{s}_d=0$) infinitesimal co-BRST (i.e. $s_d$) and anti-co-BRST (i.e. $s_{ad}$) transformations at the quantum level in a properly BRST-quantized theory. In view of these two requirements, corresponding to the infinitesimal, continuous and off-shell nilpotent ($s_d^2=0$) co-BRST symmetry transformations (3), we have the following infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-BRST transformations ($s_{ad}$), namely;

$$\begin{array}{ccc}\hfill & & s_{ad}{A}_{\mu \nu \sigma }={\epsilon }_{\mu \nu \sigma \rho }{\partial }^{\rho }C,s_{ad}{A}_{\mu }={\displaystyle \frac{1}{2}}{\epsilon }_{\mu \nu \sigma \rho }{\partial }^{\nu }{C}^{\sigma \rho },s_{ad}{C}_{\mu \nu }={\partial }_{\mu }{\beta }_{\nu }-{\partial }_{\nu }{\beta }_{\mu },\hfill \\ & & s_{ad}{\mathcal{B}}_{\mu \nu }={\partial }_{\mu }{F}_{\nu }-{\partial }_{\nu }{F}_{\mu },s_{ad}{\beta }_{\mu }={\partial }_{\mu }{C}_2,s_{ad}{\overline{\beta }}_{\mu }=-{\overline{f}}_{\mu },s_d{\tilde{\varphi }}_{\mu }=+F_{\mu },\hfill \\ & & s_{ad}{\overline{C}}_{\mu \nu }=-{\overline{\mathcal{B}}}_{\mu \nu },s_{ad}\overline{C}=-B_1,s_{ad}{\overline{C}}_2=B_5,s_{ad}{\overline{C}}_1=B_3,\hfill \\ & & s_{ad}{C}_1=+B_4,s_{ad}{f}_{\mu }=+{\partial }_{\mu }{B}_4,s_{ad}{\overline{F}}_{\mu }=+{\partial }_{\mu }{B}_3,\hfill \\ & & s_{ad}[C,C_2,B,B_1,B_2,B_3,B_4,B_5,{\overline{f}}_{\mu },F_{\mu },{\varphi }_{\mu },B_{\mu \nu },{\overline{B}}_{\mu \nu },{\overline{\mathcal{B}}}_{\mu \nu }]=0,\hfill \end{array}$$

(30)

which transform the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ to the total spacetime derivative:

$$\begin{array}{ccc}\hfill s_{ad}{\mathcal{L}}_{\left(\overline{B}\right)}& =& {\displaystyle \frac{1}{2}}{\partial }_{\mu }[{\overline{\mathcal{B}}}^{\mu \nu }{F}_{\nu }-({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\overline{\mathcal{B}}}_{\nu \sigma }-B_5{\partial }^{\mu }{C}_2\hfill \\ & +& B_3{F}^{\mu }-B_4{\overline{f}}^{\mu }+({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }){\overline{f}}_{\nu }]+{\partial }_{\mu }\left[B_1{\partial }^{\mu }C\right].\hfill \end{array}$$

(31)

The above observation renders the action integral (i.e. $S=\int d^{4}x{\mathcal{L}}_{\left(\overline{B}\right)}$), corresponding to the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$, invariant (i.e. $s_{ad}S=0$) due to Gauss’s divergence theorem where all the physical fields vanish off as $x\to \pm \infty$. Hence the infinitesimal, continuous and nilpotent anti-co-BRST transformations (30) are the symmetry transformations for our 4D BRST-quantized field-theoretic system of the Abelian 3-form and 1-form gauge theories.

We wrap-up this short (but quite important) section with the following crucial remarks. First of all, we would like to point out that the Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ are coupled because some of their fields are not independent (in the true sense of the word) because of the presence of the non-trivial CF-type restrictions on our 4D BRST -quantized theory (cf. Sec. 8 below for details). These coupled Lagrangian densities are also equivalent from the points of view of (i) the symmetry considerations (cf. Sec. 8 below), and (ii) the direct equality (see, e.g. [20] for details). Second, the nilpotency property establishes that the anti-co-BRST symmetry transformation operator $s_{ad}$ is fermionic in nature which transforms a bosonic field into its counterpart fermionic field and vice-versa. Third, the operation of the anti-co-BRST symmetry transformation operator on a field leads to the increment in the ghost number as well as the mass dimension (in the natural units: $\hslash =c=1$) by one [cf. Eq. (30)]. Fourth, the absolute anticommutativity property (i.e. $s_d{s}_{ad}+s_{as}{s}_d=0$) between the co-BRST and anti-co-BRST symmetry transformation operators encodes (i) the linear independence of these two transformation operators, and (ii) the validity of the (anti-)BRST and (anti-)co-BRST invariant CF-type restrictions (cf. Sec. 8 for details). Finally, the nilpotent co-BRST and anti-co-BRST symmetry transformation operators are quite different from the nilpotent $\mathcal{N}=2$ SUSY transformation operators because of their absolute anticommuting property. Hence, the latter property is the distinguishing (and very crucial) feature of the co-BRST and anti-co-BRST symmetry transformation operators.

6. Conserved Noether Anti-Co-BRST Charge: Comment on Its Nilpotency Property

In our previous section, we have seen that the action integral (corresponding to the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$) remains invariant because of (i) our observation in (31), and (ii) the application of the partial integration followed by the Gauss divergence theorem due to which all the physical fields vanish off as $x\to \pm \infty$. This specific observation, according to the Noether theorem, leads to the derivation of the Noether anti-co-BRST current (i.e. $J_{ad}^{\mu }$) by exploiting the following formula, namely;

$$\begin{array}{ccc}\hfill J_{\left(ad\right)}^{\mu }& =& s_{ad}{\Phi }_i{\displaystyle \frac{\partial {\mathcal{L}}_{\left(\overline{B}\right)}}{\partial \left({\partial }_{\mu }{\Phi }_i\right)}}-{\displaystyle \frac{1}{2}}[{\overline{\mathcal{B}}}^{\mu \nu }{F}_{\nu }-({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\overline{\mathcal{B}}}_{\nu \sigma }-B_5{\partial }^{\mu }{C}_2\hfill \\ & +& B_3{F}^{\mu }-B_4{\overline{f}}^{\mu }+({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }){\overline{f}}_{\nu }]-B_1{\partial }^{\mu }C,\hfill \end{array}$$

(32)

where the generic field ${\Phi }_i\equiv A_{\mu \nu \sigma },C_{\mu \nu },{\overline{C}}_{\mu \nu },A_{\mu },{\varphi }_{\mu },{\tilde{\varphi }}_{\mu },{\beta }_{\mu },{\overline{\beta }}_{\mu },C_1,{\overline{C}}_1,C,\overline{C}$ stands for all the bosonic/ fermionic dynamical fields of the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$. These fields are dynamical because of the presence of their derivative terms in ${\mathcal{L}}_{\left(\overline{B}\right)}$. The rest of the terms (ROTs), in the above equation, are the ones that are present in the square brackets of Equation (31) modulo a sign factor which is consistent with the basic tenets of Noether’s theorem. The first term, in the above Equation (32), can be precisely computed by using (i) the proper rules for the derivatives w.r.t. the fermionic and bosonic fields of our 4D BRST-quantized theory, and (ii) the infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-co-BRST symmetry transformations $s_{ad}$ [cf. Eq. (30)]. The explicit expression for the Noether anti-co-BRST current $J_{\left(ad\right)}^{\mu }\left(fp\right)$, from the FP-ghost part (i.e. ${\mathcal{L}}_{\left(fp\right)}$) of the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ plus the ROTs, turns out to be the following

$$\begin{array}{ccc}\hfill J_{\left(ad\right)}^{\mu }\left(fp\right)& =& -{\displaystyle \frac{1}{2}}[({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu })\left({\partial }_{\nu }{\beta }_{\sigma }-{\partial }_{\sigma }{\beta }_{\nu }\right)+B_5{\partial }^{\mu }{C}_2\hfill \\ & +& B_4{\overline{f}}^{\mu }-({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }){\overline{f}}_{\nu }+({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }){\partial }_{\nu }{C}_2],\hfill \end{array}$$

(33)

where the argument $\left(fp\right)$ in the above current [i.e. $J_{\left(ad\right)}^{\mu }\left(fp\right)$] denotes the fact that (i) we have taken into account only the Lagrangian density ${\mathcal{L}}_{\left(fp\right)}$, and (ii) the rest of the terms (ROTs) of Equation (31) which have been taken into account from the terms that are present in the square brackets in the transformation $s_{ad}{\mathcal{L}}_{\left(\overline{B}\right)}$ [cf. Eq. (32)]. It is very interesting to point out that the above part [i.e. $J_{\left(ad\right)}^{\mu }\left(fp\right)$] of the total anti-co-BRST current $J_{\left(ad\right)}^{\mu }$ is conserved [i.e. ${\partial }_{\mu }{J}_{\left(ad\right)}^{\mu }\left(fp\right)=0$] provided we use the following EL-EoMs

$$\begin{array}{ccc}\hfill & & \square C_2=0,{\partial }_{\mu }\left({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }\right)={\partial }^{\nu }{B}_4,{\partial }_{\mu }\left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right)=-{\partial }^{\nu }{B}_5,\hfill \\ & & {\partial }_{\mu }\left({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }\right)={\displaystyle \frac{1}{2}}\left({\partial }^{\nu }{\overline{f}}^{\sigma }-{\partial }^{\sigma }{\overline{f}}^{\nu }\right),(\partial ·\overline{f})=0,\hfill \end{array}$$

(34)

which emerge out from the Lagrangian density ${\mathcal{L}}_{\left(fp\right)}$. The contribution [i.e. $J_{\left(ad\right)}^{\mu }\left(ng\right)$] to the total anti-co-BRST current $J_{\left(ad\right)}^{\mu }$, that emerges from the non-ghost part of the Lagrangian density (i.e. ${\mathcal{L}}_{\left(ng\right)}$) of the total Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$, is as follows:

$$\begin{array}{ccc}\hfill J_{\left(ad\right)}^{\mu }\left(ng\right)& =& B_1{\partial }^{\mu }C-{\displaystyle \frac{1}{2}}[({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\overline{\mathcal{B}}}_{\nu \sigma }-B_3{F}^{\mu }-{\overline{\mathcal{B}}}^{\mu \nu }{F}_{\nu }\hfill \\ & +& {\epsilon }^{\mu \nu \sigma \rho }\left({\partial }_{\nu }C\right){\overline{B}}_{\sigma \rho }+{\epsilon }^{\mu \nu \sigma \rho }B\left({\partial }_{\nu }{C}_{\sigma \rho }\right)].\hfill \end{array}$$

(35)

It is quite interesting to point out that the above Noether current $J_{\left(ad\right)}^{\mu }\left(ng\right)$ has been derived from (i) the use of the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$ [cf. Eq. (28)], and (ii) the infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-co-BRST symmetry transformations $s_{ad}$. Just like the current $J_{\left(ad\right)}^{\mu }\left(fp\right)$, this current [i.e. $J_{\left(ad\right)}^{\mu }\left(ng\right)$], emerging out purely from the non-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$ [cf. Eq. (28)], is also conserved [i.e. ${\partial }_{\mu }{J}_{\left(ad\right)}^{\mu }\left(ng\right)=0$]. To substantiate this statement, we have to use the following EL-EoMs

$$\begin{array}{ccc}\hfill & & {\partial }_{\mu }{\overline{\mathcal{B}}}^{\mu \nu }+{\partial }^{\nu }{B}_3=0,{\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\overline{B}}_{\sigma \rho }+{\partial }^{\mu }{B}_1=0,{\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\overline{\mathcal{B}}}_{\sigma \rho }-{\partial }^{\mu }B=0,\hfill \\ & & {\partial }_{\mu }\left({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }\right)={\displaystyle \frac{1}{2}}\left({\partial }^{\nu }{F}^{\sigma }-{\partial }^{\sigma }{F}^{\nu }\right),(\partial ·F)=0,\square C=0,\hfill \end{array}$$

(36)

which are derived from the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$. We would like to lay emphasis on the following two inputs, namely;

$$\begin{array}{ccc}\hfill & & \left({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }\right){\partial }_{\mu }{\overline{\mathcal{B}}}_{\nu \sigma }=\left({\partial }^{\mu }{\overline{\mathcal{B}}}^{\nu \sigma }+{\partial }^{\nu }{\overline{\mathcal{B}}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{\mathcal{B}}}^{\mu \nu }\right){\partial }_{\mu }{C}_{\nu \sigma },\hfill \\ & & {\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\overline{\mathcal{B}}}_{\sigma \rho }-{\partial }^{\mu }B=0⟺\left({\partial }^{\mu }{\overline{\mathcal{B}}}^{\nu \sigma }+{\partial }^{\nu }{\overline{\mathcal{B}}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{\mathcal{B}}}^{\mu \nu }\right)-{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\rho }B=0,\hfill \end{array}$$

(37)

which are found to be very useful in the proof of the conservation law [i.e. ${\partial }_{\mu }{J}_{\left(ad\right)}^{\mu }\left(ng\right)=0$]. It is worthwhile to mention that the latter (i.e. bottom) entry in the above equation is nothing but the two different ways of writing the EL-EoMs w.r.t the Abelian 1-form vector field $A_{\mu }$ that is derived from non-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$.

At this juncture, we would like to point out that the total anti-co-BRST current $J_{\left(ad\right)}^{\mu }=J_{\left(ad\right)}^{\mu }\left(ng\right)+J_{\left(ad\right)}^{\mu }\left(fp\right)$ is the sum of the currents in equations (35) and (33). It is crystal clear that this total current is also conserved whose explicit expression is as follows:

$$\begin{array}{ccc}\hfill J_{\left(ad\right)}^{\mu }& =& -{\displaystyle \frac{1}{2}}[({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\overline{\mathcal{B}}}_{\nu \sigma }-B_3{F}^{\mu }-{\overline{\mathcal{B}}}^{\mu \nu }{F}_{\nu }\hfill \\ & +& {\epsilon }^{\mu \nu \sigma \rho }\left({\partial }_{\nu }C\right){\overline{B}}_{\sigma \rho }+{\epsilon }^{\mu \nu \sigma \rho }B\left({\partial }_{\nu }{C}_{\sigma \rho }\right)+B_4{\overline{f}}^{\mu }+B_5{\partial }^{\mu }{C}_2\hfill \\ & +& ({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu })\left({\partial }_{\nu }{\beta }_{\sigma }-{\partial }_{\sigma }{\beta }_{\nu }\right)\hfill \\ & -& ({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }){\overline{f}}_{\nu }+({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }){\partial }_{\nu }{C}_2]+B_1{\partial }^{\mu }C.\hfill \end{array}$$

(38)

From the above conserved current, it is straightforward to compute the expression for the conserved Noether anti-co-BRST charge (i.e. $Q_{ad}=\int d^{3}x{J}_{\left(ad\right)}^0$) as follows:

$$\begin{array}{ccc}\hfill Q_{ad}& =& \int d^{3}x[B_1\dot{C}-{\displaystyle \frac{1}{2}}\{({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i}){\overline{\mathcal{B}}}_{ij}-B_3{F}^0-{\overline{\mathcal{B}}}^{0i}{F}_i\hfill \\ & +& {\epsilon }^{0ijk}\left({\partial }_{i}C\right){\overline{B}}_{jk}+{\epsilon }^{0ijk}B\left({\partial }_i{C}_{jk}\right)+B_4{\overline{f}}^0+B_5{\dot{C}}_2\hfill \\ & +& ({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}\left)\right({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & -& ({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\overline{f}}_i+({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2\left\}\right].\hfill \end{array}$$

(39)

The above conserved Noether anti-co-BRST charge is the generator for the infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-co-BRST symmetry transformations $s_{ad}$ [cf. Eq. (30)] which can be verified in a straightforward manner by taking into account Equation (9) with the following replacements: $s_d\to s_{ad},Q_d\to Q_{ad}$.

We end this section with the following clinching remarks. First of all, we find that using the basic tenets and techniques of Noether’s theorem leads to the derivation of the conserved Noether anti-co-BRST charge $Q_{ad}$ [cf. Eq. (39)] from the Noether conserved anti-co-BRST current $J_{\left(ad\right)}^{\mu }$. Second, we observe that the Noether current $J_{\left(ad\right)}^{\mu }\left(ng\right)$ [cf. Eq. (38)], that is derived from the non-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$ [cf. Eq. (28)], is found to be conserved [i.e. ${\partial }_{\mu }{J}_{\left(ad\right)}^{\mu }\left(ng\right)=0$] provided we use the EL-EoMs [cf. Eq. (36)] that are derived from the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$. Third, we note that the the sum of the Noether current (that is derived from the ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(fp\right)}$ [cf. Eq. (33)]) and the rest of the terms (ROTs) [that have been taken into account in Equation (32) due to Noether’s theorem because of our observation in (31)] also turn out to be conserved [i.e. ${\partial }_{\mu }{J}_{\left(ad\right)}^{\mu }\left(fp\right)=0$] provided we use the EL-EoMs [cf. Eq. (34)] that have been derived from the Lagrangian density ${\mathcal{L}}_{\left(fp\right)}$. Fourth, it is interesting to point out that the ROTS in Equation (32) are not required for the proof of the conservation law ${\partial }_{\mu }{J}_{\left(ad\right)}^{\mu }\left(ng\right)=0$ (for the Noether current $J_{\left(ad\right)}^{\mu }\left(ng\right)$ that emerges out from the non-ghost sector of Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$). Finally, we find that the conserved Noether anti-co-BRST charge $Q_{ad}$ is not nilpotent of order two (i.e. $Q_{ad}^2\ne 0$). This can be verified by using the standard relationship between the continuous symmetry transformations and their generator as the Noether conserved charge. For instance, we note the following relationship, namely;

$$\begin{array}{ccc}\hfill s_{ad}{Q}_{ad}=i\left\{Q_{ad},Q_{ad}\right\}& \equiv & +{\displaystyle \frac{1}{2}}\int d^{3}x[({\partial }^0{\overline{\mathcal{B}}}^{ij}+{\partial }^i{\overline{\mathcal{B}}}^{jk}+{\partial }^j{\overline{\mathcal{B}}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & +& ({\partial }^0{\overline{f}}^i-{\partial }^i{\overline{f}}^0){\partial }_i{C}_2]\ne 0,\hfill \end{array}$$

(40)

which, ultimately, implies: $i\left\{Q_{ad},Q_{ad}\right\}=2i{Q}_{ad}^2\ne 0$. In other words, we find that the explicit computation of the quantity $s_{ad}{Q}_{ad}$ [by using equations (30) and (39)] leads to our conclusion that the Noether anti-co-BRST charge $Q_{ad}$ is not nilpotent (i.e. $Q_{ad}^2\ne 0$) of order two. As a consequence, this conserved (but non-nilpotent) Noether charge is not suitable for the discussion on the cohomological aspects of BRST formalism (see, e.g. [23] for explicit examples) and the physicality criterion w.r.t. it in the BRST-quantized version of the quantum Hilbert space of states (cf. Sec. 7 below for details).

7. Nilpotent and Conserved Anti-Co-BRST Charge from Non-nilpotent Noether Anti-co-BRST Charge

The key purpose of our present section is to derive explicitly the conserved and off-shell nilpotent (i.e. $Q_{AD}^2=0$) version of the anti-co-BRST charge $Q_{AD}$ from the non-nilpotent (i.e. $Q_{ad}^2\ne 0$) version [cf. Eq. (39)] of the Noether conserved anti-co-BRST charge $Q_{ad}$ [cf. Eq. (40)]. In this context, we shall exploit the theoretical techniques that have been proposed in our earlier work [21] where the emphasis have been laid on the use of (i) the appropriate EL-EoMs which are derived from the perfectly anti-co-BRST invariant Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$, (ii) the partial integration followed by the application of the celebrated Gauss divergence theorem, and (iii) the infinitesimal, continuous and off-shell nilpotent anti-co-BRST symmetry transformations $s_{ad}$ [cf. Eq. (30)] at appropriate places, to ensure that we finally obtain: $s_{ad}{Q}_{AD}=i\left\{Q_{AD},Q_{AD}\right\}=0\Rightarrow Q_{AD}^2=0$.

In view of our elaborate discussion on the derivation of the conserved and nilpotent version of the co-BRST charge $Q_D$ from the conserved (but non-nilpotent) version of the Noether co-BRST charge $Q_d$, we shall be quite brief in our present section as far as the derivation of the nilpotent version of the anti-co-BRST charge $Q_{AD}$ from its counterpart non-nilpotent version [cf. Eq. (40)] of the Noether anti-co-BRST charge $Q_{ad}$ is concerned. In this context, let us focus, first of all, on the fifth and sixth terms of the conserved (but non-nilpotent) anti-co-BRST charge $Q_{ad}$ [cf. Eq. (39)]. Application of the partial integration followed by the Gauss divergence theorem leads to the following

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left[\left({\partial }_{i}C\right){\overline{B}}_{jk}+B\left({\partial }_i{C}_{jk}\right)\right]=-{\displaystyle \frac{1}{2}}\int d^{3}x{\epsilon }^{0ijk}\left[C\left({\partial }_i{\overline{B}}_{jk}\right)+\left({\partial }_{i}B\right)C_{jk}\right],\end{array}$$

(41)

where we have dropped the total space derivative terms for obvious reasons. Using the following EL-EoMs (that have been derived from ${\mathcal{L}}_{\left(ng\right)}$ [cf. Eq. (28])), namely;

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\overline{B}}_{\sigma \rho }=-{\partial }^{\mu }{B}_1⟹{\displaystyle \frac{1}{2}}{\epsilon }^{oijk}{\partial }_i{\overline{B}}_{jk}=-{\dot{B}}_1,\hfill \\ & & {\displaystyle \frac{1}{2}}{\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\nu }{\overline{\mathcal{B}}}_{\sigma \rho }={\partial }^{\mu }B=0⟺\left({\partial }^{\mu }{\overline{\mathcal{B}}}^{\nu \sigma }+{\partial }^{\nu }{\overline{\mathcal{B}}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{\mathcal{B}}}^{\mu \nu }\right)={\epsilon }^{\mu \nu \sigma \rho }{\partial }_{\rho }B,\hfill \end{array}$$

(42)

we find the following form of Equation (41), namely;

$$\begin{array}{c}\hfill -\int d^{3}x{\dot{B}}_{1}C+{\displaystyle \frac{1}{2}}\int d^{3}x\left({\partial }^0{\overline{\mathcal{B}}}^{ij}+{\partial }^i{\overline{\mathcal{B}}}^{j0}+{\partial }^j{\overline{\mathcal{B}}}^{i0}\right)C_{ij},\end{array}$$

(43)

where we have used: ${\epsilon }^{0ijk}{\partial }_{k}B=({\partial }^0{\overline{\mathcal{B}}}^{ij}+{\partial }^i{\overline{\mathcal{B}}}^{jk}+{\partial }^j{\overline{\mathcal{B}}}^{i0})$. Both the above inetgrals will be present in the nilpotent version of the anti-co-BRST charge $Q_{AD}$. The first integrals of (i) the Noether non-nilpotent charge $Q_{ad}$ [cf. Eq. (39)], and (ii) the above equation can be combined together to obtain the analogue of the expression (13) (cf. Sec. 4) as follows

$$\begin{array}{c}\hfill Q_{AD}^{\left(1\right)}=\int d^{3}x\left[B_1\dot{C}-{\dot{B}}_{1}C\right],\end{array}$$

(44)

which is an anti-co-BRST invariant quantity (i.e. $s_{ad}{Q}_{AD}^{\left(1\right)}=0$). The superscript $\left(1\right)$ on $Q_{AD}^{\left(1\right)}$ denotes the fact that the auxiliary field $B_1$ is present in this expression. Toward our main goal of obtaining $s_{ad}{Q}_{AD}=0$, we have to apply the anti-co-BRST symmetry operator $s_{ad}$ on the second integral of Equation (43) which leads to the following integral:

$$\begin{array}{c}\hfill +{\displaystyle \frac{1}{2}}\int d^{3}x\left({\partial }^0{\overline{\mathcal{B}}}^{ij}+{\partial }^i{\overline{\mathcal{B}}}^{j0}+{\partial }^j{\overline{\mathcal{B}}}^{i0}\right)\left({\partial }_i{\beta }_j-{\partial }_j{\beta }_i\right).\end{array}$$

(45)

We have to modify an appropriate integral in the expression for $Q_{ad}$ [cf. Eq. (39)] so that when the transformations $s_{ad}$ act on a part of that term, the outcome should cancel out with our result in (45). Such an integral is: $-{\displaystyle \frac{1}{2}}\int d^{3}x({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)$ which can be modified in the following form:

$$\begin{array}{ccc}\hfill & & +{\displaystyle \frac{1}{2}}\int d^{3}x({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & & -\int d^{3}x({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i).\hfill \end{array}$$

(46)

It is obvious that if we apply $s_{ad}$ [cf. Eq. (30)] on the first integral of the above equation, the outcome will cancel out with our result in (45). Hence, the first integral will be present in the nilpotent version of the anti-co-BRST charge $Q_{AD}$. Toward ou main goal of obtaining $s_{ad}{Q}_{AD}=0$, we now concentrate on the second integral of (46) which is equivalent to

$$\begin{array}{c}\hfill -2\int d^{3}x({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})\left({\partial }_i{\beta }_j\right)\equiv +2\int d^{3}x{\partial }_i({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}){\beta }_j,\end{array}$$

(47)

where the r.h.s. of the above equation has been obtained after the applications of (i) the partail integration, and (ii) the Gauss divergence theorem. Using the EL-EoM: ${\partial }_{\mu }({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu })={\displaystyle \frac{1}{2}}({\partial }^{\nu }{\overline{f}}^{\sigma }-{\partial }^{\sigma }{\overline{f}}^{\nu })$ (with the choices: $\nu =0,\sigma =j$), it can be checked that the r.h.s. of Equation (47) can be re-written as

$$\begin{array}{c}\hfill +2\int d^{3}x{\partial }_i({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i}){\beta }_j=-\int d^{3}x\left({\partial }^0{\overline{f}}^i-{\partial }^0{\overline{f}}^i\right){\beta }_i,\end{array}$$

(48)

which will be present in the final expression of the nilpotent $Q_{AD}$. Toward our central goal of achieving $s_{ad}{Q}_{AD}=0$, we have to apply $s_{ad}$ on the r.h.s. of (48) which yields:

$$\begin{array}{c}\hfill -s_{ad}\left[\int d^{3}x\left({\partial }^0{\overline{f}}^i-{\partial }^i{\overline{f}}^0\right){\beta }_i\right]=\int d^{3}x\left({\partial }^0{\overline{f}}^i-{\partial }^0{\overline{f}}^i\right){\partial }_i{C}_2,\end{array}$$

(49)

where we have taken into account the fermionic (i.e. $s_{ad}{\overline{f}}_{\mu }+{\overline{f}}_{\mu }{s}_{ad}=0$) natures of $s_{ad}$ and ${\overline{f}}_{\mu }$. To cancel out the above result, an appropriate integral in the expression for $Q_{ad}$ [cf. Eq. (39)] has to be modified. Such a term is: $-{\displaystyle \frac{1}{2}}\int d^{3}x\left({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0\right){\partial }_i{C}_2$ which can be re-written in the following form:

$$\begin{array}{c}\hfill \int d^{3}x\left({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0\right){\partial }_i{C}_2-{\displaystyle \frac{3}{2}}\int d^{3}x\left({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0\right){\partial }_i{C}_2.\end{array}$$

(50)

It is clear that the application of $s_{ad}$ [cf. Eq. (30)] on the first integral would yield the result that would cancel out with the r.h.s. of (49). Hence the first integral of (50) will be present in the final nilpotent version of $Q_{AD}$. At this stage, we focus on the second integral of Equation (50) and apply (i) the partial integration, and (ii) the Gauss divergence theorem which, taken together, lead to the following result:

$$\begin{array}{c}\hfill +{\displaystyle \frac{3}{2}}\int d^{3}x{\partial }_i\left({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0\right)C_2.\end{array}$$

(51)

Application of the EL-EoM: ${\partial }_{\mu }({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu })=-{\partial }^{\nu }{B}_5$ (with the choice $\nu =0$) leads to the above integral to take the following explicit form

$$\begin{array}{c}\hfill +{\displaystyle \frac{3}{2}}\int d^{3}x{\dot{B}}_5{C}_2,\end{array}$$

(52)

which turns out to be an anti-co-BRST invariant quality [i.e. $s_{ad}\left({\dot{B}}_5{C}_2\right)=0$]. Collecting (i) all the useful and relevant terms of our exercise, and (ii) the anti-co-BRST invariant integrands of $Q_{ad}$ [cf. Eq. (39)], we obtain the off-shell nilpotent (i.e. $Q_{AD}^2=0$) version of the conserved anti-co-BRST charge $Q_{AD}$ as follows:

$$\begin{array}{ccc}\hfill Q_{AD}& =& \int d^{3}x[(B_1\dot{C}-{\dot{B}}_{1}C)+({\partial }^0{\overline{\beta }}^i-{\partial }^i{\overline{\beta }}^0){\partial }_i{C}_2-({\partial }^0{\overline{f}}^i-{\partial }^i{\overline{f}}^0){\beta }_i\hfill \\ & -& {\displaystyle \frac{1}{2}}\{({\partial }^0{C}^{ij}+{\partial }^i{C}^{j0}+{\partial }^j{C}^{0i}){\overline{\mathcal{B}}}_{ij}-B_3{F}^0-{\overline{\mathcal{B}}}^{0i}{F}_i+B_4{\overline{f}}^0\hfill \\ & -& ({\partial }^0{\mathcal{B}}^{ij}+{\partial }^i{\overline{\mathcal{B}}}^{j0}+{\partial }^j{\overline{\mathcal{B}}}^{0i})C_{ij}-({\partial }^0{\overline{C}}^{ij}+{\partial }^i{\overline{C}}^{j0}+{\partial }^j{\overline{C}}^{0i})({\partial }_i{\beta }_j-{\partial }_j{\beta }_i)\hfill \\ & -& ({\partial }^0{\beta }^i-{\partial }^i{\beta }^0){\overline{f}}_i+B_5{\dot{C}}_2-3{\dot{B}}_5{C}_2\left\}\right].\hfill \end{array}$$

(53)

It is an elementary exercise to check that $s_{ad}{Q}_{AD}=i\left\{Q_{AD},Q_{AD}\right\}=0\Rightarrow Q_{AD}^2=0$ by computing the l.h.s (i.e. $s_{ad}{Q}_{AD}$) by the direct application of the co-BRST symmetry transformation operator $s_{ad}$ [cf. Eq. (30)] on the above explicit expression for $Q_{AD}$. We would like to lay emphasis on the fact that the non-nilpotent Noether charge $Q_{ad}$ and the nilpotent version of the anti-co-BRST charge $Q_{AD}$ both are conserved quantities because we have used only the appropriate EL-EoMs and the partial integration followed by the Gauss divergence theorem which are consistent with the basic tenets of the Noether theorem.

We would like to dwell a bit on the physicality criterion w.r.t. the above nilpotent version of the anti-co-BRST charge $Q_{AD}$ and establish that the conditions on the physical states (i.e. $|phys>$) are dual to the conditions [cf. Eq. (A.12)] that have been obtained on the physical states by the nilpotent version of the anti-BRST charge $Q_{AB}$ (cf. Appendix A). A careful look at the Lagrangian densities ((28) and (29) show that the basic/auxiliary (anti-)ghost fields are decoupled from the basic/auxiliary physical fields of our theory. As a consequence, a quantum state in the Hilbert space of states is the direct product [24] of the physical states (i.e. $|phys>$) and the ghost states (i.e. $|ghost>$). Before pointing out the consequences coming out from the physicality criterion: $Q_{AD}|phys>0$, we would like to lay emphasis on the fact that every individual term in the integrand of $Q_{AD}$ [cf. Eq.(53)] (i) carry the ghost number equal to $+1$, (ii) made up of the physical fields (with ghost number equal to zero) and the basic/auxiliary (anti-)ghost fields (carrying the ghost number equal to $+1$), and (iii) constituted by only the basic/auxiliary (anti-)ghost fields. When we apply the anti-co-BRST charge operator acts on a quantum state, the ghost fields act on the ghost states (i.e. $|ghost>$) leading to the non-zero result. Thus, to satisfy the sacrosanct requirement that the physical states are those that are annihilated the conserved and nilpotent version of the anti-co-BRST charge $Q_{AD}$, it is clear that only the physical fields (i) carrying the ghost number equal to zero, and (ii) associated with the basic ghost fields (carrying the ghost number equal to $+1$) would act on the true physical states5. Thus, the requirement: $Q_{AD}|phys>0$ leads to the following conditions on the physical states (i.e. the harmonic states in the Hodge decomposed quantum states)

$$\begin{array}{ccc}\hfill & & B_1|phys>=0,{\dot{B}}_1|phys>=0,{\displaystyle \frac{1}{2}}{\overline{\mathcal{B}}}_{ij}|phys>=0,\hfill \\ & & {\displaystyle \frac{1}{2}}\left({\partial }^0{\overline{\mathcal{B}}}^{ij}+{\partial }^i{\overline{\mathcal{B}}}^{j0}+{\partial }^j{\overline{\mathcal{B}}}^{0i}\right)|phys>=0,\hfill \end{array}$$

(54)

which are dual to what we have obtained from the physicality criterion: $Q_{AB}|phys>0$ w.r.t. the conserved and nilpotent version of the anti-BRST charge $Q_{AB}$ (see Appendix A for details). In other words, our results (i.e. $B|phys>=0,\dot{B}|phys>=0,{\displaystyle \frac{1}{2}}{\overline{B}}_{ij}|phys>=0$ and ${\displaystyle \frac{1}{2}}\left({\partial }^0{\overline{B}}^{ij}+{\partial }^i{B}^{j0}+{\partial }^j{\overline{B}}^{0i}\right)|phys>=0$) that have been listed in (A.12) due to the physicality criterion: $Q_{AB}|phys>0$ w.r.t. the conserved and nilpotent version of the anti-BRST charge $Q_{AB}$ are deeply connected with conditions in Equation (54). To corroborate and substantiate this claim, we would like to point out that the non-ghost sector of the Lagrangian density ${\mathcal{L}}_{\left(ng\right)}$ [cf. Eq. (28)] respects (i.e. ${\mathcal{L}}_{\left(ng\right)}\to {\mathcal{L}}_{\left(ng\right)}$) the following discrete duality symmetry transformations:

$$\begin{array}{ccc}\hfill & & A_{\mu }⟶\mp {\displaystyle \frac{1}{3!}}{\epsilon }_{\mu \nu \sigma \rho }{A}^{\nu \sigma \rho },A_{\mu \nu \sigma }⟶\pm {\epsilon }_{\mu \nu \sigma \rho }{A}^{\rho },{\overline{B}}_{\mu \nu }\to \mp {\overline{\mathcal{B}}}_{\mu \nu },{\overline{\mathcal{B}}}_{\mu \nu }\to \pm {\overline{B}}_{\mu \nu },\hfill \\ & & B\to \pm B_1,B_1\to \mp B,B_2\to \pm B_3,B_3\to \mp B_2,{\varphi }_{\mu }\to \pm {\tilde{\varphi }}_{\mu },{\tilde{\varphi }}_{\mu }\to \mp {\varphi }_{\mu }.\hfill \end{array}$$

(55)

Thus, we note that if we choose the harmonic states (in the Hodge decomposed quantum states in the Hilbert space) to be the true physical states (i.e. $|phys>$), the conditions that emerge out from the conserved and nilpotent versions of the (anti-)BRST $Q_{AB}$ and (anti-)co-BRST $Q_{AD}$ charges are found to be deeply connected with one-another due to the presence of the discrete duality symmetry transformations in Equation (55).

We conclude this section with the final remark that the conserved and off-shell nilpotent versions of the (anti-)BRST and (anti-)co-BRST charges lead to the annihilation of the physical states by the operator forms of the (i) the first-class constraints (see, e.g. [20] and our Appendix A), and (ii) the dual versions of the first-class constraints (see, e.g. Secs. 4 and 7 for details), respectively, provided we choose the physical states to be the harmonic state in a given Hodge decomposed quantum state. The latter state, as is self-evident, is chosen from the total quantum Hilbert space of states of our BRST-quantized field-theoretic system. Our present crucial and decisive statements, connected with the emergence of the type of constraints from the physicality criteria, are true only for the field-theoretic systems which happen to be (i) the gauge theories at the classical level, and (ii) the examples for Hodge theory at the quantum level within the framework of BRST formalism.

8. Curci-Ferrari Type Restrictions: Derivations

In our present section, we derive the non-trivial (anti-)co-BRST invariant Curci-Ferrari (CF) type restrictions from two different theoretical angles. Let us, first of all, focus on the symmetry considerations where we demand that the coupled Lagrangian densities must respect the nilpotent co-BRST as well as the anti-co-BRST symmetry transformations so that their equivalence could be established. In this context, we would like to point out that the Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eqs. (1),(2)] transforms to the total spacetime derivative [cf. Eq. (4)] under the off-shell nilpotent co-BRST symmetry transformations (3) thereby rendering the action integral (corresponding to this Lagrangian density) invariant under the co-BRST symmetry transformations (3). We also demand, at this stage, the symmetry invariance of the coupled Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ [cf. Eqs. (28),(29)] under the co-BRST symmetry transformations (3). This requirement, as stated earlier, leads to the derivation of the CF-type restrictions: ${\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }=({\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu }),f_{\mu }+F_{\mu }={\partial }_{\mu }{C}_1,{\overline{f}}_{\mu }+{\overline{F}}_{\mu }={\partial }_{\mu }{\overline{C}}_1$. To corroborate this statement, as a first step, we note that the non-ghost part (i.e. ${\mathcal{L}}_{\left(ng\right)}$) of the coupled Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ transforms, under the off-shell nilpotent co-BRST symmetry transformations (3), as follows:

$$\begin{array}{ccc}\hfill s_d{\mathcal{L}}_{\left(ng\right)}& =& -{\partial }_{\mu }\left[{\epsilon }^{\mu \nu \sigma \rho }{\overline{F}}_{\nu }\left({\partial }_{\sigma }{A}_{\rho }\right)+{\displaystyle \frac{1}{2}}\left({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }\right){\mathcal{B}}_{\nu \sigma }-{\displaystyle \frac{1}{2}}{B}_3{\overline{F}}^{\mu }\right]\hfill \\ & +& B_1\square \overline{C}+{\displaystyle \frac{1}{2}}[({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\partial }_{\mu }{\overline{\mathcal{B}}}_{\nu \sigma }-{\overline{\mathcal{B}}}^{\mu \nu }\left({\partial }_{\mu }{\overline{F}}_{\nu }\right)\hfill \\ & +& \left({\partial }^{\mu }{\tilde{\varphi }}^{\nu }-{\partial }^{\nu }{\tilde{\varphi }}^{\mu }\right)\left({\partial }_{\mu }{\overline{F}}_{\nu }\right)-{\overline{F}}^{\mu }{\partial }_{\mu }{B}_3].\hfill \end{array}$$

(56)

In the next step, we apply the co-BRST symmetry transformations (3) on the FP-ghost part (i.e. ${\mathcal{L}}_{\left(fp\right)}$) of the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ which leads to the following:

$$\begin{array}{ccc}\hfill s_d{\mathcal{L}}_{\left(fp\right)}& =& {\displaystyle \frac{1}{2}}[({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\partial }_{\mu }{\mathcal{B}}_{\nu \sigma }+{\mathcal{B}}^{\mu \nu }\left({\partial }_{\mu }{\overline{f}}_{\nu }\right)+\left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right)\left({\partial }_{\mu }{F}_{\nu }\right)\hfill \\ & +& \left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right)\left({\partial }_{\mu }{f}_{\nu }\right)+\left({\partial }^{\mu }{\overline{C}}_1-{\overline{f}}^{\mu }\right){\partial }_{\mu }{B}_3-\left(f^{\mu }+F^{\mu }-{\partial }^{\mu }{C}_1\right){\partial }_{\mu }{B}_5]\hfill \\ & +& {\partial }_{\mu }{B}_1{\partial }^{\mu }\overline{C}-{\displaystyle \frac{1}{2}}{\partial }_{\mu }[{\overline{C}}^{\mu \nu }{\partial }_{\nu }{B}_3+C^{\mu \nu }{\partial }_{\nu }{B}_5+\left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right)F_{\nu }\hfill \\ & +& {\mathcal{B}}^{\mu \nu }{\overline{f}}_{\nu }-B_5{f}^{\mu }-B_4{\partial }^{\mu }{\overline{C}}_2].\hfill \end{array}$$

(57)

To observe the full consequence of the application of the co-BRST symmetry transformations (3) on the total coupled Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$, we have to add the results that have been obtained in (56) and (57) which leads to the following:

$$\begin{array}{ccc}\hfill & & s_d{\mathcal{L}}_{\left(\overline{B}\right)}={\partial }_{\mu }\left[B_1{\partial }^{\mu }\overline{C}-{\epsilon }^{\mu \nu \sigma \rho }{\overline{F}}_{\nu }\left({\partial }_{\sigma }{A}_{\rho }\right)\right]-{\displaystyle \frac{1}{2}}{\partial }_{\mu }[{\overline{C}}^{\mu \nu }{\partial }_{\nu }{B}_3+C^{\mu \nu }{\partial }_{\nu }{B}_5+{\mathcal{B}}^{\mu \nu }{\overline{f}}_{\nu }\hfill \\ & & +\left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right)F_{\nu }-B_3{\overline{F}}^{\mu }-B_5{f}^{\mu }+({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\mathcal{B}}_{\nu \sigma }-B_4{\partial }^{\mu }{\overline{C}}_2]\hfill \\ & & +{\displaystyle \frac{1}{2}}\{{\mathcal{B}}^{\mu \nu }{\partial }_{\mu }\left({\overline{f}}_{\nu }+{\overline{F}}_{\nu }-{\partial }_{\nu }{\overline{C}}_1\right)-\left({\overline{f}}^{\mu }+{\overline{F}}^{\mu }-{\partial }^{\mu }{\overline{C}}_1\right){\partial }_{\mu }{B}_3-\left(f^{\mu }+F^{\mu }-{\partial }^{\mu }{\overline{C}}_1\right){\partial }_{\mu }{B}_5\hfill \\ & & -\left[{\mathcal{B}}^{\mu \nu }+{\overline{\mathcal{B}}}^{\mu \nu }-({\partial }^{\mu }{\tilde{\varphi }}^{\nu }-{\partial }^{\nu }{\tilde{\varphi }}^{\mu }\right]\left({\partial }_{\mu }{\overline{F}}_{\nu }\right)+\left({\partial }^{\mu }{\overline{\beta }}^{\nu }-{\partial }^{\nu }{\overline{\beta }}^{\mu }\right){\partial }_{\mu }\left[f_{\nu }+F_{\nu }-{\partial }_{\nu }{C}_1\right]\hfill \\ & & +({\partial }^{\mu }{\overline{C}}^{\nu \sigma }+{\partial }^{\nu }{\overline{C}}^{\sigma \mu }+{\partial }^{\sigma }{\overline{C}}^{\mu \nu }){\partial }_{\mu }\left[{\mathcal{B}}_{\nu \sigma }+{\overline{\mathcal{B}}}_{\nu \sigma }-\left({\partial }_{\nu }{\tilde{\varphi }}_{\sigma }-{\partial }_{\sigma }{\tilde{\varphi }}_{\nu }\right)\right]\}.\hfill \end{array}$$

(58)

Thus, we note that the application of the nilpotent co-BRST symmetry transformation operator $s_d$ [cf. Eq. (3)] on the coupled Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ leads to the sum of (i) the total spacetime derivative terms, and (ii) the terms that contain the CF-type restrictions: ${\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }=({\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu }),f_{\mu }+F_{\mu }={\partial }_{\mu }{C}_1,{\overline{f}}_{\mu }+{\overline{F}}_{\mu }={\partial }_{\mu }{\overline{C}}_1$. In other words, the requirement of the co-BRST invariance of the perfectly anti-co-BRST invariant coupled Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ [cf. Eq. (31)] leads to the derivations of the (anti-)co-BRST and (anti-)BRST invariant CF-type restrictions. To be more precise, the coupled Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}$ respects both the off-shell nilpotent (i.e. co-BRST and anti-co-BRST) symmetry transformations provided the whole 4D BRST-quantized field-theoretic system is considered on the submanifold of the quantum fields where the above three CF-type restrictions6 are satisfied.

Against the backdrop of the above discussions, we would like to point out that the action integral, corresponding to the Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eqs. (1),(2)], remains invariant under the infinitesimal, continuous and off-shell nilpotent (i.e. $s_d^2=0$) co-BRST symmetry transformations $s_d$ [cf. Eqs. (3),(4)]. We demand, at this juncture, the invariance of this Lagrangian density ${\mathcal{L}}_{\left(B\right)}$ under the infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-co-BRST symmetry transformations $s_{ad}$ [cf. Eq. (30)], too. Under this requirement, as stated earlier, we envisage to derive of the non-trivial CF-type restrictions (i.e. ${\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }=({\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu }),f_{\mu }+F_{\mu }={\partial }_{\mu }{C}_1,{\overline{f}}_{\mu }+{\overline{F}}_{\mu }={\partial }_{\mu }{\overline{C}}_1$). Toward this goal in mind, first of all, we note that the non-ghost part (i.e. ${\mathcal{L}}_{\left(NG\right)}$) of the Lagrangian density ${\mathcal{L}}_{\left(B\right)}$ transforms, under the infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-co-BRST symmetry transformations $s_{ad}$, as:

$$\begin{array}{ccc}\hfill s_{ad}{\mathcal{L}}_{\left(NG\right)}& =& {\partial }_{\mu }\left[{\epsilon }^{\mu \nu \sigma \rho }{F}_{\nu }\left({\partial }_{\sigma }{A}_{\rho }\right)+{\displaystyle \frac{1}{2}}\left({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }\right){\mathcal{B}}_{\nu \sigma }+{\displaystyle \frac{1}{2}}{B}_3{F}^{\mu }\right]\hfill \\ & -& {\displaystyle \frac{1}{2}}[({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\partial }_{\mu }{\mathcal{B}}_{\nu \sigma }+B^{\mu \nu }\left({\partial }_{\mu }{F}_{\nu }\right)+F^{\mu }{\partial }_{\mu }{B}_3\hfill \\ & -& \left({\partial }^{\mu }{\tilde{\varphi }}^{\nu }-{\partial }^{\nu }{\tilde{\varphi }}^{\mu }\right)\left({\partial }_{\mu }{F}_{\nu }\right)]+B_1\square C.\hfill \end{array}$$

(59)

On the other hand, we note that the FP-ghost part (i.e. ${\mathcal{L}}_{\left(FP\right)}$) of the Lagrangian density ${\mathcal{L}}_{\left(B\right)}$ transforms, under the infinitesimal, continuous and off-shell nilpotent (i.e. $s_{ad}^2=0$) anti-co-BRST symmetry transformations $s_{ad}$ [cf. Eq. (30)], as:

$$\begin{array}{ccc}\hfill & & s_{ad}{\mathcal{L}}_{\left(FP\right)}={\displaystyle \frac{1}{2}}[{\overline{\mathcal{B}}}^{\mu \nu }\left({\partial }_{\mu }{f}_{\nu }\right)-({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\partial }_{\mu }{\overline{\mathcal{B}}}_{\nu \sigma }++\left({\partial }^{\mu }{C}_1-f^{\mu }\right){\partial }_{\mu }{B}_3\hfill \\ & & +\left({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }\right){\partial }_{\mu }[{\overline{f}}_{\nu }+{\overline{F}}_{\nu }-{\partial }_{\nu }{\overline{C}}_1]+\left({\overline{f}}^{\mu }+{\overline{F}}^{\mu }-{\partial }^{\mu }{\overline{C}}_1\right){\partial }_{\mu }{B}_4]+{\partial }_{\mu }{B}_1{\partial }^{\mu }C\hfill \\ & & -{\displaystyle \frac{1}{2}}{\partial }_{\mu }\left[{\overline{\mathcal{B}}}^{\mu \nu }{f}_{\nu }-{\overline{C}}^{\mu \nu }{\partial }_{\nu }{B}_4-C^{\mu \nu }{\partial }_{\nu }{B}_3+\left({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }\right){\overline{F}}_{\nu }+B_4{\overline{f}}^{\mu }+B_5{\partial }^{\mu }{C}_2\right].\hfill \end{array}$$

(60)

To obtain the full outcome of the application of the anti-co-BRST symmetry transformation operator on the Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eqs. (1),(2)], we have to take the sum of our observations in equations (59) and (60) which leads to the following:

$$\begin{array}{ccc}\hfill & & s_{ad}{\mathcal{L}}_{\left(B\right)}={\partial }_{\mu }\left[B_1{\partial }^{\mu }C+{\epsilon }^{\mu \nu \sigma \rho }{F}_{\nu }\left({\partial }_{\sigma }{A}_{\rho }\right)\right]+{\displaystyle \frac{1}{2}}{\partial }_{\mu }[C^{\mu \nu }{\partial }_{\nu }{B}_3+{\overline{C}}^{\mu \nu }{\partial }_{\nu }{B}_4-{\overline{\mathcal{B}}}^{\mu \nu }{f}_{\nu }\hfill \\ & & -\left({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }\right){\overline{F}}_{\nu }+B_3{F}^{\mu }-B_4{\overline{f}}^{\mu }+({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu })B_{\nu \sigma }-B_5{\partial }^{\mu }{C}_2]\hfill \\ & & +{\displaystyle \frac{1}{2}}\{{\overline{\mathcal{B}}}^{\mu \nu }{\partial }_{\mu }\left(f_{\nu }+F_{\nu }-{\partial }_{\nu }{C}_1\right)-\left(f^{\mu }+F^{\mu }-{\partial }^{\mu }{C}_1\right){\partial }_{\mu }{B}_3+\left({\overline{f}}^{\mu }+{\overline{F}}^{\mu }-{\partial }^{\mu }{\overline{C}}_1\right){\partial }_{\mu }{B}_4\hfill \\ & & -\left[{\mathcal{B}}^{\mu \nu }+{\overline{\mathcal{B}}}^{\mu \nu }-({\partial }^{\mu }{\tilde{\varphi }}^{\nu }-{\partial }^{\nu }{\tilde{\varphi }}^{\mu }\right]\left({\partial }_{\mu }{F}_{\nu }\right)+\left({\partial }^{\mu }{\beta }^{\nu }-{\partial }^{\nu }{\beta }^{\mu }\right){\partial }_{\mu }\left[{\overline{f}}_{\nu }+{\overline{F}}_{\nu }-{\partial }_{\nu }{\overline{C}}_1\right]\hfill \\ & & -({\partial }^{\mu }{C}^{\nu \sigma }+{\partial }^{\nu }{C}^{\sigma \mu }+{\partial }^{\sigma }{C}^{\mu \nu }){\partial }_{\mu }\left[{\mathcal{B}}_{\nu \sigma }+{\overline{\mathcal{B}}}_{\nu \sigma }-\left({\partial }_{\nu }{\tilde{\varphi }}_{\sigma }-{\partial }_{\sigma }{\tilde{\varphi }}_{\nu }\right)\right]\}.\hfill \end{array}$$

(61)

The above observation establishes that the Lagrangian density ${\mathcal{L}}_{\left(B\right)}$ respects (i) the co-BRST transformations [cf. Eqs. (3),(4)], and (ii) the anti-co-BRST transformations (30) provided we consider the entire 4D BRST-quantized theory on the submanifold of the quantum fields where the (anti-)BRST and (anti-)co-BRST invariant CF-type restrictions (i.e. ${\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }=({\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu }),f_{\mu }+F_{\mu }={\partial }_{\mu }{C}_1,{\overline{f}}_{\mu }+{\overline{F}}_{\mu }={\partial }_{\mu }{\overline{C}}_1$) are satified. In other words, from the above equation, we note that the Lagrangina density ${\mathcal{L}}_{\left(B\right)}$ transforms to a total spacetime derivative under the anti-co-BRST transformations $s_{ad}$ if we take into account the validity of the above three CF-type restrictions. Thus, the requirements of the symmetry invariance of the Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$, under the nilpotent co-BRST as well as the nilpotent anti-co-BRST transformations together, lead to the derivations of the (anti-)BRST as well as the (anti-)co-BRST invariant CF-type restrictions.

We discuss concisely the absolute anticommutativity property of the off-shell nilpotent co-BRST and anti-co-BRST transformation operators and lay emphasis on the (anti-) BRST and (anti-)co-BRST invariant CF-type restrictions. It turns out that the absolute anticommutativity property (i.e. $\{s_d,s_{ad}\}=0$) is satisfied for all the fields of our 4D BRST-quantized theory except fields $A_{\mu },C_{\mu \nu }$ and ${\overline{C}}_{\mu \nu }$ because we observe the following:

$$\begin{array}{ccc}\hfill & & \{s_d,s_{ad}\}{A}_{\mu }=-{\displaystyle \frac{1}{2}}{\epsilon }_{\mu \nu \sigma \rho }{\partial }^{\nu }\left({\mathcal{B}}^{\sigma \rho }+{\overline{\mathcal{B}}}^{\sigma \rho }\right),\hfill \\ & & \{s_d,s_{ad}\}{C}_{\mu \nu }=-{\partial }_{\mu }(f_{\nu }+F_{\nu })+{\partial }_{\nu }(f_{\mu }+F_{\mu }),\hfill \\ & & \{s_b,s_{ad}\}{\overline{C}}_{\mu \nu }=-{\partial }_{\mu }({\overline{f}}_{\nu }+{\overline{F}}_{\nu })+{\partial }_{\nu }({\overline{f}}_{\mu }+{\overline{F}}_{\mu }).\hfill \end{array}$$

(62)

However, we have the sanctity of the absolute anticommutativity property (i.e. $\{s_d,s_{ad}\}=0$) of the (anti-)co-BRST transformations for the above fields, too, provided we use the following (anti-)co-BRST as well as the (anti-)BRST invariant CF-type restrictions:

$$\begin{array}{c}\hfill {\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }={\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu },f_{\mu }+F_{\mu }={\partial }_{\mu }{C}_1,{\overline{f}}_{\mu }+{\overline{F}}_{\mu }={\partial }_{\mu }{\overline{C}}_1.\end{array}$$

(63)

We lay emphasis on the fact that the above crucial CF-type restrictions have appeared in our earlier equations (58) and (61) where we have demanded that the coupled Lagrangian densities of our theory must respect the co-BRST as well as the anti-co-BRST symmetry transformations simultaneously. Using the off-shell nilpotent (anti-)BRST transformations [cf. Eqs. (A.8),(A.1)] and off-shell nilpotent (anti-)co-BRST symmetry transformations [cf. Eqs. (30),(3)], it can be checked that all the four CF-type restrictions on our 4D BRST-quantized field-theoretic system are (anti-)BRST as well as (anti-)co-BRST invariant quantities, namely;

$$\begin{array}{ccc}\hfill & & s_{\left(a\right)b}[B_{\mu \nu }+{\overline{B}}_{\mu \nu }-({\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu })]=0,s_{\left(a\right)b}[f_{\mu }+F_{\mu }-{\partial }_{\mu }{C}_1]=0,\hfill \\ & & s_{\left(a\right)b}[{\overline{f}}_{\mu }+{\overline{F}}_{\mu }-{\partial }_{\mu }{\overline{C}}_1]=0,s_{\left(a\right)b}[{\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }-({\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu })]=0,\hfill \\ & & s_{\left(a\right)d}[B_{\mu \nu }+{\overline{B}}_{\mu \nu }-({\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu })]=0,s_{\left(a\right)d}[f_{\mu }+F_{\mu }-{\partial }_{\mu }{C}_1]=0,\hfill \\ & & s_{\left(a\right)d}[{\overline{f}}_{\mu }+{\overline{F}}_{\mu }-{\partial }_{\mu }{\overline{C}}_1]=0,s_{\left(a\right)d}[{\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }-({\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu })]=0,\hfill \end{array}$$

(64)

which corroborate our earlier claim [after Equation (62)]. Hence, these CF-type restrictions on our theory are physical and they are connected with the geometrical objects called gerbes (see, e.g. [25,26]). The coupled nature of the Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ has been demonstrated in our earlier work [20] by using the EL-EoMs that are derived from the above Lagrangian densities w.r.t the bosonic/fermionic auxiliary fields (of our 4D BRST-quantized field-theoretic system) which lead to the derivations of all the four CF-type restrictions on our theory. In other words, it is the existence of the non-trivial CF-type restrictions that are responsible for (i) the coupled nature of these Lagrangian densities, and (ii) the equivalent nature of the Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ from the point of view of the symmetry considerations. In fact, the direct equality of the BRST-quantized versions of the coupled Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ also leads to the derivations of all the four CF-type restrictions on our 4D BRST-quantized field-theoretic system (see, e.g. [20]).

9. Conclusions

In our present endeavor, we have been able to demonstrate that the coupled (but equivalent) Lagrangian densities ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ respect the infinitesimal, continuous and off-shell nilpotent co-BRST (i.e. dual-BRST) and anti-co-BRST (i.e. anti-dual-BRST) symmetry transformations under which the total gauge-fixing terms for the Abelian 3-form and 1-form gauge fields remain invariant in the description of our present 4D BRST-quantized combined field-theoretic system within the framework of the BRST invariant Lagrangian formulation. It has been clearly explained that the gauge-fixing terms for the Abelian 3-form and 1-form gauge fields owe their mathematical origin to the co-exterior (i.e. dual-exterior) derivative of differential geometry [15-19]. The vector field ${\varphi }_{\mu }$ appears in the gauge-fixing term for the Abelian 3-form gauge field because of its reducibility property. This field also remains invariant (i.e. $s_{\left(a\right)d}{\varphi }_{\mu }=0$) under the (anti-)co-BRST symmetry transformations $s_{\left(a\right)d}$ [cf. Eqs. (30),(3)]. Hence, the nomenclatures (that are associated with the infinitesimal, continuous and off-shell nilpotent (anti-)dual-BRST [i.e. (anti-)co-BRST] symmetry transformations $s_{\left(a\right)d}$) are absolutely correct.

We have laid a great deal of importance and emphasis on the discussions and derivations of the (anti-)BRST and (anti-)co-BRST invariant non-trivial CF-type restrictions [cf. Eqs. (63),(64)] because their existence is one of the decisive features of a properly BRST-quantized theory. In our earlier works (see, e.g. [25,26]), we have been able to establish that these physical restrictions, on a properly BRST-quantized theory, are connected with the geometrical objects called gerbes. We have been able to derive these non-trivial physical restrictions on our 4D BRST-quantized theory from two different theoretical angles. First of all, we have derived these CF-type restrictions from the requirements that both the Lagrangian densities ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ must respect the off-shell nilpotent co-BRST as well as the anti-co-BRST symmetry transformations [cf. Eqs. (3),(30)]. On the other hand, the requirement of the absolute anticmmutativity property (i.e. $\{s_d,s_{ad}\}=0$) between the off-shell nilpotent co-BRST and anti-co-BRST symmetry transformation operators also indicates [cf. Eq. (62)] the validity of the CF-type restrictions on our 4D BRST-quantized theory. In other words, the absolute anticommutativity property (i.e. $\{s_d,s_{ad}\}=0$) is satisfied on the submanifold of the quantum fields where the celebrated CF-type restrictions are valid.

As far as the continuous symmetry considerations are concerned, we find that (i) the anti-co-BRST insurance (i.e. $s_{ad}{\mathcal{L}}_{\left(B\right)}=0$) of the Lagrangian density ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ [cf. Eq. (61)], and (ii) the co-BRST invariance (i.e. $s_d{\mathcal{L}}_{\left(\overline{B}\right)}=0$) of the Lagrangian density ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ [cf. Eq. (57)] are valid if and only if we invoke the validity of the CF-type restrictions (i.e. ${\mathcal{B}}_{\mu \nu }+{\overline{\mathcal{B}}}_{\mu \nu }={\partial }_{\mu }{\tilde{\varphi }}_{\nu }-{\partial }_{\nu }{\tilde{\varphi }}_{\mu },f_{\mu }+F_{\mu }={\partial }_{\mu }{C}_1,{\overline{f}}_{\mu }+{\overline{F}}_{\mu }={\partial }_{\mu }{\overline{C}}_1$). The coupled nature of the Lagrangian desities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ also owe their origin to the existence of the physically important (anti-)BRST as well as the (anti-)co-BRST invariant CF-type restrictions which can be verified in the language of the EL-EoMs w.r.t. the bosonic/fermionic auxiliary fields which are derived from the above coupled Lagrangian densities ${\mathcal{L}}_{\left(B\right)}={\mathcal{L}}_{\left(NG\right)}+{\mathcal{L}}_{\left(FP\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}={\mathcal{L}}_{\left(ng\right)}+{\mathcal{L}}_{\left(fp\right)}$ (see, e.g. [20] for details). In other words, we note that the EL-EoMs from the coupled Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ are such that they imply the validity of the CF-type restrictions on our theory. The direct equality of the coupled Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ also leads to the derivation of the above three CF-type restrictions along with the fourth one as: $B_{\mu \nu }+{\overline{B}}_{\mu \nu }={\partial }_{\mu }{\varphi }_{\nu }-{\partial }_{\nu }{\varphi }_{\mu }$ which plays a crucial role in the discussion on the (anti-)BRST symmetries (see, e.g. [20]).

One of the highlights of our present investigation is the observation that the non-ghost parts [cf. Eqs. (1),(28)] of the coupled Lagrangian densities ${\mathcal{L}}_{\left(B\right)}$ and ${\mathcal{L}}_{\left(\overline{B}\right)}$ remain invariant under a set of discrete duality symmetry transformations [cf. Eqs. (27),(59)] which provide a deep connection between the conditions on the physical states (i.e. $|phys>$) that emerge out from the physicality criteria $Q_{\left(A\right)B}|phys>=0$ and $Q_{\left(A\right)D}|phys>=0$ (cf. Secs. 4 and 7 or details). To be utmost precise, we find that (i) the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)BRST charges lead to the annihilation of the physical states (i.e. $|phys>$) by the operator forms of the first-class constraints existing on our 4D classical theory, and (ii) the physicality criteria w.r.t. the nilpotent versions of the (anti-)co-BRST charges lead to the annihilation of the physical states (i.e. $|phys>$) by the dual versions of the first-class constraints7. The mathematical origin for such a crucial and decisive observation is hidden in the discrete duality symmetry transformations [cf. Eqs. (27),(59)] that exist in our 4D BRST-quantized theory.

In our present endeavor, we have focused only on (i) the infinitesimal, continuous and off-shell nilpotent (anti-)co-BRST symmetry transformations, (ii) the derivations of the non-nilpotent Noether (anti-)co-BRST conserved charges, (iii) the deductions of the off-shell nilpotent versions of the (anti-)co-BRST charges from the non-nilpotent Noether (anti-)co-BRST charges, (iv) the discussions and deliberations on the existence of the (anti-)BRST as well as the (anti-)co-BRST invariant CF-type restrictions, and (v) the physicality criteria w.r.t. the conserved and nilpotent versions of the (anti-)co-BRST charges. In our earlier work (see, e.g. [20] for details) such kinds of discussions have been performed in the context of the infinitesimal, continuous and off-shell nilpotent (anti-)BRST symmetry transformations. In our future endeavor, we envisage to discuss the (anti-)BRST and (anti-)co-BRST symmetry transformation operators, a unique bosonic symmetry transformation operator that is derived from the appropriate anticommutator(s) between the nilpotent (anti-)BRST and (anti-)co-BRST symmetry operators, the ghost-scale symmetry transformation operator along with a couple of discrete duality symmetry transformation operators. We plan to derive the extended algebra among all these discrete as well as the continuous symmetry transformation operators. The derivation of (i) the conserved Noether currents and corresponding conserved charges, (ii) the deductions of the extended BRST algebra amongst all the appropriate conserved charges, and (iii) the derivations of the mapping between the appropriate set of conserved charges and the cohomological operators of differential geometry, are yet another set of future directions which we envisage to pursue so that we can prove, in an elaborate fashion, that our present 4D BRST-quantized field-theoretic system8 is a tractable example for Hodge theory.