2 The free theory: Lagrangian, gauge symmetries and BRST differential

The starting point is a free theory in $D=5$, whose Lagrangian action is written as the sum between the Lagrangian action of an Abelian BF model with a maximal field spectrum (a single scalar field $\varphi$, two types of one-forms $H^{\mu }$ and $V_{\mu }$, two kinds of two-forms $B^{\mu \nu }$ and $\phi _{\mu \nu }$, and one three-form $K^{\mu \nu \rho }$) and the Lagrangian action of a free, massless tensor field with the mixed symmetry $(2,1)$ $t_{\mu \nu \vert\alpha }$ (meaning it is antisymmetric in its first two indices $t_{\mu \nu \vert\alpha }=-t_{\nu \mu \vert\alpha }$ and fulfills the identity $t_{[\mu \nu \vert\alpha ]}\equiv 0$)

$$S_{0}^{\mathrm{L}}\left[ \Phi ^{\alpha _{0}}\right] = \int d^{5}x\left[ H^{\mu }\partial _{\mu }\varphi +\tfrac{1}{2}B^... ...{1}{3}K^{\mu \nu \rho }\partial _{\lbrack \mu }\phi _{\nu \rho ]}\right. \notag$$ (1)

$$\left. -\tfrac{1}{12}\left( F_{\mu \nu \rho \vert\alpha }F^{\mu \nu \rho \vert\alpha }-3F_{\mu \nu }F^{\mu \nu }\right) \right] \notag$$ (2)

$$\equiv \int d^{5}x\left( \mathcal{L}_{0}^{\mathrm{BF}}+\mathcal{L}_{0}^{ \mathrm{t}}\right) ,$$ (3)

where we used the notations

$$\Phi ^{\alpha _{0}} = \left( \varphi ,H^{\mu },V_{\mu },B^{\mu \nu },\phi _{\mu \nu },K^{\mu \nu \rho },t_{\mu \nu \vert\alpha }\right) ,$$ (4)

$$F_{\mu \nu \rho \vert\alpha } = \partial _{\lbrack \mu }t_{\nu \rho ]\vert\alpha },\qquad F_{\mu \nu }=\sigma ^{\rho \alpha }F_{\mu \nu \rho \vert\alpha }.$$ (5)

Everywhere in this paper the notations $[\mu \nu \ldots \rho ]$ and $(\mu \nu \ldots \rho )$ signify complete antisymmetry and respectively complete symmetry with respect to the (Lorentz) indices between brackets, with the conventions that the minimum number of terms is always used and the result is never divided by the number of terms. It is convenient to work with the Minkowski metric tensor of `mostly plus' signature $\sigma _{\mu \nu }=\sigma ^{\mu \nu }=\mathrm{diag}\left( -++++\right)$ and with the five-dimensional Levi-Civita symbol $\varepsilon ^{\mu \nu \rho \lambda \sigma }$ defined according to the convention $\varepsilon ^{01234}=-\varepsilon _{01234}=-1$.

Action (1) is found invariant under the gauge transformations

$$\delta _{\Omega }\varphi =0,\qquad \delta _{\Omega }H^{\mu }=2\partial _{\nu }\epsilon ^{\mu \nu },$$ (6)

$$\delta _{\Omega }V_{\mu }=\partial _{\mu }\epsilon ,\qquad \delta _{\Omega }B^{\mu \nu }=-3\partial _{\rho }\epsilon ^{\mu \nu \rho },$$ (7)

$$\delta _{\Omega }\phi _{\mu \nu }=\partial _{\lbrack \mu }\xi _{\... ... _{\Omega }K^{\mu \nu \rho }=4\partial _{\lambda }\xi ^{\mu \nu \rho \lambda },$$ (8)

$$\delta _{\Omega }t_{\mu \nu \vert\alpha }=\partial _{\lbrack \mu ... ...rtial _{\lbrack \mu }\chi _{\nu ]\alpha }-2\partial _{\alpha }\chi _{\mu \nu },$$ (9)

where all the gauge parameters are bosonic, with $\epsilon ^{\mu \nu }$, $\epsilon ^{\mu \nu \rho }$, $\xi ^{\mu \nu \rho \lambda }$, and $\chi _{\mu \nu }$ completely antisymmetric and $\theta _{\mu \nu }$ symmetric. By $\Omega$ we denoted collectively all the gauge parameters as

$$\Omega ^{\alpha _{1}}\equiv \left( \epsilon ^{\mu \nu },\epsilon ... ...\mu },\xi ^{\mu \nu \rho \lambda },\theta _{\mu \nu },\chi _{\mu \nu }\right) .$$ (10)

The gauge transformations given by (4)-(7) are off-shell reducible of order three (the reducibility relations hold everywhere in the space of field history, and not only on the stationary surface of field equations). This means that:

1. there exist some transformations of the gauge parameters (8 )

   $$\Omega ^{\alpha _{1}}\rightarrow \Omega ^{\alpha _{1}}=\Omega ^{\alpha _{1}}\left( \bar{\Omega}^{\alpha _{2}}\right) ,$$ (11)

   
   
   such that the gauge transformations of all fields vanish strongly (first-order reducibility relations)

   $$\delta _{\Omega \left( \bar{\Omega}\right) }\Phi ^{\alpha _{0}}=0;$$ (12)

   
   

2. there exist some transformations of the first-order reducibility parameters $\bar{\Omega}^{\alpha _{2}}$

   $$\bar{\Omega}^{\alpha _{2}}\rightarrow \bar{\Omega}^{\alpha _{2}}=\bar{\Omega} ^{\alpha _{2}}\left( \check{\Omega}^{\alpha _{3}}\right) ,$$ (13)

   
   
   such that the gauge parameters vanish strongly (second-order reducibility relations)

   $$\Omega ^{\alpha _{1}}\left( \bar{\Omega}^{\alpha _{2}}\left( \check{\Omega} ^{\alpha _{3}}\right) \right) =0;$$ (14)

   
   

3. there exist some transformations of the second-order reducibility parameters $\check{\Omega}^{\alpha _{3}}$

   $$\check{\Omega}^{\alpha _{3}}\rightarrow \check{\Omega}^{\alpha _{3}}=\check{ \Omega}^{\alpha _{3}}\left( \hat{\Omega}^{\alpha _{4}}\right) ,$$ (15)

   
   
   such that the first-order reducibility parameters vanish strongly (third-order reducibility relations)

   $$\bar{\Omega}^{\alpha _{2}}\left( \check{\Omega}^{\alpha _{3}}\left( \hat{ \Omega}^{\alpha _{4}}\right) \right) =0;$$ (16)

   
   

4. there is no nontrivial transformation of the third-order reducibility parameters $\hat{\Omega}^{\alpha _{4}}$ that annihilates all the second-order reducibility parameters

   $$\check{\Omega}^{\alpha _{3}}\left( \hat{\Omega}^{\alpha _{4}}\right) =0\Leftrightarrow \hat{\Omega}^{\alpha _{4}}=0.$$ (17)

   
   

This is indeed the case for the model under study. In this situation a complete set of first-order reducibility parameters $\bar{\Omega}^{\alpha _{2}}$ is given by

$$\bar{\Omega}^{\alpha _{2}}\equiv \left( \bar{\epsilon}^{\mu \nu \... ...\bar{\xi},\bar{\xi}^{\mu \nu \rho \lambda \sigma },\bar{\theta}_{\mu }\right) ,$$ (18)

and transformations (9) have the form

$$\epsilon ^{\mu \nu }\left( \bar{\Omega}^{\alpha _{2}}\right) =-3\... ...lpha _{2}}\right) =4\partial _{\lambda }\bar{\epsilon}^{\mu \nu \rho \lambda },$$ (19)

$$\xi _{\mu }\left( \bar{\Omega}^{\alpha _{2}}\right) =\partial _{\... ...ha _{2}}\right) =-5\partial _{\sigma }\bar{\xi}^{\mu \nu \rho \lambda \sigma },$$ (20)

$$\theta _{\mu \nu }\left( \bar{\Omega}^{\alpha _{2}}\right) =3\par... ...bar{\Omega}^{\alpha _{2}}\right) =\partial _{\lbrack \mu }\bar{\theta}_{\nu ]},$$ (21)

with $\bar{\epsilon}^{\mu \nu \rho }$, $\bar{\epsilon}^{\mu \nu \rho \lambda }$, and $\bar{\xi}^{\mu \nu \rho \lambda \sigma }$ completely antisymmetric. Further, a complete set of second-order reducibility parameters $\check{\Omega}^{\alpha _{3}}$ can be taken as

$$\check{\Omega}^{\alpha _{3}}\equiv \left( \check{\epsilon}^{\mu \nu \rho \lambda },\check{\epsilon}^{\mu \nu \rho \lambda \sigma }\right) ,$$ (22)

and transformations (11) are

$$\bar{\epsilon}^{\mu \nu \rho }\left( \check{\Omega}^{\alpha _{3}}... ...}\right) =-5\partial _{\sigma }\check{\epsilon}^{\mu \nu \rho \lambda \sigma },$$ (23)

$$\bar{\xi}\left( \check{\Omega}^{\alpha _{3}}\right) =0,\qquad \ba... ...ht) =0,\qquad \bar{\theta}_{\mu }\left( \check{\Omega}^{\alpha _{3}}\right) =0,$$ (24)

where both $\check{\epsilon}^{\mu \nu \rho \lambda }$ and $\check{\epsilon} ^{\mu \nu \rho \lambda \sigma }$ are some arbitrary, bosonic, completely antisymmetric tensors. Next, a complete set of third-order reducibility parameters $\hat{\Omega}^{\alpha _{4}}$ is represented by

$$\hat{\Omega}^{\alpha _{4}}\equiv \left( \hat{\epsilon}^{\mu \nu \rho \lambda \sigma }\right) ,$$ (25)

and transformations (13) can be chosen of the form

$$\check{\epsilon}^{\mu \nu \rho \lambda }\left( \hat{\Omega}^{\alp... ...on}^{\mu \nu \rho \lambda \sigma }\left( \hat{ \Omega}^{\alpha _{4}}\right) =0,$$ (26)

with $\hat{\epsilon}^{\mu \nu \rho \lambda \sigma }$ an arbitrary, completely antisymmetric tensor. Finally, it is easy to check (15 ). Indeed, we work in $D=5$, such that $\partial _{\sigma }\hat{\epsilon} ^{\mu \nu \rho \lambda \sigma }=0$ implies $\hat{\epsilon}^{\mu \nu \rho \lambda \sigma }=const.$.Since $\hat{\epsilon}^{\mu \nu \rho \lambda \sigma }$ are arbitrary smooth functions that effectively depend on the spacetime coordinates, it follows that the only possible choice is $\hat{\epsilon} ^{\mu \nu \rho \lambda \sigma }=0$.

We observe that the free theory under study is a usual linear gauge theory (its field equations are linear in the fields), whose generating set of gauge transformations is third-order reducible, such that we can define in a consistent manner its Cauchy order, which is found to be equal to five.

In order to construct the BRST symmetry of this free theory, we introduce the field/ghost and antifield spectra (2) and

$$\eta ^{\alpha _{1}}=\left( C^{\mu \nu },\eta ,\eta ^{\mu \nu \rho... ..._{\mu }, \mathcal{G}^{\mu \nu \rho \lambda },S_{\mu \nu },A_{\mu \nu }\right) ,$$ (27)

$$\eta ^{\alpha _{2}}=\left( C^{\mu \nu \rho },\eta ^{\mu \nu \rho \lambda },C, \mathcal{G}^{\mu \nu \rho \lambda \sigma },S_{\mu }\right) ,$$ (28)

$$\eta ^{\alpha _{3}}=\left( C^{\mu \nu \rho \lambda },\eta ^{\mu \... ...t) ,\qquad \eta ^{\alpha _{4}}=\left( C^{\mu \nu \rho \lambda \sigma }\right) ,$$ (29)

$$\Phi _{\alpha _{0}}^{\ast }=\left( \varphi ^{\ast },H_{\mu }^{\as... ...\ast \mu \nu },K_{\mu \nu \rho }^{\ast },t^{\ast \mu \nu \vert\alpha }\right) ,$$ (30)

$$\eta _{\alpha _{1}}^{\ast }=\left( C_{\mu \nu }^{\ast },\eta ^{\a... ...G}_{\mu \nu \rho \lambda }^{\ast },S^{\ast \mu \nu },A^{\ast \mu \nu }\right) ,$$ (31)

$$\eta _{\alpha _{2}}^{\ast }=\left( C_{\mu \nu \rho }^{\ast },\eta... ...ast },\mathcal{G}_{\mu \nu \rho \lambda \sigma }^{\ast },S^{\ast \mu }\right) ,$$ (32)

$$\eta _{\alpha _{3}}^{\ast }=\left( C_{\mu \nu \rho \lambda }^{\as... ..._{\alpha _{4}}^{\ast }=\left( C_{\mu \nu \rho \lambda \sigma }^{\ast }\right) .$$ (33)

The fermionic ghosts (25) correspond to the bosonic gauge parameters (8), and therefore $C^{\mu \nu }$, $\eta ^{\mu \nu \rho }$, $\mathcal{G}^{\mu \nu \rho \lambda }$, and $A_{\mu \nu }$ are completely antisymmetric and $S_{\mu \nu }$ is symmetric. The bosonic ghosts for ghosts (26) are respectively associated with the first-order reducibility parameters (16), such that $C^{\mu \nu \rho }$, $\eta ^{\mu \nu \rho \lambda }$, and $\mathcal{G}^{\mu \nu \rho \lambda \sigma }$ are completely antisymmetric. Along the same line, the fermionic ghosts for ghosts for ghosts $\eta ^{\alpha _{3}}$ from (27) correspond to the second-order reducibility parameters (20). As a consequence, the ghost fields $C^{\mu \nu \rho \lambda }$ and $\eta ^{\mu \nu \rho \lambda \sigma }$ are again completely antisymmetric. Finally, the bosonic ghosts for ghosts for ghosts for ghosts $\eta ^{\alpha _{4}}$ from (27) are associated with the third-order reducibility parameters (23), so $C^{\mu \nu \rho \lambda \sigma }$ is also completely antisymmetric. The star variables represent the antifields of the corresponding fields/ghosts. Their Grassmann parities are obtained via the usual rule $\varepsilon \left( \chi _{\Delta }^{\ast }\right) =\left( \varepsilon \left( \chi ^{\Delta }\right) +1\right) \mathrm{mod\,}2$, where we employed the notations

$$\chi ^{\Delta }=\left( \Phi ^{\alpha _{0}},\eta ^{\alpha _{1}},\e... ... _{2}}^{\ast },\eta _{\alpha _{3}}^{\ast },\eta _{\alpha _{4}}^{\ast }\right) .$$ (34)

It is understood that the antifields are endowed with the same symmetry/antisymmetry properties like those of the corresponding fields/ghosts.

Since both the gauge generators and the reducibility functions are field-independent, it follows that the BRST differential reduces to $s=\delta +\gamma$, where $\delta$ is the Koszul-Tate differential, and $\gamma$ means the exterior longitudinal derivative. The Koszul-Tate differential is graded in terms of the antighost number ( $\mathrm{agh}$, $\mathrm{agh}\left( \delta \right) =-1$, $\mathrm{agh}\left( \gamma \right) =0$) and enforces a resolution of the algebra of smooth functions defined on the stationary surface of field equations for action (1), $C^{\infty }\left( \Sigma \right)$, $\Sigma :\delta S_{0}^{\mathrm{L} }/\delta \Phi ^{\alpha _{0}}=0$. The exterior longitudinal derivative is graded in terms of the pure ghost number ( $\mathrm{pgh}$, $\mathrm{pgh} \left( \gamma \right) =1$, $\mathrm{pgh}\left( \delta \right) =0$) and is correlated with the original gauge symmetry via its cohomology in pure ghost number zero computed in $C^{\infty }\left( \Sigma \right)$, which is isomorphic to the algebra of physical observables for this free theory. These two degrees of generators (2) and (25)-(31) from the BRST complex are valued like

$$\mathrm{pgh}\left( \Phi ^{\alpha _{0}}\right) =0,\qquad \mathrm{p... ..._{0}}^{\ast }\right) =\mathrm{pgh}\left( \eta _{\alpha _{m}}^{\ast }\right) =0,$$ (35)

$$\mathrm{agh}\left( \Phi ^{\alpha _{0}}\right) =\mathrm{agh}\left(... ... }\right) =1,\qquad \mathrm{agh}\left( \eta _{\alpha _{m}}^{\ast }\right) =m+1,$$ (36)

for $m=\overline{1,4}$. The actions of the differentials $\delta$ and $\gamma$ on the above generators read as

$$\left( \delta \Phi ^{\alpha _{0}}=0,\qquad \delta \eta ^{\alpha _... ...=0,\qquad m=\overline{1,4}\right) \Longleftrightarrow \delta \chi ^{\Delta }=0,$$ (37)

$$\delta \varphi ^{\ast }=\partial _{\mu }H^{\mu },\qquad \delta H_... ...tial _{\mu }\varphi ,\qquad \delta V^{\ast \mu }=-\partial _{\nu }B^{\mu \nu },$$ (38)

$$\delta B_{\mu \nu }^{\ast }=-\tfrac{1}{2}\partial _{\lbrack \mu }... ...\mu \nu \rho }^{\ast }=-\tfrac{1}{3}\partial _{\lbrack \mu }\phi _{\nu \rho ]},$$ (39)

$$\delta t^{\ast \mu \nu \vert\alpha }=-\tfrac{1}{2}\partial _{\rho... ... ,\qquad \delta C_{\mu \nu }^{\ast }=\partial _{\lbrack \mu }H_{\nu ]}^{\ast },$$ (40)

$$\delta \eta ^{\ast }=-\partial _{\mu }V^{\ast \mu },\qquad \delta... ... ]}^{\ast },\qquad \delta C^{\ast \mu }=2\partial _{\nu }\phi ^{\ast \mu \nu },$$ (41)

$$\delta \mathcal{G}_{\mu \nu \rho \lambda }^{\ast }=\partial _{\lb... ...,\qquad \delta A^{\ast \mu \nu }=3\partial _{\rho }t^{\ast \mu \nu \vert\rho },$$ (42)

$$\delta C_{\mu \nu \rho }^{\ast }=-\partial _{\lbrack \mu }C_{\nu ... ... \rho \lambda ]}^{\ast },\qquad \delta C^{\ast }=\partial _{\mu }C^{\ast \mu },$$ (43)

$$\delta \mathcal{G}_{\mu \nu \rho \lambda \sigma }^{\ast }=-\parti... ...^{\ast \rho \mu }\right) \equiv 2\partial _{\rho }\mathcal{C}^{\ast \rho \mu },$$ (44)

$$\delta C_{\mu \nu \rho \lambda }^{\ast }=\partial _{\lbrack \mu }... ...sigma }^{\ast }=-\partial _{\lbrack \mu }C_{\nu \rho \lambda \sigma ]}^{\ast },$$ (45)

and respectively

$$\left( \gamma \Phi _{\alpha _{0}}^{\ast }=0,\qquad \gamma \eta _{... ...d m=\overline{1,4}\right) \Longleftrightarrow \gamma \chi _{\Delta }^{\ast }=0,$$ (46)

$$\gamma \varphi =0,\qquad \gamma H^{\mu }=2\partial _{\nu }C^{\mu \nu },\qquad \gamma V_{\mu }=\partial _{\mu }\eta ,$$ (47)

$$\gamma B^{\mu \nu }=-3\partial _{\rho }\eta ^{\mu \nu \rho },\qqu... ...mma K^{\mu \nu \rho }=4\partial _{\lambda }\mathcal{G}^{\mu \nu \rho \lambda },$$ (48)

$$\gamma t_{\mu \nu \vert\alpha }=\partial _{\lbrack \mu }S_{\nu ]\... ... }A_{\mu \nu },\qquad \gamma C^{\mu \nu }=-3\partial _{\rho }C^{\mu \nu \rho },$$ (49)

$$\gamma \eta =0,\qquad \gamma \eta ^{\mu \nu \rho }=4\partial _{\lambda }\eta ^{\mu \nu \rho \lambda },\qquad \gamma C_{\mu }=\partial _{\mu }C,$$ (50)

$$\gamma \mathcal{G}^{\mu \nu \rho \lambda }=-5\partial _{\sigma }\... ..._{(\mu }S_{\nu )},\qquad \gamma A_{\mu \nu }=\partial _{\lbrack \mu }S_{\nu ]},$$ (51)

$$\gamma C^{\mu \nu \rho }=4\partial _{\lambda }C^{\mu \nu \rho \la... ... }=-5\partial _{\sigma }\eta ^{\mu \nu \rho \lambda \sigma },\qquad \gamma C=0,$$ (52)

$$\gamma \mathcal{G}^{\mu \nu \rho \lambda \sigma }=0,\qquad \gamma... ...^{\mu \nu \rho \lambda }=-5\partial _{\sigma }C^{\mu \nu \rho \lambda \sigma },$$ (53)

$$\gamma \eta ^{\mu \nu \rho \lambda \sigma }=0,\qquad \gamma C^{\mu \nu \rho \lambda \sigma }=0.$$ (54)

The overall degree that grades the BRST complex is named ghost number ( $\mathrm{gh}$) and is defined like the difference between the pure ghost number and the antighost number, such that $\mathrm{gh}\left( \delta \right) =\mathrm{gh}\left( \gamma \right) =\mathrm{gh}\left( s\right) =1$.

The BRST symmetry admits a canonical action $s\cdot =\left( \cdot ,\bar{S} \right)$, where its canonical generator ( $\mathrm{gh}\left( \bar{S}\right) =0$, $\varepsilon \left( \bar{S}\right) =0$) satisfies the classical master equation $\left( \bar{S},\bar{S}\right) =0$. The symbol $(,)$ denotes the antibracket, defined by decreeing the fields/ghosts conjugated with the corresponding antifields. In the case of the free theory under discussion the solution to the master equation takes the form

$$\bar{S}=S_{0}^{\mathrm{L}} + \int d^{5}x\left[ 2H_{\mu }^{\ast }\partial _{\nu }C^{\mu \nu }+V... ...\nu \rho }+\phi ^{\ast \mu \nu }\partial _{\lbrack \mu }C_{\nu ]}\right. \notag$$ (55)

$$+4K_{\mu \nu \rho }^{\ast }\partial _{\lambda }\mathcal{G}^{\mu \... ... _{\lbrack \mu }A_{\nu ]\alpha }-2\partial _{\alpha }A_{\mu \nu }\right) \notag$$ (56)

$$-3C_{\mu \nu }^{\ast }\partial _{\rho }C^{\mu \nu \rho }+4\eta _{... ...a }^{\ast }\partial _{\sigma }\mathcal{G}^{\mu \nu \rho \lambda \sigma } \notag$$ (57)

$$+3S^{\ast \mu \nu }\partial _{(\mu }S_{\nu )}+A^{\ast \mu \nu }\p... ...+4C_{\mu \nu \rho }^{\ast }\partial _{\lambda }C^{\mu \nu \rho \lambda } \notag$$ (58)

$$\left. -5\eta _{\mu \nu \rho \lambda }^{\ast }\partial _{\sigma }... ...o \lambda }^{\ast }\partial _{\sigma }C^{\mu \nu \rho \lambda \sigma }\right] .$$ (59)

The solution to the master equation encodes all the information on the gauge structure of a given theory. We remark that in our case solution (53) decomposes into terms with antighost numbers ranging from zero to four. Let us briefly recall the significance of the various terms present in the solution to the master equation. Thus, the part with the antighost number equal to zero is nothing but the Lagrangian action of the gauge model under study. The components of antighost number equal to one are always proportional with the gauge generators. If the gauge algebra were non-Abelian, then there would appear terms simultaneously linear in the antighost number two antifields and quadratic in the pure ghost number one ghosts. The absence of such terms in our case shows that the gauge transformations are Abelian. The terms from (53) with higher antighost numbers give us information on the reducibility functions. If the reducibility relations held on-shell, then there would appear components linear in the ghosts for ghosts (ghosts of pure ghost number strictly greater than one) and quadratic in the various antifields. Such pieces are not present in (53) since the reducibility relations (10), ( 12), and (14) hold off-shell. Other possible components in the solution to the master equation offer information on the higher-order structure functions related to the tensor gauge structure of the theory. There are no such terms in (53) as a consequence of the fact that all higher-order structure functions vanish for the theory under study.