4 Standard results

In the sequel we determine all consistent Lagrangian interactions that can be added to the free theory described by (1) and (4)-(7). This is done by means of solving the deformation equations (58)-(61), etc., with the help of specific cohomological techniques. The interacting theory and its gauge structure are then deduced from the analysis of the deformed solution to the master equation that is consistent to all orders in the deformation parameter.

For obvious reasons, we consider only analytical, local, Lorentz covariant, and Poincaré invariant deformations (i.e., we do not allow explicit dependence on the spacetime coordinates). The analyticity of deformations refers to the fact that the deformed solution to the master equation, (55), is analytical in the coupling constant $\lambda$ and reduces to the original solution, (53), in the free limit $\lambda =0$. In addition, we require that the overall interacting Lagrangian satisfies two further restrictions related to the derivative order of its vertices:

i)

the maximum derivative order of each interaction vertex is equal to two;

ii)

the differential order of each interacting field equation is equal to that of the corresponding free equation (meaning that at most one spacetime derivative can act on each field from the BF sector and at most two spacetime derivatives on the tensor field $t_{\mu \nu \vert \alpha }$).

If we make the notation $S_{1}=\int d^{5}x\,a$, with $a$ local, then equation (58) (which controls the first-order deformation) takes the local form

$$sa=\partial _{\mu }m^{\mu },\qquad \mathrm{gh}\left( a\right) =0,\qquad \varepsilon \left( a\right) =0,$$ (70)

for some local $m^{\mu }$. It shows that the nonintegrated density of the first-order deformation pertains to the local cohomology of $s$ in ghost number zero, $a\in H^{0}\left( s\vert d\right)$, where $d$ denotes the exterior spacetime differential. The solution to (62) is unique up to $s$-exact pieces plus divergences

$$a\rightarrow a+sb+\partial _{\mu }n^{\mu }.$$ (71)

If the general solution to (62) is trivial, $a=sb+\partial _{\mu }n^{\mu }$, then it can be made to vanish, $a=0$.

In order to analyze equation (62) we develop $a$ according to the antighost number

$$a=\sum\limits_{i=0}^{I}a_{i},\qquad \mathrm{agh}\left( a_{i}\righ... ...ad \mathrm{gh}\left( a_{i}\right) =0,\qquad \varepsilon \left( a_{i}\right) =0,$$ (72)

and assume, without loss of generality, that the above decomposition stops at some finite value of $I$. This can be shown for instance like in [43] (Section 3), under the sole assumption that the interacting Lagrangian at order one in the coupling constant, $a_{0}$, has a finite, but otherwise arbitrary derivative order. Inserting (64) into (62) and projecting it on the various values of the antighost number, we obtain the tower of equations (equivalent to (62))

$$\gamma a_{I} = \partial _{\mu }\overset{\left( I\right) }{m}^{\mu },$$ (73)

$$\delta a_{I}+\gamma a_{I-1} = \partial _{\mu }\overset{\left( I-1\right) }{m }^{\mu },$$ (74)

$$\delta a_{i}+\gamma a_{i-1} = \partial _{\mu }\overset{\left( i-1\right) }{m }^{\mu },\qquad I-1\geq i\geq 1,$$ (75)

for some local $\left( \overset{\left( i\right) }{m}^{\mu }\right) _{i= \overline{0,I}}$. Equation (65) can always be replaced in strictly positive values of the antighost number by

$$\gamma a_{I}=0,\qquad I>0.$$ (76)

Due to the second-order nilpotency of $\gamma$ ( $\gamma ^{2}=0$), the solution to (68) is unique up to $\gamma$-exact contributions

$$a_{I}\rightarrow a_{I}+\gamma b_{I}.$$ (77)

If $a_{I}$ reduces only to $\gamma$-exact terms, $a_{I}=\gamma b_{I}$, then it can be made to vanish, $a_{I}=0$. The nontriviality of the first-order deformation $a$ is translated at its highest antighost number component into the requirement that $a_{I}\in H^{I}\left( \gamma \right)$, where $H^{I}\left( \gamma \right)$ denotes the cohomology of the exterior longitudinal derivative $\gamma$ in pure ghost number equal to $I$. So, in order to solve equation (62) (equivalent with (68) and (66)-(67)), we need to compute the cohomology of $\gamma$, $H\left( \gamma \right)$, and, as it will be made clear below, also the local homology of $\delta$, $H\left( \delta \vert d\right)$.

From definitions (44)-(52) it is posible to show that $H\left( \gamma \right)$ is spanned by

$$F_{\bar{A}}=\left( \varphi ,\partial _{\mu }H^{\mu },\partial _{\... ...,\partial _{\mu }K^{\mu \nu \rho },R_{\mu \nu \rho \vert\alpha \beta }\right) ,$$ (78)

the antifields $\chi _{\Delta }^{\ast }$, and all of their spacetime derivatives as well as by the undifferentiated objects

$$\eta ^{\bar{\Upsilon}}=\left( \eta ,D_{\mu \nu \rho },C,\mathcal{... ...,\eta ^{\mu \nu \rho \lambda \sigma },C^{\mu \nu \rho \lambda \sigma }\right) .$$ (79)

In (70) and (71) we respectively used the notations

$$R_{\mu \nu \rho \vert\alpha \beta }=-\tfrac{1}{2}F_{\mu \nu \rho ... ...lpha ,\beta ]},\qquad D_{\mu \nu \rho }=\partial _{\lbrack \mu }A_{\nu \rho ]},$$ (80)

with $f_{,\beta }\equiv \partial _{\beta }f$. It is useful to denote by $R_{\mu \nu \vert\alpha }$ and $R_{\mu }$ the trace and respectively double trace of $R_{\mu \nu \rho \vert\alpha \beta }$

$$R_{\mu \nu \vert\alpha }=\sigma ^{\rho \beta }R_{\mu \nu \rho \ve... ...=\sigma ^{\rho \beta }\sigma ^{\nu \alpha }R_{\mu \nu \rho \vert\alpha \beta }.$$ (81)

The spacetime derivatives (of any order) of all the objects from (71) are removed from $H\left( \gamma \right)$ since they are $\gamma$ -exact. This can be seen directly from the last definition in (45), the last present in (48), the first from (49), the second in (50), the last from (51), and also using the relations

$$\partial _{\alpha }D_{\mu \nu \rho }=\gamma \left[ -\tfrac{1}{2}F... ...right) \right] \equiv \gamma \left[ \tfrac{1}{2}\mathcal{C}_{\mu \nu }\right] .$$ (82)

Let $e^{M}\left( \eta ^{\bar{\Upsilon}}\right)$ be the elements with pure ghost number $M$ of a basis in the space of polynomials in the objects (71). Then, the general solution to (68) takes the form (up, to trivial, $\gamma$-exact contributions)

$$a_{I}=\alpha _{I}\left( \left[ F_{\bar{A}}\right] ,\left[ \chi _{\Delta }^{\ast }\right] \right) e^{I}\left( \eta ^{\bar{\Upsilon}}\right) ,$$ (83)

where $\mathrm{agh}\left( \alpha _{I}\right) =I$ and $\mathrm{pgh}\left( e^{I}\right) =I$. The notation $f([q])$ means that $f$ depends on $q$ and its spacetime derivatives up to a finite order. The objects $\alpha _{I}$ (obviously nontrivial in $H^{0}\left( \gamma \right)$) will be called invariant `polynomials'. They are true polynomials with respect to all variables (71) and their spacetime derivatives, excepting the undifferentiated scalar field $\varphi$, with respect to which $\alpha _{I}$ may be series. This is why we will keep the quotation marks around the word polynomial(s). The result that we can replace equation (65) with the less obvious one (68) for $I>0$ is a nice consequence of the fact that the cohomology of the exterior spacetime differential is trivial in the space of invariant `polynomials' in strictly positive antighost numbers. These results on $H\left( \gamma \right)$ can be synthesized in the following array

$$\begin{array}{cccc} \begin{array}{c} \mathrm{BRST} \\ \mathrm... ...ha _{4}} & 4 & 0 & C^{\mu \nu \rho \lambda \sigma } \end{array}$$ (84)

where notations (2), (25)-(31), (32), and (70) should be taken into account.

Inserting (75) in (66) we obtain that a necessary (but not sufficient) condition for the existence of (nontrivial) solutions $a_{I-1}$ is that the invariant `polynomials' $\alpha _{I}$ are (nontrivial) objects from the local cohomology of Koszul-Tate differential $H\left( \delta \vert d\right)$ in antighost number $I>0$ and in pure ghost number zero,

$$\delta \alpha _{I}=\partial _{\mu }\overset{\left( I-1\right) }{j... ...I-1,\qquad \mathrm{pgh}\left( \overset{\left( I-1\right) }{j}^{\mu }\right) =0.$$ (85)

We recall that $H\left( \delta \vert d\right)$ is completely trivial in both strictly positive antighost and pure ghost numbers (for instance, see [42], Theorem 5.4, and [43]), so from now on it is understood that by $H\left( \delta \vert d\right)$ we mean the local cohomology of $\delta$ at pure ghost number zero. Using the fact that the free model under study is a linear gauge theory of Cauchy order equal to five and the general result from the literature [42,43] according to which the local cohomology of the Koszul-Tate differential is trivial in antighost numbers strictly greater than its Cauchy order, we can state that

$$H_{J}\left( \delta \vert d\right) =0\quad \mathrm{for\quad all}\quad J>5,$$ (86)

where $H_{J}\left( \delta \vert d\right)$ represents the local cohomology of the Koszul-Tate differential in antighost number $J$. Moreover, it can be shown that if the invariant `polynomial' $\alpha _{J}$, with $\mathrm{agh}\left( \alpha _{J}\right) =J\geq 5$, is trivial in $H_{J}\left( \delta \vert d\right)$, then it can be taken to be trivial also in $H_{J}^{\mathrm{inv}}\left( \delta \vert d\right)$

$$\left( \alpha _{J}=\delta b_{J+1}+\partial _{\mu }\overset{(J)}{c... ... \alpha _{J}=\delta \beta _{J+1}+\partial _{\mu }\overset{(J)}{\gamma }^{\mu },$$ (87)

with both $\beta _{J+1}$ and $\overset{(J)}{\gamma }^{\mu }$ invariant `polynomials'. Here, $H_{J}^{\mathrm{inv}}\left( \delta \vert d\right)$ denotes the invariant characteristic cohomology in antighost number $J$ (the local cohomology of the Koszul-Tate differential in the space of invariant `polynomials'). An element of $H_{I}^{\text{\textrm{inv}}}\left( \delta \vert d\right)$ is defined via an equation like (77), but with the corresponding current an invariant `polynomial'. This result together with ( 78) ensures that the entire invariant characteristic cohomology in antighost numbers strictly greater than five is trivial

$$H_{J}^{\mathrm{inv}}\left( \delta \vert d\right) =0\quad \mathrm{for\quad all} \quad J>5.$$ (88)

It is possible to show that no nontrivial representative of $H_{J}(\delta \vert d)$ or $H_{J}^{\mathrm{inv}}(\delta \vert d)$ for $J\geq 2$ is allowed to involve the spacetime derivatives of the fields [32] and [62]. Such a representative may depend at most on the undifferentiated scalar field $\varphi$. With the help of relations (35)-(43 ), it can be shown that $H^{\mathrm{inv}}\left( \delta \vert d\right)$ and $H\left( \delta \vert d\right)$ are spanned by the elements

$$\begin{array}{ccc} \mathrm{agh} & \begin{array}{c} \mathrm{No... ...a }^{\ast },S^{\ast \mu \nu },A^{\ast \mu \nu } & 0 \end{array}$$ (89)

where

$$\left( W\right) _{\mu \nu \rho \lambda \sigma } = \frac{dW}{d\varphi } C_{\mu \nu \rho \lambda \sigma }^{\ast }+\fr... ...]}^{\ast }+C_{[\mu \nu }^{\ast }C_{\rho \lambda \sigma ]}^{\ast }\right) \notag$$ (90)

$$+\frac{d^{3}W}{d\varphi ^{3}}\left( H_{[\mu }^{\ast }H_{\nu }^{\a... ..._{[\mu }^{\ast }C_{\nu \rho }^{\ast }C_{\lambda \sigma ]}^{\ast }\right) \notag$$ (91)

$$+\frac{d^{4}W}{d\varphi ^{4}}H_{[\mu }^{\ast }H_{\nu }^{\ast }H_{... ...\ast }H_{\nu }^{\ast }H_{\rho }^{\ast }H_{\lambda }^{\ast }H_{\sigma }^{\ast },$$ (92)

$$\left( W\right) _{\mu \nu \rho \lambda } = \frac{dW}{d\varphi }C_{\mu \nu \rho \lambda }^{\ast }+\frac{d^{2}... ...lambda ]}^{\ast }+C_{[\mu \nu }^{\ast }C_{\rho \lambda ]}^{\ast }\right) \notag$$ (93)

$$+\frac{d^{3}W}{d\varphi ^{3}}H_{[\mu }^{\ast }H_{\nu }^{\ast }C_{... ...phi ^{4}}H_{\mu }^{\ast }H_{\nu }^{\ast }H_{\rho }^{\ast }H_{\lambda }^{\ast },$$ (94)

$$\left( W\right) _{\mu \nu \rho }=\frac{dW}{d\varphi }C_{\mu \nu \... ...\frac{ d^{3}W}{d\varphi ^{3}}H_{\mu }^{\ast }H_{\nu }^{\ast }H_{\rho }^{\ast },$$ (95)

$$\left( W\right) _{\mu \nu }=\frac{dW}{d\varphi }C_{\mu \nu }^{\ast }+\frac{ d^{2}W}{d\varphi ^{2}}H_{\mu }^{\ast }H_{\nu }^{\ast },$$ (96)

whit $W=W\left( \varphi \right)$ an arbitrary, smooth function depending only on the undifferentiated scalar field $\varphi$.

In contrast to the spaces $\left( H_{J}(\delta \vert d)\right) _{J\geq 2}$ and $\left( H_{J}^{\mathrm{inv}}(\delta \vert d)\right) _{J\geq 2}$, which are finite-dimensional, the cohomology $H_{1}(\delta \vert d)$ (known to be related to global symmetries and ordinary conservation laws) is infinite-dimensional since the theory is free. Fortunately, it will not be needed in the sequel.

The previous results on $H(\delta \vert d)$ and $H^{\mathrm{inv}}(\delta \vert d)$ in strictly positive antighost numbers are important because they control the obstructions to removing the antifields from the first-order deformation. More precisely, we can successively eliminate all the pieces of antighost number strictly greater that five from the nonintegrated density of the first-order deformation by adding solely trivial terms, so we can take, without loss of nontrivial objects, the condition $I\leq 5$ into (64 ). In addition, the last representative is of the form (75), where the invariant `polynomial' is necessarily a nontrivial object from $H_{5}^{ \mathrm{inv}}(\delta \vert d)$.