B. No-go result for $I=4$ in $a^{\mathrm {int}}$

We have seen in Appendix A that we can always take (193) in ( 181). Consequently, the first-order deformation of the solution to the master equation in the interacting case stops at antighost number four

$$a^{\mathrm{int}}=a_{0}^{\mathrm{int}}+a_{1}^{\mathrm{int}}+a_{2}^{\mathrm{int }}+a_{3}^{\mathrm{int}}+a_{4}^{\mathrm{int}},$$ (246)

where the components on the right-hand side of (194) are subject to the equations (68) and (66)-(67) for $I=4$.

The piece $a_{4}^{\mathrm{int}}$ as solution to equation (68) for $I=4$ has the general form expressed by (75) for $I=4$, with $\alpha _{4}$ from $H_{4}^{\mathrm{inv}}(\delta \vert d)$. According to (81) at antighost number four, it follows that $H_{4}^{\mathrm{inv}}(\delta \vert d)$ is spanned by some representatives involving only BF generators. Since $a_{4}^{\mathrm{int}}$ should again mix the BF and the $(2,1)$ tensor field sectors, it follows that one should retain from the basis elements $e^{4}\left( \eta ^{\bar{\Upsilon}}\right)$ only the objects containing at least one ghost from the $(2,1)$ tensor field sector, namely $D_{\mu \nu \rho }$ or $S_{\mu }$. The general solution to (68) for $I=4$ reads as

$$a_{4}^{\mathrm{int}} = \tfrac{1}{2}\tilde{\eta}^{\ast }\left( q_{1}S_{\mu }S^{\mu }+\tfr... ...) +\left( \tilde{U} _{5}\right) ^{\mu }\eta \tilde{D}_{\mu \nu }S^{\nu } \notag$$ (247)

$$+\left( \left( \tilde{U}_{6}\right) ^{\mu }C+\left( \tilde{U}_{7}... ...de{D}^{\lambda \delta }\sigma _{\alpha (\gamma }\sigma _{\delta )\beta } \notag$$ (248)

$$-\tfrac{1}{2}\left( \tilde{U}_{9}\right) ^{\mu }D_{\mu \nu \rho }\tilde{D} ^{\nu \alpha }\tilde{D}^{\rho \beta }\eta \sigma _{\alpha \beta },$$ (249)

where each element generically denoted by $\left( \tilde{U}\right) ^{\mu }$ is the Hodge dual of an object similar to (83), but with $W$ replaced by the arbitrary, smooth function $U$, depending on the undifferentiated scalar field, $\left( U_{8}\right) _{\mu \nu \rho \lambda }$ reads as in (83) with $W\left( \varphi \right) \rightarrow U_{8}\left( \varphi \right)$, and $q_{1,2}$ are two arbitrary, real constants.

Introducing (195) in equation (66) for $I=4$ and using definitions (35)-(52), we determine the component of antighost number three from $a^{\mathrm {int}}$ in the form

$$a_{3}^{\mathrm{int}} = \tfrac{1}{2}q_{1}\tilde{\eta}^{\ast \mu }S^{\nu } \mathcal{C}_{\m... ...V_{\lambda }+\tfrac{3}{2}\tilde{F}_{\beta \nu \vert\lambda }\eta \right) \notag$$ (250)

$$+\tfrac{1}{2}\left( \tilde{U}_{5}\right) ^{\mu \nu }\left[ \left(... ...ho \lambda }\eta \tilde{D}_{\mu \rho }\mathcal{C}_{\nu \lambda } \right] \notag$$ (251)

$$-\tfrac{1}{2}\left( \tilde{U}_{6}\right) ^{\mu \nu }\left( A_{\mu... ...athcal{\tilde{G}}+\tfrac{2}{5}S_{\mu }\mathcal{ \tilde{G}}_{\nu }\right) \notag$$ (252)

$$-\tfrac{1}{2}\left( \tilde{U}_{9}\right) ^{\mu \nu }\sigma _{\alp... ...vert\nu }\tilde{D}^{\lambda \alpha }\tilde{D} ^{\rho \beta }\eta \right] \notag$$ (253)

$$-\tfrac{1}{2}\left( \tilde{U}_{8}\right) ^{\mu \tau }\varepsilon ... ... }\sigma _{\alpha (\gamma }\sigma _{\delta )\beta }+\bar{a}_{3}^{\mathrm{int}},$$ (254)

where each $\left( \tilde{U}\right) ^{\mu \nu }$ is the Hodge dual of an object of the type (84), with $W$ replaced by the corresponding function of the type $U$. Here, $\bar{a}_{3}^{\mathrm{int}}$ is the general solution to the homogeneous equation (68) for $I=3$, showing that $\bar{a}_{3}^{\mathrm{int}}$ is a nontrivial object from $H\left( \gamma \right)$ in pure ghost number three.

At this point we decompose $\bar{a}_{3}^{\mathrm{int}}$ in a manner similar to (185)

$$\bar{a}_{3}^{\mathrm{int}}=\hat{a}_{3}^{\mathrm{int}}+\check{a}_{3}^{\mathrm{ int}},$$ (255)

where $\hat{a}_{3}^{\mathrm{int}}$ is the solution to (68) for $I=3$ that ensures the consistency of $a_{3}^{\mathrm{int}}$ in antighost number two, namely the existence of $a_{2}^{\mathrm{int}}$ as solution to (67 ) for $i=3$ with respect to the terms from $a_{3}^{\mathrm{int}}$ containing the functions of the type $U$ or the constants $q_{1}$ or $q_{2}$, while $\check{a}_{3}^{\mathrm{int}}$ is the solution to (68) for $I=3$ which is independently consistent in antighost number two

$$\delta \check{a}_{3}^{\mathrm{int}}=-\gamma \check{c}_{2}+\partial _{\mu } \check{m}_{2}^{\mu }.$$ (256)

Based on definitions (35)-(52) and taking into account decomposition (197), we get by direct computation

$$\delta a_{3}^{\mathrm{int}} = \delta \left[ \hat{a}_{3}^{\mathrm{int}}- \tfrac{q_{2}}{6}\tilde{... ...ilde{U}_{5}\right) ^{\mu \nu \alpha }B_{\mu \nu }^{\ast }\right. \right. \notag$$ (257)

$$\left. +\tfrac{1}{3}\left( \tilde{U}_{5}\right) ^{\mu \nu \rho \a... ...mu \nu \rho \lambda }^{\ast }\right) \tilde{D}_{\alpha \beta }S^{\beta } \notag$$ (258)

$$+\tfrac{1}{2}\sigma _{\alpha \beta }D_{\mu \nu \rho }\tilde{D}^{\... ...\mu \lambda \sigma \gamma }\eta _{\lambda \sigma \gamma }^{\ast }\right. \notag$$ (259)

$$\left. \left. +\tfrac{1}{12}\left( \tilde{U}_{9}\right) ^{\mu \la... ... }\right) \right] +\gamma c_{2}+\partial _{\lambda }j_{2}^{\lambda }+\chi _{2},$$ (260)

where

$$c_{2} = -\check{c}_{2}+\tfrac{q_{1}}{12}\tilde{\eta}^{\ast \mu \nu }\left... ...{\rho \lambda }\mathcal{C} _{\mu \rho }\mathcal{C}_{\nu \lambda }\right) \notag$$ (261)

$$+\tfrac{q_{2}}{4}\tilde{\eta}^{\ast \lambda \sigma }\sigma ^{\mu ... ...\lambda }^{\alpha \beta }\eta \right) \tilde{F}_{\beta \nu \vert\sigma } \notag$$ (262)

$$+\tfrac{1}{2}\left( \tilde{U}_{5}\right) ^{\mu \nu \rho }\left[ V... ...de{D}_{\nu \lambda } \mathcal{C}_{\rho }^{\quad \lambda }\right) \right. \notag$$ (263)

$$\left. +\tfrac{1}{2}\eta \left( \tilde{F}_{\mu \lambda \vert\nu }... ...\tilde{D}_{\mu }^{\quad \alpha }t_{\nu \rho \vert\alpha }\right) \right] \notag$$ (264)

$$+\tfrac{1}{2}\left( \left( \tilde{U}_{5}\right) ^{\mu \nu \lambda... ...\tilde{D}_{\lambda \alpha }\mathcal{C} _{\sigma }^{\quad \alpha }\right) \notag$$ (265)

$$+\tfrac{1}{2}\left( \tilde{U}_{6}\right) ^{\mu \nu \rho }\left( A... ...cal{\tilde{G}}_{\rho }+ \tfrac{1}{4}S_{\mu }\tilde{K}_{\nu \rho }\right) \notag$$ (266)

$$+\tfrac{1}{8}\left( \tilde{U}_{8}\right) ^{\mu \varepsilon \pi }\... ...t\pi }^{\sigma \delta }\sigma _{\alpha (\gamma }\sigma _{\delta )\beta } \notag$$ (267)

$$-\tfrac{1}{8}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma }\s... ...\alpha }\tilde{F}_{\quad \vert\sigma }^{\rho \beta }\eta \right) \right. \notag$$ (268)

$$\left. +2F_{\mu \nu \rho \vert\sigma }\tilde{D}^{\nu \alpha }\lef... ...mbda }+\tilde{F}_{\quad \vert\lambda }^{\rho \beta }\eta \right) \right] \notag$$ (269)

$$-\tfrac{1}{4}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma \ga... ...^{\nu \alpha }\right) \tilde{D}^{\rho \beta }B_{\lambda \sigma }^{\ast } \notag$$ (270)

$$+\tfrac{1}{12}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma \g... ... \alpha }\right) \tilde{D}^{\rho \beta }\eta _{\lambda \sigma \gamma }^{\ast },$$ (271)

$$\chi _{2} = \tfrac{q_{1}}{6}\tilde{\eta}^{\ast \mu \nu }S^{\rho }D_{\mu \nu \... ...\rho \alpha }-3\tilde{R}_{\mu \lambda \vert\nu \rho }S^{\lambda }\right) \notag$$ (272)

$$+\tfrac{q_{2}}{6}\sigma ^{\mu \nu }\tilde{D}_{\mu \alpha }\tilde{... ...{\ast \lambda \rho }\tilde{R}_{\beta \nu \vert\lambda \rho }\eta \right] \notag$$ (273)

$$+\tfrac{1}{6}\left( \tilde{U}_{6}\right) ^{\mu \nu \rho }D_{\mu \... ...ilde{U}_{7}\right) ^{\mu \nu \rho }D_{\mu \nu \rho } \mathcal{\tilde{G}} \notag$$ (274)

$$-\tfrac{1}{2}\left( \tilde{U}_{8}\right) ^{\mu \varepsilon \pi }\... ...\lambda \gamma }\tilde{R}_{\quad \vert\varepsilon \pi }^{\sigma \delta } \notag$$ (275)

$$+\tfrac{1}{4}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma }\s... ...ert\lambda \sigma }\tilde{D}^{\nu \alpha }\right) \tilde{D}^{\rho \beta }\eta ,$$ (276)

and $j_{2}^{\mu }$ are some local currents. Reprising an argument similar to that employed in Appendix A between equations (190) and (193), we find that the existence of $a_{2}^{\mathrm{int}}$ as solution to equation (67) for $i=3$ finally implies that $\chi _{2}$ expressed by (201) must vanish. This is further equivalent to the fact that all the functions of the type $U$ must be some real constants and both constants $q_{1,2}$ must vanish

$$U_{5}\left( \varphi \right) =u_{5},\qquad U_{6}\left( \varphi \right) =u_{6},\qquad U_{7}\left( \varphi \right) =u_{7},$$ (277)

$$U_{8}\left( \varphi \right) =u_{8},\qquad U_{9}\left( \varphi \right) =u_{9},\qquad q_{1}=0=q_{2}.$$ (278)

Inserting (202) and (203) in (195), we conclude that we can safely take

$$a_{4}^{\mathrm{int}}=0$$ (279)

in (194).