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

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

$\displaystyle a_{4}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2}\tilde{\eta}^{\ast }\left( q_{1}S_{\mu
}S^{\mu }+\tfr...
...) +\left( \tilde{U}
_{5}\right) ^{\mu }\eta \tilde{D}_{\mu \nu }S^{\nu } \notag$ (247)
    $\displaystyle +\left( \left( \tilde{U}_{6}\right) ^{\mu }C+\left( \tilde{U}_{7}...
...de{D}^{\lambda \delta }\sigma _{\alpha (\gamma }\sigma
_{\delta )\beta } \notag$ (248)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{9}\right) ^{\mu }D_{\mu \nu \rho }\tilde{D}
^{\nu \alpha }\tilde{D}^{\rho \beta }\eta \sigma _{\alpha \beta },$ (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

$\displaystyle a_{3}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \tfrac{1}{2}q_{1}\tilde{\eta}^{\ast \mu }S^{\nu }
...V_{\lambda }+\tfrac{3}{2}\tilde{F}_{\beta \nu \vert\lambda }\eta
\right) \notag$ (250)
    $\displaystyle +\tfrac{1}{2}\left( \tilde{U}_{5}\right) ^{\mu \nu }\left[ \left(...
...ho \lambda }\eta \tilde{D}_{\mu \rho }\mathcal{C}_{\nu \lambda }
\right] \notag$ (251)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{6}\right) ^{\mu \nu }\left( A_{\mu...
...athcal{\tilde{G}}+\tfrac{2}{5}S_{\mu }\mathcal{
\tilde{G}}_{\nu }\right) \notag$ (252)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{9}\right) ^{\mu \nu }\sigma _{\alp...
...vert\nu }\tilde{D}^{\lambda \alpha }\tilde{D}
^{\rho \beta }\eta \right] \notag$ (253)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{8}\right) ^{\mu \tau }\varepsilon ...
... }\sigma _{\alpha
(\gamma }\sigma _{\delta )\beta }+\bar{a}_{3}^{\mathrm{int}},$ (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)

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

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

Based on definitions (35)-(52) and taking into account decomposition (197), we get by direct computation
$\displaystyle \delta a_{3}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \delta \left[ \hat{a}_{3}^{\mathrm{int}}-
...ilde{U}_{5}\right) ^{\mu \nu \alpha }B_{\mu
\nu }^{\ast }\right. \right. \notag$ (257)
    $\displaystyle \left. +\tfrac{1}{3}\left( \tilde{U}_{5}\right) ^{\mu \nu \rho \a...
...mu \nu \rho \lambda }^{\ast }\right)
\tilde{D}_{\alpha \beta }S^{\beta } \notag$ (258)
    $\displaystyle +\tfrac{1}{2}\sigma _{\alpha \beta }D_{\mu \nu \rho }\tilde{D}^{\...
...\mu \lambda \sigma \gamma }\eta _{\lambda \sigma \gamma }^{\ast }\right.
\notag$ (259)
    $\displaystyle \left. \left. +\tfrac{1}{12}\left( \tilde{U}_{9}\right) ^{\mu \la...
... }\right)
\right] +\gamma c_{2}+\partial _{\lambda }j_{2}^{\lambda }+\chi _{2},$ (260)

$\displaystyle c_{2}$ $\displaystyle =$ $\displaystyle -\check{c}_{2}+\tfrac{q_{1}}{12}\tilde{\eta}^{\ast \mu \nu }\left...
...{\rho \lambda }\mathcal{C}
_{\mu \rho }\mathcal{C}_{\nu \lambda }\right) \notag$ (261)
    $\displaystyle +\tfrac{q_{2}}{4}\tilde{\eta}^{\ast \lambda \sigma }\sigma ^{\mu ...
...\lambda }^{\alpha \beta }\eta \right) \tilde{F}_{\beta
\nu \vert\sigma } \notag$ (262)
    $\displaystyle +\tfrac{1}{2}\left( \tilde{U}_{5}\right) ^{\mu \nu \rho }\left[ V...
...de{D}_{\nu \lambda }
\mathcal{C}_{\rho }^{\quad \lambda }\right) \right. \notag$ (263)
    $\displaystyle \left. +\tfrac{1}{2}\eta \left( \tilde{F}_{\mu \lambda \vert\nu }...
...\tilde{D}_{\mu }^{\quad \alpha }t_{\nu
\rho \vert\alpha }\right) \right] \notag$ (264)
    $\displaystyle +\tfrac{1}{2}\left( \left( \tilde{U}_{5}\right) ^{\mu \nu \lambda...
...\tilde{D}_{\lambda \alpha }\mathcal{C}
_{\sigma }^{\quad \alpha }\right) \notag$ (265)
    $\displaystyle +\tfrac{1}{2}\left( \tilde{U}_{6}\right) ^{\mu \nu \rho }\left( A...
...cal{\tilde{G}}_{\rho }+
\tfrac{1}{4}S_{\mu }\tilde{K}_{\nu \rho }\right) \notag$ (266)
    $\displaystyle +\tfrac{1}{8}\left( \tilde{U}_{8}\right) ^{\mu \varepsilon \pi
...t\pi }^{\sigma \delta }\sigma _{\alpha (\gamma
}\sigma _{\delta )\beta } \notag$ (267)
    $\displaystyle -\tfrac{1}{8}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma }\s...
...\alpha }\tilde{F}_{\quad \vert\sigma }^{\rho \beta }\eta \right)
\right. \notag$ (268)
    $\displaystyle \left. +2F_{\mu \nu \rho \vert\sigma }\tilde{D}^{\nu \alpha }\lef...
...mbda }+\tilde{F}_{\quad \vert\lambda }^{\rho \beta }\eta
\right) \right] \notag$ (269)
    $\displaystyle -\tfrac{1}{4}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma \ga...
...^{\nu \alpha }\right)
\tilde{D}^{\rho \beta }B_{\lambda \sigma }^{\ast } \notag$ (270)
    $\displaystyle +\tfrac{1}{12}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma \g...
... \alpha
}\right) \tilde{D}^{\rho \beta }\eta _{\lambda \sigma \gamma }^{\ast },$ (271)

$\displaystyle \chi _{2}$ $\displaystyle =$ $\displaystyle \tfrac{q_{1}}{6}\tilde{\eta}^{\ast \mu \nu }S^{\rho }D_{\mu \nu
...\rho \alpha }-3\tilde{R}_{\mu \lambda
\vert\nu \rho }S^{\lambda }\right) \notag$ (272)
    $\displaystyle +\tfrac{q_{2}}{6}\sigma ^{\mu \nu }\tilde{D}_{\mu \alpha }\tilde{...
...{\ast \lambda \rho }\tilde{R}_{\beta \nu \vert\lambda \rho
}\eta \right] \notag$ (273)
    $\displaystyle +\tfrac{1}{6}\left( \tilde{U}_{6}\right) ^{\mu \nu \rho }D_{\mu \...
...ilde{U}_{7}\right) ^{\mu \nu \rho }D_{\mu \nu \rho }
\mathcal{\tilde{G}} \notag$ (274)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{8}\right) ^{\mu \varepsilon \pi
...\lambda \gamma }\tilde{R}_{\quad \vert\varepsilon \pi }^{\sigma \delta }
\notag$ (275)
    $\displaystyle +\tfrac{1}{4}\left( \tilde{U}_{9}\right) ^{\mu \lambda \sigma }\s...
...ert\lambda \sigma }\tilde{D}^{\nu \alpha
}\right) \tilde{D}^{\rho \beta }\eta ,$ (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

$\displaystyle U_{5}\left( \varphi \right) =u_{5},\qquad U_{6}\left( \varphi \right) =u_{6},\qquad U_{7}\left( \varphi \right) =u_{7},$ (277)
$\displaystyle U_{8}\left( \varphi \right) =u_{8},\qquad U_{9}\left( \varphi \right) =u_{9},\qquad q_{1}=0=q_{2}.$ (278)

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

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

in (194).

Ashkbiz Danehkar