C. No-go result for $ I=3$ in $ a^{\mathrm {int}}$

We have seen in the previous two Appendixes A and B that we can always take (193) and (204) in (181). Consequently, the first-order deformation of the solution to the master equation in the interacting case stops at antighost number three

$\displaystyle a^{\mathrm{int}}=a_{0}^{\mathrm{int}}+a_{1}^{\mathrm{int}}+a_{2}^{\mathrm{int }}+a_{3}^{\mathrm{int}},$ (280)

where the components on the right-hand side of (205) are subject to the equations (68) and (66)-(67) for $ I=3$.

The piece $ a_{3}^{\mathrm{int}}$ as solution to equation (68) for $ I=3$ has the general form expressed by (75) for $ I=3$, with $ \alpha _{3}$ from $ H_{3}^{\mathrm{inv}}(\delta \vert d)$. Looking at formula (76) and also at relation (81) in antighost number three and requiring that $ a_{3}^{\mathrm{int}}$ mixes BRST generators from the BF and $ (2,1)$ sectors, we find that the most general solution to (68) for $ I=3$ reads as8

$\displaystyle a_{3}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \tilde{\eta}^{\ast \mu }\left( q_{3}\eta S_{\mu
}+q_{4}S^{\nu }\t...
}D_{\mu \nu \rho }\tilde{D}^{\nu \alpha }\tilde{D}^{\rho \beta }\right)
\notag$ (281)
    $\displaystyle +q_{6}S^{\ast \mu }\eta S_{\mu }+\tfrac{1}{6}\sigma ^{\mu \nu }\l...
...) \tilde{D}_{\mu \alpha
}\tilde{D}^{\alpha \beta }\tilde{D}_{\beta \nu } \notag$ (282)
    $\displaystyle +\left( \tilde{U}_{10}\right) ^{\mu \nu }\tilde{D}_{\mu \nu }\mat...
...\nu }\sigma ^{\alpha \beta
}\eta \tilde{D}_{\mu \alpha }\tilde{D}_{\nu \beta },$ (283)

where any object denoted by $ q$ represents an arbitrary, real constant. Inserting (206) in equation (66) for $ I=3$ and using definitions (35)-(52), we can write
$\displaystyle a_{2}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle -q_{3}\tilde{\eta}^{\ast \mu \nu }\left( V_{\mu
}S_{\nu }+\tfrac{...
...e{D}_{\nu }^{\quad \rho
}+S^{\rho }\tilde{F}_{\rho \mu \vert\nu }\right) \notag$ (284)
    $\displaystyle -\tfrac{q_{5}}{4}\tilde{\eta}^{\ast \mu \varepsilon }\sigma _{\al...
...vert\varepsilon }\tilde{D}^{\nu \alpha }\right) \tilde{D}
^{\rho \beta } \notag$ (285)
    $\displaystyle -q_{6}\mathcal{C}^{\ast \mu \nu }\left( 2V_{\mu }S_{\nu }+\eta \m...
\alpha }\tilde{D}^{\alpha \beta }\tilde{F}_{\beta \nu \vert\lambda } \notag$ (286)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{10}\right) ^{\mu \nu \rho }\left( ...
...}_{\mu \nu }C_{\rho }-\tfrac{1}{2}\tilde{F}_{\mu \nu
\vert\rho }C\right) \notag$ (287)
    $\displaystyle +\tfrac{1}{2}\left( \tilde{U}_{12}\right) ^{\mu \nu \rho }\sigma ...
...alpha \mu
\vert\nu }\right) \tilde{D}_{\rho \beta }+\bar{a}_{2}^{\mathrm{int}}.$ (288)

The component $ \bar{a}_{2}^{\mathrm{int}}$ represents the solution to the homogeneous equation in antighost number two (68) for $ I=2$, so $ \bar{a}_{2}^{\mathrm{int}}$ is a nontrivial element from $ H\left( \gamma \right) $ of pure ghost number two and antighost number two. It is useful to decompose $ \bar{a}_{2}^{\mathrm{int}}$ as a sum between two terms

$\displaystyle \bar{a}_{2}^{\mathrm{int}}=\hat{a}_{2}^{\mathrm{int}}+\check{a}_{2}^{\mathrm{ int}},$ (289)

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

$\displaystyle \delta \check{a}_{2}^{\mathrm{int}}=-\gamma \check{c}_{1}+\partial _{\mu } \check{m}_{1}^{\mu }.$ (290)

Using definitions (35)-(49) and decomposition (208 ), by direct computation we obtain that

$\displaystyle \delta a_{2}^{\mathrm{int}}$ $\displaystyle =$ $\displaystyle \delta \left[ \hat{a}_{2}^{\mathrm{int}}-
\tfrac{1}{2}\left( \lef...
\beta }\tilde{D}_{\lambda \alpha }\tilde{D}_{\sigma \beta }\right] \notag$ (291)
    $\displaystyle +\gamma c_{1}+\partial _{\lambda }j_{1}^{\lambda }+\chi _{1},$ (292)

where we used the notations
$\displaystyle c_{1}$ $\displaystyle =$ $\displaystyle -\check{c}_{1}-\tfrac{1}{2}\tilde{B}^{\ast \mu \nu \rho }\left[
...rt\rho }+t_{\mu \nu \vert\alpha }
\tilde{D}_{\rho \beta }\right) \right. \notag$ (293)
    $\displaystyle \left. +\tfrac{q_{5}}{4}\sigma _{\lambda \sigma }\left( D_{\mu \a...
...ha \lambda }\right) \tilde{F}_{\quad \vert\rho }^{\beta \sigma }
\right] \notag$ (294)
    $\displaystyle +q_{6}t^{\ast \mu \nu \vert\rho }\left( 6V_{\mu }\mathcal{C}_{\nu...
...vert\rho }C_{\lambda }+\tilde{D}
_{\mu \nu }\phi _{\rho \lambda }\right) \notag$ (295)
    $\displaystyle -\tfrac{1}{4}\sigma ^{\mu \nu }\left( q_{7}\phi ^{\ast \lambda \r...
...F}_{\quad \vert\lambda }^{\alpha \beta }\tilde{F}_{\beta \nu \vert\rho }
\notag$ (296)
    $\displaystyle -\tfrac{1}{10}\left( \tilde{U}_{10}\right) ^{\mu \nu \rho \lambda...
...mbda }+\tfrac{1}{4}\tilde{D
}_{\mu \nu }\tilde{K}_{\rho \lambda }\right) \notag$ (297)
    $\displaystyle +\tfrac{1}{2}\left( \tilde{U}_{12}\right) ^{\mu \nu \rho \lambda ...
...lde{F}_{\mu \alpha \vert\nu }\tilde{F}_{\rho
\beta \vert\lambda }\right) \notag$ (298)
    $\displaystyle +\tfrac{1}{2}\left( \tilde{U}_{12}\right) ^{\mu \nu \rho \lambda ...
...u \nu }^{\ast }\tilde{D}_{\sigma \alpha }\tilde{
F}_{\rho \beta \vert\lambda },$ (299)

$\displaystyle \chi _{1}$ $\displaystyle =$ $\displaystyle \tfrac{1}{6}\tilde{B}^{\ast \mu \nu \rho }\left[ q_{3}\left(
... \beta
}+3\tilde{R}_{\mu \alpha \vert\nu \rho }S_{\beta }\right) \right. \notag$ (300)
    $\displaystyle \left. +\tfrac{3q_{5}}{2}\sigma _{\lambda \sigma }\left( R_{\mu \...
...\nu \vert\rho }\left( \partial _{\lbrack \mu }V_{\nu
]}\right) S_{\rho } \notag$ (301)
    $\displaystyle +\tfrac{1}{2}\sigma ^{\mu \nu }\left( q_{7}\phi ^{\ast \rho \lamb...
... \rho \lambda }\tilde{R}_{\mu \nu
\vert\rho \lambda }\mathcal{\tilde{G}} \notag$ (302)
    $\displaystyle -\tfrac{1}{2}\left( \tilde{U}_{11}\right) ^{\mu \nu \rho \lambda ...
...a \beta }\eta \tilde{D}_{\mu \alpha }\tilde{R
}_{\nu \beta \vert\rho \lambda },$ (303)

and $ j_{1}^{\mu }$ are some local currents. It is easy to see that $ \chi
_{1} $ given in (212) is a nontrivial object from $ H\left( \gamma \right) $ in pure ghost number two, which obviously does not reduce to a full divergence. Then, since (210) requires that it is $ \gamma $ -exact modulo $ d$, it must vanish, which further implies that all the functions of the type $ U\left( \varphi \right) $ are some real constants and all the constants denoted by $ q$ vanish

$\displaystyle U_{10}\left( \varphi \right) =u_{10},\qquad U_{11}\left( \varphi \right) =u_{11},\qquad U_{12}\left( \varphi \right) =u_{12},$ (304)
$\displaystyle q_{3}=q_{4}=q_{5}=q_{6}=q_{7}=q_{8}=0.$ (305)

Inserting conditions (213) and (214) into (206), we conclude that we conclude that we can safely take

$\displaystyle a_{3}^{\mathrm{int}}=0$ (306)

in (205).

Ashkbiz Danehkar