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

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

$$a_{3}^{\mathrm{int}} = \tilde{\eta}^{\ast \mu }\left( q_{3}\eta S_{\mu }+q_{4}S^{\nu }\t... ... }D_{\mu \nu \rho }\tilde{D}^{\nu \alpha }\tilde{D}^{\rho \beta }\right) \notag$$ (281)

$$+q_{6}S^{\ast \mu }\eta S_{\mu }+\tfrac{1}{6}\sigma ^{\mu \nu }\l... ...) \tilde{D}_{\mu \alpha }\tilde{D}^{\alpha \beta }\tilde{D}_{\beta \nu } \notag$$ (282)

$$+\left( \tilde{U}_{10}\right) ^{\mu \nu }\tilde{D}_{\mu \nu }\mat... ...\nu }\sigma ^{\alpha \beta }\eta \tilde{D}_{\mu \alpha }\tilde{D}_{\nu \beta },$$ (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

$$a_{2}^{\mathrm{int}} = -q_{3}\tilde{\eta}^{\ast \mu \nu }\left( V_{\mu }S_{\nu }+\tfrac{... ...e{D}_{\nu }^{\quad \rho }+S^{\rho }\tilde{F}_{\rho \mu \vert\nu }\right) \notag$$ (284)

$$-\tfrac{q_{5}}{4}\tilde{\eta}^{\ast \mu \varepsilon }\sigma _{\al... ...vert\varepsilon }\tilde{D}^{\nu \alpha }\right) \tilde{D} ^{\rho \beta } \notag$$ (285)

$$-q_{6}\mathcal{C}^{\ast \mu \nu }\left( 2V_{\mu }S_{\nu }+\eta \m... ...\mu \alpha }\tilde{D}^{\alpha \beta }\tilde{F}_{\beta \nu \vert\lambda } \notag$$ (286)

$$-\tfrac{1}{2}\left( \tilde{U}_{10}\right) ^{\mu \nu \rho }\left( ... ...}_{\mu \nu }C_{\rho }-\tfrac{1}{2}\tilde{F}_{\mu \nu \vert\rho }C\right) \notag$$ (287)

$$+\tfrac{1}{2}\left( \tilde{U}_{12}\right) ^{\mu \nu \rho }\sigma ... ...alpha \mu \vert\nu }\right) \tilde{D}_{\rho \beta }+\bar{a}_{2}^{\mathrm{int}}.$$ (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

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

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

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

$$\delta a_{2}^{\mathrm{int}} = \delta \left[ \hat{a}_{2}^{\mathrm{int}}- \tfrac{1}{2}\left( \lef... ...alpha \beta }\tilde{D}_{\lambda \alpha }\tilde{D}_{\sigma \beta }\right] \notag$$ (291)

$$+\gamma c_{1}+\partial _{\lambda }j_{1}^{\lambda }+\chi _{1},$$ (292)

where we used the notations

$$c_{1} = -\check{c}_{1}-\tfrac{1}{2}\tilde{B}^{\ast \mu \nu \rho }\left[ q... ...rt\rho }+t_{\mu \nu \vert\alpha } \tilde{D}_{\rho \beta }\right) \right. \notag$$ (293)

$$\left. +\tfrac{q_{5}}{4}\sigma _{\lambda \sigma }\left( D_{\mu \a... ...ha \lambda }\right) \tilde{F}_{\quad \vert\rho }^{\beta \sigma } \right] \notag$$ (294)

$$+q_{6}t^{\ast \mu \nu \vert\rho }\left( 6V_{\mu }\mathcal{C}_{\nu... ...vert\rho }C_{\lambda }+\tilde{D} _{\mu \nu }\phi _{\rho \lambda }\right) \notag$$ (295)

$$-\tfrac{1}{4}\sigma ^{\mu \nu }\left( q_{7}\phi ^{\ast \lambda \r... ...F}_{\quad \vert\lambda }^{\alpha \beta }\tilde{F}_{\beta \nu \vert\rho } \notag$$ (296)

$$-\tfrac{1}{10}\left( \tilde{U}_{10}\right) ^{\mu \nu \rho \lambda... ...mbda }+\tfrac{1}{4}\tilde{D }_{\mu \nu }\tilde{K}_{\rho \lambda }\right) \notag$$ (297)

$$+\tfrac{1}{2}\left( \tilde{U}_{12}\right) ^{\mu \nu \rho \lambda ... ...lde{F}_{\mu \alpha \vert\nu }\tilde{F}_{\rho \beta \vert\lambda }\right) \notag$$ (298)

$$+\tfrac{1}{2}\left( \tilde{U}_{12}\right) ^{\mu \nu \rho \lambda ... ...u \nu }^{\ast }\tilde{D}_{\sigma \alpha }\tilde{ F}_{\rho \beta \vert\lambda },$$ (299)

$$\chi _{1} = \tfrac{1}{6}\tilde{B}^{\ast \mu \nu \rho }\left[ q_{3}\left( \eta... ... \beta }+3\tilde{R}_{\mu \alpha \vert\nu \rho }S_{\beta }\right) \right. \notag$$ (300)

$$\left. +\tfrac{3q_{5}}{2}\sigma _{\lambda \sigma }\left( R_{\mu \... ...\nu \vert\rho }\left( \partial _{\lbrack \mu }V_{\nu ]}\right) S_{\rho } \notag$$ (301)

$$+\tfrac{1}{2}\sigma ^{\mu \nu }\left( q_{7}\phi ^{\ast \rho \lamb... ... \rho \lambda }\tilde{R}_{\mu \nu \vert\rho \lambda }\mathcal{\tilde{G}} \notag$$ (302)

$$-\tfrac{1}{2}\left( \tilde{U}_{11}\right) ^{\mu \nu \rho \lambda ... ...a \beta }\eta \tilde{D}_{\mu \alpha }\tilde{R }_{\nu \beta \vert\rho \lambda },$$ (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

$$U_{10}\left( \varphi \right) =u_{10},\qquad U_{11}\left( \varphi \right) =u_{11},\qquad U_{12}\left( \varphi \right) =u_{12},$$ (304)

$$q_{3}=q_{4}=q_{5}=q_{6}=q_{7}=q_{8}=0.$$ (305)

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

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

in (205).