5 Computation of first-order deformation

In the case $I=5$ the nonintegrated density of the first-order deformation (see (64)) becomes

$$a=a_{0}+a_{1}+a_{2}+a_{3}+a_{4}+a_{5}.$$ (97)

We can further decompose $a$ in a natural manner as a sum between two kinds of deformations

$$a=a^{\mathrm{BF}}+a^{\mathrm{int}},$$ (98)

where $a^{\mathrm{BF}}$ contains only fields/ghosts/antifields from the BF sector and $a^{\mathrm {int}}$ describes the cross-interactions between the two theories.⁶ The piece $a^{\mathrm{BF}}$ is completely known [32]. It is parameterized by seven smooth, but otherwise arbitrary functions of the undifferentiated scalar field, $\left( W_{a}\left( \varphi \right) \right) _{a=\overline{1,6} }$ and $\bar{M}(\varphi )$. In the sequel we analyze the cross-interacting piece, $a^{\mathrm {int}}$.

Due to the fact that $a^{\mathrm{BF}}$ and $a^{\mathrm {int}}$ involve different types of fields and that $a^{\mathrm{BF}}$ separately satisfies an equation of the type (62), it follows that $a^{\mathrm {int}}$ is subject to the equation

$$sa^{\mathrm{int}}=\partial ^{\mu }m_{\mu }^{\mathrm{int}},$$ (99)

for some local current $m_{\mu }^{\mathrm{int}}$. In the sequel we determine the general solution to (88) that complies with all the hypotheses mentioned in the beginning of section 4.

In agreement with (86), the general solution to the equation $sa^{ \mathrm{int}}=\partial ^{\mu }m_{\mu }^{\mathrm{int}}$ can be chosen to stop at antighost number $I=5$

$$a^{\mathrm{int}}=a_{0}^{\mathrm{int}}+a_{1}^{\mathrm{int}}+a_{2}^{\mathrm{int }}+a_{3}^{\mathrm{int}}+a_{4}^{\mathrm{int}}+a_{5}^{\mathrm{int}}.$$ (100)

We will show in Appendixes A, B and C that we can always take $a_{5}^{\mathrm{int}}=a_{4}^{\mathrm{int}}=a_{3}^{\mathrm{int} }=0$ into decomposition (89), without loss of nontrivial contributions. Consequently, the first-order deformation of the solution to the master equation in the interacting case can be taken to stop at antighost number two

$$a^{\mathrm{int}}=a_{0}^{\mathrm{int}}+a_{1}^{\mathrm{int}}+a_{2}^{\mathrm{int }},$$ (101)

where the components on the right-hand side of (90) are subject to equations (68) and (66)-(67) for $I=2$.

The piece $a_{2}^{\mathrm{int}}$ as solution to equation (68) for $I=2$ has the general form expressed by (75) for $I=2$, with $\alpha _{2}$ from $H_{2}^{\mathrm{inv}}(\delta \vert d)$. Looking at formula (76) and also at relation (81) in antighost number two and requiring that $a_{2}^{\mathrm{int}}$ mixes BRST generators from the BF and $(2,1)$ sectors, we get that the most general solution to (68) for $I=2$ reads as⁷

$$a_{2}^{\mathrm{int}} = q_{9}\eta ^{\ast \mu \nu \rho }\eta D_{\mu \nu \rho }+\left( q_{1... ...t \mu }\right) S_{\mu }+q_{12}A^{\ast \mu \nu }\eta \tilde{D}_{\mu \nu } \notag$$ (102)

$$+\tfrac{q_{13}}{2}\tilde{\eta}^{\ast \mu \nu }\sigma ^{\alpha \be... ...tilde{D}_{\nu \beta }+S^{\ast }\left( k_{1}C+k_{2} \mathcal{\tilde{G}}\right) ,$$ (103)

where all quantities denoted by $q$ or $k\$are some real, arbitrary constants.

In the above and from now on we will use a compact writing in terms of the Hodge duals

$$\tilde{\Psi}^{\nu _{1}\ldots \nu _{j}}=\tfrac{1}{\left( 5-j\right... ..._{1}\ldots \nu _{j}\mu _{1}\ldots \mu _{5-j}}\Psi _{\mu _{1}\ldots \mu _{5-j}}.$$ (104)

Consequently $\tilde{\eta}^{\ast \mu \nu }$, $\mathcal{\tilde{G}}^{\ast \varepsilon }$ and $\mathcal{\tilde{G}}_{\rho }$ are the Hodge duals of $\eta _{\rho \lambda \sigma }^{\ast }$, $\mathcal{G}_{\mu \nu \rho \lambda }^{\ast }$, and respectively $\mathcal{G}^{\mu \nu \lambda \sigma }$.

Substituting (91) in (66) for $I=2$ and using definitions ( 35)-(52), we determine the solution $a_{1}^{\mathrm{int}}$ under the form

$$a_{1}^{\mathrm{int}} = -3q_{9}B^{\ast \mu \nu }\left( V^{\rho }D_{\mu \nu \rho }+\tfrac{... ...ilde{K }^{\ast \mu \nu }+q_{11}\phi ^{\ast \mu \nu }\right) A_{\mu \nu } \notag$$ (105)

$$-3q_{12}t^{\ast \mu \nu \vert\rho }\left( V_{\rho }\tilde{D}_{\mu... ...^{\alpha \beta }\tilde{F}_{\mu \alpha \vert\rho } \tilde{D}_{\nu \beta } \notag$$ (106)

$$-2t_{\mu }^{\ast }\left( k_{1}C^{\mu }-\tfrac{k_{2}}{5}\mathcal{\tilde{G}} ^{\mu }\right) +\bar{a}_{1}^{\mathrm{int}},$$ (107)

where $\tilde{F}_{\lambda \mu \vert\alpha }$ is the Hodge dual of $F_{\qquad \vert\alpha }^{\nu \rho \sigma }$ defined in (3) with respect to its first three indices

$$\tilde{F}_{\lambda \mu \vert\alpha }=\tfrac{1}{3!}\varepsilon _{\lambda \mu \nu \rho \sigma }F_{\qquad \vert\alpha }^{\nu \rho \sigma }.$$ (108)

In the last formulas $\tilde{K}_{\lambda \sigma }$ is the dual of the three-form $K^{\mu \nu \rho }$ from action (1), $\tilde{B}^{\ast \rho \lambda \sigma }$ and $\tilde{K}^{\ast \lambda \sigma }$ represent the duals of the antifields $B_{\mu \nu }^{\ast }$ and respectively $K_{\mu \nu \rho }^{\ast }$ from (28).

In the above $\bar{a}_{1}^{\mathrm{int}}$ is the solution to the homogeneous equation (68) in antighost number one, meaning that $\bar{a}_{1}^{\mathrm{int}}$ is a nontrivial object from $H\left( \gamma \right)$ in pure ghost number one and in antighost number one. It is useful to decompose $\bar{a}_{1}^{\mathrm{int}}$ like in (208)

$$\bar{a}_{1}^{\mathrm{int}}=\hat{a}_{1}^{\mathrm{int}}+\check{a}_{1}^{\mathrm{ int}},$$ (109)

with $\hat{a}_{1}^{\mathrm{int}}$ the solution to (68) for $I=1$ that ensures the consistency of $a_{1}^{\mathrm{int}}$ in antighost number zero, namely the existence of $a_{0}^{\mathrm{int}}$ as solution to (67) for $i=1$ with respect to the terms from $a_{1}^{\mathrm{int}}$ containing the constants of the type $q$ or $k\$, and $\check{a}_{1}^{\mathrm{int}}$ the solution to (68) for $I=1$ that is independently consistent in antighost number zero

$$\delta \check{a}_{1}^{\mathrm{int}}=-\gamma \check{c}_{0}+\partial _{\mu } \check{m}_{0}^{\mu }.$$ (110)

With the help of definitions (35)-(52) and taking into account decomposition (208), we infer by direct computation

$$\delta a_{1}^{\mathrm{int}} = \delta \left[ \hat{a}_{1}^{\mathrm{int} }+\left( 2k_{1}K^{\ast \m... ...k_{2}}{30}\tilde{K}^{\ast \mu \nu \rho }\right) D_{\mu \nu \rho }\right] \notag$$ (111)

$$+\gamma c_{0}+\partial _{\lambda }j_{0}^{\lambda }+\chi _{0},$$ (112)

where

$$c_{0}=-\check{c}_{0}+\tfrac{q_{13}}{16}\tilde{V}^{\mu \nu \rho \l... ..._{1}\phi ^{\mu \nu }-\tfrac{k_{2}}{20}\tilde{K}^{\mu \nu }\right) F_{\mu \nu },$$ (113)

$$\chi _{0} = -3q_{9}\left[ \left( \partial ^{\lbrack \mu }V^{\nu ]}\right) V^{... ...( \partial ^{\nu }\eta \right) F_{\mu \nu }-V^{\mu }\eta R_{\mu }\right] \notag$$ (114)

$$+\tfrac{1}{18}\left( q_{10}\tilde{\phi}^{\mu \nu \rho }+6q_{11}K^{\mu \nu \rho }\right) D_{\mu \nu \rho } \notag$$ (115)

$$-\tfrac{3q_{12}}{4}\left[ \partial _{\rho }\left( F^{\rho \mu \nu... ...alpha }\tilde{D}_{\mu \nu }+\eta \tilde{F}_{\mu \nu \vert\alpha }\right) \notag$$ (116)

$$+\tfrac{q_{13}}{8}\varepsilon ^{\mu \nu \rho \lambda \sigma }\sig... ...V_{\rho } \tilde{D}_{\nu \beta }+\eta \tilde{F}_{\nu \beta \vert\rho }\right) ,$$ (117)

and $j_{0}^{\lambda }$ are some local currents. In the above $\tilde{V}^{\mu \nu \rho \lambda }$ and $\tilde{\phi}^{\mu \nu \lambda }$ represent the Hodge duals of the one-form $V_{\sigma }$ and respectively of the two-form $\phi _{\rho \sigma }$ from (2) and $\tilde{R}_{\lambda \sigma \vert\alpha \beta }$ is nothing but the Hodge dual of the tensor $R_{\qquad \vert\alpha \beta }^{\mu \nu \rho }$ defined in (72) with respect to its first three indices, namely

$$\tilde{R}_{\lambda \sigma \vert\alpha \beta }=\tfrac{1}{3!}\varep... ...n _{\lambda \sigma \mu \nu \rho }R_{\qquad \vert\alpha \beta }^{\mu \nu \rho }.$$ (118)

Inspecting (97), we observe that equation (67) for $i=1$ possesses solutions if and only if $\chi _{0}$ expressed by (99) is $\gamma$-exact modulo $d$. A straightforward analysis of $\chi _{0}$ shows that this is not possible unless

$$q_{9}=q_{10}=q_{11}=q_{12}=q_{13}=0.$$ (119)

Now, we insert conditions (101) in (91) and identify the most general form of the first-order deformation in the interacting sector at antighost number two

$$a_{2}^{\mathrm{int}}=S^{\ast }\left( k_{1}C+k_{2}\mathcal{\tilde{G}}\right) .$$ (120)

The same conditions replaced in (97) enable us to write

$$\hat{a}_{1}^{\mathrm{int}}=-\left( 2k_{1}K^{\ast \mu \nu \rho }+\tfrac{k_{2} }{30}\tilde{\phi}^{\ast \mu \nu \rho }\right) D_{\mu \nu \rho }.$$ (121)

Introducing (103) in (95) and then the resulting result together with (101) in (93), we obtain

$$a_{1}^{\mathrm{int}}=-2t_{\mu }^{\ast }\left( k_{1}C^{\mu }+\tfra... ...i}^{\ast \mu \nu \rho }\right) D_{\mu \nu \rho }+ \check{a}_{1}^{\mathrm{int}}.$$ (122)

Next, we determine $\check{a}_{1}^{\mathrm{int}}$ as the solution to the homogeneous equation (68) for $I=1$ that is independently consistent in antighost number zero, i.e. satisfies equation (96). According to (75) for $I=1$ the general solution to equation (68) for $I=1$ has the form

$$\check{a}_{1}^{\mathrm{int}} = t^{\ast \mu \nu \vert\rho }\left( L_{\mu \nu \vert\rho }\eta +L_{... ...st }M_{\mu \nu \rho }^{\alpha }+\varphi ^{\ast }M_{\mu \nu \rho }\right. \notag$$ (123)

$$\left. +H_{\alpha }^{\ast }\bar{M}_{\mu \nu \rho }^{\alpha }+B_{\... ...{\ast }M_{\mu \nu \rho }^{\alpha \beta \gamma }\right) D^{\mu \nu \rho } \notag$$ (124)

$$+\left( V_{\alpha }^{\ast }N^{\alpha }+\varphi ^{\ast }N+H_{\alph... ...lpha \beta }+\phi _{\alpha \beta }^{\ast }\bar{N}^{\alpha \beta }\right. \notag$$ (125)

$$\left. +K_{\alpha \beta \gamma }^{\ast }N^{\alpha \beta \gamma }\right) \eta ,$$ (126)

where all the quantities denoted by $L$, $M$, $N$, $\bar{M}$, or $\bar{N}$ are bosonic, gauge-invariant tensors, and therefore they may depend only on $F_{\bar{A}}$ given in (70) and their spacetime derivatives. The functions $L_{\mu \nu \vert\rho }$ and $L_{\mu \nu \vert\rho }^{\alpha \beta \gamma }$ exhibit the mixed symmetry $(2,1)$ with respect to their lower indices and, in addition, $L_{\mu \nu \vert\rho }^{\alpha \beta \gamma }$ is completely antisymmetric with respect to its upper indices. The remaining functions, $M$ , $\bar{M}$, $N$, and $\bar{N}$, are separately antisymmetric (where appropriate) in their upper and respectively lower indices.

In order to determine all possible solutions (105) we demand that $\check{a}_{1}^{\mathrm{int}}$ mixes the BF and $(2,1)$ sectors and (for the first time) explicitly implement the assumption on the derivative order of the interacting Lagrangian discussed in the beginning of section 4 and structured in requirements i) and ii). Because all the terms involving the functions $N$ or $\bar{N}$ contain only BRST generators from the BF sector, it follows that each such function must contain at least one tensor $R_{\mu \nu \rho \vert\alpha \beta }$ defined in (72), with $F$ as in (3). The corresponding terms from $\check{a}_{1}^{\mathrm{int}}$, if consistent, would produce an interacting Lagrangian that does not agree with requirement ii) with respect to the BF fields and therefore we must take

$$N^{\alpha }=N=\bar{N}^{\alpha }=N^{\alpha \beta }=\bar{N}^{\alpha \beta }=N^{\alpha \beta \gamma }=0.$$ (127)

In the meantime, requirement ii) also restricts all the functions $M$ and $\bar{M}$ to be derivative-free. Since the undifferentiated scalar field is the only element among $F_{\bar{A}}$ and their spacetime derivatives that contains no derivatives, it follows that all $M$ and $\bar{M}$ may depend at most on $\varphi$. Due to the fact that we work in $D=5$ and taking into account the various antisymmetry properties of these functions, it follows that the only eligible representations are

$$M_{\mu \nu \rho }^{\alpha }=M_{\mu \nu \rho }=\bar{M}_{\mu \nu \rho }^{\alpha }=0,$$ (128)

$$M_{\mu \nu \rho }^{\alpha \beta }=U_{13}\varepsilon _{\quad \mu \... ...lta _{\lbrack \mu }^{\alpha }\delta _{\nu }^{\beta }\delta _{\rho ]}^{\gamma },$$ (129)

with $U_{13}$, $U_{14}$, and $U_{15}$ some real, smooth functions of $\varphi$. The same observation stands for $L_{\mu \nu \vert\rho }$ and $L_{\mu \nu \vert\rho }^{\alpha \beta \gamma }$, so their tensorial behaviour can only be realized via some constant Lorentz tensors. Nevertheless, there is no such constant tensor in $D=5$ with the required mixed symmetry properties, and hence we must put

$$L_{\mu \nu \vert\rho }=0,\qquad L_{\mu \nu \vert\rho }^{\alpha \beta \gamma }=0.$$ (130)

Inserting results (106)-(109) in (105), it follows that the most general (nontrivial) solution to equation (68) for $I=1$ that complies with all the working hypotheses, including that on the differential order of the interacting Lagrangian, is given by

$$\check{a}_{1}^{\mathrm{int}}=\varepsilon ^{\mu \nu \rho \lambda \... ...\right) D_{\rho \lambda \sigma }+U_{15}K^{\ast \mu \nu \rho }D_{\mu \nu \rho }.$$ (131)

By acting with $\delta$ on (110) and using definitions (35 )-(52) we infer

$$\delta \check{a}_{1}^{\mathrm{int}} = \gamma \left[ \left( -3U_{14}\tilde{K} ^{\mu \nu }+2U_{15}\phi ^{\mu \nu }\right) F_{\mu \nu }\right] \notag$$ (132)

$$+\partial _{\alpha }\left( \varepsilon ^{\mu \nu \rho \lambda \al... ...^{\rho \lambda \sigma }+U_{14}\phi _{\mu \nu }D^{\alpha \mu \nu }\right) \notag$$ (133)

$$+\varepsilon _{\mu \nu \rho \lambda \sigma }\left[ -\left( \parti... ...alpha }U_{14}\right) K^{\alpha \mu \nu }\right] D^{\rho \lambda \sigma } \notag$$ (134)

$$-\left( \partial _{\mu }U_{15}\right) \phi _{\nu \rho }D^{\mu \nu... ... }\mathcal{\tilde{G}} _{\nu ]}-U_{15}\partial _{\lbrack \mu }C_{\nu ]}\right) .$$ (135)

Comparing (111) with (96), we conclude that function $U_{13}$ reduces to a real constant and meanwhile functions $U_{14}$ and $U_{15}$ must vanish

$$U_{13}=u_{13},\qquad U_{14}=0=U_{15},$$ (136)

so (110) becomes

$$\check{a}_{1}^{\mathrm{int}}=\varepsilon ^{\mu \nu \rho \lambda \sigma }u_{13}B_{\mu \nu }^{\ast }D_{\rho \lambda \sigma },$$ (137)

wich produces trivial deformations because it is a trivial element from $H_{1}(\delta \vert d)$

$$\check{a}_{1}^{\mathrm{int}}=\delta \left( \varepsilon ^{\mu \nu ... ... \nu \rho \lambda \sigma }u_{13}B_{\nu \rho }^{\ast }A_{\lambda \sigma }\right)$$ (138)

and by further taking

$$\check{a}_{1}^{\mathrm{int}}=0.$$ (139)

As a consequence, we can safely take the nontrivial part of the first-order deformation in the interaction sector in antighost number one, (104 ), of the form

$$a_{1}^{\mathrm{int}}=-2t_{\mu }^{\ast }\left( k_{1}C^{\mu }+\tfra... ...}+ \tfrac{k_{2}}{30}\tilde{\phi}^{\ast \mu \nu \rho }\right) D_{\mu \nu \rho }.$$ (140)

In addition, (115) leads to

$$\check{c}_{0}=0,\qquad \check{m}_{0}^{\mu }=0$$ (141)

in (96). Replacing now (101) and (117) in (97), we are able to identify the piece of antighost number zero from the first-order deformation in the interacting sector as

$$a_{0}^{\mathrm{int}}=\left( k_{1}\phi ^{\mu \nu }-\tfrac{k_{2}}{20}\tilde{K} ^{\mu \nu }\right) F_{\mu \nu }+\bar{a}_{0}^{\mathrm{int}},$$ (142)

where $\bar{a}_{0}^{\mathrm{int}}$ is the solution to the `homogeneous' equation in antighost number zero

$$\gamma \bar{a}_{0}^{\mathrm{int}}=\partial _{\mu }\bar{m}_{0}^{\mu }.$$ (143)

We will prove in Appendix D that the only solution to (119) that satisfies all our working hypotheses, including that on the derivative order of the interacting Lagrangian, is $\bar{a}_{0}^{\mathrm{int}}=0$, such that the nontrivial part of the first-order deformation in the interaction sector in antighost number zero reads as

$$a_{0}^{\mathrm{int}}=\left( k_{1}\phi ^{\mu \nu }-\tfrac{k_{2}}{20}\tilde{K} ^{\mu \nu }\right) F_{\mu \nu }.$$ (144)

The main conclusion of this section is that the general form of the first-order deformation of the solution to the master equation as solution to (58) for the model under study is expressed by

$$S_{1}=\int d^{5}x\left( a^{\mathrm{BF}}+a^{\mathrm{int}}\right) ,$$ (145)

where $a^{\mathrm{BF}}$ can be found in [32] and

$$a^{\mathrm{int}} = a_{0}^{\mathrm{int}}+a_{1}^{\mathrm{int}}+a_{2}^{\mathrm{ int}} \notag$$ (146)

$$= S^{\ast }\left( k_{1}C+k_{2}\mathcal{\tilde{G}}\right) -2t_{\mu }... ... }\left( k_{1}C^{\mu }+\tfrac{k_{2}}{5}\mathcal{\tilde{G}}^{\mu }\right) \notag$$ (147)

$$-\left( 2k_{1}K^{\ast \mu \nu \rho }+\tfrac{k_{2}}{30}\tilde{\phi... ...{1}\phi ^{\mu \nu }-\tfrac{ k_{2}}{20}\tilde{K}^{\mu \nu }\right) F_{\mu \nu }.$$ (148)

It is now clear that the first-order deformation is parameterized by seven arbitrary, smooth functions of the undifferentiated scalar field ( $\left( W_{a}\left( \varphi \right) \right) _{a=\overline{1,6} }$ and $\bar{M}(\varphi )$ corresponding to $a^{\mathrm{BF}}$ and by two arbitrary, real constants ($k_{1}$ and $k_{2}$ from $a^{\mathrm {int}}$). We will see in the next section that the consistency of the deformed solution to the master equation in order two in the coupling constant will restrict these functions and constants to satisfy some specific equations.