8 Some solutions to the consistency equations

Equations (130)-(135) and (141)-(142), required by the consistency of the first-order deformation, possess the following classes of solutions, interesting from the point of view of cross-couplings between the BF field sector and the tensor field with the mixed symmetry $(2,1)$.

I.

The real constants $k_{1}$ and $k_{2}$ are arbitrary ( $k_{1}^{2}+k_{2}^{2}>0$), functions $\bar{M}$ and $W_{2}$ are some arbitrary, real, smooth functions of the undifferentiated scalar field, and

$$W_{1}\left( \varphi \right) =W_{3}\left( \varphi \right) =W_{4}\left( \varphi \right) =W_{5}\left( \varphi \right) =0,$$ (208)

$$W_{6}\left( \varphi \right) =-\frac{k_{2}}{5!k_{1}}W_{2}\left( \varphi \right) .$$ (209)

The above formulas allow one to infer directly the solution in the general case $k_{2}=0$. This class of solutions can be equivalently reformulated as: the real constants $k_{1}$ and $k_{2}$ are arbitrary ( $k_{1}^{2}+k_{2}^{2}>0$ ), functions $\bar{M}$ and $W_{6}$ are some arbitrary, real, smooth functions of the undifferentiated scalar field, and

$$W_{1}\left( \varphi \right) =W_{3}\left( \varphi \right) =W_{4}\left( \varphi \right) =W_{5}\left( \varphi \right) =0,$$ (210)

$$W_{2}\left( \varphi \right) =-\frac{5!k_{1}}{k_{2}}W_{6}\left( \varphi \right) .$$ (211)

The last formulas are useful at writing down the solution in the particular case $k_{1}=0$.

II.

The real constants $k_{1}$ and $k_{2}$ are arbitrary ( $k_{1}^{2}+k_{2}^{2}>0$), functions $\bar{M}$ and $W_{5}$ are some arbitrary, real, smooth functions of the undifferentiated scalar field, and

$$W_{1}\left( \varphi \right) =W_{2}\left( \varphi \right) =W_{6}\left( \varphi \right) =0,$$ (212)

$$W_{3}\left( \varphi \right) =-\frac{k_{2}}{60k_{1}}W_{5}\left( \v... ...\right) =\left( \frac{k_{2}}{5!k_{1}} \right) ^{2}W_{5}\left( \varphi \right) .$$ (213)

The above formulas allow one to infer directly the solution in the general case $k_{2}=0$. This class of solutions can be equivalently reformulated as: the real constants $k_{1}$ and $k_{2}$ are arbitrary ( $k_{1}^{2}+k_{2}^{2}>0$ ), functions $\bar{M}$ and $W_{4}$ are some arbitrary, real, smooth functions of the undifferentiated scalar field, and

$$W_{1}\left( \varphi \right) =W_{2}\left( \varphi \right) =W_{6}\left( \varphi \right) =0,$$ (214)

$$W_{3}\left( \varphi \right) =-2\cdot 5!\frac{k_{1}}{k_{2}}W_{4}\l... ...\right) =\left( \frac{5!k_{1}}{ k_{2}}\right) ^{2}W_{4}\left( \varphi \right) .$$ (215)

The last formulas are useful at writing down the solution in the particular case $k_{1}=0$.

III.

The real constants $k_{1}$ and $k_{2}$ are arbitrary ( $k_{1}^{2}+k_{2}^{2}>0$), functions $W_{1}$ and $W_{5}$ are some arbitrary, real, smooth functions of the undifferentiated scalar field, and

$$W_{2}\left( \varphi \right) =W_{6}\left( \varphi \right) =\bar{M}\left( \varphi \right) =0,$$ (216)

$$W_{3}\left( \varphi \right) =-\frac{k_{2}}{60k_{1}}W_{5}\left( \v... ...\right) =\left( \frac{k_{2}}{5!k_{1}} \right) ^{2}W_{5}\left( \varphi \right) .$$ (217)

The above formulas allow one to infer directly the solution in the general case $k_{2}=0$. This class of solutions can be equivalently reformulated as: the real constants $k_{1}$ and $k_{2}$ are arbitrary ( $k_{1}^{2}+k_{2}^{2}>0$ ), functions $W_{1}$ and $W_{4}$ are some arbitrary, real, smooth functions of the undifferentiated scalar field, and

$$W_{2}\left( \varphi \right) =W_{6}\left( \varphi \right) =\bar{M}\left( \varphi \right) =0,$$ (218)

$$W_{3}\left( \varphi \right) =-2\cdot 5!\frac{k_{1}}{k_{2}}W_{4}\l... ...\right) =\left( \frac{5!k_{1}}{ k_{2}}\right) ^{2}W_{4}\left( \varphi \right) .$$ (219)

The last formulas are useful at writing down the solution in the particular case $k_{1}=0$.

For all classes of solutions the emerging interacting theories display the following common features:

1.

there appear nontrivial cross-couplings between the BF fields and the tensor field with the mixed symmetry $(2,1)$;

2.

the gauge transformations are modified with respect to those of the free theory and the gauge algebras become open (only close on-shell);

3.

the first-order reducibility functions are changed during the deformation process and the first-order reducibility relations take place on-shell.

Nevertheless, there appear the following differences between the above classes of solutions at the level of the higher-order reducibility:

a)

for class I the second-order reducibility functions are modified with respect to the free ones and the corresponding reducibility relations take place on-shell. The third-order reducibility functions remain those from the free case and hence the associated reducibility relations hold off-shell;

b)

for class II both the second- and third-order reducibility functions remain those from the free case and hence the associated reducibility relations hold off-shell;

c)

for class III all the second- and third-order reducibility functions are deformed and the corresponding reducibility relations only close on-shell.