# 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 .

I.
The real constants and are arbitrary ( ), functions and are some arbitrary, real, smooth functions of the undifferentiated scalar field, and (208) (209)

The above formulas allow one to infer directly the solution in the general case . This class of solutions can be equivalently reformulated as: the real constants and are arbitrary ( ), functions and are some arbitrary, real, smooth functions of the undifferentiated scalar field, and (210) (211)

The last formulas are useful at writing down the solution in the particular case .

II.
The real constants and are arbitrary ( ), functions and are some arbitrary, real, smooth functions of the undifferentiated scalar field, and (212) (213)

The above formulas allow one to infer directly the solution in the general case . This class of solutions can be equivalently reformulated as: the real constants and are arbitrary ( ), functions and are some arbitrary, real, smooth functions of the undifferentiated scalar field, and (214) (215)

The last formulas are useful at writing down the solution in the particular case .

III.
The real constants and are arbitrary ( ), functions and are some arbitrary, real, smooth functions of the undifferentiated scalar field, and (216) (217)

The above formulas allow one to infer directly the solution in the general case . This class of solutions can be equivalently reformulated as: the real constants and are arbitrary ( ), functions and are some arbitrary, real, smooth functions of the undifferentiated scalar field, and (218) (219)

The last formulas are useful at writing down the solution in the particular case .

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.
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.

Ashkbiz Danehkar
2018-03-26