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

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

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

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

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

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

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.

2018-03-26