A. No-go result for in

In agreement with (86), the general solution to the equation can be chosen to stop at antighost number

where the components on the right-hand side of (181) are subject to the equations (68) and (66)-(67) for .

The piece
as solution to equation (68) for has the general form expressed by (75) for , with
from
. According to (81)
at antighost number five, it follows that
is spanned by the generic representatives (82). Since
should effectively mix the BF and the tensor field
sectors in order to produce cross-couplings and (82) involves only
BF generators, it follows that one should retain from the basis elements
only the objects containing at
least one ghost from the tensor field sector, namely
or . Recalling that we work precisely in , we obtain
that the general solution to (68) for reduces to

Each tilde object from the right-hand side of (182) means the Hodge dual of the corresponding non-tilde element, defined in general by formula ( 92). The elements are dual to as in (82), with respectively replaced by the smooth function depending only on the undifferentiated scalar field .

Introducing (182) in equation (66) for and recalling
definitions (35)-(52), we obtain

In (183) are dual to (83 ), with . In addition, is implicitly defined by formula (74) so it is a ghost field of pure ghost number one without definite symmetry/antisymmetry property, is its associated antifield, defined such that the antibracket is equal to the `unit'

The nonintegrated density stands for the solution to the homogeneous equation (68) for , showing that can be taken as a nontrivial element of in pure ghost number equal to four.

At this stage it is useful to decompose as a sum between two components

where is the solution to (68) for which is explicitly required by the consistency of in antighost number three (ensures that (67) possesses solutions for with respect to the terms from (183) containing the functions of the type ) and signifies the part of the solution to (68) for that is independently consistent in antighost number three

Using definitions (35)-(52) and decomposition (185 ), by direct computation we obtain that

where we made the notations

and are some local currents. In (187)-(189) and denote the duals of (84) and (85) with . In addition, represents the dual of and the dual of . Inspecting (187), it follows that the consistency of in antighost number three, namely the existence of as solution to (67) for , requires the conditions

and

where we made the notations and . Nevertheless, from (189) it is obvious that is a nontrivial element from in pure ghost number four, which does not reduce to a full divergence, and therefore (190) requires that , which further imply that all the functions of the type must be some real constants

Based on (192), it is clear that given by ( 182) vanishes, and hence we can assume, without loss of nontrivial terms, that

in (181).

2018-03-26