B. No-go result for in

We have seen in Appendix A that we can always take (193) in ( 181). Consequently, the first-order deformation of the solution to the master equation in the interacting case stops at antighost number four

where the components on the right-hand side of (194) 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 four, it follows that
is spanned by some representatives involving only BF generators. Since
should again mix the BF and the tensor field
sectors, 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 . The general solution to (68) for reads
as

where each element generically denoted by is the Hodge dual of an object similar to (83), but with replaced by the arbitrary, smooth function , depending on the undifferentiated scalar field, reads as in (83) with , and are two arbitrary, real constants.

Introducing (195) in equation (66) for and using
definitions (35)-(52), we determine the component of
antighost number three from
in the form

where each is the Hodge dual of an object of the type (84), with replaced by the corresponding function of the type . Here, is the general solution to the homogeneous equation (68) for , showing that is a nontrivial object from in pure ghost number three.

At this point we decompose in a manner similar to (185)

where is the solution to (68) for that ensures the consistency of in antighost number two, namely the existence of as solution to (67 ) for with respect to the terms from containing the functions of the type or the constants or , while is the solution to (68) for which is independently consistent in antighost number two

Based on definitions (35)-(52) and taking into account decomposition (197), we get by direct computation

where

and are some local currents. Reprising an argument similar to that employed in Appendix A between equations (190) and (193), we find that the existence of as solution to equation (67) for finally implies that expressed by (201) must vanish. This is further equivalent to the fact that all the functions of the type must be some real constants and both constants must vanish

Inserting (202) and (203) in (195), we conclude that we can safely take

in (194).

2018-03-26