C. No-go result for in

We have seen in the previous two Appendixes A and B that we can always take (193) and (204) in (181). Consequently, the first-order deformation of the solution to the master equation in the interacting case stops at antighost number three

where the components on the right-hand side of (205) 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
. Looking at formula (76)
and also at relation (81) in antighost number three and
requiring that
mixes BRST generators from the BF and sectors, we find that the most general solution to (68) for reads as^{8}

where any object denoted by represents an arbitrary, real constant. Inserting (206) in equation (66) for and using definitions (35)-(52), we can write

The component represents the solution to the homogeneous equation in antighost number two (68) for , so is a nontrivial element from of pure ghost number two and antighost number two. It is useful to decompose as a sum between two terms

with the solution to (68) for that ensures the consistency of in antighost number one, namely the existence of as solution to (67) for with respect to the terms from containing the functions of the type or the constants denoted by , and the solution to (68) for that is independently consistent in antighost number one

Using definitions (35)-(49) and decomposition (208
), by direct computation we obtain that

where we used the notations

and are some local currents. It is easy to see that given in (212) is a nontrivial object from in pure ghost number two, which obviously does not reduce to a full divergence. Then, since (210) requires that it is -exact modulo , it must vanish, which further implies that all the functions of the type are some real constants and all the constants denoted by vanish

Inserting conditions (213) and (214) into (206), we conclude that we conclude that we can safely take

in (205).

2018-03-26