Theorem: Exact Ward-normalization compatibility

Under ass:ward, let

Then:

1. a counterterm $c$ with $a(c)=0$ exists exactly when $[a]=0$ in ${H_{\mathrm{adm}}}^1$;

2. if $c_0$ is one solution, all solutions are $c_0+\ker L$;

3. replacing the reference prescription by an admissible change preserves the obstruction class.

Here $\mathrm{im}L$ is the actual image, not its closure.

This is a restricted algebraic obstruction quotient. Identifying it with ambient BRST cohomology requires an identification of $C,L,d_1$, their domains, and their relations with that complex. The theorem does not provide the unresolved interacting quantum BRST construction of X-G.

Proof. Since $d_{1}L=0$, the quotient is defined. The equation $a+Lc=0$ is equivalent to $a\in \mathrm{im}L$, proving (i).

Given $c_0$, the equation for $c$ reduces to $L(c-c_0)=0$, proving (ii). A reference change sends $a$ to $a+L{c}_{\ast }$, preserving its quotient class, proving (iii).