I am trying to show that in a cohomological TQFT, given a physical operator $\phi^{(0)}$, one can construct a chain of non-local physical operators. In doing so, I need to show that a certain set of operators $\phi^{(n)}$ obey the so called topological descent equations. I am following these notes, as well as these ones. This unanswered StackExchange question is similar, but somewhat more general and unanswered. I will be following the conventions of my first reference.
The setup is as follows. We consider a TQFT with a nilpotent fermionic (supercharge), operator $Q$. Physical operators are those for which $[Q,\mathcal{O}\}=0$, where $[\cdot,\cdot\}$ is the graded commutator (the ordinary commutator unless both arguments are fermionic). Being a TQFT, we also have that the momentum operator is $P_{\mu}=\{Q,G_{\mu}\}$ for some fermionic operator $G_{\mu}$.
We take a (for simplicity), bosonic physical operator $\phi^{(0)}$ , so that $[Q,\phi^{(0)}]=0$. From here, we define descendent operators: \begin{align*} & \phi^{(1)}=\phi_{\mu}^{(1)}dx^{\mu}=[G_{\mu},\phi^{(0)}]dx^{\mu} \\ & \phi^{(2)}=\phi_{\mu\nu}^{(2)}dx^{\mu}\wedge dx^{\nu}=\{G_{\mu},[G_{\nu},\phi^{(0)}]\}dx^{\mu}\wedge dx^{\nu}=\{G_{\mu},\phi^{(1)}_{\nu}\}dx^{\mu}\wedge dx^{\nu} \end{align*} and so on. The topological descent equations are: $$ d\phi^{(n)}=i[Q,\phi^{(n+1)}\}\text{ , for all }n\geq 0\text{.} $$
For $n=0$, things are relatively easy because $\phi^{(0)}$ is physical. My second reference shows this case in detail. However, I am having trouble showing the the next-simplest case, $n=1$. I am able to obtain $d\phi^{(1)}=i[Q,\phi^{(2)}]$ up to a second term, but cannot see why the second term vanishes. I will repeat my calculation here.
\begin{align*} d\phi^{(1)} &= \frac{\partial\phi_{\mu}^{(1)}}{\partial x^{\nu}}dx^{\nu}\wedge dx^{\mu} \\ &= i[P_{\nu},\phi^{(1)}_{\mu}]dx^{\nu}\wedge dx^{\mu} \\ &= i[\{Q,G_{\nu}\},\phi^{(1)}_{\mu}]dx^{\nu}\wedge dx^{\mu} \\ &= i \left( - [\{G_{\nu},\phi^{(1)}_{\mu}\},Q] - [\{\phi^{(1)}_{\mu},Q\},G_{\nu}] \right) \\ &= i[Q,\phi^{(2)}]-i[\{\phi^{(1)}_{\mu},Q\},G_{\nu}]dx^{\nu}\wedge dx^{\mu} \end{align*}
Where in the second equality I have used that momentum is the generator of translations, and in the fourth equality I have used the Jacobi-like identity $[\{A,B\},C]=\{[B,C],A\}-\{[C,A],B\}$.
The first term is as desired, but I can see no reason why the term $-i[\{\phi^{(1)}_{\mu},Q\},G_{\nu}]dx^{\nu}\wedge dx^{\mu}$ should vanish. I have fiddled around with Jacobi-like identities and tried to relate it to $0=d^{2}\phi^{(0)}$, but with no luck. Any pointers would be much appreciated.
