The Topological form is the Pfaffian form

For a given graph $G$, Budzik, Gaiotto, Kulp, Wang, Williams, Wu, Yu, and the first author studied a ‘‘topological’’ differential form ${\alpha }_G$, which expresses violations of BRST-closedness of a quantum field theory along a single topological direction. In a seemingly unrelated context, Brown, Panzer, and the second author studied a ‘‘Pfaffian’’ differential form ${\phi }_G$, which is used to construct cohomology classes of the odd commutative graph complex. We give an explicit combinatorial proof that ${\alpha }_G$ coincides with ${\phi }_G$. We also discuss the equivalence of several properties of these forms, which had been established independently for both contexts in previous work.