The topological form is the Pfaffian form

Abstract

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 ${\varphi }_G$, which is used to construct cohomology classes of the odd commutative graph complex and of ${\mathsf{GL}}_{2n}(\mathbb{Z})$. We give an explicit combinatorial proof that ${\alpha }_G$ coincides with ${\varphi }_G$. We also discuss the equivalence of several properties of these forms which had been established independently for both contexts in previous work.