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The Topological form is the Pfaffian form

Paul-Hermann Balduf, Simone Hu

TL;DR

The paper proves an explicit equivalence between two differential forms associated to a connected graph: the topological form $\alpha_G$, arising from BRST-type quantum corrections in a 1D topological QFT, and the Pfaffian form $\phi_G$, used to construct cocycles in the odd graph complex $\mathsf{GC}_3$. The main result shows $\alpha_G = \frac{\det(\mathcal{C}\,|\,\mathcal{P})}{2^{\ell}}\;\phi_G$, where $\ell$ is the loop number and $\det(\mathcal{C}\,|\,\mathcal{P})$ is a sign depending on labelling and the chosen cycle basis, independent of the path matrix. The authors develop a Dodgson-polynomial framework to express both forms in a common combinatorial basis and deduce the equivalence by comparing spanning-tree-based expansions, up to the global sign. This unifies BRST/topological data with graph-complex cohomology, clarifies how quadratic/Stokes relations and dipole structures translate between the two viewpoints, and suggests deep connections to formality theorems and TQFT. Practically, it provides a concrete, computable bridge between parametric Feynman integrals and graph-cohomology constructions, enabling transfer of results and techniques across contexts. The work also analyzes implications for dipole graphs, Moyal products, and the Maurer–Cartan framework within GC$_3$, with potential extensions to holomorphic settings and higher-dimensional topological theories.

Abstract

For a given graph $G$, Budzik, Gaiotto, Kulp, Wang, Williams, Wu, Yu, and the first author studied a ''topological'' differential form $α_G$, 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 $φ_G$, which is used to construct cohomology classes of the odd commutative graph complex. We give an explicit combinatorial proof that $α_G$ coincides with $φ_G$. We also discuss the equivalence of several properties of these forms, which had been established independently for both contexts in previous work.

The Topological form is the Pfaffian form

TL;DR

The paper proves an explicit equivalence between two differential forms associated to a connected graph: the topological form , arising from BRST-type quantum corrections in a 1D topological QFT, and the Pfaffian form , used to construct cocycles in the odd graph complex . The main result shows , where is the loop number and is a sign depending on labelling and the chosen cycle basis, independent of the path matrix. The authors develop a Dodgson-polynomial framework to express both forms in a common combinatorial basis and deduce the equivalence by comparing spanning-tree-based expansions, up to the global sign. This unifies BRST/topological data with graph-complex cohomology, clarifies how quadratic/Stokes relations and dipole structures translate between the two viewpoints, and suggests deep connections to formality theorems and TQFT. Practically, it provides a concrete, computable bridge between parametric Feynman integrals and graph-cohomology constructions, enabling transfer of results and techniques across contexts. The work also analyzes implications for dipole graphs, Moyal products, and the Maurer–Cartan framework within GC, with potential extensions to holomorphic settings and higher-dimensional topological theories.

Abstract

For a given graph , Budzik, Gaiotto, Kulp, Wang, Williams, Wu, Yu, and the first author studied a ''topological'' differential form , 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 , which is used to construct cohomology classes of the odd commutative graph complex. We give an explicit combinatorial proof that coincides with . We also discuss the equivalence of several properties of these forms, which had been established independently for both contexts in previous work.

Paper Structure

This paper contains 17 sections, 19 theorems, 80 equations.

Table of Contents

  1. Introduction and results
  2. The main result
  3. The topological form $\alpha_G$
  4. The Pfaffian form $\phi_G$
  5. Acknowledgements
  6. Discussion
  7. Immediate algebraic properties
  8. Dipole / multiedge graphs and the Moyal product
  9. Quadratic / Stokes relations
  10. Outlook
  11. Definitions and proofs
  12. Incidence and path matrices
  13. Laplacians
  14. Concatenated matrices and signs
  15. Dodgson polynomials
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G$ be a connected graph with loop number $\ell \geq 0$, and let $\phi_G$ be its Pfaffian form (def:pff-form) and $\alpha_G$ its topological form (def:alpha). Then Here, $\left( \mathcal{C} \,\vert\, \mathcal{P} \right)$ is the concatenation of the cycle incidence matrix $\mathcal{C}$ (def:cycle_incidence_matrix) used to define $\phi_{G}$ and a path matrix $\mathcal{P}$ (def:pathmatrix). The

Theorems & Definitions (40)

  • Theorem 1
  • Definition 1
  • Proposition 1: Theorem 1 of balduf_combinatorial_2024
  • Definition 2
  • Lemma 2: Lemma 2.2, Theorem 6.7 of brown_unstable_2024
  • Lemma 3: Lemma 2.2, Lemma 5.6, Corollary 5.7 of brown_unstable_2024
  • Corollary 4
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 30 more