Feynman symmetries of the Martin and $c_2$ invariants of regular graphs

TL;DR

The paper introduces the Martin invariant M(G) for 2k-regular graphs, showing it equals the diagonal coefficient of Ψ_{G\setminus v}^k and counts partitions of G\setminus v into k spanning trees. It proves that M(G) encodes all known symmetries of Feynman period integrals (including product, twist, duality, and Fourier-split) and that the extended graph permanent and the c2-invariant at primes are determined by the Martin sequence via duplications G^{[r]}. Two complementary proofs establish c2 for all primes and connect permanents to Martin via mod p relations; a Prüfer-code–based bijection underpins the diagonal interpretations, with extensive computer data supporting the conjecture that Martin sequences classify Feynman periods. The work unifies combinatorics of spanning-tree partitions, graph polynomials, and quantum-field-theory amplitudes, offering a robust toolkit for understanding period symmetries through a single invariant and suggesting avenues toward period equivalence from Martin sequences.

Abstract

For every regular graph, we define a sequence of integers, using the recursion of the Martin polynomial. This sequence counts spanning tree partitions and constitutes the diagonal coefficients of powers of the Kirchhoff polynomial. We prove that this sequence respects all known symmetries of Feynman period integrals in quantum field theory. We show that other quantities with this property, the $c_2$ invariant and the extended graph permanent, are essentially determined by our new sequence. This proves the completion conjecture for the $c_2$ invariant at all primes, and also that it is fixed under twists. We conjecture that our invariant is perfect: Two Feynman periods are equal, if and only if, their Martin sequences are equal.

Figures (30)

• Figure 1: The periods and Martin invariants of primitive $\phi^4$ graphs Schnetz:Census with $9$ vertices (7 loops decompleted). The dashed curve is a fit $\mathcal{P}(G\setminus v)\approx 32150\cdot \operatorname{\mathsf{M}}(G)^{-3.015}$.
• Figure 2: The three circuit partitions of a 2-rose and its Martin polynomial.
• Figure 3: Splitting off a pair $(va,vb)$ of edges.
• Figure 4: Decomposition of two totally decomposable graphs into 3 copies of $C_3^{[2]}$.
• Figure 5: Illustration for the proof of \ref{['lem:cyclic-bigger']}: A non-admissible pair $(va,vb)$ implies the existence of a forbidden cut $C$ in $G$.

Theorems & Definitions (147)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Theorem 1.8
• Corollary 1.9: Completion invariance at primes
• Conjecture 1.10