IAS Amplitudes Group Meeting

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Abstract: In this talk, I will give an elementary introduction and explain surprising relations between combinatorics and Feynman integrals, from the paper https://arxiv.org/abs/2304.05299 with Karen Yeats. For every regular graph, we define a sequence of integers, using the recursion of the Martin polynomial, which appears in physics as the symmetry factor of the O(N) vector model. Our sequence counts spanning tree partitions and constitutes the diagonal coefficients of powers of the Kirchhoff and Symanzik polynomials. We prove that this sequence respects all known symmetries of Feynman period integrals in perturbative scalar quantum field theory, and we apply this to prove the completion conjecture for the c2 invariant at all primes. We conjecture that our invariant is perfect: Two Feynman periods are equal, if and only if, their Martin sequences are equal. Extensive calculations give further evidence to conjectured identities of Feynman periods contributing to the 8 loop beta function.