Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers

Abstract

We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for $\mathcal{A}$-hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

51 page, 5 figures, matches published version

References in corpus (7)

• Lectures on differential equations for Feynman integrals
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Duals of Feynman Integrals, 2: Generalized Unitarity
• GKZ-hypergeometric systems for Feynman integrals
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• Four Lectures on Euler Integrals
• Self-duality from twisted cohomology
• Real time lattice correlation functions from differential equations
• Analytic continuations and numerical evaluation of the Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$ and Lauricella-Saran $F_S^{(3)}$ and their Application to Feynman Integrals
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• GKZ hypergeometric systems of the three-loop vacuum Feynman integrals
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• GKZ hypergeometric systems of the four-loop vacuum Feynman integrals