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.

arXiv comment51 page, 5 figures, matches published version

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