An Algorithm of Computing Cohomology Intersection Number of Hypergeometric Integrals

Abstract

We show that the cohomology intersection number of a twisted Gauss–Manin connection with regularization condition is a rational function. As an application, we obtain a new quadratic relation associated to period integrals of a certain family of K3 surfaces.

From positive geometries to a coaction on hypergeometric functions

“…We remark that a similar degenerate limit was taken in ref. [59] in the context of computing intersection numbers in period integrals associated to K3 surfaces.…”

Co-homology of Differential Forms and Feynman diagrams

“…Recent mathematical literature on intersection numbers of twisted cycles and co-cycles include application to Gel'fand-Kapranov-Zelevinski systems [97,82,98] and to bilinear relations [54,55,56,88,89,70,71,99].…”

“…with on-shell internal lines, that admit a one-fold integral representations [59,61]. The entire decomposition of Feynman integrals in terms of a complete set of MIs, including the ones corresponding to subdiagrams, as well as the derivation of contiguity relations for special functions admitting multi-fold integral representation [61,63,65], required the use of multivariate intersection numbers [75,76,77,78,79,80,81,46,82]. A recursive algorithm for computing intersection numbers for meromorphic n-forms was proposed in [60] and later refined and applied to a few paradigmatic cases of Feynman integral decomposition [63,65,66].…”

Status of Intersection Theory and Feynman Integrals

“…Equivalent definitions can be found even in the earlier literature, most notably in the work of Deligne and Mostow [12] and Saito [13] in the 1980's, though the focus of these works was less practical. Later mathematical literature on computing intersection numbers in various contexts includes [47][48][49][50][51][52].…”