Intersection Numbers from Higher-order Partial Differential Equations

Abstract

We propose a new method for the evaluation of intersection numbers for twisted meromorphic $n$-forms, through Stokes' theorem in $n$ dimensions. It is based on the solution of an $n$-th order partial differential equation and on the evaluation of multivariate residues. We also present an algebraic expression for the contribution from each multivariate residue. We illustrate our approach with a number of simple examples from mathematics and physics.

37 pages, matches published version, added ancillary files

References in corpus (8)

• Resolution of singularities for multi-loop integrals
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Duals of Feynman Integrals, 2: Generalized Unitarity
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• Correlation functions on the lattice and twisted cocycles
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory

Cited by in corpus (11)

• Cosmology meets cohomology
• Reduction to master integrals via intersection numbers and polynomial expansions
• Canonical Differential Equations Beyond Genus One
• On one-loop corrections to the Bunch-Davies wavefunction of the universe
• Three-loop banana integrals with four unequal masses
• Intersection Numbers from Companion Tensor Algebra
• Twisted Riemann bilinear relations and Feynman integrals
• The recursive structure of Baikov representations II: the top-down reduction with intersection theory
• General Relativity from Intersection Theory
• Differential equations for tree--level cosmological correlators with massive states
• Canonical differential equations and intersection matrices