Four Lectures on Euler Integrals

Abstract

These lecture notes provide a self-contained introduction to Euler integrals, which are frequently encountered in applications. In particle physics, they arise as Feynman integrals or string amplitudes. Our four selected topics demonstrate the diverse mathematical techniques involved in the study of Euler integrals, including polyhedral geometry, very affine varieties, differential equations, and computational algebra.

arXiv comment48 pages, journal version

References in corpus (24)

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• Scattering Equations and KLT Orthogonality
• Lectures on differential equations for Feynman integrals
• Feynman Integrals and Intersection Theory
• Positive Geometries and Canonical Forms
• Scattering Amplitudes from Intersection Theory
• Collider Physics at the Precision Frontier
• Critical points and number of master integrals
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Aspects of Scattering Amplitudes and Moduli Space Localization
• The maximum likelihood degree of a very affine variety
• Cosmological Polytopes and the Wavefunction of the Universe
• Feynman integral relations from parametric annihilators
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• Tropical Monte Carlo quadrature for Feynman integrals
• Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers
• Tree-level amplitudes from the pure spinor superstring
• Tropical Feynman integration in the Minkowski regime
• Hepp's bound for Feynman graphs and matroids
• Coaction and double-copy properties of configuration-space integrals at genus zero
• Quadratic relations between periods of connections
• Relative twisted homology and cohomology groups associated with Lauricella's $F_D$
• Bayesian Integrals on Toric Varieties
• Twisted cohomology and likelihood ideals

Cited by in corpus (5)

• Landau Singularities Revisited: Computational Algebraic Geometry for Feynman Integrals
• Principal Landau Determinants
• Cutting-Edge Tools for Cutting Edges
• 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
• Lorentzian contours for tree-level string amplitudes