Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry

TL;DR

This paper introduces algebraic geometry techniques applied to multi-loop scattering amplitude calculations, focusing on integrand reduction, residue analysis, and IBP reduction, with examples and exercises for clarity.

Contribution

It presents a comprehensive overview of algebraic geometry methods tailored for multi-loop integral reduction in quantum field theory.

Findings

Effective algebraic geometry methods for integrand reduction

Residue computation techniques in unitarity analysis

Application of IBP reduction in multi-loop integrals

Abstract

These notes are for the author's lectures, "Integral Reduction and Applied Algebraic Geometry Techniques" in the School and Workshop on Amplitudes in Beijing 2016. I introduce the applications of algebraic geometry methods on multi-loop scattering amplitudes, for instance, integrand reduction, residue computation in unitarity analysis and Integration-by-parts reduction. Illustrative examples and exercises are included in these notes.