Co-homology of Differential Forms and Feynman diagrams

TL;DR

This review explores the deep connection between Feynman integrals in physics and cohomology theories in mathematics, offering tools to better understand and compute complex integrals in quantum field theory and related areas.

Contribution

It introduces mathematical concepts like twisted cohomology and intersection products to physicists, framing Feynman integrals within a rigorous cohomological context.

Findings

Feynman integrals can be viewed as elements of a vector space with a scalar product.

Interpreting these spaces via twisted cohomology clarifies the role of intersection products.

Provides mathematical tools for advanced analysis of Feynman integrals in physics.

Abstract

In the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in the light of the recent developments. Feynman integrals enter in several perturbative methods for solving non linear PDE, starting from Quantum Field Theories and including General Relativity and Condensed Matter Physics. Precision calculations involve several loop integrals, one strategy to address which is to bring them back in terms of linear combinations of a complete set of integrals (the Master Integrals). In this sense Feynman integrals can be thought as defining a sort of vector space to be decomposed in term of a basis. Such a task may be simpler if the vector space is endowed with a scalar product. Recently, it has been discovered that, interpreting these spaces in terms of twisted cohomology, the role of a scalar product is played by intersection products. The present review is meant to provide the mathematical tools, usually familiar to mathematicians but often not in the standard baggage of physicists, like singular, simplicial and intersection (co)homologies, hodge structures etc., apt to restate this strategy on precise mathematical grounds. It is thought to be both an introduction for beginners interested to the topic, as well as a general reference providing helpful tools for tackling the several still open problems.