Quantization of gauge fields, graph polynomials and graph cohomology

TL;DR

This paper explores the algebraic and topological structures of 3-regular graphs to derive Feynman integrands for non-abelian gauge theories, introducing a new graph polynomial that simplifies covariant quantization without ghosts.

Contribution

It introduces the corolla polynomial and cycle homology, providing a novel algebraic framework for gauge field quantization using graph polynomials and cohomology.

Findings

Derived n-loop Feynman integrands from scalar integrands using graph polynomials.

Introduced the corolla polynomial to incorporate ghost sector signs.

Proposed a covariant quantization method without ghosts using graph cohomology.

Abstract

We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands for connected 3-regular graphs, obtained from the two Symanzik polynomials. The transition to the full gauge theory amplitude is obtained by the use of a third, new, graph polynomial, the corolla polynomial. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by the corolla polynomial -we call it cycle homology- and by graph homology.