Locality Galois groups of meromorphic germs in several variables

TL;DR

This paper develops a framework for analyzing meromorphic germs in several variables using locality structures, defining locality Galois groups, and applying these concepts to number theory and quantum field theory, including renormalization.

Contribution

It introduces the concept of locality Galois groups for meromorphic germs and explicitly describes their structure for classes arising in number theory and physics.

Findings

Locality Galois groups are explicitly characterized for specific classes of meromorphic germs.

Locality polynomial bases are given by locality Lyndon words.

The framework provides a mathematical interpretation of Speer's analytic renormalization.

Abstract

Meromorphic germs in several variables with linear poles naturally arise in mathematics in various disguises. We investigate their rich structures under the prism of locality, including locality subalgebras, locality transformation groups and locality characters. The key technical tool is the dependence subspace for a meromorphic germ with which we define a locality orthogonal relation between two meromorphic germs. We describe the structure of locality subalgebras generated by classes of meromorphic germs with certain types of poles. We also define and determine their group of locality transformations which fix the holomorphic germs and preserve multivariable residues, a group we call the locality Galois group. We then specialise to two classes of meromorphic germs with prescribed types of nested poles, arising from multiple zeta functions in number theory and Feynman integrals in perturbative quantum field theory respectively. We show that they are locality polynomial subalgebras with locality polynomial bases given by the locality counterpart of Lyndon words. This enables us to explicitly describe their locality Galois group. As an application, we propose a mathematical interpretation of Speer's analytic renormalisation for Feynman amplitudes. We study a class of locality characters, called generalised evaluators after Speer. We show that the locality Galois group acts transitively on generalised evaluators by composition, thus providing a candidate for a renormalisation group in this multivariable approach.