TY - JOUR

T1 - Locality Galois Groups of Meromorphic Germs in Several Variables

AU - Guo, Li

AU - Paycha, Sylvie

AU - Zhang, Bin

N1 - Publisher Copyright:
© 2024, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2024/2

Y1 - 2024/2

N2 - 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 groups. 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.

AB - 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 groups. 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.

UR - http://www.scopus.com/inward/record.url?scp=85183701225&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85183701225&partnerID=8YFLogxK

U2 - 10.1007/s00220-023-04915-2

DO - 10.1007/s00220-023-04915-2

M3 - Article

AN - SCOPUS:85183701225

SN - 0010-3616

VL - 405

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 2

M1 - 28

ER -