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Locality Galois Groups of Meromorphic Germs in Several Variables

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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 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.

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Acknowledgements

The second author is grateful to the Perimeter Institute in Waterloo where she was hosted on an Emmy Noether fellowship. This research is supported by the National Natural Science Foundation of China (11890663 and 11821001).

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, USA

    Li Guo

  2. Institute of Mathematics, University of Potsdam, 14469, Potsdam, Germany

    Sylvie Paycha

  3. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Bin Zhang

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  1. Li Guo
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  2. Sylvie Paycha
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  3. Bin Zhang
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Correspondence to Li Guo.

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Communicated by C. Schweigert.

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Guo, L., Paycha, S. & Zhang, B. Locality Galois Groups of Meromorphic Germs in Several Variables. Commun. Math. Phys. 405, 28 (2024). https://doi.org/10.1007/s00220-023-04915-2

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