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Workshop Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory

ICMAT-MZV 2014: Periods in Quantum Field Theory and Arithmetic pp 293–325Cite as

q-Analogues of Multiple Zeta Values and Their Application in Renormalization

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 314))

Abstract

In this paper we report on recent developments on q-analogues of multiple zeta values (MZVs), which are power series in a formal parameter q that reduce to classical MZVs in the limit \(q\rightarrow 1\). First of all, we systematically develop the double shuffle relations of three q-models, whose shuffle products rely on a description of iterated Rota–Baxter operators. In the second part we use two of these q-models to construct solutions to the renormalization problem of MZVs, i.e., a systematic extension of MZVs to negative integers. In one case the renormalized MZVs satisfy the quasi-shuffle relations and in the other case the shuffle relations are verified.

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Acknowledgements

Some results presented in this work are based on a project which was carried out during the Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory (September–December 2014) at ICMAT, Madrid. I have greatly enjoyed the hospitality and the nice and stimulating atmosphere at ICMAT. Furthermore I am very grateful to Andreas Knauf for his comments which significantly improved the paper.

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Authors and Affiliations

  1. Department Mathematik, Friedrich–Alexander–Universität Erlangen–Nürnberg, Cauerstraße 11, 91058, Erlangen, Germany

    Johannes Singer

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Correspondence to Johannes Singer .

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Editors and Affiliations

  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

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Singer, J. (2020). q-Analogues of Multiple Zeta Values and Their Application in Renormalization. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_11

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