# Perturbative Quantum Field Theory Meets Number Theory

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## Abstract

Feynman amplitudes are being expressed in terms of a well structured family of special functions and a denumerable set of numbers—*periods*, studied by algebraic geometers and number theorists. The periods appear as residues of the poles of regularized primitively divergent multidimensional integrals. In low orders of perturbation theory (up to six loops in the massless \(\varphi ^4\) theory) the family of hyperlogarithms and multiple zeta values (MZVs) serves the job. The (formal) hyperlogarithms form a double shuffle differential graded Hopf algebra. Its subalgebra of single valued multiple polylogarithms describes a large class of euclidean Feynman amplitudes. As the grading of the double shuffle algebra of MZVs is only conjectural, mathematicians are introducing an abstract graded Hopf algebra of *motivic zeta values* whose weight spaces have dimensions majorizing (hopefully equal to) the dimensions of the corresponding spaces of real MZVs. The present expository notes provide an updated version of 2014’s lectures on this subject presented by the author to a mixed audience of mathematicians and theoretical physicists in Sofia and in Madrid.

## Keywords

Residue Transcendental Polylogarithm Shuffle Stuffle product Formal multizeta values Single-valued hyperlogarithm## Notes

### Acknowledgements

It is a pleasure to thank Francis Brown and Herbert Gangl for enlightening discussions at different stages of this work and Kurusch Ebrahimi-Fard for his invitation to the 2014 ICMAT Research Trimester. I thank IHES for its hospitality during the completion of these notes (January, 2016). The author’s work has been supported in part by Grant DFNI T02/6 of the Bulgarian National Science Foundation.

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