© 2020

Periods in Quantum Field Theory and Arithmetic

ICMAT, Madrid, Spain, September 15 – December 19, 2014

  • José Ignacio Burgos Gil
  • Kurusch Ebrahimi-Fard
  • Herbert Gangl
Conference proceedings ICMAT-MZV 2014

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

Table of contents

  1. Front Matter
    Pages i-x
  2. Erik Panzer
    Pages 29-44
  3. S. Stieberger
    Pages 45-76
  4. Oliver Schlotterer
    Pages 77-103
  5. Nils Matthes
    Pages 105-132
  6. Luise Adams, Christian Bogner, Stefan Weinzierl
    Pages 133-143
  7. Henrik Bachmann, Ulf Kühn
    Pages 237-258
  8. Nikolay M. Nikolov
    Pages 327-343
  9. Claudia Malvenuto, Frédéric Patras
    Pages 377-398
  10. Adriana Salerno, Leila Schneps
    Pages 399-430
  11. Kurusch Ebrahimi-Fard, W. Steven Gray, Dominique Manchon
    Pages 445-468
  12. Dominique Manchon
    Pages 469-481
  13. Loïc Foissy, Frédéric Patras
    Pages 483-540

About these proceedings


This book is the outcome of research initiatives formed during the special "Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory'' at the ICMAT (Instituto de Ciencias Matemáticas, Madrid) in 2014. The activity was aimed at understanding and deepening recent developments where Feynman and string amplitudes on the one hand, and periods and multiple zeta values on the other, have been at the heart of lively and fruitful interactions between theoretical physics and number theory over the past few decades. 

In this book, the reader will find research papers as well as survey articles, including open problems, on the interface between number theory, quantum field theory and string theory, written by leading experts in the respective fields. Topics include, among others, elliptic periods viewed from both a mathematical and a physical standpoint; further relations between periods and high energy physics, including cluster algebras and renormalisation theory; multiple Eisenstein series and q-analogues of multiple zeta values (also in connection with renormalisation); double shuffle and duality relations; alternative presentations of multiple zeta values using Ecalle's theory of moulds and arborification; a distribution formula for generalised complex and l-adic polylogarithms; Galois action on knots. Given its scope, the book offers a valuable resource for researchers and graduate students interested in topics related to both quantum field theory, in particular, scattering amplitudes, and number theory.


11M32, 17B81, 20E08, 11G09 periods multiple zeta values Feynman amplitudes polylogarithms elliptic dilogarithm q-multiple zeta values Conference Proceedings renormalization string amplitudes motivic Galois group rooted trees Lie algebras shuffle algebras Ecalle's mould calculus

Editors and affiliations

  • José Ignacio Burgos Gil
    • 1
  • Kurusch Ebrahimi-Fard
    • 2
  • Herbert Gangl
    • 3
  1. 1.Institute of Mathematical Sciences (ICMAT)Spanish National Research Council (CSIC)MadridSpain
  2. 2.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway
  3. 3.Department of Mathematical SciencesDurham UniversityDurhamUK

About the editors

José Ignacio Burgos Gil is a Spanish mathematician working in Arithmetic Geometry, and focusing on Arakelov theory and the theory of periods. He completed his Ph.D. at the University of Barcelona and is currently a researcher at the ICMAT (Instituto de Ciencias Matematicas), Madrid. With more than 50 scientific papers to his credit, he regularly lectures at institutes and conferences around the world and is a member of the Academia Europaea.

Kurusch Ebrahimi-Fard is a German mathematician working as an associate professor of mathematics at the Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU) in Trondheim, Norway. After receiving his Ph.D. in theoretical physics from Bonn University in Germany, he has held postdoctoral positions at e.g. the IHES (Bures-sur-Yvette, France), Max Planck Institute for Mathematics (Bonn, Germany), and the Instituto de Ciencias Matematicas (Madrid, Spain). 

Herbert Gangl is a German mathematician working on polylogarithms and explicit algebraic K-theory, and an associate professor of pure mathematics at Durham University, UK. After completing his Ph.D. at Bonn University, he spent most of his pre-tenure career at the Max Planck Institute for Mathematics, also in Bonn, complemented by a long-term stay at the Institute for Experimental Mathematics in Essen and a substituting professorship at the University of Siegen, both Germany, and as an assistant at Kyushu University in Fukuoka, Japan. He is also a passionate designer of 3D printed geometric jewellery. 


Bibliographic information

  • Book Title Periods in Quantum Field Theory and Arithmetic
  • Book Subtitle ICMAT, Madrid, Spain, September 15 – December 19, 2014
  • Editors José Ignacio Burgos Gil
    Kurusch Ebrahimi-Fard
    Herbert Gangl
  • Series Title Springer Proceedings in Mathematics & Statistics
  • Series Abbreviated Title Springer Proceedings in Mathematics & Statistics
  • DOI
  • Copyright Information Springer Nature Switzerland AG 2020
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-030-37030-5
  • Softcover ISBN 978-3-030-37033-6
  • eBook ISBN 978-3-030-37031-2
  • Series ISSN 2194-1009
  • Series E-ISSN 2194-1017
  • Edition Number 1
  • Number of Pages X, 630
  • Number of Illustrations 400 b/w illustrations, 32 illustrations in colour
  • Topics Algebraic Geometry
    Mathematical Applications in the Physical Sciences
    Number Theory
  • Buy this book on publisher's site
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