# Motives, Quantum Field Theory, and Pseudodifferential Operators

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*Alan Carey; David Ellwood; Sylvie Paycha; Steven Rosenberg*

A co-publication of the AMS and Clay Mathematics Institute

This volume contains articles related to the conference
“Motives, Quantum Field Theory, and Pseudodifferntial
Operators” held at Boston University in June 2008, with partial
support from the Clay Mathematics Institute, Boston University, and
the National Science Foundation. There are deep but only partially
understood connections between the three conference fields, so this
book is intended both to explain the known connections and to offer
directions for further research.

In keeping with the organization of the conference, this book
contains introductory lectures on each of the conference themes and
research articles on current topics in these fields. The introductory
lectures are suitable for graduate students and new Ph.D.'s in both
mathematics and theoretical physics, as well as for senior
researchers, since few mathematicians are expert in any two of the
conference areas.

Among the topics discussed in the introductory lectures are the
appearance of multiple zeta values both as periods of motives and in
Feynman integral calculations in perturbative QFT, the use of Hopf
algebra techniques for renormalization in QFT, and regularized traces
of pseudodifferential operators. The motivic interpretation of
multiple zeta values points to a fundamental link between motives and
QFT, and there are strong parallels between regularized traces and
Feynman integral techniques.

The research articles cover a range of topics in areas related to
the conference themes, including geometric, Hopf algebraic, analytic,
motivic and computational aspects of quantum field theory and mirror
symmetry. There is no unifying theory of the conference areas at
present, so the research articles present the current state of the art
pointing towards such a unification.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in algebraic geometry, quantum field theory, and pseudodifferential operators, and the connections between these areas in mathematics.