ESI Program
on
Number Theory and Physics

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Overview

Recent research in mathematical physics has revealed profound connections between many subjects previously thought unrelated on the surface. The most noteworthy connection is perhaps to number theory. In recent years, we have witnessed, for instance, the appearances of automorphic forms (modular forms, quasimodular forms, bimodular forms, etc), motives, zeta-functions and L-series, Galois representations, and the geometric Langlands program in string theory and perturbative quantum field theory.

At this time the connections between all of these areas are still in a embryonic phase. This has led to a realization that time is ripe to assess the implications for both number theory and physics of these apparent connections. There is an overwhelming consensus from researchers working at the crossroads of number theory and physics to have this kind of program. One of the principal aims of such a workshop is to bring together researchers from both sides, and to make a serious attempt to overcome some conceptual barriers between experts in the two camps. This synthesis should prove to be extremely powerful and beneficial in enhancing understanding the subjects of common interests for both parties involved.

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Organisation

There will a two week instructional program for students and postdoctoral fellows. The support of good expositors who have some experience in the field will result in an accessible introduction. This will be followed by at least one advanced workshop with the explicit role of looking to the future of the physics/number theory interactions. We anticipate an expanded interest in the area and the realization that experts and students in each field can indeed work in the other. Similar earlier workshops were at IHES in Spring 2005 and another will be held at Boston University in June 2008 (http://math.bu.edu/qftconference). We believe that by 2009 there will be an emergence of new themes and additional activity. Thus a longer meeting containing a general education component will be of great service for the European mathematical community.

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Discussion

Perturbative QFT:

Here we take QFT to mean perturbative quantum field theory, whose main subject is the computation of Feynman integrals. These are finite dimensional but typically divergent integrals whose suitably renormalized answers can be matched with physical experiment, and do so with remarkable precision for e.g. quantum electrodynamics. The implications for mathematics of this remarkable agreement with experiment has been a subject of speculation for a long time and led to the creation of constructive QFT.

In the late 1990s, Kreimer and Connes-Kreimer uncovered the fundamental Hopf algebra structure in Feynman rules, and in fact realized the universal nature of the Hopf algebra of Feynman diagrams. As a result, many previously intractable Feynmann integrals could be more efficiently organized and computed by implementing the Hopf algebra structure. More importantly from the theoretical viewpoint, algebraic machinery became available in QFT, so that e.g. classical identities in QFT (Slavnov-Taylor identities) were interpreted as a Hochschild one-cocyle on this Hopf algebra. It is fair to say that this approach was a driving force behind the rejuvenation of QFT during this period, particularly in Europe. At present, algebraists with expertise in Hopf algebras are playing a role in the development of QFT, and ideas from QFT, suitably abstracted, are filtering into the Hopf algebra literature.

Around the same time, it was realized that the actual values computed by Feynman integrals had unexpected number theoretic significance. In particular, these integrals often (but not always!) produced rational multiples of multiple zeta values (MZVs) for no apparent a priori reason. A conjectural link between Feynman integrals and MZVs is given by the shuffle-type relations common to both the Kreimer Hopf algebra and MZVs. While this has been worked out to some extent, the shuffle relations seem to be more of a common occurence than an underlying explanation. It is likely that a deeper explanation for the appearance of MZVs in QFT is the theory of motives, where a major success is the explanation of relations in the algebra of MZVs via calculating dimensions of certain motivic cohomology groups. As a first step towards uncovering a QFT-motivic connection, Bloch-Kreimer-Esnault developed a motivic interpretation of some simple Feynmann diagrams in terms of a blow-up procedure familiar as a standard example of motivic methods.

These connections will certainly be extended by the time of the proposed ESI program, yielding a new relationship between two major fields of active research. A major obstacle towards progress is the historical divide between these two fields. Both QFT and Grothendieck style algebraic geometry require a vast background, but in very different areas, with QFT usually based on analytic and/or operator algebra techniques, and motives based on the GAGA approach.

ΨDO, multizeta functions and QFT

The relation between QFT and pseudodifferential operator (ΨDO) theory is much more direct: Feynmann integrals are iterated integrals of Green's functions, which are kernels of classical ΨDOs. Certain locality properties essential to QFT should have explanations in terms of known asymptotic expansions for ΨDOs. In addition, in work of Paycha and others, the symbol calculus for zeta functions of elliptic operators is being extended to produce multizeta functions for elliptic operators. In this program, renormalization techniques common in QFT are used, and the shuffle relations mentioned above are still valid. So once again, the Hopf algebraic structure seems to interact with analytic calculations. Although a direct motivic-ΨDO link is premature speculation at this point, it may not be by 2009.

Noncommutative geometry

Noncommutative geometry (NCG) enters the picture in several ways. First multizeta functions also arise in QFT on noncommutative spaces. While this theory still lacks a convincing physical application much more will be known by 2009 about its relevance to physics. Second, NCG is linked to all of the topics QFT, motives and ΨDOs, although the precise nature of the linkage is not entirely clear. Connes and Marcolli have produced papers relating noncommutative geometry both to motives and to QFT, and ΨDO techniques lie at the heart of the Connes-Moscovici local index theorem. This is another piece in the still emerging puzzle that deserves further investigation.

Gauge theory and geometric Langlands

A further development which may be ripe for exposition in 2009 are number theoretic results arising from recent work connecting the geometric Langlands program with gauge theory. A thorough discussion of this relationship would require a whole workshop on its own which is not intended. At this time we are interested in number theoretic implications of the fundamental observation of Kapustin-Witten that the Wilson operators in gauge theory correspond to twisted D -modules of the sigma model corresponding to M and the t'Hooft operators of gauge theory correspond to Hecke operators.

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Proposed Participants

Yves Andre	Ecole Normale Superieure, Paris
Abishek Banerjee	Johns Hopkins
David Ben-Zvi	University of Texas at Austin
Christoph Bergbauer	Freie Universitat Berlin
David Broadhurst	Open University, Milton Keynes
Spencer Bloch	University Of Chicago
Louis Boutet de Monvel	Paris-Jussieu
Francis Brown	Ecole Normale Superieure, Paris
Ulrich Bunke	Regensburg
Alan Carey	Australian National University
Alain Connes	IHES
Katia Consani	Johns Hopkins
Helene Esnault	Duisburg-Essen
Edward Frenkel	Berkeley
Alexander Goncharov	Brown
Harald Grosse	Vienna
Dirk Kreimer	IHES
Marcelo Laca	University of Victoria
Dominic Manchon	Clermont-Ferrand
Matilde Marcolli	MPIM-Bonn
Louise Nyssen	Universite Montpellier
Tony Pantev	University of Pennsylvania
Sylvie Paycha	Clermont-Ferrand
Frederic Paugam	Paris Jussieu
Jorge Plazas	IHES
Abhijnan Rej	MPIM-Bonn
Vincent Rivasseau	Orsay
Steve Rosenberg	Boston University
A Schneider	Regensburg
Raimar Wulkenhaar	Muenster
Karen Yeats	Boston University
Noriko Yui	Queen's University
Don Zagier	MPIM-Bonn

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