Number Theory and Physics at the Crossroads (08w5077)

Arriving in Banff, Alberta Sunday, September 21 and departing Friday September 26, 2008


(University of Alberta, Canada)

Sergei Gukov (California Institute of Technology)

Helena Verrill (Louisiana State University)

(Queen's University, Canada)

Don Zagier (Max-Planck Institute Bonn)


Physical duality symmetries relate special limits of the various
consistent string theories (Types I, II, Heterotic string and
their cousins, including F-theory) one to another. By comparing
the mathematical descriptions of these theories, one reveals
often quite deep and unexpected mathematical conjectures.
The best known string duality to mathematicians, Type IIA/IIB
duality also called {it mirror symmetry}, has inspired many
new developments in algebraic and arithmetic geometry, number
theory, toric geometry, Riemann surface theory, and infinite
dimensional Lie algebras. Other string dualities such as
Heterotic/Type II duality and F-Theory/Heterotic string
duality have also, more recently, led to series of
mathematical conjectures, many involving elliptic curves,
K3 surfaces, and modular forms.

In recent years, we have witnessed that modular forms,
quasi-modular forms and automorphic forms play a central role
in mirror symmetry, in particular, as generating functions
counting the number of curves on Calabi--Yau manifolds and
describing Gromov--Witten invariants. This has led to a
realization that time is ripe to assess the role of number
theory, in particular that of modular forms, in mirror symmetry
and string dualities in general. Indeed, there have been at
least two efforts along this line.

One is the the first five-day workshop on ``Modular Forms and
String Duality'' which was held at BIRS in June 2006. It brought
together mathematicians and physicists working on problems
inspired by string theory. Many researchers in string theory
and number theory, working on the same or related problems from
different angles came together at BIRS. This synthesis proved
powerful and beneficial to both parties involved; simply put,
the workshop was a huge success. There was an overwhelming
concensus from researchers working at the crossroads of number
theory and physics to organize this kind of workshop more
frequently. In particular, many researchers are extremely
eager to have a second five-day workshop at BIRS in 2008 in
the subject area of number theory and physics.

This brings to the second effort: A new research journal
``Communications in Number Theory and Physics'' (published
by International Press of Boston) has been launched, specifically
devoted to subject areas at the crossroads of number theory
and physics. The editors-in-chief of this new journal are
Robert Dijkgraaf, David Kazdhan, Maxim Kontsevich, and Shing-Tung
Yau. The editorial board consists of prominent mathematicians
and theoretical physicists, many who will be taking part in
the workshop.

One of the principal goals of this workshop is to look at
automorphic forms, zeta-functions, $L$-series, Galois
representations, arising from Calabi--Yau manifolds or
more general varieties with the aim of interpreting duality
symmetries in string theory in terms of arithmetic invariants
associated to the varieties in question. The subject area
of interest might be roughly divided into the following

-- Modular, quasimodular, bimodular forms and their applications.
These have begun to play an important role in the
Type II/Heterotic string duality.

-- Topological string theory, and modular forms.

-- Modularity of Galois representations, and arithmetic questions.
Generalizations of the Shimura-Taniyama conjecture for Calabi-Yau
varieties are of particular interest.

-- Mirror symmetry. There are various versions of ``mathematical
mirror symmetry'' (e.g., Kontsevich's Homological Mirror Symmetry
Conjecture and the proposal of Strominger-Yau-Zaslow). Of
particular interest for this workshop are the arithmetic
aspects of each of these.

-- Toric geometry, and combinatorial methods for describing
the geometry and topology of Calabi-Yau manifolds
(e.g., work of Batyrev-Borisov and Doran-Morgan).

-- Differential equations (e.g., Picard-Fuchs, quantum DE, etc.);
periods of differential forms on Calabi-Yau manifolds.

-- Conformal field theory, and modular forms. Relationship
with modular moonshine.

-- Arithmetic, geometric, and Hodge theoretic aspects of
the F-theory/Heterotic string duality, especially the
role played by complex multiplication.

-- Other topics at the crossroads of number theory and physics.