# Number Theory and Physics at the Crossroads (11w5001)

Arriving in Banff, Alberta Sunday, May 8 and departing Friday May 13, 2011

## Organizers

Victor Batyrev (University of Tuebingen)

Sergei Gukov (California Institute of Technology)

Don Zagier (Max-Planck Institute Bonn)

## Objectives

In the last decades and half, the world
has seen explosive interactions between
Number Theory, Arithmetic and Algebraic
Geometry, and Theoretical Physics (in
particular, String Theory). To name a few,
the classical modular forms, quasi-modular
forms, and Jacobi forms appear in many
areas of Physics, e.g., in mirror symmetry,
topological quantum field theory, Gromov-Witten
invariants, conformal field theory,
Calabi-Yau manifolds and in block holes.
Also modularity questions of Calabi-Yau varieties
and other higher dimensional varieties in
connection with Langlands Program are getting
considerable attention and feedbacks from physics.
Zeta-functions and L-series enter scenes at
various places in physics. Via renormalization,
Feynman integrals are related to multiple zeta-values,
and purportedly to motives. Calculations of the
energy and charge degeneracies of black holes
lead surprisingly to Jacobi forms and Siegel
modular forms.

Though mathematicians (number theorists) and
physicists (string theorists) have been working
on modular forms, quasimodular forms,
Jacobi forms and more generally automorphic
forms in their respective fields, there have
been very little interactions between the two sets
of researchers, although with some exceptions.
In other words, both camps have been living in
parallel universes.

There have been strong desires among mathematicians
and physicists for more workshops directed to the areas
of number theory and physics at the crossroads.
A series of workshops have been organized responding
to that demand, and in fact, the proposed
workshop is the fifth in this series. The last
workshop at BIRS in 2008 brought together researchers
in number theory, algebraic geometry, and physics
(string theory) whose common interests are centered
around modular forms. We witnessed very active and
intensive interactions of both camps from early
mornings to late nights. We all felt that all things
modular have came together at BIRS from both sides:
number theory and physics (in particular, string
theory). At the end of the workshop, all participants
felt that both camps have finally crossed boundaries
and established relatively comfortable rapport.
This has led to very strong desire to have
the next BIRS five-day follow-up workshop in 2011.

One of the principal goals of this
workshop is to compare new developments
since the last one, and give directions
to future researches in the interface
of number theory and physics. We ontinue
to look at various modular forms, zeta-functions,
$L$-series, Galois representations, arising from
Calabi--Yau manifolds, conformal field theory,
quantum field theory, $4D$ gauge theory, and
Feynman diagrams and integrals.
The subject area of interest might be classified
into not clearly disjoint sets of the following
subjects:
(a) Modular, quasimodular, Siegel, and Jacobi
modular forms, and their applications to physics.
They have begun to play an important role in
the Type II/Heterotic string duality.

(b) Topological string theory, and modular forms.

(c) Modularity of Galois representations, and
arithmetic questions. Generalizations of the
modularity of the Shimura--Taniyama conjecture
for Calabi--Yau varieties of higher dimension
are of particular interest.

(d) 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.

(e) Conformal field theory, and modular forms.
Relationship with monstrous moonshine.

(f) Holomorphic anomaly equations.

(g) Differential equations, in particular,
Picard--Fuchs differential equations associated
to Calabi--Yau families, and periods of differential
forms of Calabi--Yau manifolds.

(h) Wall-crossing formula, Black holes, and
Jacobi forms.

(i) Feymnan diagrmas and integrals.

(j) Toric geometry, and combinatorial methods
for describing the geometry and topology of
Calabi--Yau manifolds.

(k) Other topics at the crossroads of number
theory and physics.