# 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)

Charles Doran (University of Alberta, Canada)

Sergei Gukov (California Institute of Technology)

Noriko Yui (Queen's University, Canada)

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.

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.