# Number Theory and Physics at the Crossroads (08w5077)

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

## Organizers

Charles Doran (University of Alberta, Canada)

Sergei Gukov (California Institute of Technology)

Helena Verrill (Louisiana State University)

Noriko Yui (Queen's University, Canada)

Don Zagier (Max-Planck Institute Bonn)

## Objectives

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

sub-categories:

-- 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.

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

sub-categories:

-- 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.