Integrable Systems in Quantum Theory

Description

Many physical theories such as quantum field theory and string theory

exhibit important connections with integrable systems.

We mention two instances where they are. The first amazing connection

is from the Witten conjectures in two-dimensional theories, which relate

the generating function of intersection numbers of Morita-Mumford classes,

matrix models and classical integrable systems of Khadomtsev-

Petviashvilii type. For the Korteweg de Vries equation the first step of

this conjecture was proved by Kontsevich and the second by Kharchev,

Marshakov, Mironov, Morozov and Zabrodin. The second striking relation

became visible when it was conjectured and proved that another class of

integrable systems, the so-called Toda hierarchies, lay at the foundation

of the Gromov-Witten invariants of projective space. It forms a key element

in Givental's proof of mirror symmetry for these spaces. Now mirror symmetry

is a duality, where two seemingly different physical theories can be shown to

be isomorphic by taking quantum corrections into account. As such it is a rich 

common research area for both mathematicians and physicists. Integrability

can be of great use, e.g., to verify connections between different theories such

as the AdS/CFT- correspondence that states that a string theory on Anti-de

Sitter space is equivalent to a Conformal Field Theory on its boundary. 

Aims

At the conference we will discuss a number of aspects of the rich interaction

between integrable systems and quantum theory. More in particular, we want

to focus on the following connections:

1)     Progress in the AdS/CFT- correspondence and integrable systems

2)     New developments around the role of integrable systems in Seiberg-Witten theory

3)     Algebraic geometric aspects of the relevant spaces and