# Aspects of Finite Gap Integration

## Abstracts

Leonid Chekov: Multi-cut solutions to matrix models

Solutions of matrix models in asymptotic (1/N) expansion are based on
complex curves related to large-N eigenvalue distribution. These curves
are hyperelliptic (for one-matrix model) or general (for two-matrix
model); their genera related to the potential and to the number of
eigenvalue distribution intervals. We propose the technique for
constructing 1/N-expansion of the matrix model free energy in terms of
quantities (canonically normalized bidifferentials and the 1-form dS)
defined on the spectral curve; this technique pertains to a special
Feynman diagrammatic technique.

Nick Dorey & Benoit Vicedo:
Finite gap solutions of classical string theory and the AdS/CFT correspondence

In this two part talk we will review the construction of finite gap solutions
in classical string theory on certain curved backgrounds.
We will discuss the role of these solutions in testing the AdS/CFT correspondence which
relates string theory on AdS_5 x S^5 to N=4 supersymmetric Yang-Mills theory.

Victor Enolskii: Aspects of Finite Gap Integration of the su(2) Bogomolny Equation

The theory of su(2) monopoles pioneered by Nahm and Hitchin is well developed. We discuss
an addendum to this theory that seeks explicit solutions in terms of finite-gap integration.
We recall and extend the work of Ercolani-Sinha and then consider in detail the case of
symmetric three monopoles. We also discuss the reconstruction of solutions to the
Bogomolny equation from Nahm data.

John Gibbons:
Sigma-functions of cyclic trigonal curves and Benney reductions

We discuss the theory of generalized Weierstrass sigma and wp
functions defined on a trigonal curve of genus four, following
earlier work on the genus three case. The specific example of
the ``purely trigonal'' (or ``cyclic trigonal'') curve
y^3=x^5+\lambda_4 x^4 +\lambda_3 x^3+\lambda_2 x^2 +\lambda_1 x+\lambda_0
is discussed in detail, including an expansion of sigma, a list of some of the
associated partial differential equations satisfied by the wp
functions, and the derivation of 2 kinds of addition formula.
This work also gives an explicit form for Schwartz-Christoffel
maps associated with a class of reductions of the Benney hierarchy.