A Riemann–Hilbert approach to the Akhiezer polynomials

In this paper, we study those polynomials, orthogonal with respect to a particular weight, over the union of disjoint intervals, first introduced by N. I. Akhiezer, via a reformulation as a matrix factorization or Riemann–Hilbert problem. This approach complements the method proposed in a previous paper, which involves the construction of a certain meromorphic function on a hyperelliptic Riemann surface. The method described here is based on the general Riemann–Hilbert scheme of the theory of integrable systems and will enable us to derive, in a very straightforward way, the relevant system of Fuchsian differential equations for the polynomials and the associated system of the Schlesinger deformation equations for certain quantities involving the corresponding recurrence coefficients. Both of these equations were obtained earlier by A. Magnus. In our approach, however, we are able to go beyond Magnus' results by actually solving the equations in terms of the Riemanni Θ -functions. We also show that the related Hankel determinant can be interpreted as the relevant τ -function.

PREPOTENTIAL OF N=2 SUPERSYMMETRIC YANG–MILLS THEORIES IN THE WEAK COUPLING REGION

We show how to obtain the explicit form of the low energy quantum effect action for N=2 supersymmetric Yang–Mills theory in the weak coupling region from the underlying hyperelliptic Riemann surface. This is achieved by evaluating the integral representation of the fields explicitly. We calculate the leading instanton corrections for the group SU (Nc), SO (N) and SP (2N) and find that the one-instanton contribution of the prepotentials for these groups coincide with the one obtained recently by using the direct instanton calculation.

NONPERTURBATIVE EFFECTIVE ACTIONS OF (N=2)-SUPERSYMMETRIC GAUGE THEORIES

We elaborate on our previous work on (N=2)-supersymmetric Yang-Mills theory. In particular, we show how to explicitly determine the low energy quantum effective action for G=SU(3) from the underlying hyperelliptic Riemann surface, and calculate the leading instanton corrections. This is done by solving Picard-Fuchs equations and asymptotically evaluating period integrals. We find that the dynamics of the SU(3) theory is governed by an Appell system of type F4, and compute the exact quantum gauge coupling explicitly in terms of Appell functions.

SL(n + 1)-invariant equations which reduce to equations of Korteweg-de Vries type

It is shown how to derive SL(n + 1)-invariant equations which reduce to scalar Lax equations for an operator of order n + 1. The existence of these systems explains the Miura transformation between modified Lax and scalar Lax equations. In particular we study an SL(2)-invariant system with a certain space of solutions lying over the solution space of a Korteweg-de Vries equation described by G. B. Segal and G. Wilson. This enables us to write down some solutions of this SL(2)-invariant system in terms of θ-functions of a hyperelliptic Riemann surface.

On the Hyperelliptic Riemann Surfaces of Infinite Genus with Absolutely Convergent Riemann’s Theta Functions

The Riemann’s theta functions associated with a closed Riemann surface are absolutely convergent. In the present paper, we shall show an example of an hyperelliptic Riemann surface of infinite genus such that the Riemann’s theta functions associated with are absolutely convergent.