Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions

Marco Bertola

Abstract:

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices. These orbits are the phase space of generalized integrable lattices of Toda type.

For two of these minimal orbits the polynomials reduce to the ordinary classes of (generalized) orthogonal polynomials and Laurent biorthogonal polynomials, which appear in the solution of the Hermitean and Unitary matrix models. The polynomials associated to the other orbits naturally interpolate between the above two cases and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices.

The 2x2 system of Differential-Difference-Deformation equations satisfied by the polynomials and second type solutions, as well as the associated Riemann-Hilbert problem, are analyzed in the most general setting of pseudomeasures with arbitrary rational logarithmic derivative supported on curve segments in the complex plane. The corresponding isomonodromic tau function is explicitly related to the shifted Toeplitz (or Hankel) determinants of the moments of the pseudo-measure. The results imply that any (shifted) Toeplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.

(Based on joint work with M. Gekhtman, extending earlier joint work with J. Harnad and B. Eynard)