# 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)