Fredholm Pfaffian ${\tau}$-functions for orthogonal isospectral and isomonodromic systems

TL;DR

This paper extends the framework of ${ au}$-functions as Widom constants to orthogonal hierarchies and isomonodromic systems, revealing Pfaffian structures and providing explicit examples of polynomial ${ au}$-functions.

Contribution

It introduces a novel extension of ${ au}$-function theory to orthogonal systems, showing their Pfaffian nature and connection to determinantal ${ au}$-functions.

Findings

${ au}$-functions expressed as sums of correlators with finite determinants

Orthogonal case reduces correlators to Pfaffians and ${ au}$-functions to squares of Pfaffian-type functions

Explicit polynomial ${ au}$-functions for orthogonal Drinfeld-Sokolov and isomonodromic systems

Abstract

We extend the approach to ${\tau}$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the ${\tau}$-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and the dependence on the flow (or deformation) parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the determinantal ${\tau}$-functions, both in the Drinfeld-Sokolov and isomonodromic case, become squares of ${\tau}$-functions of Pfaffian type. The results are illustrated by several examples, consisting of polynomial ${\tau}$-functions of orthogonal Drinfeld-Sokolov type and of isomonodromic ones with four regular singular points.