Differential algebras with Banach-algebra coefficients II: The operator cross-ratio tau-function and the Schwarzian derivative

TL;DR

This paper extends the theory of operator-valued tau-functions and Schwarzian derivatives within an infinite-dimensional Grassmannian framework, linking them to integrable PDEs, C*-algebras, and complex geometry.

Contribution

It introduces operator-valued tau-functions and cross-ratios in a Banach algebra setting, connecting them to KP hierarchy and projective structures on Riemann surfaces.

Findings

Defined operator-valued tau-functions in Banach *-algebra context

Extended the operator cross-ratio and Schwarzian derivative to this setting

Linked these structures to Fay's identity and KP hierarchy

Abstract

Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable Hilbert spaces are generalized to a class of polarized Hilbert modules, in particular the Baker and tau-functions, which become operator-valued. Following from Part I we produce a pre-determinant structure for a class of tau-functions defined in the setting of the similarity class of projections of a certain Banach *-algebra. This structure is explicitly derived from the transition map of a corresponding principal bundle. The determinant of this map gives a generalized, operator-valued tau-function that takes values in a commutative C*-algebra. We extend to this setting the operator cross-ratio which had been used to produce the scalar-valued tau-function, as well as the associated notion of a Schwarzian derivative along curves inside the space of similarity classes. We link directly this cross-ratio with Fay's trisecant identity for the tau-function (equivalent to the KP hierarchy). By restriction to the image of the Krichever map, we use the Schwarzian to introduce the notion of operator-valued projective structure on a compact Riemann surface: this allows a deformation inside the Grassmannian (as it varies its complex structure). Lastly, we use our identification of the Jacobian of the Riemann surface in terms of extensions of the Burchnall-Chaundy C*-algebra (Part I) to describe the KP hierarchy.