Loops in SU(2), Riemann Surfaces, and Factorization, I

TL;DR

This paper generalizes the factorization of SU(2) loops from the circle to Riemann surfaces, linking root subgroup factorizations with semistable holomorphic bundles and spin Toeplitz operators.

Contribution

It extends loop factorization theory to Riemann surfaces using Krichever-Novikov expansions, connecting multiloop factorizations with bundle stability and operator determinants.

Findings

Multiloops with root subgroup factorizations define semistable SL(2,C) bundles.

Factorizations for determinants of spin Toeplitz operators are established.

Generalization from disks to Riemann surfaces broadens the scope of loop factorization theory.

Abstract

In previous work we showed that a loop $g\colon S^1 \to {\rm SU}(2)$ has a triangular factorization if and only if the loop $g$ has a root subgroup factorization. In this paper we present generalizations in which the unit disk and its double, the sphere, are replaced by a based compact Riemann surface with boundary, and its double. One ingredient is the theory of generalized Fourier-Laurent expansions developed by Krichever and Novikov. We show that a ${\rm SU}(2)$ valued multiloop having an analogue of a root subgroup factorization satisfies the condition that the multiloop, viewed as a transition function, defines a semistable holomorphic ${\rm SL}(2,\mathbb C)$ bundle. Additionally, for such a multiloop, there is a corresponding factorization for determinants associated to the spin Toeplitz operators defined by the multiloop.