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

Estelle Basor a and Doug Pickrell b
a) American Institute of Mathematics, 600 E. Brokaw Road, San Jose, CA 95112, USA
b) Mathematics Department, University of Arizona, Tucson, AZ 85721, USA

Received October 24, 2015, in final form March 02, 2016; Published online March 08, 2016

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.

Key words: loop group; factorization; Toeplitz operator; determinant.

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