Loops in SU(2), Riemann Surfaces, and Factorization, I
Abstract
In previous work we showed that a loop g\colon S^1 to 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 FourierLaurent expansions developed by Krichever and Novikov. We show that a 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 SL(2,C) bundle. Additionally, for such a multiloop, there is a corresponding factorization for determinants associated to the spin Toeplitz operators defined by the multiloop.
 Publication:

SIGMA
 Pub Date:
 March 2016
 DOI:
 10.3842/SIGMA.2016.025
 arXiv:
 arXiv:1504.00715
 Bibcode:
 2016SIGMA..12..025B
 Keywords:

 loop group;
 factorization;
 Toeplitz operator;
 determinant;
 Mathematics  Representation Theory;
 22E67
 EPrint:
 SIGMA 12 (2016), 025, 29 pages