### Abstract

In previous work we showed that a loop g: S1 ! 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 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.

Original language | English (US) |
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Article number | 025 |

Journal | Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) |

Volume | 12 |

DOIs | |

State | Published - Mar 8 2016 |

### Keywords

- Determinant
- Factorization
- Loop group
- Toeplitz operator

### ASJC Scopus subject areas

- Analysis
- Geometry and Topology
- Mathematical Physics