Loops in SU(2) and Factorization

Pickrell, Doug

doi:10.48550/arxiv.0903.4983

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2016

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“…(a) In the classical case, all three conditions are equivalent (with z = z), and there are no constraints on y * or x * in (I. 3) and (II.3), respectively (see [15]).…”

Section: Introductionmentioning

confidence: 99%

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

Basor

Pickrell²

2016

SIGMA

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Abstract. In previous work we showed that a loop g : S 1 → 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.

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“…(a) In the classical case, all three conditions are equivalent (with z = z), and there are no constraints on y * or x * in (I. 3) and (II.3), respectively (see [15]).…”

Section: Introductionmentioning

confidence: 99%

“…if and only if g has a triangular factorization, a generic condition, and these factorizations are unique (see [15]). In our more general context we ask similar questions: can we characterize the set of g which have a factorization g = k * 1 diag(e χ , e −χ )k 2 , is the relevant condition generic, and is the factorization unique?…”

Section: Introductionmentioning

confidence: 99%