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

Abstract

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.

Loops in SL(2, ℂ) and root subgroup factorization

“…The proof of the following result is essentially the same as in the unitary case, see Theorem 5.3 of [1]. We include the proof, because we bungled the last line in the proof of Theorem 5.3 of [1].…”

“…where f ± is holomorphic in the interior of Σ (Σ * , respectively), with appropriate boundary behavior, depending on the smoothness of f , f + ((0)) = 0, f − ((∞)) = 0, and f 0 is the restriction to S of a meromorphic function which belongs to a genus( Σ) + 1 dimensional complementary subspace, which we refer to as the vector space of zero modes (see Proposition 2.3 of [1]). In the classical case f 0 is the zero mode for the Fourier series of f .…”

“…Remark. It is very unlikely that there exists a converse to this statement, because it appears that there is a more general condition on a multi-loop g also implies that E(g) is semi-stable; see Theorem 4.3 of [1]. We will not include it here because it is more difficult to state, we lack specific examples, and we have not established the converse.…”

“…In [5] and [1] we initiated the study of root subgroup factorization for loops in the compact group SU (2, C). The purpose of this paper is to show that there is a simple way to extend the broad outline of this theory to loops in the complex group SL(2, C).…”

“…the unit disk, which for unitary loops is the context of [5]. In Section 4 we will consider 'the general case', in which the disk is replaced by a compact Riemann surface with boundary, which is the context of [1].…”