On the Verlinde Formulas for So(3)-Bundles

Abstract

This paper computes the quantization of the moduli space of flat SO(3)-bundles over an oriented surface with boundary, with prescribed holonomies around the boundary circles. The result agrees with the generalized Verlinde formula conjectured by Fuchs and Schweigert.

Multiplicative vector fields on bundle gerbes

“…A prequantization of a quasi-Hamiltonian G-space (M, ω, Φ) is a relative G-equivariant bundle gerbe (Q, G) for Φ-i. For related work on the integrality conditions required for prequantization of the moduli space of flat Gbundles, see [19,20,22,25].…”

Verlinde Formulas for Nonsimply Connected Groups

“…Some special cases of (4) were already known in the algebrogeometric context: Pantev [26] obtained the Verlinde numbers for G ′ = SO(3), and Beauville [7] for G ′ = PU(n) for n prime. In the symplectic framework, the case of G ′ = SO(3) had been worked out for an arbitrary number of boundary components in [23,24].…”

Prequantization of the Moduli Space of Flat PU(p) -Bundles with Prescribed Boundary Holonomies

“…Let C * = q(D * ) be the corresponding conjugacy class in PU(p). Therefore, we obtain the following Corollary (cf [17, . Lemma 2.3]).…”

“…ThenŇ is naturally a quasi-Hamiltonian G-space with moment mapΦ. The following proposition from [17] and its Corollary summarize some properties of this construction. Hence to identify the components of (4.1), it suffices to identify the components ofŇ //G, where N = D(G/Z) h × C 1 × · · · × C s -namely, X j //G, where X j ranges over the components ofŇ .…”

“…Moreover integrality of k is not sufficient to guarantee a prequantization even in the absence of markings/prescribed boundary holonomies: if Σ is closed and has genus at least 1, then k must be a multiple of an integer l 0 (G) (computed in [15] for each G). If Σ has boundary with prescribed holonomies, only the case G = SO(3) ∼ = PU (2) has been fully resolved [17].…”