Pre-quantization of the moduli space of flat G-bundles over a surface

Abstract

MSC: 53D50 57T10Keywords: Quantization Moduli space of flat connections Riemann surface Lie group a b s t r a c t For a simply connected, compact, simple Lie group G, the moduli space of flat G-bundles over a closed surface Σ is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction -namely a certain cohomology class in H 3 (G 2 ; Z)-that places further restrictions on the underlying level. The levels that admit a prequantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G.

Verlinde Formulas for Nonsimply Connected Groups

“…• For A l with Z = Z m , where m divides l + 1, the basic level k 0 is the smallest natural number such that k 0 (l + 1) ∈ m 2 Z, while k 1 = m. (In particular, k 1 = k 0 if and only if m and (l + 1)/m are relatively prime.) • For D l , there are three different subgroups Z ⊆ Z(G) isomorphic to Z 2 ; with the labelings of fundamental coroots as in [10], they are generated by the exponentials of the coroots The following result was proved in [22] using a case-by-case examination; we will give a new proof based on the criterion in Section 3.2. Proof.…”

“…1 (Krepski[22]). The quasi-Hamiltonian G-space D(G/Z) is prequantizable at level k if and only if k is a multiple of the basic level k 0 .…”

Groupoid Equivariant Prequantization

“…Theorem 5.5 verifies a Morita invariance property of prequantization, showing that prequantization respects this correspondence. As a corollary (Corollary 5.6), we recover the equivalence of prequantizations for Hamiltonian loop group actions and quasi-Hamiltonian group actions, without the assumption of simple-connectivity on the underlying Lie group in [11,Theorem A.7].…”

“…As an application of Theorem 5.5 to the equivalence (see [3,Theorem 8.3] and [32,Corollary 4.28]) between Hamiltonian LG-actions with proper moment map and quasi-Hamiltonian G-actions with group-valued moment map, we obtain the following corollary (cf. [11,Theorem A.7] which assumes G is simply connected).…”

Basic equivariant gerbes on non-simply connected compact simple Lie groups

“…The order ofφ * z was computed directly in [11] (see also [15, [13]), the basic level ℓ b usually refers to the smallest level at which the loop group L(G/Z) admits a central U(1)-extension. That ℓ b coincides with the order of the classφ * z, described in the paragraph preceding Theorem 4.2, was noted in [11] as a coincidence-that paper computes computes the obstruction classφ * z as the obstruction to the existence of a prequantization of the moduli space of flat G/Zbundles on a closed Riemann surface. A construction of a prequantization of this moduli space using loop groups appears in [12].…”