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

Using the framework of quasi-Hamiltonian actions, we compute the obstruction to prequantization for the moduli space of flat PU(p)-bundles over a compact orientable surface with prescribed holonomies around boundary components, where p>2 is prime.

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Date:	2014
Main Author:	Krepski, D.
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Language:	English
Published:	Інститут математики НАН України 2014
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Cite this:	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies / D. Krepski // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ.

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spelling	irk-123456789-1464052019-02-10T01:24:56Z Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies Krepski, D. Using the framework of quasi-Hamiltonian actions, we compute the obstruction to prequantization for the moduli space of flat PU(p)-bundles over a compact orientable surface with prescribed holonomies around boundary components, where p>2 is prime. 2014 Article Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies / D. Krepski // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D50; 53D30 DOI:10.3842/SIGMA.2014.109 http://dspace.nbuv.gov.ua/handle/123456789/146405 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Using the framework of quasi-Hamiltonian actions, we compute the obstruction to prequantization for the moduli space of flat PU(p)-bundles over a compact orientable surface with prescribed holonomies around boundary components, where p>2 is prime.
format	Article
author	Krepski, D.
spellingShingle	Krepski, D. Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Krepski, D.
author_sort	Krepski, D.
title	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies
title_short	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies
title_full	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies
title_fullStr	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies
title_full_unstemmed	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies
title_sort	prequantization of the moduli space of flat pu(p)-bundles with prescribed boundary holonomies
publisher	Інститут математики НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/146405
citation_txt	Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies / D. Krepski // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 23 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT krepskid prequantizationofthemodulispaceofflatpupbundleswithprescribedboundaryholonomies
first_indexed	2025-07-10T23:55:22Z
last_indexed	2025-07-10T23:55:22Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 109, 13 pages Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies? Derek KREPSKI Department of Mathematics, University of Manitoba, Canada E-mail: Derek.Krepski@umanitoba.ca URL: http://server.math.umanitoba.ca/~dkrepski/ Received August 05, 2014, in final form November 28, 2014; Published online December 05, 2014 http://dx.doi.org/10.3842/SIGMA.2014.109 Abstract. Using the framework of quasi-Hamiltonian actions, we compute the obstruction to prequantization for the moduli space of flat PU(p)-bundles over a compact orientable surface with prescribed holonomies around boundary components, where p > 2 is prime. Key words: quantization; moduli space of flat connections; parabolic bundles 2010 Mathematics Subject Classification: 53D50; 53D30 1 Introduction Let G be a compact connected simple Lie group and Σ a compact oriented surface with s boundary components. Given conjugacy classes C1, . . . , Cs, let M = MG(Σ; C1, . . . , Cs) denote the moduli space of flat G-bundles on Σ with prescribed boundary holonomies in the conjugacy classes Cj . Alternatively, M may be described as the character variety of the fundamental group of Σ, M = HomC1,...,Cs(π1(Σ), G)/G. Here, HomC1,...,Cs(π1(Σ), G) consists of homomorphisms ρ : π1(Σ) → G whose restriction to (the homotopy class of) the j-th boundary circle of Σ lies in Cj , and G acts by conjugation. Recall that M is a (possibly singular) symplectic space, where the symplectic form is defined by a choice of invariant inner product on the Lie algebra g of G [5, 12]. This paper considers the obstruction to the existence of a prequantization of M – that is, a prequantum (orbifold) line bundle L → M (see Section 5.2 for details) – by expressing the corresponding integrality condition on the symplectic form in terms of the choice of inner product on the simple Lie algebra g, which is hence a certain multiple k of the basic inner product. If the underlying structure group G is simply connected, the moduli space M is connected and the obstruction to prequantization is well known – a prequantization exists if and only if k ∈ N and each conjugacy class Cj corresponds to a level k weight (e.g., see [4, 6, 18]). If G is not simply connected, M may have multiple components. 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 l0(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]. ?This paper is a contribution to the Special Issue on Poisson Geometry in Mathematics and Physics. The full collection is available at http://www.emis.de/journals/SIGMA/Poisson2014.html mailto:Derek.Krepski@umanitoba.ca http://server.math.umanitoba.ca/~dkrepski/ http://dx.doi.org/10.3842/SIGMA.2014.109 http://www.emis.de/journals/SIGMA/Poisson2014.html 2 D. Krepski In this paper, we describe the connected components of M for non-simply connected struc- ture groups G/Z in Corollary 4.2 and Proposition 4.3 (where G is simply connected and Z is a subgroup of the centre of G). The decomposition into components makes use of an action of the centre Z(G) on a fundamental Weyl alcove ∆ in t, the Lie algebra of a maximal torus. The action is described concretely in [23] for classical groups and Appendix A records the action for the two remaining exceptional cases. Finally, we compute the obstruction to prequantization in Theorem 5.8 in the case G = PU(p) (p > 2, prime) for any number of boundary components s. We work within the theory of quasi- Hamiltonian group actions with group-valued moment map [2], where the moduli space M is a central example. In quasi-Hamiltonian geometry, quantization is defined as a certain element of the twisted K-theory of G [19], analogous to Spinc quantization for Hamiltonian group actions on symplectic manifolds. In this context, the obstruction to the existence of a prequantization is a cohomological obstruction (see Definition 5.1). The obstruction for other cases of non-simply connected structure group does not follow from the approach here (see Remark 5.7) and will be considered elsewhere. 2 Preliminaries Notation. Unless otherwise indicated, G denotes a compact, simply connected, simple Lie group with Lie algebra g. We fix a maximal torus T ⊂ G and use the following notation: t – Lie algebra of T ; t∗ – dual of the Lie algebra of T ; W = N(T )/T – Weyl group; I = ker expT – integer lattice; P = I∗ ⊂ t∗ – (real) weight lattice; Q ⊂ t∗ – root lattice; Q∨ ⊂ t – coroot lattice; P∨ ⊂ t – coweight lattice. Recall that since G is simply connected, I = Q∨. Moreover, the coroot lattice and weight lattice are dual to each other, as are the root lattice and coweight lattice. A choice of simple roots α1, . . . , αl (with l = rank(G)) spanning Q, determines the fundamental coweights λ∨1 , . . . , λ ∨ l spanning P∨, defined by 〈αi, λ∨j 〉 = δi,j . We let 〈−,−〉 denote the basic inner product, the invariant inner product on g normalized to make short coroots have length √ 2. With this inner product, we will often identify t ∼= t∗. Given a subgroup Z of the centre Z(G) of G, we shall abuse notation and denote by q : G→ G/Z the resulting covering(s). Finally, let {e1, . . . , en} denote the standard basis for Rn, equipped with the standard inner product that will also be denoted with angled brackets 〈−,−〉. Quasi-Hamiltonian group actions. We recall some basic definitions and facts from [2]. (For the remainder of this section, we may take G to be any compact Lie group with invariant inner product 〈−,−〉 on g.) Let θL, θR denote the left-invariant, right-invariant Maurer–Cartan forms on G, and let η = 1 12〈θ L, [θL, θL]〉 denote the Cartan 3-form on G. For a G-manifold M , and ξ ∈ g, let ξ] denote the generating vector field of the action. The Lie group G is itself viewed as a G-manifold for the conjugation action. Definition 2.1 ([2]). A quasi-Hamiltonian G-space is a triple (M,ω,Φ) consisting of a G- manifold M , a G-invariant 2-form ω on M , and an equivariant map Φ: M → G, called the moment map, satisfying: i) dω + Φ∗η = 0, Prequantization of the Moduli Space of Flat PU(p) Bundles 3 ii) ιξ]ω + 1 2Φ∗((θL + θR) · ξ) = 0 for all ξ ∈ g, iii) at every point x ∈M , kerωx ∩ ker dΦx = {0}. We will often denote a quasi-HamiltonianG-space (M,ω,Φ) simply by the underlying spaceM when ω and Φ are understood from the context. The fusion product of two quasi-Hamiltonian G-spaces Mj with moment maps Φj : Mj → G (j = 1, 2) is the product M1×M2, with the diagonal G-action and moment map Φ : M1×M2 → G given by composing Φ1 × Φ2 with multiplication in G. The symplectic quotient of a quasi-Hamiltonian G-space is the symplectic space M//G = Φ−1(1)/G, which is a symplectic orbifold whenever the group unit 1 ∈ G is a regular value. If 1 is a singular value, then the symplectic quotient is a singular symplectic space as defined in [20]. The conjugacy classes C ⊂ G, with moment map the inclusion into G, are basic examples of quasi-Hamiltonian G-spaces. Another important example is the double D(G) = G × G, equipped with diagonal G-action and moment map Φ(g, h) = ghg−1h−1, the group commutator. These two families of examples form the building blocks of the moduli space of flat G-bundles over a surface Σ with prescribed boundary holonomies. (See Section 4 for a sketch of this construction.) 3 Conjugacy classes invariant under translation by central elements This section describes the set of conjugacy classes D ⊂ G that are invariant under translation by a subgroup ZD of the centre Z(G) of G. We begin with the following Lemma, which identifies such a subgroup ZD with the fundamental group of a conjugacy class in G/Z, where Z ⊂ Z(G). Lemma 3.1. Let Z be a subgroup of the centre Z(G) of G and let C ⊂ G/Z be a conjugacy class. For any conjugacy class D ⊂ G covering C, the restriction q\|D : D → C is the universal covering projection and hence the fundamental group π1(C) ∼= ZD = {z ∈ Z : zD = D}. Proof. The inverse image q−1(C) is a disjoint union of conjugacy classes in G that cover C. Since conjugacy classes in a compact simply connected Lie group are simply connected and ZD acts freely on D, the lemma follows. � Recall that every element in G is conjugate to a unique element exp ξ in T , where ξ lies in a fixed (closed) alcove ∆ ⊂ t of a Weyl chamber. Therefore, the set of conjugacy classes in G is parametrized by ∆. Since the Z(G)-action commutes with the conjugation action, we obtain an action Z(G)×∆→ ∆. Next we identify this description of the action of Z(G) on an alcove ∆ with a more concrete description of a Z(G)-action on ∆ given in [23, Section 4.1]. (See also [7, Section 3.1] for a similar treatment.) Let {α1, . . . , αl} be a basis of simple roots for t∗, with highest root α̃ =: −α0. Let ∆ ⊂ t be the alcove ∆ = {ξ ∈ t : 〈ξ, αj〉 ≥ 0, 〈ξ, α̃〉 ≤ 1}. Recall that the exponential map induces an isomorphism Z(G) ∼= P∨/Q∨, and that the non- zero elements of the centre have representatives λ∨i ∈ P∨ given by minimal dominant coweights. By [23, Lemma 2.3] the non-zero minimal dominant coweights λ∨i are dual to the special roots αi, which are those roots with coefficient 1 in the expression α̃ = ∑ miαi. In Proposition 4.1.4 of [23], Toledano-Laredo provides a Z(G)-action on ∆ defined by z · ξ = wiξ + λ∨i , 4 D. Krepski where z = expλ∨i , and wi ∈ W is a certain element of the Weyl group. The element wi ∈ W is the unique element that leaves ∆ ∪ {α0} invariant (i.e., induces an automorphism of the extended Dynkin diagram) and satisfies wi(α0) = αi (see [23, Proposition 4.1.2]). The following proposition shows these actions coincide. Proposition 3.2. The translation action of Z(G) on G induces an action Z(G)×∆→ ∆ and is given by the formula z · ξ = wiξ + λ∨i , where z = expλ∨i and wi is the unique element in W that leaves ∆ ∪ {α0} invariant and satisfies wi(α0) = αi. Proof. Observe that for any element w in W , wλ∨i − λ∨i ∈ I = Q∨ since w expλ∨i = expλ∨i . Therefore, wiξ + λ∨i = wi(ξ + λ∨i + (w−1 i λ∨i − λ∨i )). In other words, wiξ + λ∨i = ŵ(ξ + λ∨i ) for some ŵ in the affine Weyl group. Letting z = expλ∨i , this shows that z exp ξ = exp(ξ + λ∨i ) is conjugate to exp(wiξ + λ∨i ), which proves the proposition. � In fact, as the next proposition shows, the automorphism of the Dynkin diagram induced by wi encodes the resulting permutation of the vertices of the alcove ∆. Proposition 3.3. Let v0, . . . , vl denote the vertices of ∆ with vj opposite the facet parallel to kerαj. Then expλ∨i · vj = vk whenever wiαj = αk, where wi is as in Proposition 3.2. Proof. Let v0, . . . , vl denote the vertices of ∆, where the vertex vj is opposite the facet (codi- mension 1 face) parallel to kerαj . That is, v0 = 0 and for j 6= 0, vj satisfies: 〈α0, vj〉 = −1 and 〈αr, vj〉 = 0 if and only if 0 6= r 6= j. (Hence, for j 6= 0 we have 〈αj , vj〉 = 1 mj , where mj is the coefficient of αj in the expression α̃ = ∑ miαi.) Suppose that wiα0 = αi and let wiαj = αk (where k depends on j). Consider expλ∨i ·v0. Since 〈α0, wiv0 +λ∨i 〉 = 〈α0, λ ∨ i 〉 = −1, and (for r 6= 0) 〈αr, wiv0 +λ∨i 〉 = 〈αr, λ∨i 〉 = δr,k, we have expλ∨i · v0 = vi. Next, consider expλ∨i · vj = wivj + λ∨i , where j 6= 0. If k = 0 so that wiαj = α0 then αj = w−1 i α0 is a special root (i.e. mj = 1) since w−1 i = wj . Therefore, 〈α0, wivj + λ∨i 〉 = 〈w−1 i α0, vj〉 − 1 = 〈αj , vj〉 − 1 = 0. And if r 6= 0, 〈αr, wivj + λ∨i 〉 = 〈w−1 i αr, vj〉+ 〈αr, λ∨i 〉. (3.1) If r 6= i, w−1 i αr is a simple root other than αj ; therefore, each term above is 0. Moreover, if r = i, then the above expression becomes 〈α0, vj〉 + 〈αi, λ∨i 〉 = −1 + 1 = 0. Hence we have expλ∨i · vj = v0 whenever wiαj = α0. On the other hand, if k 6= 0 so that wiαj = αk is a simple root, then 〈α0, wivj + λ∨i 〉 = 〈w−1 i α0, vj〉 + 〈α0, λ ∨ i 〉 = 0 − 1 = −1 since the simple root w−1 i α0 6= αj . And if r 6= 0, we consider again the expression (3.1) and find (for the same reason as above) that (3.1) is trivial whenever r 6= k. If r = k, (3.1) becomes 〈αk, wivj +λ∨i 〉 = 〈w−1 i αk, vj〉+ 〈αk, λ∨i 〉 = 〈αj , vj〉 6= 0. Hence we have that expλ∨i · vj = vk, as required. � The Z(G)-action on ∆ is explicitly described in [23] for all classical groups. (In Appendix A, we record the action of the centre on the alcove for the exceptional groups E6 and E7, the remaining compact simple Lie groups with non-trivial centre.) Conjugacy classes in SU(n). We now specialize to the case G = SU(n) and consider the action of the centre on the alcove. Identify t ∼= t∗ ⊂ Rn as the subspace {x = ∑ xjej : ∑ xj = 0} and recall that the basic inner product coincides with (the restriction of) the standard inner product on Rn. The roots are the vectors ei − ej with i 6= j. Taking the simple roots to be αi = ei − ei+1 (i = 1, . . . , n− 1) and the resulting highest root α̃ = e1 − en gives the alcove ∆ = {x ∈ t : x1 ≥ x2 ≥ · · · ≥ xn, x1 − xn ≤ 1}. Prequantization of the Moduli Space of Flat PU(p) Bundles 5 Its vertices are v0 = 0 and vj = j∑ i=1 ei − j n n∑ i=1 ei, j = 1, . . . , n. The centre Z(SU(n)) ∼= Z/nZ is generated by (exp of) the minimal dominant coweight λ∨1 = e1 − 1 n n∑ i=1 ei corresponding to the special root α1 = e1 − e2. Since the element w1 inducing an automorphism of the extended Dynkin diagram for SU(n) satisfies w1α0 = α1, by Proposition 3.3 the permutation of the vertices of ∆ induced by the action of expλ∨1 is the n-cycle (v0v1 · · · vn−1) (since vj is the vertex opposite the facet parallel to kerαj). It follows that the only point in ∆ fixed by the action of Z(G) is the barycenter ζ∗ = 1 n n−1∑ j=0 vj = n− 1 2n e1 + n− 3 2n + · · ·+ 1− n 2n en. Hence there is a unique conjugacy class in SU(n) that is invariant under translation by the centre – namely, matrices in SU(n) with eigenvalues z1, . . . , zn, the distinct n-th roots of (−1)n+1. As the next proposition shows, however, restricting the action to a proper subgroup Z ∼= Z/νZ (ν\|n) of the centre results in larger Z-fixed point sets in ∆. Proposition 3.4. Let n = νm and consider the subgroup Z/νZ ⊂ Z/nZ ∼= Z(SU(n)). The Z/νZ-fixed points in the alcove ∆ for SU(n) consist of the convex hull of the barycenters of the faces spanned by the orbits of the vertices v0, . . . , vm−1 of ∆. Proof. Write x = ∑ tivi in ∆ in barycentric coordinates (with ti ≥ 0 and ∑ ti = 1). Then a generator of Z/νZ sends x to ∑ t′ivi, with t′i = ti−m mod n. Therefore x is fixed if and only if ti = ti−m mod n, and in this case we may write, x = t0 ν−1∑ j=0 vjm + t1 ν−1∑ j=0 v1+jm + · · ·+ tm−1 ν−1∑ j=0 vm−1+jm = νt0 1 ν ν−1∑ j=0 vjm + νt1 1 ν ν−1∑ j=0 v1+jm + · · ·+ νtm−1 1 ν ν−1∑ j=0 vm−1+jm, which exhibits a fixed point in the desired form. � To illustrate, consider the subgroup Z ∼= Z/2Z of the centre Z(SU(4)) ∼= Z/4Z, which acts by transposing the vertices v0 ↔ v2 and v1 ↔ v3. The barycenters ζ0, ζ1 of the edges v0v2 and v1v3, respectively, are fixed and thus the Z-fixed points are those on the line segment joining ζ0 and ζ1 (see Fig. 1). Figure 1. Alcove for SU(4). The indicated line segment through the barycenter parametrizes the set of conjugacy classes invariant under translation by Z/2Z ⊂ Z(SU(4)). 6 D. Krepski 4 Components of the moduli space with markings In this section we recall the quasi-Hamiltonian description of the moduli space of flat bundles over a compact orientable surface with prescribed boundary holonomies. We refer to the original article [2] for the details regarding the construction sketched below. Let Σ be a compact, oriented surface of genus h with s boundary components. For conjugacy classes C1, . . . , Cs in G/Z, let MG/Z(Σ; C1, . . . , Cs) be the moduli space of flat G/Z-bundles over Σ with prescribed boundary holonomies lying in the conjugacy classes Cj (j = 1, . . . , s). Points in MG/Z(Σ; C1, . . . , Cs) are (gauge equivalence classes of) principal G/Z-bundles over Σ equipped with a flat connection whose holonomy around the j-th boundary component lies in the conjugacy class Cj . This moduli space is an important example in the theory of quasi- Hamiltonian group actions, where it is cast a symplectic quotient of a fusion product, MG/Z(Σ; C1, . . . , Cs) = ( D(G/Z)h × C1 × · · · × Cs ) //(G/Z), (4.1) which may have several connected components if Z is non-trivial. Extending the discussion in [17, Section 2.3], we describe the connected components of (4.1) as symplectic quotients of an auxiliary quasi-Hamiltonian G-space. As in [17, Section 2.2], given a quasi-Hamiltonian G/Z-space N with group-valued moment map Φ : N → G/Z, let Ň be the fibre product defined by the Cartesian square, Ň �� Φ̌ // G q �� N Φ // G/Z (4.2) Then Ň is naturally a quasi-Hamiltonian G-space with moment map Φ̌. The following proposi- tion from [17] and its Corollary summarize some properties of this construction. Proposition 4.1 ([17, Proposition 2.2]). Let Ň be the fibre product defined by (4.2), where Φ : N → G/Z is a group-valued moment map. i) We have a canonical identification of symplectic quotients Ň//G ∼= N//(G/Z). ii) For a fusion product N = N1× · · · ×Nr of quasi-Hamiltonian G/Z-spaces, the space Ň is a quotient of Ň1 × · · · × Ňr by the group { (c1, . . . , cr) ∈ Zr \| r∏ j=1 cj = e } . iii) If Φ: N → G/Z lifts to a moment map Φ′ : N → G, thus turning N into a quasi- Hamiltonian G-space then Ň = N × Z. Corollary 4.2. Let Ň be the fibre product defined by (4.2), where Φ : N → G/Z is a group- valued moment map, and write Ň = ⊔ Xj as a union of its connected components. Then the components of N//(G/Z) can be identified with the symplectic quotients Xj//G. Proof. The restrictions Φ̌j = Φ̌\|Xj are G-valued moment maps whose fibres are connected by [2, Theorem 7.2]. Since Φ̌−1(e) = ⊔ Φ̌−1 j (e), it follows that Ň//G = Φ̌−1(e)/G = ⊔ Φ̌−1 j (e)/G =⊔ Xj//G. The result follows from Proposition 4.1(i). � Hence to identify the components of (4.1), it suffices to identify the components of Ň//G, where N = D(G/Z)h × C1 × · · · × Cs – namely, Xj//G, where Xj ranges over the components of Ň . In particular, we may view the moduli space (4.1) as a union of symplectic quotients of quasi-Hamiltonian G-spaces (as opposed to G/Z-spaces), which will be very important for the approach taken in Section 5. Prequantization of the Moduli Space of Flat PU(p) Bundles 7 With this in mind, choose conjugacy classes Dj ⊂ G covering Cj (j = 1, . . . , s) and let Ñ = D(G)h ×D1 × · · · × Ds. Let Γ = { (γ1, . . . , γs) ∈ ZD1 × · · · × ZDs : ∏ γj = 1 } ⊂ Zs (4.3) (cf. Lemma 3.1). We show next that the components of Ň are all homeomorphic to Ñ/(Z2h×Γ) (generalizing the decomposition appearing in [17, Lemma 2.3] for G = SO(3)). Proposition 4.3. Let N = D(G/Z)h×C1×· · ·×Cs for conjugacy classes Cj ⊂ G/Z (j = 1, . . . , s) and let Ň be the fibre product defined by (4.2). Then Ň may be written as a union of its connected components, Ň ∼= ⊔ Z/(ZD1 ...ZDs ) D(G/Z)h × (D1 × · · · × Ds)/Γ, where Dj ⊂ G are conjugacy classes covering Cj (j = 1, . . . , s) and Γ is as in (4.3). Proof. This is a straightforward application of the properties (ii) and (iii) listed in Proposi- tion 4.1. By property (iii), Ď(G/Z)h = D(G/Z)h × Z, and by Lemma 3.1, Čj = Dj × Z/ZDj . Therefore, by property (ii), Ň ∼= D(G/Z)h × (Z ×D1 × Z/ZD1 × · · · × Ds × Z/ZDs)/Λ, where Λ = {(c0, . . . , cs) ∈ Zs+1 : c0 · · · cs = 1}. Since (Z ×D1 × Z/ZD1 × · · · × Ds × Z/ZDs)/Λ ∼= (Z ×D1 × · · · × Ds)/Γ′, where Γ′ = {(γ0, . . . , γs) ∈ Z × ZD1 × · · · × ZDs : ∏ γj = 1}, we see that the components of Ň are in bijection with Z/(ZD1 · · ·ZDs). Consider the component corresponding to z̄ ∈ Z/(ZD1 · · ·ZDs) in which each point is of the form (~g, [(z, x1, . . . , xs)]Γ′), where [ ]Γ′ denotes a Γ′-orbit. (Note that there is always a rep- resentative of this form with z in the first coordinate.) This component is homeomorphic to D(G/Z)h × (D1 × · · · × Ds)/Γ by the map (~g, [(x1, . . . , xs)]Γ) 7→ (~g, [(z, x1, . . . , xs)]Γ′). � Remark 4.4. The decomposition in Proposition 4.3 is consistent with Theorem 14 in Ho–Liu’s work [13] on the connected components of the moduli space for any compact connected Lie group G. For the case G/Z = SU(p)/(Z/pZ) = PU(p), where p is prime, the decomposition above simplifies. In particular, there is only one conjugacy class D∗ = SU(p) · exp ζ∗, corresponding to the barycenter ζ∗ ∈ ∆, invariant under the action of the centre. Let C∗ = q(D∗) be the corresponding conjugacy class in PU(p). Therefore, we obtain the following Corollary (cf. [17, Lemma 2.3]). Corollary 4.5. Let p be prime and let N = D(PU(p))h × C1 × · · · × Cs for conjugacy classes Cj ⊂ PU(p) (j = 1, . . . , s) and let Ň be the fibre product defined by (4.2). Then, Ň ∼= { D(PU(p))h × (D1 × · · · × Ds)/Γ if ∃ j : Cj = C∗, D(PU(p))h ×D1 × · · · × Ds × Z otherwise, where Dj ⊂ SU(p) are conjugacy classes covering Cj (j = 1, . . . , s) and Γ is as in (4.3). In particular, if (after re-labelling) Cj = C∗ for all j ≤ r (r > 0), then we obtain Ň ∼= D(PU(p))h × (D∗)r/Γ×Dr+1 × · · · × Ds, (4.4) where, in this case, Γ = {(γ1, . . . , γr) ∈ Zr : ∏ γj = 1}. 8 D. Krepski 5 Obstruction to prequantization 5.1 Prequantization for quasi-Hamiltonian group actions We recall some definitions and properties regarding prequantization of quasi-Hamiltonian group actions. Recall that the Cartan 3-form η ∈ Ω3(G) is integral – in fact, [η] ∈ H3(G;R) is the image of a generator x ∈ H3(G;Z) ∼= Z under the coefficient homomorphism induced by Z→ R. Condition (i) in Definition 2.1 says that the pair (ω, η) defines a relative cocycle in Ω3(Φ), the algebraic mapping cone of the pull-back map Φ∗ : Ω∗(G) → Ω∗(M), and hence a cohomology class [(ω, η)] ∈ H3(Φ;R). (See [8, Chapter I, Section 6] for the definition of relative cohomology.) Definition 5.1 ([15, 19]). Let k ∈ N. A level k prequantization of a quasi-Hamiltonian G-space (M,ω,Φ) is an integral lift α ∈ H3(Φ;Z) of the class k[(ω, η)] ∈ H3(Φ;R). Remark 5.2. The definition of prequantization in Definition 5.1 uses the assumption in this paper that G is simply connected. The general definition of prequantization [19, Definition 3.2] (with G semi-simple and compact) requires an integral lift in H3 G(Φ;Z) of an equivariant ex- tension of the class k[(ω, η)]. When G is simply connected, [15, Proposition 3.5] shows that the definition above is equivalent. Our main goal is to apply the quasi-Hamiltonian viewpoint on prequantization to the moduli space of flat bundles with prescribed holonomies; therefore, by Corollary 4.2 it suffices to work on each component of the quasi-Hamiltonian G-space Ň in Proposition 4.3 using Definition 5.1. We list some basic properties of level k prequantizations that we shall encounter. (a) If M1 and M2 are pre-quantized quasi-Hamiltonian G-spaces at level k, then their fusion product M1 ×M2 inherits a prequantization at level k. Conversely, a prequantization of the product induces prequantizations of the factors. See [15, Proposition 3.8]. (b) A level k prequantization of M induces a prequantization of the symplectic quotient M//G, equipped with the k-th multiple of the symplectic form. (c) The long exact sequence in relative cohomology gives a necessary condition kΦ∗(x) = 0 for the existence of a level k-prequantization. If H2(M ;R) = 0, kΦ∗(x) = 0 is also sufficient [15, Proposition 4.2] to conclude a level k-prequantization exists. The following examples relate to the moduli space of flat bundles with prescribed boundary holonomies. Example 5.3. The double D(G) = G × G with moment map Φ : D(G) → G equal to the group commutator admits a prequantization at all levels k ∈ N. For non-simply connected groups, the double D(G/Z) with moment map Φ : D(G/Z)→ G the canonical lift of the group commutator admits a level k-prequantization if and only if k is a multiple of l0 ∈ N, where l0 is a positive integer depending on G/Z computed for all compact simple Lie groups in [15]. For G/Z = PU(n), l0 = n. Example 5.4. Conjugacy classes D ⊂ G admitting a level k-prequantization are those D = G · exp ξ (ξ ∈ ∆) with (kξ)[ ∈ P [18], where (kξ)[ = 〈kξ,−〉 (i.e., a level k weight). For simply laced groups (such as G = SU(n)), under the identification t ∼= t∗, P∨ ∼= P . Therefore, in this case, D admits a level k-prequantization if and only if kξ ∈ P∨. Since exp−1 Z(G) = P∨, we see that D admits a level k-prequantization if and only if gk ∈ Z(G) for all g ∈ D. (So in particular if k is a multiple of the order of D [6, Definition 5.76], then D admits a level k prequantization.) Prequantization of the Moduli Space of Flat PU(p) Bundles 9 5.2 Quasi-Hamiltonian prequantization and symplectic quotients To provide some context, we further elaborate on property (b) following Definition 5.1 since we view the moduli space of flat bundles as a symplectic quotient (4.1) in quasi-Hamiltonian geometry. By [15, Proposition 3.6], a level k prequantization of a quasi-Hamiltonian G-space (M,ω,Φ) gives an integral lift of the equivariant cohomology class [j∗ω] ∈ H2 G(Φ−1(1) ;R), where j : Φ−1(1)→M denotes inclusion. Hence, there is a G-equivariant line bundle L→ Φ−1(1) with connection of curvature j∗ω. If 1 is a regular value, the symplectic quotient M//G = Φ−1(1)/G is a symplectic orbifold [2] and the G-equivariant line bundle over the level set descends to a prequantum orbifold line bundle L/G → M//G. (See [1, Example 2.29] for a discussion of orbifold vector bundles in this context.) Remark 5.5. The orbifold line bundle L/G → M//G need not be an ordinary line bundle over the underlying topological space M//G (i.e., the coarse moduli space of the orbifold). (For orbifolds that arise as quotients X/G of a smooth, proper, locally free action of a Lie group G on a smooth manifold X, this distinction is apparent from the observation that H2 G(X ;Z) is not necessarily isomorphic to H2(X/G ;Z).) Some works in the literature require a prequanti- zation to be an ordinary line bundle, and hence obtain a further obstruction to the existence of a prequantization (e.g. [14, Theorem 4.2], [10, Theorems 4.12 and 6.1], [21, Lemme 3.2]). We gratefully acknowledge the referee’s comments that led to this important clarification. 5.3 The obstruction to prequantization for the moduli space of PU(p) bundles, p prime Let p be an odd prime. In this section we obtain the obstruction to prequantization for the quasi-Hamiltonian SU(p)-space Ň , where N = D(PU(p))h × C1 × · · · × Cs for conjugacy classes Cj ⊂ PU(p) (j = 1, . . . , s). Let M ⊆ Ň be a connected component (by Corollary 4.5), M = D(PU(p))h × (D1 × · · · × Ds) /Γ, where Γ is as in (4.3). As we shall see in the proof of Theorem 5.8, we will find property (a) in Section 5.1 very useful in order to proceed ‘factor by factor’, using the decomposition (4.4). To begin, we establish the following proposition which allows us to use property (c) in Sec- tion 5.1 to compute the obstruction to prequantization for the factor (D∗)r/Γ in (4.4). Proposition 5.6. Let D∗ ⊂ SU(p) denote the conjugacy class of the barycenter ζ∗ of the alcove ∆ and let Γ = {(γ1, . . . , γr) ∈ Zr : ∏ γj = 1} with r > 1. Then H2((D∗)r/Γ;R) = 0. Proof. Since (D∗)r → (D∗)/Γ is a covering projection, H2((D∗)r/Γ;R) ∼= H2((D∗)r;R)Γ. By the Künneth Theorem, H2((D∗)r;R) ∼= ⊕ H2(D∗,R). Since the Γ-action factors through Zm, H2((D∗)r;R)Γ = ⊕ H2(D∗;R)Z . Recall that since ζ∗ lies in the interior of the alcove, the centralizer SU(p)exp ζ∗ = T and hence D∗ ∼= SU(p)/T . Moreover, we have H∗(D∗;R) ∼= R[t1, . . . , tp]/(σ1, . . . , σp), where σi’s are the elementary symmetric polynomials. In particular, we may write H2(D∗;R) ∼= (Rt1 ⊕ · · · ⊕ Rtp)/(t1 + · · ·+ tp = 0). The Z-action on D∗ corresponds to an action on SU(p)/T by a cyclic subgroup of the Weyl group (e.g., see the proof of Proposition 3.2). Since the Weyl group (i.e., symmetric group Σp) acts by permuting the ti, Z acts by a p-cycle on the ti. Therefore, H2(D∗;R)Z = 0, which establishes the result. � Remark 5.7. The analogue of Proposition 5.6 for the factors (D1× · · · ×Ds)/Γ that appear in the decomposition in Proposition 4.3 need not hold when considering other non-simply connected structure groups G/Z. 10 D. Krepski Theorem 5.8. The quasi-Hamiltonian SU(p)-space M = D(PU(p))h×(D1×· · ·×Ds)/Γ admits a level k-prequantization if and only if the following conditions are satisfied: i) if h ≥ 1, then k ∈ pN; ii) gk ∈ Z(SU(p)) for every g ∈ D1 ∪ · · · ∪ Ds. Moreover, if in addition the identity matrix 1 ∈ SU(p) is a regular value of the restriction of the group-valued moment map Φ̌ : M → SU(p), the prequantization descends to a prequantization of the corresponding component of the moduli space M = MPU(p)(Σ; C1, . . . , Cs), where Cj = q(Dj) ⊂ PU(p). Proof. By property (a) in Section 5.1, M admits a level k-prequantization if and only if each factor does. Since D(PU(p)) admits a level k-prequantization if and only if condition (i) is satisfied (see Example 5.3), we may assume from now on h = 0. We first verify the necessity of condition (ii). A prequantization of M = (D1 × · · · × Ds)/Γ induces a prequantization of its universal cover M̃ = D1 × · · · × Ds, and hence each Dj must admit a prequantization, which is equivalent to condition (ii). Next we verify that condition (ii) is sufficient for a level k-prequantization of M (with h = 0). As in the decomposition (4.4), write (possibly after re-labelling) M = (D∗ × · · · × D∗︸︷︷︸ r factors )/Γ×Dr+1 × · · · × Ds Using property (a) in Section 5.1 again, it suffices to consider the case 1 < r = s. (Note that if s = r = 1, Γ is trivial.) In this case, condition (ii) is simply that D∗ admit a level k prequantization. Since D∗ consists of matrices in SU(p) conjugate to exp ζ∗ = diag ( exp (p−1 p π √ −1 ) , exp (p−3 p π √ −1 ) , . . . , exp (1−p p π √ −1 )) , D∗ admits a level k-prequantization if and only if (exp ζ∗) k is a scalar matrix; if and only if k is a multiple of p. By property (c) in Section 5.1 and Proposition 5.6, it suffices to show that p · Φ̌∗x = 0, where Φ̌ : M → SU(p) is the group-valued moment map. By Corollary 7.6 in [3], h∨Φ̌∗x = W3(M), the third integral Stiefel–Whitney class, where h∨ denotes the dual Coxeter number. Recall that W3(M) = βw2(M), where β : H2(M ;Z/2Z) → H3(M ;Z) is the (integral) Bockstein homomorphism and w2(M) is the second Stiefel–Whitney class. Since Γ has odd order, H2(M ;Z/2Z) ∼= H2((D∗)r;Z/2Z)Γ, which is trivial (by an argu- ment similar to the proof of Proposition 5.6). Since h∨ = p, we are done. As discussed in Section 5.2, a quasi-Hamiltonian prequantization descends to a prequantiza- tion of the symplectic quotient M. � A The action of the centre on the alcove of exceptional Lie groups Below we record the action of the centre Z(G) on an alcove for the exceptional Lie groups G = E6 and G = E7. (The action for classical groups appears in [23].) The vertices of the alcove were obtained using polymake [11], which outputs the vertices of a polytope presented as an intersection of half-spaces. The relevant Weyl group element from Proposition 3.2 – one which gives an automorphism of the extended Dynkin diagram – was found with the help of John Stembridge’s coxeter-weyl package for Maple [22]; a direct calculation then shows that this element has the desired properties in Proposition 3.2. Prequantization of the Moduli Space of Flat PU(p) Bundles 11 Let {e1, . . . , e8} denote the standard basis in R8, equipped with the usual inner product. Given a vector α in R8, sα : R8 → R8 denotes reflection in the subspace orthogonal to α, sα(v) = v − 2〈α, v〉〈α, α〉 α. The notation used below is consistent with that found in [9, Planches V-VI]. G = E6. Let t ∼= t∗ ∼= {(x1, . . . , x8) ∈ R8 : x6 = x7 = −x8}. The simple roots α1, . . . , α6 and highest root α̃ determine the half-spaces whose intersection is the alcove ∆ ⊂ t. The vertices of ∆ (opposite the facets parallel to the corresponding root hyperplanes) are given in Table 1. Table 1. Alcove data for E6. Simple or dominant root Opposite vertex α1 = ( 1 2 ,− 1 2 ,− 1 2 ,− 1 2 ,− 1 2 ,− 1 2 ,− 1 2 , 1 2 ) v1 = ( 0, 0, 0, 0, 0,−2 3 ,− 2 3 , 2 3 ) α2 = (1, 1, 0, 0, 0, 0, 0, 0) v2 = ( 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ,− 1 4 ,− 1 4 , 1 4 ) α3 = (−1, 1, 0, 0, 0, 0, 0, 0) v3 = ( −1 4 , 1 4 , 1 4 , 1 4 , 1 4 ,− 5 12 ,− 5 12 , 5 12 ) α4 = (0,−1, 1, 0, 0, 0, 0, 0) v4 = ( 0, 0, 1 3 , 1 3 , 1 3 ,− 1 3 ,− 1 3 , 1 3 ) α5 = (0, 0,−1, 1, 0, 0, 0, 0) v5 = ( 0, 0, 0, 1 2 , 1 2 ,− 1 3 ,− 1 3 , 1 3 ) α6 = (0, 0, 0,−1, 1, 0, 0, 0) v6 = ( 0, 0, 0, 0, 1,−1 3 ,− 1 3 , 1 3 ) α̃ = ( 1 2 , 1 2 , 1 2 , 1 2 , 1 2 ,− 1 2 ,− 1 2 , 1 2 ) v0 = 0 The non-zero elements of the centre Z(E6) ∼= Z/3Z are given by (exp of) the minimal dominant coweights λ∨1 = 2 3(e8 − e7 − e6) and λ∨6 = e5 + 1 3(e8 − e7 − e6). The corresponding elements w1 and w6 of the Weyl group (as in Proposition 3.2), inducing automorphisms of the extended Dynkin diagram are: w1 = sα1sα3sα4sα2sα5sα4sα3sα1sα6sα5sα4sα2sα3sα4sα5sα6 , w6 = sα6sα5sα4sα2sα3sα1sα4sα3sα5sα4sα2sα6sα5sα4sα3sα1 . The permutation of the vertices induced by the action of exp(λ∨1 ) (encoded by the automor- phism w1 of the underlying extended Dynkin diagram) is shown schematically in Fig. 2. Figure 2. Permutation induced by action of expλ∨1 on the vertices of the alcove for E6. G = E7. Let t ∼= t∗ ∼= {(x1, . . . , x8) ∈ R8 : x7 = −x8}. The simple roots α1, . . . , α7 and highest root α̃ determine the half-spaces whose intersection is the alcove ∆ ⊂ t. The vertices of ∆ (opposite the facets parallel to the corresponding root hyperplanes) are given in Table 2. The non-zero element of the centre Z(E7) ∼= Z/2Z is given by (exp of) the minimal domi- nant coweight λ∨7 = e6 + 1 2(e8 − e7). The corresponding element w7 of the Weyl group (as in 12 D. Krepski Table 2. Alcove data for E7. Simple or dominant root Opposite vertex α1 = ( 1 2 ,− 1 2 ,− 1 2 ,− 1 2 ,− 1 2 ,− 1 2 ,− 1 2 , 1 2 ) v1 = ( 0, 0, 0, 0, 0, 0,−1 2 , 1 2 ) α2 = (1, 1, 0, 0, 0, 0, 0, 0) v2 = ( 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ,− 1 2 , 1 2 ) α3 = (−1, 1, 0, 0, 0, 0, 0, 0) v3 = ( −1 6 , 1 6 , 1 6 , 1 6 , 1 6 , 1 6 ,− 1 2 , 1 2 ) α4 = (0,−1, 1, 0, 0, 0, 0, 0) v4 = ( 0, 0, 1 4 , 1 4 , 1 4 , 1 4 ,− 1 2 , 1 2 ) α5 = (0, 0,−1, 1, 0, 0, 0, 0) v5 = ( 0, 0, 0, 1 3 , 1 3 , 1 3 ,− 1 2 , 1 2 ) α6 = (0, 0, 0,−1, 1, 0, 0, 0) v6 = ( 0, 0, 0, 0, 1 2 , 1 2 ,− 1 2 , 1 2 ) α7 = (0, 0, 0, 0,−1, 1, 0, 0) v7 = ( 0, 0, 0, 0, 0, 1,−1 2 , 1 2 ) α̃ = (0, 0, 0, 0, 0, 0,−1, 1) v0 = 0 Proposition 3.2), inducing an automorphism of the extended Dynkin diagram is w7 = sα7sα6sα5sα4sα2sα3sα1sα4sα3sα5sα4sα2sα6sα5sα4sα3sα1 × sα7sα6sα5sα4sα2sα3sα4sα5sα6sα7 . The permutation of the vertices induced by the action of exp(λ∨7 ) (encoded by the automor- phism w7 of the underlying extended Dynkin diagram) is shown schematically in Fig. 3. Figure 3. Permutation induced by action of expλ∨7 on the vertices of the alcove for E7. Acknowledgements A portion of this work is a revised version of (previously unpublished) results from the author’s Ph.D. Thesis [16], supervised by E. Meinrenken and P. Selick. I remain grateful for their guidance and support. My sincere thanks as well to the referees for their helpful suggestions for clarification. References [1] Adem A., Leida J., Ruan Y., Orbifolds and stringy topology, Cambridge Tracts in Mathematics, Vol. 171, Cambridge University Press, Cambridge, 2007. [2] Alekseev A., Malkin A., Meinrenken E., Lie group valued moment maps, J. Differential Geom. 48 (1998), 445–495, dg-ga/9707021. [3] Alekseev A., Meinrenken E., Dirac structures and Dixmier–Douady bundles, Int. Math. Res. 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Prequantization of the Moduli Space of Flat PU(p)-Bundles with Prescribed Boundary Holonomies

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