Prequantization of the Moduli Space of Flat ${\rm PU}(p)$-Bundles with Prescribed Boundary Holonomies

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


Introduction
Let G be a compact connected simple Lie group and Σ a compact oriented surface with s boundary components.Given conjugacy classes C 1 , . . ., C s , let M = M G (Σ; C 1 , . . ., C s ) denote the moduli space of flat G-bundles on Σ with prescribed boundary holonomies in the conjugacy classes C j .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 [4].This paper considers the obstruction to the existence of a prequantization of M (i.e.prequantum line bundle L → M), 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 C j corresponds to a level k weight (e.g.see [2,5,13]).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 l 0 (G) (computed in [10] for each G).If Σ has boundary with prescribed holonomies, only the case G = SO(3) ∼ = PU(2) has been fully resolved [12].
In this paper, we describe the connected components of M for non-simply connected structure 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 [17] for classical groups and Appendix A records the action for the two remaining exceptional cases.
Finally, we compute the obstruction to prequantization in Theorem 5.6 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 [1], 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 [14], analogous to Spin c 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 5.5) and will be considered elsewhere.

Acknowledgements. A portion of this work is a revised version of (previously unpublished) results
from the author's Ph.D. thesis [11], supervised by E. Meinrenken and P. Selick.I remain grateful for their guidance and support.

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 exp T -integer 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 α 1 , . . ., α l (with l = rank(G)) spanning Q, determines the fundamental coweights λ ∨ 1 , . . ., λ ∨ l spanning P ∨ , defined by α i , λ ∨ j = δ i,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 {e 1 , . . ., e n } denote the standard basis for R n , 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 [1].(For the remainder of this section, we may take G to be any compact Lie group with invariant inner product −, − on g.) Let θ L , θ R denote the left-invariant, right-invariant Maurer-Cartan forms on G, and let η = 1  12 θ L , [θ L , θ L ] 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.[1] 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: We will often denote a quasi-Hamiltonian G-space (M, ω, Φ) simply by the underlying space M when ω and Φ are undestood from the context.
The fusion product of two quasi-Hamiltonian G-spaces M j with moment maps Φ j : M j → G (j = 1, 2) is the product M 1 × M 2 , with the diagonal G-action and moment map Φ : 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 is a regular value.If 1 is a singular value, then the symplectic quotient is a singular symplectic space as defined in [15] 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 −1 h −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.)

Conjugacy classes invariant under translation by central elements
This section describes the set of conjugacy classes in G that are invariant under translation by a subgroup of the centre Z(G) of G.We begin with a Lemma that relates the central subgroups leaving conjugacy classes (of G) invariant with conjugacy classes in G/Z with Z ⊂ Z(G).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 Z D 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 [17,Section 4.1].(See also [6, Section 3.1] for a similar treatment.) Let {α 1 , . . ., α l } be a basis of simple roots for t * , with highest root α =: Recall that the centre Z(G) ∼ = P ∨ /Q ∨ (induced by the exponential map), and that the nonzero elements of the centre have representatives λ ∨ i ∈ P ∨ given by minimal dominant coweights.By [17,Lemma 2.3] the non-zero minimal dominant coweights λ ∨ i are dual to the special roots α i , which are those roots with coefficient 1 in the expression α = m i α i .In Proposition 4.1.4 of [17], Toledano-Laredo provides a Z(G)-action on ∆ defined by i , and w i ∈ W is a certain element of the Weyl group.The element w i ∈ W is the unique element that leaves ∆ ∪ {α 0 } invariant (i.e.induces an automorphism of the extended Dynkin diagram) and satisfies w i (α 0 ) = α i (see [17,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
, which proves the Proposition.
In fact, as the next Proposition shows, the automorphism of the Dynkin diagram induced by w i encodes the resulting permutation of the vertices of the alcove ∆.Proposition 3.3.Let v 0 , . . ., v l denote the vertices of ∆ with v j opposite the facet parallel to ker α j .Then exp λ ∨ i • v j = v k whenever w i α j = α k , where w i is as in Proposition 3.2.
Proof.Let v 0 , . . ., v l denote the vertices of ∆ where the vertex v j is opposite the facet (codimension 1 face) parallel to ker α j .That is, v 0 = 0 and for j = 0, v j satisfies: α 0 , v j = −1, and α r , v j = 0 if and only if 0 = r = j.
(Hence, for j = 0 we have α j , v j = 1 m j where m j is the coefficient of α j in the expression α = m i α i .) Suppose that w i α 0 = α i and let w i α j = α k (where k depends on j).
i α r is a simple root other than α j ; therefore, each term above is 0.Moreover, if r = i, then the above expression becomes α 0 , v j + α i , λ On the other hand, if k = 0 so that And if r = 0, we consider again the expression (1) and find (for the same reason as above) that ( 1) is trivial whenever The Z(G)-action on ∆ is explicitly described in [17] for all classical groups.(In Appendix A, we record the action of the centre on the alcove for the exceptional groups E 6 and E 7 , 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 * ⊂ R n as the subspace {x = x j e j : x j = 0} and recall that the basic inner product coincides with (the restriction of) the standard inner product on R n .The roots are the vectors e i − e j with i = j.Taking the simple roots to be α i = e i − e i+1 (i = 1, . . ., n − 1) and the resulting highest root α = e 1 − e n gives the alcove (since v j is the vertex opposite the facet parallel to ker α j ).
It follows that the only point in ∆ fixed by the action of Z(G) is the barycenter Hence there is a unique conjugacy class in SU(n) that is invariant under translation by the centrenamely, matrices in SU(n) with eigenvalues z 1 , . . ., z n , 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 ∆.Proof.Write x = t i v i in ∆ in barycentric coordinates (with t i ≥ 0 and t i = 1).Then a generator of Z/νZ sends x to t ′ i v i , with t ′ i = t i−m mod n .Therefore x is fixed if and only if t i = t i−m mod n , and in this case we may write, 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 v 0 ↔ v 2 and v 1 ↔ v 3 .The barycenters ζ 0 , ζ 1 of the edges v 0 v 2 and v 1 v 3 , respectively, are fixed and thus the Z-fixed points are those on the line segment joining ζ 0 and ζ 1 .(See Figure 1).

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 [1] 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 C 1 , . . ., C s in G/Z, let M G/Z (Σ; C 1 , . . ., C s ) be the moduli space of flat G/Z-bundles over Σ with prescribed boundary holonomies lying in the conjugacy classes C j (j = 1, . . ., s).Points in M G/Z (Σ; C 1 , . . ., C s ) 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 C j .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, which may have several connected components if Z is non-trivial.Extending the discussion in [12, Section 2.3], we describe the connected components of (2) as symplectic quotients of an auxiliary quasi-Hamiltonian G-space.
As in [12, 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, Then Ň is naturally a quasi-Hamiltonian G-space with moment map Φ.The following Proposition from [12] and its Corollary summarize some properties of this construction.(ii) For a fusion product Corollary 4.2.Let Ň be the fibre product defined by ( 3) where Φ : N → G/Z is a group-valued moment map, and write Ň = X j as a union of its connected components.Then the components of N/ /(G/Z) can be identified with the symplectic quotients X j / /G.Hence to identify the components of (2), it suffices to identify the components of Ň / /G, where

Proof. The restrictions
where X j ranges over the components of Ň.With this in mind, choose conjugacy classes D j ⊂ G covering C j (j = 1, . . ., s) and let (cf.Lemma 3.1).We show next that the components of Ň are all homeomorphic to N /(Z 2h × Γ).
and let Ň be the fibre product defined by (3).Then Ň may be written as a union of its connected components, where D j ⊂ G are conjugacy classes covering C j (j = 1, . . ., s) and Γ is as in (4).
Proof.This is a straightforward application of the properties (ii) and (iii) listed in Proposition 4.1.
By property (iii), Ď(G/Z) h = D(G/Z) h × Z, and by Lemma 3.1, Čj = D j × Z/Z D j .Therefore, by property (ii), Consider the component corresponding to z ∈ Z/(Z D 1 • • • Z Ds ) in which each point is of the form ( g, [(z, x 1 , . . ., x s )] Γ ′ ), where [ ] Γ ′ denotes a Γ ′ -orbit.(Note that there is always a representative of this form with z in the first coordinate.)This component is homeomorphic to 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.[12,Lemma 2.3]).
Corollary 4.4.Let p be prime and let s) and let Ň be the fibre product defined by (3).Then, where D j ⊂ SU(p) are conjugacy classes covering C j (j = 1, . . ., s) and Γ is as in (4).
In particular, if (after re-labelling) C j = C * for all j ≤ r (r > 0), then we obtain where, in this case, Γ = {(γ 1 , . . ., γ r ) ∈ Z r : γ j = 1}.Example 5.2.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 l 0 ∈ N, where l 0 is a positive integer depending on G/Z computed for all compact simple Lie groups in [10].For G/Z = PU(n), l 0 = n.
Example 5.3.Conjugacy classes D ⊂ G admitting a level k-prequantization are those D = G•exp ξ (ξ ∈ ∆) with (kξ) ♭ ∈ P [13], 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 g k ∈ Z(G) for all g ∈ D. (So in particular if k is a multiple of the order of D [5, Definition 5.76], then D admits a level k prequantization.)5.2.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 where Γ is as in (4).As we shall see in the proof of Theorem 5.6, we will find Property (a) in Section 5.1 very useful in order to proceed 'factor by factor,' using the decomposition (5).
To begin, we establish the following Proposition which allows us to use Property (c) in Section 5.1 to compute the obstruction to prequantization for the factor (D * ) r /Γ in (5).
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 Σ p ) acts by permuting the t i , Z acts by a p-cycle on the t i .Therefore, H 2 (D * ; R) Z = 0, which establishes the result.G = E 7 .Let t ∼ = t * ∼ = {(x 1 , . . ., x 8 ) ∈ R 8 : x 7 = −x 8 }.The simple roots α 1 , . . ., α 7 and highest root α 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.

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 π 1 (C) ∼ = Z D = {z ∈ Z : zD = D}.
j = 1, . . ., n).The centre Z(SU(n)) ∼ = Z/nZ is generated by (exp of) the minimal dominant coweight λ ∨ 1 = e 1 − 1 n n i=1 e i corresponding to the special root α 1 = e 1 − e 2 .Since the element w 1 inducing an automorphism of the extended Dynkin diagram for SU(n) satisfies w 1 α 0 = α 1 , by Proposition 3.3 the permutation of the vertices of ∆ induced by the action of exp λ

Proposition 3 . 4 .
Let n = νm and consider the subgroup Z/νZ ⊂ Z/nZ ∼ = Z(SU(n)).The Z/νZfixed 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 v 0 , . . ., v m−1 of ∆.

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 Cartan3-form η ∈ Ω 3 (G) is integral-in fact, [η] ∈ H 3 (G; R) isthe image of a generator x ∈ H 3 (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 Ω 3 (Φ), the algebraic mapping cone of the pull-back map Φ * : Ω * (G) → Ω * (M ), and hence a cohomology class [(ω, η)] ∈ H 3 (Φ ; R). (See [7, Ch.I, Sec.6] for the definition of relative cohomology.)Definition 5.1.[10, 14] Let k ∈ N. A level k prequantization of a quasi-Hamiltonian G-space (M, ω, Φ) is an integral lift α ∈ H 3 (Φ ; Z) of the class k[(ω, η)] ∈ H 3 (Φ ; R).The definition of prequantization in 5.1 uses the assumption in this paper that G is simply connected.The general definition of prequantization [14, Definition 3.2] (with G semi-simple and compact) requires an integral lift in H 3 G (Φ ; Z) of an equivariant extension of the class k[(ω, η)].When G is simply connected, [10, Proposition 3.5] shows that the definition above is equivalent.We list some basic properties level k prequantizations that we shall encounter.(a) If M 1 and M 2 are pre-quantized quasi-Hamiltonian G-spaces at level k, then their fusion product M 1 × M 2 inherits a prequantization at level k.Conversely, a prequantization of the product induces prequantizations of the factors.See [10, 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 H 2 (M ; R) = 0, kΦ * (x) = 0 is also sufficient [10, 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.

Figure 2 .
Figure 2. Permutation induced by action of exp λ ∨ 1 on the vertices of the alcove for E 6 .

Figure 3 .
Figure 3. Permutation induced by action of exp λ ∨ 7 on the vertices of the alcove for E 7 .

Table 1 .
Alcove data for E 6 .
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 5.2), we may assume from now on h = 0.