whereV[sp]σ denotes the differentiation ofsp atσalongV. ∇ˆL can be seen to be compatible with the Hermitian structureˆhL.
We recall the definition of a connection∇ˆT in T → T ×M in the following way. For directions tangent toM let
( ˆ∇TXY)(σ,p):= ((∇Tσ)XYσ)p,
where ∇T is the connection induced by the Levi-Civita connection, Y a section ofT and X ∈TpM. For directions tangent toT, letV ∈TσT be any vector onT, define
( ˆ∇TVY)(σ,p):=πσ1,0V[Yp]σ,
whereV[Yp]σis the differentiation ofYpin the trivial bundleT ×TpMCandπ1,0σ :T ×T MC→Tσ
is the projection.
Then∇ˆT induces a connection∇ˆK inK=Vm
T∗, which induces a connection∇ˆδ inδ.
Which, with the help of∇ˆL, induces a connection∇ˆrin Lˆk⊗δ Definition 4.5 (Reference connection). The connection
∇ˆr= ( ˆ∇L)⊗k⊗Id + Id⊗∇ˆδ in Lˆk⊗δ→ T ×M is calledthe reference connection.
The Hitchin connection
LetD(M,Lk⊗δσ)denote the space of differential operators onLk⊗δσ. These form a bundle D(M,ˆ Lk⊗δσ)overT. We seek a one formuδ ∈Ω1(T,D(M,ˆ Lk⊗δσ))such that∇δ= ˆ∇+uδ preserves the subspaces Hδ,σ(k) inside each fiber H(k)δ,σ. Such a connection is called a Hitchin connection.
Lemma 4.6 ([5] Lemma 5.1). The connection ∇δ is a Hitchin connection if and only if the one form uδ satisfies
∇0,1uδ(V)s+ i
2ω·G(V)· ∇s+ i
4ω·δ(G(V))s= 0 for any V vector field on T, anyσ∈ T and any s∈Hσ(k).
For the general case we need a second order operator∆G, which is defined similarly
∆G:C∞(Mσ,Lk⊗δσ)−−→∇σ C∞(Mσ, T MC∗⊗ Lk⊗δσ)
G⊗id⊗Id
−−−−−−→C∞(Mσ, Tσ⊗ Lk⊗δσ)
∇˜σ⊗id+id⊗∇˜σ
−−−−−−−−−−→C∞(Mσ, Tσ∗⊗Tσ⊗ Lk⊗δσ)
−→Tr C∞(M,Lk⊗δσ),
where∇˜σ is the Levi-Civita connection onMσ induced by the metric onMσ.
Again we have to assume the family of Kähler structures is rigid. Now using rigidity and much the same calculations as in proving Equation (4.4) one can prove the following Lemma Lemma 4.7 ([5] Lemma 5.4). At every pointσ∈ T the operator ∆G(V)satisfies
∇0,1∆G(V)s=−2ikω·G(V)· ∇s−ikω·δ(G(V))s+i
2δ(ρ·G(V))s
for all vector fieldsV onT and any local holomorphic sectionssof the line bundleL ⊗δσ→M.
This Lemma applies to all k, so also fork= 0. In this case the Lemma yields
∇0,1∆G(V)s= i
2δ(ρ·G(V))s.
Now apply the∂operator on both sides, and we get 0 =∂(δ(ρ·G(V))),
thus if we assumeH0,1(M) = 0we see thatδ(ρ·G(V)) is exact with respect to the∂operator onM, and we have proven the following
Corollary 4.8([5] Cor. 5.5). Provided thatH0,1(M) = 0, we have thatδ(ρ·G(V))is exact with respect to the∂ operator onM.
For any compact Kähler manifold with H1(M,R) = 0, Hodge decomposition will give H0,1(M) = 0. Now by Corollary 4.8 there exists a smooth one formβ∈Ω1(T, C∞(M))such that
∂β(V) =−i
2δ(ρ·G(V))
for any vector fieldV onT. We can now defineu(V)satisfying the wanted equation u(V) = 1
4(∆G(V)+β(V))
Theorem 4.9 ([5] Theorem 1.2). Let (M, ω) be a prequantizable symplectic manifold with vanishing second Stiefel Whitney class. LetJ be a rigid family of Kähler structures on M parametrized by a smooth manifoldT, all satisfying H0,1(Mσ) = 0,σ∈ T. Then there exists a one formβ∈Ω1(T, C∞(M))satisfying∂β(V) =−2iδ(ρ·G(V))and the connection
∇δV = ˆ∇V + 1
4k(∆G(V)+β(V)) is a Hitchin connection onH(k)δ overT.
Note that the restrictions onM are a lot weaker than in the original setting with geometric quantization. We do not need that the first Chern class is proportional to[ω], now we just have to know that it is even, since the second Stiefel Whitney class is equal to the first Chern class modulo two.
It was proven by Gammelgaard in his thesis (Theorem 6.22), that
Theorem 4.10 ([25]). The connection∇ˆ defined in Theorem 4.9 is projectively flat, provided H0(Mσ, Tσ) = 0for all σ∈ T.
Chapter 5
Review of the moduli space of flat connections on a compact surface
The main focus of this thesis is to construct a Hitchin connection in the case of a moduli space of a surface with marked points. We will construct the moduli space and a prequantum line bundle in that case. Before we do this, we will in this Chapter review the case of the closed surface, since many of the ideas are similar to how we do it in the case of the surface with marked points. The proofs will not be rigorous in this Chapter, since it is mainly here to serve as inspiration of how to do in the case of the surface with marked points.
5.1 The moduli space of flat connections
LetΣbe a compact surface,pa fixed base-point,π1(Σ, p)the fundamental group ofΣbased at pand letGbe a compact connected Lie-group.
Definition 5.1. Therepresentation variety of Σis the set MG= Hom(π1(Σ, p), G)/G ofG-valued representations of π1(Σ, p)modulo conjugation inG.
Let us now recall the gauge theoretic description ofMG thus realizing it as the moduli space of flatG-connections onΣ.
LetP →Σ be a principalG-bundle. LetAP denote the space of all connections in the principal G-bundle. Let FP denote the subset of AP consisting of all the flat connections.
Define an equivalence relation onFP byA∼A0if and only if they are gauge-equivalent. Then we can define the moduli space of flat connections as
MP =FP/∼.
Definition 5.2 (Moduli space of flat connections). Themoduli space of flat connections on a principal G-bundleP →M is the space
MP =FP/GP.
We will assumeGto be simply connected, thus all principalG-bundles are trivializable, thus hence MP is not dependent onP, and we will assumeP = Σ×G, .
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Let us try to give a short explanation of the idea behind the proof of why these two definitions are the same; letαbe a loop,α(0) =α(1) =p. Then for p0 ∈π−1(p), we know there exist a unique horizontal curveβ, with starting point p0 andπ◦β =α. Since α is assumed to be a loop,β(0)andβ(1)are both in the fibrePp overp, so there exists agsuch thatβ(0) =β(1)·g. Thisg is called the holonomy ofAalongαwith respect top0, denoted holA,p0(α). This induces a well-defined map
hol :MP →Hom(π1(Σ, p), G)/G=MG, that sends[A]to[holA].
Conversely ifρ:π1(Σ, p)→Gis a given homomorphism, we consider the trivialGbundle P˜ = ˜Σ×G over the universal covering space Σ˜ of Σ. We can get a right action of the fundamental group on P˜ by doing the following. Let γ ∈ π1(Σ, p) and (y, g)∈ P. Define˜ (y, g)·γ= (y·γ, ρ(γ)−1g), wherey·γdenotes the natural action ofπ1(Σ, p)on the covering space. This action is free. We can also see thatP= ˜P /π1(Σ, p)is a principalGbundle over Σ. We have a free right action, so all we need is thatP˜ is locally trivializable. Letπ be the projectionπ: P→Σ. We have to show that for allx∈Σthere exists a neighborhoodU such thatπ:π−1(U)→U×Gis an equivariant diffeomorphism, which covers the identity onU. Letq∈Σ. LetΣ˜ denote the universal covering ofΣ. LetU be an open neighborhood of q such that inΣ˜ the open neighborhoods of each pre-image ofqare disjoint, this can be done since it is the universal covering. LetU˜ be one of these. P˜ = ˜Σ×Gis locally trivializable, so P|˜ U˜ = ˜U×G'U×G. HenceP /π˜ 1(Σ, p)is locally trivializable.
The trivial connection onP˜ (the pullback of the Maurer-Cartan form onG) is invariant under the action of π1(Σ, p), so it descends to a flat connection on P. Hence we have a well-defined map
MG→ MP.