The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory
aa r X i v : . [ m a t h . DG ] S e p The Hitchin-Witten Connection andComplex Quantum Chern-Simons Theory
Jørgen Ellegaard Andersen and Niels Leth GammelgaardSeptember 4, 2014
Abstract
We give a direct calculation of the curvature of the Hitchin connection, in geo-metric quantization on a symplectic manifold, using only differential geometric tech-niques. In particular, we establish that the curvature acts as a first-order operatoron the quantum spaces. Projective flatness follows if the K¨ahler structures do notadmit holomorphic vector fields. Following Witten, we define a complex variant ofthe Hitchin connection on the bundle of prequantum spaces. The curvature is essen-tially unchanged, so projective flatness holds in the same cases. Finally, the resultsare applied to quantum Chern-Simons theory, both for compact and complex gaugegroups.
Since their introduction by Atiyah [Ati], Segal [Seg] and Witten [Wit1, Wit2], topolog-ical quantum field theories (TQFTs) have been studied intensely using a wide range oftechniques. The first construction in 2 + 1 dimensions was given by Reshetikhin and Tu-raev [RT2, RT1, Tur] using representation theory of quantum groups at roots of unity toconstruct link invariants and in turn derive invariants of 3-manifolds through surgery andKirby calculus. Shortly thereafter, a combinatorial construction was given by Blanchet,Habegger, Masbaum and Vogel [BHMV1, BHMV2] in the language of skein theory.A geometric realization was proposed by Witten [Wit1], suggesting the use of quantumChern-Simons theory or conformal field theory to construct the 2-dimensional part. Thegauge theoretic approach was studied independently by Axelrod, Della Pietra and Wit-ten [ADPW] and Hitchin [Hit3], proving that the quantum spaces arising from geometricquantization of Chern-Simons theory for compact gauge group are indeed independentof the conformal structure on the surface, in the sense that they are identified by paral-lel transport of a projectively flat connection over the Teichm¨uller space of the surface.These constructions have been expressed and generalized in purely differential geometricterms in [And5] and [AGL], and we shall be mainly concerned with this description in
Supported in part by the center of excellence grant “Center for quantum geometry of Moduli Spaces”from the Danish National Research Foundation) + ].For non-compact gauge group, the situation is very different. In the paper [Wit3],Witten initiated the study of Chern-Simons topological quantum field theory for complexgauge groups from a physical point of view. He proposed that the smooth sections of theChern-Simons line bundle over the moduli space of flat connections in the correspondingcompact real form of the group should be the appropriate pre-Hilbert space of this theory.He reduced the description to this model space by considering a real polarization, whichwe review in section 5, on the space of connections with values in the complex gauge group.Although this model space itself does not depend on a choice of complex structure on thesurface, the polarization, and hence the interpretation as the quantum space, does. Wittenargued that the needed infinitesimal change of polarization, under infinitesimal change ofthe complex structure on the surface, can be encoded as a connection in the trivial bundlewith the fixed model space as fiber. Furthermore, Witten provided infinite-dimensionalgauge theory arguments for the projective flatness of this connection.In this paper, we review the general differential geometric construction of the Hitchinconnection for a rigid family of complex structures on a symplectic manifold with van-ishing first Betti number and first Chern class represented essentially by the symplecticform. Inspired by Witten’s considerations, we also consider a certain one-parameter familyof connections on the prequantum spaces. For any value of the parameter, the resultingconnection will be called the Hitchin-Witten connection. The main feature of both con-nections is projective flatness, at least when the K¨ahler structures have few symmetries,and we shall establish this fact by direct curvature calculations.Let us briefly introduce the setting and state the main results. The basic assumptionsand their implications will be explored in greater detail in the following sections. Con-sider a symplectic manifold ( M, ω ), with vanishing first Betti number and first Chern classgiven by c ( M, ω ) = λ (cid:2) ω π (cid:3) , for some integer λ . Futhermore, let T be a complex manifoldparametrizing a holomorphic family J : T → C ∞ ( M, End(
T M )) of integrable almost com-plex structures on (
M, ω ), none of which admit non-constant holomorphic functions on M .The variation of the K¨ahler structure, along the holomorphic part of a vector field on V on T , is encoded by a section G ( V ) ∈ C ∞ ( M, S ( T ′ M )), defined by V ′ [ J ] = G ( V ) · ω , of thesecond symmetric power of the holomorphic tangent bundle, and we will assume that thefamily J is rigid in the sense that the bivector field G ( V ) defines a holomorphic section G ( V ) ∈ H J ( M, S ( T ′ M )).By the assumption on the Chern class, the symplectic manifold ( M, ω ) admits a Hermi-tian line bundle L with a compatible connection of curvature F ∇ = − iω . For any σ ∈ T ,2he space H ( k ) σ = H σ ( M, L k ) of holomorphic setions is the quantum space, at level k ∈ N ,arising from geometric quantization using the complex structure J σ . These spaces sit in-side the prequantum space H ( k ) = C ∞ ( M, L k ) of smooth sections, and as σ ∈ T varies,they form a sub-bundle of the trivial bundle ˆ H ( k ) = T × H ( k ) . As proved in [And5], thissub-bundle is preserved by the explicitly given connection, ∇ V = ∇ T V + 14 k + 2 n (∆ G ( V ) + 2 ∇ G ( V ) · dF − λV ′ [ F ]) + V ′ [ F ] , where ∆ G ( V ) is a second-order operator with symbol G ( V ), and F ∈ C ∞ ( T × M ) isthe Ricci potential, expressing the relation between the Ricci form ρ and the symplecticform through ρ = λω + 2 i∂ ¯ ∂F . This connection will be called the Hitchin connection,and generalizes the connections studied by [Hit3] and [ADPW] in the setting of quantumChern-Simons theory for compact gauge group.Inspired by the work of Witten [Wit3], we may also use the family of K¨ahler structuresto define another connection on ˆ H ( k ) by the expression,˜ ∇ V = ∇ T V + 12 t (∆ G ( V ) +2 ∇ G ( V ) · dF − λV ′ [ F ]) − t (∆ ¯ G ( V ) +2 ∇ ¯ G ( V ) · dF − λV ′′ [ F ])+ V [ F ] , for any complex number t ∈ C with real part equal to k . We shall refer to this as theHitchin-Witten connection at level k . Of course it also depends on the imaginary part of t ,so for each level k we get family of theories, parametrized by one real parameter.Unlike the Hitchin connection, the Hitchin-Witten connection will not preserve thesub-bundle of quantum spaces. Both connections do, however, share another meritoriousfeature, captured by the main theorem of the paper. Theorem 1.1.
If all complex structures in the family have zero-dimensional symmetrygroup, then both the Hitchin and Hitchin-Witten connections are projectively flat.
Since the moduli spaces have no holomorphic vector fields, which is equivalent to thecondition stated in the theorem, we get the following immediate corollary
Theorem 1.2.
The Hitchin and Hitchin-Witten connections for the moduli spaces of flatconnections on a closed oriented surface are projectively flat.
This result on the Hitchin connection, for the moduli spaces of flat SU( n ) connections,is due to Hitchin, but his proof uses algebraic geometric properties of these moduli spaces.As mentioned above, Witten gave an infinite-dimensional gauge-theoretic argument for thisresult on the Hitchin-Witten connection, for the same moduli spaces. In this paper, weprovide a purely differential geometric finite-dimensional argument for projective flatness,which applies in the more general setting we have described.As explained above, the quantum representation of the mapping class groups from theReshetikhin-Turaev TQFT for SU( n ) coincides with the representation obtained from theHitchin connection. We expect that the same will hold for the Hitchin-Witten connectionand quantum Chern-Simons theory for the complex gauge group SL( n, C ). In the paper[And7], the first author has computed explicitly the resulting representation of the mappingclass group for genus 1, the gauge group SL(2 , C ) and all integer levels k .3nly recently has the whole TQFT been rigorously constructed by the first author ofthis paper and Kashaev in [AK1] and [AK2] for the case of PSL(2 , C ) and level k = 1,using quantum Teichm¨uller theory and the Faddeev quantum dilogarithm. In the paper[AK3], the same authors have constructed quantum Chern-Simons theory for PSL(2 , C )and all non-negative integer levels k and further understood how it relates to the geometricquantization of the PSL(2 , C )-moduli spaces. In fact, they have proposed a very generalscheme which just requires a Pontryagin self-dual locally compact group, which we expectwill lead to the construction of the SL( n, C ) for all non-negative integer levels k . Thisshould be seen in parallel to the developments on indeces [Gar, DG1], which should berelated to the level k = 0 theory. In the physics literature, the complex quantum Chern-Simons theory has been discussed from a path integral point of view in a number of papers[Dim1, DGG, DGLZ, DG2, GM, Guk, Hik1, Hik2, Wit4, BNW] and latest by Dimofte[Dim2] using the more advanced 3d-3d correspondence. Outline
Let us briefly outline the organization of the paper. Section 2 discusses general aspects offamilies of K¨ahler structures on a fixed symplectic manifold. In particular, a rather seriousholomorphicity condition, called rigidity (Definition 2.4), on the infinitesimal deformationof the K¨ahler structure will be discussed. We derive a symmetry result (Proposition 2.6) forcertain tensor fields associated with such families, which will prove crucial in the calculationof the curvature of the Hitchin connection.In section 3, we briefly recall the basics of geometric quantization, described as a twostage process where the prequantum space H ( k ) is first constructed as sections of a linebundle, and the quantum space H ( k ) is then defined as the subspace of polarized sectionswith respect to some choice of auxiliary K¨ahler polarization. One way of understandingthe influence of this choice on quantization is by studying the infinitesimal behaviourof a family of such polarizations, and this is exactly the approach employed with theHitchin connection, which relates the quantum spaces through parallel transport. Followingthe initial discussion, we calculate the commutators of general second-order differentialoperators acting on the prequantum spaces. These results will be useful when calculatingthe curvature of the Hitchin connection.The Hitchin connection is the subject of section 4, which contains the main results.After briefly reviewing the differential geometric construction of the Hitchin connection, wemove on to the straightforward but rather lengthy calculation of its curvature, culminatingwith the first major result in Theorem 4.8. The fact that the curvature acts as a differentialoperator of order at most one is a crucial point. Building on these computations andinspiration from Witten’s work on quantum Chern-Simons theory for complex gauge group[Wit3], we then consider the Hitchin-Witten connection defined in (44) as well as above.The second major result is the calculation of its curvature in Theorem 4.10. The expressionsturn out to be essentially equal to the curvature of the Hitchin connection in Theorem 4.8,and in particular it acts as a differential operator of order at most one. This property,shared by both connections, entails projective flatness if, for instance, the family of complexstructures does not admit holomorphic vector fields. This is the content of Theorem 4.11,leading ultimately to Theorem 1.1. 4he final section applies the results to quantum Chern-Simons theory, both for com-pact and complex gauge groups. The Hitchin connection was originally studied in thissetting, with compact gauge group SU( n ), by Hitchin [Hit3] and Axelrod, Della Pietra andWitten [ADPW]. Our results provide another proof of projective flatness, which still relieson the absence of holomorphic vector fields, but uses general properties of rigid families ofK¨ahler structures to prove the vanishing of higher-order symbols. As mentioned, the caseof complex gauge group was studied by Witten in [Wit3], where he used a real polarizationto reduce the quantum space from complex to unitary connections. The real polarization,and hence the reduction, depends on the conformal structure on the surface, and Wittenderived a formula for an analogue of the Hitchin connection in this model, arriving atexactly the expression (44). We recall the necessary theory and connect it with the re-sults of previous sections, providing a differential geometric and purely finite-dimensionalconstruction of a projectively flat connection, the Hitchin-Witten connection, in Witten’smodel of quantum Chern-Simons theory with complex gauge group. Before we recall the construction of the Hitchin connection and calculate its curvature,we will explore the properties of families of K¨ahler structures on a symplectic manifold.Such families are central to the notion of a Hitchin connection, and the results obtainedwill play a fundamental role in subsequent parts. The section serves to introduce notation,establish conventions and provide a number of basic results for later reference. The resultis somewhat lengthy and can be read swiftly on first reading.Let (
M, ω ) be a symplectic manifold. If T is a manifold, we say that a smooth map, J : T → C ∞ ( M, End(
T M )) , is a family of K¨ahler structures on ( M, ω ) if it defines an integrable and ω -compatiblealmost complex structure for every point σ ∈ T . Smoothness of J means that it defines asmooth section of the pullback bundle π ∗ M End(
T M ) over
T × M , where π M : T × M → M denotes the projection.For any point σ ∈ T , the almost complex structure J σ induces a splitting, T M C = T ′ M σ ⊕ T ′′ M σ , of the complexified tangent bundle of M into the two eigenspaces of J σ , with associatedsubspace projections π , σ : T M C → T ′ M σ and π , σ : T M C → T ′′ M σ , explicitly given by π , σ = (Id − iJ σ ) and π , σ = (Id + iJ σ ) . We denote by X = X ′ σ + X ′′ σ the associated splitting of a vector field X on M σ . In general,the subscript σ indicates dependence on the complex structure, but we shall typicallyommit it when the dependence is obvious and the formula is valid for any point in T .The K¨ahler metric associated with the complex structure J is given by g = ω · J, g and ω are denoted by ˜ g and˜ ω , respectively, and they are the unique symmetric, respectively anti-symmetric, bivectorfields satisfying g · ˜ g = ˜ g · g = Id and ω · ˜ ω = ˜ ω · ω = Id . As above, a dot will be used to denote contraction of tensors, and the placement of thetensors relative to each other indicates which entries to contract. We will, however, alsoencounter more complicated expressions, where the entries to be contracted cannot beindicated by simply placing the tensors next to each other. In such cases, we will useabstract indices to denote the entries of each tensor. The indices only name the entries ofa tensor, and do not represent a choice of local coordinates. As usual, subscript indicesrefer to covariant entries of a tensor, whereas superscript indices refer to contravariantentries, and following the Einstein convention, repeated indices will indicate contraction.If the two contracted indices are both either subscript or superscript, the K¨ahler metric isused for contraction. With these conventions, the above identities become g ab = ω au J ub and g au ˜ g ub = ω au ˜ ω ub = Id ba . In a few places, we will also need to apply the projections π , and π , to the entries ofthe tensor. In the index notation, composition with π , will be indicated by a prime onthe index, whereas composition with π , will be indicated by two primes. As an examplewe can write ω ab = ω a ′ b ′′ + ω a ′′ b ′ for the symplectic form, which is of type (1,1) with respect to a compatible complexstructure.Associated with the K¨ahler metric g , we have the Levi-Civita connection ∇ g , and asusual its curvature is defined by R ( X, Y ) Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [ X,Y ] Z, for any vector fields X, Y and Z on M . In abstract index notation, the curvature writes R dabc , and we can use the metric to lower the upper index and get the curvature tensor R abcd = R rabc g rd . The Ricci curvature is the symmetric J -invariant tensor r defined by r ( X, Y ) = Tr( Z R ( Z, X ) Y ) , and its corresponding skew-symmetric two-form is the Ricci form ρ = J · r , which wouldcorrespond to r ab = R uuab = R uabu = R auub and ρ ab = J ua r ub = 12 R abuv ˜ ω uv in index notation. Finally, the scalar curvature s is the metric trace of the Ricci curvature s = r uu = r uv ˜ g uv = ρ uv ˜ ω vu . E is a vector bundle over M and D ∈ D ( M, E ) is a differential operator of orderat most n , we can assign the principal symbol σ P ( D ) ∈ C ∞ ( M, S n ( T M )), which is asymmetric section of the n ’th tensor power of the tangent bundle. If the principal symbolvanishes, then D is of order at most n −
1. In general, there is no good notion of lower ordersymbols of differential operators, but a connection on E and can be combined with theLevi-Civita connection on M to define symbols of all orders. For vector fields X , . . . , X n on M , we consider the inductively defined differential operator on sections of E , ∇ nX ,...,X n s = ∇ X ∇ n − X ,...,X n s − X j ∇ n − X ,..., ∇ X X j ,...,X n s, (1)with the obvious induction start given by the covariant derivative. It is easily verified thatthis expression is tensorial in the vector fields, so we get a map ∇ n : C ∞ ( M, T M n ) → D ( M, E ) . For any tensor field T n ∈ C ∞ ( M, T M n ), the symbol of ∇ nT n is given by the symmetrization S ( T n ) ∈ C ∞ ( M, S n ( T M )) of T n . If D ∈ D ( M, E ) is an operator, of order at most n , withprincipal symbol σ P ( D ) = S n ∈ C ∞ ( M, S n ( T M )), then the operator D − ∇ nS n is of orderat most n −
1, since its principal symbol vanishes. Inductively, it follows that the operator D can be written uniquely in the form D = ∇ nS n + ∇ n − S n − + · · · + ∇ S + S , (2)where S d ∈ C ∞ ( M, S d ( T M )) is called the symbol of order d and gives rise to a map σ d : D ( M, E ) → C ∞ ( M, S d ( T M )) . Any finite order differential operator on E is uniquely determined by the values of thesesymbol maps. In fact, through the expression (2), a choice of symbols specifies a differentialoperator on any vector bundle with connection, and in particular on functions.We will also need the notion of divergence of vector fields and more general contravarianttensors. Recall that the divergence δX of a vector field X on M is defined in terms ofthe Lie derivative and volume form by the equation L X ω m = ( δX ) ω m . Although thedivergence of a vector field only depends on the symplectic volume, and not on the K¨ahlermetric itself, a simple computation reveals that the divergence can be calculated using theLevi-Civita connection by the formula δX = Tr ∇ X = ∇ a X a , (3)in which the independence of the K¨ahler structure is perhaps not so evident. The Laplace-de Rham operator on functions can be expressed in terms of the divergence by∆ f = − iδX ′ f , (4)where X ′ f = ¯ ∂f · ˜ ω denotes the (1,0)-part of the Hamiltonian vector field associated withthe function f ∈ C ∞ ( M ).The formula (3) generalizes to tensors of higher degree. For vector fields X , . . . , X n om M , we define δ ( X (cid:15) · · · (cid:15) X n ) = δ ( X ) X (cid:15) · · · (cid:15) X n + X j X (cid:15) · · · (cid:15) ∇ X X j (cid:15) · · · (cid:15) X n . δ : C ∞ ( M, T M n ) → C ∞ ( M, T M n − ), also called the divergence, whichdoes depend on the K¨ahler structure.The generalization of divergence to sections of the endomorphism bundle of the tangentbundle will also be convenient. If α ∈ Ω ( M ) is a one-form and X is a vector field, wedefine δ ( X (cid:15) α ) = δ ( X ) α + ∇ X α, which gives a map δ : C ∞ ( M, End(
T M )) → Ω ( M ).Finally, for any bivector field B ∈ C ∞ ( M, T M ), we introduce the second-order differ-ential operator, ∆ B = ∇ B + ∇ δB , which will appear repeatedly throughout the paper. Infinitesimal Deformations
For a smooth family of K¨ahler structures, we can take its derivative along a vector field V on T to obtain a map V [ J ] : T → C ∞ ( M, End(
T M )) . Differentiating the identity J = − Id, we see that V [ J ] and J anti-commute, V [ J ] J + JV [ J ] = 0 , (5)so V [ J ] σ interchanges types on the K¨ahler manifold M σ . Therefore, it splits as V [ J ] = V [ J ] ′ + V [ J ] ′′ , (6)where V [ J ] ′ σ ∈ C ∞ ( M, T ′ M σ (cid:15) T ′′ M ∗ σ ) and V [ J ] ′′ σ ∈ C ∞ ( M, T ′′ M σ (cid:15) T ′ M ∗ σ ) is its conjugate.Notice that this splitting occurs for any infinitesimal deformation of an almost complexstructure on M and defines an almost complex structure on the space of almost complexstructures on M .Differentiating the integrability condition on J , expressed through the vanishing of theNijenhuis tensor, reveals that V [ J ] ′ ∈ Ω , ( M, T ′ M ) satisfies the holomorphicity condition¯ ∂V [ J ] ′ = 0, and the associated cohomology class in H ( M, T ′ M ) is the Kodaira-Spencerclass of the deformation (see [Kod]).Define a bivector field ˜ G ( V ) ∈ C ∞ ( M, T M C (cid:15) T M C ) by the relation V [ J ] = ˜ G ( V ) · ω, for any vector field V on T . Differentiating the identity ˜ g = − J · ˜ ω along V , we get V [˜ g ] = − V [ J ] · ˜ ω = − ˜ G ( V ) , (7)and since ˜ g is symmetric, this implies that ˜ G ( V ) is a symmetric bivector field. Furthermore,the combined types of V [ J ] and ω yield a decomposition,˜ G ( V ) = G ( V ) + ¯ G ( V ) , G ( V ) σ ∈ C ∞ ( M, S ( T ′ M σ )) and ¯ G ( V ) σ ∈ C ∞ ( M, S ( T ′′ M σ )). In other words, thereal symmetric bivector field ˜ G ( V ) has no (1,1)-part. The variation of the K¨ahler metricis obtained by differentiating the identity g = ω · J , which yields V [ g ] = ω · V [ J ] = ω · ˜ G ( V ) · ω = g · ˜ G ( V ) · g. We shall also need the variation of the Levi-Civita connection, which is the tensor field V [ ∇ g ] ∈ C ∞ ( M, S ( T M ∗ ) (cid:15) T M )given by (see [Bes] Theorem 1.174)2 g ( V [ ∇ g ] X Y, Z ) = ∇ X ( V [ g ])( Y, Z ) + ∇ Y ( V [ g ])( X, Z ) − ∇ Z ( V [ g ])( X, Y ) , for any vector fields X, Y and Z on M . In index notation, this translates to2 V [ ∇ g ] cab = ∇ a ˜ G ( V ) cu g ub + g au ∇ b ˜ G ( V ) uc − g au ˜ g cw ∇ w ˜ G ( V ) uv g vb , (8)and we remark that the trace V [ ∇ g ] xxb of this tensor vanishes. Indeed, we get that V [ ∇ g ] xxb = ∇ x ˜ G ( V ) xu g ub + g xu ∇ b ˜ G ( V ) ux − g xu ˜ g xw ∇ w ˜ G ( V ) uv g vb = 0 , (9)where the first and last term cancel, and the middle term vanishes because ˜ G ( V ) has nopart of type (1,1), which is the type of the metric.We will also need to know the variation of the Ricci curvature, which will result from theBianchi identity of a certain line bundle associated with the family of complex structures. The Canonical Line Bundle of a Family
For a family J of K¨ahler structures, we can consider the vector bundle,ˆ T ′ M → T × M, with fibers ˆ T ′ M ( σ,p ) = T ′ p M σ given by the holomorphic tangent spaces of M . Throughoutthe paper, we shall generally use a hat in the notation to indicate that we are workingover the product T × M . Following this convention, the exterior differential on T × M isdenoted by ˆ d , whereas the differential on T is denoted by d T and by d on M .The K¨ahler metric induces a Hermitian structure ˆ h T ′ M on ˆ T ′ M , and the Levi-Civitaconnection gives a compatible partial connection along the directions of M . We can extendthis partial connection to a full connection ˆ ∇ T ′ M on ˆ T ′ M in the following way. If Z ∈ C ∞ ( T × M, ˆ T ′ M ) is a smooth family of sections of the holomorphic tangent bundle, and V is a vector field on T , then we defineˆ ∇ V Z = π , V [ Z ] . In other words, we regard Z as a smooth family of sections of the complexified tangentbundle T M C , and then we simply differentiate Z along V in this bundle, which does notdepend on the point in T , and project the result back onto the holomorphic tangent bundle.9learly, the connection ˆ ∇ T ′ M preserves the Hermitian structure in the directions of M ,since it is induced by the Levi-Civita connection. Moreover, if V is a vector field on T ,and X and Y are sections of ˆ T ′ M , we get that V [ˆ h T ′ M ( X, Y )] = V [ g ( X, ¯ Y )] = V [ g ]( X, ¯ Y ) + g ( V [ X ] , ¯ Y ) + g ( X, V [ Y ])= h ( ˆ ∇ V X, Y ) + h ( X, ˆ ∇ V Y ) , since the (1 , V [ g ] vanishes. It follows that ˆ ∇ T ′ M preserves the Hermitian struc-ture on ˆ T ′ M .Now consider the line bundleˆ K = V m ˆ T ′ M ∗ → T × M, which will be referred to as the canonical line bundle of the family of K¨ahler structures. Asusual, the Hermitian structure and connection on ˆ T ′ M induce a Hermitian structure ˆ h K and a compatible connection ˆ ∇ K on ˆ K . The curvature of ˆ ∇ K was calculated in [AGL] andwill be recalled below, but before stating it, we introduce the following important notation.For any vector fields V and W on T , we define Θ ∈ Ω ( T , S ( T M )) byΘ(
V, W ) = S ( ˜ G ( V ) · ω · ˜ G ( W )) , where S denotes symmetrization. We also give a name to the metric trace of the symmetricbivector field Θ( V, W ) and define θ ( V, W ) = − g (Θ( V, W )) = − g uv Θ( V, W ) uv . (10)Clearly, this defines a real two-form θ ∈ Ω ( T , C ∞ ( M )) on T with values in smoothfunctions on M .We note that the two-form Θ over T is exact. To see this, we take the variation of G ( V ) = π , ( ˜ G ( V )) = ( π , (cid:15) π , ) ˜ G ( V ) along W to get2 W [ G ( V )] = iG ( V ) · ω · ¯ G ( W ) − i ¯ G ( W ) · ω · G ( V ) − π , ( W V [˜ g ])= 2 i S ( G ( V ) · ω · ¯ G ( W )) − π , ( W V [˜ g ]) , (11)which in turn shows that V [ G ( W )] − W [ G ( V )] = − i S ( ¯ G ( V ) · ω · G ( W )) − i S ( G ( V ) · ω · ¯ G ( W )) = − i Θ( V, W ) , (12)for commuting vector fields V and W on T . This can be rephrased as d T G = − i Θ , (13)where G is viewed as a one-form in Ω ( T , S ( T ′ M )).The following proposition gives the curvature of the canonical line bundle of a familyof K¨ahler structures and is proved in [AGL]. Proposition 2.1.
The curvature of ˆ ∇ K is given by F ˆ ∇ K ( X, Y ) = iρ ( X, Y ) , F ˆ ∇ K ( V, X ) = i δ ˜ G ( V ) · ω · X, F ˆ ∇ K ( V, W ) = iθ ( V, W ) , for any vector fields X, Y on M and V, W on T .
10y applying the Bianchi identity to the connection ˆ ∇ K , and using the formulas ofProposition 2.1, we get three useful results. The first is the fact that the two-form θ ∈ Ω ( T , C ∞ ( M )) is closed, which is a trivial reformulation of the Bianchi identity for threevector fields on T . By applying the Bianchi identity to two vector fields V and W on T ,and one vector field on M , we get dθ ( V, W ) = 12 W [ δ ˜ G ( V )] · ω − V [ δ ˜ G ( W )] · ω. (14)Finally, the Bianchi identity for two vector fields on M and one on T gives followingimportant formula for the variation of the Ricci form, V [ ρ ] = 12 d ( δ ˜ G ( V ) · ω ) , (15)for any vector field V on T . As an immediate consequence of (15), we get the followingsimple formula for the variation of the scalar curvature V [ s ] = V [ ρ ] uv ˜ ω vu = 12 ∇ u ( δ ˜ G ( V )) w ω wv ˜ ω vu − ∇ v ( δ ˜ G ( V )) w ω wu ˜ ω vu = δδ ˜ G ( V ) , (16)for any vector field V on T . Holomorphic Families of K¨ahler Structures
In case the manifold T is itself a complex manifold, we can require the family J to be aholomorphic map from T to the space of complex structures. This is made precise by thefollowing definition, which uses the splitting (6) of V [ J ]. Definition 2.2.
Suppose that T is a complex manifold, and that J is a family of complexstructures on M , parametrized by T . Then J is holomorphic if V ′ [ J ] = V [ J ] ′ and V ′′ [ J ] = V [ J ] ′′ , for any vector field V on T . If I denotes the integrable almost complex structure on T induced by its complexstructure, then we get an almost complex structure ˆ J on T × M defined byˆ J ( V ⊕ X ) = IV ⊕ J σ X, V ⊕ X ∈ T ( σ,p ) ( T × M ) . The following proposition gives another characterization of holomorphic families [AGL].
Proposition 2.3.
The family J is holomorphic if and only if ˆ J is integrable. By this proposition, a holomorphic family induces a complex structure on the productmanifold
T × M . Clearly, the projection π T : T × M → T is a holomorphic map, andits differential is the projection dπ T : ˆ T ′ T ⊕ ˆ T ′ M → T ′ T , where ˆ T ′ T is the pullback of T ′ T by π T . Since the bundle ˆ T ′ M over T × M is the kernel of this map, it has thestructure of a holomorphic vector bundle, and it is easily verified that the connectionˆ ∇ T ′ M is compatible with this holomorphic structure. Since the connection also preservesthe Hermitian structure, it must be the Chern connection.11olomorphicity has several useful consequences. First of all, it implies that˜ G ( V ′ ) = V ′ [ J ] · ˜ ω = V [ J ] ′ · ˜ ω = G ( V ) , and similarly ˜ G ( V ′′ ) = ¯ G ( V ). This means that G , viewed as a one-form over the complexmanifold T , has type (1,0) and that Θ and θ have type (1,1). These important facts willbe used without reference going forward.Finally, for commuting vector fields V ′ and W ′′ , the identity (12) reduces to W ′′ [ G ( V )] = i G ( V ) · ω · ¯ G ( W ) − i G ( W ) · ω · G ( V ) = i Θ( V ′ , W ′′ ) . (17)This expression for the second-order variation of the complex structure will prove veryuseful in later calculations, but we emphasize the fact that it only holds for commutingvector fields. Rigid Families of K¨ahler Structures
The following rather serious assumption on a family of K¨ahler structures turns out tobe crucial to the construction of the Hitchin connection as well as the calculation of itscurvature.
Definition 2.4.
A family of K¨ahler structures is called rigid if ∇ X ′′ G ( V ) = 0 , (18) for all vector fields V on T and X on M . In other words, the family J is rigid if G ( V ) is a holomorphic section of S ( T ′ M ), forany vector field V on T . For examples of rigid families in a basic setting, we refer to [AGL].By differentiating the rigidity condition (18) along T , we get the following crucial result. Proposition 2.5.
Any rigid family of K¨ahler structures satisfies the symmetry property S ( G ( V ) ·∇ G ( W )) = S ( G ( W ) ·∇ G ( V )) , (19) for any vector fields V and W on T .Proof. Throughout the proof, let V and W be commuting vector fields on T . By differen-tiaing the holomorphicity condition on the bivector field G ( V ) along W , we obtain0 = W [ ∇ a ′′ G ( V ) bc ]= W [ π , ] ua ∇ u G ( V ) bc + ∇ a ′′ W [ G ( V )] bc + W [ ∇ ] ba ′′ u G ( V ) uc + W [ ∇ ] ca ′′ u G ( V ) bu . (20)Let us work out each of these terms individually. Using rigidity, the first term reduces to2 W [ π , ] ua ∇ u G ( V ) bc = i ˜ G ( W ) uv ω va ∇ u G ( V ) bc = − g av G ( W ) uv ∇ u G ( V ) bc . ∇ a ′′ ( W [ G ( V )] bc )= −∇ a ( ¯ G ( W ) bu ) g uv G ( V ) vc − G ( V ) bu g uv ∇ a ( ¯ G ( W ) vc ) − ∇ a ′′ (( π , ) bcuv W V [˜ g uv ]) . Finally, for the last two terms of (20), we can apply (8) in combination with rigidity of thefamily to obtain2 W [ ∇ ] ba ′′ u G ( V ) uc = g av ∇ u ( G ( W ) vb ) G ( V ) uc + ∇ a ( ¯ G ( W ) bv ) g vu G ( V ) uc . Substituting these expressions into (20), and cancelling terms, we get that0 = g av ∇ u ( G ( W ) vb ) G ( V ) uc + g av ∇ u ( G ( W ) vc ) G ( V ) ub − g av G ( W ) uv ∇ u G ( V ) bc − ∇ a ′′ (( π , ) bcuv W V [˜ g uv ]) . Raising an index, this implies that3 S ( G ( V ) ·∇ G ( W )) abc = G ( V ) au ∇ u G ( W ) bc + G ( W ) au ∇ u G ( V ) bc + ˜ g aw ∇ w ′′ (( π , ) bcuv W V [˜ g uv ]) , and clearly this is symmetric in V and W because these were chosen to commute. Thisproves the proposition.Repeatedly taking the divergence of the identity (19) yields similar identities for con-travariant tensors of lower degree. These will play a crucial role in the calculation of thecurvature of the Hitchin connection, so we introduce special notation for them. Inspiredby Proposition 2.5, we define a two-form Γ on T , with values in sections of the thirdsymmetric power of the holomorphic tangent bundle over M , byΓ ( V, W ) = S ( G ( V ) ·∇ G ( W )) , for any vector fields V and W on T . Then Proposition 2.5 ensures that Γ actually definesa symmetric two-form on T .Clearly, the symmetry of the two-form over T is not affected by taking the divergenceof the tri-vector field part of Γ over M . In other words, δ Γ defines a symmetric two-formon T with values in the second symmetric power of the holomorphic tangent bundle on M . It is given by3 δ Γ ( V, W ) = ∆ G ( V ) G ( W ) + 2 S ( G ( V ) ·∇ δG ( W )) + 2 S ( ∇ u G ( V ) av ∇ v G ( W ) ub ) , for any vector fields V and W on T . In contrast to the first two terms of this ex-pression, the last term is obviously symmetric in V and W . This leads us to defineΓ ∈ Ω ( T , C ∞ ( M, S ( T ′ M )) byΓ ( V, W ) = ∆ G ( V ) G ( W ) + 2 S ( G ( V ) ·∇ δG ( W )) , which is also a symmetric two-form on T and encodes the interesting part of the symmetrystatement for δ Γ . 13epeatedly taking the divergence, and removing obviously symmetric parts, we get thefollowing important proposition Proposition 2.6.
The four two-forms Γ j on T , with values in symmetric contravarianttensors on M , defined by Γ ( V, W ) = S ( G ( V ) ·∇ G ( W ))Γ ( V, W ) = 2 S ( G ( V ) ·∇ δG ( W )) + ∆ G ( V ) G ( W )Γ ( V, W ) = 2∆ G ( V ) δG ( W ) + G ( V ) · dδδG ( W ) + ∇ w ( G ( V ) uv ) ∇ uv ( G ( W ) wa )Γ ( V, W ) = ∆ G ( V ) δδG ( W ) + ∇ w ( G ( V ) uv ) ∇ uv δG ( W ) w , are all symmetric in the vector fields V and W on T . Families of Ricci Potentials
Suppose that the first Chern class of (
M, ω ) is represented by the symplectic form, that is c ( M, ω ) = λ [ ω π ] , (21)for some integer λ ∈ Z . Now the first Chern class is also represented by the Ricci form ρ π , so the difference between the forms ρ and λω is exact. A smooth real function F ∈ C ∞ ( T × M ), which can be viewed as a smooth map F : T → C ∞ ( M ), is called a familyof Ricci potentials if it satisfies ρ σ = λω + 2 i∂ σ ¯ ∂ σ F σ , (22)for every point σ ∈ T .Any two Ricci potentials differ by a global pluriharmonic real function. If we assumethat H ( M, R ) vanishes, such a function is globally the real part of a holomorphic function,so a family of Ricci potential is then uniquely determined up to a function on T if thefamily of K¨ahler structures does not allow any non-constant holomorphic functions on M .The existence of a Ricci potential is clearly a global issue over M , as the local ∂ ¯ ∂ -lemma ensures local existence around any point on M when (21) holds. If the manifold M is compact, the global ∂ ¯ ∂ -lemma from Hodge theory ensures the existence of a Riccipotential, using the fact that the K¨ahler form ω is harmonic. In this case, the Riccipotential is certainly unique up to a function on T , and by imposing zero average over M , Z M F ω m = 0 . (23)we can fix it uniquely.The following proposition gives an important identity, involving the variation of a familyof Ricci potentials. Proposition 2.7.
Suppose that M is a symplectic manifold with H ( M, R ) = 0 and c ( M, ω ) = λ [ ω π ] , and let J be a holomorphic family of K¨ahler structures on M , none ofwhich admit non-constant holomorphic functions on M . Then i ¯ ∂V ′ [ F ] = δG ( V ) · ω + 2 dF · G ( V ) · ω, (24) for any family of Ricci potentials F and any vector field V on T . roof. By differentiating the identity (22) in the direction of V ′ , we get V ′ [ ρ ] = − d ( dF · G ( V ) · ω ) + 2 i∂ ¯ ∂V ′ [ F ] , and by using (15) on the left-hand side, this yields d ( δG ( V ) · ω ) + 2 d ( dF · G ( V ) · ω ) − id ¯ ∂V ′ [ F ] = 0 . On one hand, it follows that the one-form δG ( V ) · ω + 2 dF · G ( V ) · ω − i ¯ ∂V ′ [ F ]is closed, and hence exact by the assumption H ( M, R ) = 0. On the other hand, it is oftype (0,1), so it cannot be exact unless it is zero, because we assumed that none of theK¨ahler structures admit non-constant holomorphic functions. This proves the lemma.Another equivalent form of (24) is the following,4 iV ′ [ ¯ ∂F ] = δG ( V ) · ω, and if we combine this with (14), for commuting vector fields V ′ and W ′′ on T , we get2 dθ ( V ′ , W ′′ ) = W ′′ [ δG ( V ) · ω ] − V ′ [ δ ¯ G ( W ) · ω ]= 4 iW ′′ V ′ [ ¯ ∂F ] − iV ′ W ′′ [ ∂F ]= 4 iV ′ W ′′ [ dF ]= 4 i∂ T ¯ ∂ T F ( V ′ , W ′′ ) . Since the family is holomorphic, the form θ has type (1,1) on T , so we have shown Proposition 2.8.
In the setting of Proposition 2.7, any family of Ricci potentials satisfies θ − i∂ T ¯ ∂ T F ∈ Ω , ( T ) . In other words, the form takes values in constant functions on M . This ends the general discussion of families of K¨ahler structures. In the next section, wediscuss general aspects of geometric quantization, and in particular the need for a choice ofauxiliary polarization in the construction. Understanding the effects of this choice naturallyleads us to consider families of K¨ahler structures, and ultimately to the Hitchin connectionrelating the different choices. The results from this section will play a fundamental role inthe discussion.
In very broad terms, geometric quantization concerns the passage from classical mechanicsto quantum mechanics. It aims to produce a Hilbert space of quantum states from aclassical phase space, in the form of a symplectic manifold, and a self-adjoined operatorfrom a classical observable, in the form of a function on the classical phase space.15n the following, we shall briefly review the basic notions from geometric quantizationrelevant to us. A key role is played by an auxiliary choice of polarization, which is oftenchosen to be K¨ahler. For a broader treatment of geometric quantization, the reader isreferred to [Woo] and [AE].After reviewing the elements of geometric quantization, we calculate the commutatorsof certain differential operators acting on the prequantum spaces. These will be relevantin later discussions of the Hitchin connection and its curvature.
Prequantization
In geometric quantization, the Hilbert space of quantum states arises as sections of acertain Hermitian line bundle over the classical phase space. As a model for this classicalphase space, we consider a symplectic manifold (
M, ω ) of dimension 2 m . A prequantumline bundle over the M is a complex line bundle L endowed with a Hermitian metric h anda compatible connection ∇ of curvature F ∇ = − iω. A symplectic manifold admitting a prequantum line bundle is called prequantizable.Evidently, this is not the case for every symplectic manifold. Indeed, the real first Chernclass of a prequantum line bundle is given by c ( L ) = (cid:2) ω π (cid:3) , leading us to the followingnecessary condition for prequantizability, called the prequantum condition, (cid:2) ω π (cid:3) ∈ Im (cid:0) H ( M, Z ) → H ( M, R ) (cid:1) . (25)This is, in fact, also sufficient to ensure the existence of a prequantum line bundle, and theinequivalent prequantum line bundles over M are parametrized by H ( M, U(1)).For any natural number k , called the level , we consider the prequantum space , H ( k ) = C ∞ ( M, L k ) , of smooth sections of the k ’th tensor power of the line bundle L . These sections play therole of wave functions in the quantum theory. If f ∈ C ∞ ( M ) is a function on M , thecorresponding prequantum operator , acting on H ( k ) , is defined by P k ( f ) = ik ∇ X f + f, (26)where X f is the Hamiltonian vector field of the function f . The virtue of (26) is that theprequantum operators satisfy the correspondence principle (cid:2) P k ( f ) , P k ( g ) (cid:3) = ik P k ( { f, g } ) , (27)which is one of the distinctive features of a viable quantization.From a physical perspective, the wave functions in H ( k ) depend on twice the number ofvariables they should. A standard way to remedy this is to pick an auxiliary polarizationon M and consider the space of polarized sections of the line bundle. The polarization canbe by real or complex Lagrangian subspaces. In the following, we will focus on the case ofcomplex K¨ahler polarizations. 16 ¨ahler Quantization From now on, we assume that the symplectic manifold admits a K¨ahler structure, in theform of an integrable almost complex structure J on M which is compatible with thesymplectic structure. Since the K¨ahler form ω has type (1 , L defines a holomorphic structure.Therefore, we can define the quantum space to be the space of holomorphic (or polarized)sections, H ( k ) J = H ( M J , L k ) = { s ∈ H ( k ) | ∇ Z s = 0 , ∀ Z ∈ T ′′ M J } , which is a subspace of the prequantum space H ( k ) of smooth sections. If the manifold M iscompact, then H ( k ) J is a finite-dimensional space by standard theory of elliptic operators.Unfortunalely, the prequantum operators do not in general preserve the space of holo-morphic sections. A function f ∈ C ∞ ( M ) is polarized if the (1,0)-part of its correspondingHamiltonian vector field is holomorphic. In other words, the space of polarized functionsis given by C ∞ J ( M ) = { f ∈ C ∞ ( M ) | ∇ Z X f = 0 , ∀ Z ∈ T ′′ M J } . Now, the operator P k ( f ) preserves H ( k ) J if and only f is a polarized function. In fact,it is easily verified that a first-order differential operator of the form ik ∇ X + f on H ( k ) preserves the subspace H ( k ) J if and only if f is a polarized function and X ′ = X ′ f . Thisgives another justification for the choice of prequantum operators. In fact, the (0,1)-partof the vector field makes no difference to the action of P k ( f ) on H ( k ) J , so if we define apolarized variation of the prequantum operators by P k ′ ( f ) J = ik ∇ X ′ f + f, (28)then these are essentially the only first-order operators with a chance of preserving thesubspace H ( k ) J of H ( k ) . For a real polarized function f ∈ C ∞ J ( M ), the Hamiltonian vec-tor field X f must be Killing for the K¨ahler metric, effectively reducing the quantizableobservables to an at most finite-dimensional, and often trivial, space (see [Woo]).To get more quantizable observables, their quantization is modified in the following way.The space H ( k ) J is in fact a closed subspace of H ( k ) , and therefore we have the orthogonalprojection π ( k ) J : H ( k ) → H ( k ) J . For f ∈ C ∞ ( M ), we then define the corresponding quantumoperator by Q k ( f ) J = π ( k ) J ◦ P k ( f ) . These operators do not form an algebra, but they satisfy a weaker form of (27) (at least if M is compact) in the sense that (cid:13)(cid:13)(cid:13) [ Q k ( f ) , Q k ( g )] − ik Q k ( { f, g } ) (cid:13)(cid:13)(cid:13) = O ( k − ) as k → ∞ , (29)with respect to the operator norm on H ( k ) J . The proof of (29) relies on the fact that theseoperators are Toeplitz operators (see [BMS]).17lthough this quantization scheme gives a Hilbert space of the right size, it still fails toproduce the right answers on basic examples from quantum mechanics. In the end, whatreally matters is the spectrum of the operators, and if the above procedure is applied to theone-dimensional harmonic oscillator, the quantization yields a spectrum which differs fromthe correct one by a shift. To deal with this problem, the so-called metaplectic correctioncan be introduced. We shall not pursue this direction further in the present paper, butrefer the interested reader to [AGL], where the constrution of the Hitchin connection inthe metaplectic setting is discussed. Commutators of Differential Operators
The description of the Hitchin connection, and the calculation of its curvature in particular,requires the calculation of commutators of a number of differential operators on sectionson the prequantum line bundle L and its tensor powers. Since the commutators of evensecond-order operators are quite complicated, it will be convenient to encode the operatorsthrough their symbols, as in (2), and have general but explicit descriptions of the symbolsof the commutators in terms of the symbols of the commutants. The followings lemmasgive exactly such descriptions.The very definition of the curvature of the line bundle L k implies the basic relation[ ∇ X , ∇ Y ] s = ∇ [ X,Y ] s − ikω ( X, Y ) s for any vector fields X, Y on M and any smooth section s ∈ H ( k ) . Things become a littlemore complicated when second-order operators are introduced. Lemma 3.1.
For any K¨ahler structure on M , any vector field X on M , and any symmetricbivector field B ∈ C ∞ ( M, S ( T M )) , we have the symbols σ (cid:2) ∇ B , ∇ X (cid:3) = 2 S ( B ·∇ X ) − ∇ X Bσ (cid:2) ∇ B , ∇ X (cid:3) = ∇ B X − ikB · ω · X + B uv R awuv X w σ (cid:2) ∇ B , ∇ X (cid:3) = − ik ω ( B ·∇ X ) for the commutator of the operators ∇ X and ∇ B acting on H ( k ) .Proof. By straightforward calculation, we get B uv ∇ uv X x ∇ x s = B uv X x ∇ uvx s + 2 B uv ∇ v ( X x ) ∇ ux s + B uv ∇ uv ( X x ) ∇ x s = B uv X x ∇ xuv s − ikB uv ω vx X x ∇ u s − B uv X x R ruxv ∇ r s + B uv ∇ v ( X x ) ∇ ux s + B uv ∇ v ( X x ) ∇ xu s − ikB uv ω ux ∇ v ( X x ) s + B uv ∇ uv ( X x ) ∇ x s, where we used the fact that the symplectic form ω is parallel with respect to the Levi-Civitaconnection. To get an expression for the desired commutator, we subtract X x ∇ x B uv ∇ uv s = X x ∇ x ( B uv ) ∇ uv s + B uv X x ∇ xuv s, and the stated symbols can easily be extracted from the result.18aturally, things get even more complicated for two second-order operators. Lemma 3.2.
For any K¨ahler structure on M and any symmetric bivector fields A, B ∈ C ∞ ( M, S ( T M )) , we have the symbols σ (cid:2) ∇ A , ∇ B (cid:3) = 2 S ( A ·∇ B ) − S ( B ·∇ A ) σ (cid:2) ∇ A , ∇ B (cid:3) = ∇ A B − ∇ B A − ik S ( A · ω · B )+ 2 S ( A xy R auxy B ub ) − S ( B uv R axuv A xb ) σ (cid:2) ∇ A , ∇ B (cid:3) = − ikA xy ω yu ∇ x ( B ua ) + 2 ikB uv ω vx ∇ u ( A xa ) − A xy ∇ x ( R ayuv ) B uv + B uv ∇ v ( R auxy ) A xy − A xy R axuv ∇ y B uv + 43 B uv R auxy ∇ v A xy σ (cid:2) ∇ A , ∇ B (cid:3) = ik A xy J jy R xuvj B uv − ik B uv J jv R uxyj A xy for the commutator of the operators ∇ A and ∇ B acting on H ( k ) .Proof. Once again, the proof proceeds by straightforward calculation. We get A xy ∇ xy B uv ∇ uv s = A xy B uv ∇ xyuv s + 2 A xy ∇ y ( B uv ) ∇ xuv s + A xy ∇ xy ( B uv ) ∇ uv s. (30)Focusing on the first term, we commute the indices x and y past u and v . A xy B uv ∇ xyuv s = A xy B uv (cid:0) ∇ xuyv − ∇ x R ryuv ∇ r − ikω yu ∇ xv (cid:1) s = A xy B uv (cid:0) ∇ xuvy − ikω yv ∇ xu − ∇ x R ryuv ∇ r − ikω yu ∇ xv (cid:1) s = A xy B uv (cid:0) ∇ uxvy − R rxuv ∇ ry − R rxuy ∇ vr − ikω xu ∇ vy − ikω yv ∇ xu − ∇ x R ryuv ∇ r − ikω yu ∇ xv (cid:1) s = A xy B uv (cid:0) ∇ uvxy − ∇ u R rxvy ∇ r − ikω xv ∇ uy − R rxuv ∇ ry − R rxuy ∇ vr − ikω xu ∇ vy − ikω yv ∇ xu − ∇ x R ryuv ∇ r − ikω yu ∇ xv (cid:1) s, where we used the fact that ω is parallel, but otherwise just added the curvature terms. Ex-panding by the Leibniz rule, using symmetries, and collecting terms, this can be rewrittenas A xy B uv ∇ xyuv s = A xy B uv (cid:0) ∇ uvxy − R rxuv ∇ ry − R rxuv ∇ yr + R ruxy ∇ vr + R ruxy ∇ rv − ikω xu ∇ yv − ikω xu ∇ vy + ∇ u ( R rvxy ) ∇ r − ∇ x ( R ryuv ) ∇ r − ikR ruxy ω vr (cid:1) s, (31)where the order of differentiation was interchanged for the term A xy B uv R rxuy ∇ vr s . Thesecond term of (30) can be rewritten as2 A xy ∇ y ( B uv ) ∇ xuv s = 23 A xy ∇ y ( B uv ) (cid:0) ∇ xuv + 2 ∇ uxv − R rxuv ∇ r − i k ω xu ∇ v (cid:1) s = 23 A xy ∇ y ( B uv ) (cid:0) ∇ xuv + ∇ uxv + ∇ uvx − R rxuv ∇ r − i k ω xu ∇ v (cid:1) s. (32)19nalagous to (30), the other term of the commutator (cid:2) ∇ A , ∇ B (cid:3) yields B uv ∇ uv A xy ∇ xy s = A xy B uv ∇ uvxy s + 2 B uv ∇ v ( A xy ) ∇ uxy s + B uv ∇ uv ( A xy ) ∇ xy s, (33)where the second term can be rewritten as2 B uv ∇ v ( A xy ) ∇ uxy s = 23 B uv ∇ v ( A xy ) (cid:0) ∇ uxy + 2 ∇ xuy s − R ruxy ∇ r − i k ω ux ∇ y (cid:1) s = 23 B uv ∇ y ( A xy ) (cid:0) ∇ uxy + ∇ xuy + ∇ xyu − R ruxy ∇ r − i k ω ux ∇ y (cid:1) s. (34)By subtracting (33) from (30), substituting (32) and (34), and collecting terms by orderof covariant differentiation, one verifies the claimed symbols. In this section, we study the Hitchin connection and calculate its curvature. We startby recalling the differential geometric construction of the Hitchin connection in geometricquantization. The results concerning this construction are all proved in [And5], to whichthe reader is referred for further details.Consider a symplectic manifold (
M, ω ), equipped with a prequantum line bundle L ,and assume that H ( M, R ) = 0 and that the real first Chern class of ( M, ω ) is given by c ( M, ω ) = λ (cid:2) ω π (cid:3) , (35)for some integer λ ∈ Z . Further, assume that M is of K¨ahler type, and let J be a rigidand holomorphic family of K¨ahler structures on ( M, ω ), parametrized by some complexmanifold T . Finally, assume that the family of K¨ahler structures admits a family of Riccipotentials F , and that it does not admit any non-constant holomorphic functions on M .The prequantum space H ( k ) = C ∞ ( M, L k ) forms the fiber of a trivial, infinite-rankvector bundle over T , ˆ H ( k ) = T × H ( k ) . If ∇ T denotes the trivial connection on ˆ H ( k ) , we consider a connection of the form ∇ = ∇ T + a, (36)where a ∈ Ω ( T , D ( M, L k )) is a one-form on T with values in the space of differentialoperators on sections of L k , and we seek an a for which the connection ∇ preserves thequantum subspaces H ( k ) σ = H ( M σ , L k ) of holomorphic sections inside each fiber of ˆ H ( k ) . Definition 4.1. A Hitchin connection on the bundle ˆ H ( k ) is a connection of the form (36) which preserves the fiberwise subspaces H ( k ) .
20t turns out that, with the assumptions made above, an explicit construction of aHitchin connetion can be given. For any vector field V on T , the operator a ( V ) is of ordertwo, with principal symbol G ( V ) and lower order symbols given in terms of G ( V ) and theRicci potential. The precise statement is contained in the following theorem from [And5]. Theorem 4.2.
Let ( M, ω ) be a prequantizable symplectic manifold with H ( M, R ) = 0 and c ( M, ω ) = λ (cid:2) ω π (cid:3) . Further, let J be a rigid, holomorphic family of K¨ahler structureson M , parametrized by a complex manifold T , admitting a family of Ricci potentials F butno non-constant holomorphic functions on M . Then the expression ∇ V = ∇ T V + 14 k + 2 λ (∆ G ( V ) + 2 ∇ G ( V ) · dF + 4 kV ′ [ F ]) defines a Hitchin connection in the bundle ˆ H ( k ) over T . The characterizing feature of the operator-valued one-form a is the fact that it satisfies (cid:2) ∇ , , a ( V ) (cid:3) s = − i ω · G ( V ) ·∇ s, (37)for any section s of ˆ H ( k ) . In fact, this property, and the fact that the Hitchin connectionpreserves the quantum subspaces H ( k ) inside H ( k ) , implies that these subspaces form abundle ˆ H ( k ) over T , and this is part of the statement in Theorem 4.2.Having reviewed the explicit differential geometric construction of a Hitchin connectionin geometric quantization, we turn to the calculation of its curvature. Curvature of the Hitchin Connection
Suppose the assumptions of Theorem 4.2 are satisfied, ensuring the existence of the Hitchinconnection, and let us calculate its curvature. For this calculation, it will be convenient torewrite the Hitchin connection slightly as ∇ V = ∇ T V + 14 k + 2 λ b ( V ) + V ′ [ F ] with b ( V ) = ∆ G ( V ) + 2 ∇ G ( V ) · dF − λV ′ [ F ] , (38)essentially splitting the operator a ( V ) into orders of k . In particular, the one-form b doesnot involve the level k .We shall divide the calculation of the curvature into a number of propositions. Thefirst relies on Lemma 3.1 and Lemma 3.2, in combination with Proposition 2.6, to computea commutator of fundamental importance to the curvature calculation. Proposition 4.3.
For any rigid family of K¨ahler structures, the commutator of ∆ ˜ G ( V ) and ∆ ˜ G ( W ) , acting on H ( k ) , has the symbols σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = 0 σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = − ik Θ( V, W ) σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = ∆ G ( V ) δG ( W ) − ∆ G ( W ) δG ( V ) + ∆ ¯ G ( V ) δ ¯ G ( W ) − ∆ ¯ G ( W ) δ ¯ G ( V ) − ik δ (Θ( V, W )) − δ ( ˜ G ( V ) · r ) · ˜ G ( W ) + δ ( ˜ G ( W ) · r ) · ˜ G ( V ) σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = − ik δδ Θ( V, W ) + ik r (Θ(
V, W )) , for any vector fields V and W on T . roof. We verify that the third-order symbol vanishes. Using Proposition 2.6 and itsconjugated version, which rely heavily on rigidity on the family, we get σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = σ (cid:2) ∇ G ( V ) , ∇ G ( W ) (cid:3) = 2 S ( ˜ G ( V ) ·∇ ˜ G ( W )) − S ( ˜ G ( W ) ·∇ ˜ G ( V ))= 2 S ( G ( V ) ·∇ G ( W )) − S ( G ( W ) ·∇ G ( V ))+ 2 S ( ¯ G ( V ) ·∇ ¯ G ( W )) − S ( ¯ G ( W ) ·∇ ¯ G ( V ))= 2Γ ( V, W ) − ( W, V ) + 2Γ ( V, W ) − ( W, V )= 0 . To calculate the second-order symbol, we first notice that σ (cid:2) ∆ G ( V ) , ∆ G ( W ) (cid:3) = σ (cid:2) ∇ G ( V ) , ∇ G ( W ) (cid:3) + σ (cid:2) ∇ G ( V ) , ∇ δG ( W ) (cid:3) − σ (cid:2) ∇ G ( W ) , ∇ δG ( V ) (cid:3) = ∇ G ( V ) G ( W ) + ∇ δG ( V ) G ( W ) + 2 S ( G ( V ) ·∇ δG ( W )) − ∇ G ( W ) G ( V ) − ∇ δG ( W ) G ( V ) − S ( G ( W ) ·∇ δG ( V ))= Γ ( V, W ) − Γ ( W, V )= 0 . For the mixed-type terms of the second-order symbol, we first observe that G ( V ) ·∇ δ ¯ G ( W ) = G ( V ) xy ∇ y ∇ u ¯ G ( W ) uv = G ( V ) xy R ryru ¯ G ( W ) uv + G ( V ) xy R ryuv ¯ G ( W ) uv = − G ( V ) · r · ¯ G ( W ) + G ( V ) xy R ryuv ¯ G ( W ) uv . (39)Using this, we calculate that σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) = σ (cid:2) ∇ G ( V ) , ∇ G ( W ) (cid:3) + σ (cid:2) ∇ G ( V ) , ∇ δ ¯ G ( W ) (cid:3) − σ (cid:2) ∇ G ( W ) , ∇ δG ( V ) (cid:3) = − ik S ( G ( V ) · ω · ¯ G ( W )) + 2 S ( G ( V ) xy R auxy ¯ G ( W ) ub ) − S ( ¯ G ( W ) uv R axuv G ( V ) xb )+ 2 S ( G ( V ) ·∇ δ ¯ G ( W )) − S ( ¯ G ( W ) ·∇ δG ( V ))= − ik S ( G ( V ) · ω · ¯ G ( W )) , where the last equality follows by inserting (39) and its conjugate. In total, this meansthat the second-order symbol is given by σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = − ik S ( G ( V ) · ω · ¯ G ( W )) − ik S ( ¯ G ( V ) · ω · G ( W ))= − ik S ( ˜ G ( V ) · ω · ˜ G ( W ))= − ik Θ( V, W ) . Next, we find the first-order symbol. First of all, we get σ (cid:2) ∆ G ( V ) , ∆ G ( W ) (cid:3) = ∆ G ( V ) δG ( W ) − ∆ G ( W ) δG ( V ) . M , we get the following computation, σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) = σ (cid:2) ∇ G ( V ) , ∇ G ( W ) (cid:3) + σ (cid:2) ∇ δG ( V ) , ∇ δ ¯ G ( W ) (cid:3) + σ (cid:2) ∇ G ( V ) , ∇ δ ¯ G ( W ) (cid:3) − σ (cid:2) ∇ G ( W ) , ∇ δG ( V ) (cid:3) = − ikG ( V ) xy ω yu ∇ x ( ¯ G ( W ) ua ) + 2 ik ¯ G ( W ) uv ω vx ∇ u ( G ( V ) xa ) − G ( V ) xy ∇ x ( R ayuv ) ¯ G ( W ) uv + ¯ G ( W ) uv ∇ v ( R auxy ) G ( V ) xy − G ( V ) xy R axuv ∇ y ¯ G ( W ) uv + ¯ G ( W ) uv R auxy ∇ v G ( V ) xy − δG ( V ) · r · ¯ G ( W ) + δG ( V ) x R axuv ¯ G ( W ) uv + δ ¯ G ( W ) · r · G ( V ) − δ ¯ G ( W ) u R auxy G ( V ) xy + ∇ G ( V ) δ ¯ G ( W ) − ikG ( V ) · ω · δ ¯ G ( W ) + G ( V ) xy R auxy δ ¯ G ( W ) u − ∇ G ( W ) δG ( V ) + 2 ik ¯ G ( W ) · ω · δG ( V ) − ¯ G ( W ) uv R axuv δG ( V ) x = − G ( V ) xy ∇ x ( R ayuv ) ¯ G ( W ) uv + ¯ G ( W ) uv ∇ v ( R auxy ) G ( V ) xy − δG ( V ) · r · ¯ G ( W ) + δ ¯ G ( W ) · r · G ( V ) − G ( V ) xy ∇ x ( r yu ) ¯ G ( W ) ua + G ( V ) xy ∇ x ( R ayuv ) ¯ G ( W ) uv − ikG ( V ) · ω · δ ¯ G ( W )+ ¯ G ( W ) uv ∇ u ( r vx ) G ( V ) xa − ¯ G ( V ) uv ∇ u ( R avxy ) G ( V ) xy + 2 ik ¯ G ( W ) · ω · δG ( V )= − δG ( V ) · r · ¯ G ( W ) + δ ¯ G ( W ) · r · G ( V ) − G ( V ) xy ∇ x ( r yu ) ¯ G ( W ) ua − ikG ( V ) · ω · δ ¯ G ( W )+ ¯ G ( W ) uv ∇ u ( r vx ) G ( V ) xa + 2 ik ¯ G ( W ) · ω · δG ( V )= − δ ( G ( V ) · r ) · ¯ G ( W )+ δ ( ¯ G ( W ) · r ) · G ( V )+2 ikδ ( ¯ G ( W ) · ω · G ( V )) − ikδ ( G ( V ) · ω · ¯ G ( W ))= − δ ( G ( V ) · r ) · ¯ G ( W ) + δ ( ¯ G ( W ) · r ) · G ( V ) − ikδ S ( G ( V ) · ω · ¯ G ( W )) , where we have indicated cancelling terms. Analagously, we compute σ (cid:2) ∆ ¯ G ( V ) , ∆ G ( W ) (cid:3) = δ ( G ( W ) · r ) · ¯ G ( V ) − δ ( ¯ G ( V ) · r ) · G ( W ) − ikδ S ( ¯ G ( V ) · ω · G ( W )) , so that finally σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = − iδ (Θ( V, W )) − δ ( ˜ G ( V ) · r ) · ˜ G ( W ) + δ ( ˜ G ( W ) · r ) · ˜ G ( V )+ ∆ G ( V ) δG ( W ) − ∆ G ( W ) δG ( V ) + ∆ ¯ G ( V ) δ ¯ G ( W ) − ∆ ¯ G ( W ) δ ¯ G ( V ) . For the symbol of order zero, we first observe that σ (cid:2) ∆ G ( V ) , ∆ G ( W ) (cid:3) = 0 . For the mixed terms, we get σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) = σ (cid:2) ∇ G ( V ) , ∇ G ( W ) (cid:3) + σ (cid:2) ∇ δG ( V ) , ∇ δ ¯ G ( W ) (cid:3) + σ (cid:2) ∇ G ( V ) , ∇ δ ¯ G ( W ) (cid:3) − σ (cid:2) ∇ G ( W ) , ∇ δG ( V ) (cid:3) = − kG ( V ) xy R xuvy ¯ G ( W ) uv − ikδG ( V ) · ω · δ ¯ G ( W ) − ikω ( G ( V ) ·∇ δ ¯ G ( W )) + ikω ( ¯ G ( W ) ·∇ δG ( V ))= − kG ( V ) xy R xuvy ¯ G ( W ) uv − ikδδ ( G ( V ) · ω · ¯ G ( W )) − ikω ( G ( V ) ·∇ δ ¯ G ( W ))= − ik δδ ( G ( V ) · ω · ¯ G ( W )) + ik r ( G ( V ) · ω · ¯ G ( W )) , σ (cid:2) ∆ ˜ G ( V ) , ∆ ˜ G ( W ) (cid:3) = − ik δδ Θ( V, W ) + ik r (Θ(
V, W )) . This proves the proposition.Before stating the next proposition, we introduce a one-form c ∈ Ω , ( T , C ∞ ( M )), withvalues in smooth functions on M , which will play a central role in the calculations. It isdefined by the expression, c ( V ) = − ∆ G ( V ) F − dF · G ( V ) · dF − λV ′ [ F ] . (40)This one-form serves, in fact, as the zero-order part of the Hitchin connection in metaplecticquantization, studied in [AGL], but as we will see, it also appears in the curvature of theHitchin connection in the setting considered here. In the metaplectic case, the crucialproperty satisfied by this one-form is the following relation,¯ ∂c ( V ) = i δ ( G ( V ) · ρ ) , (41)which will also be useful to us. This is verified through the calculation2 ¯ ∂ ∆ G ( V ) F = 2 ¯ ∂δ ( G ( V ) · dF )= − iρ · G ( V ) · dF + 2 δ ( G ( V ) · ∂ ¯ ∂F )= − iλω · G ( V ) · dF + 4 ∂ ¯ ∂F · G ( V ) · dF − iδ ( G ( V ) · ρ ) + iλδG ( V ) · ω = − λ ¯ ∂V ′ [ F ] − ∂ ( dF · G ( V ) · dF ) − iδ ( G ( V ) · ρ ) , where we applied (22) twice for the third equality and (24) for the last equality.As hinted above, the exterior derivative of the one-form c over T appears in the curva-ture of the Hitchin connection. To see how, we must be able to recognize this derivative. Proposition 4.4.
The exterior derivative of the one-form c defined by (40) is given by ∂ T c ( V, W ) = b ( V ) W ′ [ F ] − b ( W ) V ′ [ F ]¯ ∂ T c ( V, W ) = i δδ Θ( V, W ) − i r (Θ( V, W )) − iλθ ( V, W ) − λ∂ T ¯ ∂ T F ( V, W ) for any vector fields V and W on T .Proof. For the first statement, choose V and W so that V ′ and W ′ commute. Using theidentity ∆ G ( V ) F = δ ( G ( V ) · dF ) and the fact that the divergence operator on vector fieldsdoes not depend on the K¨ahler structure, we get ∂ T c ( V, W ) = V ′ [ c ( W )] − W ′ [ c ( V )]= − ∆ V ′ [ G ( W )] F − ∆ G ( W ) V ′ [ F ] − dF · G ( W ) · dV ′ [ F ]+ ∆ W ′ [ G ( V )] F + ∆ G ( V ) W ′ [ F ] + 2 dF · G ( V ) · dW ′ [ F ]= b ( V ) W ′ [ F ] − b ( W ) V ′ [ F ] , where we used the the fact that W ′ [ G ( V )] = − W ′ V ′ [˜ g ] = V ′ [ G ( W )], since the vector fieldswere chosen to commute. 24or the second statement, choose V and W so that V ′ and W ′′ commute. Once againusing ∆ G ( V ) F = δ ( G ( V ) · dF ), the identities (17) and (24) give us that4 W ′′ [∆ G ( V ) ( F )]= 4 δ ( W ′′ [ G ( V )] · dF ) + 4 δ ( G ( V ) · W ′′ [ F ])= iδ ( δ ¯ G ( W ) · ω · G ( V )) − iδ ( ¯ G ( W ) · ω · G ( V ) · dF )= iδδ ( ¯ G ( W ) · ω · G ( V )) − iδ ¯ G ( W ) · ω · G ( V ) · dF + 2 i∂ ¯ ∂F ( ¯ G ( W ) · ω · G ( V )) , where rigidity of the family of K¨ahler structures was used for the last equality. The Riccipotential satisfies the equation ρ = λω + 2 i∂ ¯ ∂F , so we get that2 i∂ ¯ ∂F ( ¯ G ( W ) · ω · G ( V )) = ρ ( ¯ G ( W ) · ω · G ( V )) − λω ( ¯ G ( W ) · ω · G ( V ))= ir (Θ( V ′ , W ′′ )) + 4 iλθ ( V ′ , W ′′ ) . Finally, the identities (17) and (24) can be used to verify that4 W ′′ [ dF · G ( V ) · dF ] = 2 iδ ¯ G ( W ) · ω · G ( V ) · dF. Combining the identities above, we get4 ¯ ∂ T c ( V ′ , W ′′ )= − W ′′ [∆ G ( V ) F + dF · G ( V ) · dF + 2 nV ′ [ F ]]= iδδ Θ( V ′ , W ′′ ) − ir (Θ( V ′ , W ′′ )) − iλθ ( V ′ , W ′′ ) − λ∂ T ¯ ∂ T F ( V ′ , W ′′ ) . For the first term of the last equality, we used the fact that repeated application of thedivergence operator to a bivector field only depends on its symmetric part. This finishesthe proof of the proposition.We will also need the following
Proposition 4.5.
The one-form c defined in (40) satisifes X ′ ∂ T c ( V,W ) = i G ( W ) δG ( V ) − i G ( V ) δG ( W ) + i G ( W ) · dc ( V ) − i G ( V ) · dc ( W ) for any vector fields V and W on T .Proof. As usual, the Hamiltonian vector field in the statement is determined by X ′ ∂ T c ( V,W ) · ω = ¯ ∂ ( ∂ T c ( V, W )) . To calculate the right-hand side of this, we use the fact that c satisfies (41) to get i W ′ [ δ ( G ( V ) · ρ )] = W ′ [ ¯ ∂c ( V )] = ¯ ∂W ′ [ c ( V )] − i ω · G ( W ) · dc ( V ) .
25n the other hand, we calculate W ′ [ δ ( G ( V ) · ρ )] = W ′ [ ∇ x ( G ( V ) xy ρ xa )]= W ′ [ ∇ g ] zzx G ( V ) xy ρ xa − W ′ [ ∇ g ] zxa G ( V ) xy ρ yz + δ ( W ′ [ G ( V ) · ρ ])= δ ( W ′ [ G ( V )] · ρ ) + δ ( G ( V ) · W ′ [ ρ ]) , where the last equality uses (8) and (9) and type considerations. Applying (15), we get2 δ ( G ( V ) · W ′ [ ρ ]) = δ ( G ( V ) · d ( δG ( W ) · ω )) = δ ( G ( V ) ·∇ δG ( W )) · ω = ∆ G ( V ) δG ( W ) · ω. Combining the previous three identities, we find¯ ∂∂ T c ( V, W ) = ¯ ∂V ′ [ c ( W )] − ¯ ∂W ′ [ c ( V )]= i ω · G ( V ) · dc ( W ) − i ω · G ( W ) · dc ( V ) + i G ( W ) δG ( V ) · ω − i G ( V ) δG ( W ) · ω, for commuting vector fields V ′ and W ′′ . Raising the index with ω ends the proof.The curvature calculation for the Hitchin connection proceeds with expressions forcommutators involving the one-form b defined in (38). Proposition 4.6.
The commutator of the operators b ( V ) and b ( W ) , acting on sections of ˆ H ( k ) , is a first-order operator with symbols given by σ (cid:2) b ( V ) , b ( W ) (cid:3) = 4 iX ′ ∂ T c ( V,W ) σ (cid:2) b ( V ) , b ( W ) (cid:3) = − λ∂ T c ( V, W ) , for any vector fields V and W on T .Proof. The vanishing of the third-order symbol is essentially Proposition 4.3, σ (cid:2) b ( V ) , b ( W ) (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ G ( W ) (cid:3) = − ik Θ( V ′ , W ′ ) = 0 . Vanishing of the second-order symbol is seen through the following calculation usingLemma 3.1 and Lemma 3.2, σ (cid:2) b ( V ) , b ( W ) (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ G ( W ) (cid:3) + 2 σ (cid:2) ∆ G ( V ) , ∇ G ( W ) · dF (cid:3) − σ (cid:2) ∆ G ( W ) , ∇ G ( V ) · dF (cid:3) = 2 σ (cid:2) ∇ G ( V ) , ∇ G ( W ) · dF (cid:3) − σ (cid:2) ∇ G ( W ) , ∇ G ( V ) · dF (cid:3) = 4 S ( G ( V ) ·∇ ( G ( W ) · dF )) − S ( G ( W ) ·∇ ( G ( V ) · dF )) − dF · G ( W ) ·∇ G ( V ) + 2 dF · G ( V ) ·∇ G ( W )= 6 S ( G ( V ) ·∇ G ( W )) · dF − S ( G ( W ) ·∇ G ( V )) · dF = 6Γ ( V, W ) · dF − ( W, V ) · dF = 0 . σ (cid:2) ∆ G ( V ) , ∇ G ( W ) · dF (cid:3) − σ (cid:2) ∆ G ( W ) , ∇ G ( V ) · dF (cid:3) = σ (cid:2) ∇ G ( V ) , ∇ G ( W ) · dF (cid:3) − σ (cid:2) ∇ G ( W ) , ∇ G ( V ) · dF (cid:3) + σ (cid:2) ∇ δG ( V ) , ∇ G ( W ) · dF (cid:3) − σ (cid:2) ∇ δG ( W ) , ∇ G ( V ) · dF (cid:3) = ∇ G ( V ) ( G ( W ) · dF ) − ∇ G ( W ) ( G ( V ) · dF ) + [ δG ( V ) , G ( W ) · dF ] − [ δG ( W ) , G ( V ) · dF ]= ∆ G ( V ) ( G ( W ) · dF ) − ∆ G ( W ) ( G ( V ) · dF ) − dF · G ( W ) ·∇ δG ( V ) + dF · G ( V ) ·∇ δG ( W )= dF · G ( V ) ·∇ δG ( W ) − dF · G ( W ) ·∇ δG ( V )+ ∆ G ( V ) ( G ( W )) · dF + 2 G ( V ) xy ∇ y G ( W ) au ∇ xu F + G ( W ) · d ∆ G ( V ) F − G ( W ) au ∇ u ( G ( V ) xy ) ∇ xy F − G ( W ) ·∇ δG ( V ) · dF − ∆ G ( W ) ( G ( V )) · dF − G ( W ) uv ∇ v G ( V ) ax ∇ xu F − G ( V ) · d ∆ G ( W ) F + G ( V ) ax ∇ x ( G ( W ) uv ) ∇ uv F + G ( V ) ·∇ δG ( W ) · dF = 3Γ ( V, W ) axy ∇ xy F − ( W, V ) axy ∇ xy F + Γ ( V, W ) · dF − Γ ( W, V ) · dF + G ( W ) · d ∆ G ( V ) F − G ( V ) · d ∆ G ( W ) F = G ( W ) · d ∆ G ( V ) F − G ( V ) · d ∆ G ( W ) F. Here Proposition 2.6, once again, played a central role. Similarly, we find that2 σ (cid:2) ∇ G ( V ) · dF , ∇ G ( W ) · dF (cid:3) = 2 (cid:2) G ( V ) · dF, G ( W ) · dF (cid:3) = 3 dF · Γ ( V, W ) · dF − dF · Γ ( W, V ) · dF + G ( W ) · d ( dF · G ( V ) · dF ) − G ( V ) · d ( dF · G ( W ) · dF )= G ( W ) · d ( dF · G ( V ) · dF ) − G ( V ) · d ( dF · G ( W ) · dF ) . Combining the last two computations, we see that σ (cid:2) b ( V ) , b ( W ) (cid:3) = ∆ G ( V ) δG ( W ) − ∆ G ( V ) δG ( W )+ 2 G ( W ) · d ∆ G ( V ) F − G ( V ) · d ∆ G ( W ) F + 2 G ( W ) · d ( dF · G ( V ) · dF ) − G ( V ) · d ( dF · G ( W ) · dF )+ 4 λG ( W ) · dV ′ [ F ] − λG ( V ) · dW ′ [ F ]= ∆ G ( V ) δG ( W ) − ∆ G ( V ) δG ( W )+ 2 G ( V ) · dc ( W ) − G ( W ) · dc ( V ) , where the last equation follows by recalling (40).Finally, for the zeroth-order symbol, we rely on Proposition 4.4 to get that σ (cid:2) b ( V ) , b ( W ) (cid:3) = 2 λb ( W ) V ′ [ F ] − λb ( V ) W ′ [ F ] = − λ∂ T c ( V, W ) . This finishes the proof of the proposition. 27inally, to calculate the curvature of the Hitchin connection, we need the extoriorderivative of the one-form b over T . This is calculated in the following proposition. Proposition 4.7.
The two-form d T b ∈ Ω ( T , D ( M, L k )) is given by d T b ( V, W ) = − i ∆ Θ( V,W ) − i ∇ Θ( V,W ) · dF − ∇ G ( V ) · dW [ F ] − G ( W ) · dV [ F ] + 2 λ∂ T ¯ ∂ T F ( V, W ) on sections of ˆ H ( k ) , and by d T b ( V, W ) = 2 ∇ G ( W ) · V ′ [ F ] − ∇ G ( V ) · W ′ [ F ] − ikθ ( V, W ) + 2 λ∂ T ¯ ∂ T F ( V, W ) when restricted to sections of ˆ H ( k ) .Proof. For any section s of ˆ H ( k ) over T , we have the identity W [∆ G ( V ) ] s = W [ ∇ x G ( V ) xy ∇ y ] s = W [ ∇ g ] zzx G ( V ) xy ∇ y + ∇ x W [ G ( V )] xy ∇ y s = ∆ W [ G ( V )] s, (42)where the last equality follows from (9). In particular, for commuting V and W , we get V [∆ G ( W ) ] − W [∆ G ( V ) ] = ∆ V [ G ( W )] − ∆ W [ G ( V )] = ∆ d T G ( V,W ) = − i ∆ Θ( V,W ) , where (13) was used for the last equation. Similarly, we calculate W [ ∇ G ( V ) · dF ] s = ∇ G ( V ) · W [ F ] s + ∇ W [ G ( V )] · dF s, so that finally d T b ( V, W ) = V [ b ( W )] − W [ b ( V )]= − i ∆ Θ( V,W ) − i ∇ Θ( V,W ) · dF − ∇ G ( V ) · dW [ F ] − G ( W ) · dV [ F ] + 2 λ∂ T ¯ ∂ T F ( V, W ) . (43)This proves the first statement of the proposition.For the second statement, observe that the stated formulas on sections of ˆ H ( k ) andˆ H ( k ) agree for ∂ T b . For the case of ¯ ∂ T b , suppose that s is a section of ˆ H ( k ) and observethat 2 ∇ V ′ ,W ′′ ) s = −∇ G ( W ) · ω · G ( V ) s = ikω ( ¯ G ( W ) · ω · G ( V )) s = 4 kθ ( V ′ , W ′′ ) s. Using the conjugated version of (24), we also get that2 ∇ δ Θ( V ′ ,W ′′ ) s + 4 ∇ Θ( V ′ ,W ′′ ) · dF s = ∇ G ( V ) · ω · δ ¯ G ( W ) s + 2 ∇ G ( V ) · ω · ¯ G ( W ) · dF s = 4 i ∇ G ( V ) · dW ′′ [ F ] s. Finally, plugging these two expressions into (43), we get d T b ( V ′ , W ′′ ) s = − ∇ G ( V ) · W ′ [ F ] s − ikθ ( V ′ , W ′′ ) s + 2 λ∂ T ¯ ∂ T F ( V ′ , W ′′ ) s, which finishes the proof of the proposition. 28ith the above propositions at hand, calculating the curvature of the Hitchin connec-tion is a straightforward matter. Theorem 4.8.
The curvature of the Hitchin connection acts by F , ∇ = k (2 k + λ ) P k ( ∂ T c ) F , ∇ = − ik k + λ ( θ − i∂ T ¯ ∂ T F ) F , ∇ = 0 , on sections of the bundle ˆ H ( k ) .Proof. For commuting vector fields V and W , we get F ∇ ( V, W ) = (cid:2) ∇ V , ∇ W (cid:3) = (cid:2) b ( V ) , b ( W ) (cid:3) (4 k + 2 λ ) + d T b ( V, W ) + (cid:2) b ( V ) , W ′ [ F ] (cid:3) − (cid:2) b ( W ) , V ′ [ F ] (cid:3) k + 2 λ − ∂ T ¯ ∂ T F ( V, W ) . Now Proposition 4.4 can be applied to see that (cid:2) b ( V ) , W ′ [ F ] (cid:3) − (cid:2) b ( W ) , V ′ [ F ] (cid:3) = 2 ∇ G ( V ) · W ′ [ F ] − ∇ G ( W ) · V ′ [ F ] + ∂ T c ( V, W ) , and combining the above with Proposition 4.6 and Proposition 4.7 yields the followingexpression for the (2,0)-part of the curvature F , ∇ ( V, W ) = (cid:2) b ( V ) , b ( W ) (cid:3) (4 k + 2 λ ) + ∂ T c ( V, W )4 k + 2 λ = k (2 k + λ ) P k ( ∂ T c ( V, W )) , where P k denotes the prequantum operator defined in (26). For the (1,1)-part of thecurvature, Proposition 4.7 yields F , ∇ ( V, W ) = ¯ ∂ T b ( V, W )4 k + 2 λ − ∂ T ¯ ∂ T F ( V, W ) = − ik k + λ ( θ − i∂ T ¯ ∂ T F ) . Finally, the (0,2)-part of the curvature clearly vanishes, and the theorem is proved.The fact that the (1,1)-part of the curvature is an zeroth-order operator, combinedwith the fact that the Hitchin connection preserves the subbundle of quantum spaces ˆ H ( k ) implies that the (1,1)-part must take values in holomorphic and hence constant functionson M . This was, however, already known from Proposition 2.8 for the particular expressionwe found in Theorem 4.8. Also, being a first-order operator, it is no surprise that the (2,0)-part of the curvature acts by prequantum operator. After all, prequantum operators ofthe form (28) are the only first-order operators which preserve the holomorphic sections. The Hitchin-Witten Connection
In the previous section, we saw how the higher-order symbols of the curvature of the Hitchinconnection vanished and left a first-order operator with a relatively simple expression. Forthe (2,0)-part, the higher order symbols vanished for general reasons, related to rigidity ofthe family of K¨ahler structures, but for the (1,1)-part, the curvature reduced to a zero-orderoperator only when restricted the subbundle ˆ H ( k ) of quantum spaces.29t turns out that, with a slight modification of the formula, we can achieve a cancellationof higher-order terms for the (1,1)-part of the connection defined on the whole bundle ˆ H ( k ) of prequantum spaces over T , but still maintain the vanishing of higher-order symbols forthe (2,0) and (0,2)-parts.In this section, we let t ∈ C be any complex number with integer real part, k = Re( t ) ∈ Z , and we consider the connection on ˆ H ( k ) given by˜ ∇ V = ∇ T V + 12 t b ( V ) − t ¯ b ( V ) + V [ F ] , (44)where ¯ b ( V ) has the conjugated symbols of b ( V ) so that b ( V ) = ∆ G ( V ) + 2 ∇ G ( V ) · dF − λV ′ [ F ] and ¯ b ( V ) = ∆ ¯ G ( V ) + 2 ∇ ¯ G ( V ) · dF − λV ′′ [ F ] . We will refer to ˜ ∇ as the Hitchin-Witten connection. It is a generalization of the connectionfor quantum Chern-Simons theory with complex gauge group SL( n, C ) discussed by Wittenin [Wit3], where he arrives at exactly the formula (44). This relation will be furtherexplored in the final section of the paper.We will prove that the curvature of ˜ ∇ V , acting on sections of ˆ H ( k ) , has essentiallythe same expression as the curvature of ∇ , acting on sections of ˆ H ( k ) . We start with thefollowing. Proposition 4.9.
The commutator of the operators b ( V ) and ¯ b ( W ) , acting on sections of ˆ H ( k ) , is a second-order operator with symbols given by σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = − ik Θ( V ′ , W ′′ ) σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = − ik δ Θ( V ′ , W ′′ ) − ik Θ( V ′ , W ′′ ) · dFσ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = − ik δδ Θ( V ′ , W ′′ ) − ik r (Θ( V ′ , W ′′ )) − ikλ θ ( V ′ , W ′′ ) − ik dF · Θ( V ′ , W ′′ ) · dF − ik δ Θ( V ′ , W ′′ ) · dF, and furthermore σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = 2¯ t b ( V ) W ′′ [ F ] + 2 t ¯ b ( W ) V ′ [ F ] − ikλ θ ( V ′ , W ′′ ) , for any vector fields V and W on T .Proof. The third-order symbol vanishes, as Proposition 4.3 readily gives σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) = 0 . The calculation of the second-order symbol is also straightforward, σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) + 2 σ (cid:2) ∇ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) − σ (cid:2) ∇ G ( W ) , ∆ G ( V ) · dF (cid:3) = − ik Θ( V ′ , W ′′ ) + 4 S ( G ( V ) · ∂ ¯ ∂F · ¯ G ( W )) − S ( ¯ G ( W ) · ¯ ∂∂F · G ( V ))= − ik Θ( V ′ , W ′′ ) . σ (cid:2) ∆ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) . We get2 σ (cid:2) ∇ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) = 2 ∇ G ( V ) ( ¯ G ( W ) · dF ) − ikG ( V ) · ω · ¯ G ( W ) · dF + 2 G ( V ) xy R auxy ¯ G ( W ) uv dF v and 2 σ (cid:2) ∇ δG ( V ) , ∇ ¯ G ( W ) · dF (cid:3) = 2 ∇ δG ( V ) ( ¯ G ( W ) · dF ) − dF · ¯ G ( W ) ·∇ δG ( V )= 2 ∇ δG ( V ) ( ¯ G ( W ) · dF ) + 2 idF · ¯ G ( W ) · ρ · G ( V ) − dF u ¯ G ( W ) uv R avxy G ( V ) xy , and when combining these, we get2 σ (cid:2) ∆ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) = 2∆ G ( V ) ( ¯ G ( W ) · dF ) − ikG ( V ) · ω · ¯ G ( W ) · dF − iG ( V ) · ρ · ¯ G ( W ) · dF. The first term of this expression can be rewritten as2∆ G ( V ) ( ¯ G ( W ) · dF ) = 2 δ ( G ( V ) ·∇ ( ¯ G ( W ) · dF )) = 2 δ ( G ( V ) · ∂ ¯ ∂F · ¯ G ( W ))= − iδ ( G ( V ) · ρ · ¯ G ( W )) + iλδ ( G ( V ) · ω · ¯ G ( W )) . For the first-order symbol, we must also understand σ (cid:2) ∇ G ( V ) , W ′′ [ F ] (cid:3) , which yields2 σ (cid:2) ∇ G ( V ) , W ′′ [ F ] (cid:3) = 4 G ( V ) · dW ′′ [ F ] = − iG ( V ) · ω · δ ¯ G ( W ) − iG ( V ) · ω · ¯ G ( W ) · dF. Altogether, the previous identities yield2 σ (cid:2) ∆ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) − λ σ (cid:2) ∇ G ( V ) , W ′′ [ F ] (cid:3) = 2∆ G ( V ) ( ¯ G ( W ) · dF ) − ikG ( V ) · ω · ¯ G ( W ) · dF − iG ( V ) · ρ · ¯ G ( W ) · dF − iλδ ( ¯ G ( W ) · ω · G ( V )) + 2 iλG ( V ) · ω · ¯ G ( W ) · dF = − iδ ( G ( V ) · ρ · ¯ G ( W )) − ikG ( V ) · ω · ¯ G ( W ) · dF + 4 G ( V ) · ∂ ¯ ∂F · ¯ G ( W ) · dF + 2 iλ δ Θ( V ′ , W ′′ )= δ ( G ( V ) · r ) · ¯ G ( W ) − ikG ( V ) · ω · ¯ G ( W ) · dF + 4 dF · ¯ G ( W ) ·∇ ( G ( V ) · dF )+ 2 iλ δ Θ( V ′ , W ′′ ) . A completely analogous computation shows2 σ (cid:2) ∆ ¯ G ( W ) , ∇ G ( V ) · dF (cid:3) − λ σ (cid:2) ∇ G ( W ) , V ′ [ F ] (cid:3) = δ ( ¯ G ( W ) · r ) · G ( V ) − ik ¯ G ( W ) · ω · G ( V ) · dF + 4 dF · G ( V ) ·∇ ( ¯ G ( W ) · dF )+ 2 iλ δ Θ( V ′ , W ′′ ) . σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) + 4 (cid:2) G ( V ) · dF, ¯ G ( W ) · dF (cid:3) + 2 σ (cid:2) ∆ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) − λ σ (cid:2) ∇ G ( V ) , W ′′ [ F ] (cid:3) − σ (cid:2) ∆ ¯ G ( W ) , ∇ G ( V ) · dF (cid:3) + 2 λ σ (cid:2) ∇ G ( W ) , V ′ [ F ] (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) − ik Θ( V ′ , W ′′ ) · dF + δ ( G ( V ) · r ) · ¯ G ( W ) − δ ( ¯ G ( W ) · r ) · G ( V )= − ik δ Θ( V ′ , W ′′ ) − ik Θ( V ′ , W ′′ ) · dF It only remains to calculate the zero-order symbol. First we compute2 σ (cid:2) ∇ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) = − ik ω ( G ( V ) ·∇ ( dF · ¯ G ( W )))= − ik ω ( G ( V ) · ∂ ¯ ∂F · ¯ G ( W ))= − ik ∂ ¯ ∂F ( G ( V ) · ω · ¯ G ( W ))= − kρ ( G ( V ) · ω · ¯ G ( W )) + kλ ω ( G ( V ) · ω · ¯ G ( W ))= − ik r (Θ( V ′ , W ′′ )) − ikλ θ ( V ′ , W ′′ ) , and similarly, we have2 σ (cid:2) ∇ G ( W ) , ∇ G ( V ) · dF (cid:3) = ik r (Θ( V ′ , W ′′ )) + 4 ikλ θ ( V ′ , W ′′ ) . Furthermore, we calculate2 σ (cid:2) ∇ δG ( V ) , ∇ ¯ G ( W ) · dF (cid:3) − σ (cid:2) ∇ δ ¯ G ( W ) , ∇ G ( V ) · dF (cid:3) = − ikδG ( V ) · ω · ¯ G ( W ) · dF + 2 ikδ ¯ G ( W ) · ω · G ( V ) · dF = − ik δ Θ( V ′ , W ′′ ) · dF, and finally 4 σ (cid:2) ∇ G ( V ) · dF , ∇ ¯ G ( W ) · dF (cid:3) = − ik dF · Θ( V ′ , W ′′ ) · dF. All of the above contribute to the zero-order symbol, but we also need to calculate b ( V ) W ′′ [ F ] and ¯ b ( W ) V ′ [ F ]. First we compute4∆ G ( V ) W ′′ [ F ]= 4 δ ( G ( V ) · dW ′′ [ F ])= − iδ ( G ( V ) · ω · δ ¯ G ( W )) − iδ ( G ( V ) · ω · ¯ G ( W ) · dF )= iδδ ( ¯ G ( W ) · ω · G ( V )) − i∂ ¯ ∂F ( G ( V ) · ω · ¯ G ( W )) − iδG ( V ) · ω · ¯ G ( W ) · dF = − iδδ (Θ( V ′ , W ′′ )) − ir (Θ( V ′ , W ′′ )) − iλ θ ( V ′ , W ′′ ) − iδG ( V ) · ω · ¯ G ( W ) · dF, where the last equality used the fact that the double divergence of a bivector field onlydepends on its symmetric part. Similarly, we find4∆ ¯ G ( W ) V ′ [ F ] = iδδ Θ( W ′′ , V ′ ) + ir (Θ( W ′′ , V ′ )) + 4 iλ θ ( W ′′ , V ′ ) + 2 iδ ¯ G ( W ) · ω · G ( V ) · dF.
32e also have4 ∇ G ( V ) · dF W ′′ [ F ] = − idF · G ( V ) · ω · δ ¯ G ( W ) − idF · Θ( V ′ , W ′′ ) · dF ∇ ¯ G ( W ) · dF V ′ [ F ] = + idF · ¯ G ( W ) · ω · δG ( V ) + 2 idF · Θ( W ′′ , V ′ ) · dF, so finally, we conclude that4 b ( V ) W ′′ [ F ] = − iδδ (Θ( V ′ , W ′′ )) − i r (Θ( V ′ , W ′′ )) − iλ θ ( V ′ , W ′′ ) − iδ Θ( V ′ , W ′′ ) · dF − idF · Θ( V ′ , W ′′ ) · dF − λV ′ [ F ] W ′′ [ F ] . Using this identity, conjugation yields¯ b ( W ) V ′ [ F ] = b ( W ) V ′′ [ F ] = b ( V ) W ′′ [ F ] , so the contributions of ¯ b ( W ) V ′ [ F ] and b ( V ) W ′′ [ F ] to the zero-order symbol of the com-mutator cancel. This finally gives σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = σ (cid:2) ∆ G ( V ) , ∆ ¯ G ( W ) (cid:3) + 2 σ (cid:2) ∆ G ( V ) , ∇ ¯ G ( W ) · dF (cid:3) − σ (cid:2) ∆ ¯ G ( W ) , ∇ G ( V ) · dF (cid:3) + 4 σ (cid:2) ∇ G ( V ) · dF , ∇ ¯ G ( W ) · dF (cid:3) − λ b ( V ) W ′′ [ F ] + 2 λ ¯ b ( W ) V ′ [ F ]= − ik δδ Θ( V ′ , W ′′ ) − ik r (Θ( V ′ , W ′′ )) − ikλ θ ( V ′ , W ′′ ) − ik dF · Θ( V ′ , W ′′ ) · dF − ki δ Θ( V ′ , W ′′ ) · dF, as claimed in the proposition. To prove the final statement of the proposition, we notefrom the above that2¯ t b ( V ) W ′′ [ F ] + 2 t ¯ b ( W ) V ′ [ F ]= − ikδδ (Θ( V ′ , W ′′ )) − ik r (Θ( V ′ , W ′′ )) − ikλ θ ( V ′ , W ′′ ) − ik dF · Θ( V ′ , W ′′ ) · dF − ik δ Θ( V ′ , W ′′ ) · dF − λkV ′ [ F ] W ′′ [ F ] , which gives σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) = 2¯ t b ( V ) W ′′ [ F ] + 2 t ¯ b ( W ) V ′ [ F ] − ikλ θ ( V ′ , W ′′ ) + 4 λkV ′ [ F ] W ′′ [ F ]= 2¯ tσ (cid:2) b ( V ) , W ′′ [ F ] (cid:3) + 2 t σ (cid:2) ¯ b ( W ) , V ′ [ F ] (cid:3) − ikλ θ ( V ′ , W ′′ ) . This proves the proposition.With the bulk of computations encoded in Proposition 4.9, we can calculate the curva-ture of the Hitchin-Witten connection. This is the content of the following theorem.
Theorem 4.10.
The curvature of the Hitchin-Witten connection ˜ ∇ acts as a first-orderoperator with symbols σ F , ˜ ∇ = it X ′ ∂ T c σ F , ˜ ∇ = 0 σ F , ˜ ∇ = − i ¯ t X ′′ ¯ ∂ T ¯ c σ F , ˜ ∇ = t − λ t ∂ T c σ F , ˜ ∇ = ikλt ¯ t (cid:0) θ − i∂ T ¯ ∂ T F (cid:1) σ F , ˜ ∇ = − ¯ t + λ t ¯ ∂ T ¯ c on sections of the bundle ˆ H ( k ) over T . roof. The calculation of the (2,0)-part proceeds as in the proof of Theorem 4.8 and yieldsa first-order operator with symbols given by σ (cid:0) F , ˜ ∇ ( V, W ) (cid:1) = it X ′ ∂ T c ( V,W ) and σ (cid:0) F , ˜ ∇ ( V, W ) (cid:1) = t − λ t ∂ T c ( V, W ) , as claimed in the theorem.The calculation of the (0,2)-part is completely analogous. We have the conjugate state-ments to Proposition 4.6, σ (cid:2) ¯ b ( V ) , ¯ b ( W ) (cid:3) = − iX ′′ ¯ ∂ T ¯ c ( V,W ) and σ (cid:2) ¯ b ( V ) , ¯ b ( W ) (cid:3) = − λ ¯ ∂ T ¯ c ( V, W ) , and Proposition 4.7, d T ¯ b ( V, W ) = i ∆ Θ( V,W ) + 2 i ∇ Θ( V,W ) · dF − ∇ ¯ G ( V ) · dW [ F ] − ¯ G ( W ) · dV [ F ] − λ∂ T ¯ ∂ T F ( V, W ) , and these can be combined with the conjugate of Proposition 4.4 to compute σ (cid:0) F , ˜ ∇ ( V, W ) (cid:1) = − i ¯ t X ′′ ¯ ∂ T ¯ c ( V,W ) and σ (cid:0) F , ˜ ∇ ( V, W ) (cid:1) = − ¯ t + λ t ∂ T c ( V, W ) . It only remains to compute the (1,1)-part, and we get4 t ¯ t F , ˜ ∇ ( V ′ , W ′′ ) = − (cid:2) b ( V ) , ¯ b ( W ) (cid:3) + 2¯ t d T b ( V ′ , W ′′ ) − t d T ¯ b ( V ′ , W ′′ )+ 2¯ t (cid:2) b ( V ) , W ′′ [ F ] (cid:3) + 2 t (cid:2) ¯ b ( W ) , V ′ [ F ] (cid:3) . By Proposition 4.9 and the above expression for d T ¯ b , the second-order symbol vanishes,4 t ¯ t σ (cid:0) F , ˜ ∇ ( V ′ , W ′′ ) (cid:1) = − σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) + 2¯ t σ (cid:0) d T b ( V ′ , W ′′ ) (cid:1) − t σ (cid:0) d T ¯ b ( V ′ , W ′′ ) (cid:1) = 4 ik Θ( V ′ , W ′′ ) − i ( t + ¯ t ) Θ( V ′ , W ′′ )= 0 . The same is true for the first-order symbol4 t ¯ t σ (cid:0) F , ˜ ∇ ( V ′ , W ′′ ) (cid:1) = − σ (cid:2) b ( V ) , ¯ b ( W ) (cid:3) + 2 i ¯ t σ (cid:0) d T b ( V ′ , W ′′ ) (cid:1) − it σ (cid:0) d T ¯ b ( V ′ , W ′′ ) (cid:1) + 2¯ t σ (cid:2) b ( V ) , W ′′ [ F ] (cid:3) + 2 t σ (cid:2) ¯ b ( W ) , V ′ [ F ] (cid:3) = 4 ik δ Θ( V ′ , W ′′ ) + 8 ik Θ( V ′ , W ′′ ) · dF − i ( t + ¯ t ) δ Θ( V ′ , W ′′ ) − i ( t + ¯ t )Θ( V ′ , W ′′ ) · dF − tG ( V ) · W ′′ [ F ] − t ¯ G ( W ) · dV ′ [ F ]+ 4¯ tG ( V ) · W ′′ [ F ] + 4 t ¯ G ( W ) · dV ′ [ F ]= 0 . Finally, the last statement of Proposition 4.9 gives the zeroth-order part4 t ¯ t σ (cid:0) F , ˜ ∇ ( V ′ , W ′′ ) (cid:1) = 4 ikλ θ ( V ′ , W ′′ ) + 4 λ (¯ t + t ) ∂ T ¯ ∂ T F ( V ′ , W ′′ )= 4 ikλ (cid:0) θ ( V ′ , W ′′ ) − i∂ T ¯ ∂ T F ( V ′ , W ′′ ) (cid:1) . This completes the proof of the theorem. 34 rojective Flatness
In case the Hitchin connection ∇ in the bundle ˆ H ( k ) over T is projectively flat, the paralleltranslation maps along homotopic curves are equal up to scale. Thus, if the parameterspace T is simply connected, the Hitchin connection gives a canonical identification of theprojectivized quantum spaces associated with different complex structures. In this sense,the quantization is independent of the complex structure.The following theorem is an immediate consequence of the explicit curvature calcula-tions of Theorem 4.8 and Theorem 4.10. Theorem 4.11.
The connections ∇ and ˜ ∇ , acting on ˆ H ( k ) and ˆ H ( k ) , respectively, areboth projectively flat if and only if the holomorphic vector field on M given by X ′ ∂ T c ( V,W ) = i G ( W ) δG ( V ) − i G ( V ) δG ( W ) + i G ( W ) · dc ( V ) − i G ( V ) · dc ( W ) vanishes, for all vector fields V on T .Proof. Projective flatness amounts to the curvature being a two-form on T with valuesin constant functions on M . In particular, it takes values in zeroth-order differentialoperators. Being proportional to the first-order symbol of the (2,0)-part of the curvatureof both ∇ and ˜ ∇ , the vector field X ′ ∂ T c ( V,W ) will of course vanish if either of these areprojectively flat.Conversely, suppose that X ′ ∂ T c ( V,W ) vanishes, which means that the curvature of ∇ isa zeroth-order operator, and since it preserves the subbundle ˆ H ( k ) of holomorphic sectionsof L k it takes values in holomorphic functions on M . By assumption, such functions areconstants, which proves that ∇ is projectively flat. In particular, the form ∂ T c takes valuesin constant functions on M , which is also the case for θ − i∂ T ¯ ∂ T F , as already noted inProposition 2.8. By Theorem 4.10, this proves that the Hitchin-Witten connection ˜ ∇ isprojectively flat.We observe that our main Theorem 1.1 follows as an immediate corollary of the theoremabove. This ends the general discussion of the Hitchin and the Hitchin-Witten connection.In the final section, we will apply the results to the setting originally motivating the studyof these connections. Is this section, we apply the results of previous sections to Chern-Simons theory witheither compact or complex gauge group. To be specific, let G denote the real Lie groupSU( n ), sitting as the maximal compact subgroup of its complexification G C , which canbe identified with SL( n, C ). Indeed, if g denotes the Lie algebra of G , consisting of skew-Hermitian traceless matrices, then its complexification g C can be identified with the Liealgebra sl n ( C ) of traceless matrices through the unique splitting of any complex matrixinto Hermitian and skew-Hermitian parts. Furthermore, let h· , ·i be an invariant innerproduct on g C , such as − π Tr, which is normalized so that h ϑ ∧ [ ϑ ∧ ϑ ] i represents anintegral generator of H ( G, R ), where ϑ is the Maurer-Cartan form on G .35et Σ be a closed surface of genus g ≥
2, and let Σ p be the surface obtained bypuncturing ¯Σ at a point p ∈ Σ. Fix an element d ∈ Z /n Z and a small loop γ around thepuncture, and consider the moduli spaces M = Hom d ( π (Σ p ) , G ) /GM C = Hom + d ( π (Σ p ) , G C ) /G C , (45)of representations of the fundamental group mapping γ to the element e πid/n I in thecommon center of G and G C . For M C , we restrict to reductive representations, whichis indicated by a plus in the notation. The subspaces M s and M s C corresponding toirreducible connections are of particular relevance to us. For these spaces, we have enembedding M s ⊂ M s C , since any irreducible unitary representation can only be conjugateto another such by a unitary transformation.The spaces M and M C also have a gauge theoretic realization as moduli spaces of flatconnections. Let P denote the trivial principal bundle over Σ p with structure group G .Fix an element a d ∈ g such that exp(2 πa d ) = e πid/n I , and denote by F the space of flatconnections on P which are all equal to a d dθ for some fixed polar coordinate system, withangular coordinate θ , in a disc around the puncture. These connections are said to be intemporal gauge near the puncture and clearly have holonomy e πit/n Id around γ . If G denotes the space of gauge transformations equal to the identity in a neighbourhood of thepuncture, then the holonomy representations provide an identification M ∼ = F / G . Similarly, let F + C be the space of flat reductive connections on the trivial G C -bundle P C over Σ p , and let G C be the space of gauge transformations, both spaces with restrictionssimilar to the above around the puncture. The holonomy representation then gives anidentification M C ∼ = F + C / G C . By a more careful selection of connections and gauge transformations, using exponentialdecay in weighted Sobolev norms around the puncture [DW], the subsets M s and M s C cor-responding to irreducible connections can be endowed with the structure of smooth mani-folds. The tangent space T [ A ] M s C at a connection A , is given by the compactly supportedfirst cohomology H A (Σ p , g C ), with values in the adjoint bundle and exterior derivative d A induced by A . We can then define a complex symplectic form ω C by ω C ([ α ] , [ β ]) = − π Z Σ h α ∧ β i , (46)In complete analogy, the tangent space T [ A ] M s is given by the cohomology H A (Σ p , g ). Theformula (46) defines in this case a real symplectic form on M s ⊂ M s C , which is of coursejust the restriction of ω C .If t ∈ C is a complex number with integer real part, k = Re t ∈ Z , we wish to quantizethe space M s C with respect to the real symplectic form ω t = 12 ( tω C + tω C ) , as well as the space M s equipped with the restriction of ω t , which is given by kω .36uppose that A ∈ F + C is a connection, and that g : Σ p → G C is a gauge transformationin G C . Since G C is simply connected and hence 2-connected, we can choose a homotopy˜ g : Σ p × [0 , → G C from the trivial gauge transformation to g , keeping identity valuesfixed. If π : Σ p × [0 , → Σ p denotes projection onto the first factor, we can pull back A to a connection ˜ A = π ∗ A on Σ p × [0 ,
1] and consider the Chern-Simons form α C ( ˜ A ) = h ˜ A ∧ F ˜ A i − h ˜ A ∧ [ ˜ A ∧ ˜ A ] i , which will of course be complex-valued in general. Using the complex number t ∈ C frombefore, we shall consider the real form α t ( ˜ A ) = 12 ( tα C ( ˜ A ) + tα C ( ˜ A )) , and the Chern-Simons cocycle given byΘ t ( A, g ) = exp (cid:16) πi Z Σ × [0 , α t ( ˜ A ˜ g ) (cid:17) ∈ U(1) , where ˜ A ˜ g denotes the gauge-transformed connection. Since A is flat and in temporalgauge, the Chern-Simons form vanishes in a neighbourhood of { p } × [0 ,
1] and the integralconverges. Furthermore, since the real part of t is an integer, the expression is independentof the choice of homotopy ˜ g and defines a map Θ : F C × G C → U (1), which can be shownto satisfy the cocycle relationΘ t ( A g , h )Θ t ( A, g ) = Θ t ( A, gh ) . This cocycle can be used to lift the action by the gauge group G C on F + C to the trivial linebundle F + C × C by ( A, z ) · g = ( A g , Θ t ( A, g ) z ) , and the quotient defines a Hermitian line bundle, L t C → M s C , over the smooth part of the moduli space. Furthermore, this bundle comes with a unitaryconnection ∇ , given on the trivial bundle F + C × C by the one-form B t ( α ) = 2 πi Z Σ h tA ∧ α + tA ∧ α i , for a tangent vector α ∈ T A F ∼ = Ω (Σ , g C ) at A ∈ F + C . It is easily verified that thecurvature of this connection is given by F ∇ = − iω t , so that L t C defines a prequantum line bundle over M s C , with symplectic form ω t , and clearlythis restricts to a prequantum line bundle L k over M s , equipped with kω as its symplecticstructure. 37o perform geometric quantization, we must introduce a polarization. This will comefrom a choice of Riemann surface structure on Σ, which amounts to a Hodge star-operator ∗ : Ω (Σ) → Ω (Σ), satisfying ∗ = − Id for dimensional reasons. Using Hodge theoryto identify H A (Σ , g C ) with the space of harmonic forms, we define an almost complexstructure J on M s C by the expression Jα = −∗ α (47)on harmonic representatives, the space of which is preserved by ∗ . It is easily checked that J is compatible with ω t . There is another obvious almost complex structure I on M s C ,which is simply given by Iα = iα , and clearly this anti-commutes with J . Both of thesealmost complex structure are in fact integrable, as seen through the correspondence withHiggs bundles discussed below, giving M s C the structure of a hyperk¨ahler manifold.The K¨ahler structure J only depends on the Riemann surface structure up to isotopy,so in fact we get a family J : T Σ → C ∞ ( M s C , End(
T M s C )) of K¨ahler structures parametrizedby the Teichm¨uller space T Σ of the surface. At a unitary connection A , the family J clearlypreserves the real subspace H A (Σ , g ) tangent to M s , so J also defines a family of K¨ahlerstructures on M s . This is indeed the polarization we shall use for quantizing M s , but for M s C we shall instead rely on a certain real polarization, considered by Witten in [Wit3],which also depends on the Riemann surface structure of Σ. Let us first recall how thequantization of the moduli space M s for compact gauge group proceeds.The moduli space M s is simply connected, so in particular H ( M s , R ) vanishes, andfurthermore H ( M s , Z ) = Z (see [AB2, DW, AHJ + ]). For a given point σ ∈ T Σ inTeichm¨uller space, the properties of M sσ as a complex manifold can be understood throughthe classical work of Narasimhan and Seshadri [NS], which identifies M with the modulispace of S-equivalence classes of semi-stable holomorpic vector bundles, over the Riemannsurface Σ, of rank n , degree d and fixed determinant. This space has the structure ofan normal projective algebraic variety which is typically singular but contains the modulispace of stable bundles as an open smooth subvariety, corresponding exactly to the manifold M s of irreducible connections. In this picture, the holomorphic tangent space T [ E ] M s at astable holomorphic bundle E is given by the cohomology H (Σ , End E ), with values in thetraceless endomorphisms. As we saw above, the moduli space M s admits a prequantumline bundle L , which is just the determinant line bundle in this picture. It generates thePicard group, so in particular [ ω π ] is a generator of H ( M s , Z ), and c ( M s ) = λ [ ω π ] with λ = 2 gcd( r, d ) as proved in [DN].If the rank n and the degree d are coprime, there are no strictly semi-stable bundles,so M = M s is a compact K¨ahler manifold. In this case, clearly H ( M s , O ) = C , butas observed by Hitchin [Hit3], this holds even for non-coprime n and d by the Hartogstheorem, since the complement of M s in M has codimension at least 2. Hitchin also notesthat the family J of K¨ahler structures on M s parametrized by Teichm¨uller space, which isitself in a canonical way a contractible complex manifold, is holomorphic and rigid in thesense of Definition 2.2 and Definition 2.4. Finally, the K¨ahler metric on the moduli spaceof stable bundles was studied by Zograf and Takhtajan in [ZT], where they give a Riccipotential in terms of the determinant of the Laplacian on the endomorphism bundle of astable bundle.Altogether, the discussion above demonstrates that the moduli space M s satisfies all theconditions of Theorem 4.2 to ensure the existence of a Hitchin connection. Furthermore, in38he case of coprime n and d , Narasimhan and Ramanan [NR] have shown that M s does notadmit any holomorphic vector fields, so Theorem 1.1 implies that the Hitchin connectionmust be projectively flat.Projective flatness also holds in the non-coprime case, which similarly does not admitholomorphic vector fields. Hitchin [Hit3] proves this by regarding a holomorphic vectorfield on M s as a holomorphic function on the cotangent bundle T ∗ M s , which sits inside themoduli space of semi-stable Higgs bundles M , as discussed below. Once again appealingto the Hartogs theorem, the function, which is homogeneous of degree 1 in the action of C ∗ and constant along fibers of the Hitchin fibration, can be extended to M , which isquite easily seen not to support such functions. The proof does not apply to the specialsituation when the genus g and the rank n are both equal to 2, and the holonomy around γ is trivial, because the moduli space M C is isomorphic to C P and clearly has holomorphicvector fields.In summary, we have proved projective flatness of the Hitchin connection for the modulispaces as claimed in Theorem 1.2. Our proof of projective flatness follows the original proofby Hitchin [Hit3] from the point where the curvature is known to be at most order one,but vanishing of the higher-order terms was established by Hitchin through particularproperties of the Hitchin integrable system [Hit1], whereas we derive it from rigidity of thefamily of K¨ahler structures.We now turn to the problem of quantizing the moduli space M s C of irreducible G C -connections. The theory of Higgs bundles naturally enters in this discussion as well. Sup-pose again that Σ is endowed with a Riemann surface structure and recall that a Higgsbundle is a pair ( E, Φ), where E → Σ is a holomorphic vector bundle and the Higgs fieldΦ ∈ H (Σ , End E (cid:15) K ) is a holomorphic one-form with values in the traceless endomor-phisms of E . Stability for Higgs bundles is defined by imposing the usual stability conditionon the slope of subbundles E ′ ⊂ E , but only the invariant ones satisfying Φ( E ′ ) ⊂ E ′ (cid:15) K .This leads to the moduli spaces M and M s , of semi-stable, respectively stable, Higgsbundles of rank n , degree d and fixed determinant. Nitsure [Nit] proves that M is a quasi-projective algebraic variety, which contains M s as an open smooth subvariety. Throughthe work of Hitchin, Simpson, Donaldson and Corlette [Hit2, Sim1, Sim2, Don, Cor],the moduli space M can be identified, via the Hitchin equations and non-abelian Hodgetheory, with the moduli space M C of flat reductive G C -connections, with stable bundlescorresponding to irreducible connections.If E is itself already a stable bundle, then obviously stability as a Higgs bundle isimplied for any Higgs field Φ. In particular, by simply taking the Higgs field to be zero,the moduli space M s sits canonically inside M s ∼ = M s C , as we discussed already at thelevel of representations. But in fact, the entire cotangent bundle of M s can be embeddedin M s C , although the embedding depends crucially on the Riemann surface structure of Σ.Indeed, we get the following identification by Serre duality, H (Σ , End E (cid:15) K ) ∼ = H (Σ , End E ) ∗ , where we recognize the right-hand side as the holomorphic cotangent space of the modulispace M s . In other words, the Higgs fields on stable bundles can be viewed as cotangentvectors to M s through Serre duality. The Higgs bundle model M s of M s C illuminates thehyperk¨ahler structure but it also carries a natural action of C ∗ by scaling the Higgs field.39o quantize the modulu space M s C , we follow Witten [Wit3] and define a family of realpolarizations, parametrized by the Teichm¨uller space of Σ, in the following way. For anyRiemann surface structure on Σ, in the form of a star-operator ∗ as usual, we can useHodge theory to split the space H A (Σ , g C ) into types, H A (Σ , g C ) = H , A (Σ , g C ) ⊕ H , A (Σ , g C ) . Although this uses the complex structure on H A (Σ , g C ), it defines a splitting of the under-lying real space, which is a model for the real tangent space T [ A ] M s C . Since the symplecticform ω C is invariant under the Hodge-star, the same of course applies to ω t , so that eachof the summands are Lagrangian.We shall take the subspaces H , A (Σ , g C ) as a polarization, and define the quantumspace to be the polarized sections of L C , which are covariantly constant along its leaves.Notice that the polarization at a unitary connection A is transverse to the tangent space T [ A ] M s = H A (Σ , g ), simply because the only real form of type (1,0) is the zero form. Thismeans that a polarized section is determined by its values on the space M s , or in otherwords, that the quantum space is identified with the prequantum space C ∞ ( M s , L k ) ofsmooth sections over M s , although the identification depends on the complex structure onΣ. Surely this identification requires covariantly constant sections to exist on the leaves ofthe foliation, which would be guaranteed for instance if they were simply connected, but weshall not deal with this question here. Instead, the discussion can be taken as motivationfor using C ∞ ( M s , L k ) as a model for the quantum space of M s C .To understand how the identification between the quantum space of M s C and the pre-quantum space C ∞ ( M s , L k ) depends on the Riemann surface structure, we consider thelatter as the fiber of a trivial bundle over Teichm¨uller space, T Σ × C ∞ ( M s , L k ) → T Σ . Then the expression (44) defines a connection ˜ ∇ V on this bundle, which is projectively flatby Theorem 1.1, once again due to the fact that the moduli space M s does not admit anyholomorphic vector fields for any of the K¨ahler structures in the family J parametrized byTeichm¨uller space. This establishes the second statement of Theorem 1.2 on the Hitchin-Witten connection. We also expand the statement in the following theorem. Theorem 5.1.
For any t ∈ C with k = Re( t ) ∈ Z , the trivial bundle T Σ × C ∞ ( M s , L k ) over Teichm¨uller space, with fiber given by the smooth sections of the Chern-Simons linebundle over the moduli space of flat SU( n ) connections, has a projectively flat connectiongiven by ˜ ∇ V = ∇ T V + 12 t (∆ G ( V ) +2 ∇ G ( V ) · dF − λV ′ [ F ]) − t (∆ ¯ G ( V ) +2 ∇ ¯ G ( V ) · dF − λV ′′ [ F ])+ V [ F ] , for any vector field V on Teichm¨uller space. In this way, the quantum spaces arising from different real polarizations as above areprojectively identified through the parallel transport of the Hitchin-Witten connection.40 eferences [AE] S. T. Ali and M. Engliˇs. Quantization methods: a guide for physicists andanalysts.
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