Genus-one complex quantum Chern--Simons theory
Jørgen Ellegaard Andersen, Alessandro Malus?, Gabriele Rembado
aa r X i v : . [ m a t h . QA ] D ec GENUS-ONE COMPLEX QUANTUM CHERN–SIMONS THEORY
JØRGEN ELLEGAARD ANDERSEN, ALESSANDRO MALUSÀ, GABRIELE REMBADOA bstract . We consider the geometric quantisation of Chern–Simons theory forclosed genus-one surfaces and semisimple complex groups. First we introducethe natural complexified analogue of the Hitchin connection in Kähler quantisa-tion, with polarisations coming from the nonabelian Hodge hyper-Kähler geom-etry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, andwe identify it with the complexified Hitchin connection using a version of theBargmann transform on polarised sections over the moduli spaces. C ontents Introduction and main results 1Acknowledgements 61. Moduli spaces 72. Polarisations 103. Prequantisation and geometric quantisation 134. Complexified Hitchin connection 165. Hitchin–Witten connection 196. The Bargmann transform 207. Coordinates and frames 228. The L -connection 279. Conjugation of the Hitchin–Witten connection 2910. Identifications of the connections on the moduli spaces 34References 37I ntroduction and main results One of the many mathematical approaches to the quantisation of the Chern–Simons gauge theory [20, 21] consists in applying geometric quantisation to themoduli spaces of flat connections on a closed oriented smooth surface, comple-menting the original quantum field theoretic approach of [40]. This has been car-ried out for the structure group SU ( n ) by Hitchin [25] and Axelrod, Della Pietra,and Witten [8], and for its complexification SL ( n , C ) by Witten [41]. In both casesquantisation relies on the choice of a conformal/complex structure on the sur-face, resulting in families of quantum Hilbert spaces parametrised by Teichmüllerspace and identified, up to projective factors, via the holonomy of projectively flatconnections—the Hitchin connection for SU ( n ) and the Hitchin–Witten connec-tion for SL ( n , C ) (later reformulated in a more general setting in [2, 3]). However, the two constructions employ polarisations of rather different kinds: in the com-pact case, a Kähler structure on the moduli space is used, while the complexifiedcase involves a Lagrangian foliation. It is nonetheless the case that a family ofKähler polarisations also exists for SL ( n , C ) , coming from the hyper-Kähler non-abelian Hodge structure on the moduli space (also relying on a Riemann surfacestructure on the base). Hence one may try to define a complexified analogueof the Hitchin connection in this setting, which stands as an open problem forarbitrary genus.In this paper we unify these two viewpoints on the quantisation of complexChern–Simons theory for genus-one closed surfaces, by considering the naturalcomplexified analogue of the Hitchin connection, and relating it to the Hitchin–Witten connection.To draw a more precise picture let us briefly recall some aspects of the generalhigh-genus theory. For integers n , g > K = SU ( n ) , K C = SL ( n , C ) ,and let Σ be a smooth genus- g closed oriented surface. The moduli space M fl of isomorphism classes of irreducible flat K -connections on Σ may be obtainedby symplectic reduction on the affine space A of connections on Σ × K → Σ ,interpreting the curvature as a moment map for the action of the gauge group [7].Using the Chern–Simons action functional one may lift the gauge action to thetrivial Hermitian line bundle A × C → A (cf. § 3), and define a prequantum linebundle L → M fl .If, moreover, Σ is endowed with a Riemann surface structure, then M fl is en-riched with a Kähler structure, coming from (the principal bundle analogue of)the theorem of Narasimhan–Seshadri [28, 13], i.e. the identification M fl ≃ Bun K C with the moduli space of isomorphism classes of topologically trivial stable holo-morphic K C -bundles on Σ —and the Chern–Simons line bundle is then identifiedwith the Quillen determinant bundle [29]. One can then apply Kähler quantisa-tion for every level k ∈ Z > , resulting in the space of holomorphic sections of L ⊗ k → M fl (the space of nonabelian theta functions). Further, the moduli spaceof irreducible flat connections sits as a low-codimension smooth locus inside the compact singular moduli space of flat connections, and Hartog’s theorem can beused to extend holomorphic sections [25, § 5].Once the conformal structure on Σ varies, these spaces fit into a vector bundleover the Riemann moduli space of genus- g closed Riemann surfaces, on whichthe Hitchin connection is defined; equivalently they form a vector bundle overthe Teichmüller space of the underlying topological surface, equipped with amapping class group invariant projectively flat connection.Starting with the complex group K C instead yields a richer geometric struc-ture. In this case the Atiyah–Bott symplectic form is complex-valued, resultingin the complex symplectic de Rham space M dR of isomorphism classes of irre-ducible flat K C -connections, whose complex structure essentially comes from thatof Lie ( K C ) . The Chern–Simons action functional now depends on the choice ofa complex quantum level t = k + is , whose real part matches up with the levelin compact Chern–Simons theory, and consequently so does the lift of the gaugegroup action. This yields a Hermitian line bundle L C t → M dR with connection,prequantising a t -dependent real symplectic structure ω t on the moduli space. ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 3
A real polarisation can be introduced, relying again on a conformal/complexstructure on Σ , and the geometry of the integral Lagrangian foliation allows us toidentify polarised sections with their restriction to M fl ⊆ M dR . The T -dependenceof this construction is finally resolved by introducing the Hitchin–Witten connec-tion [41, 3].On the other hand Kähler polarisations can be obtained via nonabelian Hodgetheory [24, 14, 12, 35], also coming from the Riemann surface structure of Σ . Thereis a non-biholomorphic diffeomorphism M dR ≃ M Dol with the Dolbeault space ofisomorphism classes of stable K C -Higgs bundles on Σ , the nonabelian Hodge cor-respondence, endowing M dR with a second complex structure which can be com-pleted to a hyper-Kähler triple. This is a hyper-Kähler rotation within the Hitchinspace M of isomorphism classes of stable solutions of the self-duality/Hitchinequations on Σ [24], and in this viewpoint ω t ∈ Ω ( M dR , R ) becomes a Kählerform for a complex structure extracted from the hyper-Kähler sphere of M dR ≃ M , which finally defines Kähler polarisations. Moreover taking monodromy dataof topologically trivial stable local K C -systems on Σ yields a non-algebraic biholo-morphism M dR ≃ M B with the Betti space (an instance of the Riemann–Hilbertcorrespondence): the Betti space is the K C -character variety of Σ , completing thetriple of complex-valued nonabelian cohomologies on Σ [36, 37]. It yields themost useful description of the moduli spaces for our purpose.Hereafter we consider the exceptional case g =
1, where all these construc-tions need to be adjusted, mainly since the irreducible locus is now empty; how-ever, after the due changes the above theory goes through. In a trivialisation T ∗ Σ ≃ Σ × R one may consider the finite-dimensional subspace A ( C ) ⊆ A ( C ) of translation-invariant connections whose connection forms lie in the Lie subal-gebra of diagonal matrices. All the connections in these subspace are completelyreducible and flat, and they intersect all the gauge orbits of flat connections.Furthermore, two elements of A ( C ) are gauge-equivalent if and only if they arerelated by the action of a finitely-generated discrete subgroup K ⊆ K of the compact gauge group, so there are finite-dimensional descriptions M fl ≃ A (cid:14) K ֒ → A C (cid:14) K ≃ M dR .With some care this description applies to more general algebraic/Lie groupsthan SU ( n ) ⊆ SL ( n , C ) . Suppose K is a compact connected simply-connectedreal Lie group, and K ֒ → K C a complexification. Then the moduli space M fl offlat K -connections can be described exactly as above, up to replacing diagonalmatrices with elements of a maximal toral subalgebra t ⊆ Lie ( K ) . For the com-plexified group one can still define the finite-dimensional space A C using thecomplexification of t , but in general taking monodromy data yields an identifi-cation of the quotient A C (cid:14) K with the normalisation of the Betti space—viewedas a complex algebraic variety [38]. This is the viewpoint we take since it lendsitself to quantisation. Note the normalisation map is an isomorphism for all clas-sical groups [34] (hence including SL ( n , C ) for n > Hereafter a superscript “ ( C ) ” denotes presence/absence of a superscript “ C ”. JØRGEN ELLEGAARD ANDERSEN, ALESSANDRO MALUSÀ, GABRIELE REMBADO
The next goal is to introduce (projectively) flat connections to relate the quan-tum Hilbert spaces from different polarisations. For the more general case of thefamily of all linear
Kähler polarisations on a finite-dimensional symplectic vec-tor space, there are two natural constructions [8, 42]. The first, a close analogueof the Hitchin connection and a generalisation of [30, Chap. 4], is defined as acovariant differential whose potential is essentially the variation of the Laplace–Beltrami operator. The second relies on the fact that, among all square-summablesections, the holomorphic ones form a L -closed subspace: the Segal–Bargmannspace [10, 32]. Consequently a connection is defined by the trivial connectionand the orthogonal projections on the Hilbert subspaces (see [8, Eq. 1.42] andbelow). These two connections are known to agree and to be projectively flat. Wewould like to add that there are related, but different works in a similar contextin [9, 16, 17, 18, 26, 39].For the purpose of the quantisation of M dR we consider these two connectionson the covering space A C , and restrict them to the families of Kähler polarisationsarising from Teichmüller space. We will refer to them as the lifted complexifiedHitchin connection and the L -connection, respectively. Importantly, the formernaturally descends to a complexified Hitchin connection on the moduli space,since it is defined via differential operators constructed intrinsically from theLaplace–Beltrami operator. Then we show the following, by establishing algebraicrelations between those differential operators. Theorem 1 (§ 4) . The complexified Hitchin connection preserves holomorphicity. More-over it is flat and mapping class group invariant.
Next we investigate the relations between this Kähler quantisation scheme andWitten’s approach with real polarisations, i.e. with the genus-one Hitchin–Wittenconnection. Over the covering space the quantum Hilbert spaces originating fromgeometric quantisation are isometric through an explicit integral transform: theBargmann transform [23, 42]. In our case, having a T -family of polarisations ofeach kind, we introduce a T -family of Bargmann transforms, each identifying thetwo corresponding quantum Hilbert spaces. We lift the Hitchin-Witten connec-tion to act on the T -families of sections of the line bundle L k → A , and establishthe following—passing through the L -connection. Theorem 2 (§ 9) . The Bargmann transform intertwines the covariant derivatives ofthe lifted Hitchin–Witten and complexified Hitchin connections, on regular T -families ofsmooth sections. We move on to relate the quantum connections themselves, as intrinsicallydefined on the moduli space. To do this we first consider an analogue of theBargmann transform defined on the spaces of sections over the moduli spaces,using their identification with K -equivariant sections over the covers. Then weconsider the subspaces S and S C of Schwartz-class sections of the line bundles L k → A and L C t → A C (complex-analytic in the latter case). They are preservedby the Bargmann transform, so its transpose maps the spaces S ′ C and S ′ of tem-pered distributions into each other. Then we use the ideas of [1, § 3.5], i.e. thatsquare-summable polarised sections over the moduli spaces define regular tem-pered distributions, to obtain embeddings of the quantum Hilbert spaces inside S ′ C and S ′ , and to show that the transpose Bargmann transform reduces to the ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 5 moduli-space theoretic version of the Bargmann transform. Finally we considerthe dual versions of the Hitchin–Witten and complexified Hitchin connections,acting on T -families of tempered distributions, and we show the following. Theorem 3 (§ 10) . The dual connections are intertwined by the transpose Bargmanntransform, and they agree with the Hitchin–Witten and complexified Hitchin connectionson the subspaces of square-summable polarised sections over the moduli spaces.
Combining this with Thm. 2 we conclude that the (restricted) transpose of theBargmann transform intertwines the Hitchin–Witten and complexified Hitchinconnections, thereby unifying the two approaches to the quantisation of genus-one complex Chern–Simons theory. Moreover in the setup of Thm. 3 the regular-ity assumption of Thm. 2 are automatically satisfied.
Plan of the exposition.
The layout of this paper is the following.In § 1 we introduce the moduli spaces of flat connections, and give intrin-sic descriptions of their symplectic structure: first using the C ∞ /topologicalapproach à la Atiyah–Bott [7], then the algebraic approach à la
Goldman [22].The results are a real symplectic space ( M fl , ω ) , which is the quotient of a finite-dimensional real symplectic vector space ( A , ω ) , and a complex symplectic space (cid:0) M dR , J , ω C (cid:1) , which is the quotient of a finite-dimensional complex symplectic vec-tor space (cid:0) A C , J , ω C (cid:1) .We denote by M C fl the underlying real space of the de Rham space, which willin general be equipped with different complex structures than the de Rham one.In § 2 to § 5 we attach various objects to the moduli spaces and their covers A ( C ) ։ M ( C ) fl : polarisations, prequantum line bundles, (pre)quantum Hilbertspaces, and flat connections. The notation will be the same for the objects definedon the moduli spaces and their covers when the former are reductions of thelatter; we will however keep distinct notation for the functional Hilbert spaces.In § 2 we introduce Kähler polarisations in the compact and the complexifiedcase using a conformal structure on the surface and some nonabelian Hodgehyper-Kähler geometry; then we consider the real polarisations of [41] in thecomplexified case, depending on an almost-complex structure on the surface.The results are a family of complex structures I τ in the compact case, a family ofcomplex structures I τ , t in the complexified case, and finally a family of integrableLagrangian distributions P τ , also in the complexified case.In § 3 we prequantise all spaces using the Chern–Simons action functional [20,21]. The results are prequantum line bundles L C t and L k over the moduli spacesand their covers. Then with the polarisations of § 2 we define the standard geo-metric quantisation spaces. In the Kähler-polarised case we get Hilbert spaces H C τ , t ⊆ L (cid:0) M C fl , L C t (cid:1) , e H C τ , t ⊆ e L (cid:0) A C , L C t (cid:1) , H k ⊆ L ( M fl , L k ) and e H k ⊆ e L ( A , L k ) .Then we interpret the spaces L ( M fl , L k ) and e L ( A , L k ) as quantum Hilbertspaces for the real polarisations.In § 4 we define the complexified analogue of the Hitchin connection [25],following the formulation of [2, 30], and prove Thm. 1. The result are two con-nections, acting on T -families of holomorphic sections of the prequantum linebundles over M C fl and its cover via second order differential operators ∆ G C . JØRGEN ELLEGAARD ANDERSEN, ALESSANDRO MALUSÀ, GABRIELE REMBADO
In § 5 we give an explicit expression for the Hitchin–Witten connection [41],following the formulation of [3, 27]. The result are again two connections, actingon T -families of smooth sections of the prequantum line bundle over M fl and itscover via second order differential operators ∆ G ( V ) and ∆ G ( V ) .In §§ 6–9 we work on flat space.In § 6 we consider an abstract finite-dimensional affine symplectic space and fixnotation/conventions for the Bargmann transform [10], with values in the Segal–Bargmann space [32, 10], which we geometrically see as relating the quantumHilbert spaces for the real and Kähler polarisations [23]. The results are Hilbertspaces e H C h and e H P , which constitute models for the quantum spaces of sectionsover the cover A ( C ) ։ M ( C ) fl , and a unitary isomorphism B : e H P → e H C h .In § 7 we introduce coordinates on A ( C ) adapted to its affine euclidean struc-ture, we give local formulæ for the previously intrinsically defined tensors andpolarisations, and we compute the variation of some T -families of tensors.In § 8 we define the natural L -connection, acting on T -families of holomorphicsections of L C t → A C , and we give a proof of the fact it coincides with the liftedcomplexified Hitchin connection.In § 9 we use the T -family of Bargmann transforms derived from § 6 to relatethe L -connection of § 8 with the lifted Hitchin–Witten connection of § 5, therebyproving Thm. 2 (using the material of § 7).Finally in § 10 we introduce the dual pairs of Schwartz spaces ( S k , S ′ k ) and ( S τ , t , C , S ′ τ , t , C ) for sections of the line bundles over the covers A ( C ) , and embed e L k ֒ → S ′ k and e H C τ , t ֒ → S ′ τ , t , C . Then we consider the dual of the connections of §§ 4and 5, acting on T -families on sections of tempered distributions, as well as thetranspose Bargmann transform t B τ : S ′ τ , t , C → S ′ , and we prove Thm. 3.A cknowledgements All authors thank the former Centre for Quantum Geometry of Moduli Spaces(QGM) at the University of Aarhus, and the new Centre for Quantum Mathemat-ics (QM) at the University of Southern Denmark for hospitality and support.The first-named author was supported in part by the Danish National ScienceFoundation Center of Excellence grant, Centre for Quantum Geometry of Modulispaces, DNRF95 and by the ERC-SyG project, Recursive and Exact New QuantumTheory (ReNewQuantum) which receives funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 research and innovationprogramme under grant agreement No 810573.The second-named author thanks for their support the University of Toronto,the University of Saskatchewan, the Pacific Institute for the Mathematical Sciences(PIMS), the Centre for Quantum Topology and its Applications (quanTA), andNSERC, through the Discovery Grant of Steven Rayan, who also deserves creditfor many discussions.The third-named author was supported by the grant number 178794 of theSwiss National Science Foundation (SNSF), and by the National Centre of Com-pentence in Research SwissMAP, of the SNSF.
ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 7
1. M oduli spaces
Let Σ = T be the standard real 2-torus, K a compact connected simply-connected real Lie group and K ֒ → K C a complexification of K . Thus K C is aconnected simply-connected semisimple complex algebraic group, the Lie alge-bra k C := Lie (cid:0) K C (cid:1) is a complexification of its (semisimple) compact real form k := Lie ( K ) , and K is mapped to a maximal compact subgroup of K C —which weidentify with K .1.1. Flat connections.
As recalled in the introduction, classical Chern–Simonstheory [20, 21] attaches spaces of fields to the pair (cid:0) Σ , K ( C ) (cid:1) via symplectic re-duction of the affine space A ( C ) of connections on the trivial smooth principal K ( C ) -bundle P ( C ) = Σ × K ( C ) → Σ , with respect to the action of the smooth gaugegroup K ( C ) ≃ Ω (cid:0) Σ , K ( C ) (cid:1) [7]. In brief fixing the trivial connection as origin iden-tifies A ( C ) with the vector space of smooth differential 1-forms on Σ with valuesin the adjoint vector bundle Ad P ( C ) → Σ , and flat/integrable connections thencorrespond to 1-forms A ∈ A ( C ) such that F A := dA + [ A ∧ A ] = ∈ Ω (cid:0) Σ , k ( C ) (cid:1) , (1)where [ · ∧ · ] is the natural bracket on Lie algebra valued differential forms. Thenwe define the subspaces A ( C ) fl := (cid:10) A ∈ A ( C ) (cid:12)(cid:12)(cid:12) F A = (cid:11) ⊆ A ( C ) of flat connec-tions, as well as the quotient space M ( C ) fl = M fl (cid:0) Σ , K ( C ) (cid:1) := A ( C ) fl (cid:14) K ( C ) ,with respect to the flatness-preserving affine right gauge action A · g := Ad g − ( A ) + g ∗ θ ( C ) , for A ∈ A ( C ) , g ∈ K ( C ) , (2)where θ ( C ) ∈ Ω (cid:0) K ( C ) , k ( C ) (cid:1) is the left Maurer–Cartan form.The translation- and gauge-invariant symplectic structure we want to reduce isdefined on the tangent spaces of A ( C ) by the following nondegenerate alternatingbilinear form: e ω ( C ) ( A , B ) := Z Σ h A ∧ B i k ( C ) , for A , B ∈ Ω (cid:0) Σ , k ( C ) (cid:1) , (3)where h· ∧ ·i k ( C ) is the contraction with a suitable multiple of the Cartan–Killingform of k ( C ) [7]. Namely, the negative-definite pairing on the compact Lie al-gebra is such that the cohomology class of the canonical 3-form on K lies in H ( K , 2 π Z ) , and the pairing on the complexified Lie algebra is by definition thecomplexification of that on the compact form. We denote ω ( C ) the reduced forms, living on symplectic spaces with singularpoints corresponding to degenerate gauge orbits. Below we will describe themexplicitly. Then the imaginary exponential of the Chern–Simons action is gauge-invariant on closed 3-manifolds [20, § 2] (cf. § 3.1).
JØRGEN ELLEGAARD ANDERSEN, ALESSANDRO MALUSÀ, GABRIELE REMBADO
Remark . The space A C comes with a linear com-plex structure e J , induced from the complex structure of k C , for which the sym-plectic form (3) is of type (
2, 0 ) : e J ( α ⊗ X ) := α ⊗ ( iX ) , for α ∈ Ω ( Σ , R ) , X ∈ k C . (4)Suppose now A · K ( C ) ⊆ A ( C ) is the nondegenerate gauge orbit of the element A ∈ A ( C ) , corresponding to a smooth point [ A ] ∈ M ( C ) fl . Then a model for thetangent space at that point isT [ A ] M ( C ) fl = H A (cid:0) Σ , k ( C ) (cid:1) := H (cid:0) Ω • (cid:0) Σ , Ad P ( C ) (cid:1) , d A (cid:1) , (5)where d A is the connection on the adjoint vector bundle induced by A .Since e J commutes with d A , it induces a complex structure J on (5): it is the deRham complex structure. We will write M dR := (cid:0) M C fl , J (cid:1) the de Rham space, whichis the moduli space of flat K C -connections with its natural complex structure.Hence the notation M C fl refers to the underlying space (possibly equipped withdifferent complex structures, cf. Rem. 2.2). △ Betti viewpoint.
Flat connections on P ( C ) → Σ may be described as topo-logically trivial local K ( C ) -systems on Σ . These are classified by monodromy, sothere are identification with compact and complex character varieties of Σ : M fl ≃ Hom (cid:0) π ( Σ ) , K (cid:1)(cid:14) K , M dR ≃ M B = M B (cid:0) Σ , K C (cid:1) := Hom (cid:0) π ( Σ ) , K C (cid:1) (cid:12) K C ,(6)where the choice of a base point for the fundamental group is immaterial, andwhere on the right one takes a GIT quotient for complex algebraic varieties (asnot all K C -orbits are closed).To give a more explicit description introduce further Lie-theoretic data. Fixa maximal torus T ⊆ K , set t := Lie ( T ) ⊆ k , and let T C ⊆ K C be the connected(Cartan) subgroup with Lie algebra t C := t ⊗ R C ⊆ k C —a Cartan subalgebra.Denote N K ( T ) ⊆ K the normaliser of T in K and W := N K ( T ) (cid:14) T the Weyl group.Up to choosing a base point and two generating loops on Σ , a morphism π ( Σ ) → K ( C ) is uniquely determined by a pair of commuting elements in K ( C ) .In the case of K two such elements must sit in a maximal torus, which after aconjugation can be taken inside T ; then the residual conjugation amounts to thediagonal action of the Weyl group [11]. Using (6) thus yields a homeomorphism: M fl ≃ T × T (cid:14) W , (7)which yields a finite-dimensional description of the moduli space for the compactgroup. Moreover the reduction ω of (3) is now described as the reduction of thetranslation- and Weyl-invariant symplectic form on the Lie group T , which isdefined on the Lie algebra t ≃ T ∗ t via the Cartan–Killing duality t ≃ t ∗ . Thenotation will thus not distinguish the two.In the case of the complex group instead we explicitly restrict to representa-tions with values in the prescribed maximal algebraic torus T C , which in particu-lar have closed K C -orbits. This yields completely reducible representations, whereasthere are no irreducible representations [34] (see e.g. [33] for definitions). Proposition 1.1 (Normalisation of the Betti space) . The composition T C × T C ≃ Hom (cid:0) Z , T C (cid:1) −→ Hom (cid:0) Z , K C (cid:1) −→ M B ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 9 factorises through the (set-theoretic) quotient e M B := (cid:0) T C × T C (cid:1)(cid:14) W for the diagonalaction of the Weyl group. The resulting arrow χ : e M B → M B is an embedding and anormalisation map. Proof.
By [38, 34] χ is an embedding, it surjects on the irreducible componentof the trivial representation, and yields a normalisation of the irreducible compo-nent; but since K C is semisimple and simply-connected M B itself is irreducible [31]. (cid:3) Further e M B carries the reduction of the translation- and Weyl-invariant com-plex symplectic form on the Lie group T C × T C , just as in the compact case. Thismatches up with the genus-one Goldman symplectic structure, defined on anopen dense subset of M B , as the normalisation map is symplectic [34]. Similarlythe de Rham complex structure J on M dR ≃ M B matches up with the reductionof the Weyl-invariant complex structure on the Lie group T C × T C .Hereafter we work on the normal singular variety e M B ֒ → M B , with its givencomplex structure and complex symplectic form. For the sake of simplicity thenotation will not distinguish between the de Rham/Betti spaces and their normal-isation, neither for the spaces nor for the complex and symplectic structures.Thus we have the following finite-dimensional description of the complex sym-plectic space for the complexified group: M dR ≃ T C × T C (cid:14) W , (8)i.e. the complexified version of (7) from the compact case. Remark.
There are natural embedding/projection arrows ι : M fl ⇄ M C fl : π , sincethe diagonal Weyl group action on T ( C ) × T ( C ) commutes with the factorwiseprojection T C ։ T and inclusion T ֒ → T C . △ Finite-dimensional de Rham viewpoint.
We now give a finite-dimensionalpresentation of the moduli spaces in the de Rham viewpoint, which matches upwith (7) and (8).Let A ( C ) ⊆ A ( C ) fl be the subspace of connections with translation-invariant t ( C ) -valued connection form. The space A ( C ) contains strongly flat connections,i.e. connections A ∈ A ( C ) such that both d A = [ A ∧ A ] = t ( C ) × t ( C ) once atrivialisation T ∗ Σ ≃ Σ × R is chosen.It is easy to see that taking monodromy data along the two generating loopssurjects A ( C ) on the normalisation of the Betti space, via the exponential mapexp : t ( C ) → T ( C ) . The loss of injectivity is thus controlled by the action of the Z -submodule Λ := Ker ( exp ) ≃ H (cid:0) T ( C ) , Z (cid:1) , and by the residual diagonal W -action.Setting r = rk ( K ) := dim R (t) one has Λ ≃ Z r , sitting inside t ≃ R r as a (full-rank)lattice and t C ≃ t ⊕ i t as a half-rank Z -submodule.Hence it is possible to describe the moduli space as a quotient of the finite-dimensional vector space A ( C ) with respect to the action of Λ and W . To this end It is known [34] that χ is an isomorphism when K C is a classical group, in which cases one neednot discard any representation. for λ = ( λ , λ ) ∈ Λ consider the function f λ : ( x , y ) e xλ + yλ , well definedfrom Σ to T . Definition 1.1.
The restricted gauge group is the subgroup K ⊆ K generated by(1) the maps f λ : Σ → T above, for λ ∈ Λ ;(2) constant maps Σ → N K ( T ) ⊆ K .It follows from (2) that the K -action on A ( C ) corresponds to the natural affineaction of Λ ⋊ W on t ( C ) × t ( C ) —taking the semidirect product with respect tothe diagonal action of the Weyl group on Λ ⊆ t .Hence there is a new finite-dimensional presentation M ( C ) fl ≃ A ( C ) (cid:14) K . Inparticular the tangent bundle to the smooth locus is isomorphic to the trivialvector bundle with fibre A ( C ) , so one can make sense of translation-invarianttensors on the moduli spaces. The symplectic form ω ( C ) and complex structure J are translation invariant, as they come from gauge-invariant tensors on A ( C ) .Finally note the form (3) restricts on A ( C ) ≃ T ∗ t ( C ) to the canonical sym-plectic structure, and the complex structure (4) restricts on A C ≃ t C × t C to thenatural linear complex structure t C . We will thus denote ω ( C ) and J the symplec-tic/complex structures on the covering space A ( C ) ։ M ( C ) fl .No further notation will be introduced for the smooth loci of M fl and M C fl .2. P olarisations Let T = T Σ be the Teichmüller space of the real oriented closed surface Σ .We identity T with the upper-half complex plane H ⊆ C by considering thediffeomorphisms Σ ≃ X τ , where X τ = C (cid:14) ( Z ⊕ τ Z ) is a complex torus and τ ∈ H .Fix then τ ∈ T throughout this section.2.1. Kähler polarisations.
Hyperkähler structures.
The space of smooth k C -valued 1-forms on X τ carriesa linear complex structure defined by the C -antilinear extension ∗ : Ω p , q ( X τ , C ) → Ω − q ,1 − p ( X τ , C ) of the Hodge star operator ∗ : Ω ( X τ , R ) → Ω ( X τ , R ) of the flat torus—for { p , q } = {
0, 1 } . Moreover the contraction of e ω against ∗ (resp. of R e (cid:0) e ω C (cid:1) against ∗ ) yields a negative-definite symmetric bilinear form on A (resp. on A C ). Hence e I τ := − ∗ and e I ( C ) τ := − ∗ define linear complex structures on A and A C with as-sociated Kähler metrics e g ( C ) τ = e ω ( C ) · e I ( C ) τ —meaning the contraction of the leftindex of e I ( C ) τ with the right one of e ω ( C ) .Then we induce complex structures on the tangent spaces at smooth pointsof the moduli spaces, following Rem.1.1. Namely (abelian) Hodge theory on X τ yields an identification H A (cid:0) X τ , k ( C ) (cid:1) ≃ Ker ( ∆ A ) ,where ∆ A := d A d ∗ A + d ∗ A d A and d ∗ A := − ∗ d A ∗ . This space of harmonic forms is ∗ -stable, hence a reduced complex structure I ( C ) τ is intrinsically defined, and we ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 11 denote g ( C ) τ the associated Kähler metric. By construction I C τ anticommutes withthe de Rham complex structure J in the complexified case. Remark . Moreover I ( C ) τ is translation-invariant, whencethe Kähler metric too. Thus the Levi–Civita connection of the parallelisable Rie-mannian manifold (cid:0) M ( C ) fl , g ( C ) τ (cid:1) is trivial. △ In the finite-dimensional description of § 1.3 there are a natural restricted linearcomplex structure and a Kähler metric on A ( C ) ֒ → A ( C ) , denoted I ( C ) τ and g ( C ) τ (see (24) for formulæ in coordinates).On the whole there is a fibre bundle M fl → T of complex manifolds with fibre ( M fl , I τ ) for τ ∈ T , with a global trivialisation M fl ≃ M fl × T → T as smooth fibrebundle. It is a fibrewise quotient of the complex vector bundle A → T , whosefibre over τ ∈ T is the complex vector space (cid:0) A , I τ (cid:1) , with a global trivialisation A ≃ A × T → T as rank-2 r real vector bundle. (Recall we consider the smoothlocus within the quotient.)As recalled for the high-genus case in the introduction, the two anticommutingcomplex structures on M C fl can be completed to a hyper-Kähler triple with thecomplex structure K τ := I C τ · J , and the Kähler metric upgraded to a hyper-Kählermetric; equivalently, they can be seen as two complex structures on the hyper-Kähler Hitchin moduli space M τ = M (cid:0) X τ , K C (cid:1) . In genus one M τ can be realisedas the quotient of a finite-dimensional hyper-Kähler vector space, analogously to§ 1.3, and the hyper-Kähler metric and the complex structures will be explicitlydescribed by translation-invariant tensors in § 7.Hence in the complexified case there is no natural Kähler polarisation comingfrom a Riemann surface structure on Σ , but rather a hyper-Kähler 2-sphere worthof them: C P τ := (cid:10) aI C τ + bJ + cK τ (cid:12)(cid:12)(cid:12) q = ( a , b , c ) ∈ S ⊆ R (cid:11) ⊆ Ω (cid:0) M τ , End ( T M τ ) (cid:1) . (9)2.1.2. Extraction from hyper-Kähler spheres.
To get a genuine Kähler polarisation fixa complex number t = k + is ∈ R > × i R with positive real part, and considerthe t -deformed real symplectic form e ω t := R e ( t e ω C ) = k R e ( e ω C ) − s I m ( e ω C ) , (10)on A C . This is a Kähler form for the complex structure e I τ , t := k ′ e I C τ − s ′ e K τ ,where e K τ := e I C τ · e J and k ′ + is ′ = sgn ( t ) := t | t | ∈ U ( ) . The result is a newKähler vector space (cid:0) A C , e I τ , t , e ω t (cid:1) Kähler metric e ω t · e I τ , t = | t | e g C τ , and the complexstructure that corresponds to e I τ , t on A ⊆ A C is e I τ —with Kähler form k e ω , therestriction of e ω t to A . Definition 2.1 (Kähler polarisations) . We let I τ , t be the complex structure on thesmooth locus of M C fl , induced from the linear complex structure e I τ , t .The complex structure I τ , t is extracted from the hyper-Kähler sphere (9), andcorresponds to the point q = ( k ′ , 0, − s ′ ) ∈ S . Hence its Kähler form is the K C -reduction ω t of (10), whereas the Kähler metric is t -independent and alwayscoincides with the hyper-Kähler metric g C τ . Further I τ , t and ω t are the K -reduction of the restricted complex/symplecticstructures I τ , t and ω t on A C ⊆ A C . Remark . If s = I τ , t = I C τ : this is the Dolbeault structure onthe moduli space M Dol, τ = M Dol (cid:0) X τ , K C (cid:1) of (polystable) K C -Higgs bundles.If s = ± I C τ excluded. We denote (cid:0) M C fl , I τ , t , ω t (cid:1) the resulting Kähler manifold. △ Hence for every choice of t = k + is ∈ R > × i R with nonvanishing imaginarypart there is a local system M C fl, t → T of complex manifolds with fibres ( M C fl , I τ , t ) for τ in Teichmüller space, with a global trivialisation M C fl, t ≃ M C fl × T → T assmooth real fibre bundle. It is a fibrewise quotient of the complex vector bundle A C t → T , whose fibre over τ ∈ T is the complex vector space (cid:0) A C , I τ , t (cid:1) , with aglobal trivialisation A C t ≃ A C × T → T as rank-4 r real vector bundle.In case of vanishing imaginary part we have the fibre bundle M Dol → T ofDolbeault spaces. Its fibres are the complex manifolds (cid:0) M Dol, τ , I C τ (cid:1) , and again aglobal trivialisation is given by the underlying smooth manifold. This latter is thesetup of [4], which however considers the moduli stack. Remark.
One may symmetrically fix τ ∈ T and consider all Kähler polarisationscoming from (9) at once. One way is to consider the twistor space Z τ → C P of the hyper-Kähler Hitchin moduli space, a fibre bundle of complex manifoldsparametrised by the complex projective line. (That is, one may vary the Kählerpolarisation within hyper-Kähler spheres, and not between hyper-Kähler spheres.)We plan to pursue this viewpoint in subsequent work. △ Real polarisations.
The Riemann surface Σ ≃ X τ carries an (integrable)almost-complex structure, which defines a splitting of the space of k C -valuedsmooth differential forms: A C ≃ Ω (cid:0) X τ , k C (cid:1) ⊕ Ω (cid:0) X τ , k C (cid:1) .Both the subspace e P τ := Ω (cid:0) X τ , k C (cid:1) and its complement are e ω t -isotropic, where e ω t is as in (10) for some level t . This implies they are both Lagrangian, so e P τ defines a linear real polarisation on the real symplectic vector space ( A C , e ω t ) .If [ A ] ∈ M C fl is a smooth point, the Hodge decomposition for X τ yieldsT [ A ] M C fl = H A (cid:0) X τ , k C (cid:1) ≃ H A (cid:0) X τ , k C (cid:1) ⊕ H A (cid:0) X τ , k C (cid:1) ,using the A -twisted Dolbeault operator and Rem. 1.1. Hence e P τ ⊆ A C is reducedto the ω t -isotropic subspace P τ , [ A ] := H A (cid:0) X τ , k C (cid:1) ⊆ T [ A ] M C fl , which is La-grangian by dimension count. Further, since it comes from the linear polarisation e P τ , the distribution of such subspaces is translation-invariant, hence smooth andintegrable. Definition 2.2 (Real polarisations) . We denote P τ the real polarisation thus in-duced from the subspace e P τ ⊆ A C .We will denote P τ the linear real polarisation on the subspace A C ⊆ A C , whose K -reduction also yields P τ (see 7 for formulæ in coordinates). ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 13
Remark . Since Ω ( X τ , C ) ⊕ Ω ( X τ , R ) = Ω ( X τ , C ) one has e P τ ⊕ A = A C , so the subspace of K -connections is a complement to thereal polarisation e P τ . It follows that at a smooth point [ A ] ∈ M fl ⊆ M C fl one has P τ , [ A ] ∩ T [ A ] M fl = H A (cid:0) X τ , k C (cid:1) ∩ H A ( X τ , k) = ( ) ,so the tangent bundle T M fl ⊆ T M C fl is transverse to the real polarisation P τ , andby dimension count T M C fl ≃ T M fl ⊕ P τ . Hence the moduli space for the compactgroup is a global symplectic transverse to the real polarisation P τ . △ Remark (Mapping class group I) . The mapping class group Γ = Γ Σ of Σ actsnaturally on Teichmüller space. In our case Γ ≃ SL ( Z ) , and the action amountsto that of the modular group on the upper-half complex plane T ≃ H .The polarisations constructed in this section only depend on Γ -orbits of Teich-müller elements, i.e. on the (unmarked) Riemann surface structure chosen on Σ . In the Kähler-polarised setting this also holds for the Kähler metrics g C τ and g τ , as they are obtained from the complex structures via the contraction with a T -independent symplectic form. △
3. P requantisation and geometric quantisation
Choose again a complex number t = k + is ∈ R > × i R with positive real part.3.1. Reduction of prequantum data.
There are Chern–Simons theoretic prequan-tum data for the real symplectic manifolds ( M fl , kω ) and ( M C fl , ω t ) , compati-ble with the inclusion M fl ֒ → M C fl [20, 21]. Here we briefly recall the infinite-dimensional approach to their construction, for an easier description of the map-ping class group action (cf. Rem. 4.1).Let e L ( C ) := A ( C ) × C → A ( C ) be the trivial Hermitian line bundle on A ( C ) , andconsider the symplectic potential of (3) given by e α ( C ) A ( B ) := e ω ( C ) ( A , B ) , for A ∈ A ( C ) , B ∈ T A A ( C ) . (11)Now let e α t := R e (cid:0) t e α C (cid:1) = k R e (cid:0)e α C (cid:1) − s I m (cid:0)e α C (cid:1) ,which is a real 1-form on A C , and a symplectic potential for (10). It restrictsto k e α on A , which is a symplectic potential for k e ω . A parallel connection withcurvature − i e ω t is then defined on e L C by e ∇ t := d − i e α t , and with curvature − i e ω on e L by e ∇ := d − i e α .To reduce these prequantum data we consider an action of the gauge groupon the line bundles by fibrewise unitary maps, lifting the action on the base. Thisamounts to a map Θ C t : A C × K C −→ U ( ) satisfying cocycle conditions: Θ C t ( A , gg ′ ) = Θ C t ( A , g ) · Θ C t ( A · g , g ′ ) , for A ∈ A C fl , g , g ′ ∈ K C ,and the restriction to A × K will lift the K -action to e L → A .The cocycle is defined as Θ C t ( A , g ) := exp (cid:16) i · R e (cid:0) tS CS (cid:0) e A · e g (cid:1)(cid:1)(cid:17) ∈ U ( ) , (12) where S CS denotes the complex Chern–Simons action of the 3-manifold with bound-ary M := Σ × [
0, 1 ] , e g ∈ Ω (cid:0) M , K C (cid:1) is a homotopy between g ∈ K C and the iden-tity, and e A is the pull-back of A ∈ A C along the canonical projection M → Σ . Theintegrality condition on the nondegenerate pairing k C ⊗ k C → C assures (12) onlydepends on the boundary terms.Since S CS takes real values on A ⊆ A C (and coincides there with the action in compact Chern–Simons theory), (12) reduces to the following for ( A , g ) ∈ A × K : Θ C t ( A , g ) = exp (cid:16) ik · S CS (cid:0) e A · e g (cid:1)(cid:17) = (cid:0) Θ ( A , g ) (cid:1) k , (13)where Θ : A × K → U ( ) is in turn defined by Θ ( A , g ) := exp (cid:16) i · S CS (cid:0) e A · e g (cid:1)(cid:17) . Definition 3.1 (Prequantum line bundles) . We let L C t := A C fl × Θ C t C → M C fl bethe reduction of e L C with respect to the t -dependent action defined by (12), and L k := A fl × Θ k C → M fl be the reduction of e L with respect to the k -dependentaction defined by (13).By construction L k ≃ L ⊗ k , where L = A fl × Θ C → M fl , and since the ac-tions Θ C t and Θ are by fibrewise isometries there are naturally induced Hermitianmetrics on L C t and L k . The prequantum connection e ∇ t is Θ C t -invariant only for k ∈ Z > , in which case it induces a parallel connection ∇ t on L C . Similarly in thatcase e ∇ is Θ -invariant, so Θ induces the parallel connection ∇ —and Θ k induces k ∇ . The curvature of the reduced connections are F ∇ t = − iω t and F k ∇ = − ikω ,respectively (in our convention the form π ω is integral). The condition on thereal part of the quantum level thus becomes the prequantisation condition, andwe will refer to complex numbers t = k + is ∈ Z > × i R as admissible levels.Moreover the pull-back of the prequantum triple on M C fl along the embedding M fl ֒ → M C fl yields a Hermitian line bundle with parallel connection, whose firstChern class is k π [ ω ] ∈ H ( M fl , Z ) ; hence it is isomorphic to L k as Hermitianline bundle with connection, and we identify the two.As above there is a finite-dimensional presentation, in terms of the K -reductionof the pulled-back prequantum data on A ( C ) ⊆ A ( C ) . The resulting line bundleson the covering space will also be denoted L C t → A C and L k → A . As A ( C ) is contractible, these are trivial as hermitian line bundles, albeit equipped withnontrivial prequantum connections ∇ t and k ∇ .3.2. Prequantum spaces.
As customary, for an admissible level t = k + is we con-sider the pre-Hilbert spaces of square-integrable smooth sections of the prequan-tum line bundles, using the Liouville volume forms µ t := ω ∧ rt ( r ) ! ∈ Ω top (cid:0) M C fl , R (cid:1) and µ := ω ∧ r r ! ∈ Ω top ( M fl , R ) . Their completions are L C t = L (cid:0) M C fl , L C t (cid:1) andL k = L ( M fl , L k ) , respectively.Then with a view towards geometric quantisation we consider the trivial Hilbertbundles with these fibres, i.e. the productsLLL C t := L C t × T −→ T , LLL k := L k × T −→ T .Analogous Hilbert spaces/bundles are defined for sections of the prequantumline bundles over the covering space A ( C ) ։ M ( C ) fl , using the Liouville volume ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 15 forms µ t and µ of ( A C , ω t ) and ( A , ω ) . The resulting Hilbert spaces are written e L C t = L (cid:0) A C , L C t (cid:1) and e L k = L ( A , L k ) , and the resulting trivial Hilbert bundlesare e LLL C t := e L C t × T −→ T , e LLL k := e L k × T −→ T .3.3. Kähler quantisation.
Let further τ be a variable in T , and let I τ , t and I τ bethe complex structures on M C fl and M fl of §§ 2.1.1– 2.1.2 (recall for s = (
0, 1)-partof the prequantum connections square to zero and define holomorphic structureson the prequantum line bundles L C t and L of § 3, and we let H C τ , t := (cid:14) ϕ ∈ H (cid:0) M C fl , L C t (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z M C fl | ϕ | µ t < + ∞ (cid:15) ⊆ L C t , (14)and H τ , k := (cid:14) ϕ ∈ H (cid:0) M fl , L k (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z M fl | ϕ | µ < + ∞ (cid:15) ⊆ L k , (15)where norms of sections are taken with respect to the Hermitian structure of theline bundles. These are closed subspaces of the Hilbert spaces of L -sections,hence are Hilbert spaces themselves. They may be considered curved versions ofSegal–Bargmann spaces [10, 32] (to be introduced in § 6 in the flat case).Now the Teichmüller parameter may vary, and we consider smooth T -familiesof holomorphic sections. In the compact case this is the data of smooth maps ϕ : M fl × T → L k whose fibrewise restriction ϕ (cid:12)(cid:12) M fl × { τ } : ( M fl , I τ ) → L k is an I τ -holomorphic section for τ ∈ T ; similarly in the complexified case we considersmooth maps ϕ : M C fl × T → L C t such that ϕ (cid:12)(cid:12) M C fl × { τ } : (cid:0) M C fl , I τ , t (cid:1) → L C t is an I τ , t -holomorphic section. Hence intuitively we consider fibrewise Kähler quantisationof the fibre bundle M C fl, t and of its sub-bundle M fl, k .As above, analogous Hilbert spaces and T -families of holomorphic sections aredefined for the A ( C ) → M ( C ) fl . The resulting quantum Hilbert spaces are e H C τ , t and e H τ , k , defined analogously to (14) and (15), respectively.3.4. Real quantisation.
In the complexified case we also have the real polarisa-tions P τ . By Rem. 2.3 P τ is everywhere transverse to T M fl ⊆ T M fl C , so pullingback ∇ t -covariantly constant sections along M fl ֒ → M C fl yields an isomorphismwith the vector space of smooth sections of L k → M fl , and using this T -dependentidentification we let L k = L (cid:0) M fl , L k (cid:1) be the quantum space associated to P τ .Then the bundle arising from the fibrewise real quantisation of M C fl is the trivialHilbert bundle LLL k = L k × T → T . The same construction applies verbatim for thelinear real polarisation P τ on the covering space A C → M C fl . Remark.
The trivial bundle carries the trivial flat connection, but its trivialisationdepends on the T -dependent splitting T M C fl ≃ T M fl ⊕ P τ : hence one needs toconstruct a canonical projectively flat connection, as done in [41, 3] (see § 5; explicitformulæ are in § 9). △
4. C omplexified H itchin connection In this section we consider the natural genus-one complexified analogue of[25, 8].Fix an admissible quantum level t = k + is ∈ Z > × i R , and for τ ∈ T de-note T = T M C fl (resp. T = T M C fl ) the I τ , t -holomorphic cotangent bundle(resp. the I τ , t -holomorphic tangent bundle), and similarly for the antiholomor-phic parts. Set also T C := T ∗ M C fl ⊗ C (resp. T C := T M C fl ⊗ C ) for the complex-ified cotangent bundle (resp. complexified tangent bundle), so that T ⊆ T C ,T ⊆ T C , etc.If V is a vector field on T , differentiating the identity I τ , t = − Id shows thatthe derivative V (cid:2) I τ , t (cid:3) is a section of End ( T C ) swapping T and T . Then usingT C = T ⊕ T decompose V (cid:2) I τ , t (cid:3) = V (cid:2) I τ , t (cid:3) ′ + V (cid:2) I τ , t (cid:3) ′′ . (16)Since ω t is a non-degenerate form, for V as above there exists a unique bi-vector field e G C ( V ) such that e G C ( V ) · ω t = | t | V [ I τ , t ] .Using the fact that ω t has bi-degree (
1, 1 ) with respect to I τ , t , the decomposi-tion (16) yields an analogous splitting e G C ( V ) = G C ( V ) + G C ( V ) ,with G C ( V ) ∈ Ω (cid:0) M C fl , T ⊗ T (cid:1) and G C ( V ) ∈ Ω (cid:0) M C fl , T ⊗ T (cid:1) . Remark. If e g C τ denotes the inverse of the Kähler metric we have ω t · I τ , t = | t | g C τ ,which can be written e g C τ · ω t = − | t | I τ , t . Differentiating both sides of the lastrelation yields V [ e g C τ ] · ω t = − | t | V [ I τ , t ] , since ω is T -independent; hence we maywrite e G C ( V ) = − V (cid:2)e g C τ (cid:3) .In particular e G ( V ) is a symmetric tensor, and therefore so are G C ( V ) and G C ( V ) .One may then think of V G C ( V ) as a Sym ( T ) -valued 1-form on T . △ Remark.
The symplectic form and the complex structure are translation-invariant,so the tensor e G C ( V ) and its two components are themselves translation-invariant.Hence by Rem. 2.1 all three objects are parallel, and in particular the tensor G C ( V ) is holomorphic—we say the family of complex structures (cid:8) I τ , t (cid:9) τ ∈ T is rigid . △ Now we will define a second-order differential operator acting on smooth sec-tions of L C t → M C fl , as follows. For any symmetric tensor e G ∈ Ω (cid:0) M C fl , Sym T C (cid:1) ,denoting e G : T C → T C the arrow induced by contraction, consider the followingcomposition: Ω (cid:0) M C fl , L C t (cid:1) Ω (cid:0) M C fl , T C ⊗ L C t (cid:1) Ω (cid:0) M C fl , T C ⊗ L C t (cid:1) Ω (cid:0) M C fl , T C ⊗ T C ⊗ L C t (cid:1) ∇ t e G ∇ t Tr ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 17
In this diagram we act on sections of T C ⊗ L C t → M C fl via the tensor product of(the complexification of) the Levi–Civita connection of the Riemannian manifold (cid:0) M C fl , g C τ (cid:1) and the prequantum connection—and we abusively keep the notation ∇ t for it. The resulting operator will be denoted ∆ e G := Tr (cid:0) ∇ t e G ∇ t (cid:1) , where “Tr”means contraction of all the upper indices with all the lower ones—unambiguous,since e G is symmetric.In particular there is a Laplacian operator for the holomorphic tensor G C ( V ) .Letting V vary yields a 1-form u C := − ∆ G C ( • ) on T , with values in differentialoperators acting on smooth sections of the prequantum line bundle. This correctsthe trivial connection ∇ Tr , living on the trivial bundle LLL C t ≃ L C t × M C fl → T ,which does not preserve T -families of holomorphic sections. Theorem 4.1.
The connection ∇ C := ∇ Tr − | t | u C preserves holomorphicity, and it isflat. See § 8 for formulæ in coordinates. Throughout the proof of Thm. 4.1 we willuse the following identities, valid for any (local) vector field V on T : V (cid:2) ∇ t (cid:3) = V [ ω t ] = = ∇ t ω t = ∇ t G C ( V ) , (cid:2) ∇ t , ∇ t (cid:3) = F ∇ t = − iω t = (cid:2) ∇ t , ∇ t (cid:3) = F ∇ t = − iω t . (17)They come from the following facts: ∇ t and ω t are independent of τ , ω t is aKähler form, G C ( V ) is translation-invariant, and F ∇ t = − iω t is of bidegree (
1, 1 ) . Proof.
Since G C ( V ) is of type (
2, 0 ) the corresponding operator acts as ∆ G C ( V ) = Tr (cid:16) ∇ t G C ( V ) ∇ t (cid:17) .To prove the first statement choose a T -family of holomorphic sections ϕ , a(local) vector field X on M C fl , and a (local) vector field V on T . We must establishthe identity ∇ t , X (cid:0) V [ ϕ ] (cid:1) = | t | ∇ t , X (cid:0) ∆ G C ( V ) ϕ (cid:1) .Using ∇ t ϕ = ∇ t , X ∆ G C ( V ) ϕ = Tr (cid:16)(cid:2) ∇ t , X , ∇ t (cid:3) G C ( V ) ∇ ϕ + ∇ t G C ( V ) (cid:2) ∇ t , X , ∇ t (cid:3) ϕ (cid:17) .Since the Levi–Civita connection is trivial both commutators are controlled bythe curvature − iω t of the pre-quantum line bundle, leading to the contraction − iX · ω . Then the symmetry of G C ( V ) and (17) yield ∇ t , X ∆ G C ( V ) ϕ = − iX · (cid:0) ω t · G C ( V ) (cid:1) · ∇ t ϕ = i | t | (cid:0) V (cid:2) I τ , t (cid:3) X (cid:1) · ∇ t ϕ ,where in the last passage we further use ∇ t ϕ = ∇ t ϕ , (cid:0) V (cid:2) I τ , t (cid:3) X (cid:1) ∇ t = (cid:0) V (cid:2) I τ , t (cid:3) ′ X (cid:1) ∇ t ,as in both cases the antiholomorphic parts do not contribute.For the other term, differentiating the identity ∇ t , X ϕ = V yields0 = V (cid:2) ∇ t , X (cid:3) ϕ + ∇ t , X V [ ϕ ] = i (cid:0) V (cid:2) I τ , t (cid:3) X (cid:1) · ∇ t ϕ + ∇ t , X V [ ϕ ] , Hence [30, Rem. 4.16] is vindicated. using ∇ t , X = (cid:0) ( + iI τ , t ) X (cid:1) · ∇ t , V (cid:2) ∇ t , X (cid:3) = i (cid:0) V (cid:2) I τ , t (cid:3) X (cid:1) · ∇ t ,and the desired identity is proven.For the second statement, if V ′ is a (local) vector field on T that commutes with V , the curvature reads (cid:10) F ∇ C , V ∧ V ′ (cid:11) = − | t | (cid:16) V ′ (cid:2) ∆ G C ( V ) (cid:3) − V (cid:2) ∆ G C ( V ′ ) (cid:3)(cid:17) + | t | (cid:2) ∆ G C ( V ) , ∆ G C ( V ′ ) (cid:3) ,(18)and we will show that both summands vanish.For the former, since e G ( V ) = − V (cid:2)e g C τ (cid:3) it follows that ∆ G ( V ) = − V (cid:2) ∆ e g C τ (cid:3) asdifferential operators acting on holomorphic sections of L C t . This is becausethe Levi–Civita connection is T -independent, and because the contraction of T C against T annihilates T . Hence V ′ (cid:2) ∆ G C ( V ) (cid:3) − V (cid:2) ∆ G C ( V ′ ) (cid:3) = − V ′ V (cid:2) ∆ e g C τ (cid:3) + VV ′ (cid:2) ∆ e g C τ (cid:3) = (cid:2) V , V ′ (cid:3)(cid:2) ∆ e g C τ (cid:3) = (cid:2) ∇ t G C ( V ) ∇ t , ∇ t G C ( V ′ ) ∇ t (cid:3) , which vanishes because of the identi-ties (17), and because the contractions with G C ( V ) and G C ( V ′ ) commute. Hence (cid:2) ∆ G C ( V ) , ∆ G C ( V ′ ) (cid:3) = Tr (cid:2) ∇ t G C ( V ) ∇ t , ∇ t G C ( V ′ ) ∇ t (cid:3) = ∇ C . (cid:3) The flat connection of Thm. 4.1 should be compared with the Hitchin con-nection [25, 8], which acts on sections of H k —and was originally defined for K = SU ( n ) . Namely, the action of ∆ G C ( V ) on sections of L ⊗ k ≃ ι ∗ L C t → M fl isthe same as that of the Laplacian operator ∆ G ( V ) constructed out of variations ofthe complex structure I τ on M fl , i.e. the Laplacian constructed from the tensor G ( V ) ∈ Ω (cid:0) M fl , Sym ( T M fl ) (cid:1) defined by G ( V ) · ω = V (cid:2) I τ (cid:3) ′ . Then the connec-tion ∇ C of Thm. 4.1 pulls back to the connection ∇ Tr − k u , where u ( V ) := ∆ G ( V ) .Now note the first Chern class of ( M fl , kω ) —i.e. the first Chern class of thetrivialisable complexified cotangent bundle T C M fl —vanishes, as well as the Riccipotential for the family of Riemannian manifolds ( M fl , g τ ) . Hence indeed Eq. 1of [2, Thm. 1] matches up with ∇ . Thus using [2, Thm. 13] we see that ∇ C is agenus-one complexification of the original Hitchin connection, for any semisim-ple connected simply-connected complex algebraic group. We thus refer to theflat connection of Thm. 4.1 as the complexified Hitchin connection.
Remark . Let Diff + ( Σ ) be the group of positive diffeo-morphisms of the oriented surface Σ , whose quotient by the isotopy componentof the identity is the mapping class group Γ .The Γ -action on T lifs to the fibre bundle M C fl, t → T , since the gauge class ofthe pull-back of a connection on P C → Σ along a diffeomorphism Σ → Σ onlydepends on the mapping class of this latter. Further the symplectic pairing (3) ispreserved under the full Diff + ( Σ ) -action, as well as the symplectic potential (11).It follows e.g. from [5, Lem. 3] that the trivial lift of the Diff + ( Σ ) -action on e L ( C ) combines with the lift of the gauge action of § 3 to yield a lift of the action ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 19 of Aut (cid:0) P ( C ) (cid:1) ≃ K ( C ) ⋊ Diff + ( Σ ) . In particular the trivial lift of the action ofDiff + ( Σ ) ≃ Aut (cid:0) P ( C ) (cid:1)(cid:14) K ( C ) is compatible with the lifted gauge actions of § 3.1,and it fixes the (trivial) Hermitian metric and the prequantum connection. Hencethere are induced actions on the reduced line bundles, and further the isotopycomponent of the identity acts trivially there (see [5, Prop. 3.10], which considersthe more general case of a nonnegative number of punctures on Σ ). Hence in briefthere is a Γ -action on the prequantum line bundles over the moduli spaces, liftingthe symplectic actions on the moduli spaces and preserving all prequantum data.This yields an action on sections of the prequantum line bundle, whence finallya Γ -action on T -families of sections of the prequantum line bundles.Finally the complexified Hitchin connection of Thm. 4.1 is Γ -invariant, as it isbased on the variation of the Laplace–Beltrami operator for the mapping classgroup equivariant Kähler metric (cf. [2, Lem. 6]). △ The constructions of this section can also be carried out on the covering space A C ։ M C fl . This produces a Γ -invariant flat connection acting on T -families ofholomorphic sections of L C t → A C , which we refer to as the lifted complexifiedHitchin connection, also denoted ∇ C = ∇ Tr − | t | u C .5. H itchin –W itten connection In this section we give an explicit formula for the genus-one connection of [41].Fix an admissible quantum level t = k + is ∈ Z > × i R . Recall from § 3.4that the bundle for the geometric quantisation of M C fl with respect to the realpolarisations P τ of § 2.2 is the trivial bundle LLL k = L k × T Σ → T Σ , and that wewant to construct canonical identifications between its fibres (besides the onesgiven by the T -dependent trivialisation).To this end consider the symmetric tensor e G ( V ) ∈ Ω (cid:0) M fl , Sym ( T C M fl ) (cid:1) de-fined analogously to e G C ( V ) , namely by the relation e G ( V ) · ω = V (cid:2) I τ (cid:3) = V (cid:2) I τ (cid:3) ′ + V (cid:2) I τ (cid:3) ′′ ,for a vector field V on Teichmüller space. It has a decomposition into holomorphicand antiholomorphic part, written e G ( V ) = G ( V ) + G ( V ) , and using these tensorswe define Laplacian operators ∆ G ( V ) and ∆ G ( V ) analogously to § 4: ∆ G ( V ) := Tr (cid:0) ∇ G ( V ) ∇ ) , ∆ G ( V ) := Tr (cid:0) ∇ G ( V ) ∇ (cid:1) .Then we consider the connection ∇ HW = ∇ Tr − u HW , where u HW ( V ) := t ∆ G ( V ) − t ∆ G ( V ) , (19)acting on T -families of smooth sections. As a consequence of the flatness ofthe Riemannian metric g τ (and vanishing of the Ricci tensor), the above is aparticular instance of the connection studied in [41], i.e. the genus-one analogueof [3] (which considers the higher-genus case): we refer to it as the Hitchin–Witten connection.The tensor calculus developed in [3] applies to the genus-one case as well,and can be used to deduce flatness, analogously to the proof of Thm. 4.1 (seealso [27, 6]). Finally a density argument enables to extend the parallel transportoperators of (19) to the whole of the Hilbert space L k , e.g. using the trivialisation of the connection as in Eq. 1 of [6] (which also holds when the compact group issemisimple). Remark (Mapping class group III) . Analogously to Rem. 4.1, the Hitchin–Wittenconnection (19) is invariant for the group of bundle automorphisms of LLL k → T Σ defined by the mapping class group. △ The same construction can be carried out on the covering space A ։ M fl ,producing a Γ -invariant flat connection acting on T -families of smooth section of L K → A , which we refer to as the lifted Hitchin–Witten connection also denoted ∇ HW = ∇ Tr − u HW . 6. T he B argmann transform In this section and in §§ 7–9 we work on flat space, with a view towards A ( C ) .Here in particular we summarise some fundamental facts about the Bargmanntransform [10] on an abstract finite-dimensional vector space, so as to fix notationand coefficients (see also [23, 42]).For an integer m > C m with complex coordinates z j = p j + iq j , thestandard real symplectic form ω = m X j = d p j ∧ d q j = i m X j = d z j ∧ d z j ,and the induced Liouville volume form; then take the Kähler polarisation comingfrom the natural complex structure and the real polarisation defined by the La-grangian subspace P = R m ⊆ C m . There is a prequantum line bundle L C h → C m for every level h >
0, with Hermitian metric h : L C h × L C h → C and h -compatibleconnection ∇ h , unique up to isomorphism and topologically trivial. It is cus-tomary to choose the trivialisation so that the prequantum connection reads ∇ h = d − i h α , with α ∈ Ω ( C m , R ) the invariant symplectic potential determinedas in (11), i.e. in coordinates α = i m X j = (cid:0) z j d z j − z j d z j (cid:1) = m X j = (cid:0) p j d q j − q j d p j (cid:1) .It is easily checked that the smooth functions σ := (cid:18) π h (cid:19) m/ exp (cid:18) − h | z | (cid:19) , ρ := exp − i h m X j = p j q j , (20)are covariantly constant along the Kähler and real polarisation, respectively, andnowhere vanishing, thus providing polarised frames for the prequantum linebundle.The Hilbert space e H C h for the Kähler quantisation of consists of square-summableholomorphic sections of L C h . Using the frame σ these can be identified with or-dinary holomorphic functions, square-summable with respect to the Gaussian The level thus corresponds to the inverse module 1 / | t | > ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 21 measure defined by the square norm of σ . Explicitly, the L -product reads h fσ | gσ i = ( π h ) m Z C m f ( z ) g ( z ) exp (cid:18) − h | z | (cid:19) dvol z ,for square-summable holomorphic functions f , g : C m → C .Then the annihilation/creation operators a j := h ∂∂z j and a † j := µ z j of deriva-tion/multiplication by the complex variables are mutually adjoint. As a usefulconsequence, if π f H denotes the orthogonal projection of the space of all square-summable functions to the closed subspace e H C h , one has that π f H ( z j f ) = h ∂∂z j f . (21) Remark (Functions vs sections) . The space of holomorphic functions with finiteL -norm with respect to the Gaussian measure is the Segal–Bargmann space [10,32]. We will use the two viewpoints of functions and sections at convenience,and denote elements of e H C h as f or ϕ in the respective cases, related by the frameinsertion ϕ = fσ . △ Analogously using the frame ρ one may identify P -polarised sections withcomplex-valued functions of the variables q j alone, i.e. functions on the quotient Q = C m (cid:14) P . The intrinsic definition of the quantum Hilbert space e H P for the realpolarisation uses half-forms, but up to appropriate choices it can be identifiedwith the usual L ( Q , C ) —with respect to the standard Lebesgue measure on theglobal transverse Q ≃ i R m ⊆ C m to P . In what follows we will denote by ψ either a function on Q , or its polarised extension to C m , corresponding to thepolarised section ψρ —the two agree on Q since ρ (cid:12)(cid:12) Q ≡ ( f , ψ ) as above, correspondingto a pair of polarised sections ( ϕ , ψρ ) = ( fσ , ψρ ) , the integral h fσ | ψr i = Z C m f ( z ) ψ ( q ) h ( r , σ )( z ) dvol z defines a nondegenerate sesquilinear pairing e H P ⊗ e H C h → C . Dual to this pairingthere is a linear map B : e H P → e H C h , the Bargmann transform. Since e H C h is areproducing kernel Hilbert space, i.e. because of the identity f ( z ) = ( π h ) m/ Z C m f ( w ) exp (cid:18) h z · w (cid:19) exp (cid:18) − h | w | (cid:19) dvol w ,the Bargmann transform can be written explicitly in integral form as (cid:0) B ( ψ ) (cid:1) ( z ) := ( π h ) m/ Z R m ψ ( q ) exp (cid:18) − h (cid:0) i q · z + | q | − z · z (cid:1)(cid:19) d m q , (22)where a · b := P a j b j is the C -bilinear pairing. Moreover in this normalisa-tion the Bargmann transform is a unitary isomorphism between the quantumHilbert spaces. These formulæ are equivalent to those of [23, Chap. V, § 7] (or [10,Eq. 1.4]), only that we insist in using an invariant symplectic potential and thatwe parametrise differently the complex coordinates.Regarding e H P and e H C h as modules for the m -dimensional Weyl algebra (in theSchrödinger position representation and the Fock representation, respectively), the transform (22) realises the unitary isomorphism of the Stone–Von Neumanntheorem, in the following sense. Proposition 6.1 (cf. [10], § 1.8.i) . Suppose that ψ ∈ e H P is a smooth function such that q j and ∂ψ∂q j are also square-summable for a fixed j . Then B ( ψ ) lies in the domain of theanti/self-adjoint operators a j ± a ∗ j and one has B (cid:0) q j ψ (cid:1) = i ( a j − a ∗ j ) B ( ψ ) , B ∂ψ∂q j ! = i h ( a j + a ∗ j ) B ( ψ ) . (23)Prop. 6.1 will be used in § 9 to relate the lifted Hitchin–Witten connection (19)with the L -connection (defined in § 8), hence with the lifted complexified Hitchinconnection. Remark . In the general case of a symplectic vector space endowed with twolinear polarisations, real and Kähler respectively, one may get back to the presentsetup by choosing a real basis ( X , . . . , X m ) for the Lagrangian subspace P defin-ing the real polarisation, orthonormal with respect to the Kähler metric, andregarding it as a C -basis of the whole space. △
7. C oordinates and frames
The description of the moduli spaces of § 1 allows for natural local coordinateson it. In this section we define them and fix conventions for later use. The samediscussion is presented in further detail for the archetypal case of K = SU ( ) and K C = SL ( C ) in [27, 15, 30].Consider on Σ the coordinates ( x , y ) induced by the identification Σ ≃ R / Z .Let again t ⊆ k be a maximal toral/abelian subalgebra and t C ⊆ k C its complexi-fication, and recall r = rk ( K ) is the real dimension of t . The choice of a real basis ( T , . . . , T r ) of t , hence of a complex one for t C , induces global linear coordinates w = ( w , w ) on A C , with w := ( w , . . . , w r ) and w := ( w r + , . . . w r ) , definedby w r X j = (cid:0) w j T j d x + w j + r T j + r d y (cid:1) ,which are C -linear for the natural complex structure J on A C . Correspondingly, if w j = u j + iv j , the u j ’s define real coordinates on A ; we will write w j = u j + i v j and w = u + i v . We then have that ω C ∂∂w j , ∂∂w l + r ! = Z Σ h T j dx ∧ T l dy i k C = h T j , T l i k C , for j , l ∈ {
1, . . . , r } ,and if further the basis ( T , . . . , T r ) is h· , ·i k C -orthonormal then ω C = r X j = d w j ∧ d w r + j .In other words, these coordinates trivialise the complex structure J and the sym-plectic form ω C on A C , as well as its restriction to the real symplectic subspace ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 23 A . They do not, however, trivialise the additional structure induced by a choiceof τ ∈ T : for this reason we introduce new T -dependent coordinates.Write τ = τ + iτ ∈ T ≃ H a point in Teichmüller space, corresponding to acomplex structure on Σ with complex coordinate ζ τ := x + τy . Then the inducedHodge ∗ -operator acts on 1-forms on the flat torus Σ ≃ X τ = C (cid:14) ( Z ⊕ τ Z ) as ∗ d x = τ (cid:0) τ d x + | τ | d y (cid:1) , ∗ d y = − τ (cid:0) d x + τ d y (cid:1) . (24)Further one has h ∗ τ ( d ζ τ , d ζ τ ) = τ ,where h ∗ τ is the dual of the Riemannian metric h τ on X τ . Then a orthonormal ba-sis of the real polarisation P τ ⊆ A C (as a complex vector space with the structure J ) is given by the vectors X j := T j √ τ d ζ τ for j ∈ {
1, . . . , r } . (25)Suppose now t = k + is ∈ Z > × i R is an admissible quantum level, thusselecting a Kähler structure on A C with symplectic form ω t and complex struc-ture I τ , t as in § 2. We then construct a ( t - and τ -dependent) frame for A C byconsidering the vectors X j in (25) together with X j + r := JX j for 1 j r , Y j := I τ , t X j for 1 j r . (26)Denote by ( p , q ) := ( p , p , q , q ) the corresponding linear coordinates, with p = ( p , . . . , p r ) , p = ( p r + , . . . , p r ) , q = ( q , . . . , q r ) and q = ( q r + , . . . , q r ) .Then write A ( τ , p , q ) = √ τ r X j = T j · (cid:16) ( p j + ip j + r ) d ζ τ + ( q j + iq j + r ) d ζ τ (cid:17) ,as well as z j := p j + iq j , with the usual convention for z = ( z , z ) .It is convenient to introduce the following notation. Definition 7.1.
We will denote by δδτ the coordinate vector field on T Σ × A C associated with τ in the coordinate frame ( τ , p , q ) , and similarly for δδτ . Remark . This means that δδτ = ∂∂τ − r X j = ∂p j ∂τ ∂∂p j + ∂q j ∂τ ∂∂q j ! = ∂∂τ − r X j = ∂z j ∂τ ∂∂z j + ∂z j ∂τ ∂∂z j , δδτ = ∂∂τ − r X j = ∂p j ∂τ ∂∂p j + ∂q j ∂τ ∂∂q j ! = ∂∂τ − r X j = ∂z j ∂τ ∂∂z j + ∂z j ∂τ ∂∂z j . (27)Note that differentiation of functions along these vector fields preserves polarisedfunctions/sections, because for every j ∈ {
1, . . . , 2 r } they commute with ∂∂p j and ∂∂q j , and therefore with ∂∂z j . △ Introducing the following notations will make the formulæ in the next sectionsmore compact.
Definition 7.2.
For j ∈ {
1, . . . , r } we define operators M j ψ := ( q j + iq j + r ) ψ , D j ψ := | t | ∂∂q j + i ∂∂q j + r ! ψ , µ j f := ( z j + iz j + r ) f , δ j f := | t | ∂∂z j + i ∂∂z j + r ! f .acting on smooth functions A C → C .One may check that the two operators on each row commute with one anotherfor j ∈ {
1, . . . , r } , and that (23) becomes B ◦ M j = i ( δ j − µ j ) ◦ B and B ◦ D j = i ( µ j + δ j ) ◦ B . (28) Remark . The transition between the two coordinate systems ( p , q ) and ( u , v ) can be obtained from the identity d ζ τ = d x + τ d y , and from (25) and (26). Inwhat follows we will only need the relations q j = | t | √ τ (cid:0) −( kτ − sτ ) u j + ( kτ + sτ ) v j + ku j + r − sv j + r (cid:1) , q j + r = | t | √ τ (cid:0) ( kτ + sτ ) u j + ( kτ − sτ ) v j − su j + r − kv j + r (cid:1) , (29)and v j = √ τ (cid:18) p j + r + k | t | q j − s | t | q j + r (cid:19) , v j + r = √ τ (cid:18) τ p j + τ p j + r + kτ − sτ | t | q j − sτ + kτ | t | q j + r (cid:19) ,for j ∈ {
1, . . . , r } . In particular the subspace A ⊆ A C has equations p j = s | t | q j + k | t | q j + r , p j + r = − k | t | q j + s | t | q j + r . (30)in coordinates ( p , q ) . △ The frame constructed above is of the form described in Rem. 6.1. We canthen apply the discussion there to compare the quantisation on this linear spacewith the two different polarisations, using ω t / | t | as the symplectic structure—simplifying the coefficient of the Kähler metric—and setting h = / | t | . For τ ∈ T we use (20) to define polarised frames σ τ and ρ τ , while the Bargmann transforms B τ defined as in (22) provide identifications between the two quantum Hilbertspaces. As τ varies one may informally think of the collection { B τ } τ ∈ T as amorphism of the vector bundles of quantum Hilbert spaces.7.1. Variations over Teichmüller space.
The relations in (29) are differentiated toobtain ∂q j ∂τ = − τ q j + r − t τ | t | ( p j + ip j + r ) , ∂q j + r ∂τ = τ q j − it τ | t | ( p j + ip j + r ) (31) ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 25 for j ∈ {
1, . . . , r } , and similarly ∂z j ∂τ = − it τ | t | z j − τ z j + r + t τ | t | z j + r , ∂z j ∂τ = it τ | t | z j − t τ | t | z j + r − τ z j + r , ∂z j + r ∂τ = τ z j + t τ | t | z j + it τ | t | z j + r , ∂z j + r ∂τ = − t τ | t | z j + τ z j − it τ | t | z j + r . (32)The variations in τ are easily deduced from the above by conjugation. Definition 7.3.
For j ∈ {
1, . . . , r } we set X j := √ τ ∂∂u j + τ ∂∂u j + r ! ∈ A ⊗ R C ⊆ A C ⊗ R C . Remark.
The above are defined formally in the same way as the vectors X j (cf. (25)),except they are thought of as complex objects tangent to A rather than real objectstangent to A C . As such, they are of type (
0, 1 ) (in fact anti-holomorphic) withrespect to the complex structure I τ , so in particular I τ X j = − i X j . △ Lemma 7.1. If e g τ denotes the inverse of g τ , then e G (cid:18) ∂∂τ (cid:19) = − ∂ e g τ ∂τ = − iτ r X j = X j ⊗ X j , e G (cid:18) ∂∂τ (cid:19) = − ∂ e g τ ∂τ = iτ r X j = X j ⊗ X j . Proof.
This is proven in [15, 27] for K = SU ( ) , in which case r =
1. The generalcase follows, since A can be decomposed as the orthogonal sum of r copies forthe rank-one case. (cid:3) Corollary 7.1.
The derivatives of g τ along τ and τ read ∂g τ ∂τ ( A , B ) = − iτ r X j = g (cid:16) X j , A (cid:17) g (cid:16) X j , B (cid:17) , ∂g τ ∂τ ( A , B ) = iτ r X j = g (cid:0) X j , A (cid:1) g (cid:0) X j , B (cid:1) . Proof.
By the usual formula for the derivative of the inverse matrix, we have that ∂g τ ∂τ = − g τ · ∂ e g τ ∂τ · g τ = g τ · e G (cid:18) ∂∂τ (cid:19) · g τ ,and the result follows. The derivative in τ is obtained the same way. (cid:3) Lemma 7.2.
The covariant derivative with respect to the Hitchin–Witten connection isdetermined by ∇ HW τ = ∂∂τ − i tτ r X j = ∇ X j ∇ X j , and ∇ HW τ = ∂∂τ − i tτ r X j = ∇ X j ∇ X j .(33) Proof.
Again, this can be deduced from the particular case of K = SU ( ) discussedin [15, 27]. To show it explicitly note that G ( ∂∂τ ) = G ( ∂∂τ ) , and it follows from thedefinition (19) of u HW that u HW (cid:18) ∂∂τ (cid:19) = − t ∆ G (cid:16) ∂∂τ (cid:17) and u HW (cid:18) ∂∂τ (cid:19) = t ∆ G (cid:16) ∂∂τ (cid:17) .If G τ := G ( ∂∂τ ) then it follows from Lem. 7.1 that G τ has constant coefficients, andtherefore it is covariantly constant since the Levi-Civita connection of ( A , g τ ) istrivial. Recalling the construction of ∆ G τ , its action on a section ψ of L k isobtained by contracting ∇ ψ with G τ , taking covariant differential of the result,and then contracting the remaining two indices of the result. On the other hand,since G τ is parallel, this is equivalent to taking covariant differential twice toobtain a section ∇ ψ of T A ⊗ L k , and then contracting its two indices withthose of G τ . Using again Lem. 7.1 and that X j is parallel, the above can bewritten as ∆ G τ ψ = − iτ r X j = ∇ X j ∇ X j ψ ,which, combined with the definition (19) of ∇ HW , gives the first relation of thestatement. The second is obtained analogously. (cid:3) Lemma 7.3.
The symmetric tensor G C is determined by the identities G C (cid:18) ∂∂τ (cid:19) = − itτ | t | r X j = ∂∂z j + i ∂∂z j + r ! ⊗ ∂∂z j + i ∂∂z j + r ! , G C (cid:18) ∂∂τ (cid:19) = − itτ | t | r X j = ∂∂z j − i ∂∂z j + r ! ⊗ ∂∂z j − i ∂∂z j + r ! .Once again, the corresponding identities for G C are obtained by conjugation. Proof.
Since the decomposition A C = A ⊕ J A is orthogonal, and since J is anisometry, the metric g C τ splits as the sum of two blocks g τ and J ∗ g τ = J · g τ · J .Correspondingly, its inverse also splits as the sum e g C = e g τ ⊕ (cid:0) J − · e g τ · J − (cid:1) = e g τ ⊕ (cid:0) J · e g τ · J (cid:1) .Since both J and the splitting of A C are independent of the Teichmüller parameter,the derivatives of e g τ with respect to τ and τ also decompose in a similar way,whence ∂ e g C τ ∂τ = iτ r X j = (cid:16) X j ⊗ X j + J X j ⊗ J X j (cid:17) , ∂ e g C τ ∂τ = − iτ r X j = (cid:0) X j ⊗ X j + J X j ⊗ J X j (cid:1) .Moreover a direct computation shows thatd z j (cid:0) X l (cid:1) = δ jl (cid:18) + t | t | (cid:19) , d z j + r (cid:0) X l (cid:1) = iδ jl (cid:18) + t | t | (cid:19) ,d z j (cid:0) J X l (cid:1) = − iδ jl (cid:18) − t | t | (cid:19) , d z j + r (cid:0) J X l (cid:1) = δ jl (cid:18) − t | t | (cid:19) . ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 27
Therefore the components of X j and J X j of type (
1, 0 ) with respect to I τ , t are X ′ j = (cid:18) + t | t | (cid:19) ∂∂z j + i ∂∂z j + r ! , ( J X j ) ′ = − i (cid:18) − t | t | (cid:19) ∂∂z j + i ∂∂z j + r ! ,respectively. We conclude that X j ⊗ X j + J X j ⊗ J X j = t | t | ∂∂z j + i ∂∂z j + r ! ⊗ ,and the first identity in the statement is proven. The second is shown analogously. (cid:3) We deduce the following.
Lemma 7.4.
The covariant derivative with respect to the complexified Hitchin connectionis given by ∇ C τ = ∂∂τ − i τ t r X j = (cid:16) ∇ z j + i ∇ z j + r (cid:17) , ∇ C τ = ∂∂τ − i τ t r X j = (cid:16) ∇ z j − i ∇ z j + r (cid:17) .8. T he L - connection In this section we define the natural L -connection on the bundle of L -sectionsof L C t → A C , at an admissible level t = k + is ∈ Z > × i R , and give a proof thatit coincides with the lifted complexified Hitchin connection of § 4.Consider the trivial bundle e LLL C t → T for the prequantisation of (cid:0) A C , ω t (cid:1) . For τ ∈ T , the τ -fibre contains the Hilbert subspace e H C τ , t of I τ , t -holomorphic func-tions, so there is a canonical orthogonal projection π f H τ : L C t → e H C τ , t . This can beused to project the trivial covariant derivative of a T -family of holomorphic sec-tions (which need not be holomorphic), thereby defining the L - connection ∇ L .To be more precise, suppose that U ⊂ T is an open subset, f : A C × U → C asmooth function whose fibrewise restriction f (cid:12)(cid:12) τ := f (cid:12)(cid:12) A C × { τ } : A C → C lies in e H C τ , t for every τ ∈ U . Moreover assume that V [ fσ τ ] is square-summable forvector fields V on U , that f (cid:12)(cid:12) τ lies in the domain of the creation/annihilation op-erators and their two-fold compositions, and that the partial derivatives ∂∂τ ( fσ τ ) and ∂∂τ ( fσ τ ) are also square-summable. Definition 8.1 (L -connection) . The covariant derivative of ϕ := fσ τ along V withrespect to the L -connection is ∇ L V ϕ := π f H (cid:0) V [ ϕ ] (cid:1) . Proposition 8.1 ([8], § 1a) . Suppose that ϕ = fσ is as above. Then ∇ L V ϕ = ∇ C V ϕ .We include for completeness a proof of this fact in our notation, as the localexpression of the L -connection will be used in the proof of Thm. 2. Proof.
It is enough to consider V = ∂∂τ , writing ∇ C V = ∇ C τ and ∇ L V = ∇ L τ . Thecase of ∂∂τ can be handled in complete analogy.We start by writing the derivative of ϕ as ∂∂τ ( fσ τ ) = δfδτ σ τ + r X j = ∂z j ∂τ ∂f∂z j σ τ + f ∂σ τ ∂τ . (34)We will argue that all individual addends are square-summable and computetheir projections on e H C τ , t . We shall then use the result to write ∇ L τ ϕ in the form ∂ϕ∂τ − u L τ ϕ for an appropriate operator u L τ , and finally recognise the latter to beequal to | t | u C τ = | t | u C ( ∂∂τ ) .Starting from the right-most term in (34), it follows immediately from the def-inition of σ τ in (20) that ∂σ τ ∂τ ( A ) = − | t | ∂g C τ ∂τ ( A , A ) σ τ ( A ) .The derivative of g C τ can be obtained combining Lem. 7.3 with the usual formulafor the variation of the inverse matrix, and we find ∂g C τ ∂τ ( A , A ) = − it τ | t | r X j = (cid:16)(cid:0) z j + iz j + r (cid:1) − (cid:0) z j + iz j + r (cid:1) (cid:17) ,which implies f ∂σ τ ∂τ = it τ r X j = (cid:16)(cid:0) z j + iz j + r (cid:1) − (cid:0) z j + iz j + r (cid:1) (cid:17) ϕ .By our assumptions, the term ( z j + iz j + r ) ϕ is an element of e H C τ , t for every j ,since it can be written in terms of ladder operators acting on ϕ . In a similar fash-ion, ( z j + iz j + r ) ϕ is square-summable, although not holomorphic: its projectioncan be obtained using (the argument leading to) (21), which yields π f H τ (cid:18) f ∂σ τ ∂τ (cid:19) = it τ r X j = (cid:16) δ j f − µ j f (cid:17) σ τ , (35)in the notation of Def. 7.2.Using (32), the middle term of (34) can be determined by expanding ∂z j ∂τ ∂f∂z j = − τ (cid:18) it | t | z j + z j + r − t | t | z j + r (cid:19) ∂f∂z j − (cid:18) z j + t | t | z j + it | t | z j + r (cid:19) ∂f∂z j + r ! .By the same argument as above, each addend is square-summable, and the pro-jection is computed to be π f H τ r X j = ∂z j ∂τ ∂f∂z j σ τ = − τ r X j = it δ j f + z j + r ∂f∂z j − z j ∂f∂z j + r ! σ τ . (36)Finally, it is now clear that the remaining term δfδτ σ τ is also square-summable,since the left-hand side of (34) is by hypothesis. Furthermore, this term is already ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 29 in e H C τ , t , because of Rem. 7.1. Combining this with (35) and (36) leads to ∇ L τ ϕ = δfδτ σ τ − i τ r X j = tδ j f + tµ j f − iz j + r ∂f∂z j + iz j ∂f∂z j ! σ τ We may now compare from each term in (34) with its projection to find f ∂σ τ ∂τ − π f H τ (cid:18) f ∂σ τ ∂τ (cid:19) = it τ (cid:16)(cid:0) z j + iz j (cid:1) f − δ j f (cid:17) σ τ and similarly r X j = ∂z j ∂τ ∂f∂z j σ τ − π f H τ r X j = ∂z j ∂τ ∂f∂z j σ τ == − it τ | t | (cid:0) z j + iz j + r (cid:1) ∂f∂z j + i (cid:0) z j + iz j + r (cid:1) ∂f∂z j + r − | t | δ j f ! σ τ , .Combining the two relations above we finally obtain the potential of the L -connection, which is u L τ ϕ := ∂ϕ∂τ − ∇ L τ ϕ = it τ r X j = (cid:18)(cid:16) δ j − (cid:0) z j + iz j + r (cid:1)(cid:17) f (cid:19) σ τ .After noticing that ∇ z j σ τ = − | t | z j σ τ , for j ∈ {
1, . . . , 2 r } ,the above can be written as u L τ ϕ = i τ t r X j = (cid:16) ∇ z j + i ∇ z j + r (cid:17) ϕ = | t | u C τ ϕ .Finally in a similar way one finds u L τ ϕ = i τ t r X j = (cid:16) ∇ z j − i ∇ z j + r (cid:17) ϕ = | t | u C τ ϕ . (cid:3)
9. C onjugation of the H itchin –W itten connection In this section we use the Bargmann transform to relate the two quantisationschemes on A C , using the families of polarisations given by P τ and I τ , t —at anadmissible level t = k + is ∈ Z > × i R . More precisely, consider a local section ψ of the real-polarised quantum bundle: assuming sufficient regularity, our goalis to relate f := B ( ψ ) with the Bargmann transform of the covariant derivative ∇ HW ψ with respect to the lifted Hitchin–Witten connection of § 5. Remark . We will consider the subspace Q τ := I τ , t P τ ⊆ A C .It is easily checked that Q τ is the complex conjugate of P τ , and that it is e g C τ -orthogonal to P τ . In coordinates, this subspace is the locus p =
0, which corre-sponds to Q ⊆ C m in the notations of § 6. △ Remark . The natural identification Ω ( Σ , k C ) ≃ Ω ( Σ , k) ⊗ C induces a C -linearisomorphism A C ≃ A ⊗ C for the complex structure J on the left-hand side. Under this identification, P τ and Q τ correspond to T A and T A , respectively, for the complex structure I τ on A . To the maps π ′ = ( − iI τ ) : A → T A and π ′′ = ( + iI τ ) : A → T A of decomposition into holomorphic and anti-holomorphic part correspond theorthogonal projections π Q τ := ( − J · I τ ) = ( + K τ ) : A → Q τ , π P τ := ( + J · I τ ) = ( − K τ ) : A → P τ . (37)As is easily seen, these maps are isometries up to a projective factor 2, and theinverse map π − Q τ := Q τ → A is given by − K τ .Beware the above holds only on A C , as in the natural identification A C ≃ A ⊗ C the complex structures I τ and I C τ do not match up (cf. § 2.1.1). △ Notice again that, by transversality, polarised sections are completely deter-mined by their restriction to either Q τ or A . The respective constructions of theBargmann transform and Hitchin–Witten connection are based on the identifica-tion of polarised objects with arbitrary ones on these two subspaces, and somecare is needed when relating the two objects. However, if ψ is polarised, the con-ditions of square-summability of ψρ τ restricted to Q τ or A are equivalent. Infact, since | ρ τ | = Q τ → A is a projective isometry,the L -norms of the two restrictions differ by a constant factor 2 r .The remainder of this section is organised as follows. We shall consider a T -parametrised family ψ of P τ -polarised functions on A C , corresponding to a familyof sections ψρ τ . Using the setup of § 7, we show that its (lifted) Hitchin–Wittencovariant derivative along ∂∂τ is ∇ HW τ ( ψρ τ ) = ∂ψ∂τ ρ τ + it tτ r X j = (cid:16) tD j ψ − | t | M j D j ψ + tM j ψ (cid:17) ρ τ . (38)The above are intended as relations between smooth sections on A .Next we study the polarised extension of the right-hand side to the whole of A C , and then restrict it to Q τ . We will prove that the result is the sectionExt (cid:16) ∇ HW τ ( ψρ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) Q τ = ∂ψ∂τ + it τ r X j = (cid:16) D j + M j (cid:17) ψ , (39)where Ext we denotes the unique polarised extension to A C of a section on A .Finally, we study the Bargmann transform of (39). Assuming that each addendis square-summable, B τ may be applied to each term individually, and (28) canbe used to express the result in terms of the action of the operators δ j and µ j on B τ ( ψ (cid:12)(cid:12)(cid:12) Q τ ) . Finally we will suppose that ψ is regular enough that the partialderivatives in τ and τ commute with the integral expression of the Bargmann ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 31 transform, i.e. B τ (cid:18) δψδτ (cid:19) = δ B τ ( ψ ) δτ and B τ (cid:18) δψδτ (cid:19) = δ B τ ( ψ ) δτ . (40)Then we obtain the following reformulation of Thm. 2. Theorem 9.1.
Suppose that U ⊆ T is an open subset, and that ψ : U × A C → C is asmooth family of polarised functions satisfying (40) , and such that the restriction ψ (cid:12)(cid:12) Q τ lies in the domain of all two-fold compositions of ladder operators. Then B τ (cid:18) ∇ HW τ ( ψρ ) (cid:12)(cid:12)(cid:12) Q τ (cid:19) = ∇ L τ (cid:18) B τ (cid:16) ψ (cid:12)(cid:12) Q τ (cid:17)(cid:19) , B τ (cid:18) ∇ HW τ ( ψρ ) (cid:12)(cid:12)(cid:12) Q τ (cid:19) = ∇ L τ (cid:18) B τ (cid:16) ψ (cid:12)(cid:12) Q τ (cid:17)(cid:19) . Remark.
The condition (40) will be removed in § 10, where we will compare theconnections defined on the moduli spaces themselves. △ To prove Thm. 9.1 we will use the following.
Lemma 9.1.
For j ∈ {
1, . . . , r } one has π Q τ X j = it | t | ∂∂q j − i ∂∂q j + r ! and π Q τ X j = − it | t | ∂∂q j + i ∂∂q j + r ! . Moreover, if A ∈ A , then g τ ( A , X j ) = it | t | (cid:0) q j − iq j + r (cid:1) and g τ ( A , X j ) = − it | t | (cid:0) q j + iq j + r (cid:1) . Proof of Lem. 9.1.
Using Def. 7.3 and (29), for l ∈ {
1, . . . , r } one findsd q l ( X j ) = it | t | δ jl and d q l + r ( X j ) = t | t | δ jl ,whence π Q τ X j = r X l = d q l ( X j ) ∂∂q l = it | t | ∂∂q j − i ∂∂q j + r ! ,as desired. Furthermore since A ∈ A has coordinates ( p , q ) one has g τ ( A , X j ) = g τ (cid:0) π Q τ A , π Q τ X (cid:1) = it | t | (cid:0) q j − iq j + r (cid:1) ,as π Q τ A has coordinates ( , q ) and the vectors ∂∂q j form an orthonormal basis of Q τ . The remaining relations follow by complex conjugation. (cid:3) Proof of Thm. 9.1.
It is enough to verify the statement for the derivative in τ , asthe case of τ follows by u HW τ = − (cid:16) u HW τ (cid:17) † and u L τ = − (cid:16) u L τ (cid:17) † ,together with the unitarity of B τ and condition (40) above.We start by proving (38). Using (33) we have ∇ HW τ ( ψρ τ ) = ∂ψ∂τ ρ τ + ψ ∂ρ τ ∂τ − i tτ r X j = ∇ X j ∇ X j ( ψρ τ ) . (41) To expand it we need to compute the derivative of ρ τ .To this end, for A ∈ A the argument of ρ τ ( A ) in (20) can be written − i | t | p · q = − i ω t (cid:0) π P τ A , π Q τ A (cid:1) = − i ω t ( A − K τ A , A + K τ A ) = − is g τ ( A , A ) ,where we used (37) to expand the projections. Therefore we may use Cor. 7.1 towrite ∂ρ τ ∂τ = − is ∂g τ ∂τ ( A , A ) ρ τ = − s τ r X j = (cid:16) g τ ( A , X j (cid:17) ρ τ , (42)and by Lem. 9.1 ψ ∂ρ τ ∂τ = st τ t r X j = M j ψρ τ .We now consider the right-most term in (41). For each j ∈ {
1, . . . , r } one has ∇ X j ∇ X j ( ψρ τ ) = X j h X j [ ψ ] i ρ τ + X j [ ψ ] ∇ X j ρ τ + ψ ∇ X j ∇ X j ρ τ . (43)Let us compute all the derivatives appearing on the right-hand side. Since ψ ispolarised, we can write X j [ ψ ] = (cid:16) π Q τ X j (cid:17) [ ψ ] ,and it can be seen combining Def. 7.3 and (29) that π Q τ X j = r X l = X j [ q l ] ∂∂q j = − it | t | ∂∂q j + i ∂∂q j + r ! ,whence X j [ ψ ] = − it | t | ∂∂q j + i ∂∂q j + r ! ψ = − it D j ψ .Since ψ is a polarised function , its derivatives are polarised as well, and repeatingthe argument we obtain X j h X j [ ψ ] i = − t D j ψ .Next, we use that ω t (cid:12)(cid:12) A = kω and that I τ X j = i X j to compute ∇ X j ρ τ = − t g τ ( A , X j ) ρ τ .Using that g τ is of type (
1, 1 ) with respect to I τ , while X j is anti-holomorphic, wefind that the second covariant derivative of ρ τ is ∇ X j ∇ X j ρ τ = − t X j h g τ ( A , X j ) i ρ τ − t g τ ( A , X j ) ∇ X j ρ τ = t (cid:16) g τ ( A , X j ) (cid:17) ρ τ .Using Lem. 9.1 again, we may now substitute the last four equations in (43) tofind ∇ X j ∇ X j ( ψρ τ ) = − t D j ψρ τ + t | t | M j D j ψρ τ − t t M j ψρ τ .Then substituting this and (42) in (41) yields ∇ HW τ ( ψρ τ ) = ∂ψ∂τ ρ τ + it tτ r X j = (cid:18)(cid:16) tD j − | t | M j D j + tM j (cid:17) ψ (cid:19) ρ τ , ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 33 which is equivalent to (38), as desired.Our next goal is to study the polarised extension of the expression above andestablish (39). In fact, each individual term in (38) expresses a polarised object(restricted to A ), except for the partial derivative ∂ψ∂τ which is given by δψδτ − ∂ψ∂τ = τ r X j = (cid:18) q j + r + t | t | ( p j + ip j + r ) (cid:19) ∂ψ∂q j − (cid:18) q j − it | t | ( p j + ip j + r ) (cid:19) ∂ψ∂q j + r ! ,(44)using (27) and (31) and the polarisation condition on ψ . Now the derivative δψδτ is polarised, as noted in Rem. 7.1, while using (30) we obtain p j + ip j + r = − it | t | ( q j + iq j + r ) on A , and therefore ∂ψ∂τ (cid:12)(cid:12)(cid:12)(cid:12) A = δψδτ − τ r X j = q j + r ∂ψ∂q j − q j ∂ψ∂q j + r − it | t | M j D j ψ ! .The right-hand side expresses now a polarised function, and applying (44) againon Q τ gives Ext ∂ψ∂τ (cid:12)(cid:12)(cid:12)(cid:12) A ρ τ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q τ = ∂ψ∂τ (cid:12)(cid:12)(cid:12)(cid:12) Q τ + it | t | τ r X j = M j D j ψ (cid:12)(cid:12) Q τ ,where the factor ρ τ can be omitted on the right-hand side since ρ τ | Q τ ≡
1. Com-bined with (38), this givesExt (cid:16) ∇ HW τ ( ψρ ) (cid:17)(cid:12)(cid:12)(cid:12) Q τ = ∂ψ∂τ (cid:12)(cid:12)(cid:12)(cid:12) Q τ + it τ r X j = (cid:16) D j + M j (cid:17) ψ (cid:12)(cid:12) Q τ ,which is (39).To conclude we apply the Bargmann transform to the right-hand side of thelast relation. To this end, we expand the partial derivative again and writeExt (cid:16) ∇ HW τ ( ψρ ) (cid:17)(cid:12)(cid:12)(cid:12) Q τ − δψδτ = i τ r X j = i q j + r ∂ψ∂q j − q j ∂ψ∂q j + r ! + t ( D j + M j ) ψ ,where we dropped the restrictions for notational convenience. It follows from ourhypotheses that every individual addend is square-summable, so we can applythe Bargmann transform to each term separately. Then using Prop. 6.1 yields B ◦ q j + r ∂∂q j ! = − | t | (cid:16) ( a j + r − a ∗ j + r )( a j + a ∗ j ) (cid:17) ◦ B , B ◦ q j ∂∂q j + r ! = − | t | (cid:16) ( a j − a ∗ j )( a j + r + a ∗ j + r ) (cid:17) ◦ B , whence, using the commutativity of ladder operators with different indices, weobtain B ◦ q j + r ∂∂q j − q j ∂∂q j + r ! = z j + r ∂∂z j − z j ∂∂z j + r ! ◦ B .Moreover, it also follows from (28) that B ◦ (cid:16) D j + M j (cid:17) = − (cid:16)(cid:0) δ j − µ j (cid:1) + (cid:0) δ j + µ j (cid:1) (cid:17) ◦ B = − (cid:16) δ j + µ j (cid:17) ◦ B .Using the two relations above and the condition in (40), we finally obtain B τ (cid:18) Ext (cid:16) ∇ HW τ ( ψρ ) (cid:17)(cid:12)(cid:12)(cid:12) Q τ (cid:19) == δ B τ ( ψ ) δτ + i τ r X j = i z j + r ∂ψ∂z j − z j ∂ψ∂z j + r ! − t (cid:16) δ j + µ j (cid:17) B τ ( ψ ) ,which coincides with (36) up to the multiplication by the frame σ τ . (cid:3)
10. I dentifications of the connections on the moduli spaces
In this final section we discuss how to use the previous results to identify theHitchin–Witten and complexified Hitchin connections as intrinsically defined onthe moduli spaces—rather than their cover. Strictly speaking, the methods of theprevious sections do not extend immediately, as they rely crucially on square-summability, a property that no non-trivial section on the moduli spaces enjoysonce lifted to its cover. Nonetheless, the arguments can be adapted followingthe ideas of [1, § 3.5], using duality arguments together with an analogue of theBargmann transform on the moduli spaces.Suppose ψ is a smooth K -equivariant section of L C k → A , descending to asection on the moduli space. By compactness of M fl , it is bounded, hence theintegral (cid:0) B ( ψ ) (cid:1) ( z ) := (cid:18) | t | π (cid:19) r Z R r ψ ( q ) exp (cid:18) − | t | (cid:16) | q | + i q · z + | z | − z · z (cid:17)(cid:19) dvol q .(45)is absolutely convergent and bounded in z . Note that, compared to (22), thisformula includes the holomorphic frame σ , so the result is a holomorphic sectionof L C t → A C . Then using the explicit form of the cocycle (13) on the subspace A ⊆ A one can show that the integral kernel of (45) is K -equivariant in z ,i.e. that this integral transform is compatible with quasi-periodicity and with theWeyl group action. Hence B τ ( ψ ) descends to a holomorphic section of L C t → M C fl . Further inspection shows that B ( ψ ) is a Schwartz-class function along thedirection of P τ ⊆ A , so in particular the associated section belongs to H C τ , t .This realises the desired analogue of the linear-space Bargmann transform of § 6,mapping Ω ( M fl , L k ) to H C τ , t .We shall now embed L k and H C τ , t into appropriate spaces of tempered dis-tributions. Let S = S k be the space of smooth sections of L k → A whichcorrespond to Schwartz-class complex-valued functions in the defining triviali-sation L k = A × C . Analogously, let S C = S τ , t , C be the space of Schwartz-classholomorphic sections of L C t → A C . ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 35
These spaces embed densely inside the quantum spaces e L k and e H C τ , t of §§ 3.4and 3.3, respectively, for τ ∈ T .One of the key properties of the linear-space Bargmann transform is that itintertwines the Fock structures on L k and H C τ , t . Since the cyclic vectors in thetwo spaces are Gaussian functions, it follows that the transform (22) restricts amorphism of Fréchet spaces B τ : S → S C . Hence there is a well defined transpose map t B τ : S ′ C → S ′ between the topological duals, defined by the relation (cid:0) t B τ ( T ) (cid:12)(cid:12) ψ (cid:1) = (cid:0) T (cid:12)(cid:12) B τ ( ψ ) (cid:1) , for ψ ∈ S , T ∈ S ′ C , (46)where ( · | · ) denotes the canonical pairings.Since every smooth and K -equivariant section is bounded, there is also anatural embedding Ω ( M fl , L k ) ֒ → S ′ k . Suppose now that ϕ ∈ H C τ , t is given, andfix a test section ϕ ∈ S C . If C is a constant such that (cid:12)(cid:12) ϕ ( z ) (cid:12)(cid:12) C ( + | z | ) − r − ,then Z A C (cid:12)(cid:12) ϕ ( z ) ϕ ( z ) (cid:12)(cid:12) dvol z C k ϕ k H C τ , t X g ∈ K (cid:13)(cid:13)(cid:13) ( + | q | ) − r − (cid:13)(cid:13)(cid:13) L ( g . D C ) ,using the tiling A C = ` g ∈ K g . D C , where D C ⊆ A C is a fundamental domainfor the K -action. Hence ϕ defines an element of S ′ C , resulting in an embedding H C τ , t ֒ → S ′ C .The discussion up to this point may be summarised with the diagram in Fig. 1,where the vertical arrows are embeddings.L k H C τ , t S ′ k S ′ τ , t , C B τt B τ F igure Lemma 10.1.
The diagram of Fig. 1 is commutative.Proof.
By the definition of t B τ , for ψ ∈ L k and ψ ∈ S we have (cid:16) t B τ (cid:0) B τ ( ψ ) (cid:1) (cid:12)(cid:12)(cid:12) ψ (cid:17) = (cid:0) B τ ( ψ ) (cid:12)(cid:12) B τ ( ψ ) (cid:1) ,so our goal is equivalent to showing that (cid:0) B τ ( ψ ) (cid:12)(cid:12) B τ ( ψ ) (cid:1) = ( ψ | ψ ) .Fix a fundamental domain D ⊂ A , and for every g ∈ K let χ g denote theindicator of g . D , so that ψ χ g is square-summable on A . By dominated conver-gence and unitarity of the linear-space Bargmann transform, we then have that ( ψ | ψ ) = X g ∈ K ( ψ χ g | ψ ) = X g ∈ K (cid:0) B τ ( ψ χ g ) (cid:12)(cid:12) B τ ( ψ ) (cid:1) . It is not difficult to use dominated convergence on the integral defining B τ ( ψ ) to show that X g ∈ K B τ (cid:0) ψ χ g (cid:1) = B τ ( ψ ) point-wise on A C , and furthermore that all finite partial sums are bounded inabsolute value by a common constant M . We may then apply dominated conver-gence again and obtain that X g ∈ K (cid:0) B τ ( ψ χ g ) (cid:12)(cid:12) B τ ( ψ ) (cid:1) = X g ∈ K B τ ( ψ χ g ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B τ ( ψ ) ! = (cid:0) B τ ( ψ ) (cid:12)(cid:12) B τ ( ψ ) (cid:1) ,which concludes our proof. (cid:3) Now we consider dual analogues of the Hitchin–Witten and complexifiedHitchin connections, as follows. Suppose that T is a T -family of elements of S ′ , and assume that for every test section ψ ∈ S the pairing ( T | ψ ) is smooth over T . Then set (cid:0) ˇ ∇ HW V T (cid:12)(cid:12) ψ (cid:1) := V (cid:2) ( T | ψ ) (cid:3) − (cid:0) T (cid:12)(cid:12) ∇ HW V ψ (cid:1) , (47)where V is a vector field on T . This defines the dual Hitchin–Witten connection.The dual complexified Hitchin connection ˇ ∇ C is defined analogously, with thecaveat that the test function T ∈ S C needs to be extended to a τ -dependent familyfor the right-hand side to make sense. Lemma 10.2.
The dual Bargmann transform intertwines the dual Hitchin–Witten andthe complexified Hitchin connections.Proof.
Let V be a vector field on T , T a smooth T -family of elements of S ′ C , and ψ ∈ S a fixed test section. By (46) and (47) one has (cid:16) t B τ (cid:0) ˇ ∇ C V T (cid:1) (cid:12)(cid:12)(cid:12) ψ (cid:17) = V h(cid:0) T (cid:12)(cid:12) B τ ( ψ ) (cid:1)i − (cid:0) T (cid:12)(cid:12) ∇ C V B τ ( ψ ) (cid:1) .Now since ψ is a τ -independent function the sections δψδτ and δψδτ can be ex-pressed purely in terms of derivatives along A C and multiplication by variables.Therefore, δψδτ and δψδτ are still Schwartz-class, and a standard argument usingLebesgue-dominated convergence implies that the hypotheses of Thm. 9.1 holdfor ψ . It follows that (cid:16) t B τ (cid:0) ˇ ∇ C V T (cid:1) (cid:12)(cid:12)(cid:12) ψ (cid:17) = (cid:16) ˇ ∇ HW V t B τ ( T ) (cid:12)(cid:12)(cid:12) ψ (cid:17) ,which concludes since the test function ψ is arbitrary. (cid:3) Finally we relate the dual connections to the quantum connections of §§ 4 and 5themselves.
Proposition 10.1.
All connections agree on the subspaces of (regular) distributions com-ing from polarised sections over the moduli spaces.Proof.
By (47), if ψ is a smooth T -family of K -equivariant sections on A and ψ ∈ S ′ is a fixed test function we have that (cid:16) ˇ ∇ HW V ψ (cid:12)(cid:12)(cid:12) ψ (cid:17) = V " Z A ψ · ψ dvol − Z A ψ · ∇ HW V ψ dvol , ENUS-ONE COMPLEX QUANTUM CHERN–SIMONS 37 for every real tangent vector V on T . Since ψ is bounded, uniformly in τ up to re-stricting to appropriate open subsets of T , and the same applies to its derivativesalong the direction of V .By dominated convergence then V " Z A ψ · ψ dvol = Z A V (cid:2) ψ · ψ (cid:3) dvol = (cid:0) V [ ψ ] (cid:12)(cid:12) ψ (cid:1) .On the other hand, since T is T -independent, the Hitchin–Witten connection actsvia its potential, which is the Laplacian operator associated with a purely imagi-nary symmetric tensor. We then obtain Z A ψ · ∇ HW V ψ dvol = Z A (cid:16) u HW ( V ) ψ (cid:17) · ψ dvol = (cid:0) u HW ( V ) ψ (cid:12)(cid:12) ψ (cid:1) ,by integration by parts—using the fast decay of ψ .Overall (cid:16) ˇ ∇ HW V ψ (cid:12)(cid:12)(cid:12) ψ (cid:17) = (cid:16) V [ ψ ] − u HW ( V ) ψ (cid:12)(cid:12)(cid:12) ψ (cid:17) = (cid:16) ∇ HW V ψ (cid:12)(cid:12)(cid:12) ψ (cid:17) ,which concludes the proof.For the Kähler-polarised case a test section ϕ ∈ S ′ C may not be fixed inde-pendently of τ , since the space S ′ C itself depends on the Teichmüller parameter.We may however require that δϕδτ = δϕδτ =
0, and the same argument carriesthrough. (cid:3)
Putting together the previous statements of this section we have thus proventhe following reformulation of Thm. 3.
Theorem 10.1.
The moduli-space theoretic Bargmann transform intertwines the Hitchin–Witten and the complexified Hitchin connections.
The two viewpoints on the geometric quantisation of genus-one complex Chern–Simons theory have thus been identified.R eferences
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