JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, January 1998, Pages 189–228 S 0894-0347(98)00252-5

ON HITCHIN’S CONNECTION

BERT VAN GEEMEN AND AISE JOHAN DE JONG

1. Introduction 1.1. The aim of this paper is to give an explicit expression for Hitchin’s connection in the case of rank 2 bundles with trivial determinant over curves of genus 2. We sketch the general situation. Let π : Sbe a family of projective smooth curves. Let p : Sbe the family of moduliC→ spaces of (S-equivalence classes of) rank r bundlesM→ with trivial determinant on over S.On we have a naturally C Mk defined determinant line bundle . The vector bundles p ( ⊗ )onShave a natural (projective) flat connection (suchL a connection also exists∗ forL other structure groups besides SL(r)). This connection first appeared in Quantum Field Theory (conformal blocks) and was subsequently studied by various mathematicians (see [Hi] and [BM] and the references given there). We follow the approach of Hitchin [Hi] who showed that this connection can be constructed along the lines of Welters’ theory of deformations [W]. In case the curves have genus 2 and the rank r = 2, the family of moduli spaces p : S is isomorphic (over S)toaP3-bundle M→ p : = P S. M ∼ −→ Under this isomorphism corresponds to P(1) p∗ for some line bundle on L Ok ⊗ N N S. Giving a projective connection on p ⊗ is then the same as giving a projective ∗L connection on p P(k) ∗O (k) 1 : p P(k) p P ( k ) Ω S . ∇ ∗O −→ ∗ O ⊗ 1.2. In the first part of the paper (Section 2) we give a characterization of Hitchin’s connection (strictly speaking, we define a new connection which is almost charac- terized by Hitchin’s requirements). We first develop a bit of general theory concerning heat operators. In the situa- (2) k tion above a heat operator is a certain type of map D : p∗( S) Diff ( ⊗ ). Thus T → M L k a vector field θ on S gives a second order differential operator D(θ)on ⊗ ,which k k L determines a map D(θ):p ⊗ p ⊗ . These then determine a connection on k ∗L → ∗L 2 p ⊗ . The symbol of D is a map ρD : p∗( S ) S /S. ∗LLet s be the locus of stable bundles inT .→ TheT codimensionM of the non-stable M M k k locus in is at least two if g 3sothatp ⊗ s =p ⊗ in this case. Hitchin M ≥ ∗L |M ∗L Received by the editors January 16, 1997 and, in revised form, September 4, 1997. 1991 Mathematics Subject Classification. Primary 14H60, 53C05; Secondary 20F36, 32G15, 14D20. Key words and phrases. Hitchin’s connection, moduli of vector bundles, heat equations, heat operators.

c 1998 American Mathematical Society

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k s showed for g 3 that if a heat operator D on ⊗ over determines the desired ≥ k L M connection on p ⊗ , then its symbol map: ∗L s 2 ρD : p∗( S) S s /S, T −→ T M s s mustbeequaltoamap1/(k+2)ρHitchin.ThemapρHitchin is determined by the deformation theory of a typical fibre C and of the stable bundles on C. He also s 2 proved that ρHitchin extends to ρHitchin : p∗( S) S /S for bundles of rank r on curves of genus g, except maybe for the caseT r→=2,gTM= 2 (!). s We show, using simple bundles, that ρHitchin also extends to ρHitchin in the case g = 2 (Theorem 2.6.1). From a study of heat operators and their symbols we conclude that we can find a (projective) heat operator D with the right symbol over all of for g = 2 (Corollary 2.6.3). Such a D need not be unique however. M 0 To pin it down we use the action of the two-torsion group scheme := Pic /S[2] H C on (tensoring a rank two bundle by a line bundle of order two doesn’t change M k the determinant). Then acts (projectively) on ⊗ ; this action can be linearized H Lk to the action of a central extension of on ⊗ : G H L 1 G m 0 . −→ −→ G −→ H −→ We show that there is a unique (projective) heat operator

(2) k 1 Dk : p∗( S ) Diff ( ⊗ ) with ρDk = ρHitchin T −→ M L k +2 which commutes with the action of (Proposition 2.6.4). The (projective) connec- G tion determined by Dk is called Hitchin’s connection (cf. 2.6.6).

1.3. To be able to compute Dk (and to establish some of the results mentioned 0 above), we use a map f :Pic /S over S whose image is the family of Kummer C →M surfaces S. We show that the pullback of Dk along f determines the connec- K→ k tion on q f ∗ ⊗ given by the heat equations on (abelian) theta functions. (The ∗ L k connection defined by Dk on p ⊗ does not pullback to the ‘abelian’ connection however.) ∗L We show that Dk is characterized by this property. Moreover, we express this property in terms of the (local) equation defining (2.6.7 (Eq.)). K⊂M

1.4. To write down a completely explicit connection (on p P(k)) we must now choose a certain S and a family of curves over S.WetakeS∗=O , the configuration space of 2g + 2 points in C2g+2 and takeC the obvious family ofP hyperelliptic curves over it:

2 z:y=(xzi) C⊃Ci− Q 2g+2  := C (z1,...,z2g+2): zi =zj if i = j , z Py−{ 6 } 3y In the case g = 2 this family dominates the moduli space of genus 2 curves (the induced morphism of to the stack of genus two curves is smooth and surjective). We will define connectionsP on bundles over for any g.Inthecaseg=2this connection is basically the Hitchin connectionP (see Theorem 4.4.2). It seems possible that for general g the connection is related to the Hitchin connection for the family of hyperelliptic curves above.

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For g = 2 we can see which bundle to use: The pullback of the bundle p : P along an unramified 24 : 1-Galois cover ˜ with group H =(Z/2Z)4 is trivial.→P P→P The bundle P can thus be obtained as a quotient of ˜ P3 by H (4.4.1). Therefore, P× locally (in the complex or the ´etale topology), the vector bundles p P(k)andthe 0∗O 3 trivial bundle Sk are canonically isomorphic, with Sk = H (P , (k)), the vector space of homogeneous⊗OP polynomials of degree k in 4 variables. ThusO it suffices to write down the connection on the trivial bundle which corresponds to Hitchin’s connection on p P(k). ∗O This connection on Sk (for general g)isgivenby ⊗OP 1 fdzi :Sk S k Ω , ( w f )=w df Mij (w) , ∇ ⊗OP −→ ⊗ P ∇ ⊗ ⊗ −k+2 ⊗ zi zj i=j X6 − for certain endomorphisms Mij End(Sk). We give two descriptions of the en- ∈ domorphisms Mij . One is based on the Lie algebra so(2g + 2) and its (half) spin k representations. We will in fact identify Sk = S V (ωg+1)whereV(ωg+1)isahalf spin representation of so(6). The other description uses the Heisenberg group action and we sketch that de- scription here. The family has 2g + 2 sections which are the Weierstrass points on each fibre: C

Pi : ,z(x, y)=(zi,0). P−→C 7−→ 0 2g Each Pi Pj is a section in Pic /S[2](S) = (Z/2Z) . There is a central extension G − C ∼ of this group which acts on the Sk’s (the group scheme is a twist of the constant G ˜ group scheme G, these group schemes are isomorphic over P ). Let Uij End(S1) ±2 ∈ be the endomorphisms induced by Pi Pj with the property that U is the identity. − ij Since End(S )=S S∗ and S∗ may be identified with the derivations on S := 1 1⊗ 1 1 Sk and the composition Uij Uij may be considered as a second order differential operator which acts on any Sk, the symbol (that is, the degree two part) of this L operator is, up to a factor 1/16, the desired Mij (which, in this sense, does not depend on k). − The idea for trying these Ωij ’s comes from [vGP]. There the Hitchin map, a s Hamiltonian system on the cotangent bundle T ∗ , is studied. The Hitchin map M is basically the symbol of the heat operator k (cf. [Hi], 4). Among the results of [vGP] is an explicit description of the HitchinD map for g =§ 2 curves (obtained from 3 line geometry in P ) and implicit in that description is the one for the Mij given here.

1.5. Now that the connection is determined it is interesting to consider its mon- odromy representation. In the case of rank two bundles, this has been considered by Kohno [K2]. We verify that our connection has the same local monodromy as Kohno’s representation in Subsection 4.5. Here local monodromy means “at ”, e.g., we determine the eigenvalues of the monodromy operator of a small loop ∞ around the divisor z1 = z2 in . Kohno (and the physicists) use (locally on the base it seems) the Knizhnik-ZamolodzhikovP equations (which define a flat connec- tion over using the Lie algebra sl(2) [Ka], Chapter XIX). It is not clear to us how the KZ-equationsP are related to our connection (which is based on so(2g + 2)). We do not know which representations of the braid group one obtains from our connections.

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2. Characterizing Hitchin’s connection 2.1. Introduction. We recall some deformation theory as described by Welters [W] and relate it to connections. Then we introduce heat operators and show how they define (projective) connections on certain bundles. Next we give a criterion for the existence of a heat equation with a given symbol and in Section 2.4 we recall the definition of the symbol of Hitchin’s connection. In Section 2.5 we combine these results and give criteria (in 2.6.7 and 2.6.6) which determine Hitchin’s connection.

2.2. Deformation theory.

2.2.1. Deformations. Let X be a smooth projective variety over C with tangent 0 bundle TX .LetLbe an invertible X -module on X and let s H (X, L). We are going to classify first order deformationsO of X, of the pair (X,∈ L) and of the triple (X, L, s). See [W] for proofs. 2 Isomorphism classes of first order deformations X of X (over SpecC[]/( )) 1 1 are classified by H (X, TX). We write [X] for the class in H (X, TX)ofthe deformation X; similar notation will be used throughout. The first order deformations (X,L) of the pair (X, L) are classified by 1 (1) (1) H (X, DiffX (L)). Here Diff X (L) is the sheaf of first order differential operators on L. This sheaf sits in an exact sequence: (1) σ 0 X Diff (L) T X 0 −→ O −→ X −→ −→ (1) where the map σ :DiffX (L) TX gives the symbol of the operator. This sequence 1 (1)→ 1 gives a map α : H (X, Diff X (L)) H (X, TX), which maps [(X,L)] to [X]. Finally, we consider first order deformations→ of the triple (X, L, s). The evalua- tion map d1s :Diff(1)(L) L, D Ds gives a complex: X → 7→ 1 0 Diff (1)(L) d s L 0 . −→ X −→ −→ Let H1(d1s) be the first hypercohomology group of this complex. This is the space classifying the isomorphism classes of deformations (X,L,s)of(X, L, s) ([W], Prop. 1.2). The spectral sequence connecting hypercohomology to cohomology 1 1 1 (1) gives a map β : H (d s) H (Diff (L)), which maps [(X,L,s)] to [(X,L)]. → X Any element of H1(d1s) may be represented by a Cechˇ cocycle. Let us choose an affine open covering : X = Ui and write Uij = Ui Uj .ACechˇ cocycle is U ∩ given by a pair ( ti , Dij ) satisfying the relations: { } { } S (1) ti tj = Dij (s),Djk Dik + Dij =0 (ti L(Ui),Dij Diff (L)(Uij )). − − ∈ ∈ X The maps β and α β correspond to forgetting first s and then L.Theycanbe ◦ described as follows. The cocycle ( ti , Dij ) is mapped to the 1-cocycle Dij (1) { } { } (1) { } in the sheaf Diff (L) under β. This is then mapped to the 1-cocycle σ (Dij ) X { } in the sheaf TX . Let us make explicit a deformation (X,L,s) associated to ( ti , Dij ). Write { } { } Ui = Spec(Ai)andUij = Spec(Aij ). Let Mi = L(Ui)andMij = L(Uij). The section s gives elements si Mi;notethatti Mi as well. Write θij = σ(Dij ) ∈ ∈ ∈ TX (Uij ). The 1-cocycle θij defines a scheme X over Spec(C[]) by glueing the { } rings Ai[] via the isomorphisms

Aij [] A ij [],f+g f +(θij (f)+g). −→ 7−→

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(1) Here we view θij as a derivation. The 1-cocycle Dij in Diff (L) gives a deforma- { } X tion L of L by glueing the Ai[]-modules Mi[] via the Aij []-module isomorphisms:

φij : Mij [] M ij [],mij + m0  mij +(Dij (mij )+m0 ). −→ ij 7−→ ij Note one has Dij (fs)=fDij (s)+θij (f)s by definition of the symbol σ and σ(Dij )= 1 1 0 θij . The cocycle ( ti , Dij )inH(ds) defines a deformation s H (X,L). { } { } ∈ The section s is given by the family of elements:

s := s,i ( Mi[]) with s,i := si + ti. { } ∈ The cocycle relation ti tj = Dij (s) shows that the s,i’s glue: φij (s,j)=s,i. − 2.2.2. Second order operators. The following construction is useful to obtain ele- 1 1 (2) ments in H (d s) ([W], (1.9)). Let DiffX (L) be the sheaf of second order differen- tial operators on L. The symbol gives an exact sequence which is the first row of the complex:

(1) (2) 2 0 Diff (L) Diff (L) S T X 0 −→ X −→ X −→ −→ ↓↓↓ 0L L 0 0 , −→ −→ −→ −→ where the vertical maps are evaluation on s (D Ds), thus the first column is the complex d1s considered before. From the exact7→ sequence of hypercohomology one obtains a map 0 2 1 1 δs : H (S TX ) H ( d s ) . −→ Let us give a description in terms of cocycles of this map. Let = Ui be an 0 2 U { } affine open cover of X as before. Let w H (X, S TX). We can find operators (2) (2) ∈ 2 D Diff (L)(Ui)whichmaptowU in S TX (Ui). Then i ∈ X | i (2) (2) (2) 11 δs(w)=( D (s) , D D )(H(ds)). { i } { i − j } ∈ 1 It is easy to verify that δs(w) is indeed a cocycle for the complex defined by d s. (2) (2) (2) (2) Note w = σ(Di )=σ(Dj )onUij ,thusDi Dj is a first order operator. (2) (2) (1)− Note that β(δs(w)) = Di Dj ( DiffX (L)) is independent of the choice of { − } 0∈ 2 the section s. Therefore, given w ( H (X, S TX)) we get a deformation (X,L) 0 ∈ 0 such that any section s H (X, L) is deformed to a section s H (X,L). ∈ ∈ 2.2.3. Infinitesimal connections. Let (X,L) be a deformation of (X, L). We con- sider a differential operator D of order at most two on L which locally can be written as (notations as above, cf. 2.3.2 below)

0 (2) D = Di ( H (X, Diff (L))),Di=∂+Ri:Mi[]M i [  ] { } ∈ X −→ with ∂(n + m)=mfor n, m Mi,andRi aC[]-linear map of order at most 2 ∈ on Mi.ThusRiis a second order differential operator involving only derivatives in the fibre directions (and not in the base direction ). The symbol of such a D is the second order part of the Di (and thus of the Ri). The restriction of the symbol to X ( X)is: ⊂ 0 2 ρ¯(D):= σ(R¯i) ( H (X, S TX)), where Ri = R¯i + R¯0 { } ∈ i with R¯i, R¯0 : Mi Mi[] (recall Ri is -linear). i →

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0 For any s H (X, L) we then have the hypercohomology class δs(¯ρ(D)) = ∈ 1 1 ( R¯i(si) , R¯i R¯j )( H(ds)). Let us show that the deformation (X,L)is { } { − } ∈ the deformation determined by β(δs(¯ρ(D))): − 1 (1) [(X,L)] + β(δs(¯ρ(D))) = 0 ( H (X, Diff (L))). ∈ X

This can be verified as follows. Note that βδs(¯ρ(D)) = R¯i R¯j and [(X,L)] = { − } Dij .ThefactthatDis an operator on L implies that Di (id + Dij)= { } ◦ (id+Dij) Dj. Taking the “constant” term of this equation we obtain R¯i+Dij = R¯j as desired.◦ Next we consider any deformation (X,L,s)of(X, L, s). We say that a section 0 s = si + ti H (X,L)isflatforDif { }∈

Di(si + ti) 0mod(), equivalently ti = R¯i(si); ≡ − 1 in this case we write (D)(s) = 0. Since [(X,L,s)] = ( ti , Dij )( H(d s)) we get: ∇ { } { } ∈

δs(¯ρ(D)) = [(X,L,s)] iff (D)(s)=0. − ∇ 2.2.4. Aremarkonsymbols. We use the conventions of [W] concerning differential 2 operators and symbols. In particular when we write S TX we mean symmetric 1 tensors in the sheaf of vector fields on X.NotethatifX=ACwith coordinate t, then the symbol of the operator ∂2/∂t2 ( Diff(1)) is the symmetric section 2 ∈ O 2 2 2∂/∂t ∂/∂t ( S TX). By abuse of notation we will sometimes write σ(∂ /∂t )= 2∂2/∂t⊗2. ∈

2.3. Heat operators.

2.3.1. Notation. Let p : Sbe a smooth surjective morphism of smooth vari- eties over C.Let be anX→ invertible -module over . L OX X We write (resp. S ,resp. /S) for the sheaf of vector fields on (resp. S, TX T TX X resp. over S). We denote by Diff(k)( ) the sheaf of differential operators of order X X L(k) (k) at most k on over .Further,Diff/S( ) Diff ( ) denotes the subsheaf L X X L ⊂ X L 1 (k) of p− ( S)-linear operators. We remark that Diff ( ) is a coherent (left) - O (k) X L OX module and that Diff /S( ) is a coherent submodule. For any k there is a symbol map X L σ(k) :Diff(k)( ) S k =Symk ( ). X L −→ T X OX TX (k) k It maps Diff /S( )intoS /S. L TX Consider aX point x (C), an (´etale or analytic) neighbourhood U of p(x)inS, andan(´etale or analytic)∈X neighbourhood V of x in such that p(V ) U. Assume X ⊂ we have coordinate functions t1,... ,tr S(U) and functions x1,... ,xn (V) ∈O ∈OX such that t1 = t1 p,...,tr =tr p, x1,... ,xn are coordinates on V .(Inthe´etale ◦ 1 ◦ 1 case this means that Ω = U dti and Ω = V dxi.) Then the elements U O V/U O ∂/∂xi form a basis for /S V over V . The system (V,t1,... ,tr,x1,... ,xn) will X L L be called a coordinateT patch|. We canO find coordinate patches around any point x (C). ∈X

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2.3.2. Definition. A heat operator D on over S is a (left) -module homomor- phism L OX (2) D : p∗( S ) Diff ( ) T −→ X L which satisfies the following property: for any coordinate patch (V,t1,... ,tr, x1,... ,xn) as above, and any trivialization V ∼= V , the operator D(∂/∂t1) has the following shape: L| O ∂ ∂ n ∂ ∂2 D( )=f+ + fi + fij ∂t1 ∂t1 ∂xi ∂xi∂xj i=1 1 i j n X ≤X≤ ≤ for certain f,fi,fij (V). More precisely, let ∈OX (1) (2) (2) /S( )=Diff ( )+Diff /S( ) Diff ( ) WX L X L X L ⊂ X L be the subsheaf of second order differential operators on whose symbol lies in 2 2 (1) L S ( /S) S ( ). Note that Diff /S( ) /S( ) and that there is a canon- TX ⊂ TX L ⊂WX L ical exact sequence X (1) 2 0 Diff /S( ) /S( ) p ∗ ( S ) S /S 0 . −→ X L −→ W X L −→ T ⊕ T X −→ Our condition on D above means that D is a map D : p∗( S ) /S( )whose T →WX L composition with the map /S( ) p∗( S ) is the identity. (So D(∂/∂t1)has WX L → T no terms involving ∂/∂ti except for the term ∂/∂t1 with coefficient 1.) (2) Such a map D is determined by the S -linear map p D : S p (Diff ( )). O(2) ∗ T → ∗ X L Note that S is a subsheaf of p (Diff ( )). Any S -linear map D¯ : S (2) O ∗ X L O (2) T → p (Diff ( ))/ S can locally be lifted to a map into p (Diff ( )). This means ∗ X L O ∗ X L that any point s S(C) has a neighbourhood U such that there is an U -linear ∈ (2) O map DU : S U p (Diff ( )) U which reduces to D¯ U . T | → ∗ X L | | A projective heat operator D¯ on over S is an S-linear map L O (2) D¯ : S p Diff ( ) S T −→ ∗ X L O such that any local lifting DU as above gives rise to
. a heat operator on over U . The symbol of a heat operator D is the -linear map L OX 2 ρD : p∗( S ) S /S T −→ T X given by composing D with the symbol σ(2). We note that the symbol of a projective heat operator is well-defined. 2.3.3. Heat operators and connections. We claim that a heat operator D gives rise toaconnection 1 (D):p ( ) p ( ) S Ω S . ∗ ∗ O ∇ L −→ L1 ⊗ Indeed, suppose that s p ( )(U)= (p−(U)) and that θ S(U). Then 1 ∈ ∗ L L 1 ∈T D(p− θ) is a second order differential operator on over p− (U ). Hence L 1 1 D(p− θ)(s) (p− (U)) = p ( )(U). ∈L ∗ L 1 1 By our local description of D aboveweseethatD(p− θ)(fs)=fD(p− θ)(s)+θ(f)s 1 for every f S(U). Hence the operation (θ, s) D(p− θ)(s) defines a connection ∈O 7→ 1 (D)onp over S. In the sequel we will often write D(θ) instead of D(p− θ). ∇ ∗L If we have a projective heat operator D¯, the connection (D¯)= (DU)isonly ∇ ∇ defined locally, by choosing lifts DU . The difference of two local lifts is given by a 1 local section η Ω (U), in which case the difference of the two connections (DU ) ∈ S ∇

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is multiplication by η.ThusD¯defines unambiguously a projective connection ¯ (D¯). ∇ 2.3.4. Heat operators and deformations. Let D be a heat operator on over S.Let L 0 S(C) be a point of the base. Put X = 0 and L = 0.Anelementθ T0S will∈ also be considered as a morphism θ : SpecX(C[]) S.L Bybasechangewegeta∈ → 0 2 pair (X,L)overSpec(C[]). Clearly, wθ := ρD(θ)isanelementofH (X, S TX). It can be seen by working through the definitions that [(X,L)] + β(δs(wθ)) = 0 for any section s H0(X, L). (Note that in Subsection 2.2.2 we showed that ∈ 0 β(δs(wθ)) was independent of s.) Let s H (X,L) lift s. One can show that ∈ [(X,L,s)] = δs(wθ ) if and only if the section s is horizontal for the connection (D). − ∇ 2.3.5. Projective heat operators and change of line bundle. Let D be a heat operator 0 on over S.Letg H(S, S∗ ) be an invertible function on S. Multiplication by 1L ∈ O p− (g) defines an invertible operator on , also denoted by g. For any local section L θ of S, we have the obvious relation T 1 1 g− D(θ) g = g− θ(g)+D(θ). ◦ ◦ 1 We conclude that g− D g is a heat operator and that the projective heat operators ¯ 1 ◦ ◦ 1 D and g− D g are equal. (The difference is given by the section g− dg = d log g 1 ◦ ◦ of ΩS.) Suppose that is an invertible S-module on S. The above implies that the set of projectiveM heat operators on Oover S can be identified canonically with the L set of projective heat operators on 0 = p∗( ). Indeed, choose a covering L L⊗OX M of S on whose members Ui the line bundle becomes trivial. This identifies the M heat operators on and 0 over U . The difference in the local identifications is L L 1 given by the 1-forms d log gij Ω . ∈ Uij 2.3.6. Heat operators and flatness. We say that a heat operator D on over S is L flat if D([θ, θ0]) = [D(θ),D(θ0)] for any two local sections θ, θ0 of S on U S. T ⊂ It suffices to consider local vector fields θ, θ0 on S with [θ, θ0]=0andtocheck that [D(θ),D(θ0)] = 0. We remark that in this case the operator [D(θ),D(θ0)] (3) 1 is a section of Diff /S( )overp− (U). We say that D is projectively flat if we X L have D([θ, θ0]) = h +[D(θ),D(θ0)] for some function h = hθ,θ S(U) for any 0 ∈O θ, θ0 S(U). A projective heat operator D¯ is called projectively flat if any of the ∈T local lifts DU are projectively flat. We remark that if the heat operator D is flat, then the associated connection (D)onp is flat as well. In particular, if p is a coherent S-module, then this ∇implies that∗Lp is locally free. In the same∗ veinL we have thatO a projectively flat projective heat∗L operator D¯ defines a projectively flat projective connection ¯ (D¯) on p . ∇ ∗L 2.3.7. Heat operators with given symbol. Let Sand be as above, and assume 2 X→ L that a -linear map ρ : p∗ S S /S is given. The question we are going to study isO theX following: T → TX When can we find a (projective) heat operator D on over S, D : p∗ S (2) L T → /S ( Diff ( )) with ρD = ρ? X X W ⊂ L (2) (2) So the symbol ρD (which is the composition of D with the map σ :Diff ( ) 2 X L → S /S) should be equal to the given ρ. TX

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Let us define a number of canonical maps associated to the situation. First we have the standard exact sequence

0 /S p ∗ S 0 . −→ T X −→ T X −→ T −→ Thisgivesusan S-linear map (the Kodaira-Spencer map) O 1 κ /S : S R p /S. X T −→ ∗ T X Next we have the exact sequence (2) 2 0 /S Diff /S( ) S /S 0 . −→ T X −→ X L OX −→ T X −→ This gives an S-linear map  O 2 1 µ : p S /S R p /S. L ∗ TX −→ ∗ T X 1 The invertible -module is given by an element of H ( , ∗ ). Using the map O1X L 1 1 X OX dlog: ∗ Ω /S we get an element of H ( , Ω /S), and hence a section [ ]in OX → X L H0(S, R1p (Ω1 X )). This will be called the cohomologyX class of . By cup-product ∗ /S X 2 1 L with [ ] we get another map p S /S R p /S, w w [ ]. (There is a L 2 1 ∗ TX → ∗TX 7→ ∪ L natural pairing S /S Ω /S /S.) It is shown in [W], 1.16, that we have TX ⊗ →TX µ (w)= w [ ]+µ (w).X L − ∪ L O Now let θ be a local vector field on S, i.e., θ S(U), with U affine. We want to ∈T find a D(θ) /S( ) with symbol ρ(θ). Recall that /S( ) is an extension: ∈WX L WX L (1) 2 0 Diff /S( ) /S( ) p ∗ ( S ) S /S 0 . −→ X L −→ W X L −→ T ⊕ T X −→ 1 The obstruction against finding a lift D(θ)( /S( )) of the section p− (θ) 1 2 1 ∈WX L ⊕ ρ(p− θ)ofp∗ S S /S over p− (U )isanelemento(θ, )in T ⊕ TX L 1 1 (1) 0 1 (1) H (p− (U),Diff /S( )) = H (U, R p (Diff /S( ))) X L ∗ X L (recall that U was assumed affine). There is an exact sequence (1) 0 Diff /S( ) /S 0 . −→ O X −→ X L −→ T X −→ 1 We remark that the image of the class o(θ, )inRp /S(U) is the section L ∗TX ¯ κ /S(θ)+µ (ρ(θ)). The first condition that has to be satisfied for D to exist isX therefore: L

( ) κ /S + µ ρ =0. ∗ X L ◦ 1 1 If this condition is satisfied, then we can find a lift of p− (θ) ρ(p− θ)toan ⊕ element Dθ in the sheaf /S( )/ . This element is well defined up to addition WX L OX of a section of /S. Hence we get a second obstruction in TX [ ] 1 ( ) Coker p /S L ∪R p X . ∗∗ ∗TX −→ ∗ O Let us assume for the moment that both obstructions vanish for any θ as above. 1 1 Then we can locally find a D(θ) lifting the element p− (θ) ρ(p− θ). Thus, if ⊕ we have a basis θ ,... ,θr of S over U , then we can define DU by the formula 1 T DU ( aiθi)= aiD(θi) for certain choices of the elements D(θi). If there is some 1 way of choosing the elements D(θ)uniquelyuptoanelementofp− S(U), e.g. if P P O the map in ( ) is injective and p ( )= S,thenDU determines a unique pro- jective heat operator∗∗ over U. In this∗ O caseX theO map ρ determines a unique projective heat operator on over S. L

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Here is another set of hypotheses which imply the existence of heat operators with given symbol. Let 0 S(C) be a point. Put X = and L = as in 2.3.4. ∈ X0 L0 Let θ T S and consider (X,L)overSpec(C[]) as in 2.3.4. We know from 2.3.4 ∈ 0 that [(X,L)] + β(δs(ρ(θ))) = 0 if a heat operator exists. Suppose that 1 1 1 1. R p and R p /S are locally free on S, and have fibres H (X, X)and 1 ∗OX ∗TX O H (X, TX)atanypoint0inS(C). 2. for any 0 and θ as above, we have [(X,L)] + β(δs(ρ(θ))) = 0.

If these conditions are satisfied, then we can at least find local liftings DU of ρ to a heat operator. Again, we will get a “canonical” projective heat operator if there is a “preferred” way of choosing the local lifts D(θi).

2.3.8. Heat operators on abelian schemes. In this subsection, we let Sbe an abelian scheme with zero section o : S . We assume the lineX→ bundle is relatively ample on over S, i.e., defines→X a polarization of over S. WeL will show that there existsX a unique projectiveL heat operator on overX S; see [W]. We first remark that in this case the map ( ) of 2.3.7 isL an isomorphism. This 1 (1) ∗∗ implies that Γ(p− (U), Diff ( )) = S(U) for any U S. Hence, by the discus- /S L O ⊂ sions in 2.3.7 we get a uniqueX heat operator as soon as we have a (symbol) map ρ satisfying ( ). ∗∗ 2 We have µ = 0 in this situation, as we can let elements of S /S act by translation invariantO operators (see [W], 1.20). Hence, µ ( )= [ ]TX( ). Now we use that L − −L ∪ − 2 [ ] 1 1 p ( /S) L ∪ R p /S = o∗ /S R p ∗ ⊗ TX −→ ∗ T X ∼ TX ⊗ ∗OX 1 is an isomorphism. Consider ρ := ([ ] )− κ /S. It is well known that this is a 2 L ∪ ◦ X map into p S /S and it solves ( ) by definition; it is of course the unique solution ∗ TX ∗ ¯ ¯ to ( ). The unique projective heat operator D = D with ρ = ρD¯ is called the projective∗ heat operator associated to ( /S, ). L X n L We make the remark that if = ,thenwehaveρ¯ =(1/n)ρ ¯ .Itis 0 ⊗ D0 D known that these projective heatL operatorsL are projectively flat. This can be seen (3) in a number of ways; one way is to show that p (Diff ( )) = S in this case; ∗ /S L O compare 2.3.6. X Uniqueness of the construction of D¯ implies that D¯ commutes (projectively) with the action of the theta-group ( ). In particular the projective monodromy G L ¯ ¯ group of the (projectively flat) connection (D)on(p )0 normalizes the action of (L). ∇ ∗L GRecall that the classical theta functions satisfy a heat equation. This is exactly the heat equation above where the base is the Siegel upper half space.

2.3.9. Heat operators and functoriality. Let Sand be as above, and let D be a heat operator on over S. Note thatX→ for any morphismL of smooth schemes L ϕ : S0 S there is a pullback heat operator D0 = ϕ∗(D) on the pullback 0 on → L 0 = S0 S . We leave it to the reader to give the precise definition. This pullback X × X preserves flatness. If ϕ∗(p ) ∼= p0 0, then the connection (D0) is the pullback of the connection (D).∗ InL addition∗L pullback is well defined∇ for projective heat operators and preserves∇ projective flatness. If ψ : is an ´etale morphism of schemes over S, then the heat operator D U→X induces a heat operator ψ∗(D)onψ∗( )overS. L

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2.3.10. Heat operators and compatibility. Suppose we have a second smooth sur- jective morphism q : Sand a morphism f : over S. We will assume that f is a submersion.Y→ This means that forY→X any point y (C)there ∈Y is a coordinate patch (V,t1,... ,tr,x1,... ,xc,y1,... ,ym)off(y)in such that 1 X (f − (V ),t ,... ,tr,y = y f,...,ym =ym f) is a coordinate patch for y on 1 1 1 ◦ ◦ Y and xi f =0. Let ◦ be a line bundle on as before. Let us say that a (projective) heat operatorLD on over S is (weakly)X compatible with f : if whenever we have a L Y→X coordinate patch (V,t1,... ,tr,x1,... ,xc,y1,... ,ym) as above, and a trivialization V = V , then the operator D(∂/∂t ) has the following symbol: L| ∼ O 1 ∂ m ∂2 c ρD( )= fij + xi Ξi ∂t1 ∂yi∂yj i j=1 i=1 ≤X X 2 for certain fij (V)andΞi S /S(V ). More precisely, this means there ∈OX 2 ∈ TX exists a ρ = ρf,D : q∗ S S /S such that the following diagram commutes: T → TY 2 f ∗(ρD):f∗p∗S f ∗ S /S T −→ T X || ↑ 2 ρf,D : q∗ S S /S T −→ T Y Suppose that D0 is a heat operator on f ∗ on over S.WesaythatD0 is (weakly) compatible with D if D is (weakly)L compatibleY with f and we have

ρD0 = ρf,D. In this case it is not true in general that D(θ)(s) = D0(θ)(s ). Assume that D is compatible with f. Let us write N |Yfor the generalized|Y Y X normal bundle of in , i.e., N =Coker( /S f ∗ /S). Choose a local Y 1 X Y X TY → TX coordinate patch (f − (V ),t ,... ,tr,y =y f,...,ym =ym f)asabove.Write 1 1 1 ◦ ◦ the operator D(∂/∂t1) in the form ∂ ∂ c ∂ m ∂ m ∂2 c D( )=f+ + fi + gi + fij + xi Ξi ∂t1 ∂t1 ∂xi ∂yi ∂yi∂yj i=1 i=1 i j=1 i=1 X X ≤X X for certain local functions f,fi,gi,fij and second order operators Ξi.Itcanbe seen (and we leave this to the reader) that the class of f ∂ in N (f 1(V )) i ∂xi − is independent of the choice of the local trivialization and coordinates.Y TheX upshot of this is that if D is compatible with , then thereP is a first order symbol Y→X σf,D : q∗ S N . T −→ Y X We say that D is strictly compatible with f : if D is compatible with Y→X f and the symbol σf,D is zero. We remark that this notion is well defined for a projective heat operator as well. Assume that D is strictly compatible with . It follows from the lo- cal description of our compatibility of restrictionsY→X that D induces a heat operator D on the invertible -module f ∗ over S. It is characterized by the property Y OY L D(θ)(s) = D (θ)(s ) for any local section θ of S and any local section s of . |Y Y |Y T L It is clear that the symbol of D is equal to ρf,D, hence that D is compatible Y Y with D. We say that a heat operator D0 on f ∗ is strictly compatible with D if D L is strictly compatible with f : and D0 = D . In this case the restriction map Y→X Y p q f∗ ∗L−→ ∗ L

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is horizontal for the connections induced by D and D0 = D .NotethatD is flat if D is flat. Similar remarks hold for projective heat operatorsY and projectiveY flatness.

2.4. Moduli of bundles.

2.4.1. The symbol ρHitchin intermsofmoduliofbundles.In this subsection we explain how to get a symbol map ρ as in 2.3.7 in the case of the relative moduli space of rank 2 bundles of a family of curves. Let C be a smooth projective curve over Spec(C), and let E be a stable C- O module of rank 2 with det(E) ∼= C . This gives a point [E] C(C). There are canonical identifications O ∈M 1 0 1 T C = H (C, nd (E)) and T ∗ C = H (C, nd (E) Ω ) [E]M E 0 [E]M E 0 ⊗ C where nd (E) denotes the sheaf of endomorphisms of E with trace zero. A cotan- E 0 gent vector φ T[∗E] C thus corresponds to a homomorphism of C -modules ∈ 1 M O φ : E E ΩC . (Note that this is a Higgs field on the bundle E; see [Si].) Composing→ and⊗ taking the trace gives a symmetric bilinear pairing

0 1 2 T ∗ C T ∗ C H ( C, (Ω )⊗ ) [E]M × [E]M −→ C (φ, ψ) Trace(φ ψ). 7−→ ◦ We dualize this and use Serre duality to obtain a map 1 2 ρC,E : H (C, TC ) S T C . −→ [ E ] M Let us make the observation that the construction above can be performed in families. Let π : Sbe a smooth projective family of curves over the scheme S smooth over SpecC→(C). Let p : Sbe the associated family of moduli spaces of rank 2 semi-stable bundles withM→ trivial determinant up to S-equivalence. We remark that p is a flat projective morphism, not smooth in general. However, the open part of stable bundles s is smooth over S. We denote this smooth morphism by ps : s S. M ⊂M M → Now let T S be a morphism of finite type. We denote by an index T base → change to T .Let be a locally free sheaf of rank 2 on T . For a point 0 T (C)we E C ∈ put C = 0 and E = 0 = C. Suppose that for any 0 we have (a) det(E) ∼= C , and (b) theC bundle EEis stable.E| In this case we get an S-morphism t : T O s. Analogously to the above we have a canonical isomorphism →M 1 1 t (Ω s ) (π ) ( nd ( ) Ω ). ∗ /S = T 0 T /T M ∼ ∗ E E ⊗OCT C Using the same pairing and dualities as above we get an T -linear map O 1 2 1 R π T /T S t ∗ ( s /S ) . ∗TC −→ T M   If we compose this with the pullback to T of the Kodaira-Spencer map S 1 T → R π /S of the family S,thenwegetamap ∗TC C→ 2 1 ρ / T : S T S t ∗ ( s /S) . E C T ⊗O −→ T M   If there existed a universal bundle univ over s, then we would get a “symbol” E M s s 2 ρHitchin = ρ univ :(p )∗ S S s/S. E T → TM

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s Although the universal bundle does not exist, the symbol ρHitchin does: it is the unique s -linear map such that, whenever / T is given, the pullback of ρHitchin to T agreesOM with ρs . Uniqueness and existenceE C follows from the existence of / T E C s “universal” bundles ´etale locally over and the invariance of ρ / T under auto- M E C morphisms of over T . E C 2.4.2. Hitchin’s results in genus at least 3. Let us get back to our curve C over Spec(C). We see from the above that any deformation C of the curve will give a s 2 tensor in Γ( C,S C). However, since C has points which do not correspond M TM M to stable bundles (and these points are singular points of C for g>2) such a 0 2 M deformation does not give a global tensor in H ( C ,S T C). In [Hi], Section 5, one can find a sketch of a proof that the tensor extendsM if g>M 2 (or rank > 2, a case we do not even consider, although our present considerations work in that case as well). In the situation Sthere is a canonical line bundle on ; it can be defined by the formula C→ L M univ =detR(pr2) , pr2 : S . L ∗E C× M→M It is shown in [Hi], Theorem 3.6, Section 5, that if g>2 the tensor 2/(2k+λ)ρHitchin ¯ k is the symbol of a unique projective flat heat operator DHitchin on ⊗ over S. (We remark that the extra factor 2 comes from our way of normalizingL symbols; see Subsection 2.2.4.) In fact the arguments of [Hi] prove the existence of this heat operator over the moduli space of curves of genus g>2. We will not use these results. 2.5. The case of genus two curves. 2.5.1. Notation in genus 2 case. We fix a scheme S smooth over Spec(C)anda smooth projective morphism π : S, whose fibres C are curves of genus 2. We C→ write 0 S(C) for a typical point and C = 0. ∈ C 1 1 We introduce the following objects associated to the situation. Pic =Pic /S S denotes the Picard scheme of invertible bundles of relative degree 1 on C over→ S. There is a natural morphism PicO1C over S,givenbyP C(C)mapstoC 1 1 C→ ∈ [ C(P)] in (Pic )0(C)=Pic (C). The image of this morphism is a relative divisor ΘO1 =Θ1 on Pic1 over S; of course Θ1 as S-schemes. Let us denote by /S ∼= 1C C 1 α :Pic Sthe structural morphism. Note that α Pic1 (2Θ ) is a locally free → 1 ∗O S-module of rank 4 on S. We put P = 2Θ equal to the projective space of lines inO this locally free sheaf. More precisely,| we define|

1 p : P = P om S α Pic1 (2Θ ) , S S ; H O ∗O O −→    see [Ha], page 162, for notation used. For a point 0 S(C) we put P = P0. Let T S be a morphism of finite type, and let∈ be a locally free sheaf of → E T -modules of rank 2. For a point 0 T (C) we put C = 0 and E = 0 = C. OSupposeC that for any 0 we have ∈ C E E| (a) det(E) ∼= C , (b) the bundleOE is semi-stable. 1 (Compare with Subsection 2.4.1.) We recall the relative divisor PicT associated to . Set-theoretically it has the following description: DE ⊂ E 1 1 1 DE := Pic (C)= [L] Pic (C):dimH (C, E L) 1 . DE ∩ { ∈ ⊗ ≥ }

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To construct as a closed subscheme we may work ´etale locally on T . Hence we DE 1 may assume that a Poincar´e line bundle on T T PicT exists, i.e., which has 1 L 1 C × 1 1 relative degree 1 for p2 : T T PicT PicT and induces id : PicT PicT .We consider the line bundle C × → → 1 ⊗− = detR(p2) p1∗( ) N ∗ E ⊗L 1    on PicT . This line bundle has a natural section (the theta function) θ( ), which extends the section 1 on the open schematically dense subscheme U PicE 1 over ⊂ T which the complex R(p2) p1∗( ) is trivial. We define to be the zero set ∗ E ⊗L DE 1 of the section θ( ); it is also the largest closed subscheme of PicT over which the E 1 coherent Pic1 -module R (p2) p1∗( ) has rank 1 (i.e., defined in terms of O T ∗ E ⊗L ≥ a fitting ideal of this sheaf).   One can prove that ( 2Θ1 ) is isomorphic to the pullback of an invertible N⊗O− T T -module on T . Hence the divisor defines an S-morphism of T into P: O DE ϕ(T, ) : T P . E −→ These constructions define therefore a transformation of the stack of semi-stable rank 2 bundles with trivial determinant on over S towards the scheme P. It turns out that this defines an isomorphism of theC coarse moduli scheme P MC −→ towards P; see [NR]. A remark about the natural determinant bundle on (see [Hi], page 360): The relative Picard group of P over S is Z and is generatedL MC by (1). Hence it is OP clear that = P(n) p∗( ) for some invertible S -module on S. The integer L ∼ O ⊗ N O N λ n may be determined as follows. We know by [Hi], page 360, that P (n)− ∼= KP . The integer λ = 4 in this case, hence n = 1. We are going to defineO a projective k heat operator on ⊗ over S, but we have seen in Subsection 2.3.5 that this is the same as defining aL projective heat operator on (k). Hence we will work with the OP line bundle P(k) from now on. (Note that as both P(1) and are defined on the moduli stack,O they must be related by a line bundleO comingL from the moduli stack 2 of curves of genus 2; however Pic( 2) is rather small.) M 0 0 M Let Pic =Pic/S S denote the Picard scheme of invertible bundles of relative degree 0 onC →over S. There is a natural S-morphism OC C f :Pic0 P −→ which in terms of the moduli-interpretations of both spaces can be defined as fol- lows: 0 1 Pic (C) [L] [L L− ] P(C). 3 7−→ ⊕ ∈ 1 It is clear that the divisor DE associated to E = L L− is equal to ⊕ 1 1 1 1 DE =(Θ +[L]) (Θ +[L− ]) Pic (C). C ∪ C ⊂ 0 This implies readily that f ∗ P(1) = Pic0 (2Θ ), in fact f ∗ P(1) may serve as the 0 O 0 ∼ O O 0 definition of Pic0 (2Θ )onPic. This induces an isomorpism P ∼= Pic0 (2Θ ) ∗ which identifiesO f with the natural map. Thus f factors over := Pic|O0 / 1 ,the| relative Kummer surface, and defines a closed immersion (overKS): h± i , P. K →

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Any E on C that is semi-stable but not stable is an extension of the form 1 0 L E L− 0, with L of degree zero. This bundle is S-equivalent to the → → → 1 → s bundle L L− . Thus the open subscheme P P is equal to the complement of the image⊕ of f :Pic0 P. ⊂ → 2.5.2. Automorphisms. The group scheme =Pic0[2] of 2-torsion points of Pic0 1 H over S acts on the schemes Pic . Indeed, if over T is a family of locally free E C sheaves as above and if is a line bundle on T which defines a 2-torsion point of Pic0,then is anotherA family of locally freeC sheaves satisfying (a) and (b) of E⊗A 2.5.1. This defines an action S P Pof on P, given the identification of P as the moduli scheme. It is easyH× to see→ that thisH action is induced from the natural action of on Pic1. Furthermore the morphism f :Pic0 Pis equivariant with respect toH this action. → Note that the action just defined does not lift to an action of on P(1). Let H O 0 us write for the theta group of the relatively ample line bundle Pic0 (2Θ )on Pic0 overGS; this group scheme gives the Heisenberg group for any pointO 0 S(C). We remark that fits into the exact sequence ∈ G 1 G m,S 0 . −→ −→ G −→ H −→ There is a unique action of on P(1) which lifts the action of on P and agrees G O 0 H with the defining action on f ∗ (1) over Pic . G OP 2.5.3. Test families. Let be a rank 2 free -module on T satisfying (a) and (b) of 2.5.1. We will say thatE the pair (T, )isaOtest family ifC the following conditions are satisfied: E 1. T S is smooth, → 2. all E = 0 are simple vector bundles on C = 0,and 3. the mapE R 1 pr ( nd ( )) is an isomorphism.C TT/S −→ 2 E 0 E The last condition needs some∗ clarification. The obstruction for the locally free sheaf to have a connection on T is an element (the Atiyah class) in E C 1 1 H ( S T,Ω ST nd( )). C× C× ⊗E E We can use the maps Ω1 pr Ω1 and nd( ) nd ( ) to project this ST 2∗ T/S 0 1 C×1 → E E →E E toasectionofΩT/S R pr2 ( nd 0( )), whence the map of condition 3. This ⊗ ∗ E E condition means that the deformation of E = 0 for any 0 T is versal in the “vertical direction”. E ∈ The following lemma is proved in the usual manner, using deformation theory (no obstructions!) and Artin approximation.

2.5.4. Lemma. For any 0 S (C) and any simple bundle E on C = 0 there exists a test family (T, ) such∈ that (E,C) occurs as one of its fibres. C E 2.5.5. Lemma. Let (T, ) be a test family. The morphism ϕ(T, ) : T P is ´etale. E E → Proof. Let E be a simple rank 2 bundle on a smooth projective genus two curve 1 C over Spec(C) with det(E)= C.Letη H(C, nd 0(E)). For each invertible sheaf L on C, with deg(L)= 1O and given an∈ embeddingE L E, consider the map − ⊂ nd (E) om(L, E). E 0 −→ H We have to show ( ): If for all L E as above η maps to 0 in H1(C, om(L, E)), then η =0. ∗ ⊂ H

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Indeed, the condition η 0 means that the trivial deformation L[]ofLover C[]=C Spec(C[]) can7→ be embedded into the deformation of E given by η.If × this holds for all L E, then the divisor DE does not move, and we want this to imply that the infinitesimal⊂ deformation of E is trivial. Let us prove ( ). We will do this in the case that E is not stable, as we already have the result in∗ the stable case; see Subsection 2.4.1. Thus we may assume the bundle E is simple but not stable: E is a non-trivial extension 1 0 A E A − 0 , −→ −→ −→ −→ 2 with deg A =0,A⊗ = C. In this case it is easy to see that there exist three C 6∼O line bundles L1 E, L2 E and L3 E of degree 1onCsuch that the arrow towards E in the⊂ following⊂ exact sequence⊂ is surjective:− 0 K L L L E 0 . −→ −→ 1 ⊕ 2 ⊕ 3 −→ −→ Here K is just the kernel of the surjection. For example we can take Li of the form 1 A− ( Pi)fori=1,2andL3 of the form A( P3). (This is the only thing we will need− in the rest of the argument; it should be− easy to establish this in the case of a stable bundle E also.) This property then holds for Li E sufficiently general ⊂ also; choose Li sufficiently general. Note that K = L L L as det(E)= C. ∼ 1 ⊗ 2 ⊗ 3 O Suppose we have a deformation E of E given by η as above, i.e., we have Li[] E lifting L E. This determines an exact sequence → → 0 K  L [  ] L [  ] L [  ] E  0 . −→ −→ 1 ⊕ 2 ⊕ 3 −→ −→ But detE ∼= [], hence K ∼= L1[] L2[] L3[] ∼= (L1 L2 L3)[] is trivial as well. NowO noteC that by our general⊗ position⊗ we have ⊗ ⊗ (1) dim H0(C, om(K, L )) = dim H0(C, L L )=1, H 1 2 ⊗ 3 and similarly for L2 and L3. Therefore, up to a unit in C[], there is only one map K[] Li[], lifting K Li. Thus the sequence (1) above is uniquely determined → → up to isomorphism, hence E ∼= E[] is constant, i.e., η =0.  2.6. Results. This lemma completes the preparations. In the remainder of this section we prove the desired results on the existence and the characterization of the connection.

s 2 2.6.1. Theorem. The Hitchin symbol ρHitchin : p∗ S S Ps/S defined in Sub- section 2.4.1 extends to a symbol T → T 2 ρHitchin : p∗ S S . T −→ T P /S Proof. Let (T, ) be a test family. We have the morphism ϕ = ϕ(T, ) : T P E E → which is ´etale and which induces therefore an isomorphism T/S ϕ∗ P/S .Onthe 1 T → T other hand we have the isomorphism T/S R pr2 ( nd 0( )) of the test family (T, ). We also have the pairing as inT Subsection→ 2.4.1∗ E E E pr ( nd ( ) Ω1 ) pr ( nd ( ) Ω1 ) pr ((Ω1 ) 2). 2 0 T /T 2 0 T /T 2 T /T ⊗ ∗ E E ⊗ C ⊗ ∗ E E ⊗ C −→ ∗ C 1 Note that pr2 ( nd 0( ) Ω /T ) is a locally free sheaf of T -modules as has ∗ E E ⊗ T O E simple fibres. These maps andC duality give us together with the Kodaira-Spencer map a map 2 S T S ϕ ∗ P /S. T ⊗O −→ T s 1 s We remark that on the (non-empty) stable locus T = ϕ− (P ) this map is equal s to the pullback of ρHitchin.

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s Thus, by Lemma 2.5.5, we see that we can extend ρHitchin to any point x P(C) which is the image of some simple bundle. Note that a semi-stable bundle E∈,which 1 is a non-trivial extension 0 L E L− 0 with deg(L) = 0, is simple if and 2 → → → → only if L⊗ = C . It follows from Lemma 2.5.4 and the above that we can extend s 6∼ O 0 ρHitchin to the complement of the codimension 3 locus f(Pic [2]), and hence by Hartog’s theorem it extends.  2.6.2. Proposition. Let f :Pic0 Pbe as in 2.5.1. Then f is a submersion on the locus := Pic0 Pic0[2]. There→ is a commutative diagram Y \ 2 f ∗(ρHitchin): f∗p∗S f ∗ S T −→ T P /S || ↑ 2 4ρAb : S S /S |Y T ⊗OY −→ T Y 0 where ρAb is the symbol of the abelian heat operator on 0 (2Θ )=f∗ (1). OPic OP Proof. Consider once again a simple bundle E,whichisgivenasanextension 1 0 L E L− 0 with deg(L) = 0. Clearly, the kernel of the map → → → → 1 1 1 H (C, nd (E)) H ( C, om(L, L− )) E 0 −→ H represents deformations preserving the filtration on E, i.e., tangent vectors along the Kummer surface. Dually, this corresponds to the image of the map 0 1 1 0 1 H (C, om(L− ,L) Ω ) H ( C, nd (E) Ω ). H ⊗ C −→ E 0 ⊗ C We use the Trace-form to identify nd 0(E) with its dual. Note that there is a canonical exact sequence E 2 2 0 C nd (E)/L⊗ L ⊗− 0 −→ O −→ E 0 −→ −→ induced by the filtration L E on E.Notethat ⊂ 0 2 1 dim H (C, ( nd (E)/L⊗ ) Ω )=2, E 0 ⊗ C 0 1 since if it were 3, then dim H (C, nd 0(E) Ω ) 4, contrary to the assumption that E is simple.≥ Thus we get the exactE sequence⊗ ≥ 0 1 0 2 1 H (C, Ω ) = H (C, ( nd (E)/L⊗ ) Ω ) ∼ E 0 ⊗ C 0 1 0 2 1 H ( C, nd (E) Ω ) -H (C, L⊗ Ω ) = C. ←− E 0 ⊗ C ← ⊗ C ∼ This is (canonically) dual to the sequence of cotangent spaces 0 0 T [ L ] Pic T f ([L])(P) ( N P ) [ L ] 0 . −→ −→ −→ Y −→ 2 0 We want to see that our symbols in the point [L] lie in the space S T[L] Pic . We have to study the map

0 1 0 1 0 1 2 H (C, nd (E) Ω ) H (C, nd (E) Ω ) H ( C, (Ω )⊗ ) E 0 ⊗ C × E 0 ⊗ C −→ C X η Y ω Trace(XY ) ηω. ⊗ × ⊗ 7−→ ⊗ 0 2 1 0 1 But the elements of H (C, L⊗ Ω ), resp. those of H (C, Ω ), locally look like ⊗ C C 0 10 ∗ η, resp. ω. 00⊗ 0 1⊗ −    

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0 1 Therefore the pairing just reduces to twice the multiplication pairing H (C, ΩC ) 0 1 0 1 2 × H (C, ΩC ) H (C, (ΩC )⊗ ). This multiplication→ pairing, however, corresponds exactly (via Kodaira-Spencer) to the symbol of the heat operator on the theta-divisor of the Jacobian Pic0 of over S. Since we are looking at 2Θ0, we get the desired factor 4; compare withC Subsections 2.3.8 and 2.3.7. 

2.6.3. Corollary. The symbol ρHitchin is invariant under the action of the group scheme =Pic0[2] on P and S2 . H TP/S Proof. This follows from the observation that it is true for ρAb and thus for |Y ρHitchin , the Hitchin symbol restricted to the Kummer surface. Further, one uses the|K remark that there are no non-vanishing elements of H0(P,S2 )which TP/S vanish on ,thatis,H0(P,S2 ( 4)) = 0. K TP/S −  2.6.4. Proposition. There exists a unique projective heat operator D¯ λ,k (with λ ∈ C∗ a nonzero complex number) on (k) over S with the following properties: OP ¯ 1 1. the symbol of Dλ,k is equal to 4λ ρHitchin,and 2. the operator D¯ λ,k commutes (projectively) with the action of on (k). G OP Proof. This follows readily from the discussion in Subsection 2.3.7. Indeed, both 1 obstructions mentioned there vanish in view of the vanishing of R p P/S and 1 ∗T R p P. We have a good way of choosing the elements D(θ): namely, we choose them∗O -invariantly. This is possible: first choose arbitrary lifts, and then average over liftingsG of a full set of sections of (this can be done ´etale locally over S). This H 1 determines the D(θ)uniquelyuptoanelementofp− ( S), as (p P/S )H =(0). O ∗T Hence we get the desired projective heat operator.  It follows from Proposition 2.6.2 that the heat operators D¯ λ,k are compatible ¯ with the morphism f : P. Also, for k =1andλ=1,weseethatD1,1 is Y→ 0 compatible with the abelian heat operator on Pic0 (2Θ ) restricted to (this is 1 O Y one of the reasons for the factor 4 , but see below also). 2.6.5. Hitchin’s connection for genus 2 and rank 2. The title of this subsection is somewhat misleading. As mentioned before, in the paper [Hi] there is no definition of a heat operator in the case of genus 2 and rank 2. However, we propose the following definition.

2.6.6. Definition. The (projective) Hitchin connection on p P(k) is the projective ∗O¯ connection associated to the projective heat operator Dk := D(k+2),k.

This definition makes sense for the following reason: The symbol of the operator ¯ D(k+2),k is equal to 2/(2k +4)ρHitchin which is equal to the symbol that occurs in [Hi], Theorem 3.6 (the factor 2 comes from our way of normalizing symbols; see Subsection 2.2.4). Thus the only assumption we needed in order to get Dk was the assumption that it is compatible with the action of ; see Subsection 2.6.4. G 2.6.7. How to compute the heat operators? To determine Dk we introduce heat operators Dλ,k under the following assumptions: 1. The family of Kummer surfaces Pis given by one equation K⊂ F Γ(P, (4)). ∈ OP

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2. We have chosen an integrable connection 0 on := p P(1) over S which is equivariant for the action of . ∇ F ∗O G We remark that both conditions can be satisfied on the members of an open covering of S. Also the connection in 2 is determined uniquely up to the addition of a ∇0 linear operator of the form η id for some closed 1-form η on S. We remark that 0 · k ∇ induces a connection 0 on all the locally free sheaves S = p P(k)overS, but ∇ F ∗O these should not be confused with the connections induced by the Dλ,k! In addition determines an integrable connection on P over S and a lift of this to connections ∇0 on the sheaves P(k)overP. In other words we have rigidified (P, P(k)) over S. There is a naturalO surjection O 2 2 2 S S S ∗ p S P /S. F⊗O F −→ ∗ T 2 2 which has a canonical splitting, given by decomposing S S S ∗ into irre- ducible GL( )-modules, i.e., the unique GL( )-equivariantF⊗ splitting.O F Thus we F 2 F 2 2 identify a section X of S P/S with a section X of S S S ∗.Notethatwe T F⊗O F 2 2 (2) can regard sections of S S S ∗ as sections of Diff P(k) in a natural F⊗O F P/S O manner, by considering them as second order differential operators on the affine
4-space Spec(S∗ ) invariant under the scalar action of Gm,S.WewriteEfor the F Euler-vector field, i.e., the section of ∗ that corresponds to the identity map of . F⊗F FAssume we have a -invariant section G 2 2 1 X S S S ∗ S ΩS ∈ F⊗O F ⊗O with the property 5 1 (Eq.) X F = 0(F )E in S ( ) S ∗ S ΩS, · ∇ F ⊗O F ⊗O where X F denotes contracting once (note that F canbeviewedasasectionof S4 ). · FIf we have such an X we define a heat operator

(2) Dλ,k : p∗ S Diff ( (k)) T −→ P OP by the formula (θ a local section of S) T 1 Dλ,k(θ)=θ Xθ,k. −λ

Here Xθ,k means the following: first contract X with θ to get a section Xθ of 2 2 S S S ∗ and then consider this as a second order differential operator Xθ,k F⊗O F of P(k)onXover S by the remarks above. Note that the term θ acts on P(k) usingO . O ∇0 2.6.8. Lemma. Let Dλ,k be the heat operator defined above.

1. The heat operator Dλ,k commutes with the action of on (k). G OP 2. The heat operators Dλ,k are compatible with f : P.Ifλ=k,thenDk,k is strictly compatible with f : P. Y→ Y→ Proof. The second statement holds since X is -invariant. We now verify the last G statement. Note that Dk,k is strictly compatible with Pif the operators K→ Dk,k(θ) preserve the subsheaf P(k)of P(k). This can be seen by the de- scription of strict compatiblilityIK in·O terms of localO coordinates given in Subsection

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2.3.10. We may check this condition on the affine 4-space Spec(S∗ ). Thus let k 4 F G S − ( ) be a section, let θ be a local vector field on S and consider ∈ F 1 ,θ(GF ) Xθ(GF ). ∇0 − k We have to show that this is divisible by F . We may assume that G is horizontal for 0, as we can find a horizontal basis k 4 ∇ for S − ( ) locally. Thus we get F 1 1 1 1 G ,θ(F ) Xθ(GF )=G ,θ(F ) Xθ(G)F (X F )θ(G) GXθ(F ). ∇0 − k ∇0 − k − k · − k Here we have used a general formula for the application of a second order operator like Xθ on a product like FG. Note also that X(F ) is simply the contraction of X F , hence by (Eq.)weget · 1 1 1 G ,θ(F ) Xθ(G)F ,θ(F )E(G) GE( ,θ(F )). ∇0 − k − k ∇0 − k ∇0 Next, we use that acting by E on a homogeneous polynomial gives degree times the polynomial: 1 1 1 1 G ,θ(F ) Xθ(G)F ,θ(F )(k 4)G G4 ,θ(F )= Xθ(G)F ∇0 − k − k ∇0 − − k ∇0 −k which is divisible by F .ThusDk,k is indeed strictly compatible with f : P. Y→  2.6.9. Theorem. Let X be a solution to the equation (Eq.) above and let Dλ,k be the heat operators defined in this subsection. Then

1. The projective heat operator defined by Dλ,k is the operator D¯ λ,k defined in Subsection 2.6.4. Thus Dk ,k defines the Hitchin connection on (k). +2 OP 2. The heat operators Dλ,k are projectively flat for any k and λ. 3. For λ = k, the operator Dk,k is strictly compatible with the abelian projective 0 0 heat operator on 0 ( (2kΘ )) over an open part of Pic . OPic O Proof. Consider first the case of Dk,k as defined in 2.6.7. We have seen above that it is strictly compatible with f : P. Therefore, by Subsection 2.3.10 it defines Y→ a heat operator Dk on f ∗ P(k)over . In view of Hartog’s theorem this extends O Y 0 0 uniquely to a heat operator Dk of Pic0 ( (2kΘ )) on Pic over S. We have seen in Subsection 2.3.8 that there is a uniqueO O such heat operator, whose symbol is 1/k times the symbol ρAb of D1. Combining this with the assertions in Subsection 2.6.2 we see that (1/4k)ρHitchin ρD vanishes along and hence is zero. This proves − k,k K that Dk,k agrees with the projective heat operator D¯ k,k defined in Subsection 2.6.4. Now it follows from the transformation behaviour of the symbol of D¯λ,k and Dλ,k that these agree for arbitrary λ. Thus we get the agreement stated in the theorem. To see that these heat operators are projectively flat, we argue as follows. Let θ, θ0 be two local commuting vector fields on S. We have to see that the section (3) 1 [Dλ,k(θ),Dλ,k(θ0)] of Diff ( (k)) lies in the subsheaf p− ( S ). Again, for λ = k P/S OP O this section is zero when restricted to , and again this implies that the section is zero in that case. Now, let us computeK 1 1 [Dλ,k(θ),Dλ,k(θ0)] = [θ Xθ,k,θ0 Xθ ,k] − λ − λ 0 1 1 = [θ, Xθ ,k] [θ0,Xθ,k] + [Xθ,k,Xθ ,k]. λ 0 − λ2 0  

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2 The index k is now superfluous, as we can see this as an expression in S S 2 3 3 F⊗O S ∗ S S S ∗. We know that this expression is zero for any k with λ = k F ⊕ F⊗O F considered as a third order operator on P(k). This implies that both terms are zero, hence the expression is zero for any Oλ and any k. (We remark for the doubtful reader that we will verify the flatness also by a direct computation using the explicit description of the operator.)  3. The flat connection 3.1. We introduce certain families of flat connections on (trivial) bundles over the configuration space (cf. 1.4). These are defined by representations of the Lie algebra so(2g +2). ThenP we derive a convenient form for the equation of the family of Kummer surfaces , P (in fact for any g we point out a specific section Pz of a trivial bundle overK →). In Theorem 3.4.3 we show that this equation is flat for one of our connections.P This will be important in identifying Hitchin’s connection in the next section. 3.2. Orthogonal groups.

2 2 3.2.1. The Lie algebra. Let Q = x1 + ...+x2g+2. Then the (complex) Lie algebra so(Q)=so(2g +2)is: so(2g +2)= A End(C2g+2): tA+A=0 ; { ∈ } let Fij := 2(Eij Eji)( so(2g + 2)), where Eij is the matrix whose only non-zero − ∈ entry is a 1 in the (i, j)-th position (the commutator of such matrices is [Eij ,Ekl]= δjkEil δliEkj ). The alternating matrices Fij (with 1 i

Ωij := Fij Fij mod I ( U(so(2g + 2))), note Ωij =Ωji. ⊗ ∈ In particular, for any Lie algebra representation ρ : so(2g +2) W we now have 2 → endomorphismsρ ˜(Ωij )=ρ(Fij ) : W W which satisfy the same commutation → relations as the Ωij . For λ C∗ we define a U(so(2g + 2))-valued one-form ωλ on : ∈ P

1 Ωij dzi 1 ρ˜(Ωij )dzi ωλ := λ− ;let˜ρ(ωλ):=λ−  zi zj   zi zj  i=j i=j X6 − X6 −    

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1 whereρ ˜ : U(so(2g+2)) End(W ) is a representation. Noteρ ˜(ωλ) End(W ) Ω . → ∈ ⊗ P The one-form ωλ defines a connection on the trivial bundle W C by: ⊗ OP W 1 λ : W W Ω ,fww df fρ˜(ωλ)(w)(f ,w W). ∇ ⊗OP −→ ⊗ P 7−→ ⊗ − ∈OP ∈ The covariant derivative of f w with respect to the vector field ∂/∂zi on is the composition of W with contraction:⊗ P ∇λ W ∂f 1 λ (f w) ∂/∂z = w ρ˜(Ωij )(w) ( W ). ∇ ⊗ i ⊗ ∂zi − ⊗ zi zj ∈ ⊗OP j, j=i −
X6 3.2.4. Proposition. Let ρ : so(2g +2) End(W ) be a Lie algebra representa- W −→ tion. Then λ is a flat connection on W for any λ C∗. ∇ ⊗OP ∈ W Proof. Obviously λ is a connection. It is well known (and easy to verify) that the connection defined∇ by ω is flat (so dω + ω ω = 0) if the following infinitesimal pure braid relations in U(so(2g + 2)) are satisfied:∧

[Ωij , Ωkl]=0, [Ωik, Ωij +Ωjk]=0, where i, j, k, l are distinct indices (cf. [Ka], Section XIX.2). To check these relations we use that X Y = Y X +[X, Y ]inU(so). The first relation is then obviously satisfied. We⊗ spell out⊗ the second. Consider first:

Fik Fij Fij =(Fij Fik +[Fik,Fij ]) Fij ⊗ ⊗ ⊗ ⊗ = Fij Fik Fij Fkj Fij ⊗ ⊗ − ⊗ = Fij (Fij Fik Fkj )+Fjk Fij ⊗ ⊗ − ⊗ = Fij Fij Fik +(Fij Fjk + Fjk Fij ), ⊗ ⊗ ⊗ ⊗ so we have [Fik,Fij Fij ]=D1 with D1 symmetric in the indices i and k. Similarly we get: ⊗

Fik (Fik Fij Fij )=Fik (Fij Fij Fik + D ) ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 1 =(Fik Fij Fij ) Fik + Fik D ⊗ ⊗ ⊗ ⊗ 1 = Fij Fij Fik Fik +(D Fik + Fik D ). ⊗ ⊗ ⊗ 1 ⊗ ⊗ 1 Thus [Ωik, Ωij ]=D Fik + Fik D which is antisymmetric in i and k. Therefore 1⊗ ⊗ 1 [Ωik, Ωij ]= [Ωki, Ωkj ]= [Ωik, Ωjk] − − which proves the second infinitesimal braid relation.  3.2.5. Differential operators. Given a Lie algebra representation ρ : g End(S ) → 1 where Sk S := C[... ,Xi,...] is the subspace of homogeneous polynomials of degree k,⊂ there is a convenient way to determine the representation ρ(k) : g → End(Sk) induced by ρ.Ifρ(A)=(aij ) w.r.t. the basis Xi of S1, then we define ∂P LA := aij Xi∂j ( End(S)), with ∂j (P ):= ∈ ∂X i,j j X for P S. Then obviously ρ(A)(P )=LA(P)ifP is linear and the Leibnitz rule ∈ (k) shows that ρ (A)=LA :Sk Sk for all k. The composition (in End(S→)) of two operators is given by: 2 2 (Xi∂j) (Xk∂l)=XiXk∂j∂l+δjkXi∂l, thus L =(1/2)σ(L )+L 2, ◦ A A A

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2 with symbol σ(LA)=2 i,j,k,l aij aklXiXk∂j ∂l (see our convention 2.2.4). Assume that we have Aij g such that Ωij = Aij Aij satisfy the infinitesimal braid relations (cf. proofP of the∈ previous proposition) and⊗ that in the representation 2 ρ we have ρ(Aij ) = µI with µ C. Then we conclude that the operators ∈ σ(L2 ):S S Aij −→ also satisfy the infinitesimal braid relations.

3.3. The Kummer equation.

3.3.1. Introduction. We recall how one can obtain the equation of the Kummer variety for a genus two curve, and more generally, how a hyperelliptic curve z 4 C determines in a natural way a quartic polynomial Pz S V (ωg+1)whereV(ωg+1) is a half spin representation of so(2g +2). ∈ The polynomial Pz lies in the subrepresentation of V (4ωg+1) which also occurs in S2( g+1C2g+2)whereC2g+2 is the standard representation of so(2g +2). We ∧ will exploit this fact to verify that Pz is a flat section for one of the connections introduced in Subsection 3.2.

3.3.2. The orthogonal Grassmannian. We define two quadratic forms on C2g+2 by: 2 2 2 2 Q : x + ...+x ,Qz:zx++z g x , 1 2g+2 11··· 2 +2 2g+2 and we use the same symbols to denote the corresponding quadrics in P2g+1.The quadric Q has two rulings (families of linear Pg’s lying on it). Each of these is parametrized by the orthogonal Grassmannian (spinor variety) denoted by GrSO. 1 The variety GrSO is smooth and projective of dimension 2 g(g +1). Let V (ωg+1) be a half spin representation of so(2g + 2). There is an embedding

φ : GrSO P V ( ω g )∗;letGr (1) := φ∗ V (1). −→ +1 OSO OP In fact, φ(GrSO) is the orbit of the highest weight vector. The map φ is equivariant for the action of the spin group SO (which acts through a half spin representation on V (ωg+1)). It induces isomorphisms: 0 g H (GrSO, (n)) = V (nωg ). O +1 In case g = 2, one has isomorphisms:

∼= 3 φ : GrSO P ,Sk=V(kωg )andSO = SL (C) −→ +1 6 ∼ 4 and the half spin representation is identified with the standard representation 4 g SL4(C)onC (or its dual).

3.3.3. The Pl¨ucker map. Let GrSL be the Grassmannian of g-dimensional sub- spaces of P2g+1.WedenotethePl¨ucker map by: g +1 2g+2 p : GrSL P C , −→ ∧ v ,... ,vg v ... vg = pi ...i ei ... ei . h 1 +1i7−→ 1∧ ∧ +1 1 g+1 1 ∧ ∧ g+1 The map p is equivariant for the SL(2g + 2)-actionX on both sides and induces isomorphisms of sl(2g + 2)-representations 0 g+1 2g+2 H (GrSL, Gr (n)) = V (nλg ), with V (λg ) = C . O SL +1 +1 ∼ ∧

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As a representation of so(2g +2)thespace g+1C2g+2 is reducible; one has (cf. ∧ [vG], (3.7), [FH], 19.2, Th. 19.2 and Remarks (ii)), with V (ωg)andV(ωg+1)the half spin representations:§

g+1 2g+2 C = V (2ωg) V (2ωg ). ∧ ⊕ +1 4 In particular, the irreducible component V (4ωg+1)ofSV(ωg+1)isalsoacom- ponent of S2( g+1C2g+2) (viewed as an so(2g + 2) representation). Elements of S2( g+1C2g+2∧) can be seen as quadratic forms in the Pl¨ucker coordinates, when ∧ restricted to GrSO they can be viewed as (restrictions to GrSO of) quartic polyno- mials on PV (ωg+1)∗.

3.3.4. Notation. The following lemma gives this decomposition of g+1C2g+2 ex- 2g+2 ∧ plicitly. Let ei be the standard basis of C . For any { } S B := 1, 2,... ,2g+2 , S =g+1, ⊂ { } | |

we write S = i ,i,... ,ig with i

σS (k):=ik,σS(g+1+k):=jk.

3.3.5. Lemma. The two non-trivial so(2g +2)-invariant subspaces in g+1C2g+2 are: ∧

g+1 g+1 ... ,eS +sgn(σS)i eS0 ,... S 1 and ... ,eS sgn(σS)i eS0 ,... S 1, h i 3 h − i 3 where S runs over the subsets of B with g +1 elements with 1 S. ∈ Proof. (Cf. [FH], 19.2, Remarks(iii).) The quadratic form Q defines an so- equivariant isomorphism§

g+1 2g+2 g +1 2g+2 g+1 2g+2 B : C ( C )∗ = (C )∗,eS S := i ... i ∧ −→ ∧ ∧ 7−→ 1 ∧ ∧ g+1

with ... ,j,... the basis dual to the ei. There is also a canonical (in particular so-equivariant){ isomorphism:}

g+1 2g+2 g +1 2g+2 C : C ( C )∗,α[β cα,β] ∧ −→ ∧ 7−→ 7→

with cα,β C defined by ∈

α β = cα,βe e ... e g . ∧ 1 ∧ 2 ∧ ∧ 2 +2 1 g+1 2g+2 g+1 2g+2 Thus we have an isomorphism A := B− C : C C whose eigenspaces are so-invariant. An easy computation◦ ∧ shows that→∧ the eigenvalues of A g+1 g+1 are i and the eigenvectors are eS sgn(σS)i e0 . ± ± S 

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3.3.6. A quartic polynomial. We consider the g-dimensional subspaces of P2g+1 which are tangent to Qz: g 2g+1 g B¯z := P P : rank(Qz restricted to P ) g ( GrSL). ⊂ ≤ ⊂ ¯ The subvariety Bz is defined by a quadratic polynomial Pz in the Pl¨ucker coordi- 0 nates, that is, by a section in H (GrSL, Gr (2)) ([vG], Thm 3): O SL 0 2 g+1 2g+2 B¯z = Z(Pz),PzH(GrSL, Gr (2)) S ( C ). ∈ O SL ⊂ ∧ 0 This section is in fact the unique SOz-invariant in H (GrSL, Gr (2)), in case O SL g = 2 the quartic surface in PV defined by Pz is the Kummer surface of the curve z ([DR]). C 3.3.7. Lemma. The variety B¯z is defined by 2 Pz = zSeS, S X where S runs over the subsets with g +1 elements of B = 1,... ,2g+2 and { } zS := zi zi ...zi ,eS=ei... ei if S = i ,... ,ig . 1 2 g+1 1∧∧ g+1 { 1 +1} Proof. From [vG], Theorem 30 and its proof, we know that the trivial representa- 0 tion of soz has multiplicity one in H (GrSL, GrSL(2)) = V (2λg+1). This sl2g+2 representation corresponds to the partition µ:2Og+2=2+2+...+2andisthus 2g+2 2g+2 realized as (C )⊗ cµ where cµ is the Young symmetrizer; one has in fact 2 g+1 2g+2 V (2λg ) S ( C )(see[FH], 15.5, pp. 233-236). We use the standard +1 ⊂ ∧ § tableau as in [vG] (and [FH]) to construct cµ. The trivial soz sub-representation is obtained as follows. The quadratic form Qz 2g+2 2g+2 2g+2 defines the soz-invariant tensor i=1 ziei ei C C , and we consider the g + 1-fold tensor product of this tensor,⊗ where∈ we insert⊗ the k-th factor in the positions k and g +1+k: P 2g+2 2g+2 2g+2 τz := zi zi ...zi ei ei ... ei ei ... ei ( (C )⊗ ). 1 2 g+1 1 ⊗ 2 ⊗ ⊗ g+1 ⊗ 1 ⊗ ⊗ g+1 ∈ i ,... ,i =1 1 Xg+1 Thus τz is soz-invariant and it has the advantage that τzcµ is easily determined:

τzcµ =(cst) zS eS eS, · S X with (cst) a non-zero integer and where we sum over all subsets S with g +1 elements of 1,... ,2g +2 . (In fact, cµ = aµbµ and aµ symmetrizes the indices { } 1,g+2;2,g+3; ...; g+1, 2g+ 2 of a tensor (but τz is already symmetric in these indices) and next bµ antisymmetrizes the first g + 1 and last g + 1 indices, giving the result above.)  3.4. Verifying the contraction criterium.

3.4.1. We need to determine the images of the Ωij ’s in the representation S2( g+1C2g+2). In the standard representation C2g+2 of so(2g + 2), with basis ∧ ei and dual basis i,wehave: 2g+2 2g+2 2g+2 Fij =2(ei j ej i) (C ) (C )∗ = End(C ). ⊗ − ⊗ ∈ ⊗ g+1 2g+2 On C , with basis eS = ei ... ei as before, Fij acts as ∧ 1 ∧ ∧ g+1 Fij (ei ... ei )=Fij (ei ) ei ... ei + ...+ei ... ei Fij (ei ). 1 ∧ ∧ g+1 1 ∧ 2 ∧ ∧ g+1 1 ∧ ∧ g ∧ g+1

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Define a subset of B = 1,... ,2g+2 by { } S + ij =(S i, j ) (S i, j ). ∪{ } − ∩{ } Then Fij (eS ) is, up to sign, 2eS ij if S i, j = 1 and is zero otherwise: + | ∩{ }|

Fij =2 tS,ijeS ij S + ⊗ S, S i,j =1 | ∩{X}| 2 with tS,ij = 1. Note that if S i, j =1,thenF eS = 4eS so tS,ij = tS ij,ij . ± | ∩{ }| ij − − + 3.4.2. Lemma. Let W be a complex vector space, let D W W ∗ be a linear differential operator and let P SkW . Then the contraction∈ of P⊗ with the symbol of D2 is given by: ∈

2 k 1 σ(D ) P =2D(P)D S − W W∗. · ∈ ⊗ Proof. The proof is straightforward using the conventions of 2.2.4 and 3.2.5. 

3.4.3. Theorem. The composition, still denoted by Pz:

2 g+1 2g+2 4 S C S V ( ω g ),zPz, P−→ ∧ −→ +1 7−→ 2 with Pz = S zSeS (as in Subsection 3.3.6), satisfies the differential equations: P 1 σρ˜(Ωij ) Pz +(∂ziPz)E=0 (1 i 2g+2) 16 zi zj  · ≤ ≤ j=i X6 −   with E = eS S the Euler vector field and ⊗ P ρ˜ : U(so(2g +2)) End(S2( g+1C2g+2)). → ∧ W 4 Thus Pz is a horizontal section for the connection λ with W = S V (ωg+1) and λ = 16. ∇ − Proof. For symmetry reasons it suffices to verify the equation for i = 1. By Lemma 3.4.2 2 2 σρ˜(Ω j ) e =2ρ(F j)(e )ρ(F j )=4eSρ(F j)(eS)ρ(F j ). 1 · S 1 S 1 1 1 To simplify notation we write Ωij for σρ˜(Ωij )andFij for ρ(Fij ) (as in 3.4.1). Then F1j(eS) = 0 unless S 1,j = 1. Assume 1 S and j S (so j B S)and | ∩{ }| ∈ 2 6∈ 2 ∈ − consider the contraction of Ω1j with the term zSeS + zS+ij eS+ij from Pz:

2 2 Ω j (zSe + zS je )=8(tS, jzS + tS j, j zS j)eSeS jF j 1 · S +1 S+1j 1 +1 1 +1 +1 1 =8tS, j(z z ¯ zjz ¯)eS eS jF j 1 1 S − S +1 1 2 =(z zj)Ω j z ¯e , 1− 1 · S S where S¯ := S 1 . Therefore we get: −{ }

Ω1j 2 Pz =Ω1j zS¯eS . z1 zj · ·  S 1,j S − 3X6∈  

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Summing this result over all j and changing the order of summation gives:

Ω1j 2 Pz = zS¯ Ω1j eS  z1 zj  ·  ·  j=1 S 1 j B S X6 − X3 ∈X−     =8 z¯eS tS, jeS jF j S  1 +1 1  S1 jBS X3 ∈X−   2 where, as before, B = 1,... ,2g+2 . On the other hand, since Pz = zSeS,we have: { } 2 P ∂z1 Pz = zS¯eS. S 1 X3 Thus the theorem follows if we prove, for all S with 1 S: ∈ 2eSE + tS,1jeS+1jF1j =0. j B S ∈X− With the definition of F1j we find:

tS, jeS jF j = tS, jeS j 2tT, jeT j T 1 +1 1 1 +1  1 +1 ⊗  j B S j B S T, T 1,j =1 ∈X− ∈X− | ∩{X }|   =2 tS, jtT, jeS jeT j T .  1 1 +1 +1  ⊗ T jBS, T 1,j =1 X ∈− X| ∩{ }| Note that eSE = eSeT T (with T B, T = g + 1). Comparing coefficients T ⊗ ∈ | | of T , it remains to prove that (for all S 1andallT): P 3 eSeT + tS,1jtT,1jeS+1jeT +1j =0. j B S, T 1,j =1 ∈ − X| ∩{ }| We show that the relations for 1 T follow from those with 1 T .In ∈ 6∈ fact, we only want this relation in V (4ωg+1), so we restrict ourselves to the sub- g+1 2g+2 space V (2ωg+1) C . The kernel of the restriction map is ... ,eT g+1 ⊂∧ h − sgn(eT )i eT 0 ,... (or the other space in Lemma 3.3.5; the argument we give leads to the same conclusioni in both cases). Consider a T with 1 T and j T .Then1 T +1j and after restriction: ∈ 6∈ 6∈ e = sgn(σ )ig+1e ,e= sgn(σ )i (g+1)e T T T 0 T+1j (T +1j)0 − (T +1j)0

where 0 stands for the complement in B. Since (T +1j)0 =T0 +1j, substituting these relations and multiplying throughout by ig+1 we get: g+1 ( 1) sgn(σT )eS eT + tS, jtT, jsgn(σT j)eS jeT j. − 0 1 1 0+1 +1 0+1 j B S, T 1,j =1 ∈ − X| ∩{ }| Next we observe that g+1 sgn(σT j)=( 1) tT, jtT , jsgn(σT ), 0+1 − 1 0 1 so it suffices to consider the relations with 1 T .LetT:= 1=i ,... ,ig ,T0:= 6∈ { 1 +1} j1,... ,jg+1 with j = jk and il

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l 1 On the other hand, since 1 T, j T we get F1j (eT )= 2( 1) − eT +1j,so l ∈ 6∈k 1 − −k 1 tT, j =( 1) and F j(eT ) = +2( 1) − eT j,sotT, j =( 1) − .Thisgives 1 − 1 0 − 0+1 0 1 − the formula for sgn(σT 0+1j). From now on we consider only T ’s with 1 T and S with 1 S. We show that the desired relation: 6∈ ∈

eSeT + tS,1itT,1ieS+1ieT +1i i (B S) T ∈ X− ∩ 0 is a Pl¨ucker relation. Thus it holds in H (GrSL, (2)) and therefore also upon O restriction to GrSO GrSL. ⊂ Let S = 1=i1,... ,ig+1 and consider the Zariski open subset U GrSL of { } 2g+2 ⊂ g +1-dimensional subspaces W C with Pl¨ucker coordinate pS =0.Anysuch ⊂ 6 W has a (unique) basis ... ,wi,... with: { }

W = w ,... ,wg and (wk)i = δkl h 1 +1i l (S = 1=i ,... ,ig ,i <...

where δkl is Kronecker’s delta. Let M be the (g +1) (2g + 2) matrix whose rows × are the wi.LetMibe the i-th column of MW (note Mij = fj,thej-th standard g+1 basis vector of C ). Then the Pl¨ucker coordinate pk1,... ,kg+1 is the determinant of the (g +1) (g+ 1) submatrix of M whose j-thcolumnisMk ;wewrite: × j

pk1,... ,kg+1 =det(Mk1,... ,Mkg+1 ) with k1 <...

l 1 Let j S, with il

det(fi2 ,...,fil,Mj,fil+1 ,...,fg+1) l 1 l 1 =( 1) − det(Mj ,fi ,...,fi ,fi ,...,fg )=( 1) − (Mj) . − 2 l l+1 +1 − 1 l 1 As earlier, since 1 S we have tS, j = ( 1) − , hence pS j = tS, j(Mj) . ∈ 1 − − +1 − 1 1 Let T = j ,... ,jg ,letT S0 = a,... ,aq and let j = jn = ak.Then { 1 +1} ∩ { 1 } pT=det(Mj ,... ,Mj ) is, up to sign, the determinant of a q q submatrix N of 1 g+1 × the (g +1) q matrix with columns Ma1 ,... ,Maq;wewritepT =t det(N) with t = 1. Then× · ±

pT +1j =det(f1,Mj1,... ,Mjn,... ,Mjg+1 )

jn 1 =(1) − det(Mj1 ,... ,Mjn 1,f1,Mjn+1 ,... ,Mjg+1 ) − d − jn 1 k+1 1k =(1) − ( 1) t det(N ), − − · where N 1k is the (q 1) (q 1) submatrix of N obtained by deleting the first − × − jn 1 row and k-th column. Since 1 T ,wehavetT,1j =( 1) − and thus pT +1j = t( 1)k+1 det(N 1k). 6∈ − − Substituting these expressions for pS+1j and pT +1j we get:

k 1k 1 t det(N)+ t( 1) (Mj) det(N ) · · − 1 j a ,... ,aq ∈{ X1 } which is zero in virtue of a well known formula for the determinant. 

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4. The Heisenberg group and the spin representation 4.1. We recall the basic facts on the Heisenberg group and we discuss the pro- jective representation of its automorphism group. In Subsection 4.3 we relate the Heisenberg group (in its irreducible 2g-dimensional representation) with the (half) spin representation of the orthogonal group. Combining this with previous results, we can finally write down Hitchin’s connection in Subsection 4.4.

4.2. The Heisenberg group.

4.2.1. Definitions. We introduce a variant of the Heisenberg group (cf. Subsection 2.5.2). For any positive integer g we define a (finite) Heisenberg group G by:

4 g g G = Gg := (t, x): t C,t =1,x=(ξ,ξ0) F F { ∈ ∈ 2 × 2} with identity element (1, 0) and multiplication law:

ξη0 1 ξξ0 1 (t, (ξ,ξ0))(s, (η, η0)) := (ts( 1) ,ξ+η, ξ0 + η0), (t, x)− =(( 1) t− ,x) − − g 2g+2 with ξη0 := i=1 ξiηi0 and x =(ξ,ξ0). The group G has 2 elements, it is non-abelian, in fact its center is (t, 0) . We define a symplectic form on F2g by: P { } 2 1 1 E(x,y) E(x, y)=ξη0 + ηξ0;then(t, x)(s, y)(t, x)− (s, y)− =(( 1) , 0) − where x =(ξ,ξ0),y=(η, η0).

4.2.2. Representations. Let V be the 2g-dimensional vector space of complex valued functions Fg C. It has a standard basis consisting of δ-functions 2 → g Xσ : F C ,Xσ(σ)=1,Xσ(ρ)=0 ifσ=ρ. 2 −→ 6 The Heisenberg group G has a representation U on V , the Schr¨odinger representa- tion: σξ0 (U(t, (ξ,ξ0))f)(σ):=t( 1) f(σ + ξ), − thus (σ+ξ)ξ0 U(t, (ξ,ξ0))Xσ = t( 1) Xσ ξ. − + 2g For every x F 0 choose a (tx,x) G of order two. We define: ∈ 2 −{ } ∈ 2 Ux := U(tx,x)(GL(V )), so U = I ∈ x 4 2g and define U = I.ThenIm(U)= tUx : t =1,x F . 0 { ∈ 2 } 4.2.3. Automorphisms. We define a subgroup of Aut(G)by: A(G):= φ Aut(G): φ((t, 0)) = (t, 0) t . { ∈ ∀ } For φ A(G)and(t, x) G we can then write: ∈ ∈ 2g 2 g 2g φ(t, x):=(fφ(x)t, Mφ(x)), with Mφ : F F ,fφ:F C ∗ 2 −→ 2 2−→ (note φ(t, x)=φ(t, 0)φ(1,x)=(t, 0)φ(1,x)). The map M : A(G) Aut(F2g),φ → 2 7→ M is a homomorphism, its image is Sp(F2g,E)andker(M) G/Center(G) F2g φ 2 ∼= ∼= 2 consists of interior automorphisms:

0 F 2 g A ( G ) M Sp(2g,F ) 0 . −→ 2 −→ −→ 2 −→

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4.2.4. A projective representation. The Schr¨odinger representation U of G on V is the unique irreducible representation of G in which (t, 0) acts by multiplication by t.Givenφ A(G), the representation U φ enjoys the same property. Hence by Schur’s lemma∈ we get a linear map, unique◦ up to scalar multiple,

T˜φ : V V, with T˜φU(h)=U(φ(h))T˜φ, −→ for all h G.NoteT˜φnormalizes Im(U). In this way we get a projective repre- sentation∈ of A(G):

T˜ : A(G) PGL(V),φT˜φ. −→ 7−→ ˜ When φ = Int(t,y) we may take Tφ = Uy . On the other hand, since Im(U) = G,anyT GL(V ) which normalizes Im(U) ∼ ∈ defines an automorphism φT of G. Since the center of G acts by scalar multiples of the identity, we have φT A(G). Thus we get an exact sequence ∈ 0 C ∗ Normalizer (Im(U)) A ( G ) 0 . −→ −→ GL(V ) −→ −→ 2g 4.2.5. Example. For x =(ξ,ξ0) F 0 , we define the transvection ∈ 2 −{ } 2g 2 g Tx : F F ,yy + E(y,x)x. 2 −→ 2 7−→ Then Tx Sp(2g,F2) and the (finite) symplectic group Sp(2g,F2) is generated by ∈ 2 transvections. Note that the transvections are involutions: Tx =1. g Let x =(0,ξ0) with ξ0 =(1,0,... ,0) F2. On the basis X(0,... ,0),...,X(0,τ),..., g 1∈ X ,... ,X ,... with τ F − we have: (1,0,... ,0) (1,τ) ∈ 2 I 0 I 0 Ux = ;letT˜x:= . 0 I 0 iI −  ˜ ˜ 1   2g Then it is easy to verify that TxUyTx− = tx,yUTx(y) for all y F2 and some ˜ ˜ ∈ tx,y C∗.ThusTxlies in A(G)andM(Tx)=Tx. ∈ 4.3. The spin representation. 4.3.1. The two half spin representations of so(2g + 2) are each realized on a 2g- dimensional vector space. We recall, using the Clifford algebra, how they can be constructed using the Heisenberg group. We will have to consider the Heisenberg g+1 group Gg+1 which acts on a 2 -dimensional vector space. Elements of order two in the Heisenberg group will define the spin representation of so(2g+2). Restriction to suitable subspaces will give the half spin representations and their relation with Gg (a subquotient of Gg+1). This relation between so(2g + 2) and the Heisenberg group is used in 4.4.3 to prove the Heisenberg invariance of the flat connections introduced in 3.2.4. 4.3.2. In dealing with hyperelliptic curves and the half spin representation of the 2g orthogonal group, the following (classical) notation for points in F2 is convenient ([DO], VIII.3; [M]). Let B := 1, 2,... ,2g+2 ; { } then

FB := f : B F2 : f(b)=0 0,[b 1]b B → { 7→ ∈ } ( b B ) X∈ .

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is an F -vector space of dimension 2g. For a subset T B with an even number of 2 ⊂ elements we denote by xT FB (or simply T ) the element defined by the function ∈ f with f(b)=1iffb T.NotethatxT =xT0 when T 0 is the complement of T in B, moreover ∈

xT + xS = xR with R = T + S := (T S) (T S). ∪ − ∩ 4.3.3. We fix the following isomorphism of F2-vector spaces, and identify them in this way in the remainder of the paper ([DO], VIII.3, Lemma 2, but note we interchanged (ei, 0) (0,ei)): ↔ ∼= g g FB F 2 F 2 ,x2i1,2i=(0,ei),x2i, 2i+1,... ,2g+1 =(ei,0) (1 i g). −→ × {− } { } ≤ ≤ g g The symplectic form E on F F can now be easily computed on FB by: 2 × 2 E(xT ,xS)= T S mod 2. | ∩ | 4.3.4. To accomodate both Gg and Gg+1 we extend the construction of Subsection 4.3.2. The inclusion B := 1,... ,2g+2 , B] := 1,... ,2g+4 { } → { } and extension by zero of functions on B to functions on B] induces

F˜B := f : B F : f(b)=0 , F˜ ] := f : B F : f(b0)=0 . → 2 → B  → 2  ( b B ) b B] X∈  X0∈  ˜ ] ˜ The function g0 F ] with g0(b0)=1(allb0 B ) does not lie in FB,thus ∈ B ∈   F˜B , F ] := F˜ ] / 0,g0 . → B B { } The image in F ] of the function g ( F˜B ) with g(b)=1(allb B) is the element B ∈ ∈ p0 := x 1,2,... ,2g+2 = x 2g+3,2g+4 FB] . { } { } ∈ Using the definition of the symplectic form on FB] (Subsection 4.3.3), which we denote by E], one finds that: ] F˜B =(p0)⊥, with p0⊥ := x F ] : E (x, p0)=0 and FB =(p0)⊥/ 0,p0 . { ∈ B } { } From Subsection 4.3.3 we have an identification: g+1 g+1 F ] = F F ,p0=(0,(1, 1,... ,1)) ( F ] ). B × ∈ B The Heisenberg group defined by this identification (cf. Subsection 4.2.1) will be denoted by G], its Schr¨odinger representation by U ] (on the 2g+1-dimensional ] g+1 g+1 vector space V := f : F2 C with basis of δ-functions Yσ0 , σ0 F2 ). { → } ] ] ∈ ] For any x0 (p0 )⊥ ( F ] ), the maps U and U commute. Therefore the U B p0 x0 x0 ∈ ⊂ ] with x0 (p0)⊥ act on the two eigenspaces of U which are: ∈ p0 ] ] V+ := ... ,Yσ0,... σ0 +...+σ0 =0,V:= ... ,Yσ0,... σ0 +...+σ0 =1. h i 1 g+1 − h i 1 g+1 g g g g A quotient map from (p0)⊥ ( (F F ) (F F )) to FB = F F with kernel ⊂ 2 × 2 × 2 × 2 2 × 2 0,p0 is: { } 2g (p0)⊥ F B =(p0)⊥/ 0,p0 =F ,x0:= ((a, ag ), (b, bg )) x := (a, ¯b) −→ { }∼ 2 +1 +1 7−→ ¯ with b =(b1+bg+1,... ,bg +bg+1). We define an isomorphism of vector spaces: ] V V, Yσ ,... ,σ ,σ Xσ ,... ,σ . + −→ 1 g g+1 7−→ 1 g

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4.3.5. Lemma. For any x0 (p0 )⊥ mapping to x FB there is a commutative diagram: ∈ ∈ V ] ∼= V + −→ ] U ( t, x0) U(t, x)

] ∼=  V+ V y −→ y Moreover, if 1 j, k 2g +1, then restriction of U ] to V ] = V is given by ≤ ≤ xjk + ∼ Ux . ± ij Proof. The proof consists of easy computations which we leave to the reader. 

4.3.6. The Clifford algebra. The Clifford algebra C(Q) of the quadratic form Q = 2 2 2g+2 x1 + ...+x2g+2 on VQ = C is the quotient of the tensor algebra T (VQ)bythe two-sided ideal I generated by the elements v v Q(v)forv VQ ([FH], 20.1): ⊗ − ∈ §

C(Q):=T(VQ) I=(C VQ VQ VQ ...) (... , Q(v)+v v,...)v VQ. ⊕ ⊕ ⊗ ⊕ − ⊗ ∈ . . Let e1,... ,e2g+2 be the standard basis vectors of VQ. Then the C(Q) is generated by C and the ej with relations:

ej ej =1,ejek+ekej=0 (j=k); · · · 6 here stands for the product induced by on C(Q). The (associative) algebra C(Q) becomes· a Lie algebra by defining, as usual,⊗ the Lie bracket to be [x, y]:=xy y x. The following proposition relates the Heisenberg group, the Clifford algebra· − and· the spin representation of the Lie algebra so(2g +2). 4.3.7. Proposition. With the notation as above we have: 1. The C-linear map

] ] ] γ : C(Q) End(V ),λλI, ek U −→ 7→ 7→ xk, 2g+4 (λ C,k=1,... ,2g +2) defines an isomorphism of (associative C-) algebras.∈ 2. The linear map

] ] ] ] ] ρ : so(2g +2) End(V ) = ] C(Q),Fjk U U (= γ (ej ek)) s −→ ∼γ 7−→ xj,2g+4 xk,2g+4 · is an injective homomorphism of Lie algebras. ] ] 3. The subspace V+ of V is invariant under the action of so(2g +2).TheLie algebra representation

] ] ρs : so(2g +2) End(V+) = End(V ),ρs(x):=ρs(x) V] −→ ∼ +

is an (irreducible) half spin representation of so(2g +2). In particular, V ∼= V (ωg+1). 4. We have

2 ρs(Fjk)= iUx (Fjk so(2g +2),i=1), ± jk ∈ − 2g where the Uxjk ( End(V )) with xjk F2 are in the Schr¨odinger represen- tation of the Heisenberg∈ group G. ∈

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] 2 2g+2 Proof. (1) By definition we have (U ) = I for all x0 F . Moreover, x0 ∈ 2 ] E (x k,2g+4 ,x l,2g+4 )=1 { } { } when k = l so the corresponding maps anti-commute: 6 U ] U ] = U ] U ] . xk,2g+4 xl,2g+4 − xl,2g+4 xk,2g+4 Then the map γ] preserves the relations in C(Q) and is thus an algebra homomor- ] g+1 g+1 phism. It is surjective because the matrices U ,wherex0 runs over F F , x0 2 2 are a basis of End(V ]) (the Schr¨odinger representation being irreducible) and× any U ] is a product (up to scalar multiple) of suitable U ] . Therefore γ] is an x0 xj,2g+4 isomorphism since both algebras have the same dimension. (2) This is worked out in [FH], Lemma 20.7. With their notations and our Q, we have 2 ∼= φ : VQ so(2g +2),ejek Fjk, ∧ −→ ∧7−→ 2 ] 1 ψ : VQ C ( Q ) ,ejek ej ek (j = k), and ρ = ψ φ− . ∧ −→ ∧7−→ · 6 s ◦ ] ] (3) By definition of ρs and the fact that U is a representation we have: ρ] (F )=U] U ] = c U] , s jk xj,2g+4 xk,2g+4 jk xjk ] ] with 1 j, k 2g + 2 and some cjk C∗.Thustheρ(Fjk) commute with U s p0 ≤ ] ≤ ∈ ] in End(V ). Therefore we obtain a Lie algebra representation on V+; cf. [FH], even ] Prop. 20.15 (and identify their End( W ) with our End(V+)), where also the irreducibility is proved. ∧ ] ] ] ] 2 (4) We have ρ (Fjk)=U U ,thusρ(Fjk) = I since these two s xj,2g+4 xk,2g+4 s − elements anti-commute. Therefore ρ] (F )= iU ] , which acts as iU on s jk xjk xjk ] ± ± V+.  4.4. The Hitchin connection.

4.4.1. The symmetric group S g acts on by permuting the coordinates; the 2 +2 P quotient will be denoted by . The fundamental group of is the pure braid group P P and π ( )=B g , the braid group. 1 P 2 +2 In case g =2,wehaveS6 ∼=Sp(4, F2), in fact there is a surjective map (which factors over the mapping class group) π ( )=B Sp(4, Z); 1 P 6 −→ the kernel of the composition B6 Sp(4, Z) Sp(4, F2)isπ1( ). From the theory of theta functions→ we know→ that the group schemeP is trivialized on the cover ˜ of defined by the kernel of the composition: G P P φ : π ( ) Sp(4, Z) Sp(4, Z)/Γ (2, 4) = A(G). 1 P −→ −→ 2 ∼ Here we use Igusa’s notation:

I +2A 2B Γ2(2, 4) := Sp(4, Z): (  2CI+2D∈   diagonal(B) diagonal(C) (0, 0) mod 2 . ≡ ≡ )

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Thus we consider the following diagram of ´etale Galois coverings:

:= /S , P−→P−→P P 6 and the corresponding exact sequence of covering groups: e 0 F 2 g A ( G ) S 0 . −→ 2 −→ −→ 6 −→ Recall that we have defined a projective representation T˜ : A(G) PGL(V)in 4.2.4. This induces projective representations on each SkV . →

4.4.2. Theorem. With the notation as in 4.4.1 (except for Sk)andg=2we have: k 1. There is a line bundle on such that the pullback of p ⊗ is isomorphic N k P ∗L to Sk .HereSk=SV(ωg ) and V (ωg ) is a half spin representation ⊗C N +1 +1 of so(6) ∼= sl(4) (which is thee standard representation (or its dual) of sl(4)). k 2. The Hitchin connection on the pullback of p ⊗ to is given by the pullback ∗L P to of the connection on Sk C P defined by the one-form P ⊗ O e 1 σρ˜s(Ωij )dzi e − 16(k +2) zi zj i,j i=j X6 − with ρ˜s : U(so(2g +2)) End(S ) the half spin representation (then the → 1 σρ˜s(Ωij ) give endomorphisms of each Sk; cf. 3.2.5).

3. The Hitchin connection on the bundle Sk e over descends to a projective ⊗OP P

flat connection on the the bundle (Sk e )/A(G) over ,whereA(G)acts ⊗OP P on Sk via the k-th symmetric power of its projectivee representation on S1 and A(G) acts on as Gal( / ). This descent gives the k-th Hitchin connection of the naturalP descent ofP theP family of curves described in 1.4. e e Proof. (1) For the first part it suffices to show that the pullback of p : P to →P ˜ is trivial. Since is isomorphic to the constant group scheme G (cf. 4.2.1 (with P G t C∗ here)), the uniqueness of the Schr¨odinger representation gives the global trivialization.∈ We already observed that the line bundle does not interfere with the projective connections. N (2) We use Theorem 2.6.9 so we must verify (Eq.) from 2.6.7. The covariant derivatives given are defined by the heat operator

1 dzi 1 X = σρ˜s(Ωij ) ( (S2 C S2∗) Ω ). −16(k +2) ⊗ zi zj ∈ ⊗ ⊗ P j=i X6 − The Heisenberg-invariance of this operator follows from the facts that Ωij = Fij 2 1 4 ⊗ Fij ,thatρs(Fkj )= iUxjk with i = 1andthatUyUxUy− = Ux.SinceS(˜ρs) ± − 2 2 2g+2 ± is a subrepresentation ofρ ˜ : so(2g +2) EndS ( C )wehaveσρ˜s(Ωij )= → ∧ σρ˜(Ωij )onS4. For the connection 0 onp ˜ P(1) = S1 P˜ we simply take ∇ ∗O ⊗O 0(fw)=w df. Then the equation (Eq.) is exactly the statement of Theorem ∇3.4.3. ⊗ (3) We have a smooth projective family of genus 2 curves over the scheme ¯ given by P 2 6 5 2 4 y = f(x)=x +a x +a x +... + a = (x zi), 1 6 − i Y

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where ai is the i-th symmetric function of the zj. Thus by the general theory of 3 Section 2 we have a P -bundlep ¯ : P¯ ¯ and a projective Hitchin connection D¯ k onp ¯ (k). Let ϕ : ˜ ¯be the natural−→ P map. Then there is an isomorphism ∗O P−→P 3 ( ) ϕ∗(P¯ )=P ˜ ∗ ×P which is compatible with the action of A(G) (which lies over the action of A(G) on ˜) on both sides: this isomorphism is given by the trivialization of the action of theP theta-group scheme which was mentioned earlier. The isomorphism ( ) induces an isomorphism G ∗

( ) ϕ∗(¯p O(k)) = Sk O ˜ ∗∗ ∗ ∼ ⊗ P which is compatible with the natural projective action of A(G)ontheLHS(via ) G and the action of A(G) on the RHS via the Symk of its natural representation on S1. The heat operator given by X (as in the proof of 4.4.2.2, see 2.6.7) on the LHS of ( ) is VIA ( ) compatible with the pullback of the Hitchin heat operator of the LHS∗ of ( ). But∗ both heat operators are invariant under the action of A(G). Thus the projective∗ connections on the RHS and LHS of ( ) agree and are compatible with the projective action of A(G) on both sides. In other∗∗ words the natural descent datum on the LHS of ( ) agrees with the descent datum on the RHS of ( ). ∗∗ ∗∗  4.4.3. Examples. We give some examples of the ρs(Ωij )’s in case g = 2. We identify 2 V = S , with its standard basis Xσ,σ F. A linear map with matrix (akl)is 1 ∈ 2 then also given by the linear differential operator kl aklXk∂l (since ∂l(Xm)=0 for l = m and is 1 if l = m). Recall from 4.3.3 that (for g =2): 6 P x = ((0, 0), (1, 0)), so Ux = X ∂ + X ∂ X ∂ X ∂ . 12 12 00 00 01 01 − 10 10 − 11 11 Similarly, we had x16 = ((1, 0), (0, 0)) and x26 = ((1, 0), (1, 0)) and so:

Ux16 = X10∂00 + X11∂10 + X00∂10 + X01∂10,

Ux = i( X ∂ X ∂ + X ∂ + X ∂ ). 26 − 10 00 − 11 10 00 10 01 10 2 Since Ux = iρs(Fjk)andU = I we get (cf. 3.2.5): jk ± xjk σ(˜ρs(Ω12)) = 2(X2 ∂2 + X2 ∂2 + X2 ∂2 + X2 ∂2 +2X X ∂ ∂ 2X X ∂ ∂ ). − 00 00 01 01 10 10 11 11 00 01 00 01 − 10 11 10 11 4.5. Local monodromy. 4.5.1. Introduction. We want to obtain some information on the representation of the mapping class group for genus two curves defined by Hitchin’s connection. This representation has been studied, for arbitrary g and k, by Moore and Seiberg [MS], Kohno [K2] and in the case k = 2 by Wright [Wr]. In some sense, we merely find (weaker, local) results which do agree with their results. 4.5.2. The method. We recall how to determine the local monodromy on the trivial bundle C W with a connection . We consider a holomorphic map OP ⊗ ∇ φ : D∗ , with D∗ := t C 0 : t < −→ P { ∈ −{ } | | } (for small, positive ) and pullback the one-form ω End(W ) Ω1 which defines the connection (so (fw)=wdf fω(w)). We write∈ ⊗ P ∇ − R φ∗ω = + A(t) dt, R End(W ) t ∈  

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with A(t) holomorphic for t =0,andRis called the residue at t = 0. We identify π (D∗)=Z;1 Zrepresents a small circle traversed in anti-clockwise direction. 1 ∈ The eigenvalues of the monodromy of 1 π1(D∗) are the exp( 2πiµj)weretheµi are the eigenvalues of R on W . ∈ − In case the monodromy transformations are semi-simple, the local monodromy is conjugated to exp( 2πiR) (cf. [D], p. 54). However we could not prove the semi-simplicity (but it− seems to be known to the physicists). To get the eigenvalues of the monodromy of the Hitchin connection for a γ ∈ π1( ), let n be the order of its image in Gal( / )andlet¯γbe its image in A(G)= P P P n Gal( / ). We choose a φ : D∗ in such a way that φ (1) = γ ( π1( )). P P →P ∗ 2πi ˜∈ P Then these eigenvalues are the eigenvalues of the matrix exp(−n R)Tγ¯, with R determinede as above for the corresponding connection on (cf. [K1]). Since T˜ is P a projective representation (on S1 andthusonanySk), the set of eigenvalues are only defined up to multiplication by one non-zero constant. This corresponds to the fact that we only have a projectively flat connection.

¯ 2 4.5.3. Non-separating vanishing cycle. Let γ π1( ) such that γ = φ (1) with ∈ P ∗

φ : D∗ ,t(t + z ,z ,... ,z ) −→ P 7−→ 2 2 6

wherewefixdistinctzj C.(Thenγcorresponds to a Dehn twist in a simple non- ∈ separating loop in the mapping class group.) The residue of φ∗ω for the connection k on Sk corresponding to the Hitchin connection on p ⊗ is (cf. 4.4.2 and ⊗OP1 ∗L 4.4.3): 16(−k+2) σρ˜s(Ω12),

2 ρ˜s(Ω )=R := ((X ∂ + X ∂ ) (X ∂ + X ∂ )) . − 12 12 00 00 01 01 − 10 10 11 11

4.5.4. Lemma. The eigenvalues of the monodromy of γ as in 4.5.3 on Sk are: c(c +1) λc := exp( 2πi ), mult(λc)=(k 2c+ 1)(2c +1) − k+2 − (up to multiplication by a non-zero constant independent of c), with 2c Z and ∈ 0 c k . ≤ ≤ 2 Proof. Since the eigenvalues are only determined up to a constant and since the difference between 2R12 and σ(R12) is a multiple of the Euler vector field, which acts by multiplication by k on Sk, it suffices to consider the eigenvalues of R12.For a monomial in Sk we have:

R (Xl0 Xl1 Xm0 Xm1 )=((l +l ) (m +m ))2(Xl0 Xl1 Xm0 Xm1 ). 12 00 01 10 11 0 1 − 0 1 00 01 10 11

Thus each monomial is an eigenvector. Let b := m0 + m1. Since the monomial has degree k, its eigenvalue is (k 2b)2. The multiplicity of the eigenvalue is (k b +1)(b+ 1) (the dimension of− the space of homogeneous polynomials in 2 − 2 variables of degree c is c+1). Thus the eigenvalue 2(k 2b) of 2R12 has multiplicity (k b +1)(b+1). − − ˜ The imageγ ¯ of γ in A(G)isTx12 which acts on such a monomial by (cf. 4.2.5):

˜ l0 l1 m0 m1 b l0 l1 m0 m1 Tx12 (X00X01X10 X11 )=i(X00X01X10 X11 ).

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The eigenvalues of γ on Sk are then, with c := b/2: 2πi exp(− +1 2(k 2b)2 + b ) 2 16(k+2) − 4  2 2 = exp( 2πi k 8ck +16c +8ck +16c ) − − 16(k+2)   2 c(c +1) = exp( 2πi k ) exp( 2πi ). − 16(k+2) − k+2      ¯ 4.5.5. Separating vanishing cycle. Nowweconsideraγ π1( ) such that γ = φ (1) with ∈ P ∗ φ : D∗ ,t(tz ,tz ,tz ,z ,z ,z ), −→ P 7−→ 1 2 3 4 5 6 with distinct non-zero zi’s. The residue in t = 0 of the connection on Sk corresponding to Hitchin’s connection is ⊗P 1 16(−k+2) R123, with R123 := 2σ(˜ρs(Ω12 +Ω13 +Ω23)). 4.5.6. Lemma. Let g =2.

1. There are constants λk Q such that (in End(Sk)): ∈ R =16RQ+λkI with RQ =(X X X X )(∂ ∂ ∂ ∂ ). 123 00 10 − 01 11 00 10 − 01 11 2. We define a subspace of Sk by:

Vk := Kernel(XQ := ∂00∂10 ∂01∂11 : Sk S k 2 ) . − −→ − Then the vector space Sk is a direct sum: 2 Sk = Vk QVk 2 Q Vk 4 ... . ⊕ − ⊕ − ⊕ l Moreover, each subspace Q Vk 2l is an eigenspace of RQ with eigenvalue: − 2 λl = l(k l +1) and dim Vk 2l =(k 2l+1) . − − − 3. The eigenvalues of of the monodromy of γ on Sk are

l(l +1) 2 λl := exp( 2πi ) mult(λl)=(k 2l+1) − k+2 − (up to multiplication by a constant independent of l), with l Z and 0 l k/2. ∈ ≤ ≤ Proof. (1) Recall that x corresponds to ((0, 0), (1, 0)) F2 F2 and x = 12 ∈ 2 × 2 23 x 2,3,4,5 + x 4,5 corresponds to ((1, 1), (0, 0)), thus x13 = x12 + x23 corresponds to { } { } ((1, 1), (1, 0)). Since (1, 1)(1, 0) = 1, we have a + sign in front of Ω13 below. Let 2 ρ˜s(Ω00) be the square of the Euler vector field (which acts as k I on Sk). Then:

2 ρ˜s(Ω00)=+(X00∂00 + X01∂01 + X10∂10 + X11∂11) , 2 ρ˜s(Ω )= (X∂ + X ∂ X ∂ X ∂ ) , 12 − 00 00 01 01 − 10 10 − 11 11 2 ρ˜s(Ω )= (X∂ + X ∂ + X ∂ + X ∂ ) , 23 − 11 00 10 01 01 10 00 11 2 ρ˜s(Ω )=+(X∂ + X ∂ X ∂ X ∂ ) . 13 11 00 10 01 − 01 10 − 00 11 The sum of these operators, minus the (degree one) parts which act as constants on each Sk,is4RQ. Remembering the factor 2 in R123 and in our symbols, we find the first statement.

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(2) Thisiswell-known.Letf Vn,sofis homogeneous of degree n and ∈ XQ(f) = 0. Then an elementary computation gives for l 1 (with variables Xi and Q = X X X X ): ≥ 0 1 − 2 3 (∂ ∂ ∂ ∂ )(fQl) 0 1 − 2 3 =(∂ f)(∂ Ql)+(∂ Ql)(∂ f) (∂ f)(∂ Ql) 0 1 0 1 − 2 3 (∂ Ql)(∂ f)+f(∂ ∂ ∂ ∂ )(Ql) − 2 3 0 1 − 2 3 l 1 l 1 = lQ − (X0∂0 + X1∂1 + X2∂2 + X3∂3)(f)+l(l+1)fQ − l 1 = l(n+l +1)fQ − . Writing n = k 2l we get, for all integers l with 0 2l k: − ≤ ≤ l l RQ(fQ )=l(k l+1)fQ ,fVk2l. − ∈− l Hence each Vk 2lQ is an eigenspace for RQ. By induction (the cases k =0,1 − l being trivial) we may assume that Sk 2 = Sk 2 2lQ . Therefore QSk 2 = l+1 − − − − Q Sk 2(l+1) Sk is a direct sum of eigenspaces of RQ and none of the − ⊂ L eigenvalues of RQ on this subspace is zero (k + l(l +1) > 0fork 2,l 0). L ≥ ≥ Thus ker(RQ) QSk 2 = 0 and RQ induces an isomorphism on QSk 2.There- ∩ − { } l − fore Sk = Ker(RQ) Im(RQ)= QSk 2l. The dimension of Vk is then ⊕k+3 k+1 2− dim Sk dim Sk 2 = =(k+1) . − − 3 − 3 L (3) The image of γ in A(G) is trivial. The eigenvalues of R123 =16RQ on Sk are 16l(k l + 1). Thus the
eigenvalues
of γ are (see 4.5.2): − 1 l2 l l(k +2)+2l exp( 2πi 16l(k l +1)) = exp( 2πi − − ) − 16(−k+2) − − k+2 l(l +1)  = exp( 2πi ). − k+2  4.5.7. Kohno’s results. The monodromy of the Hitchin connection has been studied by Moore-Seiberg and Kohno (among others); we will relate the results above to those contained in [K2] for g =2. Let γ be a tri-valent graph with two vertices, so there are three edges meeting in each vertex. For k Z 0 we define a finite set bk(γ) of functions on the edges of γ by: ∈ ≥ f : Edges(γ) 0 , 1 , 1 ,... ,k −→ { 2 2} with

f(ei)+f(ej)+f(ek) k, ≤  f(ei) f(ej) f(ek) f(ei)+f(ej),  | − |≤ ≤  f(ei)+f(ej)+f(ek) Z, ∈ for any three edgesei,ej,ekmeeting in a vertex. The Verlinde space Vk is the  C-vector space with basis bk(γ); it has the same dimension as Sk. The graph γ corresponds to a pants decomposition of a genus two Riemann surface. Each vertex is a pair of pants, homeomorphic to P1 minus 3 points, the edges correspond to these points, which in turn correspond to ‘vanishing cycles’ on the Riemann surface. The mapping class group has a projective representation on Vk. The Dehn twist in a vanishing cycle corresponding to an edge e of γ acts, up

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to a scalar multiple and conjugation, by the diagonal matrix with entries (cf. [K2], p. 217; p. 214, (2-1))

exp( 2πif(e)(f(e)+1)/(k+2)) (f bk(γ)). − ∈ Using this recipe we find the same results as before (see Lemmas 4.5.4 and 4.5.6): 4.5.8. Lemma. With the notation as above:

1. The eigenvalues of the Dehn twist in a non-separating cycle on Vk are exp( 2πiv(v +1)/(k+2)) with multiplicity (k 2v + 1)(2v +1) where v − − ∈ 0, 1 , 1,... ,k . { 2 2} 2. The eigenvalues of a Dehn twist in a separating cycle on the space Vk are exp( 2πil(l +1)/(k+2)) with multiplicity (k 2l +1)2 where l is an integer with−0 l k/2. − ≤ ≤ Proof. For the first statement consider the graph γ with two vertices joined by three edges e1,e2,e3.Theedgee1corresponds to a non-separating cycle and one must determine f(e1) for all f bk(γ). For the second case the recipe∈ is not explicitly given in [K2], but is similar to the one used in the first case and gives a result that agrees with Lemma 4.5.6. One now takes δ to be the graph with two vertices v1,v2 and three edges e1,e2,e3 such that the beginning and end of ei is vi and where e3 connects v1 and v2.Thustheedgee3 corresponds to a separating vanishing cycle and one has to determine f(e3) for all f bk(δ). The computations are elementary and we leave them to the reader. ∈  4.5.9. The cases k =1, 2. The case k = 1 is particularly easy since the operators Ωij are homogeneous of degree two and thus act as zero on S1. The monodromy representation then factors over the projective representation of A(G). In the case k =2,thespaceSk is a direct sum of 10 distinct, one-dimensional G-representations (this is easy to verify, see also [vG]). On each eigenspace, any σρ˜s(Ωij ) acts as scalar multiplication by an integer. A (local) basis of flat sections r of S is then given by 10 functions of the type ( (zi zj) ijm )Qm with rijm Q 2 ⊗P − ∈ and Qm S2 is a basis of the eigenspace. These eigenspaces are permuted by the projective∈ representation of A(G). Q References

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Dipartimento di Matematica, Universita` di Torino, Via Carlo Alberto 10, 10123 Torino, Italy E-mail address: [email protected] Department of Mathematics, Princeton University, Fine Hall – Washington Road, Princeton, New Jersey 08544-1000 E-mail address: [email protected]

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