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ON HITCHIN's CONNECTION 1. Introduction 1.1. the Aim of This Paper Is to Give an Explicit Expression for Hitchin's Connectio 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 189 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 190 BERT VAN GEEMEN AND AISE JOHAN DE JONG 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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON HITCHIN'S CONNECTION 191 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).
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