Frobenius manifolds, projective special geometry and Hitchin systems
aa r X i v : . [ m a t h . AG ] M a y FROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY ANDHITCHIN SYSTEMS
CLAUS HERTLING, LUUK HOEVENAARS, AND HESSEL POSTHUMAA
BSTRACT . We consider the construction of Frobenius manifolds associated toprojective special geometry and analyse the dependence on choices involved. Inparticular, we prove that the underlying F -manifold is canonical. We then applythis construction to integrable systems of Hitchin type. I NTRODUCTION
One way of formulating the mirror symmetry conjecture is in terms of Frobe-nius manifolds. On the one hand (the A -side) it is well known that the quantumcohomology product gives rise to a natural Frobenius manifold. The other, B -side,is constructed from certain variations of Hodge structures. In the case of Calabi-Yau threefolds, this B -side of the story is perhaps less well-known to mathemati-cians and appears implicitly in [BK98, Bar02, Bar01, FP04, HM04].One of the purposes of this paper is to give an elementary description of this B -side Frobenius geometry and to specify the dependence on choices one has tomake. The starting point for this will be a projective special geometry, or in otherwords, an abstract variation of Hodge structures of weight 3 of the type consid-ered in [BG83]. There are two choices involved in the construction: one is a gen-erator for the degree three subspace in the Hodge filtration, i.e., a volume form inthe case of Calabi–Yau threefolds. Second, and more importantly, a choice of an opposite filtration . Parts of this choice have a natural interpretation in terms of spe-cial geometry as choices of affine coordinate patches. Other parts lie outside therealm of special geometry. Remarkably we find that the underlying F -manifold, cf.[HM04], is independent of all choices. Also remark that in the geometric context ofCalabi–Yau threefolds, the large complex structure limit gives a specific oppositefiltration [Del97].In the second part of the paper, we apply these results to integrable systemsof Hitchin type [Hit87]. Namely, first of all, we show that the special geometry onthe base cf. [Fre99] can be made projective. Equivalently, the variation of polarizedHodge structures of weight one refines to a variation of Hodge like filtrations ofweight three as in [BG83]. Underlying this construction is a certain family of cam-eral curves and a Seiberg–Witten differential that is constructed in terms of the nat-ural C ∗ -action on the total space of the integrable system. This is closely relatedto [DDD +
06, DDP07] where in the cases of
ADE -groups a family of Calabi–Yau
Date : 19 May 2009.2000
Mathematics Subject Classification.
Key words and phrases.
Frobenius manifolds, special geometry, Calabi-Yau threefolds, F-manifolds,potentials, unfoldings of meromorphic connections, integrable systems, Hitchin systems, stablebundles. threefolds was constructed whose variation of –a priori mixed– Hodge structurescoincides with that of the Hitchin system.This brings us to the starting data of the first part of the paper, and gives usconstructions of the associated Frobenius manifolds. In this example all choices,abstractly defined in the first part, have a natural interpretation in the elementarygeometry of curves. We believe this Frobenius manifold to be of interest for thefollowing reasons: first of all, as shown in [HT03] the Hitchin integrable systemsassociated to Langlands dual groups are SYZ-mirror to one another. Second, in thegeometric transition conjecture for
ADE -fibered Calabi–Yau threefolds [DDD + UTLINE
A Kuranishi family with base manifold B of Calaby-Yau threefolds gives rise toa variation of polarized Hodge structures of weight 3 on the primitive part of thethe middle cohomology bundle with some distinguished properties. Such a VPHSinduces on the one hand projective special K¨ahler geometry on a manifold B withdim B = dim B +
1, on the other hand it induces Frobenius manifold structureson a manifold M with dim M = B +
2. Both geometries contain some flatstructures and potentials and both depend on additional choices.The purpose of the first five sections is to review the (well known) construc-tions, to discuss the dependence on choices and to give a comparison. Section 1gives definitions, section 2 treats Frobenius manifolds, section 3 shows that theunderlying F -manifold structure is independent of choices, section 4 treats a partof projective special K¨ahler geometry, and section 5 compares them.Section 6 reviews the Hitchin system, section 7 introduces the Seiberg-Wittendifferential and section 8 gives the variation of Hodge like filtrations of weightthree on the base of the Hitchin system. Finally, the results of sections 1–5 areapplied to the Hitchin system in section 9.1. S OME DEFINITIONS
In the next five sections B will be a small neighborhood of a base point 0 in acomplex manifold of dimension n . When necessary, the size will be decreased, soessentially the germ ( B , 0 ) is considered, but we will not emphasize this.1.1. Variation of polarized Hodge structures (VPHS).
A VPHS of weight w ∈ Z on B consists of data ( B , V , ∇ , V R , S , F • , w ) . Here V is a holomorphic vector bun-dle with a flat connection ∇ and a real ∇ -flat subbundle V R such that V = V R ⊗ C , S is a ( − ) w -symmetric ∇ -flat nondegenerate pairing on V with real values on V R , ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 3 the decreasing Hodge filtration F • is a filtration of holomorphic subbundles with ∇ : O ( F p ) → O ( F p − ) ⊗ Ω B (Griffiths transversality), (1.1) V = F p ⊕ F w + − p (Hodge structure), equivalent: (1.2) V = M p H p , w − p where H p , w − p : = F p ∩ F w − p , (1.3) S ( F p , F w + − p ) = i p − w S ( v , v ) > v ∈ H p , w − p − { } (1.5)(rest of the polarization).In this chapter the real structure and the conditions (1.2), (1.3) and (1.5) will playno role. Data ( B , V , ∇ , S , F • , w ) as above with (1.1) and (1.4) only will be called variation of Hodge like filtrations with pairing .The connection ∇ induces a Higgs field C on L p F p / F p + , C = [ ∇ ] : O ( F p / F p + ) → O ( F p − / F p ) ⊗ Ω B (1.6)with C X C Y = C Y C X for X , Y ∈ T B . From section 2 on, we consider only datawhich satisfy w = F w + =
0, rank F w =
1, rank F w − / F w = n ,and thus rank F / F = n , rank F / F = F = V . (cid:27) (1.7)For any λ ∈ F w − { } the map C • λ : T B → F w − / F w is an isomorphism. (cid:27) (1.8)As B is small, (1.8) extends from 0 ∈ B to all points in B . The conditions (1.7)and (1.8) together are called CY-condition . This condition was discussed in [BG83]and is weaker than the so-called H -generating condition considered in [HM04,ch. 5].1.2. Opposite filtrations. An opposite filtration U • is defined to be an increasing ∇ -flat filtration with V = L p F p ∩ U p or equivalently V = F p ⊕ U p − , (cid:27) (1.9) S ( U p , U w − − p ) =
0. (1.10)As it is ∇ -flat, U • is determined by U • V and will be identified with that filtration.The splitting in (1.9) is holomorphic, the one on (1.3) is only real analytic. Both are S -orthogonal in the sense S ( F p ∩ U p , F q ∩ U q ) = = S ( H p , w − p , H q , w − q ) if p + q = w . (1.11) S and U • induce a symmetric and nondegenerate pairing g U on V by g U ( a , b ) : = ( − ) p S ( a , b ) for a ∈ O ( F p ∩ U p ) , b ∈ O ( V ) . (1.12)The splitting in (1.9) is also g U -orthogonal in the sense of (1.11).Now the connection ∇ decomposes into ∇ = ∇ U + C U , where ∇ U is a con-nection on each subbundle F p ∩ U p and C U is O B -linear and maps F p ∩ U p to CLAUS HERTLING, LUUK HOEVENAARS, AND HESSEL POSTHUMA ( F p − ∩ U p − ) ⊗ Ω B . The flatness of ∇ is equivalent to ∇ U being flat, C U being aHiggs field and the potentiality condition ∇ U ( C U ) =
0, more explicitly: ∇ UX ( C UY ) − ∇ UY ( C UX ) − C U [ X , Y ] = X , Y ∈ T B . (1.13)Because of (1.11) and the ∇ -flatness of S , both S and g U are ∇ U -flat, and g U satis-fies g U ( C UX a , b ) = g U ( a , C UX b ) for X ∈ T B , a , b ∈ O ( V ) . (1.14)That is: C U is selfadjoint with respect to g U . Define the endomorphismˆ V U : V → V , ˆ V U : = ∑ p ( p − w ) id | F p ∩ U p .Then ˆ V U is ∇ U -flat, and ( ˆ V U ) ∗ = − ˆ V U where ∗ denotes the adjoint with respectto g U , and [ C U , ˆ V U ] = C U .In the case w =
3, the combination of the CY-condition (1.7) & (1.8) and of thechoice of an opposite filtration U • leads to a Frobenius manifold, and the combi-nation of the CY-condition and of the part U of an opposite filtration leads to (apart of) projective special K¨ahler geometry. This will be discussed in the sections2 and 4.1.3. Frobenius manifolds and F-manifolds. An F-manifold ( M , ◦ , e , E ) [HM99,Her02] is a complex manifold of dimension ≥ T M , a unit field e ∈ T M and an Euler field E ∈ T M with the following two properties: Lie E ( ◦ ) = X ◦ Y ( ◦ ) = X ◦ Lie Y ( ◦ ) + Y ◦ Lie Y ( ◦ ) . (1.15)(1.15) implies Lie e ( ◦ ) = Frobenius manifold ( M , ◦ , e , E , g ) [Dub96] is an F-manifold together with asymmetric nondegenerate O M -bilinear pairing g on T M with the following prop-erties: its Levi-Civita connection ∇ g is flat; there is a potential Φ ∈ O M such thatfor ∇ g -flat vector fields X , Y , Zg ( X ◦ Y , Z ) = XYZ ( Φ ) ; (1.16)the unit field e is ∇ g -flat; the Euler field E satisfies Lie E ( g ) = ( − w ) · g for some w ∈ C .In fact, the potentiality condition and the flatness imply (1.15), cf. [HM99,Her02]. They also imply that the metric is multiplication invariant, g ( X ◦ Y , Z ) = g ( X , Y ◦ Z ) for X , Y , Z ∈ T M . (1.17)It turns out that ∇ g • E is a flat endomorphism of the tangent bundle.A Frobenius manifold M with a base point 0 ∈ M is called semihomogeneous if w ∈ N and if there are integers 0 = p ≤ p ≤ ... ≤ p dim M − ≤ p dim M = w andflat coordinates t i centered at 0 such that e = ∂∂ t and E = dim M ∑ i = ( − p i ) t i ∂∂ t i . (1.18)Then the numbers p i are unique, because the numbers p i − ∇ g • E . The coordinates t i are called semihomogeneous. ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 5
2. F
ROBENIUS MANIFOLDS FROM
VPHS
OF WEIGHT (( B , 0 ) , V , ∇ , S , F • , w = ) of Hodge like filtrations with pairing (see 1.1 forthis notion) of weight 3 with CY-condition (1.7)&(1.8) and n = dim B is fixed.Theorem 2.2 (a) gives a construction of Frobenius manifolds from it and an ad-ditional choice. This construction is well known, but usually hidden within muchricher structures [BK98, Bar02, Bar01, FP04, HM04]. We will give a proof whichwill make the comparison with projective special K¨ahler geometry easy.Theorem 2.2 (b) shows that the underlying F-manifold with Euler field is inde-pendent of the additional choice, contrary to the flat structure and the metric. Thisresult is new. It will be proved in section 3.6 (see also remark 2.3).The section starts with lemma 2.1 which discusses the geometry of the variationof Hodge like filtrations of weight 3 with CY-condition and an opposite filtration.It yields coordinates t , ..., t n + and a prepotential Ψ ∈ O B . It is complementedby lemma 2.4 which constructs the initial data of lemma 2.1 out of coordinates t , ..., t n + on B and an arbitrary function Ψ ∈ O B . It shows that the prepoten-tial Ψ in lemma 2.1 is not subject to any hidden conditions. Finally, proposition2.5 discusses the automorphisms of the F-manifold which underlies the Frobeniusmanifolds in Theorem 2.2. Lemma 2.1.
Additionally to the variation of Hodge like filtrations of weight 3 with CY-condition fixed above, choose the following data: (1)
An opposite filtration U • . By section (1.2) it induces a flat connection ∇ U on eachsubbundle of the splitting V = L p = F p ∩ U p ; and the pairing S is ∇ -flat and ∇ U -flat. (2) A ∇ U -flat basis v , ..., v n + of V which is compatible with the splitting of V andthe pairing S, and v ∈ F ; v , ..., v n + ∈ F ∩ U ; v n + , ..., v n + ∈ F ∩ U ; v n + ∈ U ; (cid:27) (2.1) S ( v , v n + ) = − S ( v k , v l ) = δ k + n , l for 2 ≤ k ≤ n +
1. (2.2)
Let v , ..., v n + be the ∇ -flat (here ∇ , not ∇ U ) extension of v ( ) , ..., v n + ( ) ∈ V .Then there are unique coordinates t , ..., t n + on B and there is a unique function Ψ ∈O B which satisfy Ψ ( ) = andv = v + n + ∑ i = t i v i + n + ∑ i = ∂ Ψ ∂ t i · v n + i + n + ∑ k = t k ∂∂ t k − ! ( Ψ ) · v n + . (2.3) CLAUS HERTLING, LUUK HOEVENAARS, AND HESSEL POSTHUMA
They also satisfy for i , j , k ∈ {
2, ..., n + } and a ∈ { n +
2, ..., 2 n + }∇ ∂∂ ti v = v i , (2.4) v i = v i + ∑ j ∂ Ψ ∂ t i ∂ t j v n + j + ∑ k t k ∂∂ t k − ! ( ∂ Ψ ∂ t i ) · v n + , (2.5) ∇ ∂∂ ti v j = ∑ k ∂ Ψ ∂ t i ∂ t j ∂ t k v n + k , (2.6) v a = v a + t a − n · v n + , (2.7) ∇ ∂∂ ti v a = δ i + n , a · v n + , (2.8) v n + = v n + , (2.9) ∇ ∂∂ ti v n + =
0. (2.10)
The function Ψ and the flat structure on B from the coordinates t , ..., t n + depend onlyon the choice of U • and v ( ) .Proof. Here and later the following convention for indices will be used: i , j , k ∈ {
2, ..., n + } , a , b ∈ { n +
2, ..., 2 n + } , α ∈ {
1, ..., 2 n + } . (2.11)Define p = p i = p a = p n + =
0, then v α ∈ O ( F p α ∩ U p α ) . The sections v α and v α satisfy the following properties: v α ∈ O ( U p α ) ( because U p α is ∇ -flat ) , ∇ U v α = − C U v α ∈ O ( U p α − ) ⊗ Ω B , v α ≡ v α mod O ( U p α − ) , (2.12) ∇ v α = C U v α ∈ O ( F p α − ∩ U p α − ) ⊗ Ω B . (2.13)(2.12) gives (2.9) and (2.10). There are unique functions t i , κ i , κ n + ∈ O B such that v = v + n + ∑ i = t i v i + n + ∑ i = κ i · v n + i + κ n + · v n + . (2.14)The CY-condition (1.7) & (1.8) shows that t , ..., t n + are coordinates on B . Theyare centered at 0 because v ( ) = v . From now on denote ∂ i : = ∂∂ t i . Derivation of(2.14) gives ∇ ∂ i v = v i + ∑ j ∂ i κ j · v n + j + ∂ i κ n + · v n + (2.15) ≡ v i ≡ v i mod O ( U ) .This and (2.13) show (2.4).Now we will use two times the pairing S , first for (2.7) & (2.8), second for theexistence of the function Ψ . Equation (2.13) shows ∇ ∂ i v a ∈ O B · v n + . The coef-ficient δ i + n , a in (2.8) is determined by0 = ∂ i ( ) = ∂ i S ( v , v a ) = S ( ∇ ∂ i v , v a ) + S ( v , ∇ ∂ i v a )= S ( v i , v a ) + S ( v , ∇ ∂ i v a ) = δ i + n , a + S ( v , ∇ ∂ i v a ) ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 7 and S ( v , v n + ) = −
1. This shows (2.7) & (2.8). Condition (2.2) also holds with v α replaced by v α because S is ∇ -flat and v α ( ) = v α ( ) . This, together with (2.4) and(2.15) give0 = S ( v i , v j ) = ∂ j κ i · S ( v i , v n + i ) + ∂ i κ j · S ( v n + j , v j ) = ∂ j κ i − ∂ i κ j .Therefore there is a unique function Ψ ∈ O B with Ψ ( ) = ∂ i Ψ = κ i . (2.16)Derivation by ∂ i of (2.15) and (2.4) gives ∇ ∂ i v j = ∑ k ∂ i ∂ j ∂ k Ψ · v n + k + ∂ i ∂ j κ n + · v n + (2.17) ≡ ∑ k ∂ i ∂ j ∂ k Ψ · v n + k ≡ ∑ k ∂ i ∂ j ∂ k Ψ · v n + k mod O ( U ) .This and (2.13) show (2.6). Using (2.7) this gives ∇ ∂ i v j = ∑ k ∂ i ∂ j ∂ k Ψ · v n + k = ∑ k ∂ i ∂ j ∂ k Ψ · v n + k + ∑ k ∂ i ∂ j ∂ k Ψ · t k · v n + .With (2.17) it implies ∂ i ∂ j κ n + = ∑ k t k · ∂ i ∂ j ∂ k Ψ = ∑ k t k ∂ k ! ∂ i ∂ j Ψ = ∂ i ∂ j ∑ k t k ∂ k − ! Ψ .We can conclude κ n + = ∑ k t k ∂ k − ! Ψ (2.18)because we know ( ∂ i κ n + ) ( ) = h ⇐ v i ( ) = v i and (2.15) i , ∂ i ∑ k t k ∂ k − ! Ψ ! ( ) = ∑ k t k ∂ k − ! ∂ i Ψ ! ( )= − κ i ( ) = h ⇐ v ( ) = v i , κ n + ( ) = h ⇐ v ( ) = v i , ∑ k t k ∂ k − ! Ψ ! ( ) = − Ψ ( ) = Ψ and the flat structure on B from t , ..., t n + are independent of the choice of v α except v ( ) , we suppose that a second choice e v α is made with e v ( ) = v ( ) . All its data are denoted using a tilde. The basechange from the base ( v α ) to the base ( e v α ) is constant, because both bases are flatwith respect to ∇ U . Now e v = v because both are ∇ U -flat extensions of e v ( ) = v ( ) . With S ( v , v n + ) = − = S ( e v , e v n + ) we obtain also e v n + = v n + . CLAUS HERTLING, LUUK HOEVENAARS, AND HESSEL POSTHUMA
Suppose ( e v i ) = ( v i ) · A , where i ∈ {
2, ..., n + } and ( e v i ) , ( v i ) are row vectors, and A ∈ Gl ( n , C ) . Then ( t i ) = ( e t i ) · A tr , ( e ∂ i ) = ( ∂ i ) · A , ( e κ i ) = ( κ i ) · A and thus e Ψ = Ψ .This shows the desired independencies of the flat structure and Ψ .In fact, rescaling of v ( ) ∈ F − { } leads to a rescaling of Ψ , but it does notaffect the flat structure on B . That depends only on U • . (cid:3) Theorem 2.2.
A variation (( B , 0 ) , V , ∇ , S , F • ) of Hodge like filtrations of weight 3 withpairing and with CY-condition (1.7) & (1.8) and n = dim B is fixed. (a) Any choice of an opposite filtration U • and a generator λ ∈ F leads in a canon-ical way (described below in the proof) to a Frobenius manifold M U , λ ⊃ C × B of dimension n + . It is semihomogeneous with integers ( p , ..., p n + ) =(
0, 1, ..., 1, 2, ..., 2, 3 ) ( and each n times). (b) The manifold M U , λ is canonically isomorphic to the manifold M = C × B whichis constructed in section 3.6. All Frobenius manifolds induce the same unit fielde, Euler field E and multiplication ◦ on M, so the same F-manifold structure. Butin the general the metrics and flat structures differ.The unit field is e = ∂∂ t if t is the coordinate on C of a coordinate system which respectsthe product M = C × B . The manifold B comes equipped with a projection p : B → B , the fibers are isomorphic to C n + as affine algebraic manifolds. The Euler field inducesa good C ∗ -action on the fibers with weights (
1, ..., 1, 2 ) . Part (b) will follow from the results in section 3.6.
Proof of part (a).
All the data in lemma 2.1 will be used, and also the convention(2.11). The section v is chosen such that v ( ) = λ . Define M U , λ = C × B × C n + with coordinates ( t , ..., t n + ) and the flat connection defined by these coordi-nates. Here ( t , ..., t n + ) extend the coordinates on B from lemma 2.1. Denote ∂ α = ∂∂ t α for α =
1, ..., 2 n +
2. Define a potential Φ ( t , ..., t n + ) : = Ψ ( t , ..., t n + ) + t t n + + t n + ∑ i = t i t n + i . (2.19)Define a symmetric nondegenerate flat bilinear form g on TM U , λ by g ( ∂ , ∂ α ) = δ α ,2 n + , g ( ∂ i , ∂ α ) = δ i + n , α , g ( ∂ n + i , ∂ α ) = δ i , α , g ( ∂ n + , ∂ α ) = δ α . (2.20)Then Φ , g and formula (1.16) give the following multiplication ◦ on T U , λ M . ∂ ◦ = id; ∂ i ◦ ∂ j = ∑ k ∂ i ∂ j ∂ k Ψ · ∂ n + k , ∂ i ◦ ∂ a = δ i + n , a · ∂ n + , ∂ i ◦ ∂ n + = ∂ a ◦ ∂ b = ∂ a ◦ ∂ n + = ∂ n + ◦ ∂ n + =
0. (2.21)So, it respects the grading L p = ( L p α = p O U , λ M · ∂ α ) of T U , λ M , and the multiplicationcoefficients depend at most on t , ..., t n + . We claim that it is commutative andassociative. Commutativity of this multiplication is clear. The only nontrivial partof the associativity is given by g (( ∂ i ◦ ∂ j ) ◦ ∂ k , ∂ ) = g ( ∂ i ◦ ∂ j , ∂ k ) = Ψ ijk = Ψ jki = g ( ∂ j ◦ ∂ k , ∂ i ) = g ( ∂ i ◦ ( ∂ j ◦ ∂ k ) , ∂ ) . ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 9
Define the Euler field E as E = n + ∑ α = ( − p α ) · t α ∂∂ t α . (2.22)Then E Φ = E Ψ =
0, Lie E ∂ α = ( p α − ) ∂ α , and Lie E ( ◦ ) = ◦ and Lie E ( g ) = ( − ) g follow immediately. Thus ( M U , λ , ◦ , e , E , g ) is a semihomogeneous Frobeniusmanifold with semihomogeneous coordinates ( t , ..., t n + ) .In order to show that this Frobenius manifold is independent of the choice of v α except v ( ) = λ , we continue the argument from the end of the proof of lemma2.1, with the same second choice e v α and the same matrix A . Define the isomor-phism e M → M U , λ by e t = t , e t n + = t n + , ( e t n + i ) = ( t n + i ) · A . Then e g = g , e Φ = Φ , e E = E , so one obtains the same Frobenius manifold. (cid:3) Remark 2.3.
Different choices in lemma 2.1 and theorem 2.2 lead to different co-ordinate systems on B and on M and different flat structures. The coordinatechanges are very complicated when U is changed. Trying to prove theorem 2.2(b) by controlling these coordinate changes looks hard.But when U is fixed and only U and U = ( U ) ⊥ S are changed, the coordinatechanges are much simpler. This is addressed in section 5.2 which shows that all thecoordinates and functions Ψ for fixed U and varying U & U have a nice commonorigin from projective special geometry. We now sketch the common origin of theFrobenius algebra at the level of tangent spaces: considerGr ( F • ) = F ⊕ (cid:16) F / F (cid:17) ⊕ (cid:16) F / F (cid:17) ⊕ (cid:16) F / F (cid:17) The pairing S on V induces a bilinear form on Gr ( F • ) whose symmetrization g (theconstruction is similar to g U in (1.12)) gives the pairing of the Frobenius algebra. Inorder to define the multiplication we recall the definition of the Higgs field C andthe isomorphism T B ∼ = F / F resulting from the Calabi-Yau condition togetherwith a choice of nonzero λ ∈ F .We define the following commutative, associative, unital and graded multipli-cation ◦ on Gr ( F • ) : λ ◦ V = V ∀ V ∈ Gr ( F • ) C X λ ◦ C Y λ = C X C Y λ ∀ X , Y ∈ T B C X λ ◦ W = C X W ∀ X ∈ T B , W ∈ F / F Other multiplications are zero unless they are required for commutativity. As-sociativity (and commutativity) of ◦ follows immediately from the fact that theHiggs field gives commuting endomorphisms: for instance, associativity followsfrom C X ( C Y C Z λ ) = C Z ( C X C Y λ ) Together with g , this multiplication gives a Frobenius algebra which is to be com-pared with the one given in (2.20), (2.21). A different choice of λ simply gives arescaling of the multiplication.Given the part U of an opposite filtration, one findsGr ( F • ) ∼ = F ⊕ F / F ⊕ U and the multiplication ◦ and bilinear form g can be transferred to the right handside. It is possible to identify this space with the tangent space of a manifold, in the following way. Consider the holomorphic vector bundle U → B . Using theline bundle ρ : F → B we can pull back this bundle to ρ ∗ U → F . This gives theisomorphism T (( c ) , v ) ρ ∗ U ∼ = F ⊕ T B ⊕ U ∼ = F ⊕ F / F ⊕ U where c ∈ F , v ∈ U . So we can view Gr ( F • ) as a tangent space to (the totalspace of) ρ ∗ U . It is true that refining U to a full opposite filtration U • allowsone to use the bilinear form g U together with ◦ to define a Frobenius manifoldstructure on ρ ∗ U . However, from these considerations it does not follow that allthese manifolds are isomorphic as F -manifolds to one and the same M . This is thesubject of section 3. Lemma 2.4.
Let ( B , 0 ) be a germ of a manifold with coordinates t , ..., t n + centered at0, i.e. t ( ) = ... = t n + ( ) , and let Ψ ∈ O B be an arbitrary function with Ψ ( ) = .Furthermore, let V → B be a holomorphic vector bundle with two bases v , ..., v n + andv , ..., v n + of sections which are related by (2.3) , (2.5) , (2.7) and (2.9) . (a) Let ∇ be the unique flat connection on V with flat sections v , ..., v n + . Then (2.4) , (2.6) , (2.8) and (2.10) hold. (b) Define two filtrations F • and U • on V by (2.1) and an antisymmetric pairing Sby (2.2) andS ( v α , v β ) = ( α , β ) / ∈{ (
1, 2 n + ) , ( n +
2, 1 ) }∪ { ( i , i + n ) , ( i + n , i ) | i =
2, ..., n + } . (2.23) Then (( B , 0 ) , V , ∇ , S , F • ) is a Hodge like filtration with pairing of weight w = and with CY-condition (1.7) & (1.8) , and U • is an opposite filtration.Proof. Again we use the convention (2.11) and write ∂ i = ∂∂ t i .(a) (2.4), (2.8) and (2.10) are obvious, (2.6) follows from ∇ ∂ i v j = ∑ k ∂ i ∂ j ∂ k Ψ · v n + k + ( ∑ k t k ∂ k ) ∂ i ∂ j Ψ · v n + = ∑ k ∂ i ∂ j ∂ k Ψ · ( v n + k + t k · v n + ) = ∑ k ∂ i ∂ j ∂ k Ψ · v n + k .(b) ∇ and F • satisfy Griffiths transversality (1.1) because of (2.4), (2.6), (2.8) and(2.10). For the same reason U • is ∇ -flat. By definition S satisfies (1.4) and (1.11).The definition of S in (2.2) and (2.23) and the formulas (2.3), (2.5), (2.7) and (2.9)show (2.2) and (2.23) for v α instead of v α . Therefore S is ∇ -flat.Finally, the CY-conditions also hold, (1.7) is built-in, (1.8) follows from (2.4) (cid:3) Proposition 2.5.
Consider the same data as in theorem 2.2 (a) and the Frobenius manifoldconstructed there in its proof, including the additional choice of coordinates ( t , ..., t n + ) .Consider the group Aut ( M , B , ◦ , e , E ) : = { ϕ : ( M , 0 ) → ( M , 0 ) biholomorphic | ϕ | B = id | B , ϕ respects multiplication,unit field and Euler field } of automorphisms of the underlying F-manifold which fix the submanifold B . We use thesame convention for the indices as in the proof of theorem 2.1 (a):i , j , k ∈ {
2, ..., n + } , a , b ∈ { n +
2, ..., 2 n + } , α ∈ {
1, ..., 2 n + } . ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 11 (a) For any automorphism ϕ ∈ Aut ( M , B , ◦ , e , E ) there exist β ∈ C ∗ and γ ab ∈ O B with ϕ = t , ϕ i = t i , ϕ a = β · t a , ϕ n + = β · t n + + ∑ a , b γ ab ( t , ..., t n + ) · t a t b (2.24) and = ( ∂ i ◦ ∂ j )( ϕ n + ) " = ( ∑ k ∂ i ∂ j ∂ k Ψ ∂ k + n )( ϕ n + ) . (2.25) (b) In the case when all ∂ i ∂ j ∂ k Ψ = then (2.25) is empty, and β and the γ ab can bechosen freely.(c) If some ∂ i ∂ j ∂ k Ψ = then β = , but the γ ab are only subject to condition (2.25) .Proof. Consider an automorphism ϕ : ( M , 0 ) → ( M , 0 ) . The three conditions ϕ ∗ ( e ) = e , ϕ ∗ ( E ) = E and ϕ | B = id | B are equivalent to the following: ϕ = t , ϕ i = t i , ϕ a = ∑ b β ab ( t , ..., t n + ) · t b for some β ab ∈ O B with det ( β ab ) ∈ O ∗ B , ϕ n + = ∑ a , b γ ab ( t , ..., t n + ) · t a t b + β ( t , ..., t n + ) · t n + for some γ ab ∈ O B , β ∈ O ∗ B . (2.26)Then the coordinate vector fields and their images under ϕ ∗ satisfy ϕ ∗ ( ∂ ) = ∂ , ϕ ∗ ( ∂ n + ) = ∂ n + ( ϕ n + ) · ∂ n + = β · ∂ n + , ϕ ∗ ( ∂ a ) = ∑ b β ba · ∂ b + ∂ a ( ϕ n + ) · ∂ n + , ϕ ∗ ( ∂ i ) = ∂ i + ∑ a ∂ i ( ϕ a ) · ∂ a + ∂ i ( ϕ n + ) · ∂ n + . (2.27)The additional condition that ϕ respects the multiplication reduces in view of(2.21) and (2.27) to the conditions ϕ ∗ ( ∂ i ) ◦ ϕ ∗ ( ∂ a ) = ϕ ∗ ( ∂ i ◦ ∂ a ) and ϕ ∗ ( ∂ i ) ◦ ϕ ∗ ( ∂ j ) = ϕ ∗ ( ∂ i ◦ ∂ j ) . (2.28)The first one is equivalent to δ i + n , a · β · ∂ n + = δ i + n , a · ϕ ∗ ( δ n + ) = ϕ ∗ ( ∂ i ◦ ∂ a )= ϕ ∗ ( ∂ i ) ◦ ϕ ∗ ( ∂ a ) = ∑ b β ba · δ i + n , b · ∂ n + = β i + n , a · ∂ n + .This is equivalent to β ab = δ ab · β and to ϕ a = β · t a . (2.29) Taking this into account, the second equation in (2.28) becomes ∑ k ∂ i ∂ j ∂ k Ψ · ( β · ∂ k + n + ∂ k + n ( ϕ n + ) · ∂ n + )= ∑ k ∂ i ∂ j ∂ k Ψ · ϕ ∗ ( ∂ k + n ) = ϕ ∗ ( ∂ i ◦ ∂ j ) = ϕ ∗ ( ∂ i ) ◦ ϕ ∗ ( ∂ j )= ∑ k ∂ i ∂ j ∂ k Ψ · ∂ k + n + ∑ a ∂ i ◦ ∂ j ( β ) · t a ∂ a + ∑ a ∂ i ( β ) · t a ∂ a ◦ ∂ j = ∑ k ∂ i ∂ j ∂ k Ψ · ∂ k + n + (cid:2) t i + n · ∂ j ( β ) + t j + n · ∂ i ( β ) (cid:3) · ∂ n + .This is equivalent to ∂ i ∂ j ∂ k Ψ · β = ∂ i ∂ j ∂ k Ψ for all i , j , k (2.30)and ( ∂ i ◦ ∂ j )( ϕ n + ) " = ∑ k ∂ i ∂ j ∂ k Ψ ∂ k + n ( ϕ n + ) = t i + n · ∂ j ( β ) + t j + n · ∂ i ( β ) . (2.31) ∂ i ∂ j ∂ k Ψ = : then (2.30) is empty and (2.31) becomes ∂ i ( β ) =
0. Inthis case β is an arbitrary constant in C ∗ and γ ab are arbitrary. ∂ i ∂ j ∂ k Ψ = : then (2.30) says β =
1. Now (2.31) becomes ( ∂ i ◦ ∂ j )( ϕ n + ) = (cid:3)
3. (TEP)-
STRUCTURES
Definitions.
For the proof of theorem 2.2 (b) we need a datum which is be-tween the Frobenius manifold and its F-manifold, namely the (TEP)-structure on ( π ∗ TM ) | C × M where π : P × M → M is the projection. We will show that thisstructure does not depend on ( U • , λ ) . Then theorem 2.2 (b) will follow easily.A (TEP)-structure of weight w ∈ Z consists of data ( H → C × M , ∇ , S ) . Here M is a complex manifold, H → C × M is a holomorphic vector bundle, ∇ is a flatconnection on H | C ∗ × M with a pole of Poincar´e rank 1 along { } × M , and P is a ∇ -flat ( − ) w -symmetric nondegenerate pairing P : H ( z , t ) × H ( − z , t ) → C for ( z , t ) ∈ C ∗ × M which extends with j : ( z , t ) ( − z , t ) to a nondegenerate pairing P : O ( H ) ⊗ j ∗ O ( H ) → z w O C × M . (3.1)A (TLEP)-structure of weight w ∈ Z is an extension of the bundle H → C × M of a (TEP)-structure to a holomorphic vector bundle b H → P × M such thatthe pole along { ∞ } × M is logarithmic and such that P extends to an everywherenondegenerate pairing from O ( b H ) ⊗ j ∗ O ( b H ) to z w O P × M .A (trTLEP)-structure is a (TLEP)-structure such that b H is a family of trivial bun-dles on P .A (TEP)-structure induces a Higgs field C = [ z ∇ ] : O ( H |{ }× M ) → O ( H |{ }× M ) ⊗ Ω M ,an endomorphism U = [ z ∇ ∂ z ] : O ( H |{ }× M ) → O ( H |{ }× M ) ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 13 with [ C , U ] = g = [ z − w P ] : O ( H |{ }× M ) × O ( H |{ }× M ) → O M with C ∗ = C and U ∗ = U .A (trTLEP)-structure is equivalent to a differential geometric structure on H |{ }× M containing C and g and more data, which is called Frobenius type struc-ture in [HM04, ch. 4], the equivalence is stated in [Sab02, VI 7], [Her03, § § H ∞ : = { global flat multivalued sections in H | C ∗ × M } ( M is small and contractible), cf.[Sab02, III.1.4] and [Her02, § Two examples.
For the constructions in this paper, the following two exam-ples of (TEP)-structures play an important role:(i) If M is a Frobenius manifold with Lie E ( g ) = ( − d ) · g , d ∈ C , and π : P × M → M is the projection, then π ∗ T M is canonically equipped with a(trTLEP)-structure of (any) weight w ∈ Z (Dubrovin, Manin, e.g. [HM04, ch. 4]): P : = z w · ( id, j ) ∗ g , (3.2) ∇ : = π ∗ ∇ g + z C + (cid:18) − z E ◦ −∇ g E + − d + w (cid:19) d zz , (3.3)where C is the Higgs field on TM with C X = X ◦ .(Compared to [HM04, ch. 4], here we changed the sign in C X = X ◦ and used − z instead of z in (3.3), in order to be compatible with section 2. The signs thereare chosen to make the comparison with projective special geometry in section 5smoother.)(ii) Let ( M , V , ∇ V , S , F • , w ) be a variation of Hodge like filtrations with pairing( M is small and contractible). Let π C : C × M → M be the projection and π ∗ C ∇ V be the flat connection on π ∗ C V whose flat sections are the pull backs of ∇ V -flatsections in V . Define a bundle H → C × M with H | C ∗ × M = π ∗ C V | C ∗ × M by O ( H ) : = ∑ p ∈ Z z w − p · O ( π ∗ C F p ) , (3.4)a flat connection ∇ on H | C ∗ × M by ∇ : = π ∗ C ∇ V (3.5)and a pairing P : ( π ∗ C V ) ( z , t ) × ( π ∗ C V ) ( − z , t ) → C for ( z , t ) ∈ C ∗ × M by P ( π ∗ C a , π ∗ C b ) : = ( π i ) w · S ( a , b ) . (3.6) Claim:
Then ( H → C × M , ∇ , P ) is a (TEP)-structure of weight w. On the one hand, this follows by unwinding the construction behind corollary7.14 (b) in [Her03] (an extra factor ( π i ) w in (3.6) makes the definitions here com-patible with [Her03]). On the other hand, it can be seen directly as follows. (3.4), (3.5) and the Griffithstransversality (1.1) show z ∇ ∂ z O ( H ) ⊂ O ( H ) and z ∇ X O ( H ) ⊂ O ( H ) for X ∈ T M .This gives the pole of Poincar´e rank 1 along { } × M (even z ∇ ∂ z O ( H ) ⊂ O ( H ) would be sufficient).The conditions (1.4), (3.4), (3.6) and the nondegenerateness of S show that P maps O ( H ) ⊗ O ( H ) to z w O C × M and that this map is nondegenerate. Obviously, P is ( − ) w -symmetric and ∇ -flat.3.3. F-manifolds from (TEP)-structures.
There is a construction of Frobeniusmanifolds from meromorphic connections which goes back to the construction ofFrobenius manifolds in singularity theory by M. Saito [Sai89] and K. Saito. It isformalized in [Sab02, Th´eor`eme VII.3.6][Bar02][Bar01] and [HM04, theorems 4.2and 4.5]. In [HM04] the initial data are a (trTLEP)-structure with a distinguishedsection and an isomorphy condition. The following result gives the constructionof a weaker datum, an F-manifold, from a weaker initial datum, a (TEP)-structurewith an isomorphy condition. The proof relies on [Her03, 4.1].
Theorem 3.1.
Let ( H → C × M , ∇ , P ) be a (TEP)-structure (actually, the pairingP will not be used) with Higgs field C = [ z ∇ ] and endomorphism U = [ z ∇ ∂ z ] on O ( H |{ }× M ) . Then O ( H |{ }× M ) is a T M -module. Suppose that the following isomorphycondition holds: O ( H |{ }× M ) is a free T M -module of rank 1. (3.7) Then there is a unique multiplication ◦ on T M with C X ◦ Y = C X C Y and a unique unitfield e. The multiplication is commutative and associative. The unit field satisfies C e = id .There is also a unique vector field E with C E = − U. The tuple ( M , ◦ , e , E ) is an F-manifold with Euler fieldProof. The first part of the proof follows [Her03, lemma 4.1]. Locally a section ξ in H |{ }× M is chosen such that C • ξ : TM → H |{ }× M is an isomorphism. Themultiplication ◦ and the vector fields e and E are defined by C X ◦ Y ξ = C X C Y ξ , C e ξ = ξ , C E ξ = − U ξ .The multiplication is commutative and associative, and e is a unit field.Because of C X ◦ Y C Z ξ = C X C Y C Z ξ , C e C Z ξ = C Z ξ , C E C Z ξ = − UC Z ξ the multiplication and the vector fields e and E are independent of the choice of ξ and satisfy C X ◦ Y = C X C Y , C e = id, C E = − U .The proof that they give an F-manifold with Euler field will use [Her03, lemma4.3]. In order to apply it, it would be nice to extend the (TEP)-structure to a(trTLEP)-structure. That is not always possible, but by [HM04, lemma 2.7] onecan change and extend it (locally in M ) to the following weaker structure: A holo-morphic vector bundle b H → P × M such that b H | ( C −{ } ) × M = H | ( C −{ } ) × M , suchthat the connection ∇ has logarithmic poles along { } × M and { } × M and suchthat b H is a family of trivial bundles on P (here M is supposed to be small). Be-cause of the last condition O ( b H |{ }× M ) ∼ = O ( b H |{ ∞ }× M ) ∼ = π ∗ O ( b H ) (3.8) ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 15 and C and U on b H |{ }× M as well as the residual connection ∇ res on b H |{ ∞ }× M areshifted to the isomorphic sheaves. There are two further endomorphisms V and W (with V + W = − residue endomorphism on H |{ ∞ }× M ) such that for fiberwiseglobal sections σ ∈ π ∗ O ( b H ) ∇ σ = (cid:18) ∇ res + z C + ( z U + V + zz − W ) d zz (cid:19) σ . (3.9)The flatness of ∇ yields ∇ res ( C ) = ∇ res ( U ) − [ C , V ] + C =
0. Thereforelemma 4.3 in [Her03] applies and shows that ( M , ◦ , e , E ) is an F-manifold withEuler field. (cid:3) The classifying space ˇ D PHS . For the rest of this section, a variation of Hodgelike filtrations (( B , 0 ) , V , ∇ , S , F • ) of weight w = B is a (sufficiently small) representative of a germ ( B , 0 ) ofa manifold of dimension n . For b ∈ B the filtration is denoted F • b . By abuse ofnotation we also denote its ∇ -flat shift to the fiber V by F • b .There is a classifying space ˇ D PHS for all Hodge like filtrations with the samediscrete data as F • ,ˇ D PHS : = (cid:8) filtrations F • on V | S ( F p , F − p ) = = F ⊂ F ⊂ F ⊂ F ⊂ F = V ,dim F = = dim F / F ,dim F / F = n = dim F / F (cid:9) . (3.10)It goes back to work of Griffiths and Schmid (see also [BG83]). It is a complexhomogeneous space and a projective manifold. More concretely, it is a bundleover the lagrangian Grassmannianˇ D lag = { F ⊂ V | dim F = n + S ( F , F ) = } (3.11)with fibers P ( F ) ∼ = P n . The base ˇ D lag has dimension n ( n + ) /2, the fibers con-tain the possible choices of F ⊂ F and F = ( F ) ⊥ S . The natural period map Π : B → ˇ D PHS , b F • b , (3.12)is horizontal. Because of the CY-condition it is an embedding. It determines thevariation of Hodge like filtrations.3.5. The classifying space ˇ D BL . There is a classifying space ˇ D BL for certain (TEP)-structures with a natural projection π BL : ˇ D BL → ˇ D PHS . In a more generalsetting such spaces have been constructed in [Her99] and taken up again in[HS08a][HS08b]. Here we restrict to the special case which we need. Before defin-ing and discussing ˇ D BL , some notations have to be established. V is a 2 n + S . The vector bundle H ′ : = V × C ∗ comes equipped with thetrivial flat connection ∇ and a pairing P : H ′ z × H ′− z → C for z ∈ C ∗ (3.13) ( a , b ) S ( a , b ) , here a , b ∈ H ′ z = V = H ′− z . It is ∇ -flat, antisymmetric and nondegenerate. The space of global flat sectionsin H ′ is denoted C . It is identified with V . For α ∈ Z and a ∈ V the section ( z z α · a ( z )) is denoted z α a , the space of such sections is denoted C α = z α · C .The space V α : = C { z } · C α is the germ at 0 of the Deligne extension of H ′ → C ∗ to a vector bundle on C with logarithmic pole at 0 with α as the only eigenvalueof the residue endomorphism [ ∇ z ∂ z ] . Together the spaces V α , α ∈ Z , form theKashiwara-Malgrange V -filtration. Of course Gr α V ∼ = C α canonically.Any (TEP)-structure ( H → C , ∇ , P ) with H | C ∗ = H ′ is determined by the germ H : = O ( H ) at 0. We are interested in the regular singular (TEP)-structures,i.e. those with H ⊂ ∑ α V α . The spectrum of such a (TEP)-structure is the tuple ( α , ..., α n + ) ∈ Z n + with α ≤ ... ≤ α n + and ♯ ( i | α i = α ) = dim Gr α V H / Gr α V z H . (3.14)The (TEP)-structure induces a decreasing filtration F • ( H ) on V by F p ( H ) = F p ( H ) : = z p − Gr − pV H ⊂ C = V . (3.15)The classifying space ˇ D BL of (TEP)-structures relevant for us isˇ D BL = (cid:8) regular singular (TEP)-structures ( H , ∇ , P ) of weight 3 with H | C ∗ = H ′ and spectrum ( α , ..., α n + ) = (
0, 1, ..., 1, 2, ..., 2, 3 ) (cid:9) (3.16)with 1 and 2 each n times. Theorem 3.2. ˇ D BL is an algebraic manifold and a bundle on ˇ D PHS via π BL : ˇ D BL → ˇ D PHS , H F • ( H ) . (3.17) The fibers are isomorphic to C n + as affine algebraic manifolds and carry a good C ∗ -actionwith weights (
1, ..., 1, 2 ) . The corresponding zero section ˇ D PHS ֒ → ˇ D BL is given by the(TEP)-structures defined as in (3.4) .Proof. This theorem is a special case of [Her99, theorem 5.6], but here the proofsimplifies. In the following we present the proof, as it provides useful explicitcontrol on ˇ D BL . Lemma 3.3. F • ( H ) ∈ ˇ D PHS if H ∈ ˇ D BL .Proof. F • ( H ) is decreasing because F p + ( H ) = z p + − Gr − ( p + ) V H = z p − Gr − pV z H ⊂ z p − Gr − pV H = F p ( H ) .Because of the spectral numbers ( dim F p ( H ) | p =
3, 2, 1, 0 ) = ( n +
1, 2 n +
1, 2 n + ) .If a ∈ F p ( H ) and a ∈ F − p ( H ) then there are sections σ ∈ H ∩ ( z − p a + V − p ) and σ ∈ H ∩ ( z − ( − p ) a + V − ( − p ) ) .The z -coefficient of P ( σ , σ ) ∈ z C { z } vanishes. This shows S ( a , a ) =
0. There-fore S ( F p ( H ) , F − p ( H )) = F • ( H ) ∈ ˇ D PHS (cid:3)
ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 17
Now the fiber π − BL ( F • ) for an arbitrary F • shall be determined. The (TEP)-structures in this fiber will be described by certain distinguished sections in them.For that we make the same choices as in lemma 2.1, a filtration U • which is oppo-site to F • and a basis v , ..., v n + of V which satisfies (2.1) and (2.2). We use againthe convention (2.11) for indices i , j , k , a , b , α . We define sections s = v ∈ C , s i = zv i ∈ C , s a = z v a ∈ C , s n + = z v n + ∈ C , (cid:27) (3.18)and we define ( p , p i , p a , p n + ) = (
3, 2, 1, 0 ) so that v α ∈ F p α ∩ U p α and s α ∈ z − p α · F p α ∩ U p α . The following picture illustrates this and the next lemma. ✲ Gr V H Gr V H Gr V z H Gr V H Gr V H C C C C V V V V s s i s a s n + U U U F F F ✲ z Lemma 3.4. (a) For any H ∈ π − BL ( F • ) there exist unique sections σ α ∈ H with σ α − s α ∈ ∑ β > − p α z β · U − β . They form a C { z } -basis of H . Explicitly, they take the form σ = s + ∑ a y a · z − · s a + y n + · z − · s n + , (3.19) σ i = s i + y n + i · z − · s n + , σ a = s a , σ n + = s n + , with some y a ∈ C , y n + ∈ C . (b) The other way round, for any y a ∈ C and y n + ∈ C, these sections generate over C { z } the germ H of a (TEP)-structure in π − BL ( F • ) .(c) Therefore π − BL ( F • ) ∼ = C n + as an affine algebraic manifold, andy n + , ..., y n + , y n + are coordinates on it.Proof. (a) Because of the spectral numbers H = ( H ∩ ( C + C + C )) ⊕ V .Because of F • ( H ) = F • there exist sections in H ∩ ( s α + V − p α ) . Existence anduniqueness of the sections σ α is now an easy argument in linear algebra. It is also clear that they form a C { z } -basis of H . A priori they take the form σ = s + ∑ a y a · z − · s a + x n + · z − · s n + (3.20) + y n + · z − · s n + , σ i = s i + x n + i · z − · s n + , σ a = s a , σ n + = s n + ,with y a , x n + , y n + , x a ∈ C . The germ H satisfies z ∇ z ∂ z H ⊂ H , (3.21) P : H × H → z C { z } nondegenerate. (3.22)On the other hand, the sections σ α satisfy z ∇ z ∂ z σ = ∑ a y a · σ a + x n + · z − · σ n + + y n + · σ n + , (3.23) z ∇ z ∂ z σ i = z · σ i + x n + · σ n + , , (3.24) z ∇ z ∂ z σ a = z · σ a , (3.25) z ∇ z ∂ z σ n + = z · σ n + , , (3.26) P ( σ α , σ β ) = zx n + z ( y n + i − x n + i ) z z ( x n + i − y n + i ) z z z . (3.27)with α ∈ { i , a , 2 n + } and β ∈ { j , b , 2 n + } . Both (3.23) and (3.27) show x n + =
0, (3.27) shows also x a = y a . This proves part (a).(b) The sections σ α generate over C { z } the germ H of the sections of a vectorbundle H → C which extends H ′ → C ∗ . Because of (3.23) - (3.27) H satisfies(3.21)&(3.22). Therefore ( H , ∇ , P ) is a (TEP)-structure. It is in π − BL ( F • ) .(c) It is now also clear. (cid:3) There is a natural C ∗ -action on ˇ D BL which respects the fibers of π BL . It is defined(coordinate independently) as follows. For any r ∈ C ∗ define π r : C → C , z r · z .Then ( H , ∇ , P ) ∈ π − BL ( F • ) is mapped by r ∈ C ∗ via the C ∗ -action to π ∗ r ( H , ∇ , P ) ∈ π − BL ( F • ) .This action works as follows on the sections and coordinates in the last lemma.If a ∈ C and α ∈ Z then π ∗ r ( z α · a ) = r α · z α · a , so π ∗ r σ = s + ∑ a r · y a · z − · s a + r · y n + · z − · s n + , π ∗ r σ i = r · (cid:16) s i + r · y n + i · z − · s n + (cid:17) , π ∗ r σ a = r · σ a , π ∗ r σ n + = r · σ n + ,and the C ∗ -action on π − BL ( F • ) is given in the coordinates ( y a , y n + ) by r . ( y a , y n + ) = ( r · y a , r · y n + ) . This finishes the proof of the theorem. (cid:3) ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 19
Remark 3.5. (i) Sections like the σ α above were used first in [Sai89, ch. 3].(ii) The vector bundle V × ˇ D PHS carries the trivial flat connection ∇ V × D andthe tautological filtration F • . The filtration is a family of Hodge like filtrations,but not a variation, because the Griffiths transversality is violated. Nevertheless,a filtration U • which is opposite to a reference filtration F •∗ ∈ ˇ D PHS and then alsoto all filtrations nearby, induces a decomposition ∇ V × D = ∇ U + C U into a flatconnection ∇ U : O ( F p ∩ U p ) → O ( F p ∩ U p ) ⊗ Ω ( nbhd of F •∗ ) and a tensor C U : O ( F p ∩ U p ) → O ( U p − ) ⊗ Ω ( nbhd of F •∗ ) .A basis v ( ∗ ) , ..., v n + ( ∗ ) of vectors with (2.1) and (2.2) for F •∗ extends to a ∇ V × D -flat basis of sections of V × ˇ D PHS with (2.1) and (2.2). The formulas(3.18) and (3.19) extend to these sections and yield a trivialization of the bundle π BL : ˇ D BL → ˇ D PHS on ( nbhd of F •∗ ) ⊂ ˇ D PHS , with fiber coordinates y a , y n + .(iii) The vector field on ˇ D BL which generates the canonical C ∗ -action is denoted E BL . It is tangent to the fibers of π BL . In local coordinates as in (ii) it is E BL = ∑ a y a ∂∂ y a + y n + ∂∂ y n + . The zero section ˇ D PHS ֒ → ˇ D BL consists of the (TEP)-structures with H = ∑ p C { z } · z − p · F p , F • ∈ ˇ D PHS (as in (3.4)).(iv) Any (TEP)-structure in ˇ D BL is determined by L : = H ∩ ( C + C + C ) . The z -coefficient of P restricts to a symplectic form P ( ) on C + C + C . The multi-plication by z restricts to a nilpotent endomorphism µ z : C + C + C → C + C with C µ z → C µ z → C µ z →
0. We leave it to the reader to show that the classi-fying space ˇ D BL can be identified with the following classifying space of certainlagrangian subspaces, e D BL = (cid:8) L ⊂ C + C + C | µ z ( L ) ⊂ L , µ z ( ∇ z ∂ z L ) ⊂ L , P ( ) ( L , L ) =
0, dim L = n + L ∩ ( C + C ) = n +
2, dim L ∩ C = n + (cid:9) .3.6. The canonical (TEP)-structure with isomorphy condition.
As in section 3.4a variation of Hodge like filtrations (( B , 0 ) , V , ∇ V , S , F • ) of weight w = B is identified withits image Π ( B ) ⊂ ˇ D PHS under the period map Π : B → ˇ D PHS in (3.12). Define B : = π − BL ( Π ( B )) and M : = C × B . (3.28)The coordinate on the factor C in C × B is denoted y . The tautological family of(TEP)-structures on ˇ D BL restricts to a family of (TEP)-structures on B . We extendit to a family ( H → C × B , ∇ , P ) of (TEP)-structures on M by twisting all sectionswith e y / z . Theorem 3.6.
With these definitions: (a)
This is a (TEP)-structure on M with isomorphy condition (3.7) . Theorem 3.1applies and gives M a canonical F-manifold structure. The unit field is e = ∂∂ y ,the Euler field is E = y ∂∂ y − ( E BL ) | B (E BL is defined in remark 3.5 (iii)). (b) For any of the Frobenius manifolds in Theorem 2.2, the underlying manifold M U , λ is canonically isomorphic to M. The isomorphism respects the F-manifold struc-ture and the Euler field. Proof.
It will be proved in several steps. For the rest of the section an oppositefiltration U • and a vector λ ∈ F − { } as in theorem 2.2 are chosen. Furthermore,sections v , ..., v n + on V as in lemma 2.1 and with v ( ) = λ are chosen. Lemma2.1 yields coordinates t , ..., t n + on B and a prepotential Ψ ∈ O B . Then theformulas (3.18) and (3.19) in section 3.5 provide sections σ α which generate thetautological family of (TEP)-structures on B ⊂ ˇ D BL . Lemma 3.7. (a) For i , j , k ∈ {
2, ..., n + } and a ∈ { n +
2, ..., 2 n + } z ∇ σ = ∑ i σ i d t i + ∑ a σ a d y a + σ n + d y n + + ∑ a y a σ a + y n + σ n + ! d zz , (3.29) z ∇ σ i = ∑ j ∑ k ∂ i ∂ j ∂ k Ψ · σ n + k ! · d t j + σ n + · d y n + i + y n + i · σ n + · d zz + σ i d z , (3.30) z ∇ σ a = σ n + · d t a − n + σ a · d z , (3.31) z ∇ σ n + = σ n + · d z . (3.32) (b) The family of tautological (TEP)-structures has a pole of Poincar´e rank 1 along { } × M and is therefore a (TEP)-structure on B . Its bundle is denoted H B → C × B .(c) The sections σ α define an extension to a (trTLEP)-structure on B .Proof. (a) These formulas follow from (3.18), from the formulas in lemma 2.1, from(3.23) - (3.26) and from derivating the sections σ α with ∇ ∂∂ ya and ∇ ∂∂ y n + .(b) Obvious.(c) This follows from (a) and (3.27). (cid:3) Lemma 3.8.
The bundle H → C × M whose sheaf is O ( H ) = e y / z · pr ∗ O ( H B ) (wherepr : C × B → B is the projection) is a (TEP)-structure withz ∇ ( e y / z σ α ) = e y / z · z ∇ ( σ α ) + e y / z · σ α d y − y · e y / z · d zz . (3.33) It satisfies the isomorphy condition (3.7) . The sections e y / z · σ α define an extension to a(trTLEP)-structure.Proof. (3.33) shows that the pole along { } × M is of Poincar´e rank 1. The pairing P satisfies P ( e y / z · σ α , e y / ( − z ) · σ β ) = P ( σ α , σ β ) ∈ z · C ,so it is the pairing of a (TEP)-structure. By (3.29) - (3.33) the sections e y / z · σ α define an extension to a (trTLEP)-structure. The Higgs field of the (TEP)-structuresatisfies the isomorphy condition (3.7) because of (3.29) and (3.33). (cid:3) Now theorem 3.1 applies and gives a canonical F-manifold structure. (3.33)shows C ∂∂ y = id, therefore e = ∂∂ y . In the following calculation, [ . ] denotes the ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 21 restriction to H |{ }× M , C y ∂ − ∑ a y a ∂ a − y n + ∂ n + [ e y / z · σ ]= y [ e y / z · σ ] − ∑ a y a [ e y / z · σ a ] − y n + [ e y / z · σ n + ]= − U [ e y / z · σ ] : = − [ z ∇ z ∂ z e y / z · σ ] .Because [ e y / z · σ ] generates O ( H |{ }× M ) as a T M -module, this is sufficient to see C y ∂ − ∑ a y a ∂ a − y n + ∂ n + = − U and y ∂∂ y − ( E BL ) | B = E . Part (a) of theorem 3.6 isproved.It rests to prove part (b). The choice of the sections v , ..., v n + yields co-ordinates ( t , ..., t n + ) on B , coordinates ( t , ..., t n + ) on M U , λ and coordinates ( y , t , ..., t n + y n + , ..., y n + ) on M .Of course, the most natural isomorphism between M U , λ and M is by identifyingthese coordinates. At the end of the proofs of lemma 2.1 and theorem 2.2 it wasdiscussed how the coordinates ( t , ..., t n + ) change if ( v , ..., v n + ) are changed,but U • and λ are fixed.One sees easily from (3.19) that the coordinates ( y , t , ..., t n + , y n + , ..., y n + ) change in the same way. Therefore the isomorphism M U , λ ∼ = M above is canonical.Obviously it respects unit field and Euler field.It also respects the multiplication. To see this, one chooses the section ξ : =[ e y / z · σ ] in H | { }× M and observes that the isomorphism C • ξ : TM → H | { }× M maps e to ξ , ∂ i to [ e y / z · σ i ] , ∂∂ y a to [ e y / z · σ a ] and ∂∂ y n + to [ e y / z · σ n + ] . The Higgsfield of the multiplication on TM is mapped to the Higgs field C on H | { }× M . Onecan extract the Higgs field C from (3.29) - (3.33). Comparison with (2.21) showsthat the multiplications coincide. This proves part (b) of theorem 3.6. (cid:3) Remark 3.9. (i) The isomorphism C • ξ : TM → H |{ }× M above with ξ : = [ e y / z · σ ] lifts to an isomorphism from the (trTLEP)-structure on π ∗ TM in section 3.2 (i) tothe (trTLEP)-structure on M in the last lemma, with global sections [ e y / z · σ α ] .(ii) In the beginning of section 3.3 a standard construction of Frobenius man-ifolds from meromorphic connections was mentioned. It can be applied to theFrobenius manifolds in theorem 2.2 and theorem 3.6. There it uses the (trTLEP)-structure with isomorphy condition constructed in the last lemma and the isomor-phism C • ξ : TM → H | { }× M with ξ as above.4. P ROJECTIVE SPECIAL (K ¨
AHLER ) GEOMETRY
This section presents some aspects of projective special geometry in a form whichwill make the comparison with Frobenius manifolds easy. It does not offer newresults, and it neglects some aspects, for example the role of the pairing andan induced hermitian metric. Because of that we put the “K¨ahler” in brack-ets. Projective special (K¨ahler) geometry has a purely holomorphic part, the spe-cial coordinates, which are related to Frobenius manifolds, and a part involv-ing the real structure, which is not related to Frobenius manifolds, but to tt ∗ -geometry [Her03]. We will touch the latter part only in the last part 4.4 of thissection. More complete accounts, different aspects and motivation are provided in[Fre99, Cor98, ACD02, BCOV94]. The setting and two period maps.
Let ( B , V , ∇ , S , F • ) be a variation ofHodge like filtrations with pairing of weight w = n = dim B . As before, B is a small neighborhoodof a base point 0 ∈ B . When necessary, the size of B will be decreased, so essen-tially the germ ( B , 0 ) is considered.The most important manifold in this section is B = F − { zero section } , to-gether with the natural projection p : B → B . The fibers F b − { } ∼ = C ∗ comeequipped with a C ∗ -action from the vector space structure, the correspondingvector field on B is denoted ε . Points in B are denoted ( δ , b ) where b ∈ B and δ ∈ F b − { } . The pull back with p yields on B a variation of Hodge like filtra-tions with pairing ( p ∗ V , p ∗ ∇ , p ∗ S , p ∗ F • ) of weight 3. The bundle p ∗ V carries thetautological generating section σ taut with σ taut ( δ , b ) = δ . It satisfies ( p ∗ ∇ ) ε σ taut = σ taut . (4.1)There are two natural and related period maps: P : B → V , (4.2) ( δ , b )
7→ ∇ -flat shift of δ ∈ F b ⊂ V b to V , P : TB → p ∗ F , (4.3) X ( p ∗ ∇ ) X σ taut .Here TB = T B is the holomorphic tangent bundle. Only in the last section 4.4also T B , T C B = T B ⊕ T M and T R B will be used.Because B is small and B is a C ∗ -bundle on B , the flat connection ∇ inducesthe trivialization τ : V ∼ = → V × B of the vector bundle V , and p ∗ ∇ induces thetrivialization τ : p ∗ V ∼ = → V × B of the vector bundle p ∗ V . Lemma 4.1.
The period maps P and P satisfy the following properties: (a) P and P are related by τ ◦ P = ( P ) ∗ : TB → P ∗ T V = P ∗ ( V × V ) = V × B . (4.4)(b) P is an embedding. (c) P is an embedding and thus (locally in B) an isomorphism of vector bundles.Proof. (a) It follows from the definitions.(b) The restriction of P to p − ( ) = F − { } ⊂ B is the tautological embedding F − { } → F ⊂ V . Because of this and because B is small, for P being anembedding it is sufficient to show that its differential ( P ) ∗ is injective at points ( δ , 0 ) ∈ p − ( ) . At such points ( P ) ∗ = P : T ( δ ,0 ) B → F ⊂ V . This map P : T ( δ ,0 ) B → F is an isomorphism because of (4.1) and the CY-condition (1.7)&(1.8).(c) Because B is small and B is a C ∗ -bundle on B , also this follows from thefact that the map P : T ( δ ,0 ) B → F is an isomorphism for any ( δ , 0 ) ∈ p − ( ) . (cid:3) Flat structure.
The same situation as in section 4.1 is considered. Now addi-tionally a ∇ -flat subbundle U ⊂ V of rank n + S ( U , U ) = opposite subbundle if F + U = V , equivalent: F + U = F ⊕ U , alsoequivalent: F ∩ U = { zero section } . Because B is small, these conditions arealso equivalent to their restrictions to the zero fiber V . Lemma 4.2. (a) The following three conditions are equivalent:
ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 23 ( i ) U is an opposite subbundle. ( ii ) The composition pr ◦ P : B → V → V / ( U ) is an embedding, so locally anisomorphism. Here pr : V → V / ( U ) is the projection. ( iii ) The composition pr : p ∗ F → p ∗ V → p ∗ V / p ∗ U of embedding and projectionis an isomorphism of vector bundles. Then also pr ◦ P : TB → p ∗ V / p ∗ U isan isomophism of vector bundles.(b) Suppose that (i)–(iii) hold. The vector space structure on V / ( U ) induces a flatstructure on B with flat and torsion free connection ∇ U . The flat connection p ∗ ∇ onp ∗ V induces a flat connection on the quotient bundle p ∗ V / p ∗ U because p ∗ U is a flatsubbundle, and that connection induces via ( pr ◦ P ) ∗ a flat connection ∇ ′ on TB. Then ∇ ′ = ∇ U .Proof. (a) (i) ⇐⇒ (iii) is trivial. The flat connection on p ∗ V / p ∗ U which is inducedfrom p ∗ ∇ on p ∗ V , yields the trivialization τ : p ∗ V / p ∗ U ∼ = → V / ( U ) × B of thevector bundle p ∗ V / p ∗ U . Then τ ◦ ( pr ◦ P ) = ( pr ◦ P ) ∗ (4.5)as maps from TB to ( pr ◦ P ) ∗ T ( V / ( U ) ) = V / ( U ) × B .(ii) is equivalent to two conditions: First, that the restriction of pr ◦ P to p − ( ) ,which is just the map pr ◦ P : F − { } ֒ → F ֒ → V → V / ( U ) ,is an embedding, and second that the differential ( pr ◦ P ) ∗ at points of p − ( ) is an isomorphism. The first condition is equivalent to F ∩ ( U ) = { } which ispart of (i), and the second condition is equivalent to (iii) and thus to (i), because of(4.5).(b) This follows from (4.5). (cid:3) Special coordinates.
The same situation as in 4.1 is considered. The flatstructure ∇ U on B from an opposite subbundle can be enriched by an additionalchoice, which leads to certain flat coordinates, the special coordinates .Now a , ..., a n + , b , ..., b n + are ∇ -flat sections of V which form a symplecticbasis everywhere. Then U = h b , ..., b n + i and V = h a , ..., a n + i are ∇ -flat sub-bundles of rank n + S ( U , U ) = = S ( V , V ) .By abuse of notation we write also a i for p ∗ a i and b i for p ∗ b i . There are uniquefunctions z i , w i ∈ O B , i =
1, ..., n +
1, with σ taut = n + ∑ i = z i · a i + n + ∑ i = w i · b i . (4.6) Lemma 4.3.
The following properties hold true: (a) ε ( z i ) = z i , ε ( w i ) = w i . If z , ..., z n + are coordinates on B, they are ∇ U -flat.Then ε = ∑ n + i = z i ∂ z i . Furthermore,z , ..., z n + are coordinates on B ⇐⇒ U is an opposite subbundle, (4.7) w , ..., w n + are coordinates on B ⇐⇒ V is an opposite subbundle. (4.8) If U is an opposite subbundle then z , ..., z n + are called special coordinates. Ifadditionally V is an opposite subbundle then w , ..., w n + are called adjoint spe-cial coordinates. (b) Suppose that U is an opposite subbundle. Then there is a unique function Ψ U , V ∈ O B with ∂ Ψ U , V ∂ z i = w i for i =
1, ..., n +
1, (4.9)and ε ( Ψ U , V ) = · Ψ U , V . (4.10) It depends only on U and V , not on the symplectic basis. It is called a prepoten-tial. (c) Suppose that U is an opposite subbundle and that V ∈ L ( V , U ) where L ( V , U ) : = (cid:8) V ⊂ V | ∇ -flat subbundle of rank n + S ( V , V ) = V = U ⊕ V (cid:9) . C [ z , ..., z n + ] denotes the polynomials homogeneous of degree 2. Then Ψ U , V ∈ Ψ U , V + C [ z , ..., z n + ] , and the map L ( V , U ) → Ψ U , V + C [ z , ..., z n + ] , V Ψ U , V , is a bijection. (d) The class Ψ U , V + C [ z , ..., z n + ] of prepotentials in (c) is characterized by thethird derivatives XYZ Ψ U , V where X , Y , Z ∈ L n + i = C · ∂ z i are flat vector fields,and these third derivatives are given by − S ( σ taut , ∇ X ∇ Y ∇ Z σ taut ) = XYZ Ψ U , V . (4.11) This is a coordinate free characterization of this class of prepotentials. The class ofprepotentials depends only on the flat structure ∇ U on B.Proof. (a) (4.1) gives ε ( z i ) = z i , ε ( w i ) = w i . The map pr ◦ P from section 4.2 isnow explicitly pr ◦ P : B → V / ( U ) , ( δ , b ) n + ∑ i = z i · [ a i ] . (4.12)It is an embedding iff z , ..., z n + are coordinates on B . Lemma 4.2 applies andgives (4.7).If z , ..., z n + are coordinates, they are ∇ U -flat because of (4.12). In that case ε = ∑ n + i = ε ( z i ) ∂ z i = ∑ n + i = z i ∂ z i . (4.8) is analogous to (4.7).(b) ∇ ∂ zi σ taut = a i + ∑ n + j = ∂ w j ∂ z i · b j is a section in p ∗ F , and S ( F , F ) =
0, so0 = S ( ∇ ∂ zi σ taut , ∇ ∂ zj σ taut ) = ∂ w i ∂ z j S ( a i , b i ) + ∂ w j ∂ z i S ( b j , a j ) = ∂ w i ∂ z j − ∂ w j ∂ z i .There exists a function Ψ ∈ O B with ∂ Ψ ∂ z i = w i . It is unique up to addition of aconstant. It is claimed that there is exactly one function Ψ U , V in this class with ε ( Ψ U , V ) = · Ψ U , V . Obviously there exists at most one such function. For theexistence observe ∂ z i ε ( Ψ ) = [ ∂ z i , ε ]( Ψ ) + ε∂ z i ( Ψ ) = ∂ z i ( Ψ ) + ε ( w i ) = w i .Therefore ε ( Ψ ) is also in the class. Because of ε ( Ψ ) = Ψ + constant , ε ( ε ( Ψ )) = ε ( Ψ ) , so ε ( Ψ ) is the desired function Ψ U , V . ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 25
For the independence of the symplectic basis, consider a symplectic base changewhich fixes U and V , ( a ′ , ..., a ′ n + ) = ( a , ..., a n + ) · A , ( b ′ , ..., b ′ n + ) = ( b , ..., b n + ) · ( A tr ) − with A ∈ GL ( n + C ) .Then ( z ′ , ..., z ′ n + ) = ( z , ..., z n + ) · ( A tr ) − , ( w ′ , ..., w ′ n + ) = ( w , ..., w n + ) · A , ( ∂ z ′ , ..., ∂ z ′ n + ) = ( ∂ z , ..., ∂ z n + ) · A , and thus ( ∂ z ′ , ..., ∂ z ′ n + )( Ψ ) = ( w ′ , ..., w ′ n + ) , so Ψ ′ = Ψ . Therefore Ψ U , V depends only on U and V , not on the symplectic basis.(c) Suppose that a , ... a n + , b , ..., b n + , U = h b , ..., b n + i and V = h a , ..., a n + i are given, with U an opposite subbundle. For any V ′ ∈ L ( V , U ) there are unique a ′ , ..., a ′ n + ∈ V ′ such that a ′ , ..., a ′ n + , b , ..., b n + are a symplectic basis and ( a ′ , ..., a ′ n + ) = ( a , ..., a n + ) + ( b , ..., b n + ) · A .Then A = A tr .The corresponding map { A ∈ M (( n + ) × ( n + ) , C ) | A = A tr } → V is a bijection. This and the following formulas give the claimed 1-1 correspon-dence, σ taut = n + ∑ i = z i a i + n + ∑ i = w i b i = n + ∑ i = z i a ′ i + n + ∑ i = ( w i − n + ∑ j = A ji z j ) b i , w ′ i = w i − n + ∑ j = A ji z j , Ψ U , V ′ = Ψ U , V − ∑ i , j A ij z i z j .(d) Derivation of 0 = S ( σ taut , ∇ ∂ zj ∇ ∂ zk σ taut ) by ∂ z i gives − S ( σ taut , ∇ ∂ zi ∇ ∂ zj ∇ ∂ zk σ taut ) = S ( ∇ ∂ zi σ taut , ∇ ∂ zj ∇ ∂ zk σ taut )= S a i + n + ∑ m = ( ∂ z i ∂ z m Ψ ) · b m , n + ∑ l = ( ∂ z j ∂ z k ∂ z l Ψ ) · b l ! = ∂ z i ∂ z j ∂ z k Ψ .This completes the proof. (cid:3) Data involving the real structure.
Now let ( B , V , ∇ , V R , S , F • ) be a VPHSof weight w = U and a , ..., a n + , b , ..., b n + .For the sake of completeness here we discuss another datum, a connection ∇ psg on T C B which involves the real structure. A third aspect, a hermitian pairingfrom the polarization will not be discussed here. In the following TB = T B , T C B = T B ⊕ T B and T R B will be used. The period map P and the real structure V R induce an extended period map P : T C → p ∗ V , (4.13) P = P : T B → p ∗ F , P = ′′ P ′′ : T B → p ∗ F , X P ( X ) . Lemma 4.4.
For this period map we have: (a) P is an isomorphism of C -vector bundles. It respects the real structures, i.e. itmaps T R B to p ∗ V R . (b) Let ∇ psg be the connection on T C B induced by p ∗ ∇ via P . It is flat and thusgives T C B the structure of a holomorphic vector bundle. Of course, the subbundlesP ∗ ( p ∗ F ) , P ∗ ( p ∗ F ) = T B and P ∗ ( p ∗ F ) of T C B are holomorphic subbundleswith respect to this holomorphic structure. The connection ∇ psg is torsion free.Proof. Part (a) is obvious after lemma 4.1 (c), in part (b) only the torsion freenessof ∇ psg is nontrivial. As it is classical and we will not use it, we leave the proof tothe reader. (cid:3) Remark 4.5.
Let J : T R B → T R B with J = − id give the complex structure on B .The condition that T B ⊂ T C B is a holomorphic subbundle with respect to theholomorphic structure on T C B from ∇ psg is equivalent to (cid:16) ∇ psgX J (cid:17) ( Y ) = (cid:16) ∇ psgY J (cid:17) ( X ) for X , Y ∈ T C B (4.14)[Her03, Lemma 3.6]. The condition (4.14) is often used as defining condition foraffine special geometry. Thus affine special geometry on a manifold M meansthat there is a torsion free and flat connection which together with the (Hodge)decomposition T C M = T M ⊕ T M and the real subbundle T R M yields a vari-ation of Hodge structures of weight 1 on the complex tangent bundle T C M [Her03,Proposition 3.7]. Of course, in the present situation this holds, projective specialgeometry includes affine special geometry. The Hitchin system on the other handexhibits the opposite behaviour: we will show that the natural affine special ge-ometry refines to a projective one.5. C OMPARISON
Let ( B , V , ∇ , S , F • ) be a variation of Hodge like filtrations with pairing of weight3 which satisfies the CY-condition (1.7) & (1.8), with n = dim B . As always, B issupposed to be small, a germ of a manifold at a base point 0 ∈ B .In section 2 & 3 we discussed a manifold M ⊃ B of dimension 2 n + ( U • , λ ) , where U • isan opposite filtration and λ ∈ F − { } .In section 4 we discussed a manifold B of dimension n + C ∗ -bundleon B , and a holomorphic aspect of projective special geometry, a flat structure(and special coordinates) depending on a choice of an opposite subbundle U .Now the constructions and data will be compared.5.1. Choice of U and U . In the first lemma we start with B and a choice of thesubbundles U and U of an opposite filtration U • . In the second lemma U willbe added. ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 27
Lemma 5.1.
Let U and U be flat subbundles of V withU = ( U ) ⊥ S , U = ( U ) ⊥ S , rank U =
1, rank U = n +
1. (5.1)(a)
Then F + U = V ⇐⇒ F + U = V . (b) Suppose that F + U = V . The flat connection on the quotient bundle V / U and the isomorphism F ֒ → V → V / U yield a flat connection on F and atrivialization τ : F → F × B . This restricts to a trivializationP U : B → ( F − { } ) × B (5.2) of the C ∗ -bundle B = F − { zero section } . (c) The additional choice λ ∈ F − { } distinguishes a hypersurface ( P U ) − ( { λ } × B ) ∼ = B in B.Proof. For (a) remark ( F ∩ U ) ⊥ S = ( F ) ⊥ S + U ⊥ S = F + U and F + U = V ⇐⇒ F ∩ U = { zero section } . (b) and (c) are clear. (cid:3) Lemma 5.2.
Let ( B , V , ∇ , S , F • ) be a variation of Hodge like filtrations with pairing ofweight 3 satisfying the CY-condition. (a) The choice of U and U with (5.1) and F + U = V and the choice of an oppositesubbundle U are together just the choice of an opposite filtration U • . (b) Suppose that such a choice is made. Then the hypersurfaces ( P U ) − ( { λ } × B ) ⊂ B, λ ∈ F − { } , are ∇ U -flat hyperplanes of B, and they all induce thesame flat structure on B .Proof. (a) is trivial. (b) By the embedding pr ◦ P : B → V / ( U ) in lemma4.2 (a)(ii), the fibration of B by hyperplanes ( P U ) − ( { λ } × B ) , λ ∈ F − { } , ismapped to the fibration of V / ( U ) by the affine hyperplanes [ λ ] + ( U ) / ( U ) . (cid:3) Flat structures and (pre)potentials.
In the proof of theorem 2.2 (a), the choice ( U • , λ ) with U • an opposite filtration and λ ∈ F − { } led to a Frobenius mani-fold structure on M ⊃ B with potential Φ = Ψ + ... as in (2.19) and Ψ ∈ O B . Theadditional choice of v , ..., v n + lead to flat coordinates t , ..., t n + with t , ..., t n + flat coordinates on B ⊂ M .In lemma 4.3 the choice of an opposite subbundle U and another subbundle V led to a prepotential Ψ U , V ∈ O B and a flat structure on B . The additionalchoice of a symplectic basis a , ..., a n + , b , ..., b n + with U = h b , ..., b n + i and V = h a , ..., a n + i led to flat special coordinates z , ..., z n + on B . These data will becompared now. Theorem 5.3.
Choose ( U • , λ ) as above. Choose v , ..., v n + as in lemma 2.1, withv = λ . Choose a i = v i (i =
1, ..., n + ) and b i = v n + i (i =
2, ..., n + ) andb = − v n + . Then a , ..., a n + , b , ..., b n + are a symplectic basis, and V = F . (a) Then ( P U ) − ( { λ } × B ) = { z = } ⊂ B , (5.3) and B is embedded into B as this hyperplane. The flat structure on B from theFrobenius manifold coincides with the flat structure which B inherits from B bythis embedding. (b) The following equalities hold true:t i = z i | { z = } for i =
2, ..., n +
1, (5.4) Ψ = Ψ U , F | { z = } (5.5)(c) The potential Φ of the Frobenius manifold can be changed by adding any elementof C [ t , ..., t n + ] ≤ (where the index means degree ≤ ) without changing theFrobenius manifold.All the prepotentials in the class Ψ U , F + C [ z , ..., z n + ] from lemma 4.3 (d) give via (5.5) and (2.19) ( Φ = Ψ + ... ) all the Frobenius manifold potentials in the class Φ + C [ t , ..., t n + ] ≤ .Proof. Compare (2.3) and (4.6), v = v + n + ∑ = t i · v i + n + ∑ = ∂ i Ψ · v n + i + (( n + ∑ k = t k ∂ k − ) Ψ ) · v n + , (5.6) σ taut = z · v + n + ∑ i = z i · v i + n + ∑ i = w i · v n + i − w · v n + . (5.7)On the hyperplane { z = } the section σ taut restricts to v , with t i = z i |{ z = } , ∂ i Ψ = w i |{ z = } for i =
2, ..., n + ( n + ∑ k = t k ∂ k − ) Ψ = − w |{ z = } .The equations t i = z i |{ z = } show part (a) and (5.4). The equations ∂ i Ψ = w i |{ z = } for i =
2, ..., n + ∂∂ t i (( Ψ U , F ) |{ z = } ) = ( ∂∂ z i Ψ U , F ) |{ z = } = w i |{ z = } = ∂∂ t i Ψ .This shows ( Ψ U , F ) |{ z = } = Ψ + constant. In order to see that this constant is0, we use ( ∑ n + k = t k ∂ k − ) Ψ = − w |{ z = } and v ( ) = v , which gives the firstequality in the following equations,0 = − (( n + ∑ k = t k ∂ k − ) Ψ )( ) = w ( z = z i = ) ( i =
2, ..., n + )= ∂ Ψ U , F ∂ z ! ( z = z i = )= (cid:16) ε Ψ U , F (cid:17) ( z = z i = )= · Ψ U , F ( z = z i = ) .As Ψ ( ) =
0, this shows (5.5). Part (c) is clear. (cid:3)
Remark 5.4.
The theorem says that the Frobenius manifold structures on M withchoices ( U • , λ ) with fixed U , but varying ( U , U , λ ) have a nice common geo-metric origin. The flat structures on B come from different embeddings of B as ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 29 affine hyperplanes in the flat manifold B . The parts Ψ of the Frobenius manifoldpotentials Φ = Ψ + ... arise via restriction of the same prepotential Ψ U , F .6. H ITCHIN SYSTEMS
The remainder of this paper is devoted to the application of the theory devel-oped thus far to certain integrable systems as constructed in [Hit87]. These areexamples of so-called algebraically completely integrable systems , which in turn areknown to give variations of Hodge structures of weight one on their base space.We will show that this can be refined in a natural way to a variation of Hodge likefiltrations of weight three as described in the first part of the paper, which allowsus to apply the results formulated there. We begin with a brief review of theseintegrable systems.6.1.
The moduli space of Higgs bundles.
Let C be a complex curve of genus g ( C ) ≥
2, and fix a complex reductive group G with Lie algebra g . A principalHiggs bundle is a pair ( P , Φ ) , where P → C is a holomorphic principal G -bundleover C , and Φ –called the Higgs field– is an element of H ( C , ad ( P ) ⊗ K C ) , that is,a holomorphic one-form with values in the adjoint bundle ad ( P ) of P .Recall that a principal G -bundle P is said to be stable if the adjoint bundle isa stable vector bundle, i.e., for every proper subbundle F ⊂ ad ( P ) , we havedeg ( F ) / rk ( F ) < deg ( ad ( P )) / rk ( ad ( P )) . As proved in [Ram75], the modulispace M of stable principal G -bundles is a smooth quasi-projective complex va-riety of dimension dim M = dim G ( g ( C ) − ) + dim Z ( G ) , where Z ( G ) is thecenter of G . Its tangent space is given by T [ P ] M ∼ = H ( C , ad ( P )) ,so by Serre-duality, a Higgs bundle whose underlying principal bundle is stabledetermines a unique point in T ∗ M .The complex manifold X : = T ∗ M forms an open dense subspace of the fullmoduli space of Higgs bundles. As a cotangent bundle, it carries a canonical holo-morphic symplectic form ω can : the tangent space to X at the point [ P , Φ ] fits intoan exact sequence0 → H ( C , ad ( P ) ⊗ K C ) → T [ P , Φ ] X → H ( C , ad ( P )) → hy-perk¨ahler nature of X . We shall not be concerned in this paper with this enrichedstructure except for the existence of a K¨ahler form ω K on X which is of type (
1, 1 ) with respect to the canonical complex structure as a cotangent bundle to a complexmanifold.We will now describe Hitchin’s fibration p : X → ˜ B : = k M i = H ( C , K ⊗ d i C ) ,where k = rank ( g ) . Choose a basis of invariant polynomials p , . . . p k ∈ C [ g ] G ,where p i has degree d i . Each of these p i defines a map p i : H ( C , ad ( P ) ⊗ K C ) → H ( C , K ⊗ d i C ) . Now p is simply induced by the map p ( P , Φ ) : = ∑ ki = p i ( Φ ) . The fundamentaltheorem of Hitchin [Hit87] states that the map p defines an algebraic integrablesystem on X . This means that i ) p is the restriction of a proper holomorphic map to an open dense subspacewhose generic fibers are Lagrangian with respect to the holomorphic sym-plectic form ω can , ii ) the K¨ahler form ω K restricts to each fiber to define a positive polarization.6.2. Cameral curves and abelianization.
Let ∆ ⊂ ˜ B be the discriminant of themap p above and define B : = ˜ B \ ∆ . By Hitchin’s result stated above, the fiber X b : = p − ( b ) ⊂ X is a dense open subset of a compact polarized abelian varietyof dimension dim G ( g ( C ) − ) + dim Z ( G ) for each b ∈ B . It can be identified as ageneralized Prym variety of a branched cover C b of C , called the cameral cover .Fix a maximal torus T ⊆ G with Lie algebra t ⊆ g , a Borel subgroup H of G which contains T , and denote the associated Weyl group by W . By Cheval-ley’s theorem, restriction of polynomials induces an isomorphism C [ g ] G ∼ = C [ t ] W .Consider now the quotient map t → t / W . Twisted with the canonical bun-dle K C this defines a Galois covering t ⊗ K C → ( t ⊗ K C ) / W , and observe that ( t ⊗ K C ) / W ∼ = L ki = K ⊗ d i C . With this, the cameral cover for b ∈ B is defined as C b : = b ∗ ( t ⊗ K C ) ⊂ t ⊗ K C .The projection of the bundle t ⊗ K C to the base C induces a projection π b : C b → C .By construction, this defines a W -Galois covering of C , where the Weyl group actsby the restriction of the action on t . Remark 6.1.
For the classical groups, it is sometimes more convenient to use thesmaller spectral covers which are associated to representations of G , or rather theirhighest weights. Let us explain this for the case G = GL ( n , C ) and the fundamen-tal representation. In this case the underlying moduli space M is of course simplythe moduli space of stable vector bundles of rank n . Let λ ∈ Λ be the weight ofthe fundamental representation of GL ( n , C ) on C n , and denote its stabilizer underthe action of the Weyl group by W λ . The spectral cover is defined as the quotient C b / W λ . Typically however, spectral covers suffer from singularities and it is easierto use the cameral.The abelianization procedure is the following: for any principal G -bundle P over C , the structure group of the pull-back π ∗ b P has a canonical reduction to H .The T -bundle associated to the projection H → T may not be W -invariant, butchoosing a theta-divisor on C gives a canonical twist to a W -invariant T -bundle[Sco98]. With this one proves: Theorem 6.2 (Abelianization, see [Don93, Fal93, Hit87, Sco98]) . i ) Locally around a point ( P , Φ ) ∈ X b , the moduli space of Higgs bundles X is iso-morphic to the moduli space of pairs ( ˜ C , ˜ P ) , where ˜ C is a W-invariant deformationof the cameral cover C b , and ˜ P is a W-invariant T-bundle over it.ii ) With this isomorphism, the projection ( ˜ C , ˜ P ) → ˜ C defines a Lagrangian foliationof an open subset of X . Weyl group invariant infinitesimal deformations of C b in t ⊗ K C are given byelements in H ( C b , N C b ) W , where N C b → C b is the normal bundle to C b ֒ → t ⊗ K C . ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 31
The symplectic form on K C defines an isomorphism N C b ∼ = t ⊗ K C b so that ii ) abovegives the exact sequence0 → H ( C b , t ⊗ O C b ) W → T ( P , Φ ) X → H ( C b , t ⊗ K C b ) W →
0. (6.8)In view of the Hitchin map this gives an identification T b B ∼ = H ( C b , t ⊗ K C b ) W .7. T HE S EIBERG –W ITTEN DIFFERENTIAL
In this section we will define the Seiberg–Witten differential on the cameralcurves associated to the Hitchin system and study its properties. In particular, wewill relate the differential to the C ∗ -action on the moduli space of Higgs bundles.7.1. The C ∗ -action. Let ( P , Φ ) be a Higgs bundle over the curve C with P a stable G -bundle. For ξ ∈ C ∗ , we can scale the Higgs field to ξ Φ to obtain another Higgsbundle and this induces a holomorphic action ϕ ξ ( P , Φ ) : = ( P , ξ Φ ) on the modulispace X . Of course, this is simply the canonical action of C ∗ on the cotangentbundle T ∗ M , from which one immediately deduces that ϕ ∗ ξ ω can = ξω can ,i.e., the canonical symplectic form is conformal with respect to the C ∗ -action. Let E be the generating (holomorphic) vector field of this action, and define the Liouvilleform as α : = ι E ω can . By the conformal property of the symplectic form above wehave Lie E ω can = ω can and therefore d α = ω can .Let b ∈ B and consider the restriction α b : = α | p − ( b ) , a holomorphic one-formon the fiber p − ( b ) . Recall that Hitchin’s result stated in section 6.1 identified thisfiber as a dense open subset of an Abelian variety. Lemma 7.1.
The holomorphic one-form α b is translation invariant.Proof. As above, let ( p , . . . , p k ) denote the components of the Hitchin map p : X → B . Standard symplectic geometry shows that the Hamiltonian vector fields X i of p i for i =
1, . . . , k are tangential to the fibers of p and precisely generate the affinesymmetry the fiber p − ( b ) exhibits as an Abelian variety. Let i b : p − ( b ) ֒ → X bethe canonical inclusion. Then we haveLie X i α b = ( d ι X i + ι X i d ) i ∗ b α = i ∗ b d ι X i ι E ω can + ι X i i ∗ b d α = − i ∗ b d ι E dp i + ι X i i ∗ b ω can = − d i i ∗ b ( dp i )= p : X → B is Lagrangian, i.e., i ∗ b ω can = p i are homogeneous of degree d i . (cid:3) Introduce the following C ∗ -action on the base B of the Hitchin system: ξ · ( b , . . . , b k ) = ( ξ d b , . . . , ξ d k b k ) ,where ξ ∈ C ∗ and b = ( b , . . . , b k ) ∈ B with b i ∈ H ( C , K ⊗ d i C ) . Obviously,equipped with this action, the Hitchin map p : X → B is C ∗ -equivariant. In thefollowing, we denote the generating vector field of this action on B by E . Definition and properties.
A translation invariant one-form on an Abelianvariety determines a unique element in the linear dual of the tangent space at ageneric point. Consulting the short exact sequence (6.8), this means an element in H ( C b , t ⊗ K C b ) W for the case at hand, viz. the fiber X b : = p − ( b ) of the Hitchinmap: Definition 7.2.
The Seiberg–Witten differential on the cameral curve λ SW ∈ H ( C b , t ⊗ K C b ) W is the holomorphic one-form determined by the translation in-variant one-form α b , the restriction of the Liouville form to the fiber X b .There is an alternative definition of this differential as follows: Recall that thecameral curve C b is canonically embedded in the total space of the vector bundle t ⊗ K C . There is a holomorphic action of C ∗ by scaling along the fibers of thisbundle. As a holomorphic cotangent bundle, K C carries a canonical holomorphicsymplectic form. On the tensor product t ⊗ K C , this can be interpreted as an t -valued symplectic form, denoted ω K C . Let ∂ ξ be the generator of the C ∗ -action.Once again, the contraction θ : = ι ∂ ξ ω K C , called the Liouville form, is a potentialfor this symplectic form. Proposition 7.3.
The Seiberg–Witten form is equal to the restriction of the Liouville form: λ SW = θ | C b Proof.
This is a consequence of the abelianization of Higgs bundles as described insection 6.2. Recall that the Hitchin map p : X → B is C ∗ -equivariant, and projectsthe generating vector field E to E . Let ( P , Φ ) ∈ X b . Because the Hitchin mapdefines a Lagrangian fibration, and the Seiberg–Witten differential λ SW is definedby restricting ι E ω can to the fiber X b over b ∈ B , it follows from the exact sequence(6.8) that it is given by λ SW = E ( b ) ∈ T b B = H ( C b , t ⊗ K C b ) W .Let ξ ∈ C ∗ . It is an easy consequence of the definitions that C ξ · b = ξ · { C b } ,where on the right hand side we use the canonical C ∗ -action on K C and the em-bedding C b ֒ → t ⊗ K C b . The generator ∂ ξ of this action therefore defines a W -invariant deformation of C b in t ⊗ K C b which corresponds to E using the isomor-phism H ( C b , N C b ) W ∼ = H ( C b , t ⊗ K C b ) W . As explained below Theorem 6.2, thisisomorphism is induced by contracting with the symplectic form on K C . But theLiouville form is precisely defined as ι ∂ ξ ω K C , so the result now follows. (cid:3) Some of the information about the cameral cover is conveniently encoded in thezero divisor D λ SW of λ SW . The previous proposition clarifies where these zeroesare: using the fact that ω K C is nondegenerate one finds for any vector field v that ι v λ SW ( p ) = p ∈ C b if and only if θ ( p ) = v ( p ) = c · ∂ ξ ( p ) for someconstant c . The first set of points are the intersections of C b with C while the secondset consists of the branch points of the covering map π b : C b → C . We split D λ SW = D int + D br into the intersection and branch points accordingly and calculate theirdegrees, cf. [Ksi01]. The map π b has degree | W | , the order of the Weyl group. Bydefinition, deg ( D int ) = deg ( C b ∩ s ) with s the zero section of K C → C . This isthe same as the intersection degree with any other section s ∈ H ( C , K C ) deg ( D int ) = deg ( C b ∩ s ) = deg ( π b ) deg ( s ) = | W | · | K C | ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 33
We now turn our attention to the branch points. Since the cameral cover is thepull-back via b of the W -Galois cover t → t / W , we are interested in the branchpoints of the latter. If σ α denotes the reflection in the root α , then σ α h = h ⇔ α ( h ) = t t / W has branch points exactly on the zero divisor of the map h → ∏ α α ( h ) . This gives a degree ∆ hypersurface H ⊂ t ⊗ K C , where ∆ denotes thenumber of roots of g . The branch divisor of the cameral cover is the intersectiondivisor of b with H and therefore has degree | ∆ | · | K C | where | K C | = g ( C ) − ( D br ) = | ∆ | · | K C | This is consistent with the Riemann-Hurwitz formula, which in this case reads g ( C b ) = | W | · | K C | + | K C | · | ∆ | + ( D λ SW ) = g ( C b ) − = | K C | ( | W | + | ∆ | ) = deg ( D int ) + deg ( D br ) .The multiplicities of the points in D λ SW may depend on the point in the base B ,but in the generic situation C b ∩ s consists of transversal intersections (giving firstorder zeroes of λ SW ) and the branch points are all of second order (giving secondorder zeroes). From now on, we will assume to be in the generic situation.8. V ARIATIONS OF H ODGE STRUCTURES FROM C AMERAL CURVES
In this section we study a variation of Hodge structures associated to the familyof cameral curves of the Hitchin system. A priori, this is a variation of weight one;in physics terminology the base is a rigid special K¨ahler manifold, in mathemati-cal terms it is called affine special K¨ahler (cf. [Fre99, ACD02, Her03]). However, acareful analysis of the Seiberg–Witten differential in this variation shows that thereexists a canonical refinement to a variation of weight three. In the physics litera-ture this is called a local special K¨ahler manifold, in the mathematics literature onrefers to this situation as projective special K¨ahler.8.1.
The variation of weight one.
We review the variation of Hodge structures ofweight w = B using the setup as in [Del70]. The family of cameral curves f : C → B is defined such that C b : = f − ( b ) ∼ = C b . Recall that C is equippedwith an action of the Weyl group which preserves the fibers of f . Consider nowthe direct image functor of f in the category of W -equivariant sheaves f ∗ : Sh W ( C ) → Sh ( B ) ,which assigns to S ∈ Sh W ( C ) the sheaf U S ( f − ( U )) W .Its derived functors are denoted by R • f ∗ . Let Λ be the root lattice of G and denoteby Λ the associated locally constant sheaf on C equipped with the canonical W -action. Homotopy invariance of cohomology implies that the sheaf of Z -modules V Z : = R f ∗ Λ ∈ Sh ( B ) forms a local system on B whose stalk at b ∈ B equals ( V Z ) b = H ( C b , Λ ) W . Nextwe consider the tensor product V : = V Z ⊗ Z O B ,a coherent sheaf of holomorphic sections of a vector bundle over B . Because Λ ⊗ Z C ∼ = t , its fiber at b ∈ B is given by V b = H ( C b , t ) W . Obviously, the map f isproper and therefore we have isomorphisms V ∼ = R f ∗ ( t ⊗ C f ∗ O B ) ∼ = H (cid:0) f ∗ (cid:0) t ⊗ Ω • C / B (cid:1)(cid:1) Here the relative differentials are defined through the following short exact se-quence of coherent sheaves on C → f ∗ Ω • B → Ω • C → Ω • C / B →
0. (8.9)The middle term carries a natural decreasing filtration via F k = image h f ∗ Ω kB ⊗ O C Ω •− k C → Ω • C i The associated spectral sequence degenerates and leads to a filtration on ( t ⊗ Ω • C / B , d ) , the Hodge filtration. For the case at hand, this filtration has weight one; F ⊂ F = V , with F = f ∗ (cid:16) t ⊗ Ω C / B (cid:17) , i.e., F b = H ( C b , t ⊗ K C b ) W ⊂ H ( C b , t ) W .The differential ∇ : E ∼ = V → E ∼ = f ∗ Ω B ⊗ V is a flat connection on V , called the Gauss-Manin connection, whose flat sectionsare given by V Z ⊗ Z C . Finally, there is a polarization S : V × V → O B given by S b ( α , β ) = h α ∪ β , [ C b ] i , (8.10)where the cup-product includes taking the inner product of two elements in t .Since we work with the first derived functor, it is antisymmetric: S ( α , β ) = − S ( β , α ) . Furthermore, it is ∇ -flat: dS ( α , β ) = S ( ∇ α , β ) + S ( α , ∇ β ) . (8.11)The total of these data ( B , V , ∇ , V Z , S , F • ) define a variation of polarized Hodgestructures of weight w =
1, cf section 1.1.8.2.
The derivative of the Seiberg–Witten differential.
Consider the variation ofpolarized Hodge structures ( B , V , ∇ , V Z , S , F • ) of weight 1 associated to the familyof cameral curves f : C → B constructed in the previous section. By definition, theuniversal curve C comes equipped with an embedding C ֒ → t ⊗ K C × B . Pullingback the t -valued Liouville form θ on t ⊗ K C , one obtains a holomorphic one-form λ on C which restricts to the Seiberg–Witten differential λ SW on each fiber C b .In the following we write λ b for this restriction. By definition of the relative dif-ferential forms, the one-form λ SW defines a section of t ⊗ Ω C which, under theprojection to t ⊗ Ω C / B and the direct image f ∗ , defines a section λ SW ∈ F ⊂ V and restricts to the Seiberg–Witten differential on each fiber: λ b ∈ F b = H ( C b , t ⊗ K C b ) W . ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 35
The ˇCech-de Rham resolution.
To compute the derivative of the Seiberg–Witten differential under the Gauss–Manin connection, we use a ˇCech-resolutionof the relative de Rham complex ( Ω • C / B , d ) and calculate the hypercohomologyfollowing [Del70]. Define U = { x ∈ C , d π f ( x ) = } ,i.e., the complement of the branch points of the cameral cover or equivalentlythe complement of the second order zeroes of the Seiberg–Witten differential.We choose V ⊂ C such that V ∩ C b consists of a disjoint union of small disks V , . . . , V | D br | around the second order zeroes p , . . . , p | D br | ∈ C b of λ b . Here | D br | = | ∆ || K C | denotes the number of branch points, i.e., second order zeroesof λ SW . For any W -equivariant sheaf S ∈ Sh W ( C ) , write f U ∗ S ∈ Sh ( B ) short forthe composition f ∗ ( i U ) ∗ S | U , where i U : U ֒ → C is the inclusion, and similarly for f V ∗ and f U ∩ V ∗ .To compute R f ∗ we need the following part of the double complex of coherentsheaves on B : f ∗ Ω C / B ( C ) / / f U ∗ Ω C / B ⊕ f V ∗ Ω C / B ( L XU , L XV ) (cid:10) (cid:10) δ / / ι XV − XU ( ( QQQQQQQQQQQQ f U ∩ V ∗ Ω C / B f U ∗ Ω C / B ⊕ f V ∗ Ω C / B δ / / d C / B O O f U ∩ V ∗ Ω C / Bd C / B O O L XU (cid:5) (cid:5) (8.12)The vertical map d C / B is the relative de Rham differential and δ denotes the ˇCechdifferential. The notation X U , X V will be explained below. With this resolution,elements in R f ∗ will be represented as cocycles in (cid:16) Ω C / B ( U ) ⊕ Ω C / B ( V ) (cid:17)| {z } ( I ) ⊕ Ω C / B ( U ∩ V ) | {z } ( II ) (8.13)so a relative differential α is represented by a triple ( α U , α V , g α ) satisfying d C / B g α = δ ( α V , α U ) In terms of this complex, the Hodge filtration is given by ( ) and the polariza-tion S is given by a trace-residue pairing S b ( α , β ) = | D br | ∑ k = Res V k (cid:10) g α dg β (cid:11) (8.14)where h . . . i indicates the use of a pairing on t .We now describe the Gauss-Manin connection. Over U and V , one can choosesplittings of the exact sequence of sheaves0 → f ∗ Θ C / B → f ∗ Θ C → Θ B →
0. (8.15)
This provides lifts X U , X V of holomorphic vector fields X on B . Conversely, suchlifts define a splitting. The Gauss-Manin connection now has an explicit descrip-tion: ∇ X ( α U , α V , g α ) = (cid:0) L X U α U , L X V α V , L X U g α + ι X U − X V α V (cid:1) (8.16)It is well-known that the part of the Gauss–Manin connection that actually shiftsdegree in the Hodge filtration, i.e., the Higgs field C X : F → V / F ,equals taking the cup-product with the Kodaira–Spencer class: C X ( α ) = α ∪ κ ( X ) .Here α ∈ H ( C b , t ⊗ K C b ) W , κ : T b B → H ( C b , Θ C b ) is the Kodaira–Spencer mapand the notation stands short for the natural pairing H ( C b , t ⊗ K C b ) W × H ( C b , Θ C b ) → H ( C b , t ⊗ O C b ) W .In the relative ˇCech–de Rham complex, if α is represented by ( α U , α V , 0 ) then C X is given by the interior product ι X U − X V α V , which maps ( I ) to ( II ) in (8.13).8.2.2. Derivatives of λ SW . We now have the machinery to start the computation:
Lemma 8.1. ∇ X λ SW ∈ F b for all X ∈ T b B.Proof.
We have to show that the composition C X : F ∇ X −→ V −→ V / F ,applied to λ SW , is zero. Since λ is naturally defined on C , λ SW can be representedas a differential on C by ( λ | U , λ | V , 0 ) . One finds on U ∩ V that ι X U − X V λ | V = δ (cid:0) ι X U λ | U , ι X V λ | V (cid:1) .Since this is exact, it follows that C X λ SW = (cid:3) Recall, cf. (6.8), that T b B ∼ = H ( C b , t ⊗ K C b ) W . For X ∈ T b B , we write α X for theholomorphic differential associated to X by this isomorphism. Proposition 8.2.
For all X ∈ T b B, we have ∇ X λ SW = α X . Proof.
We have already seen in the previous Lemma that the part of ∇ X λ whichmaps from ( I ) to ( II ) in the ˇCech–de Rham complex (8.13), is exact. The remain-ing part, mapping ( I ) to ( I ) , is given by taking the Lie derivatives L X U , L X V withrespect to holomorphic lifts of X to U and V . By Cartan’s formula L X U λ | U = ( d ι X U + ι X U d ) λ | U .From this we see that (cid:0) L X U λ U , L X V λ V (cid:1) − d (cid:0) ι X U λ U , ι X V λ V (cid:1) = (cid:0) ι X U ω K C , ι X V ω K C (cid:1) ,where we have used that ι X U d λ | U = ι X U ω K C | U ∈ f U ∗ Ω C / B . Recall that the secondterm on the left hand side is exactly the derivative of the cocycle needed in theproof of the previous Lemma to make C X λ SW equal to zero. We now claim that on U ∩ V we have ι X U ω K C − ι X V ω K C = f U ∩ V ∗ Ω C / B . Indeed, the difference X U − X V is a section of ker f ∗ ⊂ Θ C andtherefore tangent to each fiber C b of f : C → B . But ω K C is a ( t -valued) symplectic ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 37 form, so ι X U − X V ω K C = ι X U ω K C | C and ι X V ω K C | C are the restrictions of an element of f ∗ Ω C / B which is by definition α X . (cid:3) As a corollary one finds the following rather obvious fact:
Corollary 8.3.
For the generator E of the C ∗ -action on B, we have: ∇ E λ SW = λ SW .8.3. The variation of weight three.
Consider the variation of Hodge structuresof weight one constructed in section 8.1. With the results of the previous section,we can now refine the filtration to a obtain a variation of Hodge like filtrations ofweight 3: introduce F : = O B · λ SW F : = R f ∗ Ω C / B F : = (cid:16) F (cid:17) ⊥ S F : = V ,and note that F = F . We introduce the projectivization P ( B ) with respect to the C ∗ -action and obtain Theorem 8.4.
The data ( P ( B ) , V , ∇ , V Z , S , F • ) define a variation of Hodge like filtra-tions of weight 3 satisfying the CY-condition.Proof. Clearly, F • b defines a decreasing filtration of weight 3 on the fiber V b over b ∈ B . Therefore, the only thing left to check is that the filtration satisfies Griffithstransversality with respect to the Gauss–Manin connection, i.e., ∇F • ⊆ F •− .In degree 3, this property is equivalent to Lemma 8.1. In degree 2, let α ∈ f ∗ Ω C / B and compute S b ( ∇ X α , λ SW ) = Z C b h∇ X α ∧ λ SW i = Z C b h α ∧ ∇ X λ SW i − d Z C X ( b ) h α ∧ λ SW i ! = λ SW , as well as its derivatives ∇ X λ SW are holomorphic differentials.Here h . . . i indicates that the pairing on t has been used. Since F is defined as thesymplectic complement of λ SW , this proves that ∇F ⊆ F . This completes theproof of Griffiths transversality.Finally, the CY-condition says that ∇F should generate F . But this is clearlyimplied by Proposition 8.2. (cid:3) Remark 8.5.
The polarization S has the wrong signature for a full VPHS of weight3. Since this signature is not used in sections 4 & 5 we can endow the base of theHitchin system with a projective special (K¨ahler) geometry and apply the resultsstated there. The derivative of the period map.
We give two expressions for the derivativeof the period map corresponding to the family of cameral covers f : C → B . Oneof them (theorem 8.6) is inspired by the fact that the variation of Hodge structureof weight 1 can be refined in a natural way to a variation of Hodge structure ofweight 3, which is reminiscent of a family of Calabi-Yau threefolds. The other ex-pression (theorem 8.8) is a residue formula originally due to Balduzzi [Bal06], whogeneralized a formula of Pantev. Similar formulas are known for matrix models,see e.g. [Kri94].Given a base curve and a complex reductive group, consider the family of cam-eral curves f : C → B with central fiber C b . Associated to this family is a periodmap cf. (3.11) Π : B → ˇ D lag which is given by the embedding F ⊂ V b composed with parallel transport usingthe Gauss-Manin connection. Recall from (6.8) that this is a lagrangian embeddingwith respect to the natural symplectic pairing on V b . We are interested in thederivative of the period map d Π b : ( T B ) b → Hom (cid:16) F b , (cid:16) F b (cid:17) ∗ (cid:17) In terms of the Kodaira-Spencer map and the Gauss-Manin connection a theoremof Griffiths gives d Π b ( X ) ( α , β ) = S b ( α , C X β ) Using the natural isomorphism ( T B ) b ∼ = F b given by X → ∇ X λ b the derivative d Π becomes a tensor on B : d Π b : ( T B ) b → (( T ∗ B ) b ) ⊗ which is given by d Π b ( X )( Y , Z ) = S b ( ∇ Y λ b , ∇ X ∇ Z λ b ) (8.17)Integration by parts combined with the ∇ -flatness of S b shows that (this is one ofRiemann’s bilinear relations) d Π b : ( T B ) b → Sym ( T ∗ B ) b It is well-known [DM96] that integrable systems give special period maps in thesense that d Π is a cubic d Π ∈ H (cid:16) B , Sym ( T ∗ B ) (cid:17) In the case of the Hitchin system, we can use the variation of weight 3 given in theprevious section together with flatness of ∇ to conclude that d Π ( X , Y , Z ) is indeedsymmetric in its first and last arguments: d Π b ( X , Y , Z ) − d Π b ( Z , Y , X ) = S b (cid:16) ∇ Y λ b , ∇ [ X , Z ] λ b (cid:17) = d Π which is reminiscent of a family of Calabi-Yauthreefolds, with λ b playing the role of the holomorphic three-form. Theorem 8.6.
The derivative of the period map is given by (compare with (4.11) )d Π b ( X , Y , Z ) = − Z C b h λ b ∧ ∇ X ∇ Y ∇ Z λ b i Proof.
Use integration by parts with respect to Z in (8.17), the ∇ -flatness of S b andthe symmetry in X , Y , Z . (cid:3) ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 39
Remark 8.7.
In [DDP07] a family of noncompact CY-threefolds was constructedin the case of
ADE groups whose variation of mixed Hodge structure of weight 3turns out to be pure, and in fact a Tate twist of a variation of Hodge structure ofweight 1, which is compatible with the fact that S defines an indefinite polarizationin weight 3. The authors of [DDP07] have shown that the Yukawa cubic of thisfamily of threefolds corresponds to the cubic above.The expression in theorem 8.6 is not manifestly symmetric in its arguments.There is another, more symmetric, formula due to Balduzzi [Bal06] who general-ized a result for G = SL by Pantev. We will give a different derivation of his resulthere, which uses the ˇCech-de Rham complex as described in section 8.2. We willchoose coordinates on U ∩ V suggested by the cameral cover π : C b → C : one canpull back an affine coordinate on C via π to serve as a local coordinate z U on U .For a generic point b ∈ B the cover has second order branch points, which we willview as maps p : B r / / / / V z U / / / / C Given the branch point p , a suitable holomorphic coordinate on the component of V containing r ( b ) is given by z V = q z U − p ( b ) .The Seiberg-Witten differential has a second order zero at each of the branch pointsand can be represented in the ˇCech-de Rham complex by λ b = (cid:16) f z V dz V | U , f z V dz V | V , 0 (cid:17) where f is a t -valued holomorphic function on V with f ◦ r ( b ) =
0. The horizontallifts X U , X V of a vector field X on B are determined by the chosen coordinates via X U ( z U ) = X V ( z V ) = d Π coming from the componentof V containing p . A straightforward computation using (8.16) and the fact that ι X U dz V = − L X ( p )( b ) z V now gives ∇ Y λ b = (cid:18) ∗ , ∗ , − z V f L Y ( p )( b ) (cid:19) which has a first order zero at the branch point. The first two terms will not con-tribute to (8.17), so we omit them here. Acting with ∇ X gives ∇ X ∇ Y λ b = (cid:18) ∗ , ∗ , ∗ − L X U (cid:18) z V f L Y ( p )( b ) (cid:19)(cid:19) = (cid:18) ∗ , ∗ , ∗ + f z V L X ( p )( b ) L Y ( p )( b ) (cid:19) Only the term containing a pole at the branch point is displayed and terms whichare irrelevant for (8.17) are omitted. Using (8.14), we now arrive at the followingresult d Π ( X , Y , Z ) = ∑ p ∈ D br L X ( p ) L Y ( p ) L Z ( p ) h f ◦ r , f ◦ r i where h ., . i denotes the pairing between two elements of t . For semi-simple Liegroups there is only one Weyl-invariant pairing up to a scalar, which is the Killingform. It gives an isomorphism t ∼ = t ∗ and the pairing can be expressed in terms ofthe root system R as h h , h i = ∑ α ∈ R h h , α i h α , h i (8.18)The quadratic residue Res p of a quadratic differential at a point p is defined as thecoefficient of z − dz ⊗ dz in a Laurent expansion in terms of a coordinate z centeredat p , and is independent of z . Theorem 8.8 (Balduzzi) . For semi-simple groupsd Π b ( X , Y , Z ) = ∑ p ∈ D br ∑ α ∈ R Res p (cid:20) h α , ∇ X λ b i ⊗ h α , ∇ Y λ b i ⊗ h α , ∇ Z λ b ih α , λ b i (cid:21) (8.19) Proof.
The quotient of two holomorphic differentials is a meromorphic function,so the term in brackets is a meromorphic quadratic differential. From the compu-tation of the Gauss-Manin derivatives in the ˇCech-de Rham complex given aboveone finds that the derivatives of λ have an expansion around the branch points interms of z V ∇ X λ = (cid:20) L X ( p ) f + O ( ) (cid:21) dz V Similarly h α , ∇ X λ ih α , λ i = z V (cid:20) L X ( p ) h α , f ih α , f i + O ( ) (cid:21) = z V [ L X ( p ) + O ( )] Taking the quadratic residue and using (8.18) directly gives the desired result. (cid:3)
Remark 8.9.
Replacing the root system by an orthonormal basis for the dual pair-ing gives an analogous expression for d Π in the case of reductive non-semisimplegroups. 9. T HE F ROBENIUS MANIFOLD
The results of the previous section show that the Hitchin system gives rise toprojective special geometry as in section 4 on P ( B ) . We illustrate in this case thechoices necessary to define a Frobenius manifold structure: a natural generator λ ∈ F is provided by the Seiberg-Witten differential, and a choice of oppositefiltration U • is described geometrically in terms of a choice of cycles on the cameralcurve.9.1. The opposite filtration.
Recall the discussion of opposite filtrations in section1.2. There is a natural procedure to define an opposite filtration on V , viewedas a VHS of weight 3, as follows: fix b ∈ B , and consider ( V Z ) ∗ b = H ( C b , Λ ) W .Combining the inner product on Λ with the intersection form on H ( C b , Z ) definesa symplectic form I , the dual of S , on the lattice H ( C b , Λ ) W : I ( c , c ) : = h c · c i ,for c , c ∈ H ( C b , Λ ) W . ROBENIUS MANIFOLDS, PROJECTIVE SPECIAL GEOMETRY AND HITCHIN SYSTEMS 41
Now we choose a lagrangian subspace L ⊂ H ( C b , Λ ) W and a one-dimensionalsubspace L ⊂ L of it, subject to the condition L ker λ b (9.20)We will also need the complement L = (cid:0) L (cid:1) ⊥I of L with respect to I . With thiswe define ( U ) b : = { v ∈ V b , L ⊂ ker v } ( U ) b : = { v ∈ V b , L ⊂ ker v } ( U ) b : = { v ∈ V b , L ⊂ ker v } ( U ) b : = V b We extend these subspaces by parallel transport to ∇ -flat subbundles U • in a smallneighbourhood of b ∈ B . Proposition 9.1. U • defines an opposite filtration for the variation of Hodge-like filtra-tions on V .Proof. By construction, the subbundles U • are ∇ -flat. Next, let us check that V b = F pb ⊕ (cid:0) U p − (cid:1) b ,for all p =
0, . . . , 3. For this, first observe that rk ( V ) = rk ( F p ) + rk ( U p − ) , so wejust have to verify that F pb ∩ ( U p − ) b = { } . Since F b is spanned by λ b , this followsfor p = p =
2, we have that α ∈ F ∩ U impliesthat α ∈ H ( C b , t ⊗ K C b ) W , i.e., α is a holomorphic differential, and L ⊂ ker α . ByAbel’s theorem, this implies that α has to be zero. An element α ∈ F ∩ U satisfiesby definition α ∈ (cid:16) F (cid:17) ⊥ S & (cid:16) L (cid:17) ⊥I ⊂ ker α But the fact that S and I are dual implies that (9.20) is equivalent to (cid:16) L (cid:17) ⊥I ker (cid:16) λ ⊥ Sb (cid:17) Finally, the condition that S ( U p , U − p ) = I is the dualof S and I (cid:0) L p , L − p (cid:1) = (cid:3) Special coordinates and the prepotential.
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