Quantum cohomology of twistor spaces and their Lagrangian submanifolds
aa r X i v : . [ m a t h . S G ] S e p Accepted for publication in Journal of Differential Geometry.
QUANTUM COHOMOLOGY OF TWISTOR SPACES ANDTHEIR LAGRANGIAN SUBMANIFOLDS
JONATHAN DAVID EVANS
Abstract.
We compute the classical and quantum cohomology rings ofthe twistor spaces of 6-dimensional hyperbolic manifolds and the eigen-values of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov,a monotone Lagrangian submanifold of the twistor space. In the case ofa 3-dimensional totally geodesic submanifold of a hyperbolic 6-manifoldwe compute the obstruction term m in the Fukaya-Floer A ∞ -algebraof a Reznikov Lagrangian and calculate the Lagrangian quantum ho-mology. There is a well-known correspondence between the possiblevalues of m for a Lagrangian with nonvanishing Lagrangian quantumhomology and eigenvalues for the action of c on quantum cohomologyby quantum cup product. Reznikov’s Lagrangians account for most ofthese eigenvalues but there are four exotic eigenvalues we cannot accountfor. Contents
1. Introduction 12. The homogeneous space SO (2 n ) /U ( n )
53. Twistor spaces 64. Reznikov Lagrangians 105. Topological preliminaries 126. Topological computations 167. Definition of Gromov-Witten invariants 218. The linearised theory 229. The Gromov-Witten theory of twistor lines 2510. Higher degree curves 2911. Proof of Theorem 10 3212. Floer theory of Reznikov Lagrangians 38References 421.
Introduction
The twistor space Z of a Riemannian n -manifold M is the total spaceof the bundle of orthogonal complex structures on the tangent spaces of M . Reznikov [24] wrote down a natural closed 2-form ω rez on twistor space JONATHAN DAVID EVANS and observed that if the curvature of M satisfies a certain inequality thenthis 2-form is actually symplectic. He also demonstrated that above anytotally geodesic submanifold of the middle dimension in M there is an SO ( n ) -subbundle of the twistor space on which ω rez vanishes. We will call theseReznikov Lagrangians. For instance, when M is the round 4-sphere thetwistor space is the standard symplectic CP , an equatorial geodesic 2-spherelifts to a Lagrangian RP and an equatorial geodesic torus lifts to the Cliffordtorus.An interesting class of manifolds for which the Reznikov curvature in-equality holds are the hyperbolic n -manifolds (that is compact quotientsof hyperbolic n -space by a discrete torsionfree subgroup of SO + (2 n, ).These give twistor spaces which are of a very different character from thatof the round n -sphere. For instance they are non-Kähler (when n > )by dint of their fundamental group being hyperbolic (see [5]). Nonethe-less, as discovered in [11] when n ≥ they are monotone, meaning thatthe first Chern class is positively proportional to the cohomology class of theReznikov form; moreover Reznikov Lagrangians are monotone, meaning thatthe Maslov index of a bounding disc is positively proportional to its sym-plectic area. Monotone Lagrangians in monotone manifolds are amenable tomodern pseudoholomorphic techniques without appeal to the massive ma-chines under development to deal with the general case. What is even betteris that there is a natural almost complex structure J − , first discovered byEells and Salamon [8], which is compatible with ω rez in this very special hy-perbolic setting. The J − -holomorphic curves are in a close correspondencewith branched minimal surfaces in M by projection along the twistor fibra-tion. This allows us to write down all the genus 0 holomorphic curves (see[11], Lemma 37) and all the discs with boundary on Reznikov Lagrangiansand we have a hope of computing respectively the quantum cohomology andLagrangian intersection Floer theory. Upon noticing this property of theseLagrangian submanifolds one feels like a fortunate astronomer who espies acharming and unusual galaxy perfectly angled so one can see the glory ofits disc full on. Reznikov Lagrangians in the twistor space of a hyperbolicmanifold are topologically much more complicated than the conventional ex-amples of monotone Lagrangians: they are the principal frame bundles ofhyperbolic n -manifolds.If n = 2 then the (6-dimensional) twistor space has c = 0 and the La-grangians are Maslov zero. This case is less amenable to simplistic techniquesdue to problems arising from transversality for multiple covers of Chern zerospheres and Maslov zero discs. Though the former are unlikely to cause ma-jor headaches I decided it would cloud the exposition and therefore I haverestricted computation to the simplest case, n = 3 . Theorem A.
The small quantum cohomology of the twistor space of a hy-perbolic 6-manifold M with vanishing Stiefel-Whitney classes is QH ∗ ( Z ; Λ) ∼ = H ∗ ( M ; Λ)[ α ] / ( α = 8 ατ ∗ χ + 8 qα − q ) UANTUM COHOMOLOGY OF TWISTOR SPACES 3 where α = c ( Z ) and Λ = C [ q ] . Moreover, c ( H ) = α − q , c ( H ) = α − αq . The twistor space is also uniruled.Proof. This follows directly from the classical cohomology ring computationin Section 6.1, the computation of the 3-point Gromov-Witten contributionfrom twistor lines in Corollary 5 and Theorem 9 which proves there areno other quantum corrections. Uniruledness follows from the first part ofCorollary 5. (cid:3)
The theorem probably holds with the assumption on Stiefel-Whitney classesreplaced just by orientability, but this assumption allows us to represent var-ious homology classes in the twistor spaces explicitly as submanifolds (seeSection 6.4) which simplifies the argument. Note that there is very littleloss of generality by making this assumption: by [17, Corollary 2], any com-pact hyperbolic manifold admits a finite cover whose Stiefel-Whitney classesvanish. The use of complex coefficients is needed: the cohomology of thetwistor space is additively isomorphic to the tensor product of the fibre andbase with complex coefficients by the Leray-Hirsch theorem, but the obviouscharacteristic classes do not generate the integral cohomology of the fibre.We note the following interesting corollary.
Corollary B.
The action of c ( Z ) on QH ∗ ( Z ; Λ) by quantum cup productis given with respect to the basis τ ∗ y, ατ ∗ y, α τ ∗ y, α τ ∗ y (where y runs overa basis for H ∗ ( M ; C ) ) by the matrix − q τ ∗ χ q i.e. when y has positive degree this acts as the matrix with no τ ∗ χ entry,when y = 1 this acts as the above matrix where τ ∗ χ is replaced by the number χ ( M ) . The characteristic polynomial of this action is (cid:0) λ − qλ − χ ( M ) λ + 16 q (cid:1) · (cid:0) λ − qλ + 16 q (cid:1) D − where D = dim C H ∗ ( M ; C ) . The eigenvalues associated to the second factorare ± √ q each with multiplicity D − . The eigenvalues associated to the first factorcan be quite complicated. To see the relevance of this corollary we recall some Floer theory. The book[12] explains how to associate to an arbitrary Lagrangian submanifold of asymplectic manifold a filtered A ∞ -structure on a suitable space of Q -chains.Since the Reznikov Lagrangians are monotone when n ≥ this theory sim-plifies considerably (see [2]). When n = 3 the Reznikov Lagrangians boundholomorphic discs with Maslov index 2 and hence there could be a nontrivial“obstruction” term (the m operation in the filtered A ∞ -algebra). JONATHAN DAVID EVANS
Theorem C. If Σ is an oriented totally geodesic submanifold of an orientedhyperbolic 6-manifold M and L Σ denotes the Reznikov Lagrangian lift in thetwistor space of M then m = ± √ q [ L Σ ] Moreover HF ( L Σ , L Σ ) = H ∗ ( L ; C [ q / ]) . It is well-known (see [1], Proposition 6.8) that the possible values for m on a monotone Lagrangian with nonvanishing self-Floer homology are theeigenvalues for the action of c ( Z ) by multiplication on the small quantumcohomology. Indeed, the Fukaya category splits into summands indexedby these eigenvalues. It would be intriguing to find (or to rule out theexistence of) monotone Lagrangians in the twistor space of a hyperbolic 6-manifold whose m equals one of the four “exotic” eigenvalues from CorollaryB involving the Euler characteristic of M . Remark 1.
It may seem from a cursory reading of the paper that we do notmake much use of the fact that M is hyperbolic rather than just negatively-curved, but the computations with the linearised ∂ -operator assume that thenatural metric and Eells-Salamon almost complex structure are an almostKähler pair which happens precisely when M is 4-dimensional and Einsteinself-dual or else higher-dimensional and hyperbolic. Although we have notused this, it is interesting to note that the twistor spaces of hyperbolic n -manifolds are some of the very few known non-Kähler examples of Ricci-Hermitian almost Kähler manifolds, that is to say the Ricci curvature formis a (1 , -form. These metric occur as critical points of the Nijenhuis energyon the space of ω rez -compatible almost complex structures. Outline of the paper.
We begin in Sections 2-4 by reviewing thoseaspects of the geometry and topology of twistor spaces and Reznikov La-grangians which will be of use later in the paper. Section 5 explains someof the (classical) topological tools we will use to compute both the classicaland quantum cohomology rings of twistor spaces which are then applied inSection 6 to compute the classical cohomology ring of the twistor space ofa hyperbolic 6-manifold and of the moduli space of ‘twistor lines’, the J − -holomorphic curves of lowest degree. We also explain how to push forwardclasses from the moduli space of marked twistor lines into the twistor space.The main tool is Borel-Hirzebruch theory for performing fibre integrals ofcharacteristic classes along maps between classifying spaces.In Section 7 we briefly recall the definition of Gromov-Witten invari-ants. Section 8 is dedicated to the study of the linearised ∂ -equation for J − -holomorphic curves in twistor space and the crucial result is that wecan construct elements of the cokernel bundle explicitly out of vector fieldson M . This is used in Section 9 to compute the k -point Gromov-Wittencontributions from the moduli space of twistor lines: one can compute theGromov-Witten invariant by taking the Euler class of an obstruction bundle UANTUM COHOMOLOGY OF TWISTOR SPACES 5 and pushing forward along the evaluation map. Section 10 calculates the re-maining Gromov-Witten contributions needed for calculating the quantumcohomology in the case n = 3 (when M is a hyperbolic 6-manifold). The ideais once again that there is a nonvanishing section of the obstruction bundle,but care must be taken because the moduli space is no longer compact. Anexplanation of the main technical result is postponed to Section 11.In Section 12 we prove Theorem C using similar techniques.1.2. Acknowledgements.
It is my pleasure to acknowledge that this paperbenefitted greatly from helpful conversations with Paul Biran, Joel Fine,Dusa McDuff, Jarek Kędra, Dmitri Panov (who long ago explained to mehis own argument with Joel Fine for why these spaces should be uniruled),Dietmar Salamon and Ivan Smith. Like many symplectic geometers, I firstencountered these spaces in the paper [11]. During this work I was supportedby an ETH Postdoctoral Fellowship.2.
The homogeneous space SO (2 n ) /U ( n ) The homogeneous space F := SO (2 n ) /U ( n ) parametrises orthogonal com-plex structures on R n equipped with the Euclidean metric and an orienta-tion, i.e. SO (2 n ) /U ( n ) = { ψ ∈ GL + ( R n ) | ψ = − , ψ T = − ψ } It comes equipped with a natural almost complex structure j F defined asfollows. The tangent space T ψ SO (2 n ) /U ( n ) can be translated to a subspace π ψ passing through the origin in End( R n ) and ψ acts by left multiplicationon End( R n ) preserving π ψ . In terms of coordinates ( x , . . . , x n ) ∈ R n , theresult of applying j F to a tangent vector v kℓ ∈ T ψ F ∼ = End( R n ) is [ j F ( v )] jℓ = ψ jk v kℓ This is an integrable left-invariant almost complex structure and it is (tau-tologically) compatible with the left-invariant metric g F on SO (2 n ) /U ( n ) in-duced by the Euclidean metric on R n . The corresponding 2-form ω F ( · , j F · ) = g F ( · , · ) is symplectic so we have a natural Kähler triple ( g F , j F , ω F ) .The exceptional isomorphisms in low dimensions give us SO (4) /U (2) ∼ = CP , SO (6) /U (3) ∼ = CP In general the Z -cohomology ring is ([21], Theorem 6.11) Z [ e , e , . . . , e n − ] / { e k + k − X i =1 e i e k − i = 0 } k ≥ The tautological U ( n ) -bundle has Chern classes c i = 2 e i .In particular H ( SO (2 n ) /U ( n ); Z ) = Z ; an explicit generator is given bythe subspace of complex structures preserving a given 4-plane and fixed onthe orthogonal complement, namely SO (4) × U ( n − / ( U (2) × U ( n − ∼ = SO (4) /U (2) JONATHAN DAVID EVANS
In the case n = 3 (when F = CP ) this corresponds to a line. For any n , allholomorphic curves of degree one have this form and we will call them lines by analogy. The space of lines L ( F ) is identified with the Grassmannian SO (2 n ) /SO (4) × U ( n − and the space of lines L ( F ) with a marked point is SO (2 n ) /U (2) × U ( n − Again by analogy we will write H = e ∈ H ( F ) , thinking of it as a hyper-plane class. 3. Twistor spaces
Setting.
The twistor space Z of an oriented n -dimensional Riemann-ian manifold ( M, g ) is the total space of the twistor bundle of g -orthogonalcomplex structures on the tangent spaces of M , F −−−−→ Z y τ M with fibre F p = τ − ( p ) = { J ∈ GL + ( T p M ) | J = − , J ∗ = − J } . The fibrecan be identified with the homogeneous space SO (2 n ) /U ( n ) . Remark 2.
We will be concerned with the twistor spaces of compact, closedoriented hyperbolic n -manifolds, Γ \ SO + (2 n, /SO (2 n ) where Γ ⊂ SO + (2 n, is a cocompact discrete torsionfree subgroup. In thiscase we can write Z globally (see [11] , Section 2.3.3) as Γ \ SO + (2 n, /U ( n ) The twistor bundle inherits a connection ∇ from the Levi-Civita connec-tion of g . We will write V ⊕ H for the vertical-horizontal splitting of thisconnection and use this to define some extra geometric structure on Z . Firstof all we can define a metric using τ ∗ g on the horizontal spaces and g F onthe vertical spaces. We write this g Z = g F ⊕ τ ∗ g We define almost complex structures on T ψ Z for ψ ∈ τ − ( p ) by J ± = ( ± j F ) ⊕ τ ∗ ψ (recall that ψ is a complex structure on T p M ). • The
Atiyah-Hitchin-Singer almost complex structure J + is sometimesintegrable (if and only if either n ≥ and g is conformally flat or n = 4 and g is self-dual), • The
Eells-Salamon almost complex structure J − is never integrable. UANTUM COHOMOLOGY OF TWISTOR SPACES 7
We will only be interested in J − because of the close relationship between J − -holomorphic curves and minimal surfaces (see Section 3.2). Using g Z and J ± one can define compatible nondegenerate 2-forms ω ± ω ± = ( ± ω F ) ⊕ ( τ ∗ ω ψ ) , ω ψ ( · , ψ · ) = g ( · , · ) Reznikov observed that the 2-form ω rez = ( − ω F ) ⊕ − ˆ R ( ω ψ ) is closed (where ˆ R is the Riemann curvature acting on 2-forms). We observethat if ˆ R = ± id then ∓ ω rez = ω ± and hence is a ∓ J ± -compatible symplecticform. Note that conditions for J − to tame ω rez are given in ([10], Section4.2). Remark 3.
We will work with hyperbolic manifolds, for which ˆ R = − id so J − is an ω rez -compatible almost complex structure on the twistor space. Thestructure J + is integrable but there is no compatible symplectic form: thetwistor space of a hyperbolic n -manifold M cannot be Kähler for n > bya theorem of Carlson and Toledo [5] since its fundamental group is equal to π ( M ) . Eells-Salamon twistor correspondence.Theorem 1. If u : Σ → Z is a J − -holomorphic map into twistor space thenits projection τ ◦ u is (either constant or) a conformal harmonic map. If u (Σ) is contained in a fibre (so that τ ◦ u is constant) then we say u is vertical . Let v : Σ → M be a conformal immersion and define the normaltwistor bundle ν → Σ to be the SO (2 n − /U ( n − -bundle over Σ whosefibre ν p at p ∈ Σ is the space of orthogonal complex structures on the normalbundle to v at v ( p ) . We can define a Gauss lift
Gauss ( v ) : ν → Z living over v . This map is defined in the obvious way so that Gauss ( v )( ν p ) = { ψ ∈ F p | ψ ( T Σ) = T Σ } Theorem 2.
The conformal immersion v : Σ → M is harmonic if and onlyif Gauss ( v ) is J − -holomorphic. The construction of the Gauss lift extends to the case when v has iso-lated branch points. Since weakly conformal harmonic maps Σ → M areprecisely the branched minimal immersions [14] that means we can alwayslift a weakly conformal harmonic map. We see that the (non-vertical) J − -holomorphic curves in Z are contained in the complex submanifolds whichare the Gauss lifts of branched minimal immersions. In fact [23] if v : Σ → M is a minimal surface then there exists a J − -holomorphic curve which projectsto v . We loosely refer to the following as the Eells-Salamon twistor corre-spondence (Eells and Salamon proved it in the case n = 2 , where it really isa correspondence; Salamon proved it in general in [26]). JONATHAN DAVID EVANS
Theorem 3 (Eells-Salamon twistor correspondence (ESTC)) . Let ( M, g ) be an oriented n -dimensional Riemannian manifold. Then (non-vertical) J − -holomorphic curves in the twistor space Z project to branched minimalsurfaces in M and any branched minimal surface arises this way. In the case n = 2 the Gauss lift actually provides a bijection between these objects. Since we are looking at harmonic maps into hyperbolic manifolds, werecall the following useful theorem about harmonic maps into negativelycurved manifolds, which captures the convexity of the harmonic map energyfunctional:
Theorem 4 (See Jost [15], Theorem 8.10.2) . Suppose X is a compact Rie-mannian manifold with boundary and Y is a complete Riemannian manifoldwith negative sectional curvatures. Given a map f : ∂X → Y and a homo-topy class of maps F : X → Y such that F | ∂X = f there exists a uniqueharmonic map in this homotopy class. Classification of J − -holomorphic spheres. In a hyperbolic mani-fold it is a classical fact that there are no minimal spheres. This is a conse-quence of Theorem 4 and the fact that π ( M ) = 0 for a hyperbolic manifold.Convexity implies there is a unique minimal representative of any homo-topy class, π ( M ) = 0 implies that any such map is nullhomotopic and theconstant map is the unique nullhomotopic minimal sphere. The ESTC nowtells us that any J − -holomorphic curve projects to a point via the twistorfibration τ , that is: Proposition 1 ([11], Lemma 37) . If Z is the twistor space of a hyperbolic n -manifold then the space of J − -holomorphic spheres in ( Z, J − ) is preciselythe space of vertical spheres, i.e. j F -antiholomorphic spheres in the fibres of τ . We will use this to calculate the genus 0 Gromov-Witten invariants of Z .3.4. Characteristic classes.
First Chern class.
Eells and Salamon showed ([8], Proposition 8.1)that c ( Z, J − ) = 0 when M is 4-dimensional. Fine and Panov ([11], Propo-sition 33) extended this to arbitrary dimensions for hyperbolic manifolds asfollows Proposition 2 ([11], Proposition 33) . The first Chern class of the twistorspace of a hyperbolic n -manifold is given by c ( Z, J − ) = − ( n − c ( H ) = ( n − ω ] where H is the horizontal distribution considered as a complex rank n bundleon Z with the tautological complex structure ψ at a point ( p, ψ ) ∈ Z . We see that there is a trichotomy:
UANTUM COHOMOLOGY OF TWISTOR SPACES 9 n = 1 : ( General type ) M is a hyperbolic 2-manifold. The twistor space isjust M , the Reznikov 2-form is the area form and the Eells-Salamonalmost complex structure is the unique g -orthogonal complex struc-ture. n = 2 : ( Calabi-Yau ) M is a hyperbolic 4-manifold. The twistor space issymplectically Calabi-Yau in the sense that c = 0 . Note that Z cannot actually by Calabi-Yau in the standard sense: its fundamentalgroup is isomorphic to π ( M ) which is hyperbolic and hence cannotoccur as π of a Kähler manifold. n ≥ : ( Fano ) Again, Z cannot be Kähler but it is symplectically Fano.The calculation of the first Chern class goes via the observation that thetangent bundle of twistor space splits ( U ( n ) -equivariantly) as Λ H ∗ ⊕ H where Λ H ∗ is the vertical bundle and H is the horizontal bundle, consideredwith the Eells-Salamon almost complex structure. Since c ( V ) = c (Λ H ∗ ) = − ( n − c ( H ) and since H| F is the tautological U ( n ) -bundle over F wededuce that c ( Z ) | F = − n − H while c ( F ) = − n − H (Don’t be put off by the minus signs: we’re interested in j F - antiholomorphic curves!)3.4.2. Pontryagin classes.
Another advantage of working with the twistorspaces of hyperbolic manifolds is the following theorem of Chern [6]
Theorem 5.
An orientable hyperbolic manifold has vanishing Pontryaginclasses.
The Pontryagin class will crop up very often when we perform topologicalcalculations later and this theorem will make our life significantly simpler.3.5.
Some useful formulae.
The following is a useful formula from [7].
Lemma 1. If X ∈ V is a vertical vector then ∇ X preserves the horizontal-vertical splitting, i.e. ( ∇ X Y ) H = ∇ X ( Y H )( ∇ X Y ) V = ∇ X ( Y V ) Moreover if Y = ˜ W is the horizontal lift of a vector field W on M then ∇ X ˜ W = ( ∇ ˜ W X ) H Proof.
The fact that the fibres are totally geodesic implies that ∇ X preservesthe splitting. To prove the final formula note that ∇ X ˜ W − ∇ ˜ W X = [ X, ˜ W ] The bracket [ X, ˜ W ] is vertical because ˜ W is constant in the X -direction. Thederivative ∇ X ˜ W is horizontal because ∇ X preserves the splitting. Equatinghorizontal and vertical components gives the formula. (cid:3) Another useful observation concerns antiholomorphic curves in the twistorfibre F . If ψ : Σ → F is a j F -antiholomorphic curve then in terms of localconformal coordinates on Σ (1) ∂ s ψ jℓ − ψ jk ∂ t ψ kℓ = 0 . Reznikov Lagrangians
The following construction follows Reznikov [24]. We recall it for thereader’s convenience and because it is of prime importance in what follows.In this section M may be any n -dimensional Riemannian manifold whoseReznikov 2-form is non-degenerate.4.1. Reznikov’s construction.
Let Σ be an n -dimensional submanifold of M and consider the submanifold L Σ := { ( p, ψ ) ∈ Z | p ∈ Σ , ψ ( T p Σ) ⊥ T p m Σ } living over Σ . Lemma 2.
Suppose Σ is totally geodesic. Then T L Σ contains the horizontallift g T Σ of T Σ .Proof. Since Σ is totally geodesic, a g -exponential neighbourhood of a point p ∈ Σ is contained in Σ . Parallel transport along geodesics emanating from p preserves the splitting T p M = T p Σ ⊕ ( T p Σ) ⊥ and hence preserves thecondition for an endomorphism ψ to lie in L Σ . Thus the horizontal sectionsof Z lying over Σ are contained in T L Σ . (cid:3) Lemma 3. If Σ is totally geodesic then L Σ is ω rez -Lagrangian.Proof. The 2-form ω rez is block-diagonal with respect to the splitting H ⊕ V so it suffices to check ω rez | f T Σ = 0 and ω rez | F ∩ L Σ = 0 separately.To prove horizontal vanishing of ω rez , let X and Y be horizontal lifts oftangents to Σ . Then ω ψ ( X, Y ) = g ( X, − ψ ( Y )) = 0 since ψ ∈ L Σ . Similarly ω ψ evaluates to zero on pairs of vectors orthogonal to Σ , so ω ψ is block-antidiagonal with respect to the splitting T Σ ⊕ ( T Σ) ⊥ . Since Σ is totallygeodesic, the Riemann curvature tensor is block-diagonal with respect to thissplitting and therefore ˆ R ( ω ψ ) = ω rez = 0 .To prove vertical vanishing, define the automorphism λ : T p M → T p M with respect to the splitting T p Σ ⊕ ( T p Σ) ⊥ as ⊕− and define the involutionof the twistor fibre F p = τ − ( p ) by ι Σ : ψ
7→ − λψλ − so that F p ∩ L Σ isthe fibrewise fixed locus of ι . The involution is ω F -antisymplectic, so that F p ∩ L Σ is Lagrangian in F p . (cid:3) UANTUM COHOMOLOGY OF TWISTOR SPACES 11
Clearly the proof implies that L Σ is also ω -Lagrangian, but it is easier for ω rez to be non-degenerate than for ω to be closed, so this lemma is a strongerand more useful observation. We also note the following useful corollary. Corollary 1.
There is a fibre-preserving antisymplectic involution ι Σ of τ − (Σ) whose fixed point set is precisely L Σ . We now seek to understand the fibre of L Σ . This is naturally identi-fied with SO ( n ) as follows. Let S ( O ( n ) × O ( n )) be the stabiliser of T p Σ in SO (2 n ) . This group acts transitively on L Σ ∩ F p and the stabiliser is S ( O ( n ) × O ( n )) ∩ U ( n ) = O ( n ) ∆ where O ( n ) ∆ denotes the diagonal. There-fore L Σ ∩ F p = S ( O ( n ) × O ( n )) /O ( n ) ∆ ∼ = SO ( n ) So when n = 2 , L Σ is an S -bundle; when n = 3 , L Σ is an RP -bundle.In the hyperbolic case, the base space Σ is a totally geodesic submanifoldof a hyperbolic n -manifold M so its universal cover is a linear n -subspaceof hyperbolic space, that is Σ is a hyperbolic n -manifold.4.2. Holomorphic discs.
Let Σ ⊂ M be a totally geodesic submanifold ofa hyperbolic 4-manifold M . If we want to understand J -holomorphic discswith boundary on L Σ we must first understand the relative homotopy group π ( Z, L Σ ; Z ) and the Maslov homomorphism on this group. Lemma 4.
The homotopy classes of discs with boundary on L Σ are π ( Z, L Σ ) ∼ = ( Z when n = 2 Z when n ≥ and the Maslov homomorphism is µ = 2( n − ω Proof.
The homotopy calculation just uses the long exact sequence of thefibration of L Σ over Σ with fibre SO ( n ) and the facts that Σ is hyperbolicand hence has no higher homotopy groups and that π (Σ) injects into π ( M ) because Σ is totally geodesic.It is clear from the long exact sequence that the generators for π ( Z, L Σ ) when n = 2 are the upper and lower hemispheres of the twistor fibre. When n ≥ the hemispheres of a ‘real’ twistor line (i.e. one with boundary on therelevant SO ( n ) ) are homotopic and one of them is enough to generate. Theantisymplectic involution ι Σ switches the two hemispheres of a real twistorline and reverses their orientations. In particular they have the same Maslovindex. Gluing the two discs along their common boundary gives the twistorfibre F and the Maslov indices add. However the Maslov index of this sphereis c ( Z ) · [ F ] = 2( n − ω , which gives the result. (cid:3) Using the ESTC we now describe all J -holomorphic discs with boundaryon L Σ . Proposition 3.
Let L Σ be a Reznikov Lagrangian in the twistor space of ahyperbolic n -manifold. Then the J -holomorphic discs u with boundary on L Σ are all vertical.Proof. Let u : (∆ , ∂ ∆) → ( Z, L Σ ) be a J -holomorphic disc and suppose it isnot vertical. On the interior of ∆ the local computation proving the ESTCimplies that u projects to a weakly conformal harmonic map f = τ ◦ u : ∆ → M with boundary on Σ . By Lemma 4, ∂f : ∂ ∆ → Σ is nullhomotopic in Σ .Let F : ∆ → Σ be a nullhomotopy of ∂f . Theorem 4 above ensures that thereis a harmonic representative ˆ F in the homotopy class of F with the sameboundary values. The composition of ˆ F with the totally geodesic embedding Σ → M remains harmonic ([9], Section 5). However, a harmonic map intoa negatively curved manifold is determined uniquely by its boundary values(again by Theorem 4). Therefore f is equal to ˆ F .The ESTC now implies that u is contained in the Gauss lift Gauss ( ˆ F ) (since f is weakly conformal and harmonic it has only branch point singu-larities in the interior of the disc and hence we define the Gauss lift to be theclosure of the Gauss lift of the interior). In each fibre this consists of almostcomplex structures for which ˆ F ∗ T ∆ ⊂ T Σ is preserved. But Reznikov’s La-grangian lift of Σ consists fibrewise of complex structures for which T p Σ issent to its orthogonal complement. This implies that u is contained in asubset of the twistor space disjoint from L Σ , however the boundary of u issupposed to lie on L Σ . (cid:3) Topological preliminaries
We recall some facts from topology which we will use in the computationof Gromov-Witten invariants.5.1.
Cohomological pushforward.
We recall that it is possible to push-forward a cohomology class α along continuous maps of oriented compactmanifolds by converting α into its Poincaré dual homology class, pushingthat forward and then taking the Poincaré dual. If f : X → Y is the mapand P denotes the Poincaré duality map from cohomology to homology (su-pressing the manifold on which it takes place) then this means f ! α := P − f ∗ P α The main properties of cohomological pushforward we will need are: • f ! ( α ∪ f ∗ β ) = ( f ! α ) ∪ β . • If we have a pair of fibre bundles p : E → B and p ′ : E ′ → B ′ withfibres F and F ′ respectively and oriented vertical tangent bundles UANTUM COHOMOLOGY OF TWISTOR SPACES 13 then a commutative diagram F f −−−−→ F ′ y y E e −−−−→ E ′ p y y p ′ B −−−−→ b B ′ such that the map f has degree 1 implies the equality b ∗ p ′ ! = p ! e ∗ In particular a pullback of oriented fibre bundles satisfies this condi-tion.5.2.
Diagonal decompositions.
Let ∆ k : X → X k denote the diagonalmap x ( x, . . . , x ) and let { x i } i ∈ I be an additive basis for H ∗ ( X ; C ) .We will find a formula for the cohomology class ∆ k ! (1) ∈ H ∗ ( X k ; C ) ∼ = H ∗ ( X ; C ) ⊗ k . This is what we call a decomposition of the diagonal . We firstintroduce some notation. Let g ij = Z x i ∪ x j be the Poincaré pairing and g ij its inverse matrix (so g ab g bc = δ ca ). Denoteby C ijk the coefficients of the cup product x j ∪ x k = C ijk x i If we think of ( H ∗ ( X ; C ) , g ) as an inner product space then we can raise andlower indices with g and we see that C jkℓ = g iℓ C ijk = Z x j ∪ x k ∪ x ℓ Note that we must be careful with the order of indices since cup product isonly graded-commutative. Finally, define the
Poincaré amplitude P i ··· i k = g i b g c i g a k − i k k − Y m =2 g c m i m +1 g a m − b m k − Y m =2 C a m b m c m Mnemonically, we can think of this as the ‘Feynman amplitude’ associatedto the diagram i i k ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ i ✍✍✍✍✍✍✍✍✍ i ☞☞☞☞☞☞ · · · where the incoming edges to the m -th interior vertex (from the left) arelabelled b m , c m , a m (clockwise from the topmost). Here, the Feynman rulesassociate a propagator g pq to an edge connecting downwards from p to q anda cubic interaction C a m b m c m to the m -th interior vertex. Lemma 5. ∆ (1) = X ij g ij x i ⊗ x j ∆ k ! (1) = X i ,...,i k P i ··· i k x ⊗ · · · ⊗ x k when k ≥ Proof.
The first equation is just the Alexander-Whitney formula. The secondwill follow by induction. Observe that ∆ k +1 factors as X ∆ k −−−−→ X k id k − ⊗ ∆ −−−−−−−−−→ X k − × X Assuming inductively that the lemma holds for ∆ k , we get ∆ k +1! (1) = P i ··· i k − b k − x i ⊗ · · · ⊗ x i k − ⊗ ∆ ( x b k − ) Since ∆ m is a diagonal embedding, ∆ m ! ( x b k − ) = ∆ m ! (1) ∪ ( x b k − ⊗ ⊗ m − ) so (with propitious index naming) ∆ x b k − = g a k − i k +1 ( x a k − ∪ x b k − ) ⊗ x i k +1 = g a k − i k +1 C a k − b k − c k − g c k − i k x i k ⊗ x i k +1 which completes the induction step. Deriving the case k = 3 from theAlexander-Whitney formula is elementary. (cid:3) In the sequel we will frequently use Poincaré amplitudes of different spacesand to distinguish them we will sometimes use decorations e.g. P i ··· i k X . Remark 4.
We observe that if π : A → B is a fibre bundle satisfying thehypotheses of the ( C ) Leray-Hirsch theorem (i.e. there exist C -cohomologyclasses { z i } Ni =1 on A which pull back to give a basis of the C -cohomology ofthe fibre) then the diagonal decomposition for A has the form X z i ,...,z ik A i ...i k ( z i ⊗ · · · ⊗ z i k ) ∪ π ∗ ∆ k ! ( y i ...i k ) where y i ...y k is an element of H ∗ ( B ; C ) (we have abusively written π : A k → B k ). To see this, form the pullback A ⊗ k δ −−−−→ A k ˜ π y y π B −−−−→ ∆ k B k UANTUM COHOMOLOGY OF TWISTOR SPACES 15
The pushforward along A → A k factors through this map. Note that ˜ π : A ⊗ k → B also satisfies the hypotheses of the Leray-Hirsch theorem since anycohomology class on the fibre F k is just a pullback of a product of classesfrom A k . Therefore any class in H ∗ ( A ⊗ k ; C ) can be written ( δ ∗ c ) ∪ ˜ π ∗ y pushing this class forward gives δ ! ( δ ∗ c ∪ ˜ π ∗ y ) = c ∪ δ ! ˜ π ∗ y = c ∪ π ∗ ∆ k ! ( y ) which is of the required form. Borel-Hirzebruch theory.
Let G be a compact connected Lie groupand H ⊂ G a closed subgroup containing a maximal torus T of G . Borel-Hirzebruch theory is a means of calculating fibre integrals along bundle pro-jections and takes as its starting point the diagram B (cid:15) (cid:15) BT BGBH Bκ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄ Bι / / where T → H ι → G are the inclusions and κ = ι ◦ . Fix Θ a positive systemof roots of G and Ψ ⊂ Θ a positive system of roots for H . We can think ofthese roots as element of the cohomology H ∗ ( BT ; C ) . The Borel-Hirzebruchformula ([4], Section 22.6) tells us that(2) ( Bι ) ! x = ( Bκ ) ! (( B ) ∗ x ∪ td) where td is the Todd class Y α ∈ Ψ α − e − α ∈ H ∗∗ ( BT ; C ) of the vertical tangent bundle to the H/T -bundle BT → BH . Here H ∗∗ denotes the direct product Q i ≥ H i as opposed to the direct sum. Identifying H ∗ ( BG ; C ) as the Weyl-invariant subspace of H ∗ ( BT ; C ) via ( Bκ ) ∗ , thepushforward ( Bκ ) ! can be computed using [4], Equation (6):(3) ( Bκ ) ! y = P σ ∈ W ( G ) σ ( y )sgn( σ ) Q α ∈ Θ α where sgn( σ ) denotes the determinant of σ ∈ W ( G ) considered as a matrixacting on g .We will apply Borel-Hirzebruch theory to the following situation. Let T ⊂ H ⊂ G be as before and let π : E → B be a bundle of homogeneous spaces over a closed manifold arising as a pullback G/H G/H y y E cl E −−−−→ BH π y y Bι B −−−−→ cl B BG We can evaluate fibre integrals (pushforwards along τ ) of polynomials in thecharacteristic classes of the H -bundle over Z , for if c is such a polynomialthen τ ! cl ∗ E c = cl ∗ B ( Bι ) ! c This is useful for computing the ring structure on the cohomology of Z .Henceforth we assume that there exists an additive basis C for the coho-mology of G/H coming from pulling back (along the inclusion
G/H → Z )the characteristic classes of the tautological H -bundle over Z . In this set-ting the Leray-Hirsch theorem implies that the cohomology H ∗ ( E ; C ) is afree H ∗ ( B ; C ) -module generated by these characteristic classes. In particu-lar if C = { z i } i ∈ I and { y j } j ∈ J is a basis for the cohomology of B then anycohomology class β can be written β = X i ∈ I,j ∈ J β ij z i ∪ τ ∗ y j and τ ! ( β ∪ z i ∪ · · · ∪ z i p ) = X i ∈ I,j ∈ J β ij τ ! ( z i ∪ z i ∪ · · · ∪ z i p ) ∪ y j The fibre integrals on the right-hand side can be done using Borel-Hirzebruchtheory and this gives a system of linear equations for the coefficients β ij .This can be used to compute the cup product coefficients because if we take β = ( z a ∪ τ ∗ y b ) ∪ ( z c ∪ τ ∗ y d ) then β ij = C ijab,cd . Topological computations
We now perform some fibre integrals and computations of cohomologyrings which will prove useful in the sequel.6.1.
The twistor space.
The first bundle of interest is the twistor bundle τ : Z → M In this case cl M classifies the ( SO (2 n ) ) frame bundle, cl Z classifies the almostcomplex horizontal U ( n ) -bundle H and the fibre is SO (2 n ) /U ( n ) . We may UANTUM COHOMOLOGY OF TWISTOR SPACES 17 take as a system of positive roots of SO (2 n ) and U ( n ) respectively Ψ = { x i − x j | i < j } Θ = { x i − x j , x i + x j | i < j } We now compute some fibre integrals and the cup product structure on Z inthe cases n = 3 where the fibre is diffeomorphic to CP and the cohomologyof the fibre is therefore generated by powers of the first Chern class. When n is larger the computations become unwieldy.We have ( Bι ) ! c = 8 ( Bι ) ! c c = 4( Bι ) ! c = 0 ( Bι ) ! c c = 0( Bι ) ! c = 16 p ( Bι ) ! c c = 4 p ( Bι ) ! c = 64 χ ( Bι ) ! c c = 32 χ and, since ( B ) ∗ c = x x x = ( Bκ ) ∗ χ is Weyl-invariant, we have for any ℓ ∈ H ∗ ( BU (3); C ) ( Bι ) ! ( c ∪ ℓ ) = χ ∪ ( Bι ) ! ℓ Now by Leray-Hirsch we can write any cohomology element of Z as a lin-ear combination of powers of c ( H ) with coefficients in the cohomology of H ∗ ( M ; Q ) . In particular c ( H ) = τ ∗ ( α ) + c ( H ) · τ ∗ ( β ) + c τ ∗ ( γ ) + c τ ∗ ( δ ) Fibre integration yields δ = 0 γ = 2 p = 0 β = 8 χ which determines the ring structure completely as H ∗ ( Z ; C ) = H ∗ ( M ; C )[ c ( H )] / (cid:0) c ( H ) = 8 c ( H ) τ ∗ χ (cid:1) By similar means we can express c ( H ) , c ( H ) in terms of c ( H ) : c ( H ) = 12 c ( H ) , c ( H ) = τ ∗ χ Let z i = c ( H ) i and y j be an additive basis for H ∗ ( M ; C ) with y = 1 . Interms of the basis z i ∪ τ ∗ y j for the cohomology of Z we have Poincaré pairing g Zia,jb = 8 g Mab δ i + j, + 64 χ ( M ) δ a, δ b, δ i + j, with inverse g ia,jbZ = 18 g abM δ i + j, − χ ( M ) δ y a , vol δ y b , vol δ i + j, where vol is the (unit) volume form on M (assumed to be a part of ourbasis). We also have C Zia,jb,kc = 8 δ i + j + k, C Mabc + 64 δ i + j + k, χ ( M ) δ a, δ b, δ c, From this and Lemma 5.2 we deduce
Corollary 2 (Formulae for the first few diagonal decompositions) . ∆ = 18 X i =0 ( c ( H ) i ⊗ c ( H ) − i ) ∪ τ ∗ ∆ (1 M ) − χ ( M ) τ ∗ ∆ (vol M )∆ = 164 X ≤ i,j,k ≤ i + j + k =6 ( c ( H ) i ⊗ c ( H ) j ⊗ c ( H ) k ) ∪ τ ∗ ∆ (1 M )+ χ ( M )8 (cid:0) c ( H ) ⊗ − { c ( H ) ⊗ ⊗ } (cid:1) ∪ τ ∗ ∆ (vol M ) where vol M is a volume form with R M vol M = 1 , M denotes the fundamentalclass of M , the curly brackets { z i ⊗ · · · ⊗ z i k } denote X σ ∈ S k / Stab( z ) z i σ (1) ⊗ · · · ⊗ z i σ ( k ) and Stab( z ) denotes the subgroup of permutations acting trivially on ( z i , . . . , z i k ) (which is nontrivial when z i m = z i n for some m = n ). We have abused no-tation slightly and written ∆ k also for the diagonal inclusion of M into M k and τ : Z k → M k for the product of projections. The moduli space of twistor lines.
Recall that the holomorphiccurves of minimal degree in the twistor fibre over p are the twistor lines consisting of complex structures preserving a fixed 4-plane in the tangentspace T p M and equal to some fixed complex structure on its orthogonalcomplement. We write L for the space of twistor lines and λ : L → M forthe projection taking a line in the fibre F p to the point p . The fibre of λ is thehomogeneous space L ( F ) = SO (2 n ) /SO (4) × U ( n − of lines in a singletwistor fibre. We see that λ : L → M is the pullback of the tautological SO (2 n ) /SO (4) × U ( n − -bundle over BSO (2 n ) along the classifying mapfor T M , so it fits into our Borel-Hirzebruch setup.We will compute the cohomology ring in the case n = 3 . This will laterbe used to find the Poincaré amplitudes of a decomposition of the diagonal:these amplitudes will occur as coefficients in our formula for the Gromov-Witten class associated to the moduli space of twistor lines.When n = 3 , the fibre of λ is diffeomorphic to the Grassmannian SO (6) /SO (4) × SO (2) and so its cohomology ring is C [ e, t ] / ( e = t , et = 0) where e is the Euler class of the SO (4) -bundle and t is the first Chernclass of the U (1) -bundle. To see the relations, observe that: e is Poincarédual to the point in the Grassmannian representing the 4-plane orthogonal UANTUM COHOMOLOGY OF TWISTOR SPACES 19 to a generic pair of vectors in R ; t is Poincaré dual to the point in theGrassmannian representing the 4-plane spanned by four generic vectors and et is represented by the (generically empty) cycle of 4-planes which containa given vector and are orthogonal to another. We need to calculate the fibreintegrals λ ! e a t b (where e and t now denote the corresponding characteristic classes on L ).The only interesting ones are λ ! t = λ ! e = 2 λ ! t = 2 p (= 0 when M is hyperbolic by Theorem 5 ) λ ! et = λ ! e t = 2 χ These yield the cohomology ring H ∗ ( L ; C ) = H ∗ ( M ; C )[ e, t ] / ( e = t , et = λ ∗ χ ) when M is hyperbolic.Let z i run over the set { , t, t , t , t , e } and y j be a basis for H ∗ ( M ; C ) .Then we see the Poincare pairing is Z L ( z i ∪ λ ∗ y j ) ∪ ( z k ∪ λ ∗ y ℓ ) = 2 g Mjℓ ( δ z i z k ,t ) We also have C ia,jb,kc = 2 δ i + j + k, C Mabc + 2 χ ( M ) { δ j + k, δ i,e } δ a δ b δ c (where {} denotes the sum over permutations as before) and so we deduce Corollary 3 (Formulae for the first few diagonal decompositions) . ∆ = 12 (cid:0) { ⊗ t } + { t ⊗ t } + t ⊗ t + e ⊗ e (cid:1) ∪ λ ∗ ∆ (1 M )∆ = 14 { t ⊗ e ⊗ e } + X ≤ i,j,k ≤ i + j + k =8 ( t i ⊗ t j ⊗ t k ) ∪ λ ∗ ∆ (1 M )++ 14 X ≤ i We denote by L the moduli space of twistorlines with one marked point on the domain. Note that this has a forgetfulmap ft : L → L which exhibits it as an SO (4) /U (2) -bundle and an evalu-ation map ev : L → Z which sends a twistor line and a point on the twistorline to the corresponding point in Z . From the first description we see that L is the pullback of the tautological SO (2 n ) /U (2) × U ( n − over BSO (2 n ) along the classifying map for T M . The evaluation map is induced by theinclusion U (2) × U ( n − → U ( n ) .We will need to compute cohomological pushforwards along ev of certainclasses ft ∗ c , c ∈ H ∗ ( L ; C ) . Again we will only compute the case n = 3 . Inthis case the fibre is CP ∼ = U (3) /U (2) × U (1) We will denote by A and B the tautological U (2) and U (1) -bundles respec-tively. The integrals we need to compute are a subset of ev ! c ( A ) a c ( B ) b since the cohomology classes t and e from the previous section pullback to c ( B ) and c ( A ) respectively. The integrals we are interested in are ev ! c ( B ) k = when k = 2 c ( H ) when k = 3 c ( H ) − c ( H ) = c ( H ) = c ( H ) when k = 4 (4) ev ! c ( A ) c ( B ) k = when k = 00 when k = 10 when k = 2 c ( H ) = τ ∗ χ when k = 3 c ( H ) ∪ c ( H ) = c ( H ) ∪ τ ∗ χ when k = 4 (5) ev ! c ( A ) = c ( H ) = c ( H ) (6)where H is the tautological U (3) -bundle over Z .6.4. The topology of hyperbolic manifolds. The assumption that theStiefel-Whitney classes of M vanish allows us to give a good answer to thequestion of when M admits an almost complex structure. Notice that theinclusion of SO (2 n ) into GL + (2 n ) is a homotopy equivalence and hence thequestion of whether M admits an orthogonal almost complex structure (asection of the twistor bundle) is the same as whether it admits any almostcomplex structure. Lemma 6. An 6-manifold with vanishing Stiefel-Whitney classes admits: a)an almost complex structure, b) a field ξ of tangent 2-planes.Proof. To prove a), recall ([22], Proposition 8) that the obstruction to theexistence of a lift M → BU (3) of the classifying map for the oriented tangentbundle is the Bockstein image of w ( M ) in H ( M ; Z ) . To prove b), ([27],Theorem 1.3) gives a condition for the existence of such a 2-plane field withEuler class u ∈ H ( M ; Z ) : that there exists a class v ∈ H ( M ; Z ) whichreduces to the 4th Stiefel-Whitney class such that u ∪ v [ M ] = χ ( M ) . Van-ishing of the Stiefel-Whitney classes reduces this to the condition that v is UANTUM COHOMOLOGY OF TWISTOR SPACES 21 divisible by 2. Since the Euler characteristic of a 6-manifold is even the ex-istence of such a class v follows straight from nondegeneracy of the Poincarépairing. (cid:3) The almost complex structure gives us a section of the twistor bundleand hence a submanifold representing the homology class Poincaré-dual to c ( H ) / . The 2-plane field allows us to “Gauss-lift” the whole 6-manifoldby defining a submanifold of the twistor bundle consisting of points corre-sponding to complex structures for which ξ is preserved. This submanifoldintersects each twistor fibre in a twistor line and is Poincaré dual to c ( H ) / .We can also define a submanifold Poincaré dual to c ( H ) by taking the fi-brewise cut-locus of our section (which is a hyperplane in each fibre). Letus write Σ k ( k = 0 , , , ) for the corresponding submanifold representing P (cid:16) c ( H ) k k (cid:17) (where Σ = Z ). Notice that if Y is a submanifold of M then τ − ( Y ) intersects Σ k transversely for all k .These observations allow us to visualise homology classes in the twistorspace. Given a cohomology class y in M we can represent K P ( y ) by asubmanifold Y for large K . Now we can represent K P ( c ( H ) i τ ∗ y ) by theintersection of the preimage τ − ( Y ) with the submanifolds (sections, Gauss-lifts,...) representing c ( H ) i (assuming they exist).7. Definition of Gromov-Witten invariants We recall the definition of genus 0 Gromov-Witten invariants, quantumcohomology and the Gromov-Witten potential. For more details see [20].Let Z be a N -dimensional monotone symplectic manifold (for example thetwistor space of a hyperbolic n -manifold with n ≥ , for which N = n ( n +1)2 ). Definition 1. Let J be a regular ω -tame almost complex structure on Z , β ∈ H ( Z ; Z ) a homology class and define M ,k ( Z, β, J ) := { ( u, z ) | u : S → X, ∂ J u = 0 , u ∗ [ S ] = β, z = ( z , . . . , z k ) ∈ S k ,z = 0 , z = 1 , z = ∞ , z i = z j for i = j } This is a smooth manifold of dimension N + 2 c ( Z )[ β ] + 2( k − Consider the evaluation map ev k : M ,k ( Z, β, J ) → Z k , ev k ( u ) = ( u ( z ) , . . . , u ( z k )) This is a pseudocycle which can be compactified by adding strata of stablemaps to M ,k ( Z, β, J ) . The compactified moduli space is denoted M ,k ( Z, β, J ) We define the k -point genus 0 Gromov-Witten invariant of Z in the class A to be the homology class GW Zβ,k = (ev k ) ∗ [ M ,k ( Z, β, J )] ∈ H N +2 c ( Z )+2( k − ( Z k ; C ) in the sense of pseudocycles. One can extract numerical invariants by inter-secting with pseudocycles in Z k . If a , . . . , a k are pseudocycles representing C -homology classes in Z k then we write GW Zβ,k ( a , . . . , a k ) = GW Zβ,k · ( a × · · · ⊗ a k ) where · is the pseudocycle intersection pairing. In the case of twistor spaces all holomorphic curves live in a multiple mA of the homology class of a twistor line and when n ≥ the twistor space ismonotone so we will use the Novikov coefficient ring Λ = C [ q ] (the exponentof q corresponds to the multiplicity m and monotonicity implies we only needpolynomials rather than formal power series).Pick a Z -basis x , . . . , x N for H ∗ ( Z ; Z ) with x = 1 ∈ H ( Z ; Z ) such thatevery basis element has pure degree. Definition 2. The quantum cohomology of ( Z, ω ) is a ring structure on thegraded vector space QH ∗ ( Z ; Λ) = H ∗ ( Z ; C ) ⊗ C Λ given (on elements of pure degree) by a ⋆ b = X β ∈ H ( Z ) X ν,µ GW Zβ, ( a, b, x i ) g ij x j ⊗ e β where g ij is the inverse matrix to g ij = Z Z x i ∪ x j and the grading on QH ∗ ( Z ; Λ) is QH k ( Z ; Λ) = M i H i ( Z ; C ) ⊗ C Λ k − i The linearised theory Recall that vertical J − -holomorphic curves are j F -antiholomorphic curvesin the twistor fibre. The image of an antiholomorphic curve is equal tothe image of a holomorphic curve since we can always precompose with anantiholomorphic involution. We will now prove the following. Proposition 4. For vertical J − -holomorphic curves,(i) the kernel of the linearised ∂ J − -operator is precisely the tangent spaceof the moduli space.(ii) If moreover the image of a J − -holomorphic curve is contained in a line SO (4) × U ( n − /U (2) × U ( n − in the twistor fibre corresponding to a 4-plane π ⊂ T p M then thereis a subspace of the cokernel of the linearised ∂ J − -operator naturallyisomorphic to π . UANTUM COHOMOLOGY OF TWISTOR SPACES 23 (iii) In particular, the obstruction bundle of cokernels over the moduli space L = Γ \ SO + (2 n, /SO (4) × U ( n − of (anti)twistor lines is naturally isomorphic to the tautological SO (4) -bundle. One might be concerned that using antiholomorphic curves in the twistorfibre will affect the orientation of the moduli space. Indeed the orientationof the moduli space of twistor lines is reversed along the fibre directions, butso is the orientation of the twistor fibre and therefore the computation ofGromov-Witten invariants will not be affected if we ignore this simultaneouschange of signs.Let u : CP → Z be a genus 0 vertical J − -holomorphic curve. • Let u ∗ ∇ denote the pullback of the twistor connection to u ∗ T Z . • Recall that the linearised Cauchy-Riemann operator D u : Ω ( u ∗ T Z ) → Ω , ( u ∗ T Z ) is given by the formula ( D u ξ )( X ) = ( u ∗ ∇ ) X ξ + J − ( u ∗ ∇ ) jX ξ − ( J − ∇ ξ J − ) du ( X ) We can take vertical and horizontal parts of D u using Lemma 1 and we get D u = (cid:18) D HHu D V Hu D V Vu (cid:19) : Ω ( u ∗ H ⊕ u ∗ V ) → Ω , ( u ∗ H ⊕ u ∗ V ) where D V Hu ξ = ( D u ξ H ) V etc. The D V Vu -part is just the linearised Cauchy-Riemann operator governing deformations of u as a holomorphic curve in F (the twistor fibre). But F is a homogeneous space and is therefore convexin the sense of Kontsevich [16], i.e. all genus 0 holomorphic curves in F areregular. Therefore D V Vu is surjective.The kernel of D u fits into an exact sequence → ker D V Vu a → ker D u b → ker D HHu → where a is inclusion into the vertical component and b is projection to thehorizontal component: surjectivity of b follows from surjectivity of D V Vu . Thecokernel of D u is naturally identified with the cokernel of D HHu , therefore tocompute obstruction bundles it suffices to understand D HHu .Using the projection τ ∗ and horizontal lifting ˜ · we will identify H and τ ∗ T M . Let e , . . . , e n be a local orthonormal frame on M near a point p and { ˜ e i } ni =1 the horizontal lift of this frame to a neighbourhood of τ − ( p ) in Z . Suppose ξ = P ξ i ˜ e i ∈ H . Then ( D HHu ξ )( X ) = [ D u ξ ( X )] H = X ( ξ i )˜ e i + ξ i ∇ X ˜ e i + ( jX )( ξ i ) ψ ˜ e i + ξ i ψ ∇ jX ˜ e i − ξ i ψ ( ∇ ˜ e i du ( jX ) − J − ∇ ˜ e i X ) H where ψ is the complex structure on H at the point u ( z ) of the twistor fibre.By Lemma 1 this equation is just(7) D HHu ξ ( X ) = X ( ξ i )˜ e i + ( jX )( ξ i ) ψ ˜ e i Pick conformal coordinates ( s, t ) in a patch on CP and write X = X s ∂ s + X t ∂ t . Then Equation (7) becomes D HHu ξ ( X ) = ( X s δ ij − X t ψ ij )( ∂ s ξ j + ψ jk ∂ t ξ k ) Proof of Proposition 4 (i). To find the kernel of D HHu it therefore suffices tosolve Ξ := ∂ s ξ j + ψ jk ∂ t ξ k = 0 Differentiating this with respect to s gives ∂ s Ξ j = ∂ s ξ j + ( ∂ s ψ jk )( ∂ t ξ k ) + ψ jk ∂ s ∂ t ξ k = 0 and with respect to t gives ∂ t Ξ ℓ = ∂ t ∂ s ξ ℓ + ( ∂ t ψ ℓk )( ∂ t ξ k ) + ψ ℓk ∂ t ξ k = 0 so ∂ s Ξ j − ψ jℓ ∂ t Ξ ℓ = ∇ ξ j + ( ∂ s ψ jk − ψ jℓ ∂ t ψ ℓk ) ∂ t ξ k since ∇ = ∂ s + ∂ t in local coordinates. Note that by Equation (1) ∂ s ψ jk − ψ jℓ ∂ t ψ ℓk = 0 so ξ k ∈ ker D HHu implies that the component functions ξ j are harmonicfunctions on S and hence constant.Therefore the kernel of D u consists of precisely the tangent directionsin the moduli space M ( mA, J − ) (where u has degree m ) since these areprecisely the deformations of u as a vertical curve (kernel of D V Vu ) and thedeformations of u to nearby fibres (kernel of D HHu ). (cid:3) Proof of Proposition 4 (ii). Lemma 7. The adjoint operator ( D HHu ) ∗ is given (in conformal coordinateson the domain and coordinates on u ∗ H induced from an orthonormal frame e i of T p M as before) by (( D HHu ) ∗ η ) i = − − ( ∂ s η i − ∂ t ( ψ ij η j )) where dvol = Θ ds ∧ dt is the standard area form on S expressed in confor-mal coordinates and η i = η ( ∂ s ) i (since η ∈ Ω , ( u ∗ H ) its value on any vectordetermines its value on any other). We can now write down some solutions of ( D HHu ) ∗ η = 0 immediately. Lemma 8. Let v ∈ T p M be a vector and pick conformal coordinates ( s, t ) on a patch in S . Define η v = Ω , ( u ∗ H ) by requiring η v ( ∂ s ) i = ( ∂ t ψ ij ) v j .Then ( D HHu ) ∗ η v = 0 .Proof. Since v is constant we have (( D HHu ) ∗ η v ) i = − (cid:16) ∂ s ∂ t ψ ik − ∂ t ( ψ ij ∂ t ψ jk ) (cid:17) v k which vanishes by Equation (1). (cid:3) UANTUM COHOMOLOGY OF TWISTOR SPACES 25 Note that if ψ lands in a line in F corresponding to the plane π then ∂ t ψ ij v j ∈ π . Since u is somewhere immersed the correspondence v η v is an isomorphism for v ∈ π between π and ker( D HHu ) ∗ ∼ = coker D HHu asclaimed. (cid:3) Interpretation. Let u be a J − -holomorphic curve in Z . Observe thatan infinitesimal deformation of ω -compatible almost complex structures J − J − + δJ gives us a natural choice of η ∈ Ω , ( u ∗ T Z ) defined by η ( X ) = δJ ( u ∗ ( X )) We have defined elements η v : ∂ s ( ∂ t ψ ) v, ∂ t 7→ − ( ∂ s ψ ) v in the cokernel of D u . Here ∂ t ψ = u ∗ ( ∂ t ) , ∂ s ψ = u ∗ ( ∂ s ) . We can nowunderstand these as coming from the following infinitesimal deformation of J − δ v J ( w ) = ( ( J − w ) v when w ∈ V when w ∈ H The Gromov-Witten theory of twistor lines The obstruction bundle. In this section we compute the Gromov-Witten cycles associated to the moduli space of pseudohololmorphic spheresin the homology class A . We recall that this moduli space has a very nicedescription as a homogeneous space L = M , ( Z, A, J − ) = Γ \ SO + (2 n, /SO (4) × U ( n − L = M , ( Z, A, J − ) = Γ \ SO + (2 n, /U (2) × U ( n − The key observation is that the moduli space is compact ( A is a minimalhomology class). We can therefore apply the following theorem: Theorem 6 (See [20], Proposition 7.2.3) . Let Z be a semipositive symplecticmanifold and A a homology class which is not a multiple cover of a homologyclass B with c ( Z )[ B ] = 0 . If the moduli space M , ( Z, A, J ) is compact andsmooth with tangent space at u equal to the kernel of the linearised ∂ -operator D u then the cokernels of D u form a smooth vector bundle over M , ( Z, A, J ) called the obstruction bundle Obs and the Gromov-Witten class GW ZA,k maybe computed by P (ev k ) ! ft ∗ k obs where • P : H ∗ → H ∗ denotes Poincaré duality, ! denotes cohomologicalpushforward, • ft k : M ,k ( Z, A, J − ) → M , ( Z, A, J − ) is the forgetful map, • ev k : M ,k ( Z, A, J − ) → Z k is the evaluation map and • obs is the Euler class of the obstruction bundle over M , ( Z, A, J − ) . In Section 8 we proved that we are in precisely this setting and that Theorem 7. The obstruction bundle over L is isomorphic to the tautological SO (4) -bundle. Remark 5. One may think of the zero-set of a section of the obstructionbundle as a “regularised moduli space” f M ,k ( Z, A, J − ) . There is then a veryappealing picture. Take a vector field V on M and lift it to a section of Obs by projecting it to the tautological 4-plane bundle over the moduli space L . The regularised moduli space consists of twistor lines corresponding to4-planes in T p M which are orthogonal to V ( p ) . If f M ,k ( p ) denotes that partof the regularised moduli space living in the twistor fibre at p then whenever V ( p ) = 0 , f M , ( p ) ∼ = SO (2 n − /SO (4) × U ( n − where SO (2 n − is the stabiliser of V ( p ) , and f M , ( p ) ∼ = SO (2 n − /U (2) × U ( n − ft → SO (2 n − /SO (4) × U ( n − 3) = f M , ( p ) tells us, morally, which curves will persist in that fibre after deformation of J . In the case n = 3 this reduces to the standard fibration SO (5) /U (2) = CP → S = SO (5) /SO (4) so we see (at least heuristically) that Z is uniruled if dim( M ) = 6 . The algorithm. We now give an algorithm which can be used to com-pute GW ZA,k = P (ev k ) ! ft ∗ k obs The map ev k factors as M ,k ( Z, A, J − ) rem ×···× rem k −−−−−−−−→ L × · · · × L ×···× ev −−−−−→ Z k where rem j is the map remembering only the j -th marked point and ev : L = Γ \ SO + (2 n, /U (2) × U ( n − → Γ \ SO + (2 n, /U ( n ) = Z is the 1-point evaluation map. This factorisation fits into a diagram M ,k ( Z, A, J − ) Q kj =1 rem j −−−−−−−→ L k k −−−−→ Z k ft k y y (ft ) k L −−−−→ ∆ L k which implies that we need to compute(8) P (ev) k ! ((ft ) k ) ∗ ∆ ! obs We have ∆ k ! obs = ( obs ⊗ ⊗ k − ) ∪ ∆ k ! . Recall that the projection λ : L → M is a SO (2 n ) /SO (4) × U ( n − -bundle and that, by the Leray-Hirsch theo-rem, H ∗ ( L ; C ) is a free H ∗ ( M ; C ) -module with some collection of generators UANTUM COHOMOLOGY OF TWISTOR SPACES 27 { z i } arising as characteristic classes of the tautological SO (4) and U ( n − -bundles over the moduli space. Let { y j } be a basis for H ∗ ( M ; C ) and z i λ ∗ y j the corresponding basis for H ∗ ( L ; C ) . We take a decomposition of the diag-onal (Section 5.2) ∆ k ! X i ,...,i k ,j ,...,j k P i j ,...,i k j k L ( z i λ ∗ y j ) ⊗ · · · ⊗ ( z i k λ ∗ y j k ) Now substituting in Equation (8) and using the fact that λ ◦ ft = τ ◦ ev we get Theorem 8. GW ZA,k = P X P i j ,...,i k j k L ((ev ! (ft ∗ ( z i ∪ obs ))) ∪ τ ∗ y j ) ⊗⊗ ((ev ! ft ∗ z i ) ∪ τ ∗ y j ) ⊗ · · · ⊗ ((ev ! ft ∗ z i k ) ∪ τ ∗ y j k ) (9)It remains only to find the decomposition of the diagonal and to computethe fibre integrals ev ! ft ∗ z m , ev ! (ft ∗ ( z m ∪ obs )) along the U ( n ) /U (2) × U ( n − -bundle ev which can be done using Borel-Hirzebruch theory as in Section 6.3.9.3. Examples. We will illustrate the use of this algorithm through a num-ber of elementary examples. Corollary 4. Suppose n = 2 (so M is a hyperbolic 4-manifold). Then GW ZA,k = χ ( M ) A ⊗ k ∈ H ∗ ( Z k ; C ) where A is the homology class of the twistor fibre.Proof. In this dimension ev : L → Z and λ : L → M are diffeomorphismsso { z m } = { } and (9) reduces to P X P j ...j k M (ft ∗ obs ∪ τ ∗ y j ) ⊗ τ ∗ y j ⊗ · · · ⊗ τ ∗ y j k where (by Theorem 7) obs is the Euler class of the tangent bundle of M and ft = τ so τ ∗ obs = c ( H ) = P ( A ) . Therefore a summand is nonzero if andonly if y j has nontrivial cup product with χ . This means that y j must havedegree zero and that therefore all other y j m must be of top degree in orderthat the Poincaré amplitude P j ...j k M is nonvanishing. Under Poincaré dualitythese pullback (via τ ! ) to A , the homology class of the twistor fibre, and weget GW ZA,k = A ⊗ k ∈ H ∗ ( Z k ; C ) (cid:3) Corollary 5. Suppose n = 3 (so M is a hyperbolic 6-manifold). (1) When k = 1 , GW ZA, = [ Z ] i.e. Z is uniruled.(2) When k = 2 , GW ZA, = 14 P (cid:0) { ⊗ c ( H ) } ∪ τ ∗ ∆ (1 M ) (cid:1) where H is the horizontal distribution on Z considered as a complexvector bundle.(3) When k = 3 , GW ZA, = P (cid:18) { ⊗ c ( H ) ⊗ c ( H ) } ∪ τ ∗ ∆ (1 M )++ χ ( M )2 { c ( H ) ⊗ ⊗ } ∪ τ ∗ ∆ (vol M ) (cid:19) where we have freely used the notation of Corollary 2.Proof of k = 1 : In view of (9) it suffices to compute ev ! c ( H ) where H is thetautological U (2) -bundle over L . We saw in Section 6.3 that ev ! c ( H ) = 1 = P − [ Z ] (cid:3) Proof of k = 2 : By Corollary 3, we have the following decomposition of thediagonal ∆ X i,j g abM (cid:0) λ ∗ y a ⊗ t λ ∗ y b + tλ ∗ y a ⊗ t λ ∗ y b + t λ ∗ y a ⊗ t λ ∗ y b + t λ ∗ y a ⊗ tλ ∗ y b + t λ ∗ y a ⊗ λ ∗ y b + eλ ∗ y a ⊗ eλ ∗ y b (cid:1) Now the result follows from (9) and the formulae for cohomological pushfor-ward along the evaluation map (4). (cid:3) Proof of k = 3 : Follows similarly. (cid:3) Remark 6. We can understand the case k = 2 heuristically as follows. First,notice that the class c ( H ) pulls back to the twistor fibre as the cohomologyclass of a twistor line (since c ( H ) pulls back to H ). Now consider a ho-mology class µ ∈ H k ( M ; C ) represented by an oriented submanifold N ⊂ M .As we saw in Section 6.4 we can always find an almost complex structure ψ on M and (if the homology is torsionfree) a 2-plane field ξ . Define thesubmanifolds N ′ , N ′′ , N ′′′ of Z which fibre over N with fibre F ′ , F ′′ , F ′′′ over n ∈ N equal to • F ′′ : the set of complex structures on T n Z making ξ holomorphic (likea Gauss lift, consisting of a line in the twistor fibre), • F ′ : the point ψ ( n ) , • F ′′′ : the cut locus of ψ ( n ) (a copy of CP in the twistor fibre). UANTUM COHOMOLOGY OF TWISTOR SPACES 29 Suppose that the Poincaré dual class ˇ µ can also be represented by a sub-manifold N , which intersects N exactly once transversely at some point n .Then the Gromov-Witten contribution from twistor lines to their quantumintersection is ZA, ([ N ′ ] , [ N ′′′ ]) = GW ZA, ([ N ′′′ ] , [ N ′ ]) , GW ZA, ([ N ′′ ] , [ N ′′ ]) = 0 from our formula since [ N ′ ] = H ∪ τ ∗ µ , [ N ′′′ ] = H ∪ τ ∗ ˇ µ , etc. This can beseen via our earlier heuristic picture of the regularised moduli space (Remark5) by noticing that when the perturbing vector field V is chosen with V ( n ) = 0 there is a unique twistor line in the regularised moduli space connecting thepoint F ′ with the cycle F ′′′ and that the spheres F ′′ and F ′′ will genericallyproject to non-intersecting spheres in S = f M , ( n ) and hence there are noconnecting twistor lines.One must be careful with this heuristic because it can be misleading. Atfirst sight, if one had a pair of sections of the twistor bundle then theirquantum intersection would pick up the χ ( M ) lines joining them inside theunperturbed fibres (where there is still a line through every pair of points)but a simple dimension count shows this is not the case: the Gromov-Wittenclass has dimension 14 while the pair of sections would have codimension 12in Z and we see that the chosen regularisation of the moduli space is nottransverse to such a submanifold. Higher degree curves Easy computations. Some higher degree contributions are easy tocompute for dimension reasons. Recall that the Gromov-Witten invariantlives in degree deg GW ZmA,k = n ( n + 1) + 4 m ( n − 2) + 2( k − and dim Z k = kn ( n + 1) . This gives us the trivial bound m ( n − 2) + 2( k − ≤ ( k − n ( n + 1) necessary for the nonvanishing of the invariant. For example, Corollary 6. When n = 3 , • the 1-point invariant only gets contributions from curves of degree 1, • the 2-point invariant only gets contributions from curves of degree 3or less, • the 3-point invariant only gets contributions from curves of degree 6or less. However the special geometry of the Eells-Salamon almost complex struc-ture gives us more information still. Lemma 9. Suppose n = 3 and { y i } ki =1 are cohomology classes on M withdegrees d i . If P ki =1 d i > then GW mA,k ( c i τ ∗ y ⊗ · · · ⊗ c i k τ ∗ y k ) = 0 Proof. Let us assume without loss of generality that the homology classes P ( y i ) are represented by submanifolds Y i of M (we can rescale by a largeinteger). Recall from Section 6.4 that we have submanifolds Σ p represent-ing P ( c ( H ) p ) for p = 0 , , , which transversely intersect the preimages τ − ( Y i ) so the Gromov-Witten invariant GW mA,k ( c i τ ∗ y ⊗ · · · ⊗ c i k τ ∗ y k ) counts (for a generic J ) the number of J -holomorphic curves passing throughall of the τ − ( Y ℓ ) ∩ Σ ( ℓ ) i ℓ (where Σ ( ℓ ) i ℓ is choice of section/Gauss-lift/cut-locus,not necessarily the same for each value of i ℓ for the sake of transversality).If P ki =1 d i > then P kℓ =1 deg( Y ℓ ) < k − and we can perturb thesubmanifolds Y ℓ so that the intersection T kℓ =1 Y ℓ is empty. Then the modulispace of J − -holomorphic curves which touch all of Σ ( ℓ ) ∩ τ − ( Y ℓ ) is emptysince all J − -holomorphic curves are vertical. Therefore the Gromov-Witteninvariant is zero. (cid:3) Corollary 7. Suppose n = 3 . We have GW mA,k = 0 if k < m .Proof. For degree reasons we know that 12 + 4 m + 2( k − 3) + k X ℓ =1 (6 + deg( Y ℓ ) − i ℓ ) = 12 k which gives k X ℓ =1 deg( Y ℓ ) = 4( k − m ) − k X ℓ =1 i ℓ Since i ℓ ≤ we get k X ℓ =1 deg( Y ℓ ) ≤ k − m − The inequality k < m ensures that k − m − < k − . (cid:3) Obstruction method. Now we use an ‘obstruction bundle’ argumentto deal with the case k = m . Theorem 9. Let M be a hyperbolic 6-manifold ( n = 3 ) and m ≥ aninteger. We have GW mA,m = 0 (10) GW A, = 0 (11) It then follows from the divisor equation that GW A, = 0 . The proof will proceed by observing that in these cases there is a nonvan-ishing section of the obstruction bundle over the whole moduli space. If oneis willing to appeal to a general theory of Kuranishi structures à la Fukaya-Ono [13] that is enough to prove vanishing of the Gromov-Witten invariant, UANTUM COHOMOLOGY OF TWISTOR SPACES 31 but in our setting it should suffice to perturb the almost complex structureand indeed the explicit sections of the obstruction bundle we have arise (in-finitesimally) from precisely such perturbations. Therefore we outline theproof of a slightly more general theorem from which Theorem 9 will followbelow.Before we state the theorem, recall that if u = v ∪ · · · ∪ v k is a stablecurve then the linearised operator D u is just the restriction of L ki =1 D v i to the subspace of L ki =1 W ,p ( v ∗ i T Z ) consisting of k -tuples of vector fieldswhich agree at the nodal points. In our setting the image of D u is preciselythe image of L ki =1 D v i . To see this, observe that if η = D v i ξ i ∈ im( D v i ) thenthere is a vector field ξ j ∈ ker( D v j ) for all j such that ( ξ , . . . , ξ k ) agree at thenodes: when the domain of v j has a node n connecting it with the domain of v j ′ for which ξ j has already been constructed we let ξ Vj ′ be a vertical vectorfield in the kernel of D V Vv ′ j which agrees with ξ Vj at n (which exists by thetransitive isometric action of SO (2 n ) on the twistor fibre) and ξ Hj ′ be theconstant horizontal lift of τ ∗ ξ j ( n ) . In summary: Lemma 10. The dimension of coker( D u ) for any J − -holomorphic stablecurve u in the twistor space of a hyperbolic n -manifold depends only on thehomology class it represents. Theorem 10. Let • ( Z, ω ) be a semipositive N -dimensional symplectic manifold and J be the space of ω -compatible domain-dependent almost complex struc-tures (where the domain is S ), • J − ∈ J be a particular choice of such an almost complex structure, • β be a homology class in H ( Z ; Z ) , • X , . . . , X k be a collection of submanifolds such that codim( X × · · · × X k ⊂ Z k ) = 2 N + 2 c ( β ) + 2( k − and for any J ∈ JM ( J ) := M ( Z, β, { X i } , J ) denote the moduli space of J -holomorphic curves u representing β with k marked points z , . . . , z k in the domain such that u ( z i ) ∈ X i , • M ( J ) denote the stable map compactification of M ( J ) and M T ( J ) the stratum of stable maps modelled on a bubble tree T , • exc be an even integer (the excess dimension ),such that • each stratum M T ( J − ) (modelled on a bubble tree T with e edges) isa smooth manifold of dimension exc − e whose tangent space at u is isomorphic to the kernel of the homomorphism ker( D u ) → ⊕ ki =1 ν u ( z i ) X i given by projecting a vector field onto the normal direction to thesubmanifold X i . • the dimension of coker( D u ) is exc for all u ∈ M ( J ) , • for each u ∈ M T ( J − ) with || du || L ∞ < c , each T ′ < T and eachsufficiently small gluing datum a there is a neighbourhood ν of u inthe space of stable maps modelled on T and a gluing map Gl ( u, a, c ) : ν ∩ M T ( J − , c ) → M T ′ ( J − , ∞ ) satisfying Property ( † ) of Proposition 6. • there exists a δJ ∈ T J − J such that δJ ◦ du ◦ j im( D u ) for all J − -holomorphic stable maps u ∈ M ( J ) .Then the genus 0 Gromov-Witten invariant GW Zβ,k ( X , . . . , X k ) = 0 . When we have defined Property ( † ) of Proposition 6 we will show it issatisfied in our case, see Remark 8. We postpone the proof of Theorem 10to Section 11. Proof of Theorem 9. Equation (10) concerns the equality case m = k fromthe proof of Corollary 7. Therefore the only nonvanishing Gromov-Witteninvariants are of the form GW mA,m ( c τ ∗ y ⊗ · · · ⊗ c τ ∗ y m ) where P ( y ) , . . . , P ( y m ) are represented by submanifolds Y , . . . , Y m whichintersect transversely in a collection of points S . The moduli space of J − -holomorphic curves connecting the submanifolds is now ` s ∈ S C s where C s is the space of degree m stable curves in τ − ( s ) passing through the points p ℓ = Σ ( ℓ )3 ∩ τ − ( s ) . Let us write X i = Σ ( i )3 ∩ τ − ( Y i ) and let { z i } mi =1 be acollection of distinct points in S .Let v be a vector field on M such that at every point s ∈ S , v projectsorthogonally to a nonzero vector in any 4-plane corresponding to a twistorline connecting two of the points p i above s . Now δ v J as constructed inSection 8.1 satisfies the assumptions of Theorem 10.The same argument works for Equation (11) but one must be slightlycareful because now the submanifolds Y i can intersect in something biggerthan a point. The only issue is to find a suitable vector field v which has therelevant behaviour over Y ∩ Y ∩ Y , but because this triple intersection isnot the whole of M (for dimension reasons) it is always possible to do so. (cid:3) Proof of Theorem 10 We begin by stating the relevant implicit function and gluing theorems weneed for the proof. We have made our statements as close as possible to thosein [20] for the reader’s convenience. In the sequel ( Z, ω ) will always denote UANTUM COHOMOLOGY OF TWISTOR SPACES 33 a compact symplectic manifold, J the space of C r -differentiable domain-dependent ω -compatible almost complex structures (for some r > ). Definition 3. By an ǫ -perturbation at J we mean a smooth embedding κ : B ǫ → J of a finite-dimensional compact Euclidean ǫ -ball centred at κ (0) = J .For Y ∈ B ǫ we will write J Y := κ ( Y ) , g Y for the associated almost Kählermetric and W ,pY , L pY , C rY for norms taken with respect to the metric g Y .We write K := T J κ ( B ǫ ) . If u is a W ,p -map Σ → Z from a Riemann surface (Σ , j ) then we denote by ι u : K → Ω , ( u ∗ T Z ) the map sending Y ∈ K to Y ◦ du ◦ j (we blur the distinction between K and B ǫ , writing Y for elementsof either). We will also write B Y for the Banach manifold of W ,pY -maps from Σ to Z representing some homology class β . Recall that if d vol is a volume form on a complex Riemann surface (Σ , j ) then for any p > we denote by c p ( d vol Σ ) the norm of the Sobolev em-bedding W ,p (Σ) ֒ → C (Σ) where the norm on C (Σ) is the L ∞ -norm. Wealso note that for any Riemannian vector bundle E → Σ the L ∞ -norm of asection is bounded above by c p ( d vol Σ ) times its W ,p -norm (see [20], Remark3.5.1). Proposition 5 (Implicit function theorem) . Let (Σ , j ) be a compact Rie-mann surface and p > . Let κ be an ǫ -perturbation at J . Then for ev-ery constant c > there exists a constant δ > such that the followingholds for every volume form d vol Σ on Σ satisfying c p ( d vol) ≤ c . Suppose u ∈ W ,p (Σ , Z ) and ( ξ , Y ) ∈ W ,p (Σ , u ∗ T Z ) × T J κ ( B ǫ ) satisfy || du || L p ≤ c , || ξ || W ,p ≤ δ , || Y || C r ≤ δ , || ∂ J Y (exp g Y u ( ξ )) || L p ≤ δ c Moreover suppose that Q u : L p (Σ , Λ , ⊗ J u ∗ T Z ) → W ,p (Σ , u ∗ T Z ) × T J κ ( B ǫ ) is a right inverse of D u + ι u such that ( D u + ι u ) Q u = id , || Q u || ≤ c and suppose that ι u ( K ) is a complementary subspace to im( D u ) . Then thereexists a unique ( ξ ′ , Y ′ ) = Q u η ∈ W ,p (Σ , u ∗ T Z ) such that ∂ J Y + Y ′ (exp g Y + Y ′ u ( ξ + ξ ′ )) = 0 , || ξ + ξ ′ || W ,p ≤ δ , || Y + Y ′ || C r ≤ δ . Proof. The proof is almost identical to that of ([20], Theorem 3.5.2). Weexplain the setup and state the necessary quadratic estimate, leaving therest to the enthusiastic reader. We first observe that the norms W ,pY (or L pY )are Lipschitz equivalent for different Y by compactness of Z and that theLipschitz coefficient ℓ can be chosen uniformly since the ball B ǫ is compact.For each Y let r Y denote the injectivity radius of g Y and set I := min Y ∈ B ǫ (cid:16) r Y (cid:17) (We are perhaps overly cautious, but for the proof of the quadratic estimatewe will need to work deep inside a geodesic ball for a varying metric). Nowfor any ξ ∈ W ,p ( u ∗ T Z ) with || ξ || W ,p < ǫ = Ic ℓ the exponential map ξ exp g Y u ( ξ ) is an injective continuous map V := { ξ ∈ W ,p ( u ∗ T Z ) : || ξ || W ,p < ǫ }} → B Y whose image we denote by ν Y . Write ν := S Y ∈ B ǫ ν Y and observe thatexponentiation gives a trivialisation exp : V × K → ν There is a natural Banach bundle E over ν whose fibre at v ∈ ν Y is E ( v,Y ) := L pY (Σ , Λ , ⊗ J Y v ∗ T Z ) We must now trivialise this bundle compatibly with exp . First we use paralleltransport along geodesics using the J Y -Hermitian connection ˜ ∇ Y associatedto the Levi-Civita connection ∇ g Y to construct isomorphisms Φ ( v,Y ) : E ( u,Y ) → E ( v,Y ) We must still trivialise in the Y -direction. To this end we fix a smooth vectorfield X on Σ which vanishes on a set of measure zero. Recall that X and α ∈ L p ( u ∗ T Z ) together determine η ∈ E ( u,Y ) by the condition that η ( X/ | X | ) = α almost everywhere (since then η ( jX/ | X | ) = − J Y α almost everywhere) - L pY integrability follows from L p -integrability by Lipschitz equivalence of thenorms. This gives an isomorphism ψ Y : E ( u, → E ( u,Y ) for all Y . Now thecompositions ψ Y ◦ Φ ( v,Y ) : E := E ( u, → E ( v,Y ) give a trivialisation of thebundle exp ∗ E ∼ = E × V × K → V × K compatible with the diffeomorphism V × K → ν .The natural section ∂ : ν → E taking ( u, J ) to ∂ J ( u ) pulls back to asection F : V × K → E of the trivialisation which we consider as a functionbetween Banach spaces. We observe that d (0 , F ( ξ, Y ) = D u ξ + 12 Y ◦ du ◦ j The key step in proving the implicit function theorem is the quadratic esti-mate: Lemma 11. In the setting of Proposition 5, there exists a constant C > such that the following holds for every volume form d vol Σ with c p ( d vol Σ ) ≤ c . If || ξ || L ∞ ≤ c then || d ( ξ,Y ) F − D u − Y ◦ du ◦ j || ≤ C (cid:16) || ξ || W ,p + || Y || C r (cid:17) where the norm on the left is the operator norm. The proof of the proposition now follows precisely the same lines as ([20],Theorem 3.5.2). (cid:3) UANTUM COHOMOLOGY OF TWISTOR SPACES 35 Remark 7. When u is a stable curve modelled on a bubble tree T there is anexactly analogous statement which asserts existence and uniqueness of nearbystable curves modelled on the same bubble tree. We use the notation J T forthe space of C r -smooth domain-dependent ω -compatible complex structureswhose domain is modelled on a bubble tree T . We will also write T ′ < T to indicate that a bubble tree T ′ is obtained from T by merging bubbles (andhence decreasing the number of edges). Employing the notation of the proof of our implicit function theorem wemake the following observation. Scholium 1. If u is a J -holomorphic stable map and κ : B ǫ → J T is an ǫ -perturbation at J such that ι u ( K ) is a complement for im( D u ) then thereis a small ball ∈ U ⊂ V × K such that F − (0) ∩ U is the image of a C r -smooth map ker( D u ) ⊕ D u + ι u ) → V × K of the form ( ξ, ( ξ, 0) + Qφ ( ξ ) This is precisely the space of stable maps near u which are J Y -holomorphicfor some Y near 0. Before we state the gluing theorem we introduce some further notation.Let T and T ′ be bubble trees with T ′ < T : recall that a bubble tree consists ofa configuration of marked domains Σ i , i ∈ I where some of the marked pointsare called nodes. By a node n we mean a quadruple (Σ i ( n ) , Σ j ( n ) , z i ( n ) , z j ( n ) ) consisting of two (different) domains Σ i ( n ) and Σ j ( n ) and marked points z i ( n ) ∈ Σ i ( n ) , z j ( n ) ∈ Σ j ( n ) . We write N for the set of nodes and think of thedomain of the bubble tree as Σ T = [ i ∈ I Σ i / ( z i ( n ) ∼ z j ( n ) : n ∈ N ) Label the nodes which are merged in going from T to T ′ by M ⊂ N . Foreach n ∈ M define A n = T z i ( n ) Σ i ( n ) ⊗ C T z j ( n ) Σ j ( n ) .We will now construct a metric g ′ on Σ T ′ given a metric g on Σ T anddescribe how the complex structure changes (see [19]). By a metric we meana smooth Kähler metric on each component. This will depend on a choice of a n ∈ A n for each n ∈ N ′ ; we denote this choice by a and call it gluing data .Assume that g is flat in a neighbourhood of z i ( n ) and z j ( n ) for each n ∈ M and let exp i ( n ) : T z i ( n ) Σ i ( n ) → Σ i ( n ) , exp j ( n ) : T z j ( n ) Σ j ( n ) → Σ j ( n ) denote theexponential maps. Using the map ψ n : T z i ( n ) Σ i ( n ) \ { } → T z j ( n ) Σ j ( n ) \ { } , ψ n ( x ) = a n x we can glue the domains Σ i ( n ) \ exp i ( n ) ( B √ | a n | ) and Σ j ( n ) \ exp j ( n ) ( B √ | a n | ) via the merging identification exp j ( n ) ◦ ψ n ◦ exp − i ( n ) . Choose a function χ n :(0 , ∞ ) → (0 , ∞ ) such that χ n ( s ) = 1 when s is slightly larger than p | a n | and such that χ n ( | x | ) | dx | is ψ n -invariant (note that ψ ∗ n ( | dx | ) = (cid:12)(cid:12) a n x (cid:12)(cid:12) | dx | ). The metric g ′ is defined using this invariant metric to extend g over the necksintroduced by merging bubbles.We also need to define how the family of almost complex structures κ changes under gluing. Definition 4. Given bubble trees T ′ < T and an ǫ -perturbation κ : B ǫ → J T centred at J we say κ is gluable if there exists a constant µ such that on a µ -neighbourhood of each n ∈ M in the domain the almost complex structure J Y is domain-independent for all Y ∈ B ǫ . Given a gluable ǫ -perturbationand a bubble tree T ′ < T and a choice of a we define its a -gluing to be the ǫ -perturbation κ a : B ǫ → J T ′ where, for z ∈ Σ T ′ , κ a ( Y )( z ) equals κ ( Y )(˜ z ) ,where ˜ z ∈ Σ T is sent to z under the merging identification. Note that this iswell-defined whenever | a n | < µ for all n ∈ M because κ is gluable. We do not give a proof of gluing but refer the reader to ([20], Chapter 10)for a detailed proof without varying J and [19] for a less detailed proof withvarying J . Proposition 6 (Gluing) . Let u be a stable J -holomorphic curve modelledon a bubble tree T and suppose that κ : B ǫ → J T is a gluable ǫ -perturbationcentred at J with the further property that ι u ( dκ ( K )) is a complement for im( D u ) . Let M T ( κ, c ) denote the moduli space of stable maps modelled onthe bubble tree T which are J Y -holomorphic for some Y ∈ B ǫ and satisfy || du i || L ∞ ≤ c for all components u i . Then for any T ′ < T there is an ǫ ′ < ǫ , a neighbourhood ν of u in the space of stable maps modelled on T , anon-increasing function (0 , ∞ ) → (0 , 1) : c r ( c ) and, for each ( c, a ) with < | a n | < r ( c ) , c > || du || L ∞ an embedding Gl ( u, a, c ) : ( ν × B ǫ ′ ) ∩ M T ( κ, c ) → M T ′ ( κ R , ∞ ) with the obvious analogues of properties (i)-(iv) in ( [20] , Theorem 10.1.2).In particular ( [20] , Corollary 10.1.3) ( † ) if r j → ∞ , a j is a sequence of gluing data with | a j,n | < r j and u j ∈ M T ′ ( κ j , ∞ ) is a sequence which Gromov-converges to u thenthere is a j such that for all j ≥ j we have u j ∈ im( Gl ( u, a, c )) . We now return to the proof of Theorem 10. We will denote by M univ T the universal moduli space of pseudoholomorphic stable maps modelled ona tree T representing the class β , considered as a subset of B T . Here B T := S J ∈J B J,T denotes the union over J ∈ J of the space of W ,pJ -maps modelledon a bubble tree T . Proof. Consider a smooth 1-parameter family J t ∈ J of domain- independent almost complex structures such that J = J − and ∂∂t (cid:12)(cid:12) t =0 J t = δJ . Assumethat the Gromov-Witten invariant is nonvanishing so that for each t ∈ [0 , there exists a J t -holomorphic stable map u ′ t in the class β . By Gromovcompactness we can extract a subsequence < t k → such that u t k = u ′ t k ◦ φ k converges (for some sequence of reparametrisations φ k ) to a stable UANTUM COHOMOLOGY OF TWISTOR SPACES 37 J -holomorphic map u . We may artificially add marked points to the domainto ensure there are no automorphisms.Pick an extension of J t to an ǫ -perturbation at J − κ : B exc ǫ → J (where B exc ǫ is an exc -dimensional Euclidean ball) with the property that K := im( d κ ) ⊂ T J − J satifies ι u ( K ) ∩ im( D u ) = { } The existence of this perturbation is precisely the transversality theorem([20], Theorem 3.2.1) applied to each (possibly non-simple) component of thestable curve u - transversality can be achieved for the non-simple componentsby allowing domain-dependent J s since we are in a semi-positive symplecticmanifold. Note that to define Gromov-Witten invariants we cannot achievethis transversality simultaneously over all strata because the bubbles thatdevelop have domain-independent almost complex structures. That is notour goal: we wish to find a contradiction to the existence of the particularGromov-convergent sequence u k we constructed under the assumption thatthe Gromov-Witten invariant was non-zero.By passing to a subsequence we can assume that all u k are modelledon the same bubble tree T ′ . Let us first assume that u is also modelledon T ′ . Observe that by construction D u + ι u is surjective and Fredholmand therefore admits a right inverse Q u . Scholium 1 tells us that there isa neighbourhood U of ( u, J − ) such that all J Y -holomorphic stable maps exp g Y u ( ξ ) with ( ξ, Y ) ∈ U are in the image of a C r -smooth map ( ξ, ( ξ, 0) + Q u φ ( ξ ) defined on the kernel ker( D u ) ⊕ D u + ι u ) . For k large enough u k is J Y k -holomorphic (for some Y k = 0 ) and of this form since the sequenceconverges to in the C r -topology to u . However, we know that M T ( J − ) is a smooth manifold near u with tangent space ker( D u ) . This implies that Q u ( φ ( ξ ) ∈ V × { } for ξ small enough. This contradicts the fact that Y k = 0 .Therefore u must be modelled on a different bubble tree, T ′ < T .We choose a gluable ǫ -perturbation κ : B exc ǫ → J T ′ centred at J − whichcontains the family J t and which satisfies ι u ( dκ ( K )) ∩ im( D u ) = 0 . Theexistence of this perturbation is guaranteed by transversality, for we canpick the almost complex structures domain-dependently away from the µ -neighbourhoods of the nodes. Since our original family J t consisted ofdomain-independent almost complex structures the a -gluing κ a contains J t for all a small enough. Pick c > || du || L ∞ . By assumption the gluing map Gl ( u, a, c ) lands in M T ( J − , ∞ ) for a small enough. By Proposition 6 ( u k , J k ) lies in the image of Gl ( u, a, c ) for large k , however J k = J − by assumption.This is a contradiction. (cid:3) Remark 8. It remains to show that Property ( † ) of Proposition 6 holds forour moduli spaces of J − -holomorphic curves in the twistor fibre. This follows from [25] , since J − | F is an integrable complex structure on the twistor fibreand the twistor fibre is a convex manifold (i.e. all holomorphic curves areregular). The main theorem of [25] therefore implies that the moduli spaceof stable maps into each twistor fibre is a smooth orbifold. In particular ittells us that in the neighbourhood of a stable J − -holomorphic curve u withno automorphisms the moduli space is a smooth manifold of dimension exc (which is equal to the expected dimension of u considered as a curve in thetwistor fibre). In the cases we need it is easy to check that the strata are allstill smooth upon intersecting with the explicit cycles X , . . . , X k . Floer theory of Reznikov Lagrangians Obstruction term. The following notion was introduced by Fukaya,Oh, Ohta and Ono in a more general context in their book [12]. It arises asthe first of an infinite sequence of filtered A ∞ operations on a suitable spaceof singular chains on L . Definition 5. Let L be a monotone Lagrangian. If J is regular for all modulispaces of Maslov 2 discs then the obstruction m is the chain represented bythe evaluation map from the moduli space of Maslov 2 discs with a singleboundary marked point to L . In our case ( n = 3 ) there is precisely one component in this moduli space,corresponding to the hemispheres of real algebraic lines in CP (with bound-ary on SO (3) ∼ = RP ). The expected dimension of the moduli space is n + µ − L Σ ) so the obstruction cycle is homologous to amultiple of the fundamental class. Let us write FF µ,k for the Fukaya-Floerchain ev : M µ,k → L k Σ where M µ,k denotes the moduli space of Maslov- µ discs with boundary on L Σ and k boundary marked points. We now prove the first part of TheoremC. Theorem 11. If Σ is an oriented totally geodesic submanifold of an orientedhyperbolic 6-manifold M and L Σ denotes the Reznikov Lagrangian lift in thetwistor space of M then m = ± √ q [ L Σ ] . Proof. The moduli space of Maslov 2 discs (twistor hemispheres) is compactso we can employ obstruction bundle techniques to compute FF , (notethat by definition [FF , ] = m ). Since the totally geodesic submanifold Σ is oriented it has the form Γ Σ \ SO + (3 , × SO (3) /SO (3) × SO (3) and the Reznikov lift is Γ Σ \ SO + (3 , × SO (3) /SO (3) ∆ UANTUM COHOMOLOGY OF TWISTOR SPACES 39 where SO (3) ∆ = SO (3) × SO (3) ∩ U (3) is the diagonal subgroup. A twistorline with boundary on L Σ can be specified by giving a unit vector v ∈ T p Σ and a unit normal vector w ∈ ν p Σ and taking the set of ψ preserving the2-plane h v, w i . Hence the moduli space of twistor hemispheres is a S × S =˜Gr ( RP ) -bundle over ΣΓ Σ \ SO + (3 , × SO (3) / ( SO (2) × SO (2)) Adding a marked point on the boundary we obtain Γ Σ \ SO + (3 , × SO (3) /SO (2) ∆ The linear analysis of the ∂ -operator is identical to the case of closed curvesexcept that we only allow deformations which come from vector fields on Σ (that is, H = τ ∗ ( T M ) is replaced by τ | ∗ L Σ ( T Σ) ). This implies that theobstruction bundle is 2-dimensional with Euler class equal to the Euler classof the tautological SO (2) -bundle. The 1-point invariant FF , is thereforegiven by evaluating the fibre integral of the this Euler class which gives ± .The ± comes from the choice of spin structure on L Σ (or alternativelyfrom picking a flat connection with holonomy ± around the nontrivial loopin the SO (3) factor) and the √ q in the formula for m comes from thearea of holomorphic discs (if q = exp( − R A ω ) for a twistor line A then √ q = exp( − R h ω ) for a hemisphere h ).Note that L Σ admits a spin structure since it is diffeomorphic to Σ × SO (3) both factors of which are spin, the principal frame bundle of Σ being trivialsince Σ is an orientable 3-manifold. Changing the spin structure along thenontrivial loop in SO (3) changes the sign of m while changing the spinstructure along a loop from Σ has no effect (there are no discs with such aboundary). (cid:3) Quantum homology of Reznikov Lagrangians. We finish theproof of Theorem C by calculating the quantum homology of a ReznikovLagrangian using the techniques and definitions of [2]. We recall that thequantum homology QH ( L ) of an oriented monotone Lagrangian subman-ifold L is defined to be the homology of the pearl complex associated toa choice of Morse function F on L , metric on L (such that the gradientflow is Morse-Smale) and generic almost complex structure on Z . The chaingroups are the free C -modules on the critical points of F and the differentialcounts oriented “pearly trajectories” which are sequences of F -gradient flow-lines and J -holomorphic discs with boundary on L . Although the theory isdeveloped in [2] with Z / -coefficients the orientation issue is cleared up in([3], Appendix A). Theorem 12. Let Σ be an oriented totally geodesic 3-dimensional subman-ifold of an oriented hyperbolic 6-manifold. The quantum homology of theReznikov lift L Σ is QH ∗ ( L ) ∼ = H ∗ ( L ; C [ t ]) where we write t = q / for the Novikov parameter. We recall that the quantum homology of a monotone Lagrangian is (non-canonically) isomorphic to its self-Floer homology so this proves TheoremC. Proof. We pick a Morse function f on Σ and the standard Morse function r with four critical points on RP . We assume that the gradient flow on L Σ ∼ = Σ × SO (3) of the function F = f + r with respect to the product ofthe hyperbolic and round metrics is Morse-Smale (by suitable choice of f )and that f (and hence F ) has a unique maximum and a unique minimum.We may also assume that f is self-indexing.There is ([2], Proposition 6.1.1, Proof A) a homology spectral sequencewhose E -page is E i,j = H i − j ( L Σ ; C ) t − j and which converges to the quantum homology of L Σ . We draw the E pagebelow (denoting b (Σ) =: b ) and indicate the differentials we will show to bezero. C C t C b C t C b t C b C t C b t C b t C C b t C b t C t C b C b t C t C b t C b C t C b t C b t CC b t C b t C t C b t C t C t That the other differentials vanish follows either for degree reasons or bya combination of Poincaré duality and the Leibniz property of the higherdifferentials with respect to cup product (which is certainly true on the E -page and continues to be true on the E and E pages because the E and E differentials vanish). Let us write δ ri,j for the r -th differential whose domainis E ri,j .The differentials all vanish, but not all for the same reason. We now tacklethe reasons the differentials vanish case by case. Filtering by the value of f : We observe that the contribution to thehigher differentials from pearly trajectories joining a critical point p to acritical point q vanishes when f ( p ) > f ( q ) . To see this, let J k be a sequence UANTUM COHOMOLOGY OF TWISTOR SPACES 41 of regular almost complex structures with J k → J − as k → ∞ and supposeto the contrary that there is a nonzero differential connecting p to q . Let u k be a pearly trajectory contributing to this differential. We can extracta convergent subsequence u k ′ and the limit is a broken pearly trajectorywhose discs are J − -holomorphic and therefore contained in level sets of thefunction f . Since the gradient flow decreases f and the discs do not allowone to return to larger values of f we see that f ( p ) > f ( q ) . This argumentproves vanishing of δ , , δ , and δ , . Easy obstruction bundle methods: To prove that δ , = 0 noticethat this differential counts (for a regular J ) pearly trajectories connectinga critical point y of index 2 (which has index 2 as a critical point of f and0 as a critical point of r ) to the critical point q of index 3 correspondingto the maximum of r and the minimum of f . There is precisely one J -holomorphic disc in this trajectory and it has Maslov index 2. Such a pearlytrajectory corresponds precisely to a J -disc whose boundary intersects theunstable manifold of y and the stable manifold of q . Since the moduli spaceof Maslov 2 discs is compact by minimality of the relative homology classthe Fukaya-Floer chain FF , of Maslov 2 discs with two boundary markedpoints is a cycle: its boundary has two types of component, where the firstmarked point approaches the second from a clockwise or an anticlockwisedirection. These cancel so the boundary of FF , is the zero chain. Bya priori compactness of the moduli space we can compute the homologyclass of this cycle using the obstruction bundle techniques we used to proveTheorems 8 and 11. The moduli space M , is diffeomorphic to S × S × Σ so if we write x, y ∈ H ( S × S ; Z ) for a Z -basis with x = y = 0 then thediagonal decomposition for ∆ : M , → M , is { x ⊗ y + 1 ⊗ xy } ∪ τ ∗ ∆ (1 Σ ) The Euler class of the obstruction bundle is x (which is the Euler class of oneof the two tautological SO (2) -bundle over S × S = SO (4) /SO (2) × SO (2) ).Cupping the diagonal with x ⊗ gives { xy ⊗ x ) } ∪ τ ∗ ∆ (1 Σ ) In each fibre the moduli space of Maslov 2 discs with one marked point M , ( F ) is a circle bundle over S × S with Euler class x + y and (by theGysin sequence) it has H ( M , ( F ); Z ) = 0 Therefore the pullback of xy to M , vanishes. In particular the Fukaya-Floer cycle is nullhomologous. Since the count of pearly trajectories con-tributing to this differential is just the intersection number of this cycle withthe product of the stable manifold of q and the unstable manifold of y , thedifferential vanishes. Hard obstruction bundle methods: To prove that δ , = 0 noticethat this differential counts (for a regular J ) pearly trajectories going fromthe global minimum p of F to the critical point q of index 3 corresponding to the maximum of r and the minimum of f . (That the differential hasno contribution from pearly trajectories going from p to the critical point q ′ corresponding to the maximum of f and the minimum of r follows bythe previous filtering argument.) Such trajectories consist of a (Maslov 4) J -holomorphic disc through the global minimum whose boundary intersectsthe stable manifold of q . Assume that the differential is nonzero and thattherefore such pearly trajectories exist for any J and suppose that J t is afamily of almost complex structures obtained by exponentiating an infini-tesimal deformation δ v J at J − associated to some vector field v on M asin the proof of Theorem 10. There is a sequence of J t i -holomorphic pearlytrajectories ( J t i → J − ) which converges to some limit trajectory u . Since J − -holomorphic discs are restricted to lie within a single twistor fibre andthe stable manifold of q intersects the fibre containing p and q precisely at q we know that u is just a stable Maslov 4 disc. Now a gluing or implicitfunction theorem argument as in the proof of Theorem 10 (modified as inSection 4 of [2] to the case of holomorphic discs) shows that, for suitablechoice of v , the J t i -holomorphic pearly trajectories cannot exist for i largeenough. A similar argument proves δ , = 0 . (cid:3) As was remarked in the introduction, this tallies with the fact that aLagrangian with nonvanishing self-Floer cohomology has obstruction termequal to an eigenvalue of the first Chern class acting by quantum producton the quantum cohomology. 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