Peterson conjecture via Lagrangian correspondences and wonderful compactifications
PPETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES ANDWONDERFUL COMPACTIFICATIONS
HANWOOL BAE AND NAICHUNG CONAN LEUNGA
BSTRACT . For a simply-connected compact semisimple Lie group G and its maximal torus T ,we study the A ∞ -functor associated to the moment Lagrangian correspondence from the cotan-gent bundle T ∗ G to the square G/T − × G/T . In particular, we compute the leading term of the A ∞ -homomorphism from the wrapped Floer cohomology HW ∗ ( T ∗ e G, T ∗ e G ) of the cotangentfiber T ∗ e G to the Floer cohomology HF ∗ (∆ , ∆) of the diagonal ∆ in the square G/T − × G/T by determining the count of certain pseudo-holomorphic quilts. As a consequence, we provethat the Floer cohomologies HW ∗ ( T ∗ e G, T ∗ e G ) and HF ∗ (∆ , ∆) are isomorphic as rings after alocalization. C ONTENTS
1. Introduction 12. Geometric setup 63. Review on Fukaya categories 254. A ∞ -functor associated to the moment Lagrangian 335. A partial computation of the functor Φ C NTRODUCTION
Let G be a simply-connected compact semisimple Lie group and let T be a maximal torussubgroup of G . Then the quotient G/T is the full flag variety associated to G . The Peterson’sconjecture, which first appeared in the lecture [18], says that the homology of the based loopspace of G and the quantum cohomology of G/T are isomorphic as algebras after localizations.The conjecture was first proved combinatorially by Lam and Shimozono in [13] (See also [15]).Indeed, they made use of the quantum Schubert calculus and the affine Schubert calculus tocompute the ring structures on both algebras and compared them to get the isomorphism. As aresult, their approaches do not provide a geometric understanding behind the isomorphism.The aim of this paper is to provide a geometric proof for the Peterson’s conjecture usingFloer theoretic techniques. There are two foundational theorems our work is based on. The firsttheorem can be stated as follows: a r X i v : . [ m a t h . S G ] F e b HANWOOL BAE AND NAICHUNG CONAN LEUNG
Theorem 1.1 (Abbondandolo-Schwarz [1], Abouzaid [3]) . Let M be a compact smooth mani-fold. For any x ∈ M , there is a graded K -algebra isomorphism between H −∗ (Ω x M ) ∼ = HW ∗ ( T ∗ x M, T ∗ x M ) , where both H −∗ (Ω x M ) and HW ∗ ( T ∗ x M, T ∗ x M ) are based over a field K of arbitrary character-istic. For the second one, recall that, for any compact symplectic manifold ( Y, ω ) , the diagonal ∆ Y ⊂ Y − × Y is a Lagrangian submanifold of Y − × Y . Here Y − denotes the symplecticmanifold ( Y, − ω ) . Theorem 1.2 (Fukaya-Oh-Ohta-Ono [10]) . There is a graded ring isomorphism QH ∗ ( Y ) ∼ = HF ∗ (∆ Y , ∆ Y ) , where both QH ∗ ( Y ) and HF ∗ (∆ Y , ∆ Y ) are based over the universal Novikov ring. The proof of Theorem 1.2 suggests that the theorem still holds if the universal Novikov ring isreplaced by the group ring of π ( Y − × Y, ∆ Y ) for any monotone symplectic manifold Y .Now if there is a graded ring homomorphism Φ from HW ∗ ( T ∗ e G, T ∗ e G ) to HF ∗ (∆ G/T , ∆ G/T ) that induces an isomorphism after a localization, then one can think of the following diagram: HW ∗ ( T ∗ e G, T ∗ e G ) Theorem . (cid:15) (cid:15) Φ (cid:47) (cid:47) HF ∗ (∆ G/T , ∆ G/T ) Theorem . (cid:15) (cid:15) H −∗ (Ω e G ) QH ∗ ( G/T ) and the Peterson’s conjecture follows as a consequence.To clarify the statement above, note that the wrapped Floer cohomology HW ∗ ( T ∗ e G, T ∗ e G ) isan infinite dimensional vector space over a field K , while the Floer cohomology HF ∗ (∆ G/T , ∆ G/T ) is a finitely generated module over a Novikov ring Λ K , which is defined by the group ring of π ( G/T × G/T, ∆ G/T ) over the field K in our case. Indeed we have Theorem 1.3 (Theorem 5.6) . The wrapped Floer cohomology HW ∗ ( T ∗ e G, T ∗ e G ) is a K -vectorspace generated by the classes of Hamiltonian chords { x h | h ∈ Q ∨ } , where Q ∨ is the corootlattice of the Lie algebra of G . Theorem 1.4 (Theorem 5.9) . The Floer cohomology HF ∗ (∆ G/T , ∆ G/T ) is a free Λ K -modulegenerated by the classes of Hamiltonian chords { x w | w ∈ W } , where W is the Weyl group of G . We need the following key lemma.
Lemma 1.5 (Lemma 5.16) . There is an injective graded K -algebra homomorphism Φ : HW ∗ ( T ∗ e G, T ∗ e G ) → HF ∗ (∆ G/T , ∆ G/T ) , given by Φ( x h ) = x w h q a h + ( higher action terms ) , ∀ h ∈ Q ∨ for some w h ∈ W and a h ∈ π ( G/T − × G/T, ∆ G/T ) . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES By finding a proper filtration on the Floer cohomology HF ∗ (∆ G/T , ∆ G/T ) , we finally provethe following main theorem Theorem 1.6. (Theorem 5.25) The homomorphism Φ is injective and it induces an isomorphismafter a localization on HW ∗ ( T ∗ e G, T ∗ e G ) .Remark . A closed string analogue of Lemma 1.5 and Theorem 1.6 can be achieved usingsimilar techniques. More precisely, we have an algebra homomorphism from the symplecticcohomology of the cotangent bundle T ∗ G to the quantum cohomology of G/T − × G/T and weexpect that the homomorphism can be explicitly described as in Lemma 1.5.The homomorphism Φ is actually given as a part of the A ∞ -functor associated to the momentLagrangian correspondence C ⊂ T ∗ G − × ( G/T − × G/T ) given by C = { ( g, µ ( y ) , g · y, y ) ∈ G × g ∗ × G/T × G/T | g ∈ G, y ∈ G/T } . Indeed, Evans and Lekili([8]) studied the A ∞ -functor Φ C from the wrapped Fukaya categoryof T ∗ G to the extended Fukaya category of Y − × Y when Y admits a Hamiltonian G -action.The construction of such an A ∞ -functor was first carried out in [16] and its variation to the casewhen the domain manifold is a Liouville domain was done in [8], which we adopted in thispaper. The A ∞ -functor Φ C maps any (admissible) Lagrangian submanifold L of T ∗ G to thealgebraic composition ( L, C ) , which is quasi-isomorphic to the geometric composition L ◦ C ifit is transverse and embedded. At the level of morphisms, the A ∞ -functor consists of Φ dC : d − (cid:79) i =0 CW ( L i , L i +1 ) → CF (( L , C ) , ( L d , C )) , d ≥ . In particular, as the geometric composition T ∗ e G ◦ C is transverse and embedded, the cotangentfiber T ∗ e G is mapped to T ∗ e G ◦ C under the A ∞ -functor Φ C up to quasi-isomorphism. Since thegeometric composition T ∗ e G ◦ C is given by the diagonal ∆ G/T , we get an A ∞ -homomorphism Φ = Φ C : HW ∗ ( T ∗ e G, T ∗ e G ) → HF ∗ (∆ G/T , ∆ G/T ) as desired.At this point, the wonderful compactification provides a geometric way of understanding the A ∞ -homomorphism Φ C . Let G ad = G/C ( G ) be the quotient of G by its center and denote itscomplexification by G C ad . The wonderful compactification G C ad of the complexified Lie group G C ad is given by adding some divisors to the Lie group G C ad in such a way that the intersectionof all the divisors is isomorphic to G/T × G/T . Since the cotangent bundle T ∗ G ad and thecomplexified Lie group G C ad can be identified via the Cartan decomposition, it can be regardedas a compactification of the cotangent bundle. Accordingly, we observed that there is an openneighborhood U ad of the minimal stratum G/T − × G/T in the compactification T ∗ G ad that isidentified with a sub-bundle of the normal bundle of the minimal stratum.With the picture described in the previous paragraph in mind, the A ∞ -functor Φ C maps thecotangent fiber T ∗ e G to its “boundary at infinity” in the sense that the intersection of the minimalstratum G/T × G/T and the closure of T ∗ e G ad in the compactification is given by the diagonal ∆ G/T and that the quotient map T ∗ G → T ∗ G ad maps the cotangent fiber T ∗ e G diffeomorphicallyto the cotangent fiber T ∗ e G ad . HANWOOL BAE AND NAICHUNG CONAN LEUNG
To help understand the result of our work, let us consider the simplest case : G = SU (2) .In this case, we may take the maximal torus T to be the diagonal subgroup of G , which is ofrank . Since the center of SU (2) is {± } , the corresponding quotient Lie group is given by G ad = P SU (2) = SU (2) / {± } . Accordingly, the complexification of G ad is given by G C ad = P SL (2 , C ) = (cid:26)(cid:18) a bc d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ad − bc (cid:54) = 0 (cid:27) / C ∗ . The wonderful compactification G C ad of G C ad is given by G C ad = C P = (cid:26)(cid:18) a bc d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ( a, b, c, d ) (cid:54) = (0 , , , (cid:27) / C ∗ , which is obtained from G C ad by adding a divisor D given by D = (cid:26)(cid:18) a bc d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ad − bc = 0 , ( a, b, c, d ) (cid:54) = (0 , , , (cid:27) / C ∗ . Note that the full flag variety
G/T is isomorphic to the projective line C P and the divisor D isisomorphic to C P × C P as the divisor D coincides with the image of a (non-standard) Segreembedding C P × C P → C P given by ([ x : y ] , [ x : y ]) (cid:55)→ (cid:20) x x x y y x y y (cid:21) . Furthermore, the cotangent fiber T ∗ e G ad is given by (cid:26)(cid:18) a bb ∗ d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ad − | b | (cid:54) = 0 , a, d ∈ R , b ∈ C (cid:27) / C ∗ and its closure in the compactification G C ad intersects the divisor D in (cid:26)(cid:18) a bb ∗ d (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ad − | b | = 0 , a, d ∈ R , b ∈ C (cid:27) / C ∗ , which is the diagonal of the D = C P × C P in the identification above.As mentioned above, the wrapped Floer cohomology HW ∗ ( T ∗ e G, T ∗ e G ) is a K -vector spacegenerated by the coroot lattice Q ∨ . Here if the Lie algebra t of the maximal torus T is identifiedwith the real line R and exp − ( C ( G ) ∩ T ) = Z ⊂ R for the exponential map exp : t → T , thenthe coroot lattice is just given by Z . Hence we may write HW ∗ ( T ∗ e G, T ∗ e G ) = K (cid:104) x m | m ∈ Z (cid:105) . On the other hand, since the Floer cohomology HF ∗ (∆ G/T , ∆ G/T ) is isomorphic to the quan-tum cohomology QH ∗ ( G/T = C P ) , it is isomorphic to Λ K (cid:104) , H (cid:105) = K [ q − , q ] (cid:104) , H (cid:105) as K -vector space, where q is the formal variable of degree 4 in the Novikov ring and H denotesthe hyperplane class. To relate this with Theorem 1.4, observe that the Weyl group W of G isisomorphic to Z / . Then the hyperplane class H corresponds to ∈ { , } = Z / ∼ = W . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Then the homomorphism Φ in Lemma 1.5 is given by Φ( x m ) = (cid:40) q m m ≥ ,q − m H m < . Note that all the higher action terms vanish in this case. Also, it is straightforward to see that thehomomorphism Φ is injective. The injectivity further implies that the subset S = K (cid:104) x m | m ∈ Z ≥ (cid:105) ⊂ K (cid:104) x m | m ∈ Z (cid:105) = HW ∗ ( T ∗ e G, T ∗ e G ) is closed under multiplication. Finally if wetake the localization on the wrapped Floer cohomology with respect to the subset S , then we getthe desired isomorphism in Theorem 1.6.To point out the main difficulty we need to resolve in this work, first recall that the A ∞ -functor Φ C is defined by counting pseudo-holomorphic quilts [16, 8], which consist of pairs of(inhomogeneous) pseudo-holomorphic curves u : S → T ∗ G and u : S → G/T − × G/T ,where S and S are certain patches of a disk with boundary punctures.One crucial step of the proof is to show that the image of the map u lies in the neighborhood U for every holomorphic quilt, where U is the pre-image of U ad under the covering map T ∗ G → T ∗ G ad . Then we analyzed the projection ( (cid:101) π ◦ u , u ) of the pseudo-holomorphic quilts to theminimal stratum G/T − × G/T where (cid:101) π : U →
G/T − × G/T is the composition of the coveringmap
U → U ad and the projection U ad → G/T − × G/T , regarding U ad as a subset of the normalbundle of the minimal stratum. For this purpose, we had to require the almost complex structureson T ∗ G to have the following matrix representation(1.1) J = (cid:18) J ∗ J (cid:19) with respect to the decomposition T ( T ∗ G ) | U = Ver ⊕ Hor, where Ver is the kernel of the differ-ential of the projection and Hor is a horizontal lift of the tangent bundle of the minimal stratum.But this causes a technical issue since, a priori, it is not obvious that one can get the transversalityof the Fredholm operator associated to the pseudo-holomorphic quilts by allowing only this re-stricted family (1.1) of almost complex structures. The idea to use almost complex structures ofthe form 1.1 has been already suggested in [5, 7]. But we extend the technique to the categoricallevel in this paper.We expect that the technique developed in this work can be applied not only to Lie groups, butalso to certain symmetric spaces and consequently this would lead to a generalized Peterson’sconjecture.1.1.
Outline of paper.
In Section 2, we provide a geometric setup for the paper. This includessome preliminaries on Lie theory, a brief introduction of wonderful compactification and a de-tailed description of the Liouville structure on the cotangent bundle of a semisimple Lie group G . Moreover the moment Lagrangian is introduced in this section, which plays an important rolein this paper.In Section 3, we review the construction of the extended monotone Fukaya category and thewrapped Fukaya category. HANWOOL BAE AND NAICHUNG CONAN LEUNG
In Section 4, following the idea of [16] and [8], we review the construction of the A ∞ -functorassociated to the moment Lagrangian correspondence from the cotangent bundle of G to thesquare G/T − × G/T of the full flag variety.In Section 5, we provide an explicit description of the A ∞ -homomorphism from the endo-morphism space of the cotangent fiber T ∗ e G to that of the diagonal in G/T − × G/T , which isgiven as the first piece of the A ∞ -functor introduced in Section 4. It will be shown that the A ∞ -homomorphism induces an algebra isomorphism between a localization of the wrapped Floercohomology of the cotangent fiber and the Floer cohomology of the diagonal in the square of thefull flag variety G/T as desired.
Acknowledgements.
We would like to thank Chi Hong Chow for many helpful discussions andfor pointing out important issues. We are also grateful to Cheol-Hyun Cho, Otto van Koert,Yanki Lekili, Changzheng Li and Weiwei Wu for useful comments and encouragements. Thiswork was substantially supported by grants from the Research Grants Council of the Hong KongSpecial Administrative Region, China (Project No. CUHK14301619 and CUHK14306720), theNational Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(No.2020R1A5A1016126) and a direct grant.2. G
EOMETRIC SETUP
Preliminaries on Lie theory.
Root and weight.
Let us first introduce some notions from Lie theory that will be fre-quently used in this paper. Please refer to [11] for more detailed explanation. • G is a connected compact semisimple Lie group and g is the Lie algebra of G . • G C is the complexification of G and g C is the Lie algebra of G C . • T is a maximal torus of G and t is the Lie algebra of T . • T C < G C is the complexification of T and t C is the Lie algebra of T C .Here the Lie algebra g C (resp. t C ) can be understood as the complexification of g (resp. t .)The Killing form κ : g C ⊗ g C → C is defined by κ ( u, v ) = tr ( ad u ◦ ad v ) , ∀ u, v ∈ g C . Since the restriction of κ to g ⊗ g is negative definite, we define an Ad-invariant inner producton g by ( u, v ) = − κ ( u, v ) , ∀ u, v ∈ g . We denote the dual space of t by t ∨ . Definition 2.1 (Root and weight) . (1) An element α ∈ t ∨ is called a root if there exists a pair e α , f α ∈ g \ such that [ h, e α ] = α ( h ) f α and [ h, f α ] = − α ( h ) e α , ∀ h ∈ t . Let us denote the two dimensional subspace R (cid:104) e α , f α (cid:105) by g α .(2) An element α ∨ ∈ t is called a coroot if there exists a root α such that (cid:18) α ∨ ( α ∨ , α ∨ ) , h (cid:19) = α ( h ) , ∀ h ∈ t . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES (3) An element λ ∈ t ∨ is called a weight if λ ( α ∨ ) ∈ Z for any coroot α ∨ .(4) An element λ ∨ ∈ t is called a coweight if α ( λ ∨ ) ∈ Z is for any root α .(5) The root lattice Q is the subgroup of t ∨ generated by all roots.(6) The coroot lattice Q ∨ is the subgroup of t generated by all coroots.(7) The weight lattice P is the subgroup of t ∨ generated by all weights.(8) The coweight lattice P ∨ is the subgroup of t generated by all coweights.Since every (co)root is a (co)weight, the (co)root lattice is a sub-lattice of the (co)weightlattice.Now let (cid:101) G be the universal cover of G and let (cid:101) T be the maximal torus of (cid:101) G given by the pre-image of the maximal torus T under the covering map. Then the Lie algebra (cid:101) t of (cid:101) T is identifiedwith t naturally. Now we introduce new lattices in t as follows. • t Z is the kernel of the exponential map t → T . • (cid:101) t Z is the kernel of the exponential (cid:101) t → (cid:101) T .It is straightforward to check that (cid:101) t Z is a sub-lattice of t Z when t and (cid:101) t are identified. Lemma 2.2. If G is a compact semisimple Lie group, then we have(1) Q ∨ = (cid:101) t Z .(2) π ( G ) ∼ = t Z / (cid:101) t Z .In particular, if G is of adjoint type, that is, its center is trivial, then we have(3) P ∨ = t Z .(4) π ( G ) ∼ = P ∨ /Q ∨ ∼ = C ( (cid:101) G ) , where C ( (cid:101) G ) is the center of the universal cover (cid:101) G of G . Inparticular, the Lie group G is isomorphic to the quotient group (cid:101) G/C ( (cid:101) G ) . Let us now fix bases for these lattices. • { α , . . . , α l } ⊂ Q is a basis for the root system of g . • { α ∨ , . . . , α ∨ l } ⊂ Q ∨ forms a basis for the dual root system. • The fundamental weights { χ j } lj =1 ⊂ P are characterized by χ j ( α ∨ k ) = (cid:40) j = k otherwise. • The fundamental coweights { χ ∨ j } lj =1 ⊂ P ∨ are characterized by α k ( χ ∨ j ) = (cid:40) j = k otherwise.A root α (coroot α ∨ ) is said to be positive if all of its coefficients with respect to the basis { α , . . . , α l } (resp. { α ∨ , . . . , α ∨ l } ) are nonnegative and is said to be negative if they are non-positive. It is well-known that every root is either positive or negative. Let us denote the set ofall positive roots of and that of all negative roots by R + and R − , respectively. The sum of allpositive roots will play an important role later in this paper. HANWOOL BAE AND NAICHUNG CONAN LEUNG
Definition 2.3. ρ = (cid:88) α ∈ R + α. We will make use of the following well-known equality as well.
Lemma 2.4. ρ = (cid:88) α ∈ R + α = 2 l (cid:88) j =1 χ j . Weyl group.
Let N G ( T ) be the normalizer of T in G . Then the Weyl group of G is givenby the quotient group W = N G ( T ) /T. For future use, let us denote by ˙ w ∈ N G ( T ) a representative of w ∈ W .For each positive root α ∈ R + , let s α be the reflection on t over the hyperplane(2.1) t α = { h ∈ t | α ( h ) = 0 } , which is explicitly given by(2.2) s α ( h ) = h − α ( h ) α ∨ , ∀ h ∈ t . The Weyl group can be understood as the group of isometries on t generated by reflections s α , . . . , s α l . The Weyl group action w ( h ) is also identified with the adjoint action Ad ( ˙ w ) h .The Weyl group action naturally extends to the torus T C by w · exp( h + ih ) = exp( w ( h ) + iw ( h )) , ∀ w ∈ W , h , h ∈ t . Further, the Weyl group also acts on the dual space t ∨ from the left by w · λ = λ ◦ w − , ∀ w ∈ W , λ ∈ t ∨ . Definition 2.5.
An element h ∈ t is called regular if α ( h ) (cid:54) = 0 for all roots α ∈ R + . We denoteby t reg ⊂ t the set of all regular elements. Definition 2.6.
The positive Weyl chamber t > ⊂ t is defined by t > = { h ∈ t | α j ( h ) > , ∀ j = 1 , . . . , l } . Let us denote the closure of the positive Weyl chamber by t ≥ . Then we have
Lemma 2.7. (1) t reg = (cid:71) w ∈ W w · t > .(2) t \ t reg = (cid:91) α ∈ R + t α = (cid:91) w ∈ W l (cid:91) j =1 w t α j . We also need to understand the notion of length.
ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Definition 2.8.
The length l ( w ) of an Weyl group element w ∈ W is defined by the smallestnumber of reflections s α j ’s whose product is w .From the definition of length, we deduce l ( w ) = l ( w − ) for all w ∈ W . Besides, there isanother characterization of length. Indeed, for a given (cid:15) ∈ t > , we have Lemma 2.9. l ( w ) = { α ∈ R + | α ( w ( (cid:15) )) > } = { α ∈ R + | α ( w − ( (cid:15) )) > } . Cartan decomposition.
The Cartan decomposition theorem [12] on Lie group says thatthe map(2.3) G × g → G C , ( g, v ) (cid:55)→ g exp( iv ) is a diffeomorphism. Moreover, using the left invariant vector fields and the left invariant covec-tor fields, the tangent bundle and the cotangent bundle of G are trivialized. Then the Ad-invariantinner product ( , ) on G yields a bundle isomorphism(2.4) T ∗ G = G × g ∗ → G × g = T G.
Composing (2.3) and (2 . , we get a diffeomorphism between T ∗ G and G C .Let us identify the following spaces via these diffeomorphisms from now on. G C = T ∗ G = G × g ∗ = G × g . Similarly, we identify the following spaces T C = T ∗ T = T × t ∗ = T × t . Furthermore, the diffeomorphisms (2.3) and (2 . justify the following identifications: T x ( T ∗ G ) = g ⊕ i g = g C = T x G C , ∀ x ∈ T ∗ G = G C .T x ( T ∗ T ) = t ⊕ i t = t C = T x T C , ∀ x ∈ T ∗ T = T C . (2.5)As a consequence of the Cartan decomposition theorem, we have Lemma 2.10.
The complexified Lie group G C admits a factorization GT C G = G C . Proof.
First note that G C factors into G exp( i g ) ∼ = G × g . The map GT C G → G × g given by g exp( h + ih ) g − (cid:55)→ (cid:0) g g − exp( Ad g ( h )) , Ad g ( h ) (cid:1) , ∀ g , g ∈ G, h , h ∈ t . is a G × G -equivariant bijection since every element of g is conjugate to an element of t . (cid:3) HANWOOL BAE AND NAICHUNG CONAN LEUNG
Wonderful Compactification.
From now on, let G be a simply-connected compact semisim-ple Lie group and let G ad = G/C ( G ) the quotient group, which is of adjoint type. We have seenin Lemma 2.2 that the quotient map q : G → G ad is actually a covering map and its Decktransformation group is isomorphic to the center of G or the quotient P ∨ /Q ∨ .Then the complexification G C ad of G ad admits a compactification into a smooth projective va-riety, which is called the wonderful compactification of G C ad . It turns out that this provides us amore geometric way of understanding the moment Lagrangian C for the full flag variety G/T ,which will be introduced in Subsection 2.6.Note that G C ad carries a natural G C ad × G C ad -action given by the left and right multiplication. Thenthe wonderful compactification of G C ad can be stated as follows. Theorem 2.11 (Wonderful compactification [9]) . There exists a G C ad × G C ad -equivariant embed-ding from G C ad to a smooth projective variety X that admits a G C ad × G C ad -action such that(1) The complement X \ G C ad is the union D ∪ · · · ∪ D l of smooth divisors with normalcrossings, each of which is G C ad × G C ad -invariant. The number of the divisors is the rankof G C ad .(2) The unique closed G C ad × G C ad -orbit is given by the intersection of all divisors D i ’s and itis isomorphic to G C ad /B ad × G C ad /B − ad , where B ad is a Borel subgroup of G C ad and B − ad is its opposite.Let us denote the closed G C ad × G C ad -orbit G C ad /B ad × G C ad /B − ad by M For each ≤ j ≤ l , let O X ( D j ) be the line bundle on X associated to the divisor D j so that N j := O X ( D j ) | D j is the normal bundle of D j in X .Then the normal bundle N = N M | X of M in X is isomorphic to the direct sum of all the linebundles N j restricted to M , namely, N = l (cid:77) j =1 N j | M . We may assume that the Borel subgroup B ad contains the maximal torus T C and is positivewith respect to the fixed basis { α . . . , α l } of roots.On the other hand, denote by B the pre-image of the Borel subgroup B ad of G ad under thecovering map G → G ad . Then B is also a Borel subgroup of G and the quotient map inducesan isomorphism G C /B → G C ad /B ad . Furthermore, the inclusion map G → G C induces anisomorphism G/T → G C /B . Since the same argument works even when the Borel subgroup B ad is replaced by its opposite B − ad , denoting T ad = B ad ∩ G ad = B − ad ∩ G ad , we will identify thefollowing spaces throughout this paper: G/T = G C /B = G C ad /B ad = G ad /T ad = G C ad /B − ad . Now we describe a symplectic structure on G C /B . First the second cohomology H ( G C /B, Z ) ∼ = Pic ( G C /B ) of G C /B can be identified with the weight lattice P as abelian groups. Indeed, for ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES any weight α ∈ P , we let C α = C and define a line bundle O G C /B ( α ) on G C /B by O G C /B ( α ) = G C × B C α , where the Borel subgroup B = T C U acts on C α by tu · z = α ( t ) z for all t ∈ T C , u ∈ U and z ∈ C α . Here the exponential α ( t ) ∈ C ∗ is defined by(2.6) α ( t ) = e π ( √− α ( h )+ α ( h )) for t = exp( h + ih ) , h , h ∈ t . The correspondence α (cid:55)→ O G C /B ( α ) gives the desired isomorphism between P and Pic ( G C /B ) . Remark . Note that the way the exponential (2.6) is defined is not standard. This is a conse-quence of that negative the standard almost complex structure on G C is compatible with respectto the canonical symplectic structure on T ∗ G = G C as explained in Subsection 2.8.Alternatively, there is a de Rham theoretic description of the isomorphism above. Indeed,every α ∈ P defines a left invariant -form ˆ α on G :(2.7) ˆ α ( u, v ) = α ( p ([ u, v ])) , ∀ u, v ∈ g , where u, v ∈ g in the left hand side stands for the corresponding left invariant vector fieldson G and p : g → t is the orthogonal projection with respect to the inner product describedin Subsection 2.1.1. The 2-form ˆ α induces a well-defined 2-form on G/T since α is Ad ( T ) -invariant and p [ u, v ] = 0 if any of u and v lies in t . Let us denote the induced 2-form on G/T again by ˆ α by abuse of notation. Since the 2-form ˆ α is closed, its de Rham cohomology classdefines a nonzero cohomology class [ ˆ α ] ∈ H ( G/T, R ) . Then the first Chern class of O G C /B ( α ) is given by [ ˆ α ] for each weight α .Finally we equip G/T with a symplectic form(2.8) ω G/T = (cid:88) α ∈ R + ˆ α = ˆ ρ. Then the natural left G -action on G/T is Hamiltonian and its moment map µ : G/T → g ∗ isgiven by(2.9) µ ([ g ]) = (cid:88) α ∈ R + Ad ( g − ) ∗ α = Ad ( g − ) ∗ ρ, ∀ g ∈ G where α is regarded as an element of g ∗ via the embedding p ∗ : t ∗ → g ∗ .Moreover, if M = G/T − × G/T is oriented in such a way that the first factor
G/T and thesecond one have the opposite orientation to each other. Then we have the following lemma.
Lemma 2.13.
The tangent bundle
T M is given by
T M = π ∗ (cid:32) − (cid:77) α ∈ R + O G/T ( α ) (cid:33) ⊗ π ∗ (cid:32) (cid:77) α ∈ R + O G/T ( α ) (cid:33) , HANWOOL BAE AND NAICHUNG CONAN LEUNG where π j : M → G/T is the projection to j -th factor for j = 1 , . As a consequence, thesymplectic manifold M is monotone with monotone constant c = 1 , that is,(2.10) c ( T M ) = [ ω M ] = [( − ω G/T ) ⊕ ω G/T ] . An open neighborhood U of infinity. As mentioned in the introduction, there is an openneighborhood U ad ⊂ T ∗ G ad near the minimal stratum M = G/T × G/T in the wonderfulcompactification X = T ∗ G ad .First of all, the restriction of the line bundle N j to the minimal stratum M = G/T − × G/T = G ad /T − ad × G ad /T ad is given by(2.11) N j | M = π ∗ O G/T ( − α j ) ⊗ π ∗ O G/T ( α j ) . Since the normal bundle N is the direct sum of all N j | M ’s, we have N = l (cid:77) j =1 π ∗ O G/T ( − α j ) ⊗ π ∗ O G/T ( α j ) . This implies that the normal bundle N can be expressed as follows:(2.12) N = G ad × T ad C l × T ad G ad where T ad × T ad acts on C l by ( t , t ) · ( z , . . . , z l ) = ( α ( t ) − α ( t ) z , . . . , α l ( t ) − α l ( t ) z l ) , ∀ t , t ∈ T ad . Then we define a punctured poly-disk sub-bundle D ∗ N of N by D ∗ N = G ad × T ad ( D ∗ ) l × T ad G ad ⊂ G ad × T ad C l × T ad G ad = N, where D ∗ ⊂ C be the open unit disk with a puncture at the center Definition 2.14.
For each regular element h ∈ t reg , let w h ∈ W be the unique element of theWeyl group such that h ∈ w h t > . In particular, for any h ∈ t reg , we have α j ( w − h h ) > for all j = 1 , . . . , l .Let us consider T C ad , reg = T ad × t reg ⊂ T ad × t = T C ad . We define an open subset U ad of G C ad by U ad = G ad T C ad , reg G ad ⊂ G ad T C ad G ad = G C ad . Finally we define a map ι : U ad → G ad × T C l × T G ad = N by(2.13) ι : g tg − (cid:55)→ (cid:104) g ˙ w h , α (cid:0) w − h t (cid:1) − , . . . , α l (cid:0) w − h t (cid:1) − , g ˙ w h (cid:105) , where t = exp( h + ih ) for h ∈ t , h ∈ t reg and ˙ w h ∈ N G ( T ) is a representative of w h ∈ W .It is straightforward to check that this map is a well-defined G ad × G ad -equivariant embeddingand its image is given by D ∗ N . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Lemma 2.15.
The map ι : U ad → D ∗ N is a G ad × G ad -equivariant diffeomorphism. On the other hand, we have seen that the quotient map q : G → G ad = G/C ( G ) is a coveringmap. Since there are identifications G C = G × g = T ∗ G and G C ad = G ad × g = T ∗ G ad , it alsoextends to covering maps G C → G C ad and T ∗ G → T ∗ G ad , which we will denote by q by abuse ofnotation.We define U ⊂ G C by the pre-image U = q − ( U ad ) . To investigate the complement of the subset U in G C , first consider, for each j = 1 , . . . , l , T C ad ,j = T ad × t α j (2 . ,G C ad ,j = GT C ad ,j G. Then T C ad ,j is a submanifold of T C ad of codimension 1 and G C ad ,j is a submanifold of G C ad ofcodimension 3. By definition, the complement of U ad in G C ad is given by the union of all G C ad ,j ’s.Similarly, we define, for each j = 1 , . . . , l , T C j = q − ( T C ad ,j ) ,G C j = q − ( G C ad ,j ) . Then the complement of U in G C is the union of all G C j ’s, each of which is of codimension 3.Finally we have the following diagram of maps:(2.14) U (cid:101) ι (cid:124) (cid:124) (cid:101) π (cid:118) (cid:118) (cid:31) (cid:127) (cid:47) q (cid:15) (cid:15) T ∗ G = G C q (cid:15) (cid:15) M D ∗ N π (cid:111) (cid:111) U ad (cid:31) (cid:127) (cid:47) ι ∼ = (cid:111) (cid:111) T ∗ G ad = G C ad where (cid:101) ι is the composition ι ◦ q and (cid:101) π is the composition π ◦ ι ◦ q .Before we go further, let us first observe the following for future use. Indeed, for w ∈ W ,consider the point ([ ˙ w ] , [ ˙ w ]) ∈ G/T × G/T = M . Then we have Lemma 2.16.
The image of T C ad , reg under the embedding ι is given by ι ( T C ad , reg ) = (cid:71) w ∈ W π − ([ ˙ w ] , [ ˙ w ]) . In particular, the fiber π − ([ ˙ w ] , [ ˙ w ]) (resp. (cid:101) π − ([ ˙ w ] , [ ˙ w ]) ⊂ U ) is included in the image of thetorus T C ad (resp. T C ).Proof. For any w ∈ W , the map ι sends the any point t = exp( h + ih ) ∈ T C ad , reg with h ∈ t , h ∈ w t > to (cid:2) ˙ w, α ( w − ( t )) − , . . . , α l ( w − ( t )) − , ˙ w (cid:3) ∈ D ∗ N. It follows that ι { t ∈ T C ad | t = exp( h + ih ) , h ∈ t , h ∈ w t > } = π − ([ ˙ w ] , [ ˙ w ])) . HANWOOL BAE AND NAICHUNG CONAN LEUNG
The assertion follows since t reg = (cid:71) w ∈ W w t > . (cid:3) Remark . For a weight α ∈ P , a connection 1-form on π ∗ O G/T ( − α ) ⊗ π ∗ O G/T ( α ) can beexplicitly described. Let dθ = π d arg z and log ρ = π log | z | for z ∈ C ∗ . Consider an 1-form (cid:101) α ∈ Ω ( G ad × C ∗ × G ad ) defined by(2.15) (cid:101) α (cid:18) u, a ∂∂θ + b ∂∂ρ , v (cid:19) = − α ( p ( u )) + a + α ( p ( v )) , for u, v ∈ g and a, b ∈ R . For a T ad × T ad -action on G ad × C ∗ × G ad given by ( t , t ) · ( g , z, g ) = (cid:0) g t , α ( t ) − α ( t ) z, g t (cid:1) , (cid:101) α descends to an 1-form on the quotient space G ad × T ad C ∗ × T ad G ad , which we will still denoteby (cid:101) α by abuse of notation.Moreover, the differential of (cid:101) α is given by(2.16) d (cid:101) α = π ∗ ( π ∗ ˆ α − π ∗ ˆ α ) , where π : N → M is the projection and ˆ α ∈ Ω ( G/T ) is given as in (2.7). This shows that the -form (cid:101) α is indeed a connection one form for the bundle π ∗ O G/T ( − α ) ⊗ π ∗ O G/T ( α ) .2.4. Topology of the flag variety
G/T . To get a better understanding of the Novikov ring whichwill be introduced in Section 5, we need to investigate the second homotopy group π ( G/T ) ofthe flag variety G/T .Since the flag variety
G/T is simply-connected, the second homotopy group π ( G/T ) is iso-morphic to the second homology group H ( G/T, Z ) , which is generated by Schubert classes ofcomplex dimension .The Bruhat decomposition G C = (cid:71) w ∈ W BwB, gives a cell decomposition of G C /B = G/T as follows G C /B = (cid:71) w ∈ W X w , where X w = B ˙ wB/B ∼ = C l ( w ) , where l ( w ) is the length of w ∈ W .For each j = 1 , . . . , l , we define ˇ α j ∈ H ( G/T, Z ) = π ( G/T ) by ˇ α j = (cid:104) X s αj (cid:105) the homology class of the closure of X s αj . Then the second homotopy group π ( G/T ) is a freeabelian group generated by the Schubert classes ˇ α j ’s j = 1 , . . . , l , that is,(2.17) π ( G/T ) = Z (cid:104) ˇ α j | j = 1 , . . . , l (cid:105) Hence the group homomorphism ϕ : Q ∨ → π ( G/T ) defined by(2.18) ϕ ( α j ) = ˇ α j , ∀ j = 1 , . . . , l ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES is an isomorphism.Furthermore, if we identify H ( G/T, Z ) with the weight lattice P by mapping [ ˆ α ] (2.7) to α ∈ P , then the Poincare pairing between H ( G/T, Z ) and H ( G/T, Z ) is identified with thedual pairing between Q ∨ and P . This justifies the identifications π ( G/T ) = H ( G/T, Z ) = Q ∨ and H ( G/T, Z ) = P. Liouville structure.
A pair ( W in , λ ) of a smooth manifold W in with boundary and an 1-form λ on W in is said to be a Liouville domain if • ω = dλ is a symplectic form on W in . • λ ∧ ( dλ ) dim W in − restricts to a positive orientation form on the boundary ∂W .The latter implies that the Liouville vector field Z on W in defined by ι Z ω = λ points outwardalong the boundary. Moreover, the flow of the vector field Z defines a cylindrical end near theboundary ∂W in as follows: (0 , × ∂W in → W in , ( r, x ) (cid:55)→ ψ r ( x ) where ψ t is the time log t -flowof the Liouville vector field Z . Then W := W in ∪ ∂W in [1 , ∞ ) × ∂W in is the completion of W in equipped with a symplectic form ω W = (cid:40) w on W,d ( rλ | ∂W ) on [1 , ∞ ) × ∂W. Let us denote the symplectic form still by ω by abuse of notation.We will work with quadratic Hamiltonians H on W in the sense that(2.19) H ( r, x ) = 12 r , ( r, x ) ∈ [ r , ∞ ) × ∂W in ⊂ W. for some sufficiently large r ∈ (0 , ∞ ) . Let H ( W ) be the set of such Hamiltonians.The canonical 1-form λ can is a Liouville 1-form on T ∗ G . Let us describe the correspondingLiouville vector field and the radial coordinate for the Liouville domain ( W, λ can ) in terms of leftinvariant vector fields on T ∗ G = G × g .(1) The Liouville 1-form λ at ( g, v ) ∈ G × g = T ∗ G is given by λ ( g,v ) = ( v, ( L g − ) ∗ ( − )) . (2) The Liouville vector field Z at ( g, v ) ∈ G × g = T ∗ G is given by(2.20) Z ( g,v ) = iv ∈ i g = 0 ⊕ i g = T ( g,v ) T ∗ G. It follows that the radial coordinate r is given by r = | v | = (cid:112) ( v, v ) . (3) The Reeb vector field R at ( g, v ) ∈ G × g = T ∗ G is given by(2.21) R ( g,v ) = vr ∈ g = g ⊕ ⊂ T ( g,v ) T ∗ G. For the quotient group G ad of adjoint type, the Liouville structure on T ∗ G ad can be describedin the exactly same way as above. HANWOOL BAE AND NAICHUNG CONAN LEUNG
Moment Lagrangian.
Let ( Y, ω Y ) be a symplectic manifold that admits a Hamiltonian G -action and let µ : Y → g ∗ be its moment map. The moment Lagrangian C is a Lagrangiansubmanifold of ( T ∗ G ) − × ( Y − × Y ) defined by(2.22) C = { ( g, µ ( y ) , g · y, y ) ∈ T ∗ G × Y × Y | y ∈ Y, g ∈ G } , where the cotangent bundle T ∗ G is identified with G × g ∗ . Remark . Our definition of the moment Lagrangian is different from that given in [8]. Sucha variation was made to make the notations of this paper more consistent and easier to recognize.
Definition 2.19.
For a Lagrangian L ⊂ T ∗ G , the geometric composition L ◦ C with C is definedby L ◦ C = π Y × Y ( π − T ∗ G ( L ) ∩ C )= { ( y , y ) ∈ Y × Y |∃ x ∈ L such that ( x, y , y ) ∈ C } . A geometric composition L ◦ C is said to be transverse if π − T ∗ G ( L ) and C intersect transverselyand is said to be embedded if the projection π Y × Y restricted to the intersection is an embedding.Note that the geometric composition T ∗ e G ◦ C is transverse and embedded, which is given by T ∗ e G ◦ C = { ( y, y ) | y ∈ Y } =: ∆ Y . Now let us consider the moment Lagrangian for Y = G/T . We have seen that
G/T admits a G -Hamiltonian action and its moment map µ : G/T → g ∗ is given by (2.9). Hence the momentLagrangian C for G/T is given by(2.23) C = (cid:40)(cid:32) g, (cid:88) α ∈ R + Ad ( g − ) ∗ α, [ g ] , [ g ] (cid:33) ∈ G × g ∗ × G/T × G/T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g, g , g ∈ G with g ∈ g T g − (cid:41) . The moment Lagrangian C for G/T can be viewed as a l -torus bundle over M = G/T − × G/T . Indeed, the restriction of the projection π M : W × M → M to the moment Lagrangian C is surjective and submersive. Then the fiber over a point ([ g ] , [ g ]) ∈ G/T × G/T is identifiedwith the set { g ∈ G | [ g ] = [ g · g ] } , which is given by g T g − = { g tg − | t ∈ T } ∼ = T. Furthermore, denoting W = T ∗ G , the restriction of the projection π W : W × M → W to themoment Lagrangian C ⊂ W × M induces a diffeomorphism π W | C : C → π W ( C ) , which canbe verified by the following description of π W ( C ) : π W ( C ) = (cid:40)(cid:32) g tg − , (cid:88) α ∈ R + Ad ( g − ) ∗ α (cid:33) ∈ G × g ∗ = T ∗ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g , g ∈ G, t ∈ T (cid:41) = (cid:40)(cid:32) g tg − , l (cid:88) j =1 Ad ( g ) 4 χ ∨ j ( α ∨ j , α ∨ j ) (cid:33) ∈ G × g = T ∗ G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g , g ∈ G, t ∈ T (cid:41) . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES The equality between the first and second line comes from Lemma 2.4 and the equality (cid:18) Ad ( g ) 2 χ ∨ j ( α ∨ j , α ∨ j ) , (cid:19) = Ad ( g − ) ∗ χ j ∈ t ∗ , ∀ j = 1 , . . . , l. Since l (cid:88) j =1 χ ∨ j ( α ∨ j , α ∨ j ) ∈ t reg , the image π W ( C ) is contained in U .Therefore the moment Lagrangian C is contained in U × M . Then the description (2.23) of themoment Lagrangian implies that the map (cid:101) π × Id M : U × M → M × M maps C to the diagonal ∆ M ⊂ M × M , that is, ( (cid:101) π × Id M )( C ) = ∆ M . In summary, we have the following commutative diagram of maps such that every map fromthe second row to the third row is a torus fibration. T l (cid:18) (cid:114) (cid:36) T l (cid:19) (cid:115) (cid:37) C π M | C (cid:37) (cid:37) (cid:101) π × Id M (cid:15) (cid:15) π W | C (cid:47) (cid:47) π W ( C ) (cid:101) π (cid:15) (cid:15) ∆ M ∼ = (cid:47) (cid:47) M. Finally we are now ready to prove that the moment Lagrangian C ⊂ W × M is monotone,following the idea suggested in [21, 19].First the homotopy exact sequence associated to the torus fiber bundle C → M can be inter-preted as follows. Lemma 2.20.
The following is a commutative diagram of morphisms between abelian groups.Each row of the diagram is exact and every vertical map is an isomorphism. ker δ ∼ = Q ∨ (cid:47) (cid:47) (cid:15) (cid:15) Q ∨ × Q ∨ (cid:47) (cid:47) ϕ (cid:15) (cid:15) δ (cid:47) (cid:47) Q ∨ (cid:47) (cid:47) ϕ (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:15) (cid:15) π ( C ) ( π M | C ) (cid:47) (cid:47) π ( M ) = π ( G/T ) × π ( G/T ) δ (cid:47) (cid:47) π ( T l ) = Z l (cid:47) (cid:47) π ( C ) (cid:47) (cid:47) π ( M ) where the morphisms ϕ , ϕ , ϕ and δ , δ are described as follows: • ϕ = ϕ × ϕ for the isomorphism ϕ : Q ∨ → π ( G/T ) (2.18). • ϕ : Q ∨ → π ( T l ) = Z l is given by l (cid:88) j =1 a j α ∨ j (cid:55)→ ( a j ) lj =1 . • δ ( h , h ) = − h + h for h , h ∈ Q ∨ . • δ is the connecting map and it is given by δ ( h , h ) = ( − α j ( h ) + α j ( h )) lj =1 for h , h ∈ Q ∨ = π ( G/T ) . HANWOOL BAE AND NAICHUNG CONAN LEUNG
In particular, we have(1) π ( C ) = 0 .(2) π ( C ) ∼ = Q ∨ .Proof. The assertion follows from the description of the morphisms given above. (cid:3)
Lemma 2.21.
The moment Lagrangian C ⊂ W × M is monotone, that is, (2.24) µ C ( β ) = 2 ω ( β ) , ∀ β ∈ π ( W × M, C ) . Here the constant is given by twice the monotone constant c = 1 given in (2.10).Proof. Consider the following piece of a long exact sequence π ( W × M ) → π ( W × M, C ) → π ( C ) . Since π ( C ) is trivial, any class of π ( W × M, C ) comes from π ( W × M ) . Then the mono-tonicity of C follows from that of W − × M . Furthermore, the equality (2.24) follows from thefact that the restriction of the Maslov class µ C to π ( W × M ) is twice the first Chern class. (cid:3) Remark . It can be verified that the restriction of the map (cid:101) ι = ι ◦ q : U (cid:55)→ D ∗ N to π W ( C ) isa covering map onto the torus sub-bundle of D ∗ N ⊂ N given by (cid:101) ι ( π W ( C )) = (cid:40) [ g , ( z , . . . , z l ) , g ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g , g ∈ G ad , z j = exp (cid:18) π (cid:18) it j − α ∨ j , α ∨ j ) (cid:19)(cid:19) , t j ∈ R (cid:41) . Furthermore, its Deck transformation group is isomorphic to that of q : G → G ad .Note that the time log t -flow ψ t of the Liouville vector field Z maps the torus sub-bundle (cid:101) ι ( π W ( C )) diffeomorphically onto another torus sub-bundle given by ψ t ( (cid:101) ι ( π W ( C ))) = (cid:40) [ g , ( z , . . . , z l ) , g ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g , g ∈ G ad , z j = exp (cid:18) π (cid:18) it j − t ( α ∨ j , α ∨ j ) (cid:19)(cid:19) , t j ∈ R (cid:41) . The tuple of radii ( c ( t ) , . . . , c l ( t )) of the torus sub-bundle ψ t ( (cid:101) ι ( π W ( C ))) is given by(2.25) c j ( t ) = exp (cid:18) − πt ( α ∨ j , α ∨ j ) (cid:19) . Hamiltonian chord.
Let H ∈ H ( T ∗ G ) be a quadratic Hamiltonian on T ∗ G = G × g defined by H ( g, v ) = 12 ( v, v ) . The corresponding Hamiltonian vector field X H , i.e. ι X H ω can = − dH , is given by X H ( g, v ) = v ∈ g = g ⊕ ⊂ T ( g,v ) ( G × g ) , ∀ ( g, v ) ∈ G × g . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES As a consequence, the time- t flow of the vector field X H is given by(2.26) φ tX H ( g, v ) = ( g exp( vt ) , v ) , ∀ ( g, v ) ∈ G × g . Alternatively, one can express the quotient image of the Hamiltonian vector field X H in termsof vector fields in the punctured poly-disk subbundle of N under the quotient map q : T ∗ G → T ∗ G ad . Note that the quadratic Hamiltonian H is invariant under the covering action on T ∗ G andhence it descends to a quadratic Hamiltonian H ad ∈ H ∗ ( T ∗ G ad ) given by H ad ( g, v ) = 12 ( v, v ) , ∀ ( g, v ) ∈ G ad × g . Therefore, the Hamiltonian vector field X H in mapped to the Hamiltonian vector field X H ad via the quotient map and it is expressed as follows:(2.27) q ∗ ( X H ) = X H ad = l (cid:88) j =1 log ρ j ∂∂θ j . Hence, in each line bundle summand D ∗ N j | M , j = 1 , . . . , l of D ∗ N , the Hamiltonian flow of H ad is given by a clockwise rotation. Indeed, the time- t - flow of the vector field H ad is given by(2.28) φ tX H ad ([ g , z , . . . , z l , g ]) = (cid:104) g , z e log | z |√− t , . . . , z l e log | z l |√− t , g (cid:105) , for any [ g , z , . . . . , z l , g ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad = D ∗ N .Now let us investigate the Hamiltonian chords of the cotangent fiber L = T ∗ e G . The descrip-tion of the Hamiltonian flow (2.26) shows that, for each h ∈ t Z , the path x h : [0 , → G × g = T ∗ G given by(2.29) x h ( t ) = (exp( ht ) , h ) ∈ G × g is a Hamiltonian chord from L to itself.But these Hamiltonian chords are not nondegenerate since Ad g ( x h ) is a Hamiltonian chordfrom L to itself for every g ∈ G and they form a family of non-isolated Hamiltonian chords. Tomake them nondegenerate, we first fix a regular element (cid:15) ∈ t reg given by(2.30) (cid:15) = a l (cid:88) j =1 χ ∨ j for some < a ≤ l and we instead consider the Hamiltonian chords from L to L (cid:15) = T ∗ exp (cid:15) G .Then the path x h(cid:15) : [0 , → G × g = T ∗ G for h ∈ t Z (See Subsection 2.1.1) given by(2.31) x h(cid:15) ( t ) = (exp(( h + (cid:15) ) t ) , h + (cid:15) ) ∈ G × g is a nondegenerate Hamiltonian chord from L to L (cid:15) for each h ∈ t Z .In conclusion, the set of Hamiltonian chords from L to L (cid:15) is in one-to-one correspondencewith t Z : P ( L, L (cid:15) ; H ) = { x h(cid:15) | h ∈ t Z } ∼ = t Z . HANWOOL BAE AND NAICHUNG CONAN LEUNG
Observe that the covering map q : T ∗ G → T ∗ G ad maps the cotangent fiber T ∗ e G diffeomor-phically to the cotangent fiber T ∗ e G ad . We now investigate how the cotangent fiber T ∗ e G ∼ = T ∗ e G ad looks like in the open neighborhood U ad .The cotangent fiber T ∗ e G is identified with the subset { gtg − | g ∈ G, t = exp( ih ) , ∃ h ∈ t } ⊂ GT C G = G C . The map (cid:101) ι = ι ◦ q (2.14) maps any point g exp( ih ) g − ∈ T ∗ e G ∩ U for h ∈ t reg to [ g ˙ w h , z , . . . , z l , g ˙ w h ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad where z j = exp( − πα j ( w − h h ))) ∈ R > ∩ D ∗ ⊂ D ∗ , which shows that the intersection L ∩ U is diffeomorphic to the direct sum of real half-linebundles over the diagonal ∆ G/T ⊂ G/T × G/T = M via the map (cid:101) ι :(2.32) (cid:8) [ g, x , . . . , x l , g ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad | g ∈ G ad , x j ∈ R > , ∀ j = 1 , . . . l (cid:9) in the punctured poly-disk sub-bundle D ∗ N of N .Similarly, the cotangent fiber L (cid:15) = T ∗ exp (cid:15) G in U is mapped via (cid:101) ι diffeomorphically to (cid:8) [exp (cid:15) · g, x , . . . , x l , g ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad | g ∈ G ad , x j ∈ R > ∀ j = 1 , . . . , l (cid:9) . Therefore, the intersection of the cotangent fiber L (cid:15) and U is the direct sum of real half-linebundles over the perturbed diagonal ∆ (cid:15) ⊂ M defined by ∆ (cid:15) = { ([ g ] , [ g ]) ∈ G/T × G/T | [ g ] = [exp (cid:15) · g ] } . In particular, for g = ˙ w for some w ∈ W , the intersection (cid:101) ι ( L (cid:15) ∩ U ) ∩ ( D ∗ N ) [ ˙ w, ˙ w ] is given by(2.33) (cid:110) [ ˙ w, z , . . . , z l , ˙ w ] ∈ G ad × T ad ( D ∗ ) l × T ad G ad | z j ∈ e − π √− α j ( w − (cid:15) ) · R > , ∀ j = 1 , . . . , l (cid:111) . Hence, considering the description (2.28) of the Hamiltonian flow, the Hamiltonian chord x h(cid:15) , h ∈ t Z , lies in the fiber over ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) ∈ G/T × G/T = M and it rotates by the angle − π (cid:0) m j + α j ( w − h + (cid:15) (cid:15) ) (cid:1) in the j -th summand N j | M of N for m j = α j ( w − h + (cid:15) h ) ∈ Z ≥ .On the other hand, we observe that the homotopy group π ( W, L ) is trivial in the followinglemma. As a consequence, every Hamiltonian chord x h(cid:15) , h ∈ Q ∨ is homotopic to each others. Lemma 2.23. (1) The inclusion C ∼ = π W ( C ) (cid:44) → U induces an isomorphism π ( C ) → π ( U ) .(2) The natural map π ( U ) → π ( U , L ∩ U ) is an isomorphism.(3) The inclusion ( U , L ∩ U ) (cid:44) → ( W, L ) induces an isomorphism π ( U , L ∩ U ) → π ( W, L ) .(4) Both of the homotopy groups π ( U , L ∩ U ) and π ( W, L ) are zero. In particular, theinclusion ( U , L ∩ U ) (cid:44) → ( W, L ) induces an isomorphism π ( U , L ∩ U ) → π ( W, L ) . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Proof.
The first assertion follows from the fact that U deformation retracts to π W ( C ) . The secondassertion follows from the long exact sequence of homotopy groups associated to the inclusion L ∩ U (cid:44) → U and the fact that π ( L ∩ U ) ∼ = π ( M ) = 0 . For the third assertion, observe thatthe inclusion ( U , L ∩ U ) (cid:44) → ( W, L ) induces a map from the long exact sequence of homotopygroups associated to L ∩ U (cid:44) → U to that associated to L (cid:44) → W .For the last assertion, consider the long exact sequence associated to the inclusion L ∩ U (cid:44) → U : π ( L ∩ U ) → π ( U ) → π ( U , L ∩ U ) → π ( L ∩ U ) . First observe that the description of L ∩ U as the direct sum of real half-line bundles over thediagonal ∆ G/T (2.32) shows that the L ∩ U is homotopy equivalent to G/T and hence that thefundamental group π ( L ∩ U ) is trivial. Furthermore, the inclusion L ∩ U (cid:44) → U also induces anisomorphism on the second homotopy group π ( L ∩ U ) → π ( U ) . Once again, observe that theprojection (cid:101) π maps the intersection L ∩ π W ( C ) homeomorphically to the diagonal ∆ G/T ⊂ M .The assertion follows since both the inclusion L ∩ π W ( C ) (cid:44) → L ∩ U and L ∩ π W ( C ) (cid:44) → U induceisomorphisms on the second homotopy group. The assertion for π ( W, L ) follows from the factthat W is homotopy equivalent to G and L is contractible. (cid:3) Almost complex structure.
We introduce the notion of almost complex structure of weakcontact type.
Definition 2.24 (Almost complex structure of weak contact type) . Let W in be a Liouville domainand W be its completion. An almost complex structure J on W is of weak contact type if λ ◦ J is a positive multiple of dr on the cylindrical part [ r , ∞ ) × ∂W in ⊂ W for some r ∈ (0 , ∞ ) .The standard complex structure on G C is not suitable for our purpose since it is compatiblewith respect to the negative of the canonical symplectic form ω can under the identification G C = T ∗ G . Hence we consider the negative the standard complex structure on W = G C , which wewill denote by J . To express the almost complex structure J more explicitly, recall that thetangent space of G C is identified with g ⊕ i g as in (2.5). Then the almost complex structure J isgiven by requiring J ( v ) = − iv ∈ ⊕ i g , ∀ v ∈ g = g ⊕ at every point of G C .The equations (2.20) and (2.21) show that J maps the Liouville vector field Z to rR , where R is the Reeb vector field and r is the radial coordinate. Furthermore, J preserves the con-tact distribution since the contact distribution at a point ( g, v ) ∈ G × g = T ∗ G is spanned by (cid:104) v (cid:105) ⊥ ⊕ i (cid:104) v (cid:105) ⊥ in g ⊕ i g . Here (cid:104) v (cid:105) ⊥ is the orthogonal complement of (cid:104) v (cid:105) in g with respect to the in-ner product defined in Subsection 2.1.1. These observations imply that J is of weak contact type.We will perturb the almost complex structure J on the open neighborhood U to get thetransversality of Fredholm operators that will be introduced in Appendix A.2.Indeed, recall that the normal bundle N can be expressed by N = G ad × T ad C l × T ad G ad (2.12).Hence the tangent space of T N at a point x ∈ N is identified with T x N = g × T ad C l × T ad g . HANWOOL BAE AND NAICHUNG CONAN LEUNG
The tangent bundle
T N admits the following decomposition:(2.34)
T N = Ver ⊕ Hor , where Ver = ker dπ and Hor is a distribution in the tangent bundle such that, for every x ∈ N ,the subspace Hor x ⊂ T x N = g × T ad C l × T ad g is given by the quotient image of t ⊥ ⊕ ⊕ t ⊥ ⊂ g ⊕ C l ⊕ g . Remark . The decomposition
T N = Ver ⊕ Hor respects the symplectic structure ω | U ad on D ∗ N ∼ = U ad in the sense that both summands Ver and Hor are symplectic sub-bundles and thosetwo are symplectically orthogonal.The covering map q : U → U ad induces an isomorphism dq : T x U → T q ( x ) U ad on the tangentspace at every point x ∈ U . Hence the decomposition (2.34) can be pulled-back to the tangentbundle of U . Accordingly, we have a decomposition(2.35) T U = Ver ⊕ Hor . Here note that the projection map d (cid:101) π maps Hor isomorphically to the tangent space of the base M everywhere. (See (2.14) for the definition of the projection (cid:101) π .)The almost complex structure J preserves both distributions Ver and Hor. Hence the matrixrepresentation of J with respect to the decomposition (2.35) is block-diagonal, which we willcall split. We perturb this almost complex structure by allowing it to be non-split as follows.Note that the orthogonal complement t ⊥ decomposes into t ⊥ = (cid:77) α ∈ R + g α . Consequently, the horizontal subspace Hor at every point in U described below (2.35) admitsa decomposition(2.36) Hor = (cid:0) t ⊥ (cid:1) ⊕ (cid:0) t ⊥ (cid:1) = (cid:77) α ∈ R + g α ⊕ (cid:77) α ∈ R + g α , where the superscripts are used to distinguish the first and second summand.The restriction of the canonical symplectic structure ω to the horizontal subspace is given by(2.37) ω = (cid:88) α ∈ R + ω α + (cid:88) α ∈ R + ω α , where ω kα : g kα ⊗ g kα → R is a symplectic form on g kα for every k = 1 , and α ∈ R + . Indeed, itcan be checked that, at g = g exp( h + ih ) g − ∈ U for g , g ∈ G and h ∈ t , h ∈ t > , thesymplectic form ω is given by ω ( u , v ) = − ( h , [ u , v ]) ,ω ( u , v ) = ( h , [ u , v ]) ,ω ( u , v ) = 0 . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES for all u k ∈ g kα , v k ∈ g kβ , k = 1 , , α, β ∈ R + . The expression (2.37) can be justified since, forany u ∈ g α and v ∈ g β for α, β ∈ R + , p ([ u, v ]) (cid:54) = 0 ∈ t only if α = β .Similarly, the restriction of the pull-back (cid:101) π ∗ ω M of the symplectic form ω M to the horizontaldistribution also decomposes into (cid:101) π ∗ ω M = (cid:88) α ∈ R + (cid:101) ω α + (cid:88) α ∈ R + (cid:101) ω α , where (cid:101) ω kα is the symplectic form ˆ ρ | g kα ⊗ g kα on g kα for every k = 1 , and α ∈ R + . Indeed, thisfollows from the description of ω M in (2.8).Furthermore, for every k = 1 , and α ∈ R + , two symplectic forms ω kα and (cid:101) ω kα are related by ω kα = c α (cid:101) ω kα , for some positive function c α on U defined by c α = ( h , α ∨ ) ρ ( α ∨ ) . Let us say that an almost complex structure on Hor is split with respect to the decompositionHor = (cid:76) α ∈ R + g α ⊕ (cid:76) α ∈ R + g α if it preserves the summand g kα of Hor for each k = 1 , and α ∈ R + . The observation above implies that every almost complex structure on Hor of split typewith respect to Hor = (cid:76) α ∈ R + g α ⊕ (cid:76) α ∈ R + g α is compatible with respect to ω | Hor ⊕ Hor if and onlyif it is so with respect to (cid:101) π ∗ ω M .Consequently, for x ∈ U , if an (cid:101) π ∗ ω M -compatible almost complex structure J on Hor x is closeenough to an (cid:101) π ∗ ω M -compatible almost complex structure of split type, then it is ω | Hor ⊗ Hor -tame.This observation allows us to define the following.
Definition 2.26.
For each x ∈ U , let J x be the family of all (cid:101) π ∗ ω M -compatible almost complexstructures on Hor x that are close enough to a (cid:101) π ∗ ω M -compatible almost complex structure of splittype so that it is ω | Hor ⊗ Hor -tame.
Definition 2.27. (1) Let J ( W ) be the set of all ω -tame almost complex structures on W of weak contact type.(2) Let J ( M ) be the set of all ω M -compatible almost complex structures on M . Definition 2.28.
We define J ( W ) by the subset of J ( W ) consisting of all almost complexstructures J satisfying the three conditions below:(a) It has the following matrix representation with respect to the decomposition (2.35) on U :(2.38) J = (cid:18) J | Ver A J Hor (cid:19) for some section A ∈ Γ( U , Hom ( Hor , Ver )) and an almost complex structure J Hor on the horizontal distribution such that J Hor ,x ∈ J x , ∀ x ∈ U ,(b) There is a compact subset near π W ( C ) ⊂ U such that the section A vanishes and J Hor isequal to J | Hor outside the compact subset. HANWOOL BAE AND NAICHUNG CONAN LEUNG (c) The section A is close enough to so that the almost complex structure J is ω -tame.For an almost complex structure J given as in (2.38), the definition of J x (Definition 2.26)implies that J Hor descends to an ω M -compatible almost complex structure on M . Let us denotethe induced almost complex structure on M by π M ( J ) .Now we consider almost complex structure J on W − × M represented by(2.39) J = (cid:18) J W AB J M (cid:19) , for some sections A ∈ Γ( W × M, Hom ( T M, T W )) and B ∈ Γ( W × M, Hom ( T W, T M )) , with respect to the decomposition T ( W × M ) = T W ⊕ T M . We say that the almost complexstructure J is split if A = 0 = B .Considering that the tangent bundle T U has a further decomposition T U = Ver ⊕ Hor (2.35),the section A in (2.39) is represented by a matrix A = (cid:18) A A (cid:19) for some sections A ∈ Γ( U × M, Hom ( T M,
Ver )) and A ∈ Γ( U × M, Hom ( T M,
Hor )) and the section B is represented by a matrix B = (cid:0) B B (cid:1) for some sections B ∈ Γ( U × M, Hom ( Ver , T M ) and B ∈ Γ( U × M, Hom ( Hor , T M )) on U .Finally the almost complex structure J is written as J = J W Ver A A B J W Hor A B B J M on U × M , with respect to the decomposition(2.40) T ( W × M ) | U× M = Ver ⊕ Hor ⊕ T M.
Definition 2.29. (1) J ( W − × M ) is the set of all ( − ω ⊕ ω M ) -tame almost complex structures J on W − × M such that(a) C ⊆ Supp A, Supp B ⊂ W × M so that J is split outside Supp A ∪ Supp B .(b) For all ( x, y ) ∈ W × M \ ( Supp A ∪ Supp B ) , − J W ( x,y ) is an ω x -tame almost complex structure of weak contact type and J M ( x,y ) is an ( ω M ) y -compatible almost complex structure. ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES (2) J ( W − × M ) is the subset of J ( W − × M ) of all ( − ω ⊕ ω M ) -tame almost complexstructures J on W − × M satisfying the requirements below:(a) The almost complex structure J is represented by a matrix of the form:(2.41) J = − J | Ver A A J W Hor A B J M with respect to the decomposition (2.40), for some A and J W Hor such that the negativeof the top-left × -submatrix (cid:18) J | Ver − A − J W Hor (cid:19) satisfies (b) and (c) in Definition 2.28(b) The bottom-right × -submatrix (cid:18) J W Hor A B J M (cid:19) of (2.41), which we will denote by π M × M ( J ) , is a ( − ω M ⊕ ω M ) -compatible almost complex structure.2.9. Summary of observations.
Let us summarize the observations that will be frequently re-ferred in this paper.(A) ( (cid:101) π × Id M )( C ) = ∆ M ⊂ M × M and C is a torus bundle over ∆ M .(B) (cid:101) π ( π W ( C )) = M = G/T × G/T and π W ( C ) is a torus bundle over M .(C) (cid:101) π ( L ∩ U ) = ∆ ⊂ G/T × G/T and L ∩ U is the direct sum of half-line bundles over ∆ .(D) (cid:101) π ( L (cid:15) ∩ U ) = ∆ (cid:15) ⊂ G/T × G/T and L (cid:15) ∩ U is the direct sum of half-line bundles over ∆ (cid:15) .(E) For each h ∈ Q ∨ , the Hamiltonian chord x h(cid:15) lies in the fiber (cid:101) π − ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) .(F) The tangent bundle T U admits a decomposition T U = Ver ⊕ Hor such that the projection (cid:101) π induces an isomorphism d (cid:101) π | Hor : Hor x → T (cid:101) π ( x ) M at every point x ∈ U .3. R EVIEW ON F UKAYA CATEGORIES
Generalized Lagrangian and generalized Floer theory.
We review the construction ofFloer cohomology of generalized Lagrangians. Please refer to [16, 22] for a more detailed ex-planation.
Definition 3.1. A generalized Lagrangian L of a symplectic manifold ( M.ω ) consists of(1) A sequence of Lagrangians ( L i,i +1 ⊂ M − i × M i +1 ) m − i =0 , where { ( M i , ω i ) } mi =0 are sym-plectic manifolds with M = { pt } and M m = M .(2) A sequence of widths δ L = ( δ L ,i ) m − i =0 ⊂ R > . Definition 3.2.
A generalized Lagrangian L given as in Definition 3.1 is monotone if the follow-ing holds: • There is a monotone constant c , for which each symplectic form ω i satisfies c ( T M i ) = c [ ω i ] ∈ H ( M i , R ) . • L i,i +1 is a monotone Lagrangian submanifold in M − i × M i +1 . HANWOOL BAE AND NAICHUNG CONAN LEUNG
Let us review how to construct the Floer cohomology between two generalized Lagrangians.Here and hereafter, we assume that all generalized Lagrangians are monotone with the samemonotone constant unless mentioned otherwise.Let K = (( K i,i +1 ⊂ X − i × X i +1 ) k − i =0 , δ K ) and L = (( L j,j +1 ⊂ Y − j × Y j +1 ) l − j =0 , δ L ) be twogeneralized Lagrangian of M . For convenience, let us denote M i = (cid:40) X i i = 0 , . . . , kY l − i + k i = k + 1 , . . . , k + l and δ i = δ K ,i i = 0 , . . . , k − δ i = kδ L ,i i = k + 1 , . . . , k + l. Definition 3.3 (Floer data) . Let K and L be generalized Lagrangians given above. The Floerdata for the pair K and L consists of(1) (Hamiltonians) A sequence of Hamiltonians H = ( H i ∈ C ∞ ( M i )) k +1 i =0 for each i (2) (Almost complex structures) A sequence of time-dependent ω i -compatible almost com-plex structures J = ( J i ∈ C ∞ ([0 , δ i ] , J ( M i ))) k + li =0 .We now define the notion of generalized Hamiltonian chord, which serves as the generators ofthe Floer cochain complex. Definition 3.4. A (generalized) Hamiltonian chord x with width δ = ( δ i ) k + li =0 from K to L isdefined by (cid:110) x = ( x i : [0 , δ i ] → M i ) k + li =0 (cid:12)(cid:12)(cid:12) x i satisfies two conditions below. (cid:111) • ˙ x i ( t ) = X H i ( x i ( t )) , • ( x i ( δ i ) , x i +1 (0)) ∈ (cid:40) K i,i +1 ∀ ≤ i ≤ k − L i,i +1 ∀ k ≤ i ≤ k + l − . Let P ( K , L ) = P ( K , L ; H, δ ) be the set of all Hamiltonian chords with width δ from K to L .For a generic choice of Hamiltonians, the corresponding Hamiltonian chords are nondegener-ate. Let us assume that the Hamiltonians are chosen in such a way.Let K be a field of arbitrary characteristic. The Floer cochain complex is a module over theuniversal Novikov ring Λ univ K defined by Λ univ K = (cid:40) (cid:88) ≤ k< ∞ a k q λ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a k ∈ K, λ k ∈ Z (cid:41) , where q is a formal variable of degree 1. Note that only finite sums are allowed in our Novikovring Λ univ K , which is possible by the monotonicity. Remark . Later in Subsection 5.2, the Novkov ring for the endomorphism space of a La-grangian submanifold L ⊂ M will be enhanced to the group ring of π ( M, L ) . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Definition 3.6.
The Floer cochain complex CF ( K , L ) = CF ( K , L ; H, J, δ ) is the free Λ univ K -module generated by P ( K, L ; H, δ ) equipped with a differential µ described below.The Floer differential is defined by counting tuples u = ( u i ) k + li =0 of pseudo-holomorphicstrips. Indeed, we consider the moduli space M ( x − , x + ; H, J, δ ) of tuples ( u i ) k + li =0 of pseudo-holomorphic connecting x − , x + ∈ P ( K , L ) satisfying(a) u i : R × [0 , δ i ] → M i satisfies ∂ J,H ( u ) := 12 { ( du − X H i dt ) + J i,t ( du − X H i dt ) j } = 0 , where t denote the coordinate of [0 , δ i ] .(b) ( u i ( s, δ i ) , u i +1 ( s, ∈ (cid:40) K i,i +1 ≤ i ≤ k − L i,i +1 k ≤ i ≤ l + k − , ∀ s ∈ R (c) lim s →±∞ u i ( s, t ) = x ± i ( t ) .(d) The energy defined by E ( u ) = l + k (cid:88) i =1 (cid:90) R × [0 ,δ i ] | du − X H i dt | dsdt is finite.The following theorem is well-studied in [22]. Theorem 3.7.
For a given sequence H of nondegenerate Hamiltonians, the set of time dependentalmost complex structures J for which the corresponding linearized Fredholm operator is regularis of the second category. As a consequence of the above theorem, the moduli space M ( x − , x + ; H, J, δ ) is a smoothmanifold for such a pair of Floer datum ( H, J ) . Now we define the differential µ : CF ( K , L ) → CF ( K , L ) by(3.1) µ ( x + ) = (cid:88) x − (cid:88) u ∈M ( x − ,x + ; H.J,δ ) σ ( u ) x − , where M ( x − , x + ; HJ, δ ) is the zero dimensional component of M ( x − , x + ; HJ, δ ) and σ ( u ) ∈{− , +1 } is determined by the orientation of the corresponding moduli space induced from the(relatively) spin structure on Lagrangian submanifolds.Assuming that each of the involved Lagrangians does not bound any holomorphic disks ofMaslov index 2, then the operator µ squares to zero. We define the Floer cohomology HF ( K , L ) by the cohomology H ∗ ( CF ( K , L ) , µ ) .3.2. Wrapped Floer cohomology.
An exact Lagrangian submanifold of a completed Liouvilledomain is said to be admissible if it intersects the boundary of the Liouville domain transverselyand the restriction of the Liouville form to the Lagrangian is zero outside a compact subset.Let ( W, λ ) be a completed Liouville domain. We regard any admissible Lagrangian L of W as a generalized Lagrangian L ⊂ { pt } × W with an arbitrary positive width δ L, > . Given twoadmissible Lagrangians L , L , we choose Floer data ( H, J ) , where H ∈ H ( W ) is a quadraticfunction as defined in (2.19) and J ∈ C ∞ ([0 , , J ( W )) is a time dependent almost complexstructure of weak contact type (Definition 2.24). HANWOOL BAE AND NAICHUNG CONAN LEUNG
The wrapped Floer cochain complex CW ( L , L ; H, J ) over a field K is defined by the K -vector space generated by P ( L , L ; H, (Definition 3.4)). The differential µ is defined asin (3.1). The wrapped Floer cohomology is the cohomology with respect to the differential µ again. Note that the definition of the wrapped Floer cochain complex CW ( L , L ; H, J ) coin-cides with CF ( L , L ; H, J, given in Definition 3.6 if the coefficient ring is replaced by Λ K .We will make use of the rescaling technique introduced in [2] to define A ∞ -operations onFukaya category. Recall that ψ t is the time log t -flow of the Liouville vector field. Then we have Lemma 3.8 (Rescaling technique, see [2]) . The map P ( L , L ; H ) → P ( ψ t ( L ) , ψ t ( L ); ρH ◦ ψ t ) , x (cid:55)→ ψ t ( x ) induces a quasi-isomorphismof cochain complexes: CW ∗ ( L , L ; H, J ) ∼ = CW ∗ (cid:16) ψ t L , ψ t L ; tH ◦ ψ t , (cid:16) ψ t (cid:17) ∗ J (cid:17) , An extended monotone Fukaya category.
Following the construction in [8], we constructan extended Fukaya category for the symplectic manifold ( M, ω M ) = ( G/T × G/T , ( − ω G/T ) ⊕ ω G/T ) and the moment Lagrangian C ⊂ W − × M (2.22) where ( W, λ ) = ( T ∗ G, λ can ) .The extended Fukaya category F C ( M ) is an A ∞ -category whose objects are of either one ofthe following types:(I) a compact monotone Lagrangian in ( M, ω M ) ,(II) a monotone generalized Lagrangian L = ( L, C ) with width δ L , > where L ⊂ W is anadmissible Maslov zero Lagrangian. Remark . Every object of the (extended) Fukaya category is supposed to be equipped with abrane structure, which consists of a local system, a grading and a relative spin structure. In ourpaper, we consider trivial local systems only. Furthermore, we will make use of capping half-disks for chords to define proper gradings on the Floer cochain complexes. Finally, we restrictour attention to spin Lagrangian submanifolds since every Lagrangian we are interested in thispaper is spin. Please refer to Appendix B for this.The morphism space between two objects of F C ( M ) is given by the Floer cochain complexwith width (Definition 3.6).Since any other case can be done similarly, we only construct the A ∞ -operations on the mor-phisms spaces between objects of type (II).Let R d denote the moduli space of disks with ( d + 1) -boundary marked points, say ξ , . . . , ξ d . Definition 3.10 (Quilt data for extended monotone Fukaya category) . The quilt data for a quiltedsurface S ∈ R d consists of the following:(1) (Patches and strip-like ends) (d+2)-patches S ∗ , S , . . . , S d ⊂ S of S with complex struc-tures i ∗ , i , . . . , i d , respectively, such that • S ∗ has d -positive strip-like ends { (cid:15) j } dj =1 with width at { ξ j } and one negative strip-like end (cid:15) with width at ξ . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES • S has one positive strip-like end (cid:15) , − at ξ and one negative strip-like end (cid:15) , − at ξ .The widths of (cid:15) , − and (cid:15) , − are given by δ . • S d has one positive strip-like end (cid:15) d, + at ξ d and one negative strip-like end (cid:15) , + at ξ .The widths of (cid:15) d, + and (cid:15) , + are given by δ d . • Every S j , ≤ j ≤ d − , has two positive strip-like ends (cid:15) j, + at ξ j and (cid:15) j +1 , − at ξ j +1 .The widths of (cid:15) j, + and (cid:15) j +1 , − are given by δ j .(2) (Seams) For each j = 0 , . . . , d , I j is a curve in S adjacent to S j and S ∗ : I j = S j ∩ S ∗ . Then the seams I and I d is a curve in S connecting the boundary marked point ξ to themarked point ξ and ξ d , respectively. For each j = 1 , . . . , d − , the seam I j is a curve in S connecting the boundary marked points ξ j and ξ j +1 .The seam is required to be real analytic. It means that at each point z on a seam I j adjacent to two patches S ∗ and S j , there is an open neighborhood U of z in S such that ( U ∩ S p , U ∩ I ) embeds into the pair ( H , ∂ H ) of upper half-planes and its boundaryholomorphically for each p = ∗ , j . Please refer to [16, Definition 2.4] for more detailedexplanation for this.See Figure 1 for the case d = 2 .F IGURE
1. Quilted disk with 2 positive endsWe need to make use of non-split almost complex structures in the sense of [8, Section 5.3.1]to get the transversality for the linearized (inhomogeneous) Cauchy-Riemann operator. For thatpurpose, we have to choose an open ball centered at a point on the seam.
Definition 3.11 (Floer data for extended monotone Fukaya category) . For each S ∈ R d , theFloer data D S consists of the following:(1) (Lagrangian labels) Each seam is labelled with the moment Lagrangian C and the bound-ary component ∂S j \ I of S j is labelled with an admissible Maslov Lagrangian L j ⊂ W for each j = 0 , . . . .d . Let L i be the generalized ( L i , C ) with width δ j as described inDefinition 3.10. HANWOOL BAE AND NAICHUNG CONAN LEUNG (2) (Time-shifting map) A family of smooth functions ρ S,j : ∂S j → [1 , ∞ ) , j = 0 , . . . , d such that • every ρ S.j is constant near the ends of S , and • the constant values of ρ S,j ’s near their ends coincide if two ends converge to thesame marked point.We write w k for the value of ρ S,j ’s near the boundary marked point ξ k of S .(3) (Hamiltonians) For each j = 0 , . . . , d , the j -th patch S j is equipped with a domaindependent quadratic Hamiltonian H j ∈ C ∞ ( S j , H ( W )) for W for j = 0 , . . . , d and S ∗ is equipped with the constantly zero Hamiltonian on M .(4) (Sub-closed 1-forms) For each j = 0 , . . . , d , β j ∈ Ω ( S j ) , is an 1-form on S j such that dβ j ≤ , β j | ∂S j = 0 and (cid:15) ∗ j,k ( X H j ⊗ β j ) = w k X H ◦ ψ wj,k dt. (5) (Open balls) For each j = 0 , . . . , d , U j is an open ball in S centered at z j ∈ I j such thatthere exists a anti-biholomorphism φ j : ( U j ∩ S ∗ , i ∗ ) → ( U j ∩ S j , i j ) . The open balls U j ’sare required to be disjoint to each others and disjoint to the images of all strip-like ends.(6) (Almost complex structures)The almost complex structures J = ( J ∗ , ( J j ) j , ( J j, ∗ ) j ) ∈ C ∞ ( S ∗ \ ∪ U j , J ( M )) × d (cid:89) j =0 C ∞ ( S j \ U j , J ( W )) × d (cid:89) j =0 C ∞ ( U j ∩ S ∗ , J ( W − × M )) . are required to satisfy the following conditions: • J j, ∗ is split outside a compact set K j ⊂ U j ∩ S ∗ in the sense defined below (2.39). • J j and − J Wj, ∗ ◦ φ − j glue smoothly to an element of C ∞ ( S j \ φ j ( K j ) , J ( W )) . • J ∗ and (cid:8) J Mj, ∗ (cid:9) dj =0 glue smoothly to an element of C ∞ ( S ∗ \ K j , J ( M )) . • The pull-back ( (cid:15) ∗ , − J , (cid:15) ∗ J ∗ , (cid:15) ∗ , + J d ) coincides with the almost complex structurefrom the Floer data for the pair L and L d . (Definition 3.3). • The pull-back ( (cid:15) ∗ j, − J j − , (cid:15) ∗ j J ∗ , (cid:15) ∗ j, + J j ) coincides with the almost complex structurefrom the Floer data for the pair L j − and L j . Definition 3.12 (Conformally equivalence) . Two sets of data ( H S , β j , J S ) and ( H S , β j , J S ) are conformally equivalent if there exists a constant c such that α S = cα S J S = ( ψ c ) ∗ J S H S = ( ψ c ) ∗ H S . Recall that the compactification R d of R d is a stratified space such that its strata are indexedby the stable rooted trees with d + 1 semi-infinite ends Γ ’s. For any stratum R Γ of codimension k , there exists a gluing chart(3.2) G Γ : (0 , ∞ ] k × R Γ → R d . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Definition and Theorem 3.13 (Universal and conformally consistent choice of Floer data) . There exists a universal and conformally consistent choice of Floer data D F C ( M ) = { D µ dM } for the Fukaya category F C ( M ) in the sense that for every d ≥ , there exists a family D µ dM = { D S } S ∈R d of Floer data, which satisfies the following :(1) It varies smoothly over the compactified moduli space R d .(2) Its restriction to a boundary stratum is conformally equivalent to the product of Floerdata coming from lower dimensional moduli spaces R d (cid:48) with d (cid:48) ≤ d .(3) For every gluing chart (3.2), Floer data agree to infinite order at boundary stratum withthe Floer data obtained by gluing. Let D F C ( M ) be such Floer data. For each x j := ( x j − , x j , x j + ) ∈ CF ( L j − , L j ) , j = 1 , . . . , d and x = ( x − , x , x ) ∈ CF ( L , L d ) , we define the moduli space of pseudo-holomorphic disks: M ( x ; x , . . . , x d ; D F C ( M ) ) = { ( S, u ∗ , u , . . . , u d ) | S ∈ R d , u ∗ : S ∗ → M, u j : S j → W, j = 0 , . . . , d satisfy (a), (b), (c), (d), (e), (f) and (g) below } (a) ∂ J ∗ ( u ∗ ) = 12 ( du ∗ + J ∗ ◦ du ∗ ◦ i ∗ ) = 0 on S ∗ \ d (cid:91) i =0 U i . (b) ∂ J j ,H j ( u j ) = 12 (cid:0) du j − X H j ⊗ β j + J j ◦ ( du j − X H j ⊗ β j ) ◦ i j (cid:1) = 0 on S j \ U j . (c) The map ( u j ◦ φ j , u ∗ ) : U j ∩ S ∗ → W × M satisfies ∂ J j, ∗ ,H j ( u j ◦ φ j , u ∗ ) = 12 { d ( u j ◦ φ j , u ∗ ) − ( X H j ⊕ ⊗ φ ∗ j β j + J j, ∗ ◦ ( d ( u j ◦ φ j , u ∗ ) − ( X H j ⊕ ⊗ φ ∗ j β j ) ◦ i ∗ ) } = 0 . (d) For each j = 0 , . . . , d , u j ( z ) ∈ ψ ρ S ( z ) ( L j ) , ∀ z ∈ ∂S j \ I j (( ψ ρ S ) − u j ( z ) , u ∗ ( z )) ∈ C, ∀ z ∈ I j . (e) lim s →−∞ ( u ( (cid:15) , − ( s, t )) , u ∗ ( (cid:15) ( s, t )) , u d ( (cid:15) , + ( s, t ))) = ( ψ w ◦ x − ( t ) , x , ψ w ◦ x ( t )) (f) For each j = 1 , . . . , d , lim s →∞ ( u j − ( (cid:15) j, − ( s, t )) , u ∗ ( (cid:15) j ( s, t )) , u j ( (cid:15) j, + ( s, t ))) = ( ψ w j ◦ x j − , x j ( t ) , ψ w j ◦ x j + ( t )) . (g) The energy of ( u ∗ , u , . . . , u d ) defined by (cid:26) (cid:88) j (cid:18) (cid:90) S j \ U j | du j − X H j | J j Vol S j + (cid:90) U j ∩ S ∗ | d ( u j ◦ φ j , u ∗ ) − ( X H j ⊕ ⊗ φ ∗ j β j | J j, ∗ Vol S ∗ (cid:19) + (cid:90) S ∗ \∪ U j | du ∗ | J ∗ Vol S ∗ (cid:27) is finite.For a given Floer datum D F C ( M ) = { D µ dM } , let u ∈ M ( x ; x , . . . , x d ; D µ dM ) be a pseudo-holomorphic quilt. We introduce some related notations as follows: HANWOOL BAE AND NAICHUNG CONAN LEUNG • ind u is the Fredholm index of the associated Fredholm operator, • M u ( x ; x , . . . , x d ; D µ dM ) is the component of M ( x ; x , . . . , x d ; D µ dM ) containing u • For each q ≥ , M q ( x ; x , . . . , x d ; D µ dM ) is the union of M u ( x ; x , . . . , x d ; D µ dM ) forall u with ind u = q + 2 − d . Theorem 3.14. (Moduli space of pseudo-holomorphic quilts) For a generic choice of the almostcomplex structure J in D µ dM , the moduli space M q ( x ; x , . . . , x d ; D µ dM ) is a smooth manifold ofdimension q for all q ≤ and it admits a compactification by adding semi-stable nodal curves.In particular, the number of zero dimensional components is finite. Let D F C ( M ) be a generic choice of Floer data in the sense of the above theorem. Consideringthat each Floer cochain complex CF ( L j − , L j ) is quasi-isomorphic to CF (( ψ w j L j − , C ) , ( ψ w j L j , C )) via a rescaling map analogous to that in Lemma 3.8, we define the d -th composition µ dM : d (cid:79) j =1 CF ( L j − , L j ) → CF ( L , L d ) by(3.3) µ dM ( x ⊗ · · · ⊗ x d ) = (cid:88) x (cid:88) u ∈M (cid:16) x ; x ,...,x d ; D µdM (cid:17) σ ( u ) x , where σ ( u ) ∈ {− , +1 } is determined by the orientation of the corresponding moduli space andextend this linearly, for each d ≥ .As a consequence of 3.14, the standard argument can be applied to proving the following: Theorem 3.15 ( A ∞ -structure) . The operations { µ dM } d ≥ satisfy the A ∞ -relation. We will make use of the following theorem in Section 5.
Theorem 3.16. ( [8, 14, 16] ) Let L ⊂ W be an admissible Lagrangian. If the geometriccomposition L ◦ C is transverse and embedded (Definition 2.19), then it is quasi-isomorphic tothe generalized Lagrangian ( L, C ) with any positive width as objects of the Fukaya category F C ( M ) . In particular, their endomorphism spaces are isomorphic as A ∞ -algebras. Wrapped Fukaya category.
Let ( W, λ ) be a completed Liouville domain. The wrappedFukaya category W ( W ) is an A ∞ -category whose objects are admissible Lagrangians and mor-phism spaces between two objects are given by the wrapped Floer cochain complexes. The A ∞ -operations µ d are defined as in (3.3).Indeed, we now introduce Floer data for the wrapped Fukaya category of W . Definition 3.17 (Floer data for wrapped Fukaya category) . For each S ∈ R d , the Floer data D S consists of the following:(1) (Lagrangian labels) For each j = 0 , . . . , d , the j -th boundary component of S is labelledwith an admissible Maslov zero Lagrangian L j ⊂ W . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES (2) (Time-shifting map) A family of smooth functions ρ S : ∂S → [1 , ∞ ) , which is constantnear the ends of S . We write w k for the value of ρ S near the boundary marked point ξ k of S .(3) (Hamiltonians) A domain dependent Hamiltonian H ∈ C ∞ ( S, H ( W )) .(4) (Sub-closed 1-forms) For each j = 0 , . . . , d , β j ∈ Ω ( S j ) is an 1-form on S j such that dβ ≤ , β | ∂S = 0 and (cid:15) ∗ k ( X H ⊗ β ) = w k X H ◦ ψ wk dt. (5) (Almost complex structures) A domain dependent almost complex structure J ∈ C ∞ ( S, J ( W )) such that the pull-back (cid:15) ∗ k J coincides with the almost complex structure from the Floerdata of L k − and L k for each k = 0 , . . . , d . Here L − = L d .As done in Theorem 3.13, we have Definition and Theorem 3.18 (Universal and conformally consistent choice of Floer data) . There exists a universal and conformally consistent choice of Floer data D W ( W ) = { D µ dW } d ≥ for the Fukaya category W ( W ) in the sense that for every d ≥ , there exists a family D µ dW = { D S } S ∈R d of Floer data, which satisfies the following :(1) It varies smoothly over the compactified moduli space R d .(2) Its restriction to a boundary stratum is conformally equivalent to the product of Floerdata coming from lower dimensional moduli spaces R d (cid:48) with d (cid:48) ≤ d .(3) For every gluing chart (3.2), Floer data agree to infinite order at boundary stratum withthe Floer data obtained by gluing. Finally, we define the higher operations µ dW : d (cid:79) j =1 CW ( L j − , L j ) → CW ( L , L d ) as done in Subsection 3.3. Here we use the rescaling technique in Lemma 3.8 again.4. A ∞ - FUNCTOR ASSOCIATED TO THE MOMENT L AGRANGIAN
Let W = T ∗ G and let M = G/T − × G/T as in previous sections. In this section, wereview the construction of the A ∞ -functor Φ C from the wrapped Fukaya category W ( W ) tothe extended Fukaya category F C ( M ) associated to the moment Lagrangian C ⊂ W − × M ,following the idea of [8]. Note that the monotonicity condition [8, Definition 5.1.5] is satisfiedin our case since both the fundamental groups π ( W ) and π ( M ) are torsion and the Lagrangiansubmanifold C ⊂ W − × M is monotone. This allows us to construct such an A ∞ -functor.The A ∞ -functor maps every admissible Lagrangian L of W to the generalized Lagrangian L = ( L, C ) of M at the level of objects. To describe the functor at the level of morphisms, weneed the following preparations. HANWOOL BAE AND NAICHUNG CONAN LEUNG A ∞ -functor associated to the moment Lagrangian. We need to consider stable quilteddisks with boundary markings.
Definition 4.1.
Let d ≥ . A quilted disk with d + 1 markings is a tuple ( S, I, ξ , ξ , . . . , ξ d ) where • S is a closed 2-disk. • ( ξ , ξ , . . . , ξ d ) is a tuple of marked points on ∂S whose cyclic order is compatible withthe orientation of ∂S . • I is a curve in the interior of S which converges to the -th marked point ξ ∈ ∂S at itsboth ends.The disk S can be obtained as the union of two patches S and S which glue along I . Let S be the patch bounded by I solely and let S be the other one.Let R d, denote the moduli space of all isomorphism classes of quilted disks and let R d, be its compactification, which can be obtained as the union of R d, where Γ runs over over allcolored rooted trees with d + 1 semi-infinite ends. For a more detailed explanation, refer to [16,Subsection 5.1]. Definition 4.2 (Lower stratum R d, ) . For a colored rooted tree Γ with d + 1 semi-infinite ends,let V c be the set of all colored vertices of Γ and let V u be the set of all uncolored vertices of Γ .Then any element of the stratum R d, is a nodal quilted disk with d + 1 semi-infinite ends, whosecomponents are given by (( S v ) v ∈ V c , ( S v ) v ∈ V u ) ∈ (cid:89) v ∈ V c R | v |− , × (cid:89) v ∈ V u R | v |− such that S v and S v share a node if and only if v and v are connected by an edge. Here | v | means the valency of a vertex v .Following the notations in [16], for each colored rooted tree Γ with d + 1 semi-infinite ends,there exist a neighborhood U Γ ⊂ Z Γ of zero in the set Z Γ of balanced gluing parameters and acollar neighborhoods G Γ : R d, × U Γ → R d, . We consider quilt data analogous to Definition 3.10.
Definition 4.3 (Quilt data for functor) . The quilt data for a quilted disk S ∈ R d, consists of thefollowing:(1) (Patches and strip-like ends) Two patches S , S ⊂ S equipped with complex structures i , i , respectively, and strip-like ends as follows: • S has two negative strip-like ends (cid:15) − , (cid:15) + with width δ − = 1 = δ + , respectively, at ξ and d -positive strip-like ends { (cid:15) j } with width at { ξ j } . • S has one 1-negative strip-like end (cid:15) with width at ξ .(2) (Seam) The seam I is given by S ∩ S , which is curve in S connecting ξ to itself. It isrequired to be real analytic as in Definition 3.10. ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES F IGURE
2. Quilted disk with -positive endSee Figure 2 for the case d = 1 .Let ∂S j denote the boundary component of S between ξ j and ξ j +1 for each j = 0 , . . . , d where ξ d +1 = ξ . We need to choose the Floor data for the A ∞ -functor Φ C as follows. Definition 4.4 (Floer data for functor) . For each S ∈ R d, , the Floer data D S consists of thefollowing:(1) (Lagrangian labels) The seam I is labelled with C and the boundary component ∂S j islabelled with an admissible Maslov zero Lagrangian L j ⊂ W for each j = 0 , . . . , d .Let L be the generalized Lagrangian ( L , C ) with width δ − = 1 and L d be the gener-alized Lagrangian ( L d , C ) with width δ + = 1 .(2) (Time-shifting map) A smooth function ρ : ∂S → [1 , ∞ ) which is constant near eachend. We write w j for the value at the j -th end of S for j = 1 , . . . d and w for the valueat ξ .(3) (Hamiltonians) The patch S is equipped with a domain dependent quadratic Hamiltonian H ∈ C ∞ ( S , H ( W )) and S is equipped with the constantly zero Hamiltonian on M .(4) (Sub-closed 1-form) An 1-form β ∈ Ω ( S ) such that dβ ≤ , β | ∂S = 0 and (cid:15) ∗ ,k ( X H ⊗ β ) = w k X H ◦ ψ wk dt, k = 1 , . . . , d. (5) (Open ball) An open ball U in S centered at z ∈ I such that there exists a anti-biholomorphism φ : ( U ∩ S , i ) → ( U ∩ S , i ) . The open ball U is required to be disjoint from the imageof all strip-like ends. See Figure 3 for the case d = 1 .(6) (Almost complex structures)The almost complex structures J = ( J , J , J ) ∈ C ∞ ( S \ U, J ( W )) × C ∞ ( S \ U, J ( M )) × C ∞ ( U ∩ S , J ( W − × M )) . are required to satisfy the following conditions: • J is split outside a compact set K ⊂ U ∩ S . • J and − J W ◦ φ − glue smoothly to an element of C ∞ ( S \ φ ( K ) , J ( W )) . • J and J M glue smoothly to an element of C ∞ ( S \ K, J ( M )) . • The pull-back ( (cid:15) ∗− J , (cid:15) ∗ J , (cid:15) ∗ + J ) coincides with the almost complex structure fromthe Floer data for the pair L and L d . (Definition 3.3). HANWOOL BAE AND NAICHUNG CONAN LEUNG • The pull-back (cid:15) ∗ j J coincides with the almost complex structure from the Floer datafor the pair L j − and L j .F IGURE
3. Open ballNote that R , has exactly one element, say S . It is the union of two patches S and S and each of those patches carries a complex structure i and i , respectively. The patch S hasthree boundary components: ∂S , ∂S and the seam I . Further, it has one positive end and twonegative ends: the one between ∂S and I , and the other one between ∂S and I . We will callthe first one the lower negative end and the other one the upper negative end. We fix Floer datafor the quilted disk S . Definition 4.5 (Initial Floer data) . For each Lagrangian label, the Floer data of D Φ C consists ofthe following:(1) (Time-shifting map) A smooth function ρ S : ∂S → [1 , ∞ ) that takes the value near ξ and takes the value near ξ . We further require that ρ S is constantly on the seam.Accordingly w = and w = 1 .(2) (Hamiltonian) A fixed Hamiltonian H = r on [1 , ∞ ) × ∂W in ⊂ W .(3) (Open ball) An open ball U ⊂ S centered at a point on the seam I , which is disjoint withthe image of strip-like ends.(4) (Closed 1-form) The 1-form β ∈ Ω ( S ) given by β = df for some smooth function f on the patch S such that • f = on ∂S on I on ∂S . • f ( (cid:15) − ( s, t )) = t . • f ( (cid:15) + ( s, t )) = t + . • f is constant on U ∩ S .(5) (Almost complex structures) An almost complex structure J = ( J , J , J ) chosen as inDefinition 4.4 in such a way that • J ∈ C ∞ ( S \ U, J ( W )) . • J ∈ C ∞ ( U ∩ S , J ( W − × M )) . • J is Fredholm regular. ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES See Definition 2.28 and Definition 2.29 for the definition of J ( W ) and J ( W − × M ) ,respectively. Definition and Theorem 4.6 (Universal and conformally consistent choice of Floer data forfunctor) . There exists a universal and conformally consistent choice D Φ C = { D Φ dC } d ≥ of Floerdata for the A ∞ -functor Φ C in the sense that for every d ≥ , there exists a family D Φ dC = { D S } S ∈R d, of Floer data, which satisfies the following :(1) It varies smoothly over the compactified moduli space R d, .(2) Its restriction to each component R , in a lower stratum of R d, is conformally equiva-lent to D Φ C in Definition (4.5).(3) Its restriction to a boundary stratum R d, is conformally equivalent to the product ofFloer data D Φ d (cid:48) C coming from lower dimensional moduli spaces R d (cid:48) , with d (cid:48) < d andFloer data D µ d (cid:48) M coming from R d (cid:48) with d (cid:48) ≤ d if the corresponding vertex lies closer tothe root than the colored vertices in the graph Γ , or Floer data D µ d (cid:48) W otherwise.(4) For every gluing chart (3.2), Floer data agree to infinite order at boundary stratum withthe Floer data obtained by gluing. Let D Φ C be such Floer data. For each x i ∈ CW ( L i − .L i ) , i = 1 , . . . , d and x ∈ CF ( L , L d ) ,we define the moduli space of pseudo-holomorphic quilted disks: M ( x ; x , . . . , x d ; D Φ dC ) = { ( S, u , u ) | S ∈ R d, , u : S → W, u : S → M satisfy (a), (b), (c), (d), (e), (f) and (g) below } (4.1)(a) ∂ J ,H ( u ) = 12 ( du − X H ⊗ β + J ◦ ( du − X H ⊗ β ) ◦ i ) = 0 on S \ U. (b) ∂ J ( u ) = 12 ( du + J ◦ du ◦ i ) = 0 on S \ U. (c) The map ( u ◦ φ, u ) : U ∩ S → W × M satisfies ∂ J ,H ( u ◦ φ, u ) = 12 { d ( u ◦ φ, u ) − ( X H ⊕ ⊗ φ ∗ β )+ J ◦ ( d ( u ◦ φ, u ) − ( X H ⊕ ⊗ φ ∗ β ) ◦ i } = 0 (d) u ( z ) ∈ ψ ρ S ( z ) ( L j ) , ∀ z ∈ ∂S j and (( φ ρ S ( z ) ) − u ( z ) , u ( z )) ∈ C, ∀ z ∈ I. (e) lim s →−∞ ( u ( (cid:15) − ( s, t )) , u ( (cid:15) ( s, t )) , u ( (cid:15) + ( s, t ))) = ( ψ w ◦ x − ( t ) , x , ψ w ◦ x ( t )) . (f) For each j = 1 , . . . , d , lim s →∞ u ( (cid:15) j ( s, t )) = ψ w j ◦ x j ( t ) . (g) The energy of ( u , u ) defined by (cid:18) (cid:90) S \ U | du − X H ⊗ β | J Vol S + (cid:90) S \ U | du | J Vol S + (cid:90) S ∩ U | d ( u ◦ φ, u ) − ( X H ⊕ ⊗ φ ∗ β | J Vol S (cid:19) is finite. HANWOOL BAE AND NAICHUNG CONAN LEUNG
For a given Floer data D Φ C , let u ∈ M ( x ; x , . . . , x d ; D Φ dC ) be a pseudo-holomorphic quilt.Let us introduce some notations. • ind u is the Fredholm index of the associated Fredholm operator. • M u ( x ; x , . . . , x d ; D Φ dC ) is the component of M ( x ; x , . . . , x d ; D Φ dC ) containing u • For each q ≥ , M q ( x ; x , . . . , x d ; D Φ dC ) is the union of M u ( x ; x , . . . , x d ; D Φ dC ) forall u with ind u = q + 1 − d . Theorem 4.7. (Moduli space of pseudo-holomorphic quilts) For a generic choice of the almostcomplex structure J in D Φ dC , the moduli space M q ( x ; x , . . . , x d ; D Φ dC ) is a smooth manifoldof dimension q for q ≤ and it admits a compactification by adding semi-stable nodal quilteddisks. In particular, the number of zero dimensional components is finite. The proof of Theorem 4.7 for the case d ≥ can be done using the standard argument asdescribed in [8]. But the case d = 1 requires further arguments since there are some restrictionsimposed on the almost complex structure described in Definition 4.5. The argument will be givenin Appendix A.2.Let D Φ C be a generic choice of Floer data in the sense of the above theorem. Now we areready to define A ∞ -functor Φ C . Definition and Theorem 4.8. ( A ∞ -functor Φ C ) For each d ≥ , the d -th factor Φ dC : d (cid:79) j =1 CW ( L j − , L j ) → CF (( L , C ) , ( L d , C )) of the A ∞ -functor is defined by putting Φ dC ( x ⊗ · · · ⊗ x d ) = (cid:88) x (cid:88) u ∈M (cid:16) x ; x ,...,x d ; D Φ dC (cid:17) σ ( u ) x , where σ ( u ) ∈ {− , +1 } is determined by the orientation of the corresponding moduli space andextending this linearly. Then { Φ dC } d ≥ satisfies the equation for A ∞ -functor. (See [20, Chapter1] for the definition of A ∞ -functor.)
5. A
PARTIAL COMPUTATION OF THE FUNCTOR Φ C For the moment Lagrangian C introduced in Subsection 2.6, the A ∞ -functor Φ C maps thecotangent fiber T ∗ e G to ( T ∗ e G, C ) , an object of type (II) of the Fukaya category F C ( M ) . As aconsequence, the functor induces an A ∞ -homomorphism Φ C from the endomorphism space of T ∗ e G to that of ( T ∗ e G, C ) . Since the geometric composition T ∗ e G ◦ C is transverse and embedded,the algebraic composition ( T ∗ e G, C ) is quasi-isomorphic to the geometric composition T ∗ e G ◦ C =∆ ⊂ G/T × G/T . Hence the A ∞ -homomorphism Φ C induces an algebra homomorphismfrom the wrapped Floer cohomology of the cotangent fiber T ∗ e G to the Floer cohomology of thediagonal ∆ . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES The explicit description of the wrapped Floer cohomology of T ∗ e G and the Floer cohomologyof ∆ G/T will be given in Subsection 5.1 and Subsection 5.2. Then we provide a partial com-putation of the A ∞ -homomorphism Φ C from the wrapped Floer cohomology HW ( T ∗ e G, T ∗ e G ) in Subsection 5.3. Finally, in Subsection 5.4, we show that a localization of the wrapped Floercohomology HW ( L, L ) is isomorphic to the Floer cohomology HF (∆ , ∆) as algebras.5.1. Wrapped Floer cohomology of the cotangent fiber.
Let L = T ∗ e G and L (cid:15) = T ∗ exp (cid:15) G betwo admissible Lagrangians in W = T ∗ G as done in Subsection 2.7. We consider the wrappedFloer cohomology HW ( L, L (cid:15) ) instead of HW ( L, L ) , which makes sense since any two cotan-gent fibers are quasi-isomorphic.To define the cohomological degree of the Hamiltonian chord x h(cid:15) (2.31), we introduce thenotion of auxiliary capping half-disk for x h(cid:15) , which we will just call capping half-disk for x h(cid:15) byabuse of notation.For that purpose, let T be a closed disk D ⊂ C centered at with two boundary punctures ξ − = − ∈ ∂D and ξ + = 1 ∈ ∂D . The puncture ξ − (resp. ξ + ) is equipped with a negative (resp.positive) strip-like end (cid:15) − : ( −∞ , × [0 , → T (resp. (cid:15) + : [0 , ∞ ) × [0 , → T. ) . Then the surface T has two boundary components: ∂T = (cid:110) e s √− ∈ ∂D | s ∈ ( π, π ) (cid:111) ,∂T = (cid:110) e s √− ∈ ∂D | s ∈ (0 , π ) (cid:111) . Consider the fixed Hamiltonian chord x (cid:15) for ∈ Q ∨ . Since the homotopy group π ( U , L ∩ U ) is trivial, the following definition makes sense. Definition 5.1.
Let x : [0 , → U ⊂ W be a path such that x (0) ∈ L and x (1) ∈ L (cid:15) . A cappinghalf-disk for the chord x is a continuous map v : T → U ⊂ W such that • v ( z ) ∈ (cid:40) L z ∈ ∂T ,L (cid:15) z ∈ ∂T . • lim s →−∞ v ( (cid:15) − ( s, t )) = x ( t ) . • lim s → + ∞ v ( (cid:15) + ( s, t )) = x (cid:15) ( t ) For each h ∈ Q ∨ , the Hamiltonian chord x h(cid:15) admits a capping half-disk v h(cid:15) , which is unique upto homotopy since the homotopy group π ( U , U ∩ L ) is zero.We fix a symplectic trivialization of ( v h(cid:15) ) ∗ T W = T × E W for some symplectic vector space E W of dimension n = dim R W such that T x h(cid:15) (0) L = F and T x h(cid:15) (1) L (cid:15) = J F for some Lagrangian subspace F ⊂ E W and a complex structure J on E W compatible with thesymplectic structure on E W . Then we consider the canonical short path Λ can from T x h(cid:15) (0) L = F HANWOOL BAE AND NAICHUNG CONAN LEUNG to T x h(cid:15) (1) L (cid:15) = J F defined by Λ can ( t ) = exp( − tJ ) F, t ∈ (cid:104) , π (cid:105) . To get a path of Lagrangian subspaces connecting T x h(cid:15) (0) L and T x h(cid:15) (1) L (cid:15) , we first consider apath Λ (cid:15) : [0 , → Lag ( E W ) of Lagrangian subspaces in E W connecting T x (cid:15) (0) L and T x (cid:15) (1) L (cid:15) defined by(5.1) Λ (cid:15) ( t ) = exp( t(cid:15) ) · T x (cid:15) (0) L, t ∈ [0 , , where · means the left G -action on W . Then the concatenation of the tangent spaces of L alongthe boundary component ∂T , the path Λ (cid:15) and the tangent spaces of L (cid:15) along the boundary com-ponent ∂T gives a path of Lagrangian Λ h connecting the tangent spaces T x h(cid:15) (0) L and T x h(cid:15) (1) L (cid:15) .Finally, concatenating the path Λ h with the reversed canonical path Λ can , we get a loop of La-grangian subspaces in E W . We define the Maslov index µ ( x h(cid:15) , v h(cid:15) ) by the Maslov index of thisloop.To compute the Maslov index µ ( x h(cid:15) , v h(cid:15) ) , consider the composition (cid:101) π ◦ v h(cid:15) of the capping half-disk v h(cid:15) with the projection (cid:101) π : U → M . Recall (cid:101) π ( L ∩ U ) = ∆ and (cid:101) π ( L (cid:15) ∩ U ) = ∆ (cid:15) . Furthermore, the Hamiltonian chord x h(cid:15) projects down to ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) ∈ ∆ ∩ ∆ (cid:15) and the chord x (cid:15) projects down to ([ e ] , [ e ]) ∈ ∆ ∩ ∆ (cid:15) . Hence the composition (cid:101) π ◦ v h(cid:15) gives a capping half-disk for the intersection point ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) ∈ ∆ ∩ ∆ (cid:15) bounding the Lagrangian ∆ and ∆ (cid:15) .Therefore, as (cid:15) ∈ t approaches , the composition (cid:101) π ◦ v h(cid:15) converges to a disk bounding theLagrangian ∆ = ∆ G/T . Let us denote it by v h , namely, v h = lim (cid:15) → (cid:101) π ◦ v h(cid:15) : ( D, ∂D ) → ( M, ∆) . To compute the homotopy class of v h in the relative homotopy group π ( M, ∆) , consider thelong exact sequence of homotopy groups for the inclusion ∆ ⊂ G/T × G/T = M , π (∆) → π ( M ) = π ( G/T ) × π ( G/T ) → π ( M, ∆) → π (∆) = 0 . Since the image of the first map is given by the diagonal subgroup of π ( G/T ) × π ( G/T ) and the connecting map π ( M, ∆) → π ( G/T ) is trivial, we deduce that the homotopy group π ( M, ∆) is isomorphic to π ( G/T ) = Q ∨ (See Subsection 2.4). Indeed, one can construct anisomorphism π ( M, ∆) → π ( G/T ) as follows: Let v : ( D, ∂D ) → ( G/T − × G/T , ∆) beany continuous map and let ( v , v ) = ( π ◦ v, π ◦ v ) where π j : G/T × G/T → G/T is theprojection to the j -th factor, j = 1 , . Let v be the map v with the orientation of the domainreversed. Then v and v glue along the diagonal ∆ on the boundary ∂D and it gives a continuousmap v v : S → G/T . The correspondence(5.2) [ v ] (cid:55)→ [ v v ] ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES yields an isomorphism from π ( M, ∆) to π ( G/T ) . Lemma 5.2. (1) The correspondence (5.2) induces an isomorphism π ( M, ∆) ∼ = π ( G/T ) = Q ∨ . (2) The Maslov index of [ v ] ∈ π ( M, ∆) and the Chern number of [ v v ] ∈ π ( G/T ) arerelated by µ ∆ ([ v ]) = 2 c ( G/T )([ v v ]) . Furthermore, these two are given by ρ ([ v ]) . where [ v ] is regarded as an element of Q ∨ via the isomorphism in (1).Proof. It only remains to prove the second assertion, which follows from the way the maps v and v v are related and the symplectic structure on M described in Lemma 2.13. (cid:3) Then we have
Lemma 5.3. [ v h ] = − w − h + (cid:15) h ∈ Q ∨ = π ( M, ∆) . Proof.
Regarding the map v h as a map from ( D \ { ξ − } , ∂D \ { ξ − } ) to ( M, ∆) , it lifts to a map (cid:101) v h : ( D \ { ξ − } , ∂D \ { ξ − } ) → ( U , L ∩ U ) that is asymptotic to x h (2.29) at the end ξ − .Now recall that (cid:101) α ∈ Ω ( D ∗ N ) ∼ = Ω ( U ad ) (2.15) is a connection one form for the line bundle O G/T ( α ) for any root α . We consider its pull-back q ∗ (cid:101) α via the covering map q : U → U ad .Since the Hamiltonian chord x h rotates by the angle − πα j ( w − h + (cid:15) h ) in the j -th summand O G/T ( α j ) of the normal bundle N , we have (cid:90) ( x h ) ∗ q ∗ (cid:101) α j = − α j ( w − h + (cid:15) h ) , ∀ j = 1 , . . . , l. But, by applying the Stokes’ theorem to the lift (cid:101) v h , we have (cid:90) ( x h ) ∗ q ∗ (cid:101) α j = − (cid:90) S ( (cid:101) v h ) ∗ q ∗ d (cid:101) α j + (cid:90) ∂S ( v h(cid:15) | ∂S ) ∗ q ∗ (cid:101) α = − (cid:90) S ( (cid:101) v h ) ∗ q ∗ d (cid:101) α j = − (cid:90) S ( (cid:101) v h ) ∗ q ∗ π ∗ ( π ∗ ˆ α j ⊕ − π ∗ ˆ α j ) (2 . − π ∗ ˆ α j ⊕ π ∗ ˆ α j )([ π ◦ q ◦ (cid:101) v h ] = [ v h ])= α j ([ v h ]) . Here the second equality follows from the fact that the (cid:101) α j vanishes along the tangent space of L .Further, in the last equality follows from the identification (5.2).As a result, we have − α j ( w − h + (cid:15) h ) = α j ([ v h ]) for every j = 1 , . . . , l . The assertion followssince α , . . . , α l form a basis for t ∨ . (cid:3) HANWOOL BAE AND NAICHUNG CONAN LEUNG
Now we compute the Maslov index µ ( x h(cid:15) , v h(cid:15) ) . Lemma 5.4.
The index µ ( x h(cid:15) , v h(cid:15) ) is given by (5.3) µ ( x h(cid:15) , v h(cid:15) ) = 2 l ( w h + (cid:15) ) − ρ ( w − h + (cid:15) h ) . Proof.
Since the capping half-disk v h(cid:15) is assumed to map into U , we just need to look into thesymplectic trivialization of ( v h(cid:15) ) ∗ T U closely. Recall that U is identified with a covering space ofthe punctured poly-disk normal bundle D ∗ N (2.14) and consequently that the tangent bundle T U admits the symplectic decomposition Ver ⊕ Hor (2.35). Hence the Maslov index µ ( x h(cid:15) , v h(cid:15) ) is thesum of µ Ver and µ Hor , the contributions from the vertical and horizontal distribution respectively.First, for the contribution µ Ver from the vertical distribution, note that the normal bundle N decomposes into l -line bundles ⊕ lj =1 N j . Accordingly, the vertical distribution also decomposesinto l -sub-distributions, each of which is symplectic and symplectically orthogonal to each oth-ers. But the restrictions of each line bundle N j to the diagonal ∆ and the perturbed diagonal ∆ (cid:15) are trivial by definition of N j (2.11). Since both the Lagrangian L and L (cid:15) are parallel with re-spect to the connection one form (cid:101) α j for each j = 1 , . . . , l , we may assume that the trivializationhas been chosen in such a way that both the tangent spaces of L along the boundary component ∂T and those of L (cid:15) along ∂T are constant. Hence these do not contribute to µ Ver . For the shortpath Λ (cid:15) , the description of the action exp( t(cid:15) ) below (2.33) says that it rotates negatively by anangle less than π in the summand N j for each j = 1 , . . . , l . Hence the vertical summand of itsconcatenation with the reversed canonical short path Λ can is homotopic to the constant path. Thisshows that µ Ver = 0 .Now let us consider the contribution µ Hor from the horizontal distribution. Considering (C),(D) and (F) in Subsection 2.9, we see that the Maslov index µ Hor is equal to the Maslov indexof the loop of Lagrangian subspaces associated to the capping half-disk (cid:101) π ◦ v h(cid:15) bounding ∆ and ∆ (cid:15) . But recall the definition v h = lim (cid:15) → (cid:101) π ◦ v h(cid:15) . This implies that the difference between µ Hor and µ ∆ G/T ([ v h ]) is measured by how the tangent space T x ∆ (cid:15) rotates from T x ∆ , where x = ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) ∈ ∆ ∩ ∆ (cid:15) .To look into the difference more closely, consider a quadratic form B on the tangent space T x ∆ defined by B ( v ) = ω M ( v, Ψ( v )) , v ∈ T x ∆ where Ψ( v ) ∈ T x M is given in such a way that v + Ψ( v ) ∈ T x ∆ (cid:15) . Then one can deduce that theMaslov index µ Hor increases from the Maslov index µ ∆ ([ v h ]) by the dimension of the maximalsubspace of T x ∆ on which the quadratic form B is positive. Let us denote the dimension by λ B .Since Lemma 5.2 and Lemma 5.3 imply µ ∆ ([ v h ]) = − ρ ( w − h + (cid:15) h ) , the following lemma completes the proof. (cid:3) Lemma 5.5. λ B = 2 l ( w h + (cid:15) ) ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Proof.
Let us denote w = w h + (cid:15) in this proof. Recall that the horizontal distribution Hor at apoint x ∈ U is identified withHor x = (cid:77) α ∈ R + g α ⊕ (cid:77) α ∈ R + g α ( Please refer to (2.36) for this decomposition ) and the projection (cid:101) π : U → M . maps this isomorphically to T (cid:101) π ( x ) M .The tangent space T x ∆ ⊂ T x M = (cid:76) α ∈ R + g α ⊕ (cid:76) α ∈ R + g α is given by (cid:40) ( u, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = (cid:88) α ∈ R + a α e α + b α f α ∈ (cid:77) α ∈ R + g α (cid:41) , while the tangent space T x ∆ (cid:15) is given by (cid:40) ( Ad (exp w − (cid:15) ) u, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u = (cid:88) α ∈ R + a α e α + b α f α ∈ (cid:77) α ∈ R + g α (cid:41) . But for each α ∈ R + , the adjoint action on g α = R (cid:104) e α , f α (cid:105) is given byAd (exp w − (cid:15) ) e α = cos(2 πα ( w − (cid:15) )) e α + sin(2 πα ( w − (cid:15) )) f α Ad (exp w − (cid:15) ) f α = − sin(2 πα ( w − (cid:15) )) e α + cos(2 πα ( w − (cid:15) )) f α Since the regular element (cid:15) is chosen in such a way that < α ( (cid:15) ) < for all α ∈ R + , the following three conditions are all equivalent: ˆ α ( e α , Ad (exp( w − (cid:15) )) e α ) < , ( See (2 . for the definition of ˆ α. )ˆ α ( f α , Ad (exp( w − (cid:15) )) f α ) < ,α ( w − (cid:15) ) < . Considering that the symplectic form ω M on M = G/T − × G/T is given by (cid:32) − (cid:88) α ∈ R + ˆ α (cid:33) ⊕ (cid:88) α ∈ R + ˆ α, we deduce that the number λ B is given by the twice the number of positive roots α ∈ R + suchthat α ( w − (cid:15) ) < . But the latter one coincides with the length l ( w ) . The assertion follows. (cid:3) Finally we define the cohomological degree the Hamiltonian chord x h(cid:15) by the Maslov index µ ( x h(cid:15) , v h(cid:15) ) . Since the cohomological degrees for the Hamiltonian chords x h(cid:15) are all even, thedifferential µ on the wrapped Flocer cochain complex CW ( L, L (cid:15) ) vanishes. Hence we have Theorem 5.6 (Wrapped Floer cohomology of T ∗ e G ) . The wrapped Floer cohomology HW ( L, L (cid:15) ) is given by the K -vector space generated by the classes of the Hamiltonian chords x h(cid:15) for h ∈ Q ∨ . HANWOOL BAE AND NAICHUNG CONAN LEUNG
Remark . In the sense of Theorem 1.1, the computation of the Floer cohomology HW ( L, L (cid:15) ) in the above theorem is consistent with that of the homology of based loop spaces of Lie groupsgiven in [6].5.2. Floer cohomology of the diagonal.
Floer cohomology of the diagonal ∆ . To be consistent with that we computed the wrappedFloer cohomology HW ( L, L (cid:15) ) in Subsection 5.1, we will consider the Floer cohomology CF (∆ , ∆ (cid:15) ) where ∆ (cid:15) = L (cid:15) ◦ C . Indeed, the set ∆ (cid:15) is given by { ([ g ] , [ g ]) ∈ G/T × G/T | [ g ] = [exp (cid:15) · g ] } . It is straightforward to check that ∆ and ∆ (cid:15) intersect transversely at(5.4) ∆ ∩ ∆ (cid:15) = { ([ ˙ w ] , [ ˙ w ]) ∈ G/T × G/T | w ∈ W } . Arguing as in the proof of Lemma 5.5, we may assign, to each intersection point ([ ˙ w ] , [ ˙ w ]) ∈ ∆ ∩ ∆ (cid:15) , a cohomological degree given by(5.5) | ([ ˙ w ] , [ ˙ w ]) | = 2 l ( w ) . The perturbed diagonal ∆ (cid:15) is actually Hamiltonian isotopic to the diagonal. To see this, weconsider the Morse function f (cid:15) on G/T defined by(5.6) f (cid:15) ([ g ]) = ρ ( p ( Ad g − (cid:15) )) , ∀ g ∈ G, where p : g → t is the orthogonal projection. Note that the definition of (cid:15) (2.30) implies that theHamiltonian f (cid:15) is bounded below by − and above by , respectively.Identifying the tangent space of G/T at every point with (cid:76) α ∈ R + g α , the corresponding Hamil-tonian vector field X f (cid:15) on G/T is given by X f (cid:15) ([ g ]) = − p ⊥ ( Ad g − (cid:15) ) , ∀ g ∈ G where p ⊥ : g → t ⊥ = (cid:77) α ∈ R + g α is the orthogonal projection.Moreover, the time t -flow φ tX f(cid:15) of the Hamiltonian vector field X f (cid:15) on G/T is given by φ tX f(cid:15) ([ g ]) = [exp( − t(cid:15) ) g ] , ∀ g ∈ G. Considering the Hamiltonian f (cid:15) ◦ π on M = G/T − × G/T , we indeed observe that theperturbed diagonal ∆ (cid:15) is the image of the diagonal ∆ under the time -flow of the Hamiltonianvector field of f (cid:15) ◦ π . Then the degree (5.5) of the intersection point ([ ˙ w ] , [ ˙ w ]) coincides with dim ∆ − λ f (cid:15) ◦ π ([ ˙ w ] , [ ˙ w ]) where λ f (cid:15) ◦ π ([ ˙ w ] , [ ˙ w ]) is the Morse index of the Morse function ( f (cid:15) ◦ π ) | ∆ at ([ ˙ w ] , [ ˙ w ]) .Returning to the computation of the Floer cohomology HF (∆ , ∆ (cid:15) ) , we first define the Novikovring Λ K for the diagonal ∆ as follows. ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Definition 5.8.
The Novikov ring Λ K is defined by the group ring of π ( M, ∆ G/T ) over K . Moreconcretely, the Novikov ring ∆ K is given by Λ K = (cid:40) (cid:88) ≤ k< ∞ a k q h k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a k ∈ K, h k ∈ π ( M, ∆ G/T ) (cid:41) , where q is a formal variable.It is justified to define the cohomological degree of q h for h ∈ Q ∨ by ρ ( h ) by Lemma 5.2.The Floer complex CF (∆ , ∆ (cid:15) ) is a Λ K -module generated by ([ ˙ w ] , [ ˙ w ]) for w ∈ W . This canbe stated as follows.(5.7) CF (∆ , ∆ (cid:15) ) = Λ K (cid:104) ([ ˙ w ] , [ ˙ w ]) | w ∈ W (cid:105) Considering that the Floer cochain complex CF (∆ , ∆ (cid:15) ) (5.7) is a Λ K -module generated by ([ ˙ w ] , [ ˙ w ]) , w ∈ W , each of which has an even degree, we have Theorem 5.9 (Floer cohomology of the diagonal ∆ ) . The Floer cohomology HF (∆ , ∆ (cid:15) ) is givenby the free Λ K -module generated by the classes of the intersections points ([ ˙ w ] , [ ˙ w ]) ∈ ∆ ∩ ∆ (cid:15) for w ∈ W .Remark . The Floer data for ∆ and ∆ (cid:15) (Definition 3.3) we used in the description of theFloer cochain complex CF (∆ , ∆ (cid:15) ) consists of the constantly zero Hamiltonian on M and a timedependent almost complex structure J ∈ C ∞ ([0 , , J ( M )) .5.2.2. Floer cohomology of the generalized Lagrangian L . We first discuss the reason why theNovikov ring Λ K defined in Definition 5.8 continues to work for the Floer cohomology of thegeneralized Lagrangian ( L, C ) .Let L = ( L, C ) and L (cid:15) = ( L (cid:15) , C ) be generalized Lagrangians with width δ > . Sincethe Hamiltonian chords from L to itself are degenerate as in the case of the Lagrangian L , weconsider the Hamiltonian chords from L to L (cid:15) instead.Let T be a closed unit disk with two boundary marked point ξ − and ξ + removed as defined inSubsection 5.1. We additionally consider a seam I ⊂ T converging to ξ − at its ends. Then T becan be obtained as the union of two patches T and T which intersect along I . Let T be the onebounded by I solely and let T be the other one. We will call T the outer patch and T the innerpatch. Then the outer patch T has three boundary components: two boundary components ∂T , ∂T from those of T and the seam I . See Figure 4.Let us equip T with two negative strip-like ends (cid:15) − , (cid:15) + : ( −∞ , × [0 , δ ] → T at ξ − and one positive strip-like end (cid:15) : [0 , ∞ ) × [0 , → T at ξ + . Also equip T with a negative strip-like end (cid:15) : ( −∞ , × [0 , → T at ξ − . HANWOOL BAE AND NAICHUNG CONAN LEUNG
Definition 5.11 (Capping quilted half-disk) . Let x = ( x − , x , x + ) be a chord from L to L (cid:15) . A capping quilted half-disk for the chord x is a pair v = ( v , v ) of continuous map v : T → U ⊂ W and v : T → M such that • v ( z ) ∈ (cid:40) L, z ∈ ∂T ,L (cid:15) , z ∈ ∂T . • ( v ( z ) , v ( z )) ∈ C ⊂ W × M for all z ∈ I . • lim s →−∞ ( v ( (cid:15) − ( s, t )) , v ( (cid:15) ( s, t )) , v ( (cid:15) + ( s, t ))) = ( x − ( t ) , x ( t ) , x + ( t )) • lim s → + ∞ v ( (cid:15) ( s, t )) = x (cid:15) ( t ) .F IGURE
4. Capping quilted half-diskA Hamiltonian chord from L to L (cid:15) consists of x = ( x − , x , x + ) where x − , x + : [0 , δ ] → W are integral curves of X H and x ∈ ∆ ∩ ∆ (cid:15) such that x − (0) ∈ L, x + ( δ ) ∈ L (cid:15) and ( x − ( δ ) , x ) , ( x + (0) , x ) ∈ C. The observation (5.4) says that x = ([ ˙ w ] , [ ˙ w ]) ∈ ∆ ∩ ∆ (cid:15) for some w ∈ W .Actually the point x = ([ ˙ w ] , [ ˙ w ]) determines the paths x − and x + in this case. To describethose, first observe from the description of the moment Lagrangian C in Subsection 2.6 thatthe moment Lagrangian C projects diffeomorphically down to a submanifold π W ( C ) of W .Moreover, considering (B), (C) and (D) in Subsection 2.9, we see that each of the Lagrangian L and L (cid:15) intersects the submanifold π W ( C ) at exactly one point in the fiber (cid:101) π − ([ ˙ w ] , [ ˙ w ]) . Thenthe paths x − and x + are required to satisfy x − (0) ∈ L, x − ( δ ) ∈ π W ( C ) x + (0) ∈ π W ( C ) , x + ( δ ) ∈ L (cid:15) . Since the Hamiltonian vector field X H is tangent to the submanifold π W ( C ) , we deduce thatthe path x − lies in the fiber (cid:101) π − ([ ˙ w ] , [ ˙ w ]) . Moreover it starts at the unique intersection pointof π W ( C ) and L and ends at some point on π W ( C ) . Similarly, the path x + lies in the fiber (cid:101) π − ([ ˙ w ] , [ ˙ w ]) . Also it ends at the unique intersection point of π W ( C ) and L (cid:15) and starts at somepoint on π W ( C ) . The paths x − and x + are uniquely determined by the width δ since they areintegral curves of the Hamiltonian vector field X H . Let us denote by x w = ( x − , x , x + ) theHamiltonian chord for such a pair of paths x − , x + in W .Let ( v , v ) be a capping half-disk for the Hamiltonian chord x w = ( x − , x , x + ) for some w ∈ W . Since v is assumed to map into U , one can consider the composition of the projection ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES (cid:101) π : U → M with the map v . From (A), (C) and (D) in Subsection 2.9, we deduce that the (cid:101) π ◦ v and v glue on the seam I along the diagonal ∆ M ⊂ M × M and the glued map ( (cid:101) π ◦ v ) v maps the boundary component ∂T into ∆ and the boundary component ∂T into ∆ (cid:15) . As wedid for the capping half-disk v h(cid:15) in Subsection 5.1, we observe that the glued map ( (cid:101) π ◦ v ) v converges to a disk bounding the Lagrangian ∆ = ∆ G/T as (cid:15) ∈ t approaches . Let us denotethe disk by v w : ( D, ∂D ) → ( M, ∆) . Note that the map v w converges to x = ([ ˙ w ] , [ ˙ w ]) ∈ ∆ ∩ ∆ (cid:15) at ξ − . Definition 5.12.
A capping quilted half-disk ( v , v ) for a Hamiltonian chord x represents h ∈ Q ∨ = π ( M, ∆ G/T ) if the disk v w represents the homotopy class h .For a given pair of the Hamiltonian chord x w and its capping quilted half-disk ( v , v ) , weconsider its action and Maslov index.First, if the capping quilted half-disk ( v , v ) represents h ∈ Q ∨ = π ( M, ∆) . then the actionof the pair ( x w , ( v , v )) is given by(5.8) A f (cid:15) ( x w , ( v , v )) = ρ ( h ) + f (cid:15) ([ ˙ w ]) . Next, for the Maslov index, by considering split Lagrangian subspaces as in [22, 16], one candefine the Maslov index µ ( x w , ( v , v )) for any pair of Hamiltonian chord and its capping quiltedhalf-disk as we have done in Subsection 5.1.Following the idea of computing Maslov indices in Subsection 5.1, we have Lemma 5.13. (5.9) µ ( x w , ( v , v )) = 2 l ( w ) + 2 ρ ([ v w ]) . Lemma 5.13 allows us to identify the pair ( x w , ( v , v )) with x w q [ v w ] ∈ CF ( L , L (cid:15) ) .Once again, since the degree of every generator is even, we have Theorem 5.14 (Floer cohomology of L = ( L, C ) (c.f. Theorem 5.9)) . The Floer cohomology HF ( L , L (cid:15) ) is given by the free Λ K -module generated by the classes of x w for w ∈ W . By Theorem 5.9 and Theorem 5.14, we may identify both HF (∆ , ∆ (cid:15) ) and HF ( L , L (cid:15) ) withthe free Λ K -module generated by W . Applying the result of [14] twice, we get an isomorphismfrom CF ( L , L (cid:15) ) to CF (∆ , ∆ (cid:15) ) such that the coefficient of ([ ˙ w ] , [ ˙ w ]) q h in the image of each gen-erator x w q h is 1. This will be verified in Appendix B.Before we move on to the next subsection, we observe a specific family of capping quiltedhalf-disks for future use. Consider a capping quilted half-disk ( v , v ) for x w = ( x − , x , x + ) such that the second entry v is constant. For such a capping quilted half-disk, the concatenation x − ∗ ( v | I ) ∗ x + lies in the fiber (cid:101) π − ([ ˙ w ] , [ ˙ w ]) since the second entry v is constant. Here v | I means the restriction of v to the seam I with the reversed orientation. Then we have Lemma 5.15.
Let ( v , v ) be given as above. The covering map q : U → U ad ∼ = D ∗ N maps theconcatenation x − ∗ ( v | I ) ∗ x + to a path that rotates by the angle π ( α j ( h ) − α j ( w − (cid:15) )) in the j -th summand N j | M of the normal bundle N at the point ([ ˙ w ] , [ ˙ w ]) for some h ∈ Q ∨ if and onlyif the capping half-disk ( v , v ) represents h ∈ Q ∨ = π ( M, ∆) . HANWOOL BAE AND NAICHUNG CONAN LEUNG
Proof.
Since the second entry v is constant, the proof of Lemma 5.3 can be applied here. (cid:3) Computation of Φ C . In Subsection 5.1 we have observed that the wrapped Floer coho-mology HW ( L, L (cid:15) ) is isomorphic to the K -vector space generated by the Hamiltonian chords x h(cid:15) for h ∈ Q ∨ . On the other hand, we have shown that the Floer cohomology of the general-ized Lagrangian L = ( L, C ) is a Λ K -module generated by the generalized Hamiltonian chordsx w = ( x − , x , x + ) for w ∈ W in Subsection 5.2.2.The degree formulas in Lemma 5.4 and Lemma 5.13 suggest that the functor Φ C maps x h(cid:15) ∈ CW ( L, L ) to a linear combination of x w (cid:48) q h (cid:48) CF ( L , L ) for some w (cid:48) ∈ W and h (cid:48) ∈ Q ∨ such that(5.10) l ( w h + (cid:15) ) − ρ ( w − h + (cid:15) h ) = 2 l ( w (cid:48) ) + 2 ρ ( h (cid:48) ) since the map Φ C is supposed to preserve the degree. This will be justified by Lemma 5.18.The goal of this subsection is to prove the following key lemma. Lemma 5.16.
For any h ∈ Q ∨ , we have (5.11) Φ C ( x h(cid:15) ) = x w h + (cid:15) q − w − h + (cid:15) h + (cid:88) ( w (cid:48) ,h (cid:48) ) satisfies (5.10) and either (i) or (ii) below a h (cid:48) w (cid:48) x w (cid:48) q h (cid:48) for some a h (cid:48) w (cid:48) ∈ K , where the conditions (i) and (ii) are given by(i) l ( w h + (cid:15) ) = l ( w (cid:48) ) , w (cid:48) (cid:54) = w h + (cid:15) and f (cid:15) ([ ˙ w (cid:48) ]) > f (cid:15) ([ ˙ w h + (cid:15) ]) (5.6).(ii) l ( w h + (cid:15) ) − l ( w (cid:48) ) ∈ Z > .In particular, for h ∈ Q ∨ with l ( w h + (cid:15) ) < , we have Φ C ( x h(cid:15) ) = x w h + (cid:15) q − w − h + (cid:15) h . Most part of Lemma 5.16 will be proved practically by proving Lemma 5.23.From the equality (5.10), we observe that every other term a h (cid:48) w (cid:48) x w (cid:48) q h (cid:48) than the first term in theright hand side of (5.11) has a higher action than the first term. Indeed the action ρ ( h (cid:48) ) + f (cid:15) ([ ˙ w (cid:48) ]) (5.8) of the pair ( w (cid:48) , h (cid:48) ) is greater than that of the leading term due to the conditions (i), (ii) andthe condition (5.10). In this sense, the first term will be called the leading term and the otherswill be called higher action terms.By Theorem 4.6, we may assume that the Floer data D Φ C for the A ∞ -functor has been con-structed in such a way that the restriction of D Φ dC , d ≥ to a component R , in any lowerstratum is given by D Φ C defined in Definition 4.5. Definition 5.17.
For any h, h (cid:48) ∈ Q ∨ and x w (cid:48) ∈ P ( L , L (cid:15) ) , we define M (cid:16) x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C (cid:17) by the moduli space of all pseudo-holomorphic quilts ( u , u ) that satisfy the equations (a), (b),(c), (d), (d), (e), (f) and (g) below (4.1) for the initial Floer data D Φ C and the asymptotes x w (cid:48) and x h(cid:15) , in such a way that the glued capping quilted half-disk ( v h(cid:15) u , u ) represents h (cid:48) . (SeeDefinition 5.12.) ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES The first piece Φ C : CW ( L, L (cid:15) ) → CF ( L , L (cid:15) ) of the functor Φ C is defined by counting rigidpoints in M ( x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C ) . The dimension formula for the moduli space is given as follows. Lemma 5.18.
The dimension of the moduli space M (cid:16) x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C (cid:17) is given by l ( w (cid:48) ) − l ( w h + (cid:15) ) + 2 ρ ( h (cid:48) ) + 2 ρ ( w − h + (cid:15) h ) . Proof.
This follows from the definition of the moduli space M (cid:16) x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C (cid:17) in Definition5.17 and the Maslov index formulas (5.3) and (5.9). (cid:3) To prove Lemma 5.16, we need the following preparations. First, observe that the coveringmap q : G C → G (2.14) restricts to a covering map q : T C → T C ad .Now let ( u , u ) be an element of M ( x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C ) . Then we have Lemma 5.19.
Suppose that (cid:101) π ◦ u is constant on u − ( U ) ∩ U and that u is constant on S ∩ U ,respectively. Then the curve u maps into the torus T C and u is constant.Proof. Recall that the -form β is given by df for some function f on S in Definition 4.5. Sincethe torus T C is preserved by the Hamiltonian flow φ tX H , it suffices to show that v := φ − fX H ◦ u maps into the torus instead. Observe that the pair ( v , u ) solves the following homogeneouspseudo-holomorphic equations: dv + (cid:101) J ◦ dv ◦ i = 0 on S \ Udu + J ◦ du ◦ i = 0 on S \ Ud ( v ◦ φ, u ) + (cid:101) J ◦ d ( v ◦ φ, u ) ◦ i = 0 on U ∩ S , (5.12)where (cid:101) J = ( φ − fX H ) ∗ J and (cid:101) J = ( φ − fX H × Id M ) ∗ J .First note that the set v − ( U ) ∩ U = u − ( U ) ∩ U is a non-empty and open in S ∩ U since U ⊂ S intersects the seam I and the seam I is supposed be mapped to C ⊂ U × M under ( u , u ) . Consider the component V of v − ( U ) = u − ( U ) containing v − ( U ) ∩ U , which alsoincludes the seam and the negative ends of S .The equation (5.12) and the assumption imply that the map ( (cid:101) π ◦ v ◦ φ, u ) is a constant π M × M ( (cid:101) J ) -holomorphic map on U ∩ S , each entry of which maps constantly to ([ ˙ w (cid:48) ] , [ ˙ w (cid:48) ]) .Since the almost complex structure J is assumed to be split outside a compact subset K ⊂ U ∩ S (Definition 4.4), and two almost complex structures J and − J W ◦ φ − are glued on S \ φ ( K ) ,this means that (cid:101) π ◦ v is π M ( (cid:101) J − (cid:101) J W ◦ φ − )) -holomorphic on S \ φ ( K ) . Since (cid:101) π ◦ v is stillconstant on U ∩ S \ φ ( K ) , the identity theorem says that (cid:101) π ◦ v is constant on the component V .In other words, v maps the component V into (cid:101) π − ([ ˙ w (cid:48) ] , [ ˙ w (cid:48) ])) and hence into the torus T C dueto Lemma 2.16.On the other hand, since u is a constant map, the assumption that J ∈ C ∞ ( U ∩ S , J ( W − × M )) implies that v solves the (cid:101) J − (cid:101) J W ◦ φ − ) -holomorphic equation on V . Here note that thealmost complex structure J − J W ◦ φ − ) belongs to J ( W ) pointwisely on V (See Defini-tion 2.28 and Definition 4.5) and hence so does (cid:101) J − (cid:101) J W ◦ φ − ) since being an element of J ( W ) is unchanged after applying the Hamiltonian flow. Hence it follows that v is actually HANWOOL BAE AND NAICHUNG CONAN LEUNG J -holomorphic on V since the horizontal projection d (cid:101) π ◦ dv is zero there. Since the torus T C is a complex submanifold of G C with respect to the integrable almost complex structure J and v maps a nonempty open set into the torus, the curve v does not escape from T C . (cid:3) Now observe that Lemma 2.7 can be interpreted as follows.(5.13) T C ad ∩ U ad = T ad × t reg = T ad × (cid:32) t \ (cid:91) α ∈ R + t α (cid:33) . Besides, the torus T C ad is identified with ( C ∗ ) l by the mapping(5.14) Ψ w ( t ) = ( α ( w − t ) − , . . . , α l ( w − t ) − ) , t ∈ T C ad . for each w ∈ W . Let us denote by C ∗ j the j -th factor C ∗ and denote by D ∗ j the open disk of radius1 with a puncture at its center in C ∗ j . Then we have Lemma 5.20.
The map Ψ w maps T ad × w t > ⊂ T ad × t reg = T C ad ∩ U diffeomorphically to D ∗ × · · · × D ∗ l ⊂ ( C ∗ ) l . Lemma 5.21.
There exists a unique element ( u , u ) of M ( x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C ) such that u mapsinto the torus T C , u is constant and the Fredholm index of the pair ( u , u ) is zero. For such aunique element ( u , u ) , the following hold:(1) ( w (cid:48) , h (cid:48) ) = ( w h + (cid:15) , − w − h + (cid:15) h ) .(2) The image of u lies in U .Proof. Let us consider v = φ − fX H ◦ u instead of considering u directly. Using the identification Ψ w h + (cid:15) : T C ad → ( C ∗ ) l (5.14), the composition q ◦ v : S → T C ad = ( C ∗ ) l can be written as(5.15) q ◦ v ( z ) = ( v ( z ) , . . . , v l ( z )) , where v j is the composition of q ◦ v and the projection to the j -th factor of ( C ∗ ) l .Considering that both Lagrangian L and L (cid:15) are invariant under the Liouville flow and thedescription of the image of π W ( C ) ⊂ W under the time log t -flow ψ t of the Liouville vectorfield, we observe that the map v j is required to satisfy the followings:(a) The map v j solves the holomorphic equation given by dv j + i ◦ dv j ◦ i = 0 . (b) The map v j maps its lower boundary component to φ X H ( R > ) = (cid:110) e √− r r ∈ C ∗ | r ∈ R > (cid:111) and its upper boundary component to e − π √− α j ( w − h + (cid:15) (cid:15) ) · R > . See (2.27) for the descrip-tion of the Hamiltonian flows, and (2.32) and (2.33) for the description of the Lagrangian L and L (cid:15) . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES (c) The map v j sends the seam I (5.16) either into C = { z ∈ C ∗ || z | = c j } or into C = (cid:26) z ∈ C ∗ (cid:12)(cid:12)(cid:12)(cid:12) | z | = 1 c j (cid:27) , where c j = c j ( ) in (2.25). Note that the radius c j is less than .(d) The map v j is asymptotic to the intersection point e − π { α j ( w − h + (cid:15) ( h + (cid:15) ))+ √− α j ( w − h + (cid:15) ( h + (cid:15) )) } of φ X H ( R > ) and e − π √− α j ( w − h + (cid:15) (cid:15) ) · R > at the positive end.(e) The map v j is asymptotic to the point d j e √− dj at the lower negative end is asymptotic to the point d j e − π √− α j ( w − h + (cid:15) (cid:15) ) , t ∈ (cid:20) , (cid:21) at the upper negative end where d j = (cid:40) c j v j ( I ) ⊂ C , c j v j ( I ) ⊂ C . Since there are two possible choices for the seam condition for v j as described in the condition(c) above, the uniformization theorem says that there exist exactly two holomorphic map v j fromthe patch S to C ∗ with the prescribed the boundary condition and the asymptotic condition.Indeed, the image of each of those is just a triangle bounding φ X H ( R > ) , e − π √− α j ( w − h + (cid:15) (cid:15) ) · R > and C k for some k = 1 , . As a result, we classified all solutions q ◦ v for the above requirements andhence all solutions q ◦ u as well, since q ◦ v and q ◦ u are related by q ◦ v = q ◦ φ − fX H ◦ u = φ − fX H ad ◦ q ◦ u . This further leads to the classification of all solutions u eventually. Indeed, since the domain S is contractible, every solution q ◦ u : S → T C ad admits a topological lifting u : S → T C . Although there are | T C /T C ad | = | P ∨ /Q ∨ | -many different such topological liftings for eachsolution q ◦ u , only one among those liftings is allowed to be the desired pseudo-holomorphiccurve u due to the prescribed asymptotic condition, that is, the requirement that u is asymptoticto the Hamiltonian chord x h(cid:15) at the positive end.It still remains to show that only one among those has zero index. For that purpose, considerthe composition π t ◦ u for a given solution u , where π t : T C = T × t → t is the projection to the second factor. By the construction of the quotient map q : T C → T C ad , wehave π t ◦ u = π t ◦ q ◦ u . Hence the image Im ( π t ◦ u ) = Im ( π t ◦ q ◦ u ) connects the point h + (cid:15) ∈ w h + (cid:15) t > and the intersection point π W ( C ) ∩ w (cid:48) t > due to the asymptotic conditions of HANWOOL BAE AND NAICHUNG CONAN LEUNG u at ends. Hence it crosses at least one of the hyperplanes t α ’s (2.1) for positive roots α unless w (cid:48) = w h + (cid:15) .But (5.13) implies that the image of u lies in U if and only if the image Im ( π t ◦ u ) does notcross any hyperplanes t α . But among all solutions, there is exactly one solution u such that theimage Im ( π t ◦ u ) does not cross any hyperplanes t α . Indeed, writing the map q ◦ u as(5.17) q ◦ u = ( u , . . . , u l ) just as done for q ◦ v in (5.15), it is the one such that the corresponding j -th entry u j maps theseam I into C (5.16) for each j = 1 , . . . , l and hence the image of q ◦ u is included in D ∗ × · · · × D ∗ l = Ψ w h + (cid:15) ( T ad × w t > ) ⊂ Ψ w h + (cid:15) ( T C ad ∩ U ad ) . (Lemma 5.20)Now observe that the requirement that the image Im ( π t ◦ u ) does not cross any hyperplanes t α is equivalent to the condition w (cid:48) = w h + (cid:15) . Then the following claim and its proof complete theproof of the lemma. Claim.
The Fredholm index of ( u , u ) is greater than or equal to 2 if and only if w (cid:48) (cid:54) = w h + (cid:15) .Proof of Claim. First, note that the gluing ( v h(cid:15) u , u ) is a capping quilted half-disk for x w suchthat the second entry u is constant. Since the image of u is included in T C , one can deduce thatthe concatenated path x − ∗ ( u | I ) ∗ x + is homotopic to x h(cid:15) in T C . This means that the concatenatedpath is mapped via the covering map q to a path that rotates by the angle − πα j ( w (cid:48)− ( h + (cid:15) )) in the j -th summand N j | M of the normal bundle N at the point ([ ˙ w (cid:48) ] , [ ˙ w (cid:48) ]) . Then Lemma 5.15says that the capping quilted half-disk ( v h(cid:15) u , u ) represents − w (cid:48)− h and the Maslov index µ ( x w , ( v h(cid:15) u , u )) is given by l ( w (cid:48) ) − ρ ( w (cid:48)− h ) . But the Maslov index µ ( x h(cid:15) , v h(cid:15) ) is given by l ( w h + (cid:15) ) − ρ ( w − h + (cid:15) h ) . It follows that the Fredholm index of the pair ( u , u ) is given by the difference(5.18) l ( w (cid:48) ) − l ( w h + (cid:15) ) − ρ ( w (cid:48)− h ) + 2 ρ ( w − h + (cid:15) h ) . Hence if w (cid:48) = w h + (cid:15) , then we have h (cid:48) = − w − h + (cid:15) h and the Fredholm index is zero. Furthermore,the curve u maps into U as mentioned above. This proves one direction of the claim.For the other direction of the claim, let us show that the index (5.18) of the pair ( u , u ) isgreater than or equal to 2 if w (cid:48) (cid:54) = w h + (cid:15) .Consider the straight line segment I in t connecting the point w − h + (cid:15) ( h + (cid:15) ) ∈ t > and a point inthe Weyl chamber w (cid:48)− w h + (cid:15) t > . The line segment I intersects hyperplanes t β = { h ∈ t | β ( h ) =0 } for some β ∈ R + . Assume that it intersects m such hyperplanes t β , . . . , t β m for some m ≥ .We may assume that the roots β , . . . , β m are ordered in such a way that the intersection I ∩ t β j iscloser to the positive Weyl chamber than the intersection I ∩ t β k is so, whenever ≤ j < k ≤ m .Since the line segment I connects a point in the positive Weyl chamber to a point in the Weylchamber w (cid:48)− w h + (cid:15) t > , we deduce w (cid:48)− w h + (cid:15) = s β m . . . s β . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Furthermore, for each ≤ k ≤ m , the Lie algebra t is divided into two closed half-spaces bythe hyperplane t β k . The element s β k − . . . s β w − h + (cid:15) h belongs to the half-space { h ∈ t | β k ( h ) ≥ } since it lies in the Weyl chamber that the line segment I passes through between I ∩ t β k − and I ∩ t β k . Hence we have β k ( s β k − . . . s β w − h + (cid:15) h ) ≥ . Now Lemma 5.22 proves that the index (5.18) is not less than 2 as desired. This proves theclaim. (cid:3)(cid:3)
Lemma 5.22.
Let h ∈ Q ∨ . Let β , . . . , β m ∈ R + , m ≥ be distinct positive roots such that β k ( s β k − . . . s β w − h + (cid:15) h ) ≥ , ∀ ≤ k ≤ m. For each ≤ k ≤ m , either one of the following two holds • ρ ( s β k . . . s β w − h + (cid:15) h ) − ρ ( s β k − . . . s β w − h + (cid:15) h ) ≤ − and l ( w h + (cid:15) s β . . . s β k ) − l ( w h + (cid:15) s β . . . s β k − ) ≥ − . • ρ ( s β k . . . s β w − h + (cid:15) h ) − ρ ( s β k − . . . s β w − h + (cid:15) h ) = 0 and l ( w h + (cid:15) s β . . . s β k ) − l ( w h + (cid:15) s β . . . s β k − ) = 1 .In particular, if we write w (cid:48) = w h + (cid:15) s β . . . s β m , then we have l ( w (cid:48) ) − l ( w h + (cid:15) ) − ρ ( w (cid:48)− h ) + 2 ρ ( w − h + (cid:15) h ) ≥ . Proof.
Let us prove the assertion for an induction on k .First consider the case k = 1 . In this case, we have s β w − h + (cid:15) h = w − h + (cid:15) h − β ( w − h + (cid:15) h ) β ∨ . (2.2)If β ( w − h + (cid:15) h ) ≥ , then we have ρ ( s β w − h + (cid:15) h ) = ρ ( w − h + (cid:15) h ) − β ( w − h + (cid:15) h ) ρ ( β ∨ ) ≤ ρ ( w − h + (cid:15) h ) − l (cid:88) j =1 χ j ( β ∨ ) ≤ ρ ( w − h + (cid:15) h ) − . ( ∵ β ∨ is a positive coroot . ) Also, since the difference between l ( w h + (cid:15) s β ) and l ( w h + (cid:15) ) cannot be greater than 1, the statement l ( w h + (cid:15) s β ) − l ( w h + (cid:15) ) ≥ − holds.Otherwise, β ( w − h + (cid:15) h ) = 0 since w − h + (cid:15) h ∈ t ≥ . In this case, s β w − h + (cid:15) h = w − h + (cid:15) h . We con-sequently have ρ ( s β w − h + (cid:15) h ) = ρ ( w − h + (cid:15) h ) . But l ( w h + (cid:15) s β ) = l ( w ) + 1 since s β permutes allpositive roots other than β and we have < β ( w − h + (cid:15) ( h + (cid:15) )) = β ( w − h + (cid:15) (cid:15) ) , while > β ( s β w − h + (cid:15) ( h + (cid:15) )) = β ( s β w − h + (cid:15) (cid:15) ) . This proves the assertion for the case k = 1 .Assume that we have shown the assertion for k = l − for some ≤ l ≤ m . Now we have s β l . . . s β w − h + (cid:15) h = s β l − . . . s β w − h + (cid:15) h − β l ( s β l − . . . s β w − h + (cid:15) h ) β ∨ l HANWOOL BAE AND NAICHUNG CONAN LEUNG If β l ( s β l − . . . s β w − h + (cid:15) h ) ≥ , then we have ρ ( s β l . . . s β w − h + (cid:15) h ) = ρ ( s β l − . . . s β w − h + (cid:15) h ) − β l ( s β m − . . . s β w − h + (cid:15) h ) ρ ( β ∨ l ) ≤ ρ ( s β l − . . . s β w − h + (cid:15) h ) − l (cid:88) j =1 χ j ( β ∨ l ) ≤ ρ ( s β l − . . . s β w − h + (cid:15) h ) − . Once again, the statement about length automatically follows from the definition of length.Otherwise, β l ( s β l − . . . s β w − h + (cid:15) h ) = 0 by assumption. Then we have ρ ( s β l . . . s β w − h + (cid:15) h ) − ρ ( s β l − . . . s β w − h + (cid:15) h ) = 0 . But as in the case k = 1 , one can deduce l ( w h + (cid:15) s β . . . s β l ) = l ( ws β . . . s β l − ) + 1 . Thiscompletes the inductive proof.The last assertion follows since for each ≤ k ≤ m , we have { l ( w h + (cid:15) s β . . . s β k ) − l ( w h + (cid:15) s β . . . s β k − ) − ρ ( s β k . . . s β w − h + (cid:15) h ) + ρ ( s β k − . . . s β w − h + (cid:15) h ) } ≥ . (5.19)and l ( w (cid:48) ) − l ( w h + (cid:15) ) − ρ ( w (cid:48)− h ) + 2 ρ ( w − h + (cid:15) h ) is the sum of (5.19) for all ≤ k ≤ m . (cid:3) Finally we are ready to prove the following lemma.
Lemma 5.23. (1) For a generic choice of almost complex structure J , the zero dimensional component ofthe moduli space M (cid:16) x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C (cid:17) is nonempty only if either one of the following holds: • w (cid:48) = w h + (cid:15) and h (cid:48) = − w − h + (cid:15) h . • the pair ( w (cid:48) , h (cid:48) ) satisfies (5.10) and either (i) or (ii) given in Lemma 5.16.(2) For a generic choice of almost complex structure J , the zero dimensional component ofthe moduli space M (cid:16) x w h + (cid:15) q − w − h + (cid:15) h , x h(cid:15) ; D Φ C (cid:17) has exactly one element ( u , u ) , for which the first entry u maps into the fiber (cid:101) π − ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) and the second entry u is constant.Proof. Assume that u = ( u , u ) ∈ M ( x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C ) lies in a zero dimensional componentof the moduli space. Then Lemma 5.18 implies that(5.20) ρ ( h (cid:48) + w − h + (cid:15) h ) = l ( w h + (cid:15) ) − l ( w (cid:48) ) . On the other hand, Theorem A.6 shows that the map u does not escape from U for a genericchoice of almost complex structure. Let us assume the almost complex structure in the initialFloer data D Φ C has been chosen in such a way. This allows us to think of the quilt map into ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES M × M obtained from ( u , u ) by composing u with the projection (cid:101) π : U → M (2.14), that is,the map ( (cid:101) π ◦ u , u ) .Let J = ( J , J , J ) be the almost complex structure for S ∈ R , in Definition 4.5. Therequirements on the almost complex structures J and J given in Definition 2.28 and Definition2.29 imply that the almost complex structure π M ( J ) is a domain dependent ω M -compatiblealmost complex structure on M and π M × M ( J ) is also a domain dependent ( − ω M ) ⊕ ω M -compatible almost complex structure on M − × M . Moreover, since the Hamiltonian vectorfield X H is tangent to the fiber of (cid:101) π , it can be deduced from (a), (b) and (c) below (4.1) that theprojected quilt ( (cid:101) π ◦ u , u ) solves the Cauchy-Riemann equation for the almost complex structure(5.21) π ( J ) := ( π M ( J ) , J , π M × M ( J )) . Due to (A), (C) and (D) in Subsection 2.9, this means that the quilt map ( (cid:101) π ◦ u , u ) is an elementof the moduli space M ( w (cid:48) , w h + (cid:15) ; π ( J )) introduced in Subsection A.1.On the other hand, Lemma 5.3 shows that the capping quilted half-disk v h projects down toa disk in M bounding ∆ = ∆ G/T , which represents the homotopy class − w − h + (cid:15) h ∈ Q ∨ = π ( M, ∆) . Furthermore, the requirement on the quilt ( u , u ) given in Definition 5.17 says thatthe capping quilted half-disk ( v h(cid:15) u , u ) represents h (cid:48) . Considering that the perturbed diagonal ∆ (cid:15) is the image of the diagonal ∆ under the time -flow of the Hamiltonian vector field of f (cid:15) ◦ π (5.6), we deduce that the symplectic area of the quilt ( (cid:101) π ◦ u , u ) is given by ρ ( h (cid:48) + w − h + (cid:15) h ) + f (cid:15) ([ ˙ w (cid:48) ]) − f (cid:15) ([ ˙ w h + (cid:15) ]) .Here note that the value ρ ( h (cid:48) + w − h + (cid:15) h ) is an even integer since both h (cid:48) and w − h + (cid:15) h belong to Q ∨ , while ρ is given by ρ = 2 (cid:80) lj =1 χ j ∈ P . But since the Hamiltonian f (cid:15) takes values between − and (5.6) and the symplectic area of a holomorphic curve is nonnegative, ρ ( h (cid:48) + w − h + (cid:15) h ) cannot be negative. Hence the equality (5.20) implies that l ( w h + (cid:15) ) − l ( w (cid:48) ) is a nonnegative eveninteger.If l ( w h + (cid:15) ) > l ( w (cid:48) ) , then such a case falls into the condition (i) in Lemma 5.16.Suppose now that l ( w (cid:48) ) = l ( w h + (cid:15) ) . Then the symplectic area of the quilt ( (cid:101) π ◦ u , u ) is equalto f (cid:15) ([ ˙ w (cid:48) ]) − f (cid:15) ([ ˙ w h + (cid:15) ]) . If f (cid:15) ([ ˙ w (cid:48) ]) = f (cid:15) ([ ˙ w h + (cid:15) ]) , then the symplectic area of the quilted map ( (cid:101) π ◦ u .u ) is zero and hence both (cid:101) π ◦ u and u constantly map to ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) ∈ ∆ ∩ ∆ (cid:15) .Then Lemma 5.19 says that h (cid:48) = − w − h + (cid:15) h . Otherwise we have f (cid:15) ([ ˙ w (cid:48) ]) > f (cid:15) ([ ˙ w h + (cid:15) ]) . This casefalls into the second condition (ii) in Lemma 5.16. This proves the first statement.We have seen that if w (cid:48) = w h + (cid:15) , then both maps (cid:101) π ◦ u and u are constant. Hence Lemma5.19 and Lemma 5.21 prove the second statement. (cid:3) Let us now finish the proof of Lemma 5.16.
Proof of Lemma 5.16.
It will be shown in Theorem B.6 that the sign of of the leading term in Φ C ( x h(cid:15) ) in Lemma 5.16 is 1. Other than that, it only remains to show that Φ C ( x h(cid:15) ) = x w h + (cid:15) q − w − h + (cid:15) h . for any h ∈ Q ∨ with l ( w h + (cid:15) ) < . We prove this statement by showing that the moduli space M ( x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C ) is non-empty only if w (cid:48) = w h + (cid:15) and h (cid:48) = − w − h + (cid:15) h , in which case the modulispace has exactly one element as we observed in the proof of Lemma 5.23. HANWOOL BAE AND NAICHUNG CONAN LEUNG
First, if l ( w h + (cid:15) ) = 0 , then the assertion holds since there is no non-identity element w with l ( w ) = 0 .Now let us consider the case l ( w h + (cid:15) ) = 1 . Note that there are exactly l -many different elements s α , . . . , s α l ∈ W of length . But the Hamiltonian f (cid:15) takes the same value ρ ( s α j ( (cid:15) )) = ρ ( (cid:15) ) − a at [ s α j ] ∈ G/T for all j = 1 , . . . , l by construction (5.6). The assertion follows since thecondition (i) in Lemma 5.16 cannot be satisfied for any w (cid:48) = s α j , j = 1 , . . . , l . (cid:3) Localization.
We introduce a certain family of generators of the wrapped Floer cohomol-ogy HW ( L, L ) that forms a multiplicative set under the multiplication µ W . Then we show thatthe localization with respect to the multiplicative set is isomorphic to the Floer cohomology HF ( L , L ) as a K -algebra. For simplicity, let us identify HW ( L, L ) = HW ( L, L (cid:15) ) and denotethe cohomology class of x h(cid:15) by x h for each h ∈ Q ∨ . Similarly let us identify HF ( L , L (cid:15) ) with HF ( L , L ) .We first show that the map Φ C is injective. For that purpose, we introduce a partial order < onthe Weyl group W defined by declaring(5.22) w < w if either l ( w ) < l ( w ) or l ( w ) = l ( w ) and f (cid:15) ([ ˙ w ]) > f (cid:15) ([ ˙ w ]) . Lemma 5.16 implies that for any h ∈ Q ∨ , the Weyl group element w h + (cid:15) involved in theleading term of Φ C ( x h ) is strictly greater than any other Weyl group elements involved in thehigher action terms of Φ C ( x h ) , with respect the order < . Theorem 5.24.
The map Φ C induces an injective K -linear morphism from HW ( L, L ) to HF ( L , L ) .Proof. Let us show that the kernel of the map Φ C is zero by investigating the leading term of itsimage. Assume that(5.23) Φ C (cid:32) k (cid:88) i =1 a i x h i (cid:33) = 0 for some a i ∈ K and distinct elements h i ∈ Q ∨ , i = 1 , . . . , k . We have to show that a i = 0 forall i = 1 , . . . , k . Let us prove the assertion by an induction on the number k of terms. The case k = 1 is proven since the leading term of Φ( x h i ) is nonzero by Lemma 5.16.Let us assume that the assertion has been proven for the case k ≤ m − for some m ≥ .We consider a partial order ≺ on the set of all h i ’s involved in (5.23) induced from the order < (5.22) on W in the sense that h i ≺ h j if and only if w h i + (cid:15) < w h j + (cid:15) .Without loss of generality, we may assume that h is maximal with respect to the order ≺ .Then the way the order ≺ is defined implies that the monomial x w h (cid:15) q − w − h (cid:15) h does not appearin the image Φ C ( a i x h i ) for any i (cid:54) = 1 . Since x w h (cid:15) q − w − h (cid:15) h is the leading term of Φ C ( x h ) as mentioned in Lemma 5.16, the coefficient a of x h must be zero. The assertion for the case k = m follows from the induction hypothesis. This completes the proof. (cid:3) Let Q ∨≥ = Q ∨ ∩ t ≥ = { h ∈ Q ∨ | h + (cid:15) ∈ t > } and let S = { x h ∈ HW ( L, L ) | h ∈ Q ∨≥ } .Then the subset S of HW ( L, L ) is closed under multiplication. Indeed, the last statement ofLemma 5.16 says that x h maps to q − h via the map Φ C for every h ∈ Q ∨≥ . Therefore, for any ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES pair h , h ∈ Q ∨≥ , we have Φ C ( µ W ( x h , x h )) = µ M ( q − h , q − h ) = q − h − h . The injectivity of the map Φ C implies that µ W ( x h , x h ) = x h + h ∈ S .Let us denote by S − HW ( L, L ) the localized ring obtained by inverting the elements of S .To be more concrete, its element is represented by the fraction xx h for some x ∈ HW ( L, L ) , h ∈ Q ∨≥ , and the multiplication µ W is extended by setting µ W (cid:16) x x h , x x h (cid:17) = µ W ( x , x ) x h + h , for all x , x ∈ HW ( L, L ) and h , h ∈ Q ∨≥ .The homomorphism Φ C is naturally extended to the localized ring S − HW ( L, L ) by setting Φ C (cid:16) xx h (cid:17) = Φ C ( x ) q h for x ∈ HW ( L, L ) and h ∈ Q ∨≥ . Finally we have Theorem 5.25.
The map Φ C induces a K -algebra isomorphism S − HW ( L, L ) → HF ( L , L ) . Proof.
Since it has been already proven that the map Φ C is injective, it only remains to showthat the map is surjective. Let us show that for any w (cid:48) ∈ W and any h (cid:48) ∈ Q ∨ , x w (cid:48) q h (cid:48) lies in theimage of the map Φ C . We prove this assertion by an induction using the partial order < definedin (5.22).First consider the case l ( w (cid:48) ) ≤ . In this case, we first choose h ∈ Q ∨≥ positive enough sothat h − h (cid:48) ∈ t > . Now, for h := w (cid:48) ( h − h (cid:48) ) ∈ Q ∨ , we have w h + (cid:15) = w (cid:48) and w − h + (cid:15) h = h − h (cid:48) . Thelast statement of Lemma 5.16 implies that Φ C ( x h ) = x w (cid:48) q h (cid:48) − h and Φ C ( x h ) = q − h . Finally,we have Φ C (cid:18) x h x h (cid:19) = x w (cid:48) q h (cid:48) − h · q h = x w (cid:48) q h (cid:48) . Now assume that the assertion has been proven for any pair w (cid:48)(cid:48) ∈ W and h (cid:48)(cid:48) ∈ Q ∨ with w (cid:48)(cid:48) < w for some w ∈ W . The inductive proof will be completed if we show that x w (cid:48) q h (cid:48) liesin the image of Φ C for any pair ( w (cid:48) , h (cid:48) ) such that l ( w (cid:48) ) = l ( w ) and f (cid:15) ([ ˙ w (cid:48) ]) = f (cid:15) ([ ˙ w ]) . But,applying the same argument in the previous paragraph, for any such a pair ( w (cid:48) , h (cid:48) ) , one can find h ∈ Q ∨ and h ∈ Q ∨≥ such that the leading term of Φ C (cid:16) x h x h (cid:17) is given by x w (cid:48) x h (cid:48) . Since thehigher action terms of Φ C (cid:16) x h x h (cid:17) already lie in the image of Φ C by induction hypothesis, thiscompletes the proof. (cid:3) A PPENDIX
A. T
RANSVERSALITY
A.1.
Projection of pseudo-holomorphic quilted disk to the base manifold M . Let ( ˙ w ± , ˙ w ± ) ∈ ∆ ∩ ∆ (cid:15) be an intersection point of the diagonal and the perturbed diagonal in M = G/T − × G/T for some w ± ∈ W . HANWOOL BAE AND NAICHUNG CONAN LEUNG
As introduced right above Definition 4.5, let S and S be patches of a quilted disk S and U ⊂ S be an open ball centered at a point on the seam I . Further, let J be a triple of almostcomplex structures J = ( J , J , J ) ∈ C ∞ ( S \ U, J ( M )) × C ∞ ( S \ U, J ( M ) × C ∞ ( U ∩ S , J ( M − × M )) satisfying • J takes the form (cid:18) J M AB J M (cid:19) in such a way that both A and B vanish outside a com-pact subset of U ∩ S , where M and M denote the first and second factor of the square M × M , respectively. • J and − J M ◦ φ glue smoothly to an element of C ∞ ( S , J ( M )) , • J and J M glue smoothly to an element of C ∞ ( S , J ( M )) .Then we define M ( w − , w + ; J ) by the moduli space of J -holomorphic quilted disks that areasymptotic to ([ ˙ w ± ] , [ ˙ w ± ]) at ends and map to the diagonal Lagrangian submanifold ∆ M ⊂ M − × M along the seam I . More explicitly, it consists of pairs of maps ( v k : S k → M ) k =1 , such that(a) v k solves the J k -holomorphic equation on S k \ U , k = 1 , .(b) ( v ◦ φ, v ) : U ∩ S → M − × M solves the J -holomorphic equation.(c) v maps ∂S to ∆ and v maps ∂S to ∆ (cid:15) .(d) v ( z ) = v ( z ) for all z ∈ I (e) lim s →−∞ v ( (cid:15) − ( s, t )) = lim s →−∞ v ( (cid:15) ( s, t )) = lim s →−∞ v ( (cid:15) − ( s, t )) = ([ ˙ w − ] , [[ ˙ w − ]) and lim s →∞ v ( (cid:15) ( s, t )) = ([ ˙ w + ] , [[ ˙ w + ]) . (f) The energy of ( v , v ) is finite.Let w ∈ W and let J = ( J , J , J ) be given. Since the constant maps v and v with theconstant values ([ ˙ w ] , [ ˙ w ]) satisfy all the above conditions, the pair ( v , v ) is an element of themoduli space M ( w, w ; J ) . Lemma A.1. If ( v , v ) ∈ M ( w, w ; J ) is the pair of constant maps, then(1) its index is zero and(2) it is regular.Proof. Note that two maps v and v glue on the seam since the seam condition is just given bythe diagonal ∆ M ⊂ M − × M and the glued the map v v is a disk bounding ∆ = ∆ G/T in M . Since the index of the pair ( v , v ) is equal to that of the glued map v := v v , the firstassertion follows from the monotonicity of the Lagrangian ∆ ⊂ M .For the second assertion, we follow the argument given in [14, Proposition 7]. Once again,we only need to consider the glued map v = v v defined on S since the seam condition isthe diagonal ∆ M ⊂ M − × M . As its index is zero, it suffices to show that the kernel of thelinearized operator is zero. We may assume that the map v maps to ∈ R n and the Lagrangiansubmanifolds ∆ and ∆ (cid:15) are linear Lagrangian submanifolds that intersect transversely. Herethe space R n is viewed as the symplectic manifold with the standard symplectic form ω std = (cid:80) ni =1 dx i ∧ dy i . ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Let v (cid:48) : S → R n be an element of the kernel of the linearized Cauchy-Riemann operator ∂ J ,which has exponential decay at infinity. Since the symplectic form ω std is exact and any linearLagrangian submanifold is exact, the symplectic area (cid:82) ( v (cid:48) ) ∗ ω std = 0 .Since the almost complex structure J is not split everywhere, the energy of v is given by (cid:88) k =1 (cid:90) S k \ U | dv (cid:48) | J k Vol S k + 12 (cid:90) S ∩ U | d ( u ◦ φ, u ) | J Vol S and it equals the symplectic area (cid:82) ( v (cid:48) ) ∗ ω std = 0 . Hence v (cid:48) is a constant map. Since v (cid:48) convergesto zero at the ends of S , it follows that v (cid:48) is constantly zero. This completes the proof. (cid:3) A.2.
Proof of Transversality.
We provide a proof of Theorem 4.7 by proving the transversalityof the Fredholm operator for the case d = 1 as mentioned below the theorem.Let D Φ C be the initial Floer data described in Definition 4.5 and let J = ( J , J ., J ) ∈ J be analmost complex structure described therein. We consider the moduli space M (cid:16) x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C (cid:17) for w (cid:48) ∈ W , h (cid:48) ∈ Q ∨ and h ∈ Q ∨ .Let B be the Banach manifold of all W ,p -quilted map ( u , u ) with the boundary conditionand the seam condition on I (d) below (4.1) that converge to x h at the positive end and to x w (cid:48) atthe negative end. Let E be the Banach bundle over B whose fiber at a point ( u , u ) is given by E ( u ,u ) = L p (Λ , ⊗ u ∗ T W ) ⊕ L p (Λ , ⊗ u ∗ T M ) . The operators ∂ J ,H , ∂ J and ∂ J ,H in (a), (b) and (c) below (4.1) define a section ∂ J : B → E .Then the moduli space M ( x w (cid:48) q h (cid:48) , x h(cid:15) ; D Φ C ) can be understood as its zero set. Furthermore, itslinearization D∂ J is a Fredholm operator.Lemma 5.19 and Lemma 5.21 say that if (cid:101) π ◦ u and u are constant on u − ( U ) ∩ U and S ∩ U ,respectively and the Fredholm index of the pair ( u , u ) is zero, then there exists unique such asolution ( u , u ) such that u maps into T C ∩ U and u is constant. For such a unique solution ( u , u ) , we have Lemma A.2.
The pair ( u , u ) is regular, i.e. the linearized operator D∂ J at ( u , u ) is surjec-tive.Proof. Recall the decomposition of the tangent bundle T U into Ver ⊕ Hor (2.35). Since the map u maps into U , the codomain E ( u ,u ) of D∂ J at ( u , u ) is given by(A.1) L p (Λ , ⊗ u ∗ T U ) ⊕ L p (Λ , ⊗ u ∗ T M ) = L p (Λ , ⊗ u ∗ Ver ) ⊕ L p (Λ , ⊗ u ∗ Hor ) ⊕ L p (Λ , ⊗ u ∗ T M ) Note that the vertical space Ver x at any point x in the fiber (cid:101) π − ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) is identifiedwith the tangent space of the torus T x T C since the fiber (cid:101) π − ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) is included in thetorus T C . Furthermore, it is identified with the tangent space T q ( x ) T C ad via the quotient map q : T C → T C ad . Hence, via these identifications, we may replace the summand L p (Λ , ⊗ u ∗ Ver ) with L p (Λ , ⊗ ( q ◦ u ) ∗ T T C ad ) .But the torus T C ad is the product of 2-dimensional symplectic manifolds C ∗ . Indeed we specifiedone of such identifications Ψ w h + (cid:15) in (5.14). Accordingly, the map q ◦ u can be written as q ◦ u = HANWOOL BAE AND NAICHUNG CONAN LEUNG ( u , . . . , u ) as in (5.17) and the pseudo-holomorphic equation for q ◦ u decomposes into that ineach factor C ∗ : (cid:18) du j − log ρ j ∂∂θ j ⊗ β (cid:19) + i ◦ (cid:18) du j − log ρ j ∂∂θ j ⊗ β (cid:19) ◦ i = 0 , (2 . where z j ( ρ j , θ j ) := ρ πj exp(2 πiθ j ) ∈ C ∗ is the coordinate for the j -th factor of ( C ∗ ) l , j =1 , . . . , l . The surjectivity onto the vertical part L p (Λ , ⊗ u ∗ Ver ) ∼ = L p (Λ , ⊗ ( q ◦ u ) ∗ T T C ad ) of the differential D∂ J can be achieved by applying the automatic regularity ([20, Chapter 13])to each factor of T C ad = ( C ∗ ) l . This is available since each entry u j of the pseudo-holomorphiccurve q ◦ u is non-constant due to the classification of q ◦ v in the proof of Lemma 5.21.Now it remains to prove the surjectivity onto the direct sum L p (Λ , ⊗ u ∗ Hor ) ⊕ L p (Λ , ⊗ u ∗ T M ) in (A.1). Note that both (cid:101) π ◦ u and u are the constant maps with the constant values ( ˙ w h + (cid:15) , ˙ w h + (cid:15) ) ∈ ∆ ∩ ∆ (cid:15) and hence it is an element of M ( w h + (cid:15) , w h + (cid:15) ; π ( J )) (See SubsectionA.1). This makes sense due to Definition 2.26 and Definition 2.28. But Lemma A.1 says thatthe pair ( (cid:101) π ◦ u , u ) is regular, which means that the linearized Cauchy-Riemann operator withrespect to the almost complex structures π ( J ) (5.21) is surjective. Considering that the horizontalpart L p (Λ , ⊗ u ∗ Hor ) is mapped isomorphically to L p (Λ , ⊗ ( (cid:101) π ◦ u ) ∗ T M ) via the projectionmap d (cid:101) π , we deduce that the operator D∂ is surjective onto the direct sum L p (Λ , ⊗ u ∗ Hor ) ⊕ L p (Λ , ⊗ u ∗ T M ) . The assertion follows. (cid:3) Let J be the space of almost complex structures J = ( J , J , J ) described in Definition4.5. Once again, the (inhomogeneous) Cauchy-Riemann operators in (a), (b) and (c) below (4.1)induce a section ∂ from J × B to E given by ( J, ( u , u )) (cid:55)→ ∂ J ( u , u ) .Further, we show that, for a generic choice of almost complex structures, the first component u of any pseudo-holomorphic quilt ( u , u ) maps into U . For that purpose, we consider thefollowing map: F : J × B × S \ ∂S → E × W ( J, ( u , u ) , z ) (cid:55)→ (cid:0) ∂ ( J, ( u , u )) , u ( z ) (cid:1) (A.2)To investigate the differential of the map above, let Y be the space of infinitesimal deforma-tions of the given almost complex structure J , each component of which vanishes over the endsof each patch faster than exponentially.Let ( J, ( u , u )) ∈ B and z ∈ S \ ∂S be given. The vertical differential D F of F is given by Y ⊕ T ( u ,u ) B ⊕ T z S → E ( u ,u ) ⊕ T u ( z ) W. ( Y, ( ξ , ξ ) , v ) (cid:55)→ (cid:18) D∂ J ( ξ , ξ ) + 12 Y ◦ ( du ◦ i , du ◦ i ) , ξ ( z ) + du ( v ) (cid:19) . Following the idea given in [17, Chapter 10.5], we prove
Lemma A.3. If (cid:101) π ◦ u is non-constant on u − ( U ) ∩ U or u is non-constant on S ∩ U , then thedifferential D F is surjective.Proof. Assume (( η , η ) , X ) ∈ E ( u ,u ) ⊕ T u ( z ) W is orthogonal to the image of D F . Write η = ( η , η ) . We need to show that (( η , η ) , X ) is zero. ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Taking Y = 0 and v = 0 , we have (cid:104) D∂ J ( ξ , ξ ) , ( η , η ) (cid:105) + (cid:104) ξ ( z ) , X (cid:105) = 0 , ∀ ( ξ , ξ ) ∈ T ( u ,u ) B . This, in turn, means D∂ ∗ J ( η ) − δ z X = 0 ,D∂ ∗ J ( η ) = 0 . where D∂ ∗ J is the formal adjoint of D∂ J and δ z is the Dirac delta function at z ∈ S .As η = ( η , η ) is a solution of an elliptic operator, η vanishes everywhere if it does so on anon-empty open set in S by unique continuation theorem. Similarly, η vanishes on S \ { z } ifit does so on a non-empty open set in S \ { z } . But, for η , actually the same statement holdseven after the domain S \ { z } is replaced by S since the argument in [17, Lemma 10.5.4] canbe applied here.Now taking ( ξ , ξ ) = 0 and v = 0 , we have(A.3) (cid:104) Y ◦ ( du ◦ i , du ◦ i ) , ( η , η ) (cid:105) = 0 , ∀ Y ∈ Y . Note that the set V of points in U ∩ S where d ( (cid:101) π ◦ u ◦ φ ) or du does not vanish is open anddense in U ∩ S by assumption. Since the almost complex structure J is allowed to be non-splitas described in Definition 2.29, this implies that η vanishes on φ ( V ) ⊂ S and η vanishes on V . Hence it follows that η = 0 on S and η = 0 on S , respectively, by the argument in theprevious paragraph.Finally, since the map T ( u ,u ) B → T u ( z ) W, ξ (cid:55)→ ξ ( z ) is surjective, the vector X is also zero.This completes the proof. (cid:3) Remark
A.4 . Recall the definition of J x (Definition 2.26). The family J x is open in the space ofall (cid:101) π ∗ ω M -compatible almost complex structures on Hor x at every point x ∈ U . This allows theargument right below (A.3) in Lemma A.3 to make sense.Similarly, for the evaluation map at a point on the boundary ∂S , we consider the followingsection: F : J × B × ∂S → E × L ( J, ( u , u ) , z ) (cid:55)→ ( ∂ J ( u , u )) , u ( z )) The differential D F is given by Y ⊕ T ( u ,u ) B ⊕ T z ∂S → E ( u ,u ) ⊕ T u ( z ) L. ( Y, ( ξ , ξ ) , v ) (cid:55)→ (cid:18) D∂ J ( ξ , ξ ) + 12 Y ◦ ( du ◦ i , du ◦ i ) , ξ ( z ) + du ( v ) (cid:19) . The proof of Lemma A.3 can be slightly modified to prove the following lemma:
Lemma A.5. If (cid:101) π ◦ u is non-constant on u − ( U ) ∩ U or u is non-constant on S ∩ U , then thedifferential D F is surjective. Finally we have HANWOOL BAE AND NAICHUNG CONAN LEUNG
Theorem A.6.
For a generic choice of almost complex structure J , the following holds(1) the operator D∂ J is surjective, and(2) the pseudo-holomorphic curve u maps into U for every ( u , u ) that solves ∂ J ( u , u ) = 0 and is of index less than 2.Proof. Due to Lemma 5.19, Lemma 5.21 and Lemma A.2, we only need to consider the casewhen (cid:101) π ◦ u is non-constant on u − ( U ) ∩ U or u is non-constant on S ∩ U . Suppose that it isthe case.Let us first consider ∂ − (0) ⊂ J × B . Actually, under the assumption of the lemma, LemmaA.3 also shows that ∂ : J × B → E is transverse to the zero section since the first component ofthe map F is given by ∂ (A.2). Hence the standard argument shows that, for a generic choice ofalmost complex structure J , the operator D∂ J is surjective at any ( J, ( u , u )) ∈ ∂ − (0) . Let usdenote the set of such almost complex structures by J .Let W = W \ U , which is the union of submanifolds of codimension 3 (See Subsection 2.3).To show that every pseudo-holomorphic curve u does not map any interior point to W fora generic choice of almost complex structure, let M int ⊂ J × B × S denote the pre-image F − ( { }× W ) . Since the map F is is transverse to { }× W by Lemma A.3, it is a submanifoldof J × B × S .Then the projection map π J : M int → J is a Fredholm operator and its index is given by − as it is equal to ind D∂ J + dim S − codim W = 0 + 2 − − . Here note that we may assume that the Fredholm index of D∂ J is 0. Indeed it is assumed to beless than 2 and it must be an even integer since the Maslov indices of the involved Hamiltonianchords are even. (See Lemma 5.4 and Lemma 5.13)But, the regular values of π J are those almost complex structures J such that F | π − J ( J ) is trans-verse to { } × W . By Sard-Smale theorem, the set of such regular almost complex structures isof second category. Let us denote it by J . For any almost complex structure J ∈ J , the pre-image π − J ( J ) is empty as its expected dimension is negative. Hence the holomorphic curve u does not touch W at any interior point for every pair ( J, ( u , u )) that solves ∂ ( J, ( u , u )) = 0 for J ∈ J .A similar argument can be applied to showing that the set J of almost complex structures, forwhich every pseudo-holomorphic curve u does not map any boundary point to W , is also ofsecond category. Indeed, an argument analogous to that in the previous paragraph works evenfor this case since the intersection of W with a cotangent fiber T ∗ g G is also of codimension3 in the cotangent fiber. Once again, the expected dimension of the corresponding space isnegative, which means that the curve u does not touch W at any boundary point for every pair ( J, ( u , u )) that solves ∂ ( J, ( u , u )) = 0 for J ∈ J .Finally the assertion follows as the intersection of J , J and J is still of second category. (cid:3) Remark
A.7 . Note that the requirement on the almost complex structures given in Definition 4.5forces the (time-dependent) almost complex structures chosen as a part of Floer data in Definition3.3 to be elements of J ( W ) for any pair of admissible Lagrangians in W . In our case, any ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES failure of the transversality can be avoided even using such a restricted family of almost complexstructures.To discuss that, recall that every generator of Floer cochain complexes CW ( L, L (cid:15) ) and CF ( L , L (cid:15) ) has an even degree. Since this means that their differentials are zero, we do not haveto take care of the transversality for the pseudo-holomorphic strips of index 1 or 2. Moreover,since all involved symplectic manifolds and Lagrangian submanifolds are exact or monotonewith minimal Maslov number greater than 2, no sphere bubbles or disk bubbles occur in thecompactification of 1-dimensional moduli spaces of pseudo-holomorphic strips.Besides, we still need to verify the transversality for the (generalized) pseudo-holomorphicstrips that appear in the compactification of the moduli space M ( x ; x , . . . , x d ; D Φ dC ) (4.1)of pseudo-holomorphic quilts. For instance, it is necessary to check if there are no pseudo-holomorphic strips connecting two Hamiltonian chords between L and L (cid:15) , which is of negativeindex. But, just like the requirement on a holomorphic curve u in the assumption of Lemma5.19, if a pseudo-holomorphic strip u projects to a point in M constantly via the projection (cid:101) π on a nonempty subset, then one can show that it maps into the torus T C and its index must benon-negative. Otherwise, arguing as in the proof of Theorem A.6, for a generic choice of almostcomplex structures, we may assume that the curve u avoids the complement W of U in W . Butthe projection (cid:101) π ◦ u of the pseudo-holomorphic strip is a pseudo-holomorphic quilt connectingtwo intersection points between ∆ and ∆ (cid:15) , which is of non-negative index. This leads to thenon-negativity of the index of the curve u eventually. A more careful argument also shows thatthere are no pseudo-holomorphic strips connecting two generalized Hamiltonian chords between L and L (cid:15) , which is of negative index.Now we are ready to prove Theorem 4.7. Proof of Theorem 4.7.
As mentioned below Theorem 4.7, it only remains to prove the case d = 1 .But, as a corollary of Theorem A.6, it can be shown that the moduli space M ( x w q h (cid:48) , x h(cid:15) ; D Φ C ) is acompact smooth manifold of dimension for a generic choice of almost complex structures. (cid:3) A PPENDIX
B. O
RIENTATION
B.1.
Spin structures.
First observe that each of the cotangent fiber L = T ∗ e G , the cotangentbundle W = T ∗ G and the moment Lagrangian C has a unique spin structure since their secondStiefel-Whitney class vanish and they are simply-connected. Furthermore, the diagonal ∆ G/T also has a unique spin structure since it is simply-connected and its first Chern class is an evenmultiple of an integer cohomology class.Furthermore, recall (C) in Subsection 2.9. Considering that the projection (cid:101) π : L ∩ U → ∆ G/T is a homotopy equivalence as its fibers are contractible, we deduce that the intersection L ∩U alsohas a unique spin structure that is essentially determined by the pull-back of the spin structure ofthe diagonal ∆ G/T .Consequently, since both L and L ∩ U have a unique spin structure, we deduce that the restric-tion of the unique spin structure on L to L ∩ U coincides with the one we considered above. HANWOOL BAE AND NAICHUNG CONAN LEUNG
On the other hand, as discussed in [23, Section 4.6], the spin structures on L , W and C determine the spin structure on the geometric composition L ◦ C = ∆ G/T . Indeed consider thefiber product L × W C = π − W ( L ) ∩ C ⊂ W × M , which is mapped diffeomorphically onto thegeometric composition L ◦ C via the projection π M : W × M → M . Hence the tangent bundleof L ◦ C is isomorphic to that of the fiber product T L × T W
T C on L × W C . This, in turn, meansthat there is an isomorphism between the following two subbundles of T ( W × W × M ) | L ◦ ∆ W ◦ C :(B.1) ( T L ⊕ T ∆ W ⊕ T C ) | L ◦ ∆ W ◦ C ∼ = ( T ∆ ⊥ W ⊕ T ∆ W ⊕ T ( L ◦ C )) | L ◦ ∆ W ◦ C , where L ◦ ∆ W ◦ C = { ( x, x, y ) ∈ W × W × M | x ∈ L, ( x, y ) ∈ C } and T ∆ ⊥ W | ( x,x ) = { ( v, − v ) ∈ T x W ⊕ T x W | v ∈ T x W } for all x ∈ W .Note that each of the cotangent bundle W = T ∗ G , the flag variety G/T and the minimalstratum M = G/T − × G/T has the canonical orientation coming from its complex structure.We may assume that the cotangent fiber L = T ∗ e G and the moment Lagrangian C have beenoriented in such a way that the isomorphism (B.1) is orientation-preserving.B.2. The signs of the leading terms.
To discuss about the sign of the A ∞ -homomorphism Φ C ,we first need to recall the definition of the orientation for a (generalized) Hamiltonian chord.Please refer to [3, Appendix A] and [23] for more detailed explanation for this.Since every Lagrangian we consider is spin, we will review the definition of the orientationfor a chord assuming every involved Lagrangian to be spin.Let L and L be spin Lagrangian submanifolds of a symplectic manifold M . Let x : [0 , → M be a nondegenerate Hamiltonian chord from L to L for a Hamiltonian H on M .Let (cid:101) T be a closed disk with a boundary point removed, where it is equipped with a negativestrip-like end (cid:15) : ( −∞ , × [0 , → (cid:101) T . Additionally, we consider an 1-form β on (cid:101) T such that β vanishes near the boundary of (cid:101) T and (cid:15) ∗ β = dt .Fix a symplectic trivialization x ∗ T M = [0 , × E for some symplectic vector space E ofdimension dim M . After identifying the domain [0 , of x with a subset of (cid:101) T via the strip-like end (cid:15) − , extend the trivialization to a symplectic trivialization on the whole domain (cid:101) T anddenote the trivialization by (cid:101) T × E . As a section of Lagrangian Grassmannian of the bundle ∂ (cid:101) T × E ⊂ (cid:101) T × E , we choose a path of Lagrangian subspaces Λ connecting the tangent spaces T x (0) L and T x (1) L . We further choose a spin structure on the path Λ of Lagrangian subspaces,which restricts to the spin structure on L and L at the ends.After choosing an almost complex structure on M , we consider a Cauchy-Riemann operator D x, Λ defined by(B.2) D x, Λ ( v ) := ( dv − X H ⊗ β ) , for sections v of the bundle (cid:101) T × E on (cid:101) T such that v ( z ) ∈ Λ( z ) , ∀ z ∈ ∂ (cid:101) T . Then Definition B.1.
The orientation o x, Λ is defined by the determinant of the Cauchy-Riemann oper-ator D x, Λ .Generalizing this construction of orientation for chord, we now define the orientation for ageneralized chord from L = ( L, C ) to L (cid:15) = ( L (cid:15) , C ) . Indeed, let x = ( x − x , x + ) be a chord from L to L (cid:15) as defined in (2.31). ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES Let us consider a seam I in (cid:101) T , by which the domain (cid:101) T is divided into two patches (cid:101) T (the outerpatch) and (cid:101) T (the inner patch) just as the domain T is so above Definition 5.11. See Figure 5.Also let (cid:15) − , (cid:15) and (cid:15) + be negative strip-like ends as defined there. Then we consider an 1-form β on (cid:101) T such that β vanishes near the boundary of (cid:101) T and (cid:15) ∗± β = dt once again.F IGURE
5. Two patches (cid:101) T and (cid:101) T of (cid:101) T Fix symplectic trivializations x ∗ T M = [0 , × E M and x ∗± T W = [0 , δ ] × E W for somesymplectic vector spaces E M of dimension dim M and E W of dimension dim W .After identifying the domain [0 , δ ] of x ± (resp. [0 , of x ) with a subset of (cid:101) T (resp. (cid:101) T ) viathe strip-like end (cid:15) ± (resp. (cid:15) ), extend the trivialization to symplectic trivializations on the wholedomain (cid:101) T (resp. (cid:101) T ) and denote the trivialization by (cid:101) T × E W (resp. (cid:101) T × E M ). The direct sumof I × E − W and I × E M on the on the seam I = (cid:101) T ∩ (cid:101) T gives a symplectic bundle I × ( E − W ⊕ E M ) . As a section of Lagrangian Grassmannian of the bundle ( ∂ (cid:101) T \ I ) × E W on ∂ (cid:101) T \ I , we choosea path of Lagrangian subspaces Λ connecting the tangent spaces T x − (0) L and T x + ( δ ) L (cid:15) . Also,as a section of Lagrangian Grassmannian of the bundle I × ( E − W ⊕ E M ) , we choose a path ofLagrangian subspaces Λ connecting the tangent spaces T ( x − ( δ ) ,x (0)) C and T ( x + (0) ,x (1)) C .We further need to choose spin structures on the paths Λ and Λ of Lagrangian subspaces insuch a way that the spin structure on Λ restricts to those on L and L (cid:15) at the ends and the spinstructure on Λ restricts to that on C at the ends.After choosing almost complex structures on W and M , we consider a Cauchy-Riemann op-erator D x defined by D x , Λ , Λ ( v , v ) := (cid:0) ( dv − X H ⊗ β ) , , dv , (cid:1) for pairs ( v , v ) of a section v of (cid:101) T × E W and a section v of (cid:101) T × E M such that v ( z ) ∈ Λ ( z ) , ∀ z ∈ ∂ (cid:101) T \ I and ( v ( z ) , v ( z )) ∈ Λ ( z ) , ∀ z ∈ I . Then Definition B.2.
The orientation o x , Λ , Λ is defined by the determinant of the Cauchy-Riemannoperator D x , Λ , Λ .Let us now consider the orientations for the Hamiltonian chords involved in the A ∞ -homomorphism Φ C , that is, the orientations for the Hamiltonian chords x h(cid:15) , h ∈ Q ∨ from L = T ∗ e G to L (cid:15) = T ∗ exp (cid:15) G and those for the generalized Hamiltonian chords x w , w ∈ W from L to L (cid:15) . HANWOOL BAE AND NAICHUNG CONAN LEUNG
First consider the orientation for the Hamiltonian chord x h(cid:15) , h ∈ Q ∨ (2.31) from L to L (cid:15) . Tofind a path of Lagrangian subspaces for the chord x h(cid:15) , consider the capping half-disk v h(cid:15) : T →U ⊂ W and the symplectic trivialization ( v h(cid:15) ) ∗ T U = (cid:101) T × E W discussed in Subsection 5.1.Although the domain of v h(cid:15) is T , since it is asymptotic to the chord x (cid:15) at its positive end, onecan fill in the positive end of T by mapping it to the chord x (cid:15) and hence we may assume that v h(cid:15) is defined on (cid:101) T . Note that the symplectic trivialization of ( v h(cid:15) ) ∗ T U naturally extends that of ( x h(cid:15) ) ∗ T U .We have already constructed the path Λ h of Lagrangian subspaces by concatenating the tangentspaces of L , the short path Λ (cid:15) and the tangent spaces of L (cid:15) in Subsection 5.1. We may assumethat the path Λ h is defined on ∂ (cid:101) T by the above argumentSince the tangent bundle T U admits the decomposition T U = Ver ⊕ Hor (2.35), we mayassume that the symplectic vector space E W admits a decomposition E Ver ⊕ E Hor into symplecticsubspaces E Ver and E Hor of E W , which are symplectically orthogonal to each other. Let us denoteby π Ver : E W → E Ver the projection to the first factor and by π Hor : E W → E Hor the projectionto the second factor. Here we may identify E M with the summand E Hor of E W due to (F) inSubsection 2.9.Furthermore, since the vertical distribution Ver is identified with the tangent space of the fiberof the direct sum of the line bundles N j ’s over M , we may further assume that the verticalsubspace E Ver also decomposes into 2-dimensional symplectic subspaces E j , j = 1 , . . . , l . Each E j is spanned by the radial vector ∂∂ρ j and the angular vector ∂∂θ j just as the fiber of each linebundle N j is so.From the description of L and L (cid:15) given in Subsection 2.9, we observe(a) For every z ∈ ∂ (cid:101) T , π Ver (Λ h ( z )) is the Lagrangian subspace of E Ver spanned by the radialvectors ∂∂ρ j ’s.Considering the discussion in Subsection B.1, we deduce that the spin structure on the path Λ h is determined by that on the projection π Hor (Λ h ) to the horizontal distribution. Let o x h(cid:15) , Λ h bethe associated orientation.Secondly, we discuss the orientation for the Hamiltonian chord x w . But, before discussing itdirectly, we first consider the orientations for Hamiltonian chords from ∆ = ∆ G/T to ∆ (cid:15) . Recallthat every intersection point of ∆ and ∆ (cid:15) is of the form ([ ˙ w ] , [ ˙ w ]) ∈ ∆ ∩ ∆ (cid:15) for some w ∈ W since the perturbed diagonal ∆ (cid:15) is given by the graph of an exact 1-form d ( f (cid:15) ◦ π ) on the diagonal ∆ for some Morse function f (cid:15) ◦ π .We associate a path of Lagrangian subspace Λ w,h to the pair of an intersection point ([ ˙ w ] , [ ˙ w ]) and the Novikov variable q h for some h ∈ Q ∨ = π ( M, ∆) . For that purpose, first consider ashort path of Lagrangian subspaces connecting T ([ ˙ w ] , [ ˙ w ]) ∆ and T ([ ˙ w ] , [ ˙ w ]) ∆ (cid:15) given by Λ w ( t ) = ( Id × td ( f (cid:15) ◦ π )) T ([ ˙ w ] , [ ˙ w ]) ∆ , t ∈ [0 , . Now, for each h ∈ Q ∨ = π ( M, ∆) , we consider a disk u h : ( D, ∂D ) → ( M, ∆) representingthe homotopy class h such that u h ( −
1) = ([ ˙ w ] , [ ˙ w ]) for the boundary points − ∈ ∂D = S .Then we define the path Λ w,h by concatenating the path of Lagrangian subspaces ( u h | ∂D ) ∗ T ∆ ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES and the short path Λ w . The path Λ w,h is equipped with the spin structure derived from the spinstructure on ∆ . Let o ([ ˙ w ] , [ ˙ w ]) , Λ w,h be the associated orientation.But, considering (C), (D) and (E) in Subsection 2.9 and and [ (cid:101) π ◦ v h(cid:15) ] = − w − h + (cid:15) ( h ) (Lemma5.3), the construction of the path Λ h given below Definition 5.1 implies(b) Two paths π Hor (Λ h ) and Λ w h + (cid:15) , − w − h + (cid:15) h of Lagrangian subspaces in E M are homotopicrelative to the end points.Now we introduce some notations for the next step. The symplectic vector space E − W ⊕ E M decomposes into E − Ver ⊕ E − M ⊕ E M after identifying E Hor with E M . Let us denote the projectionsinto each summand as follows. π = π Ver ◦ π W : E − W ⊕ E M → E − Ver π = π Hor ◦ π W : E − W ⊕ E M → E − M π = π M : E − W ⊕ E M → E M . (B.3)Consider the generalized Hamiltonian chords from L = ( L, C ) to L (cid:15) = ( L (cid:15) , C ) . As we haveobserved in Subsection 5.2.2, there exists a unique generalized Hamiltonian chord x w for each w ∈ W . Hence there exists a bijection(B.4) P (∆ , ∆ (cid:15) ) ↔ P ( L , L (cid:15) ) , ([ ˙ w ] , [ ˙ w ]) (cid:55)→ x w , w ∈ W . Let ( v , v ) be a capping quilted half-disk for x w = ( x − , x , x + ) as defined in Definition 5.11.Note that the domain of the map v : T → U ⊂ W has one positive end, where it is asymptoticto the chord x (cid:15) . By filling in the positive end of T by mapping it to x (cid:15) , we may assume that thedomain of v is (cid:101) T .To find the paths Λ and Λ for the generalized Hamiltonian chord x w = ( x − , x , x + ) , firsttrivialize v ∗ T U and v ∗ T M as symplectic bundles and denote those by v ∗ T U = T × E W and v ∗ T M = T × E M . Now, since the tangent spaces of C along the seam I connect T ( x − ( δ ) ,x (0)) C and T ( x + (0) ,x (1)) C ,it gives a path Λ . Recall from Subsection 2.6 that the moment Lagrangian C maps diffeomor-phically to π W ( C ) via the projection W × M → W . Due to the condition (a) and the description(B) in Subsection 2.9, we have(c) For every z ∈ I , π (Λ ( z )) (See (B.3)) is the Lagrangian subspace of E Ver spanned byangular vectors ∂∂θ j ’s.Furthermore, due to (A) in Subsection 2.9, we may assume that the path Λ satisfies(d) For every z ∈ I , ( π ⊕ π )(Λ ( z )) = ∆ E M (See (B.3)), where ∆ E M = { ( v, v ) ∈ E − M ⊕ E M | v ∈ E M } .Similarly, we define the path Λ from T x − (0) L to T x + ( δ ) L (cid:15) by concatenating the tangent spacesof L , the short path Λ (cid:15) (5.1) and the tangent spaces of L (cid:15) along the boundary of (cid:101) T just as we didto construct the path Λ h in Subsection 5.1. Due to (C) and (D) in Subsection 2.9 again, we mayassume that the trivialization v ∗ T U = T × E W and the path Λ are chosen in such a way that(e) For some h ∈ Q ∨ , two paths π Hor (Λ ) and Λ w,h of Lagrangian subspaces in E M arehomotopic relative to the end points. HANWOOL BAE AND NAICHUNG CONAN LEUNG
The condition (e) makes sense since the tangent space of L (resp. L (cid:15) ) projects to down of thatof ∆ (resp. ∆ (cid:15) ) via (cid:101) π , which is represented by π Hor here. Let us denote the path Λ by Λ h torecord the element h ∈ Q ∨ in the condition (e). The spin structures on L and C determine thespin structures on the paths Λ h and Λ , respectively. Let o x w , Λ h , Λ be the associated orientation.Assume that v is defined on (cid:101) T as explained above. The patch (cid:101) T is homeomorphic to thestrip I × [0 , (Figure 5). From now on, we fix a capping quilted half-disk ( v , v ) for x w suchthat the map v : (cid:101) T ∼ = I × [0 , → M is constant on { z } × [0 , ⊂ I × [0 , for every z ∈ I .This implies that v ( z, t ) ∈ ( L ∪ L (cid:15) ∪ Im x (cid:15) ) ∩ π W ( C ) , ∀ ( z, t ) ∈ I × [0 ,
1] = (cid:101) T and further that v ( z ) = (cid:101) π ( v ( z, t )) , ∀ z ∈ I, t ∈ [0 , . Due to the assumption on v , we may shrink the strip (cid:101) T = I × [0 , to I × { } and mayassume that both Λ h and Λ are defined on I . This allows us to consider the path of subspacesgiven by the intersection π − W (Λ h ) ∩ Λ . Then we have Lemma B.3.
For each z ∈ I , the projection π M = π : E W ⊕ E M → E M maps the subspace π − W (Λ h ( z )) ∩ Λ ( z ) isomorphically to the Lagrangian subspace π Hor (Λ h ( z )) in E M . In particu-lar, the projection maps the path of subspaces π − W (Λ h ) ∩ Λ to a path of Lagrangian subspacesin E M that is homotopic to Λ w,h relative to the end points.Proof. For all z ∈ I , we have π − W (Λ h ( z )) ∩ Λ ( z )= π − W { π Ver (Λ h ( z )) ⊕ π Hor (Λ h ( z )) } ∩ { π (Λ ( z )) ⊕ π (Λ ( z )) ⊕ π (Λ ( z )) } = { π Ver (Λ h ( z )) ⊕ π Hor (Λ h ( z )) ⊕ E M ) } ∩ { π (Λ ( z )) ⊕ π (Λ ( z )) ⊕ π (Λ ( z )) } = { π Ver (Λ h ( z )) ⊕ π Hor (Λ h ( z )) ⊕ E M ) } ∩ { π (Λ ( z )) ⊕ ∆ E M } (( d ))= ( π Ver (Λ h ( z )) ∩ π (Λ ( z ))) ⊕ { ( π Hor (Λ h ( z )) ⊕ E M ) ∩ ∆ E M } . The projection π in (B.3) maps the last term isomorphically to π Hor (Λ h ( z )) since the Lagrangiansubspaces π Ver (Λ h ( z )) and π (Λ ( z )) intersect transversely in E Ver for every z ∈ I due to thecondition (a) and (c). Finally, the condition (e) completes the proof. (cid:3) Furthermore, the identification π M ( π − W (Λ h ) ∩ Λ ) = Λ w,h also respects the spin structuressince L and C have been oriented to ensure that as discussed in the last paragraph of SubsectionB.1. Considering that W is spin and exact, the proof of [23, Proposition 4.6.3] can be appliedhere to show the following lemma. Lemma B.4.
There is a natural identification between the orientations o ([ ˙ w ] , [ ˙ w ]) , Λ w,h and o x w , Λ h , Λ ,which is orientation preserving. To enhance the bijection in (B.4) to a (signed) chain map between CF (∆ , ∆ (cid:15) ) and CF ( L , L (cid:15) ) ,assume that we have meant the orientation o ([ ˙ w ] , [ ˙ w ]) , Λ w,h by ([ ˙ w ] , [ ˙ w ]) q h and have meant the ori-entation o x w , Λ h , Λ by x w q h in the sense of [4, 3, 20]. It was proved in [14, Section 2] that there ETERSON CONJECTURE VIA LAGRANGIAN CORRESPONDENCES is a chain map from CF ( L , L (cid:15) ) to CF (∆ , ∆ (cid:15) ) that induces an isomorphism on homologies. Theargument here can be applied to relating the orientations of ([ ˙ w ] , [ ˙ w ]) q h and x w q h and hence thecoefficient of ([ ˙ w ] , [ ˙ w ]) q h in the image of x w q h under the chain map is 1.Finally we are ready to show that the signs of the leading terms are +1 for all h ∈ Q ∨ as statedas in Lemma 5.16. First recall that the zero dimensional component of the moduli space M := M (cid:16) x w h + (cid:15) q − w − h + (cid:15) h , x h(cid:15) ; D Φ C (cid:17) has exactly one element ( u , u ) , for which the first entry u maps into the fiber (cid:101) π − ([ ˙ w h + (cid:15) ] , [ ˙ w h + (cid:15) ]) and the second entry u is constant as stated in Lemma 5.23. The tangent space of the component M can be identified with the kernel of the operator ( D∂ J ) ( u ,u ) .But the quilt ( u , u ) can be glued with the capping half disk v h(cid:15) since both of them are asymp-totic to the Hamiltonian chord x h(cid:15) at one of their ends. Accordingly, one can glue the operator ( D∂ J ) ( u ,u ) with the Cauchy-Riemann operator D x h(cid:15) , Λ h (B.2). See Figure 6. Let us denote theresulting glued operator by D .F IGURE
6. Gluing of two Cauchy-Riemann operatorsLet us compare the glued Cauchy-Riemann operator D with the Cauchy-Riemann operator D := D x wh + (cid:15) , Λ − w − h + (cid:15)h , Λ associated to the paths Λ − w − h + (cid:15) h and Λ , which is obtained from the capping quilted half-disk ( v , v ) described above Lemma B.3. Lemma B.5.
There is an continuous family of capping quilted half-disks ( v t , v t ) , t ∈ [0 , forx w such that ( v , v ) = ( v , v ) and ( v , v ) coincides with the glued quilt ( v h(cid:15) u , u ) .Proof. Both the capping quilted half-disks ( v , v ) and ( u v h(cid:15) , u ) represent − w − h + (cid:15) h ∈ Q ∨ = π ( M, ∆) . This means that (cid:101) π ◦ v ∆ M v is homotopic to ( v h(cid:15) u ) ∆ M u , where the gluing ∆ M makes sense due to the condition (A) in Subsection 2.9. Hence one may find a continuousfamily of pairs ( v t , v t ) , t ∈ [0 , of maps v t : T → M, v t : T → M such that • v t ( z ) = v t ( z ) , ∀ z ∈ I , • v t ( z ) ∈ (cid:40) ∆ , z ∈ ∂T , ∆ (cid:15) , z ∈ ∂T , • ( v , v ) = ( (cid:101) π ◦ v , v ) , • ( v , v ) = ( v h(cid:15) u , u ) . HANWOOL BAE AND NAICHUNG CONAN LEUNG
Then we lift the pairs ( v t , v t ) to a capping quilted half-disk ( v t , v t ) for x w for t ∈ [0 , in sucha way that the desired properties are satisfied. (cid:3) As a consequence of Lemma B.5, there exists an one parameter family of Cauchy-Riemannoperators connecting D and D . This implies that the determinant of these two are isomorphic.Assume that we have meant the orientation o x h(cid:15) , Λ h by x h(cid:15) . Since the determinant o x wh + (cid:15) , Λ − w − h + (cid:15)h , Λ of the Cauchy-Riemann operator D = D x wh + (cid:15) , Λ − w − h + (cid:15)h , Λ is represented by x w h + (cid:15) q − w − h + (cid:15) h , we have Theorem B.6.
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