Floer cohomologies of non-torus fibers of the Gelfand-Cetlin system
aa r X i v : . [ m a t h . S G ] S e p Floer cohomologies of non-torus fibersof the Gelfand-Cetlin system
Yuichi Nohara and Kazushi Ueda
Abstract
The Gelfand-Cetlin system has non-torus Lagrangian fibers on some of the boundarystrata of the moment polytope. We compute Floer cohomologies of such non-torus La-grangian fibers in the cases of the 3-dimensional full flag manifold and the Grassmannianof 2-planes in a 4-space.
Let P be a parabolic subgroup of GL( n, C ) and F := GL( n, C ) /P be the associated flagmanifold. The Gelfand-Cetlin system, introduced by Guillemin and Sternberg [GS83], is acompletely integrable system Φ : F −→ R (dim R F ) / , i.e., a set of functionally independent and Poisson commuting functions. The image ∆ =Φ( F ) is a convex polytope called the Gelfand-Cetlin polytope , and Φ gives a Lagrangian torusfibration structure over the interior Int ∆ of ∆. Unlike the case of toric manifolds wherethe fibers over the relative interior of a d -dimensional face of the moment polytope are d -dimensional isotropic tori, the Gelfand-Cetlin system has non-torus Lagrangian fibers over therelative interiors of some of the faces of ∆.Let ( X, ω ) be a compact toric manifold of dim C X = N , and Φ : X → R N be the toricmoment map with the moment polytope ∆ = Φ( X ). For an interior point u ∈ Int ∆, let L ( u )denote the Lagrangian torus fiber Φ − ( u ). Lagrangian intersection Floer theory endows thecohomology group H ∗ ( L ( u ); Λ ) over the Novikov ringΛ := ( ∞ X i =1 a i T λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ C , λ i ≥ , lim i →∞ λ i = ∞ ) with a structure { m k } k ≥ of a unital filtered A ∞ -algebra [FOOO09]. Let Λ and Λ + be thequotient field and the maximal ideal of the local ring Λ respectively. An odd-degree element b ∈ H odd ( L ( u ); Λ ) is said to be a bounding cochain if it satisfies the Maurer-Cartan equation ∞ X k =0 m k ( b ⊗ k ) = 0 . (1.1)A solution b ∈ H odd ( L ( u ); Λ ) to the weak Maurer-Cartan equation ∞ X k =0 m k ( b ⊗ k ) ≡ e (1.2)1s called a weak bounding cochain , where e is the unit in H ∗ ( L ( u ); Λ ). The set of weak bound-ing cochains will be denoted by c M weak ( L ( u )). The potential function is a map PO : c M weak ( L ( u )) → Λ defined by ∞ X k =0 m k ( b, . . . , b ) = PO ( b ) e . (1.3)A weak bounding cochain gives a deformed filtered A ∞ -algebra whose A ∞ -operations are givenby m bk ( x , . . . , x k ) = ∞ X m =0 · · · ∞ X m k =0 m m + ··· + m k + k ( b ⊗ m ⊗ x ⊗ b ⊗ m ⊗ · · · ⊗ x k ⊗ b ⊗ m k ) . (1.4)The weak Maurer-Cartan equation implies that m b squares to zero, and the deformed Floercohomology is defined by HF (( L ( u ) , b ) , ( L ( u ) , b ); Λ ) = Ker( m b ) (cid:14) Im( m b ) . (1.5)More generally, one can deform the Floer differential m by δ b ,b ( x ) = X k ,k ≥ m k + k +1 ( b , . . . , b | {z } k , x, b , . . . , b | {z } k ) (1.6)for a pair ( b , b ) of weak bounding cochains with PO ( b ) = PO ( b ). The Floer cohomologyof the pair (( L ( u ) , b ) , ( L ( u ) , b )) is defined by HF (( L ( u ) , b ) , ( L ( u ) , b ); Λ ) = Ker( δ b ,b )/ Im( δ b ,b ) . (1.7)If the toric manifold X is Fano, then the following hold [FOOO10]: • H ( L ( u ); Λ ) is contained in c M weak ( L ( u )). • The potential function PO on [ u ∈ Int ∆ H ( L ( u ); Λ / π √− Z ) ∼ = Int ∆ × (Λ / π √− Z ) N (1.8)can be considered as a Laurent polynomial, which can be identified with the superpo-tential of the Landau-Ginzburg mirror of X . • Each critical point of PO corresponds to a pair ( u , b ) such that the deformed Floercohomology HF (( L ( u ) , b ) , ( L ( u ) , b ); Λ) over the Novikov field Λ is non-trivial. • If the deformed Floer cohomology group over the Novikov field is non-trivial, then it isisomorphic to the classical cohomology group; HF (( L ( u ) , b ) , ( L ( u ) , b ); Λ) ∼ = H ∗ ( T N ; Λ) . (1.9) • The quantum cohomology ring QH ( X ; Λ) is isomorphic to the Jacobi ring Jac( PO ) ofthe potential function. 2n particular, the number of pairs ( L ( u ) , b ) with nontrivial Floer cohomology coincideswith rank QH ( X ; Λ) = rank H ∗ ( X ; Λ) . Nishinou and the authors [NNU10] introduced the notion of a toric degeneration of anintegrable system, and used it to compute the potential function of Lagrangian torus fibers ofthe Gelfand-Cetlin system. The resulting potential function can be considered as a Laurentpolynomial just as in the toric Fano case, which can be identified with the superpotential ofthe Landau-Ginzburg mirror of the flag manifold given in [Giv97, BCFKvS00]. In contrastto the toric case, the rank of H ∗ ( F ; Λ) is greater in general than the rank of the Jacobiring Jac( PO ), and hence than the number of Lagrangian torus fibers with non-trivial Floercohomology. In the case of the 3-dimensional flag manifold Fl(3), the potential function hassix critical points, which is equal to the rank of H ∗ (Fl(3); Λ). Similarly, the potential functionfor the Grassmannian Gr(2 ,
5) of 2-planes in C has ten critical points, which is equal to therank of H ∗ (Gr(2 , ,
4) of 2-planes in C is four, which is less than the rankof H ∗ (Gr(2 , , Theorem 1.1.
Let
Φ : Fl(3) → R be the Gelfand-Cetlin system with the Gelfand-Cetlinpolytope ∆ = Φ(Fl(3)) .1. There exists a vertex u of ∆ such that a fiber L ( u ) = Φ − ( u ) over a boundary point u ∈ ∂ ∆ is a Lagrangian submanifold if and only if u = u .2. The Lagrangian fiber L ( u ) is diffeomorphic to SU(2) ∼ = S .3. The Floer cohomology of L ( u ) over the Novikov field Λ is trivial; HF ( L ( u ) , L ( u ); Λ) = 0 . (1.10) Theorem 1.2.
Let
Φ : Gr(2 , → R be the Gelfand-Cetlin system with the Gelfand-Cetlinpolytope ∆ = Φ(Gr(2 , .1. There exists an edge of ∆ such that a fiber L ( u ) = Φ − ( u ) over u ∈ ∂ ∆ is a Lagrangiansubmanifold if and only if u is in the relative interior of the edge.2. The Lagrangian fiber L ( u ) over any point u in the relative interior of the edge is diffeo-morphic to U(2) ∼ = S × S .3. H ( L ( u ); Λ ) is contained in c M weak ( L ( u )) .4. The potential function is identically zero on H ( L ( u ); Λ ) .5. The Floer cohomology HF (( L ( u ) , b ) , ( L ( u ) , b ); Λ) of a Lagrangian U(2) -fiber L ( u ) overthe Novikov field Λ is non-trivial if and only if u is the barycenter u of the edge and b = ± π √− / e , where e is a generator of H ( L ( u ); Z ) ∼ = Z .6. If the deformed Floer cohomology group over the Novikov field is non-trivial, then it isisomorphic to the classical cohomology group; HF (( L ( u ) , ± π √− / e ) , ( L ( u ) , ± π √− / e ); Λ) ∼ = H ∗ ( S × S ; Λ) . (1.11)3 . The Floer cohomology of the pair (( L ( u ) , π √− / e ) , ( L ( u ) , − π √− / e )) is trivial; HF (( L ( u ) , π √− / e ) , ( L ( u ) , π √− / e ); Λ) = 0 . (1.12)More precise statements, which describe the Floer cohomology groups over the Novikovring Λ , are given in Theorem 4.8, Theorem 4.16, and Theorem 4.20.A symplectic manifold ( X, ω ) is monotone if the cohomology class [ ω ] is positively propor-tional to the first Chern class; ∃ λ > ω ] = λc ( X ) . (1.13)The quantum cohomology ring of a monotone symplectic manifold does not have any con-vergence issue, and hence is defined over C . A Lagrangian submanifold L is monotone if thesymplectic area of a disk bounded by L is positively proportional to the Maslov index; ∃ λ > ∀ β ∈ π ( M, L ) β ∩ ω = λµ ( β ) . (1.14)The A ∞ -operations on the Lagrangian intersection Floer complex of a monotone Lagrangiansubmanifold is defined over C . The minimal Maslov number of oriented monotone Lagrangiansubmanifold is greater than or equal to 2, so that the obstruction class m (1) can be writtenas m (1) = m ( L ) e , where m ( L ) ∈ C is the count of Maslov index 2 disks bounded by L , weighted by their symplectic areas and holonomies of a flat U (1)-bundle on L along theboundaries of the disks. The monotone Fukaya category is defined as the direct sum F ( X ) := M λ ∈ C F ( X ; λ ) , (1.15)where F ( X ; λ ) is an A ∞ -category over C whose objects are monotone Lagrangian submani-folds, equipped with flat U (1)-bundles, satisfying m ( L ) = λ . For any monotone Lagrangiansubmanifold L , there is a natural ring homomorphism QH ( X ) → HF ( L, L ) , (1.16)which is known by Auroux [Aur07], Kontsevich, and Seidel to send c ( X ) ∈ QH ( X ) to m (1) ∈ HF ( L, L ). It follows that F ( X ; λ ) is trivial unless λ is an eigenvalue of the quantumcup product by c ( X ).Now consider the case when X = Gr(2 , X = (cid:8) [ z : · · · : z ] ∈ P (cid:12)(cid:12) z = z + · · · + z (cid:9) . (1.17)The real locus X R is a monotone Lagrangian sphere, which is the vanishing cycle along adegeneration into a nodal quadric and split-generates the nilpotent summand D π F ( X ; 0) ofthe monotone Fukaya category [Smi12, Lemma 4.6]. The Floer cohomology HF ( X R , X R ) issemisimple, and carries a formal A ∞ -structure [Smi12, Lemma 4.7]. It follows that D π F ( X ; 0)is equivalent to the direct sum of two copies of the derived category D b ( C ) of C -vectorspaces. On the other hand, ( L ( u ) , ± π √− / e ) are also objects of the nilpotent sum-mand D π F ( X ; 0) of the monotone Fukaya category, which are non-zero by (1.11). Since( L ( u ) , ±√− / e ) is a pair of orthogonal non-zero objects in a triangulated category equiv-alent to D b ( C ) ⊕ D b ( C ), they split-generate the whole category: Corollary 1.3.
The pair ( L ( u ) , ± π √− / e ) split-generate D π F (Gr(2 , .Acknowledgment : We thank Hiroshi Ohta, Kaoru Ono, and Yoshihiro Ohnita for usefulconversations. Y. N. is supported by Grant-in-Aid for Young Scientists (No.23740055). K. U.is supported by Grant-in-Aid for Young Scientists (No.24740043).4 Non-torus fibers of the Gelfand-Cetlin system
For a sequence 0 = n < n < · · · < n r < n r +1 = n of integers, let F = F ( n , . . . , n r , n ) bethe flag manifold consisting of flags0 ⊂ V ⊂ · · · ⊂ V r ⊂ C n , dim V i = n i of C n . We write the full flag manifold and the Grassmannian as Fl( n ) = F (1 , , . . . , n ) andGr( k, n ) = F ( k, n ) respectively. The complex dimension of F ( n , . . . , n r , n ) is given by N = N ( n , . . . , n r , n ) := dim C F ( n , . . . , n r , n ) = r X i =1 ( n i − n i − )( n − n i ) . Let P = P ( n , . . . , n r , n ) ⊂ GL( n, C ) be the stabilizer subgroup of the standard flag ( V i = h e , . . . , e n i i ) ri =1 , where { e i } ni =1 is the standard basis of C n . The intersection of P and U( n ) isU( k ) × · · · × U( k r +1 ) for k i = n i − n i − , and F is written as F = GL( n, C ) /P = U( n ) / (U( k ) × · · · × U( k r +1 )) . We take a U( n )-invariant inner product h x, y i = tr xy ∗ on the Lie algebra u ( n ) of U( n ), andidentify the dual vector space u ( n ) ∗ of u ( n ) with the space √− u ( n ) of Hermitian matrices.For λ = diag ( λ , . . . , λ n ) ∈ √− u ( n ) with λ = · · · = λ n | {z } k > λ n +1 = · · · = λ n | {z } k > · · · > λ n r +1 = · · · = λ n | {z } k r +1 , (2.1)the flag manifold F is identified with the adjoint orbit O λ ⊂ √− u ( n ) of λ . Note that O λ consists of Hermitian matrices with fixed eigenvalues λ , . . . , λ n . Let ω (ad ξ ( x ) , ad η ( x )) = 12 π h x, [ ξ, η ] i , ξ, η ∈ u ( n )be the (normalized) Kostant-Kirillov form on O λ .For each i = 1 , . . . , r , we set P i := P (cid:0)V n i C n (cid:1) ∼ = P ( nni ) − . Then the Pl¨ucker embedding isgiven by ι : F ֒ → r Y i =1 P i , (0 ⊂ V ⊂ · · · ⊂ V r ⊂ C n ) ( V n V , . . . , V n r V r ) . Let ω P i be the Fubini-Study form on P i normalized in such a way that it represents the firstChern class c ( O (1)) of the hyperplane bundle. Then the Kostant-Kirillov form ω and thefirst Chern form c ( F ) of F are given by ω = r X i =1 ( λ n i − λ n i +1 ) ω P i and c ( F ) = r X i =1 ( n i +1 − n i − ) ω P i respectively. 5 xample 2.1. The 3-dimensional full flag manifold Fl(3) is embedded into P × P = P ( C ) × P ( V C ) ∼ = P × P as a hypersurface. The image of Fl(3) is given by the Pl¨ucker relation Z Z + Z Z + Z Z = 0 , where [ Z : Z : Z ] and [ Z : Z : Z ] are the Pl¨ucker coordinates on P and P respectively. Example 2.2.
The Grassmannian Gr(2 ,
4) of 2-plans in C is embedded into P ( V C ) ∼ = P as a hypersurface. The Pl¨ucker relation is given by Z Z − Z Z + Z Z = 0 , where [ Z : Z : Z : Z : Z : Z ] is the Pl¨ucker coordinates. For x ∈ O λ and k = 1 , . . . , n −
1, let x ( k ) denote the upper-left k × k submatrix of x . Since x ( k ) is also a Hermitian matrix, it has real eigenvalues λ ( k )1 ( x ) ≥ λ ( k )2 ( x ) ≥ · · · ≥ λ ( k ) k ( x ). Bytaking the eigenvalues for all k = 1 , . . . , n −
1, we obtain a set ( λ ( k ) i ) ≤ i ≤ k ≤ n − of n ( n − / λ λ λ · · · λ n − λ n ≥ ≥ ≥ ≥ ≥ ≥ λ ( n − λ ( n − λ ( n − n − ≥ ≥ ≥ λ ( n − λ ( n − n − ≥ ≥ ··· ··· ≥ ≥ λ (1)1 . (2.2)It follows that the number of non-constant λ ( k ) i coincides with N = dim C F . Let I = I ( n , . . . , n r , n ) denotes the set of pairs ( i, k ) such that λ ( k ) i is non-constant. Then the Gelfand-Cetlin system is defined byΦ = ( λ ( k ) i ) ( i,k ) ∈ I : F ( n , . . . , n r , n ) −→ R N ( n ,...,n r ,n ) . Proposition 2.3 (Guillemin and Sternberg [GS83]) . The map Φ is a completely integrablesystem on ( F ( n , . . . , n r , n ) , ω ) . The functions λ ( k ) i are action variables, and the image ∆ =Φ( F ) is a convex polytope defined by (2.2) . The fiber L ( u ) = Φ − ( u ) over each interior point u ∈ Int ∆ is a Lagrangian torus. (2)2(cid:21)(2)1(cid:21)(1)1 Figure 2.1: The Gelfand-Cetlin polytope for Fl(3)The image ∆ ⊂ R N ( n ,...,n r ,n ) is called the Gelfand-Cetlin polytope . The Gelfand-Cetlinsystem is not smooth on the locus where λ ( i ) k = λ ( i +1) k for some ( i, k ), or equivalently, wherethe Gelfand-Cetlin pattern (2.2) contains a set of equalities of the form λ ( i +1) k +1 = = λ ( i ) k λ ( i +1) k = = λ ( i ) k − . The image of such loci are faces of ∆ of codimension greater than two where ∆ does notsatisfy the Delzant condition. Away from such faces, each fiber Φ − ( u ) of Φ is an isotropictorus whose dimension is that of the face of ∆ containing u in its relative interior. Fl(3)
After a translation by a scalar matrix, we may assume that Fl(3) is identified with the adjointorbit of λ = diag( λ , , − λ ) for λ , λ >
0. Then the Gelfand-Cetlin polytope ∆ consists of( u , u , u ) ∈ R satisfying λ − λ ≥ ≥ ≥ ≥ u u ≥ ≥ u (2.3)as shown in Figure 2.1. The non-smooth locus of Φ is the fiber L = Φ − ( ) over the vertex = (0 , , ∈ ∆ where four edges intersect. Definition 2.4 (Evans and Lekili [EL, Definition 1.1.1]) . Let K be a compact connected Liegroup. A Lagrangian submanifold L in a K¨ahler manifold X is said to be K -homogeneous if K acts holomorphically on X in such a way that L is a K -orbit. Proposition 2.5.
The fiber L = Φ − ( ) is a Lagrangian 3-sphere given by L = z z z z λ − λ ∈ √− u (3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | z | + | z | = λ λ , hich is K -homogeneous for K = a − a a a
00 0 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a | + | a | = 1 ∼ = SU(2) . Proof.
Suppose that x ∈ L . Then λ (2)1 ( x ) = λ (2)2 ( x ) = 0 implies that x (2) = 0 and thus x hasthe form x = z z z z x for some z , z ∈ C and x ∈ R . Sincedet( λ − x ) = λ (cid:0) λ − x λ − ( | z | + | z | ) (cid:1) = 0has solutions λ = λ , , − λ , we have x = λ − λ and | z | + | z | = λ λ . Hence the fiber L is the K -orbit of √ λ λ √ λ λ λ − λ = Ad g λ − λ ∈ O λ , where g = p λ / ( λ + λ ) 0 − p λ / ( λ + λ )0 1 0 p λ / ( λ + λ ) 0 p λ / ( λ + λ ) ∈ SU(3) . Next we see that L is Lagrangian. Since K acts transitively on L , the tangent space T x L is spanned by infinitesimal actions ad ξ ( x ) of ξ ∈ k , where k = (cid:26) ξ = (cid:18) ξ (2)
00 0 (cid:19) ∈ u (3) (cid:12)(cid:12)(cid:12)(cid:12) ξ (2) ∈ su (2) (cid:27) ∼ = su (2)is the Lie algebra of K . Since x (2) = 0 for x ∈ L , we have ω (ad ξ ( x ) , ad η ( x )) = √− π tr (cid:16) x (2) [ ξ (2) , η (2) ] (cid:17) = 0for any ξ, η ∈ k .Let ι : Fl(3) → P × P = P ( C ) × P ( V C ) be the Pl¨ucker embedding and ([ Z : Z : Z ] , [ Z : Z : Z ]) be the Pl¨ucker coordinates. The Kostant-Kirillov form is given by ω = λ ω P + λ ω P . Since the Lagrangian fiber L as a submanifold in SU(3) /T consists of a − a a a
00 0 1 g = 1 √ λ + λ √ λ a −√ λ + λ a −√ λ a √ λ a √ λ + λ a −√ λ a √ λ √ λ mod T | a | + | a | = 1, the image ι ( L ) is given by ι ( L ) = ( " a : a : r λ λ , " a : a : − r λ λ | a | + | a | = 1 ) . (2.4)Define an anti-holomorphic involution τ on Fl(3) by τ ([ Z : Z : Z ] , [ Z : Z : Z ]) = (cid:18)(cid:20) Z : Z : − λ λ Z (cid:21) , (cid:20) Z : Z : − λ λ Z (cid:21)(cid:19) . (2.5) Proposition 2.6.
The Lagrangian L is the fixed point set of τ . One can easily see that τ is an anti-symplectic involution if and only if λ = λ . Gr(2 , For k < n , let e V ( k, n ) be the space of n × k matrices of rank k , and set V ( k, n ) = { Z ∈ e V ( k, n ) | Z ∗ Z = I k } . Then the Grassmannian Gr( k, n ) is given byGr( k, n ) = e V ( k, n ) / GL( k, C ) = V ( k, n ) / U( k ) . We first consider the Gelfand-Cetlin system on Gr( n, n ) for general n . Fix λ > n, n ) with the adjoint orbit O λ of λ = diag( λ, . . . , λ | {z } n , − λ, . . . , − λ | {z } n ) . The orbit O λ consists of matrices of the form 2 λZZ ∗ − λI n for Z ∈ V ( n, n ). The Gelfand-Cetlin polytope ∆ of Gr( n, n ) consists of u = ( u ( k ) i ) ( i,k ) ∈ I ∈ R n satisfying u (2 n − n ≥ ≥ λ ··· ··· − λ ≥ ≥ ≥ ≥ u ( n )1 · · · u ( n ) n ≥ ≥ ··· ··· ≥ ≥ u (1)1 . For − λ < t < λ , let L t = Φ − ( t, . . . , t ) be the fiber over the boundary point u (1)1 = · · · = u (2 n − n = t of ∆. 9 roposition 2.7. The fiber L t is a Lagrangian submanifold given by L t = (cid:26)(cid:18) tI n √ λ − t A ∗ √ λ − t A − tI n (cid:19) ∈ √− u (2 n ) (cid:12)(cid:12)(cid:12)(cid:12) A ∈ U( n ) (cid:27) ∼ = U( n ) , which is K -homogeneous for K = (cid:26) (cid:18) P I n (cid:19) ∈ U(2 n ) (cid:12)(cid:12)(cid:12)(cid:12) P ∈ U( n ) (cid:27) ∼ = U( n ) . Proof.
We write x ∈ O λ as x = 2 λZZ ∗ − λI n = λ (cid:18) Z Z ∗ − I n Z Z ∗ Z Z ∗ Z Z ∗ − I n (cid:19) for n × n matrices Z , Z with Z = (cid:18) Z Z (cid:19) ∈ V ( n, n ) . Suppose that x ∈ L t , or equivalently, λ ( n )1 ( x ) = · · · = λ ( n ) n ( x ) = t . Then the upper-left n × n block of x satisfies x ( n ) = 2 λZ Z ∗ − λI n = tI n , which means that Z ∈ p ( λ + t ) / λ U( n ). After the right U( n )-action on V ( n, n ), we mayassume that Z = p ( λ + t ) / λI n . Then the condition Z ∗ Z = I n implies that Z ∗ Z = I n − λ + t λ I n = λ − t λ I n . Hence Z has the form Z = (cid:18)p ( λ + t ) / λI n p ( λ − t ) / λA (cid:19) ∈ V ( n, n ) (2.6)for some A ∈ U( n ), which shows that x = 2 λZZ ∗ − λI n = (cid:18) tI n √ λ − t A ∗ √ λ − t A − tI n (cid:19) . The K -homogeneity is obvious from this expression. Since the tangent space T x L t is spannedby the infinitesimal action of the Lie algebra k of K , and x ( n ) = tI n is a scalar matrix, we have ω x (ad ξ ( x ) , ad η ( x )) = 12 π tr x ( n ) [ ξ ( n ) , η ( n ) ] = 0for ξ = (cid:18) ξ ( n ) (cid:19) , η = (cid:18) η ( n ) (cid:19) ∈ k , which shows that L t is Lagrangian. Corollary 2.8.
For t = 0 , the fiber L t is displaceable, i.e., there exists a Hamiltonian diffeo-morphism ϕ on Gr( n, n ) such that ϕ ( L t ) ∩ L t = ∅ . (2)2 = (cid:21)(cid:21)(2)2 = (cid:0)(cid:21) (cid:21)(2)2 = 0 u2u1 Figure 2.2: The Gelfand-Cetlin polytope for Gr(2 , Proof.
One has g ( L t ) = L − t for g = (cid:18) − I n I n (cid:19) ∈ U(2 n ) . In the rest of this subsection, we restrict ourselves to the case of Gr(2 , u , u , u , u ) = ( u (3)2 , u (2)1 , u (2)2 , u (1)1 ) for simplicity. Figure 2.2 shows the projection∆ −→ [ − λ, λ ] , u = ( u , u , u , u ) u . The non-smooth locus of Φ is the inverse image of the edge of ∆ defined by u = · · · = u .The fiber L t over ( t, t, t, t ) ∈ ∂ ∆ is a Lagrangian submanifold consists of 2 λZZ ∗ − λI n with Z = 1 √ λ (cid:18) √ λ + tI √ λ − tA (cid:19) mod U(2)for A ∈ U(2). We identify U(2) with U(1) × SU(2) ∼ = S × S byU(1) × SU(2) −→ U(2) , (cid:18) a , (cid:18) a − a a a (cid:19)(cid:19) (cid:18) a
00 1 (cid:19) (cid:18) a − a a a (cid:19) . Then the image of L t under the Pl¨ucker embedding ι : Gr(2 , → P ( V C ) ∼ = P is given by ι ( L t ) = ((cid:20)r λ + tλ − t : − a a : a : − a a : − a : r λ − tλ + t a (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | a | = | a | + | a | = 1 ) . This expression implies the following.
Proposition 2.9.
For each t ∈ ( − λ, λ ) , we define an anti-holomorphic involution τ t on Gr(2 , defined by τ t ([ Z : Z : Z : Z : Z : Z ]) = (cid:20) λ + tλ − t Z : Z : − Z : − Z : Z : λ − tλ + t Z (cid:21) (2.7) Then L t is the fixed point set of τ t . Remark 2.10.
The map τ for t = 0 is an anti-symplectic involution as well, and satisfies τ ( L t ) = L − t for each t ∈ ( − λ, λ ). 11 .5 The case of Gr(2 , We fix λ > ,
5) with the adjoint orbit O λ of diag( λ, λ, , , ∈ √− u (5).The Gelfand-Cetlin polytope ∆ is defined by λ u ≥ ≥ ≥ u u ≥ ≥ ≥ ≥ u u ≥ ≥ u (2.8)We first consider the fiber L ( s , s , t ) over a boundary point given by λ s > > > s t > = = > t t = = t . Proposition 2.11.
The fiber L ( s , s , t ) is a Lagrangian submanifold diffeomorphic to U(2) × T ∼ = S × T . Moreover, L ( s , s , t ) is K -homogeneous for K = P e √− θ e √− θ ∈ U(5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P ∈ U(2) , θ , θ ∈ R ∼ = U(2) × T . Proof.
Note that O λ consists of matrices of the form x = λZZ ∗ = λ ( z i z j + w i w j ) ≤ i,j ≤ (2.9)for Z = z w z w z w z w z w ∈ V (2 , , i.e., X i =1 | z i | = X i =1 | w i | = 1 , X i =1 z i w i = 0 . (2.10)12ince the upper-left 2 × x = λ ( z i z j + w i w j ) ∈ L ( s , s , t ) satisfies x (2) = λ (cid:18) | z | + | w | z z + w w z z + w w | z | + | w | (cid:19) = (cid:18) t t (cid:19) , (2.11)we have r λt (cid:18) z w z w (cid:19) ∈ U(2) , (2.12)and in particular, | z | + | z | = | w | + | w | = t/λ . Then the condition (2.10) implies | z | + | z | + | z | = ( λ − t ) /λ, (2.13) | w | + | w | + | w | = ( λ − t ) /λ, (2.14) z w + z w + z w = 0 . (2.15)On the other hand, the conditions tr x (3) = s + t , tr x (4) = λ + s imply | z | + | w | = ( s − t ) /λ, (2.16) | z | + | w | = ( λ − s + s − t ) /λ, (2.17) | z | + | w | = ( λ − s ) /λ. (2.18)After the right SU(2)-action on ( z, w ), we may assume that ( z , w ) = (cid:0)p ( λ − s ) /λ, (cid:1) . Then(2.13), (2.14), and (2.15) become | z | + | z | = ( s − t ) /λ, | w | + | w | = ( λ − t ) /λ,z w + z w = 0 , which mean that the 2 × z i , w i ) i =3 , has the form (cid:18) z w z w (cid:19) = (cid:18)p ( s − t ) /λ a − p ( λ − t ) /λ bc p ( s − t ) /λ b p ( λ − t ) /λ ac (cid:19) for some (cid:18) a − bb a (cid:19) ∈ SU(2) , c ∈ U(1) . Combining this with (2.16) and (2.17) we have | a | = λ − s λ − s , | b | = s − s λ − s , and hence (cid:18) z w z w (cid:19) = 1 p λ ( λ − s ) (cid:18) p ( s − t )( λ − s ) e √− θ − p ( λ − t )( s − s ) e −√− θ c p ( s − t )( s − s ) e √− θ p ( λ − t )( λ − s ) e −√− θ c (cid:19) for some θ , θ ∈ R . After the action of (cid:26) (cid:18) e √− ϕ (cid:19) ∈ U(2) (cid:12)(cid:12)(cid:12)(cid:12) ϕ ∈ R (cid:27) ∼ = U(1)13rom the right, we may assume that (cid:18) z w z w (cid:19) = 1 p λ ( λ − s ) (cid:18) p ( s − t )( λ − s ) e √− θ − p ( λ − t )( s − s ) e √− θ p ( s − t )( s − s ) e √− θ p ( λ − t )( λ − s ) e √− θ (cid:19) . Therefore Z = ( z i , w i ) i is normalized as z w ... ... z w = z w z w p ( s − t )( λ − s ) /λ ( λ − s ) e √− θ − p ( λ − t )( s − s ) /λ ( λ − s ) e √− θ p ( s − t )( s − s ) /λ ( λ − s ) e √− θ p ( λ − t )( λ − s ) /λ ( λ − s ) e √− θ p ( λ − s ) /λ with (2.12), which implies that L ( s , s , t ) is a K -orbit and diffeomorphic to U(2) × T .The assertion that L ( s , s , t ) is Lagrangian follows from the K -homogeneity as in thecases of Fl(3) and Gr( n, n ).Next we consider the fiber L ( s , s , t ) over λ t > = = t t = = > > t s > > s . Suppose that x = λ ( z i z j + w i w j ) ≤ i,j ≤ ∈ L ( s , s , t ). The condition that x (3) = λ ( z i z j + w i w j ) ≤ i,j ≤ has eigenvalues t, t, | z | + | z | + | z | = t/λ, (2.19) | w | + | w | + | w | = t/λ, (2.20) z w + z w + z w = 0 , (2.21)and hence r λλ − t (cid:18) z w z w (cid:19) ∈ U (2) . On the other hand, the conditions x (1) = s , tr x (2) = t + s , and tr x (3) = 2 t imply | z | + | w | = s /λ, | z | + | w | = ( t − s + s ) /λ, | z | + | w | = ( t − s ) /λ. Then we have the following. 14 roposition 2.12.
The fiber L ( s , s , t ) is a U(2) × T -homogeneous Lagrangian submanifolddiffeomorphic to U(2) × T ∼ = S × T . Moreover, the fibers L ( s , s , t ) and L ( s , s , t ) satisfy g ( L ( s , s , t )) = L ( λ − s , λ − s , λ − t ) for g = ··· ∈ U(5) . In particular, L ( s , s , t ) and L ( s , s , t ) are displaceable. The Hamiltonian isotopy invariance of the Floer cohomology over the Novikov field [FOOO09,Theorem G] implies the following.
Corollary 2.13.
For i = 1 , , we have HF (( L i ( s , s , t ) , b ) , ( L i ( s , s , t ) , b ); Λ) = 0 for any weak bounding cochain b . Remark 2.14.
Other boundary fibers have lower dimensions. For example, the fiber over λ t > = = t t = = = > t t = = t consists of p t/λ p t/λ z w z w mod U(2)with (cid:18) z w z w (cid:19) ∈ p ( λ − t ) /λ U(2) , which means that the fiber is diffeomorphic to U(2). Let Φ : F = F ( n , . . . , n r , n ) → ∆ be the Gelfand-Cetlin system on the flag manifold, and { θ ( k ) i } ( i,k ) ∈ I be the angle variables dual to the action variables { λ ( k ) i } ( i,k ) ∈ I . For each u =( u ( i ) k ) ( i,k ) ∈ I ∈ Int ∆, we identify H ( L ( u ); Λ ) with Λ N by b = X ( i,k ) ∈ I x ( k ) i dθ ( k ) i ∈ H ( L ( u ); Λ ) ←→ x = ( x ( k ) i ) ( i,k ) ∈ I ∈ Λ N , y ( k ) i = e x ( k ) i T u ( k ) i , ( i, k ) ∈ I,Q j = T λ nj , j = 1 , . . . , r + 1 . Theorem 3.1 ([NNU10, Theorem 10.1]) . For any interior point u ∈ Int ∆ , we have aninclusion H ( L ( u ); Λ ) ⊂ c M weak ( L ( u )) . As a function on [ u ∈ Int ∆ H ( L ( u ); Λ ) ∼ = Int ∆ × Λ N , the potential function is given by PO ( u , x ) = X ( i,k ) ∈ I y ( k +1) i y ( k ) i + y ( k ) i y ( k +1) i +1 ! , where we put y ( k +1) i = Q j if λ ( k +1) i = λ n j is a constant function. Example 3.2.
We identify the 3-dimensional flag manifold Fl(3) with the adjoint orbit of λ = diag( λ , λ , λ ). The potential function is given by PO = e − x T − u + λ + e x T u − λ + e − x T − u + λ + e x T u − λ + e x − x T u − u + e − x + x T − u + u = Q y + y Q + Q y + y Q + y y + y y . The potential function PO has six critical points given by y = y /y ,y = ± p Q ( y + Q ) ,y = p Q Q Q , e π √− / p Q Q Q , e π √− / p Q Q Q . It is easy to see that all critical points are non-degenerate and have the same valuation whichlies in the interior of the Gelfand-Cetlin polytope. Hence we have as many critical pointsas dim H ∗ (Fl(3)) = 6 in this case. One can show, using the presentation of the quantumcohomology in [GK95, Theorem 1], that the set of eigenvalues of the quantum cup product by c (Fl(3)) coincides with the set of critical values of the potential function.The Floer differential m b is trivial for each critical point ( u , x ) of PO , and the correspond-ing Floer cohomology is given by HF (( L ( u ) , b ) , ( L ( u ) , b ); Λ ) ∼ = H ∗ ( L ( u ); Λ ) ∼ = H ∗ ( T ; Λ ) . Example 3.3.
We identify Gr(2 ,
4) with the adjoint orbit of diag(2 λ, λ, , Q = T λ , the potential function is given by PO = e − x T − u +2 λ + e − x + x T − u + u + e x − x T u − u + e x T u + e x − x T u − u + e − x + x T − u + u = Qy + y y + y y + y + y y + y y . (3.1)16his function has four critical points( y , y , y , y ) = ( − i p Q , √− i r Q , √− i p Q, ( − i p Q ! for i = 0 , , ,
3, and the corresponding critical values are PO = 4 √ √− i p Q. (3.2)Since dim H ∗ (Gr(2 , H ∗ (Gr(2 , u = ( λ, λ/ , λ/ , λ ) ∈ Int ∆ . Hence there exist four weak bounding cochains b , . . . , b such that HF (( L ( u ) , b i ) , ( L ( u ) , b i ); Λ ) ∼ = H ∗ ( L ( u ); Λ ) ∼ = H ∗ ( T ; Λ )for i = 0 , , ,
3. One can show, using the presentation of the quantum cohomology in [ST97,Theorem 0.1], that the set eigenvalues of the quantum cup product by c (Gr(2 , Example 3.4.
We identify Gr(2 ,
5) with the adjoint orbit of diag( λ, λ, , , PO = Qy + y y + y y + y y + y y + y y + y + y y + y y . (3.3)This function has ten critical points defined by y = Q , Qy = y ( y − y ) , and y = Qy , y = Qy , y = Qy , y = y y . The set (cid:8) ζ i + ζ j ) Q / (cid:12)(cid:12) ζ = exp(2 π √− /
5) and 0 ≤ i < j ≤ (cid:9) (3.4)of critical values of the potential function coincides with the set of eigenvalues of the quantumcup product by c (Gr(2 , We briefly recall the construction of the A ∞ structure { m k } k ≥ , omitting various technicaldetails. Let L be a spin, oriented, and compact Lagrangian submanifold in a symplecticmanifold ( X, ω ). For an almost complex structure J compatible with ω , let M k +1 ( J, β ) be themoduli space of stable J -holomorphic maps v : (Σ , ∂ Σ) → ( X, L ) from a bordered Riemannsurface Σ in the class β ∈ π ( X, L ) of genus zero with ( k + 1) boundary marked points z , z , . . . , z k ∈ ∂ Σ. Then m k = P β ∈ π ( X,L ) T β ∩ ω m k,β : H ∗ ( L ; Λ ) ⊗ k → H ∗ ( L ; Λ ) is defined by m k,β ( x , . . . , x k ) = (ev ) ∗ (ev ∗ x ∪ · · · ∪ ev ∗ k x k ) , where ev i : M k +1 ( J, β ) → L , [ v, ( z , . . . , z k )] v ( z i ) is the evaluation map at the i th markedpoint. 17 .1 Holomorphic disks in (Fl(3) , L ) We identify Fl(3) with the adjoint orbit of diag( λ , , − λ ) for λ , λ > ω = λ ω P + λ ω P and c (Fl(3)) = 2( ω P + ω P ), respectively.Recall that the homotopy group π (Fl(3)) ∼ = Z is generated by 1-dimensional Schubertvarieties X and X , which are rational curves of bidegree (1 ,
0) and (0 ,
1) in P × P ∼ = P × P ,respectively. Since L is diffeomorphic to SU(2) ∼ = S , we have π ( L ) = π ( L ) = 0. Thelong exact sequence of homotopy groups yields π (Fl(3) , L ) ∼ = π (Fl(3)) ∼ = Z . Let β , β be generators of π (Fl(3) , L ) corresponding to X and X , respectively. Thesymplectic area of β i is given by β i ∩ ω = [ X i ] ∩ ( λ ω P + λ ω P ) = λ i . Let τ be the anti-holomorphic involution on Fl(3) defined in (2.5). For a holomorphic disk v : ( D , ∂D ) → (Fl(3) , L ), we define a new holomorphic disk τ ∗ v : ( D , ∂D ) → (Fl(3) , L )by τ ∗ v ( z ) = τ ( v ( z )) . Since L is the fixed point set of τ , one can glue v and τ ∗ v along the boundary to obtain aholomorphic curve w = v τ ∗ v : P → Fl(3). The induced involution on π (Fl(3) , L ), whichis also denoted by τ ∗ , is given by τ ∗ β = β . If v represents β or β , then [ w ] = β + β =[ X ] + [ X ], i.e., w is a rational curve of bidegree (1 , µ L : π (Fl(3) , L ) → Z be the Maslov index. If we assume λ = λ so that τ is ananti-symplectic involution, then we have µ L ( β i ) = 12 ( µ L ( β i ) + µ L ( τ ∗ β i )) = ([ X ] + [ X ]) ∩ c (Fl(3)) = 4for i = 1 ,
2. Since the symplectic form ω and the Lagrangian submanifold L depend contin-uously on λ , λ >
0, the Maslov index µ L ( β ) = µ L ( β ) = 4 is independent of λ , λ .To describe holomorphic disks with Lagrangian boundary condition, we identify the unitdisk D with the upper half plane H = H + . Proposition 4.1.
Let w : P → Fl(3) be a holomorphic curve of bidegree (1 , such that w ( R ∪ {∞} ) ⊂ L . After the SU(2) -action, we may assume w ( ∞ ) = ([1 : 0 : p λ /λ ] , [1 : 0 : − p λ /λ ]) . (4.1) We can write w (0) = (cid:16)(cid:2) a : a : p λ /λ (cid:3) , (cid:2) a : a : − p λ /λ (cid:3)(cid:17) ∈ L (4.2) for some ( a , a ) ∈ S \ { (1 , } . Then w is given by w ( z ) = (cid:16)(cid:2) cz + a : a : p λ /λ ( cz + 1) (cid:3) , (cid:2) cz + a : a : − p λ /λ ( cz + 1) (cid:3)(cid:17) with c/c = − ( a − / ( a − . emark 4.2. After the action of { g ∈ PSL(2 , R ) | g (0) = 0 , g ( ∞ ) = ∞} ∼ = R > on H , we may assume that | c | = 1. Proof.
The assumptions (4.1) and (4.2) implies that w has the form w ( z ) = (cid:16)(cid:2) c z + a : a : p λ /λ ( c z + 1) (cid:3) , (cid:2) c z + a : a : − p λ /λ ( c z + 1) (cid:3)(cid:17) for some c , c ∈ C ∗ . The Pl¨ucker relation0 = − ( c z + a )( c z + a ) − | a | + ( c z + 1)( c z + 1)= ( c − a c + c − a c ) z implies c ( a −
1) + c ( a −
1) = 0. On the other hand, the Lagrangian boundary condition w ( R ) ⊂ L implies that c x + a c x + 1 = c x + a c x + 1 , a c x + 1 = a c x + 1 , x ∈ R , which means c = c .Note that arg c is determined by a up to sign, and the sign corresponds to whether v = w | H represents β or β . Namely any holomorphic disk in the class β i satisfying (4.1) and (4.2) isuniquely determined by ( a , a ) for i = 1 , Example 4.3.
Suppose that ( a , a ) = ( − , c = ±√−
1, and the correspondingholomorphic disks are given by v ± ( z ) = (cid:16)(cid:2) z ± √− r λ λ ( z ∓ √− (cid:3) , (cid:2) z ∓ √− − r λ λ ( z ± √− (cid:3)(cid:17) . It is easy to see that the image v + ( H ) (resp. v − ( H )) is the inverse image of the edge of ∆ givenby u (1)1 = u (2)1 and u (2)2 = 0 (resp. u (1)1 = u (2)2 and u (2)1 = 0), which is the upper (resp. lower)vertical edge emanating from the vertex = (0 , , β , β of π (Fl(3) , L )are represented by v + and v − respectively. SU(2) -fiber in
Fl(3)
Let J be the standard complex structure on Fl(3). Since the fiber L is SU(2)-homogeneous,[EL, Proposition 3.2.1] implies the following. Proposition 4.4.
Any J -holomorphic disk in (Fl(3) , L ) is Fredholm regular. Hence themoduli space M reg k +1 ( J, β ) of J -holomorphic disks in the class β with k + 1 boundary markedpoints is a smooth manifold of dimension dim M reg k +1 ( J, β ) = dim L + µ L ( β ) + k + 1 − µ L ( β ) + k + 1 . In particular, we have dim M ( J, β i ) = 6 for i = 1 ,
2. Proposition 4.1 implies the following:19 orollary 4.5.
Let U = SU(2) \ { } ∼ = { ( a , a ) ∈ S | a = 1 } . Then M ( J, β i ) has an opendense subset diffeomorphic to SU(2) × U on which the evaluation map is given by SU(2) × U −→ L × L ∼ = SU(2) × SU(2) , ( g , g ) ( g , g g ) . In particular, ev : M ( J, β i ) → L × L is generically one-to-one. Since the minimal Maslov number is µ L ( β ) = µ L ( β ) = 4 anddeg m ,β ( x ) = deg x + 1 − µ L ( β ) , x ∈ H ∗ ( L ; Λ ) , the only nontrivial parts of the Floer differential are m ,β i : H ( L ) ∼ = H ( L ) −→ H ( L ) ∼ = H ( L )for i = 1 ,
2. Corollary 4.5 implies that for the class [ p ] ∈ H ( L ) of a point, we have m ,β i ([ p ]) = ev ∗ [ M ( J, β i ) ev × { p } ] = ± [ L ] . To see the sign, we use a result on the orientation of the moduli spaces of pseudo-holomorphicdisks by Fukaya, Oh, Ohta, and Ono [FOOO, Theorem 1.5]. The following statement is aslightly weaker version of the result, which is sufficient for our purpose.
Theorem 4.6.
Let ( X, ω ) be a compact symplectic manifold, and τ an anti-symplectic invo-lution on X whose fixed point set L = Fix( τ ) is non-empty, compact, connected, and spin.Then m k,β and m k,τ ∗ β satisfy m k,β ( P , . . . , P k ) = ( − ǫ m k,τ ∗ β ( P k , . . . , P ) , where ǫ = µ L ( β )2 + k + 1 + X ≤ i We have m ,β = m ,β for general λ , λ > .Proof. If λ = λ , then τ is anti-symplectic, and thus Theorem 4.6 implies m ,β = ( − µ L ( β ) / m ,τ ∗ β = m ,β . (4.3)Corollary 4.5 implies that M ( J, β i ) depends continuously on λ , λ , and hence its orientationis independent of λ , λ . Thus (4.3) holds for general λ , λ .Then we have m ([ p ]) = X i =1 m ,β i ([ p ]) T β i ∩ ω = ± ( T λ + T λ )[ L ] , which implies the following. Theorem 4.8. The Floer cohomology of L over the Novikov ring Λ is HF ( L , L ; Λ ) ∼ = Λ /T min { λ ,λ } Λ . Theorem 1.1 is an immediate consequence of Theorem 4.8.20 .3 Holomorphic disks in (Gr(2 , , L t ) We identify Gr(2 , 4) with the adjoint orbit of diag( λ, λ, − λ, − λ ) for λ > 0. Note that theKostant-Kirillov form and the first Chern class are given by ω = 2 λω FS , c (Gr(2 , ω FS , respectively, where ω FS is the Fubini-Study form on P ( V C ).Recall that π (Gr(2 , ∼ = Z is generated by a 1-dimensional Schubert variety X , which isa rational curve of degree one in P ( V C ). Since π (Gr(2 , π ( L t ) = 0 and π ( L t ) ∼ = Z ,the exact sequence0 −→ π (Gr(2 , −→ π (Gr(2 , , L t ) −→ π ( L t ) −→ π (Gr(2 , , L t ) ∼ = Z . Let β , β be generators of π (Gr(2 , , L t ) such that β + β = [ X ] ∈ π (Gr(2 , Example 4.9. Consider a holomorphic curve w : P → Gr(2 , 4) of degree one defined by w ( z ) = "r λ + tλ − t ( z − √− 1) : 0 : z − √− − z − √− r λ − tλ + t ( z + √− . (4.4)Since w maps R ∪ {∞} to L t , the restrictions v + = w | H + : ( H + , ∂ H + ) −→ (Gr(2 , , L t ) ,v − = w | H − : ( H − , ∂ H − ) −→ (Gr(2 , , L t )to the upper and lower half planes give holomorphic disks representing β and β . We define β = [ v + ] and β = [ v − ]. It is easy to see that the symplectic areas of v ± are given by ω ( β ) = Z H + v ∗ + ω = λ + t, ω ( β ) = Z H − v ∗− ω = λ − t. In the case where t = 0, the disk v + sends √− ∈ H to v + ( √− 1) = [0 : 0 : 0 : − − ( u ) over the point u ∈ ∆ defined by u (2)1 = u (1)1 = λ and u (2)2 = 0(see Figure 2.2). On the other hand, v − ( −√− 1) = [1 : 0 : 1 : 0 : 0 : 0] lies on the fiber overthe point u ∈ ∆ defined by u (2)2 = u (1)1 = − λ and u (2)1 = 0.Let τ t be the anti-holomorphic involution on Gr(2 , 4) defined in (2.7). Note that ( τ t ) ∗ is given by ( τ t ) ∗ v ( z ) = τ t ( v ( − z )) for v : ( H , ∂ H ) → (Gr(2 , , L t ). Since ( τ t ) ∗ v + = v − , theinduced involution on π (Gr(2 , , L t ) is given by ( τ t ) ∗ β = β . Then the Maslov index of β i is given by µ L t ( β i ) = 12 ( µ L t ( β i ) + µ L t (( τ t ) ∗ β i )) = [ X ] ∩ c (Gr(2 , i = 1 , w : P → Gr( n, n ) of degree one such that w ( R ∪ {∞} ) iscontained in the Lagrangian fiber L t . Proposition 4.10 below is taken from [Sot01, Theorem2.1], which is well-known in control theory (cf. e.g. [Ros70]).21 roposition 4.10. Suppose that a holomorphic curve w : P → Gr( k, n ) = e V ( k, n ) / GL( k, C ) of degree d is given by w : z (cid:18) I k F ( z ) (cid:19) mod GL( k, C ) for a rational function F ( z ) with values in ( n − k ) × K matrices. Then there exist matrixvalued polynomials P ( z ) , Q ( z ) of size k × k and ( n − k ) × k respectively such that1. F ( z ) = Q ( z ) P ( z ) − , i.e., the curve w is given by w : z (cid:18) P ( z ) Q ( z ) (cid:19) mod GL( k, C ) , P ( z ) and Q ( z ) are coprime in the sense there exist matrix valued polynomials X ( z ) , Y ( z ) such that X ( z ) P ( z ) + Y ( z ) Q ( z ) = I k , and3. deg(det P ( z )) = d .Such P ( z ) and Q ( z ) are unique up to multiplication of elements in GL( k, C [ z ]) . Note that (2.6) implies that the U( n )-fiber L t ⊂ Gr( n, n ) = e V ( n, n ) / GL( n, C ) consistsof (cid:18) I n p ( λ − t ) / ( λ + t ) A (cid:19) mod GL( n, C )for A ∈ U( n ). Proposition 4.11. Let w : P → Gr( n, n ) be a holomorphic curve of degree one such that w ( R ∪ {∞} ) ⊂ L t , and let F ( z ) denote the corresponding rational function with values in n × n matrices. By the U( n ) -action, we assume that F ( ∞ ) = r λ − tλ + t I n ∈ r λ − tλ + t U( n ) , (4.5) and set F (0) = r λ − tλ + t A (4.6) for A ∈ U( n ) . Then there exist a = a ... a n ∈ S n − /S = P n − and c ∈ C \ R such that A = I n + (cid:18) c | c | − (cid:19) aa ∗ , and F ( z ) = r λ − tλ + t z − c ( zI n − cA ) = r λ − tλ + t (cid:18) I n − c − cz − c aa ∗ (cid:19) . (4.7)22 roof. Let F ( z ) = Q ( z ) P ( z ) − be the factorization given in Proposition 4.10. Then theassumptions (4.5), (4.6), and deg(det P ( z )) = 1 imply that F ( z ) has the form F ( z ) = r λ − tλ + t z − c ( zI n − cA )for some c ∈ C . The Lagrangian boundary condition w ( R ∪ {∞} ) ⊂ L t implies that1 x − c ( xI n − cA ) ∈ U( n )for any x ∈ R , which means cA + cA ∗ = ( c + c ) I n , or equivalently, cA − Re( c ) I n is skew-hermitian. Hence cA − Re( c ) I n has pure imaginary eigenvalues √− α , . . . , √− α n , and canbe diagonalized by some g ∈ U( n ); g ∗ ( cA − Re( c ) I n ) g = diag( √− α , . . . , √− α n ) . Since g ∗ Ag = diag (cid:18) Re( c ) + √− α c , . . . , Re( c ) + √− α n c (cid:19) ∈ U( n )has eigenvalues of unit norm, we have α i = ± Im( c ) for i = 1 , . . . , n . After the action of apermutation matrix, we may assume that g ∗ Ag has the form g ∗ Ag = diag( c/c, . . . , c/c | {z } k , , . . . , | {z } n − k ) =: C (4.8)for some k . Then F ( z ) is given by F ( z ) = r λ − tλ + t z − c g ( zI n − cC ) g ∗ = r λ − tλ + t g diag (cid:18) z − cz − c , . . . , z − cz − c , , . . . , (cid:19) g ∗ In particular, we have det F ( z ) = (cid:18) λ − tλ + t (cid:19) n/ (cid:18) z − cz − c (cid:19) k . The condition deg(det P ( z )) = 1 implies that k = 1, i.e., C = diag( c/c, , . . . , 1) = ( c/c − E + I n , where E = diag(1 , , . . . , ∈ gl ( n, C ). Let a ∈ S n − ⊂ C n be the first column of g . Thenwe have A = g (cid:18)(cid:18) c | c | − (cid:19) E + I n (cid:19) g ∗ = (cid:18) c | c | − (cid:19) aa ∗ + I n , which proves the proposition. Remark 4.12. 1. The equation (4.8) (with k = 1) implies that det A = c/c = c / | c | .2. After the R > -action on the domain, we may assume that | c | = 1.We now assume that n = 2. The sign of Im( c ) = Im √ det A corresponds to the homotopyclass of the holomorphic disk v = w | H . The curve w corresponding to a = [1 : 0] and c = −√− w | H = v + represents β . Thus v = w | H represents β (resp. β ) when Im( c ) = Im √ det A < c ) > .4 Floer cohomologies of the U(2) -fibers in Gr(2 , Since the minimal Maslov number of the U (2)-fiber L t is µ L t ( β i ) = 4, we have the followingby degree reason. Lemma 4.13. The potential function PO : H ( L t ; Λ ) → Λ for L t is trivial: PO ≡ . The cohomology of L t ∼ = S × S is given by H ∗ ( L t ) ∼ = H ∗ ( S ) ⊗ H ∗ ( S ) . Let e ∈ H ( L t ; Z ) ∼ = H ( S ; Z ) and e ∈ H ( L t ; Z ) ∼ = H ( S ; Z ) be the generators, and write b = x e ∈ H ( L t ; Λ ). Since deg m b ,β = 1 − µ L t ( β ) and the minimal Maslov number is four,the only nontrivial parts of the Floer differential m b are m b ,β i : H ( L t ) ∼ = H ( S ) ⊗ H ( S ) −→ H ( L t ) ∼ = H ( S ) , m b ,β i : H ( L t ) ∼ = H ( S ) −→ H ( L t ) ∼ = Λ for i = 1 , , , L t ) is U(2)-homogeneous, any J -holomorphic disk is Fredholm regular forthe standard complex structure J by [EL, Proposition 3.2.1]. Hence one has dim M ( J, β i ) = 7for i = 1 , 2. Now Proposition 4.11 implies the following: Corollary 4.14. Define f : (0 , π ) × P → U(2) by f ( θ, a ) = ( e √− θ − aa ∗ + I . For i = 1 , ,the moduli space M ( J, β i ) has an open dense subset diffeomorphic to U(2) × (0 , π ) × P suchthat the evaluation map is given by U(2) × (0 , π ) × P −→ L t × L t ∼ = U(2) × U(2) , ( g, θ, a ) ( g, g · f ( θ, a )) . Note that e √− θ = det f ( θ, a ) is related to c ∈ S in Proposition 4.11 by c = exp( √− θ/ π )) or c = exp( √− θ/ 2) corresponding to i = 1 , M k + l +2 ( J, β i ). For a rational curve w : P → Gr(2 , 4) given by (4.7), thecomposition det ◦ w | ∂ H : ∂ H = R → L t ∼ = U(2) → S is given by x x − cx − c . Hence each boundary point x ∈ ∂ H is determined by the argument of det w ( x ) = ( x − c ) / ( x − c ).Fixing the 0-th and ( k + 1)-st boundary marked points, we have the following. Corollary 4.15. The moduli space M k + l +2 ( J, β i ) has an open dense subset diffeomorphic to (cid:26) ( g, θ, a, ( t i ) , ( s j )) ∈ U(2) × (0 , π ) × P × R k × R l (cid:12)(cid:12)(cid:12)(cid:12) < t < · · · < t k < θ,θ < s < · · · < s l < π (cid:27) on which the evaluation maps ev : M k + l +2 ( J, β i ) → L t ∼ = U(2) satisfy (ev , ev k +1 ) : ( g, θ, a, ( t i ) , ( s j )) ( g, g · f ( θ, a )) and det ev i ( g, θ, a, ( t i ) , ( s j )) = e √− t i det g, i = 1 , . . . , k,e √− θ det g, i = k + 1 ,e √− s i − k − det g, i = k + 2 , . . . , k + l + 2 . heorem 4.16. For b = x e ∈ H ( L ; Λ / π √− Z ) ∼ = Λ / π √− Z , the deformed Floerdifferential m b is given by m b ( e ) = e x T λ + t + e − x T λ − t , (4.9) m b ( e ⊗ e ) = ( e x T λ + t + e − x T λ − t ) e . (4.10) Hence the Floer cohomology of ( L t , b ) is HF (( L t , b ) , ( L t , b ); Λ ) ∼ = ( H ∗ ( L ; Λ ) if t = 0 and x = ± π √− / , (Λ /T min { λ − t,λ + t } Λ ) otherwise . The Floer cohomology over the Novikov field is given by HF (( L t , b ) , ( L t , b ); Λ) ∼ = ( H ∗ ( L ; Λ) if t = 0 and x = ± π √− / , otherwise . Recall that e , e ∈ H ∗ (U(2)) are given by e = 12 π √− g − dg ) = 12 π √− d log(det g ) , e = 124 π tr (cid:2) ( g − dg ) (cid:3) , where g − dg is the left-invariant Maurer-Cartan form on U(2). Lemma 4.17. For f ( θ, a ) = ( e √− θ − aa ∗ + I , we have f ∗ e = 12 π tr( f − df ) = dθ π , (4.11) f ∗ e = 124 π tr( f − df ) = (1 − cos θ ) dθ π ∧ ω P , (4.12) where ω P is the Fubini-Study form on P normalized in such a way that Z P ω P = 1 . Proof. The first assertion (4.11) follows from det f = e √− θ . Since f is SU(2)-equivariant withrespect to the natural action on P and the adjoint action on U(2), it suffices to show (4.12)at a = [1 : 0] ∈ P . A direct calculation gives f − df = (cid:18) √− dθ − ( e −√− θ − da ( e √− θ − da (cid:19) , so that tr( f − df ) = 3(2 − e √− θ − e −√− θ ) √− dθ ∧ da ∧ da at a = [1 : 0]. On the other hand, the Fubini-Study form on P is given by ω P = √− π da ∧ da at a = [1 : 0], which proves (4.12). 25 roof of Theorem 4.16. Note that for m : U(2) × U(2) → U(2), ( g , g ) g g , we have m ∗ e i = π ∗ e i + π ∗ e i for i = 1 , 3, where π , π : U(2) × U(2) → U(2) are the projections to thefirst and the second factors. Then ev ∗ j e i are given byev ∗ i e = 12 π dt i + g ∗ e , i = 1 , . . . , k, ev ∗ k +1+ i e = 12 π dt i + g ∗ e , i = 1 , . . . , l, ev ∗ k +1 e = f ∗ e + g ∗ e = (1 − cos θ ) dθ π ∧ ω P + g ∗ e , where g ∗ e i is the pull-back of e i by the projectionU(2) × (0 , π ) × P −→ U(2) , ( g, θ, a ) g to the first factor. For θ ∈ (0 , π ), set D ( θ ) = { ( t , . . . , t k ) ∈ R k | < t < · · · < t k < θ } ,D ( θ ) = { ( s , . . . , s l ) ∈ R l | θ < s < · · · < s l < π } . Taking a suitable orientation on M k + l +2 ( β , J ), we have from Corollary 4.15 that m k + l +1 ,β ( b, . . . , b | {z } k , e , b, . . . , b | {z } l )= Z (0 , π ) × P (cid:18)Z D ( θ ) (cid:16) x π (cid:17) k dt ∧ · · · ∧ dt k (cid:19) (cid:18)Z D ( θ ) (cid:16) x π (cid:17) l ds ∧ · · · ∧ ds l (cid:19) (1 − cos θ ) dθ π ∧ ω P = Z (0 , π ) k ! (cid:18) θ π · x (cid:19) k l ! (cid:18)(cid:18) − θ π (cid:19) x (cid:19) l (1 − cos θ ) dθ π . (4.13)Hence m b ,β ( e ) = Z π X k,l ≥ k ! (cid:18) θ π · x (cid:19) k l ! (cid:18)(cid:18) − θ π (cid:19) x (cid:19) l (1 − cos θ ) dθ π = Z π e ( θ/ π ) x e (1 − θ/ π ) x (1 − cos θ ) dθ π = Z π e x (1 − cos θ ) dθ π = e x . The same argument as the proof of Corollary 4.7 gives m k + l +1 ,β ( b, . . . , b | {z } k , e , b, . . . , b | {z } l ) = ( − k + l m k + l +1 ,β ( b, . . . , b | {z } l , e , b, . . . , b | {z } k )= m k + l +1 ,β ( − b, . . . , − b | {z } l , e , − b, . . . , − b | {z } k ) , so that m b ,β ( e ) = e − x . m b ( e ) = X i =1 m b ,β i ( e ) T β i ∩ ω = e x T λ + t + e − x T λ − t . Next we compute m b ( e ⊗ e ) ∈ H ( L ). Note thatev k +1 ( e ⊗ e ) = ( g ∗ e + f ∗ e ) ⊗ ( g ∗ e + f ∗ e ) = g ∗ e ⊗ f ∗ e + . . . . Since only the term g ∗ e ⊗ f ∗ e contribute to m k + l +1 ,β i ( b, . . . , b, e ⊗ e , b, . . . , b ) by degreereason, we have m k + l +1 ,β i ( b, . . . , b | {z } k , e ⊗ e , b, . . . , b | {z } l ) = m k + l +1 ,β i ( b, . . . , b | {z } k , e , b, . . . , b | {z } l ) g ∗ e . Hence we obtain m b ( e ⊗ e ) = X i =1 m b ,β i ( e ⊗ e ) T β i ∩ ω = X i =1 m b ,β i ( e ) T β i ∩ ω e = ( e x T λ + t + e − x T λ − t ) e . Remark 4.18. Iriyeh, Sakai, and Tasaki [IST13] computed Floer cohomologies HF ( L, L ′ ; Z / Z )of real forms in a compact Hermitian symmetric space, i.e., fixed point sets L = Fix( τ ), L ′ = Fix( τ ′ ) of anti-holomorphic and anti-symplectic involutions τ , τ ′ . In particular, theFloer cohomology of the U(2)-fiber L = Fix( τ ) with coefficients in Z / Z is given by HF ( L , L ; Z / Z ) ∼ = H ∗ ( L ; Z / Z ) ∼ = ( Z / Z ) . On the other hand, (4.9) and (4.10) implies that HF ( L , L ; Λ Z ) ∼ = (Λ Z / T λ Λ Z ) , where Λ Z = ( ∞ X i =1 a i T λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ Z , λ i ≥ , lim i →∞ λ i = ∞ ) is the Novikov ring over Z . Remark 4.19. Here we consider a Lagrangian U( n )-fiber L t in Gr( n, n ) for general n . Theone-parameter subgroup g θ = exp( θξ ) of U(2 n ) given by ξ = (cid:18) − E E (cid:19) ∈ u (2 n )sends x = t x . . . x n . . . ... ... t x n . . . x nn x . . . x n − t ... ... . . . x n . . . x nn − t ∈ L t 27o Ad g θ ( x ) ∈ O λ whose upper-left n × n block is given by(Ad g θ ( x )) ( n ) = t (1 − θ ) − ( x + x ) sin θ cos θ − x sin θ . . . − x n sin θ − x n sin θ t ... . . . − x n sin θ t . If Ad g θ ( x ) is still in L t , i.e., ( g θ xg ∗ θ ) ( n ) = tI n , then we have x = · · · = x n = 0 and Re x = − t tan θ . Since | Re x | ≤ √ λ − t , one has g θ ( L t ) ∩ L t = ∅ if | θ | > arctan r λ − t t . Note that the moment map µ : O λ → u (2 n ) of the U(2 n )-action is given by µ ( x ) = ( √− / π ) x in our setting. Hence the Hamiltonian of g θ is given by H ( x ) = √− π h x, ξ i . Since max O λ H = λ/π and min O λ H = − λ/π , the norm of g θ is given by Z θ (cid:16) max O λ H − min O λ H (cid:17) dθ = 2 λπ θ. Hence the displacement energy of L t is bounded from above by h ( t ) = 2 λπ arctan r λ − t t . Note that h ( t ) is a concave function on [ − λ, λ ] such that h ( ± λ ) = 0, h (0) = λ , and h ( t ) > min { λ − t, λ + t } for t = 0 , ± λ . Theorem 4.20. The Floer cohomology of the pair ( L , π √− / e ) , ( L , − π √− / e ) is givenby HF (( L , ± π √− / e ) , ( L , ∓ π √− / e ); Λ ) ∼ = (Λ /T λ Λ ) . In particular, the Floer cohomology over the Novikov field is trivial; HF (( L , ± π √− / e ) , ( L , ∓ π √− / e ); Λ) = 0 . Proof. For b = √− π/ e ∈ H ( L ; Λ ), we have from (4.13) that m k + l +1 ,β i ( b, . . . , b | {z } k , e , − b, . . . , − b | {z } l )= Z (0 , π ) k ! (cid:18) √− θ (cid:19) k l ! (cid:18) √− θ − π √− (cid:19) l (1 − cos θ ) dθ π . δ b, − b ( e ) = X i =1 , X k,l ≥ m k + l +1 ,β i ( b, . . . , b | {z } k , e , − b, . . . , − b | {z } l ) T β i ∩ ω = 2 T λ Z π X k,l ≥ k ! (cid:18) √− θ (cid:19) k l ! 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MR 2929086[Sot01] Frank Sottile, Rational curves on Grassmannians: systems theory, reality, andtransversality , Advances in algebraic geometry motivated by physics (Lowell,MA, 2000), Contemp. Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001,pp. 9–42. MR 1837108 (2002d:14080)[ST97] Bernd Siebert and Gang Tian, On quantum cohomology rings of Fano manifoldsand a formula of Vafa and Intriligator , Asian J. Math. (1997), no. 4, 679–695.MR 1621570 (99d:14060)Yuichi NoharaFaculty of Education, Kagawa University, Saiwai-cho 1-1, Takamatsu, Kagawa, 760-8522,Japan. e-mail address : [email protected] UedaDepartment of Mathematics, Graduate School of Science, Osaka University, Machikaneyama1-1, Toyonaka, Osaka, 560-0043, Japan. e-mail addresse-mail address