Computation of Quantum Cohomology from Fukaya Categories 2

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Theorem 1.1 (Theorem 4.10). Assume the existence of Fuk(X) and open-closed maps p, q. Let A be a subcategory of Fuk(X). Suppose that the Mukai pairing on HH (A ) is perfect. Then A • π there exists an idempotent e such that H•( ) split generates the derived category D Fuk(X)e. A +n Moreover, p induces an isomorphism HH ( ) ∼= QH• (X)e and q induces a ring isomorphism A • QH•(X)e ∼= HH•( ). Remark 1.2. A similar statement has independently been obtained by Ganatra [24].

1.2. Applications. As an application of the main result, we construct field factors of QH•(X) from the existence of certain Lagrangian submanifolds. We first consider an oriented Lagrangian torus with a spin structure L such that all non-constant holomorphic disks bounded by L have Maslov indices at least two (Condition 4.15). By counting holomorphic disks with Maslov index two, we can define a potential function WL. A critical point of WL gives a Maurer-Cartan element b such that the Floer cohomology HF •(L,b) is not equal to zero. Proposition 1.3 (Proposition 5.3 and Proposition 5.5). Let L be a torus with a spin structure which satisfies Condition 4.15. Let b be a Maurer-Cartan element∈ corresponding L to a non-degenerate critical point of WL. Then there exists an idempotent e with QH•(X)e ∼= Λ. Moreover, if e′ is an idempotent corresponding to another non-degenerate critical point, then the quantum product e e′ is equal to zero. ∗ We next consider an even dimensional oriented rational homology Lagrangian sphere with a spin structure. We note that such a Lagrangian is weakly unobstructed and 0 is a unique Maurer-Cartan element [17]. Proposition 1.4 (Proposition 5.8). Let L be an even dimensional oriented spin rational homol- ∈ L ogy sphere with the Maurer-Cartan element b =0. Suppose that HF •(L, 0) = Λ Λ as a ring. Then + ∼ × there exist idempotents e± such that QH•(X)e± = Λ and e e− =0. Moreover these idempotents L L ∼ L ∗ L are eigenvectors of the operator c1 with the same eigenvalue, here c1 denotes the first Chern class of the Tangent bundle of X. ∗ 1.3. Examples. As a first example, we consider a compact toric Fano manidold X. In [33], Ritter has proved that the torus fibers of X split generate the Fukaya category of X if the potential functions of the torus fibers are Morse. We give another proof of this statement (see Theorem 6.1). As a next example we consider a blow-up of CP 2 at four points with a monotone symplectic structure. By a result of Li-Wu [30], we can construct many Lagrangian spheres. Moreover, Pascaleff- Tonkonog [31] construct a monotone Lagrangian torus and compute its potential function. Using these results, we prove the following: Theorem 1.5 (Theorem 6.2). Let X be a blow-up of CP 2 at four points with a monotone symplectic structure. Then there exist four Lagrangian spheres and two monotone Lagrangian tori with Maurer Cartan elements such that these objects split generate the Fukaya category. Moreover QH•(X) is semi-simple. Remark 1.6. Semi-simplicity of the quantum cohomology has already been known ([5], [12]). But our proof does not use explicit computation of Gromov-Witten invariants. 1.4. Plan of the paper. In 2, we recall some definitions on A categories and Hochschild invariants. Moreover, we show a§ split generation criterion for cyclic ∞A categories. In 3, we review Lagrangian intersection Floer theory. In 4, we give some assumptions∞ on Fukaya categories§ and open-closed maps. Using these assumptions,§ we show the main result. In 5, we construct field factors of the quantum cohomology by using Floer theory of Lagrangian submanifolds.§ In 6, we compute the Fukaya category and the quantum cohomology of a blow-up of CP 2 at four points.§ COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 3

1.5. Acknowledgements. The author would like to express his gratitude to his supervisor Profes- sor Shinobu Hosono. Some parts of this paper is a extended version of the author’s Ph.D. thesis. The author would like to thank Yuuki Shiraishi for explaining to the author the importance of Shklyarov’s pairing, Sheel Ganatra for informing the author that he also has obtained a version of the main result after the authors talk at Kyoto([34]). The author also would like to thank Professor Hiroshi Ohta, Professor Kaoru Ono, Daiske Inoue, Tatsuki Kuwagaki, Makoto Miura for helpful discussions. This work was supported by JSPS KAKENHI Grant Number 12J09640, JSPS Grant-in-Aid for Scientiﬁc Research number 23224002, and JSPS Grant-in-Aid for Scientiﬁc Research number15H02054.

2. Preliminaries on A categories ∞ In 2, we recall some definitions about A categories and Hochschild invariants. We mainly follow§ [37]. Throughout this section, K denotes∞ a field of characteristic zero. Complexes (chain or cochain) mean 2-periodic complexes, i.e., Z/2Z-graded vector spaces with odd degree morphisms d such that d2 = 0. For a homogeneous element x of a Z/2Z-graded vector space, x denotes the | | degree of x. Set x ′ := x 1. | | | |− 2.1. A categories. We first recall the definition of A categories. ∞ ∞ Definition 2.1. (1) A curved A category A consists of the following data: A set of∞ objects ObA . • Z/2Z-graded K-vector spaces homA• (X, Y ) for X, Y ObA . • K-linear maps ∈ • A ms : homA• (X0,X1) homA• (X1,X2) homA• (Xs 1,Xs) homA• (X0,Xs) ⊗ ⊗···⊗ − → of degree 2 s (0 s) which satisfy the A -relations, i.e., − ≤ ∞ ′ ′ x1 + + xi A A ( 1)| | ··· | | ms j+1(x1,...,xi,mj (xi+1,...,xi+j ), xi+j+1,...,xs)=0, − − 0 i,j s i≤+Xj ≤s ≤

here xi are homogeneous elements of homA• (Xi 1,Xi). − (2) A curved A category A is called finite if homA• (X, Y ) is a finite dimensional K-vector space for each∞ X, Y . (3) A curved A category A is called (strictly) unital if for each X ObA , there exists an ∞ ∈ even element 1 homA• (X,X) called a unit of X such that X ∈ A A x A ms (x1,... 1X ,...xs 1)=0 for s =2, m2 (1X , x) = ( 1)| |m2 (x, 1X )= x. − 6 − A (4) An A category is a curved A category with m0 =0. ∞ ∞ A A A A Let be an A category. Then, by the A -relations, we see that m1 m1 = 0 and m2 naturally induce maps∞ ∞ ◦ A [m ]: H•homA• (X ,X ) H•homA• (X ,X ) H•hom• (X ,X ), 2 0 1 ⊗ 1 2 → A 0 2 A where the cohomology of homA• (X, Y ) with respect to m1 is denoted by H•homA• (X, Y ). We define the cohomology A category H•(A ) of A by the following: ∞ ∞ ObH•(A ) := ObA . • ∞ • homH• (A )∞ (X, Y ) := H•homA• (X, Y ). • A A The A structure maps ms of H•( ) are zero for s = 2 and [m2 ] for s = 2. • ∞ ∞ 6 COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 4

We can easily see that these data define an A category. If A is unital, then we can also define the ∞ cohomology category H•(A ). The objects are the same as H•(A ) and morphisms are defined by ∞ • • homH• (A )(X, Y ) := homH• (A )∞ (Y,X).

The unit 1X defines the identity morphism of X, which is also denoted by 1X . The composition of homogeneous morphisms g hom• • A (X, Y ) and f hom• • A (Y,Z) is defined by ∈ H ( ) ∈ H ( ) f g + f A (2.1) f g := ( 1)| || | | |[m ](f,g). ◦ − 2 The product f g := f g and the unit 1 gives a ring structure on H•homA• (X,X) · ◦ X Remark 2.2. The definition of the category H•(A ) is different from [37]. 2.2. Hochschild cohomology. We recall the definitions of Hochschild cohomology and its ring structure. Let A be a unital A category. We define the Hochschild cochains of degree k Z/2Z and length s by ∞ ∈ k s k s CC (A ) := HomK− homA• (X0,X1) homA• (Xs 1,Xs), homA• (X0,Xs) , ⊗···⊗ − X ,X ,...,X 0 Y1 s k s here HomK− denotes the space of K-linear morphisms of degree k s. Set − s CC•(A ) := CC•(A ) . s 0 Y≥ For ϕ CC•(A ), the length s part of ϕ is denoted by ϕs and homA• (X,X) part of ϕ0 is denoted ∈ 1 by ϕX . The Hochschild differential M is defined as follows; ′ ′ ′ 1 ϕ ( x1 + + xi ) A (2.2) M (ϕ)(x1,...,xs) := ( 1)| | | | ··· | | ms j+1(x1,...,ϕj (xi+1,... ), xi+j+1,...,xs) − − ′ ′ A X ϕ + x1 + + xi + ( 1)| | | | ··· | | ϕs j+1(x1,...,mj (xi+1,... ), xi+j+1,...,xs) − − 1 The complex CC•(A ),M Xis called the Hochschild cochain complex and its cohomology is called the Hochscild cohomology. The Hochschild cohomology of A is denoted by HH (A ). • For ϕ, ψ CC•(A ), the product ϕ ψ is defined as follows; ∈ ∪ ϕ ψ + ϕ 2 (2.3) ϕ ψ := ( 1)| || | | |M (ϕ, ψ), ∪ − 2 # A (2.4) M (ϕ, ψ)(x1,...,xs) := ( 1) m (x1,...,ϕ (xi+1,... ),...,ψ (xj+1,... ),...,xs), − ∗ ∗ ∗ 0 0 here # = ϕ ′( x ′ + + x ′)+Xψ ′( x ′ + + x ′). We define 1 CC (A ) by | | | 1| ··· | i| | | | 1| ··· | j | Hoch ∈ (1Hoch)X := 1X .

These data makes HH•(A ) into a Z/2Z-graded ring with the unit 1Hoch. The graded center Z•(A ) is deﬁned by the set of length 0 cocycles of CC•(A ). A 1 A Let ϕ be a cocycle of CC•( ). Then M (ϕ)X = 0 implies m1 (ϕX ) = 0. [ϕX ] denotes the cohomology class of ϕX . We easily see that [ϕX ] only depends on the cohomology class [ϕ]. Moreover, 1 M (ϕ)1 = 0 implies ϕ x (2.5) [ϕ ] [x] = ( 1)| |·| |[x] [ϕ ] X · − · Y for [x] H•homA• (X, Y ). Let ∈B A , i.e., ObB ObA and B is equipped with the natural unital A structure induced ⊂ ⊂ ∞ from A . By restriction, we have a natural morphism rB : CC•(A ) CC•(B), which induces a morphism on cohomology → [rB]: HH•(A ) HH•(B). → COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 5

Similarly, for X A , we have a morphism rX : CC•(A ) homA• (X,X), which induces a morphism on cohomology ∈ → [r ]: HH•(A ) H•homA• (X,X). X → We note that [rB] and [rX ] are ring homomorphisms. The following lemma is used in 4.2. § Lemma 2.3. Let [ϕ] HH•(A ) be an idempotent such that [ϕX ]=0 for all X. Then, for each ∈N A 1 N N N N Z 0, there exists ϕ CC•( ) such that M (ϕ )=0, [ϕ ] = [ϕ], and ϕs =0 for s N. ≥ ∈ ∈ A ≤ Proof. By assumption, there exists ψX such that ϕX = m1 (ψX ) for each X. Set 0 1 s ψ(x1,...,xs)= ≤ , ( ψX s =0 1 then (ϕ M ψ)X = 0. Hence we can assume ϕX =Q 0, which implies the N = 0 case of the lemma. Assume− that the lemma holds for N. Then ϕN = ϕN ϕN +M 1(ψ) for some ψ. By the definition of N N ∪ N 1 , we see that (ϕ ϕ )(x1,...,xs)=0 for s N +1. Hence ϕ M (ψ) satisfies the conditions of∪ ϕN+1, which implies∪ the lemma by induction.≤ − 2.3. Hochschild homology. Let A be a unital A category. We define the Hochschild chains as follows: ∞

CC (A ) := homA• (X0,X1) homA• (X1,X2) homA• (Xs 1,Xs) homA• (Xs,X0) • ⊗ ⊗···⊗ − ⊗ X0,XM1,...Xs An element

x0 x1 xs homA• (X0,X1) homA• (X1,X2) homA• (Xs 1,Xs) homA• (Xs,X0) ⊗ ⊗···⊗ ∈ ⊗ ⊗···⊗ − ⊗ is also denoted by x0[x1 x2 xs] or X if it is considered as an element of CC (A ). The degree | | · · · | j • of x0[x1 x2 xs] is defined by x0 + x1 ′ + + xs ′. Set X ′i := xi ′ + xi+1 ′ + + xj ′ for i j. The| | differential · · · | b is defined| by| | | ··· | | | | | | | | ··· | | ≤ X ′i A (2.6) b(x0[x1 xs]) := ( 1)| | 0 x0[ m (xi+1,... ) xs] | · · · | − · · · | ∗ | · · · | X ′i X ′s A + X( 1)| | 0·| | i+1 m (xi+1,...,x0,...xj )[xj+1 xi]. − ∗ | · · · | Then CC (A ),b is a chain complexX and its cohomology HH (A ) is called the Hochschild homology. • • 1 1 We define b | : CC•(A ) CC (A ) CC (A ) by ⊗ • → • 1 1 # A (2.7) b | (ϕ; X) := ( 1) m (xi+1,...,ϕ (xj+1,... ),...,x0,...,xk)[xk+1 xi], − ∗ ∗ | · · · | i s j here # = X ′ X ′ X+ ϕ ′ X ′ . We also define | | 0 · | | i+1 | | · | | i+1 2 1 2 b | : CC•(A )⊗ CC (A ) CC (A )[ 1] ⊗ • → • − by 2 1 # A b | (ϕ, ψ; X) := ( 1) m (xi+1,...,ϕ (xj+1,... ),...,ψ (xk+1 ... ),...,x0,...,xl)[xl+1 xi], − ∗ ∗ ∗ | · · · | i s j k 1 2 1 1 2 1 here # = X ′ XX ′ + ϕ ′ X ′ + ψ ′ X ′ . Then the operators M ,M ,b,b | ,b | satisfy | | 0 · | | i+1 | | · | | i+1 | | · | | i+1 the following relations ([27, Theorem 1.9], see also [37]): ′ 1 1 ϕ 1 1 1 1 1 (2.8) b(b | (ϕ; X)) + ( 1)| | b | (ϕ; b(X)) + b | (M (ϕ); X)=0, − ′ ′ ′ 2 1 ϕ 1 1 1 1 ϕ + ψ 2 1 (2.9) b(b | (ϕ, ψ; X)) + ( 1)| | b | (ϕ; b | (ψ; X)) + ( 1)| | | | b | (ϕ, ψ; b(X)) − −′ 1 1 2 2 1 1 ϕ 2 1 1 +b | (M (ϕ, ψ); X)+ b | (M (ϕ), ψ; X) + ( 1)| | b | (ϕ, M (ψ); X)=0. − COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 6

ϕ X + ϕ 1 1 Set ϕ X := ( 1)| |·| | | |b | (ϕ; X). Then induces an operator of cohomology ∩ − ∩ : HH•(A ) HH (A ) HH (A ), ∩ ⊗ • → • which makes HH (A ) into a Z/2Z-graded left HH•(A )-module. • 1 2 1 1 2 1 Remark 2.4. The deﬁnitions of the operators M ,M ,b,b | ,b | are the same as [37]. But and are diﬀerent from [37]. ∪ ∩ Let B A be a subcategory. cB denotes the inclusion CC (B) CC (A ), which induces a morphism⊂ on cohomology • ⊂ • [cB]: HH (B) HH (A ). • → • Similarly, for X A , the inclusion homA• (X,X) CC (A ) is denoted by cX . The morphism cX induces a morphism∈ on cohomology ⊂ •

[cX ]: H•homA• (X,X) HH (A ). → • 1 1 By the definition of b | , we easily see that 1 1 1 1 (2.10) b | (ϕ; cB(X)) = cB b | (rB(ϕ); X), ◦ here ϕ CC•(A ) and X CC (B). ∈ ∈ • 2.4. The Mukai pairing. In 2.4, we consider a finite unital A category A . We recall the definition of the Mukai pairing on§ HH (A ) ([39] for dg cases, [37, Proposition∞ 5.22] for A cases). • A ∞ For X = x0[x1 xs], X′ = x0′ [x1′ xt′ ] CC ( ), we define X, X′ Muk by | · · · | | · · · | ∈ • h i # A A tr(x′′ ( 1) m (xi+1,...,x0,...,xj ,m (xj+1,...,xi, x′′, xk′ +1,...,x0′ ,...,xl′), xl′+1,...,xk′ ), → − ∗ ∗ X s j i s k t here #=1+ X ′ + X ′ + x′′ X′ + X ′ X ′ + X′ ′ X′ ′ . This pairing satisfy | | i+1 | | 0 | | · | | | | 0 · | | i+1 | | 0 · | | k+1 X b(X), X′ + ( 1)| | X,b(X′) =0 h iMuk − h iMuk A A and induces a pairing , Muk on HH ( ) of degree zero. For example, if X = x0, X′ = x0′ HH ( ), then we have h i • ∈ • ′′ ′ x x + x x (2.11) X, X′ = str x′′ ( 1)| 0|·| | | 0|·| 0|x x′′ x′ , h iMuk → − 0 ◦ ◦ 0 where str is the supertrace of the Z/2Z-graded vector space H•homA• (X0,X 0′ ). By definition, for B A , we see that cB preserves the Mukai pairing. We recall the following theorem: ⊂ Theorem 2.5 ([39] for dg cases, [37, Proposition 5.24] for A cases). Let A be a finite unital ∞ A category. Suppose that A is smooth (see, e.g., [29] for the definition). Then , Muk is a non- degenerate∞ pairing on HH (A ). h i • Remark 2.6. For a smooth finite unital A category, the Hochschild homology is finite dimensional. Hence , is perfect. ∞ h iMuk The next lemma is used in 4.2. § Lemma 2.7. Let A be a finite unital A category and A1, A2 A . Suppose that ∞ ⊂ H•homA• (X1,X2)=0 for Xi Ai. Let Xi HH (Ai). Then X1, X2 Muk = 0. Note that Xi are naturally considered as elements∈ of HH (A )∈. • h i • Proof. By taking a minimal model of A , we can assume that homA• (X1,X2)=0 for Xi Ai. Then the lemma easily follows from the definition of , . ∈ h iMuk COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 7

2.5. Cyclic symmetry. In 2.5, we refer the reader to [16]. § Deﬁnition 2.8. Let A be a unital ﬁnite curved A category and n be a natural number. A is called n-cyclic if, for each X, Y , there exists a non-degenerate∞ pairing

, : homA• (X, Y ) homA• (Y,X) K h icyc ⊗ → of degree n which satisfy the following relations: ′ ′ ′ A xs ( x1 + + xs−1 ) A (2.12) ms 1(x1,...,xs 1), xs cyc = ( 1)| | | | ··· | | ms 1(xs, x1,...,xs 2), xs 1 cyc. h − − i − h − − − i Combined with unitality, we have ′ ′ 1+ x x (2.13) x , x = ( 1) | 1| ·| 2| x , x . h 1 2icyc − h 2 1icyc By Equations (2.12) and (2.13), we easily see that

A x1 A (2.14) ms 1(x1, x2,...,xs 1), xs cyc = ( 1)| | x1,ms 1(x2, x3,...,xs) cyc h − − i − h − i A x1 A Especially, we have m1 (x1), x2 cyc = ( 1)| | x1,m1 (x2) cyc. Hence we see that , cyc induces a non-degenerate pairingh i − h i h i H•homA• (X, Y ) H•homA• (Y,X) K, ⊗ → which is also denoted by , cyc. Therefore H•(A ) is also an n-cyclic A category. We deﬁne a h i ∞ ∞ morphism A : CC (A ) K of degree n by • → −

R 1X0 , x0 cyc if s =0 (2.15) (x0[x1 xs]) := h i . A | · · · | 0 if s =0 Z ( 6 A We call A a trace map. We note that A m2 (x0, x1) = x0, x1 cyc. Using this relation, we easily see that h i R R (2.16) b(x0[x1 xs]) = 0. A | · · · | Z Hence induces a morphism on cohomology, which is also denoted by . For ϕ CC•(A ) and A A ∈ X = x0[x1 xs] CC (A ), we define ϕ, X cyc by R | · · · | ∈ • h i R 1 1 (2.17) ϕ, X cyc := b | (ϕ; X). h i A Z We note that if ϕ, X homA• (X,X) then ϕ, X cyc is equal to the pairing on homA• (X,X). By straight forward calculation,∈ we see that h i ′ ′ x X s ϕ, X = ( 1)| 0| ·| | 1 ϕ(x ,...,x ), x . h icyc − h 1 s 0icyc Hence , cyc induces an isomorphism CC•(A ) = CC (A )[n]∨. Moreover , cyc satisfies ∼ • h i ′ h i 1 ϕ M (ϕ), X + ( 1)| | ϕ, b(X) =0 h icyc − h icyc by Equations (2.8) and (2.16). Hence , induces a pairing of degree n h icyc HH•(A ) HH (A ) K, ⊗ • → A A which is also denoted by , cyc. Moreover , cyc induces an isomorphism HH•( ) ∼= HH ( )[n]∨. By Equation (2.9), we easilyh i see that h i • n ψ (2.18) ϕ ψ, X = ( 1) ·| | ϕ, ψ X . h ∪ icyc − h ∩ icyc We define Z : CC (A ) CC•(A )[n] by the formula • → X, X′ = Z(X), X′ , h iMuk h icyc COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 8 which induces a morphism [Z]: HH (A ) HH•(A )[n]. Set ZX := rX Z and [ZX ] := [rX ] [Z]. By Theorem 2.5, [Z] is an isomorphism• if →A is smooth. ◦ ◦

2.6. Split generation. Let A be an n-cyclic A category and B A be a subcategory. For each X,Y ∞ ⊂ α X, Y A , take a homogeneous basis e of homA• (X, Y ). We deﬁne e homA• (Y,X) ∈ { α } { Y,X} ⊂ α X,Y α A by the formula eY,X ,e cyc = δβ . For an element x homA• (X,X) CC ( ), we have h β i ∈ ⊂ • Z (X), x = Z(X), x = X, x . Using Equation (2.14), we see h X icyc h icyc h iMuk X,x h iMuk # A A = tr x′ ( 1) m (xi+1,...,x0,...,xj ,m (xj+1,...,xi, x′, x) → − ∗ ∗ A A X # α Xi+1,X = ( 1) eX,Xi ,m (xi+1,...,x0,...,xj ,m (xj+1,...,xi,eα , x)) cyc − h +1 ∗ ∗ i α #+ eX,X A α A X ,X X | i+1 | i+1 = ( 1) m (eX,Xi , xi+1,...,x0,...,xj ),m (xj+1,...,xi,eα , x)) cyc − h ∗ +1 ∗ i α ′ #+ eX,X +# A A α X ,X X | i+1 | i+1 = ( 1) m (m (eX,Xi , xi+1,...,x0,...,xj ), xj+1,...,xi,eα ), x cyc, − h ∗ ∗ +1 i X

s j Xi+1,X i s α s j here # = 1+ X ′ + X ′ + eα x + X ′ X ′ and #′ = e ′ + X ′ + X ′ . | | i+1 | | 0 | | · | | | | 0 · | | i+1 | X,Xi+1 | | | i+1 | | 0 Thus we have A A X # α Xi+1,X (2.19) ZX ( )= ( 1) m (m (eX,Xi , xi+1,...,x0,...,xj ), xj+1,...,xi,eα ), − ∗ ∗ +1

Xi+1,X X i s here # = eα X + X ′ X ′ . | | · | | | | 0 · | | i+1 For K A , we define a chain complex Y r B Y l as follows: ∈ K ⊗ K Y r Y l K B K := homA• (K,X0) homA• (X0,X1)[1] homA• (Xs 1,Xs)[1] homA• (Xs,K) ⊗ ⊗ ⊗···⊗ − ⊗ X0,...,Xs B M0 s ∈ ≤ with the differential A d(m x1 xs n) := m (m, x1,...,xi) xi+1 xs n ⊗ ⊗···⊗ ⊗ − ∗ ⊗ ⊗···⊗ ⊗ ′ ′ m + x1 + + xi A + X( 1)| | | | ··· | | m m (xi+1,...,xj ) n − ⊗···⊗ ∗ ⊗···⊗ ′ ′ m + x1 + + xi A + X( 1)| | | | ··· | | m m (xi+1,...,xs,n) − ⊗···⊗ ∗ K r l X We define m : Y B Y homA• (K,K) by K ⊗ K → mK (m x x n) := mA (m, x ,...,x ,n). ⊗ 1 ⊗···⊗ s ⊗ s+2 1 s K A K By the A relations, we see m d = m1 m . The morphism between cohomology induced by mK is denoted∞ by [mK ]. We recall◦ the following◦ result which is due to Abouzaid [2]. Theorem 2.9. If 1 Im([mK ]), then K is split generated by B, i.e., K Dπ(B) Dπ(A ). K ∈ ∈ ⊂ Remark 2.10. We briefly explain the definition of the (split-closed ) derived category Dπ(A ) of a unital A category A . Let Ch(K) be the dg category of chain complexes over K. The A category ∞ ∞ of one-sided twisted complexes over A K Ch(K) is denoted by Tw(A ) (see,e.g., [3, Section 8]) and its split closure is denoted by ΠTw(⊗A ) (see,e.g., [35, Section 4]). Then Dπ(A ) is defined by H0(ΠTw(A )). For B A , we say that B split generates A if the inclusion Dπ(B) Dπ(A ) gives an equivalence. ⊂ ⊂ COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 9

B Y r Y l We define CC (∆) : CC ( ) K B K by • • → ⊗ A X # α Xi+1,K CC (∆)( ) := ( 1) m (eK,Xi , xi+1,...,x0,...,xj ) xj+1 xi eα , • − ∗ +1 ⊗ ⊗···⊗ ⊗ Xi+1,K X i s here # = eα X + X ′ X ′ . | | · | | | | 0 · | | i+1 Lemma 2.11. d CC (∆) + ( 1)nCC (∆) b =0. ◦ • − • ◦ Proof. By the A relations, we have ∞ (d CC (∆) + ( 1)nCC (∆) b)(X)= ◦ • − • ◦ # A A α Xi+1,K ( 1) m (m (eK,Xi , xi+1,...,xj ), xj+1,...,x0,...,xk) xk+1 xi eα − ∗ ∗ +1 ⊗ ⊗···⊗ ⊗ ′ A A X # α Xi+1,K + ( 1) m (eK,Xi , xi+1,...,xj ) m (xk+1,...,xi,eα ), − ∗ +1 ⊗···⊗ ∗ XXi+1,K i s α j here # = eα X + X ′ X ′ and #′ =#+ e ′ + X ′ . | | · | | | | 0 · | | i+1 | K,Xi+1 | | | i+1 Set A Xj+1,K β Xi+1,K m (xi+1,...,xj ,eα )= mαeβ . ∗ Xβ β j Xj+1,K Xi+1,K Note that m = 0 if X ′ + eα +1 = e . Then we have α | | i+1 | | 6 | β | A α A α Xj+1,K β m (eK,Xi+1 , xi+1,...,xj )= m (eK,Xi+1 , xi+1,...,xj ),eβ eK,X ∗ h ∗ i j+1 β X α eK,X α A Xj+1,K β | i+1 | = ( 1) eK,Xi+1 ,m (xi+1,...,xj ,eβ ) eK,X − h ∗ i j+1 Xβ β X ′j eK,X + i+1+1 α β = ( 1)| i+1 | | | m e . − β K,Xj+1 Xβ Using this equality, we see that (d CC (∆) + ( 1)nCC (∆) b)(X) = 0. ◦ • − • ◦ By this lemma, we see that CC (∆) induces a morphism on cohomology. By construction, we K • have m CC (∆) = ZK cB. Combined with Theorem 2.9, we have the following theorem, which is a version◦ of• Abouzaid’s◦ generating criterion [2]. Theorem 2.12. Let A be an n-cyclic A category and B A be a subcategory. Let K be an ∞ ⊂ object of A . Suppose that 1 Im([Z ] [cB]). Then K is split generated by B. K ∈ K ◦ 3. Preliminaries on Floer cohomology and Quantum cohomology In 3, we review Floer cohomology of Lagrangian submanifolds ([17]). § 3.1. Floer cohomology of Lagrangian submanifolds. Let (X,ω) be a compact symplectic manifold of dimension 2n. We fix an ω-compatible almost complex structure on X. Let L be a compact oriented Lagrangian submanifold of (X,ω) with a spin structure and ρ : π1(L) C× be a group 2 → homomorphism which is considered as a local system on L. Let µL H (X,L; Z) be the Maslov class. We denote by Λ the universal Novikov field over C. Namely, ∈

∞ λi Λ= aiT ai C λi R lim λi = . ( ∈ ∈ i ∞) i=0 →∞ X This is an algebraically closed valuation ﬁeld. The valuation is deﬁned by

∞ val( a T λi ) := min λ a =0 . i { i | i 6 } i=0 X COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 10

We denote by Λ0 the valuation ring of Λ and by Λ+ the maximal ideal of Λ0. Let Ms+1(L; β) be the moduli space of stable disks bounded by L of homology class β H2(X,L; Z) with s +1 counterclockwise ordered boundary marked points. ∈ Fukaya-Oh-Ohta-Ono construct C-linear maps can s m : H•(L; C)⊗ H•(L; C) s,β → for s Z 0 and β H2(X,L; Z) by using Ms+1(L; β) (see, e.g., [16], [17]). Let 1L be the unit of ∈ ≥ ∈ H•(L; C) and , be the Poincar´epairing, i.e., h iL x , x := x x . h 1 2iL 1 ∪ 2 ZL Set x1 x2 + x1 x1, x2 cyc := ( 1)| |·| | | | x1, x2 L. can h i − h i By construction, ms,β satisfy the following conditions: Condition 3.1 ([16, Corollary 12.1]). can (1) m is degree 2 s µ (β) with respect to the cohomological Z grading of H•(L; C). s,β − − L (2) If mcan =0, then there exist β ,...,β such that β = β + + β and M (L; β ) = . s,β 6 1 m 1 ··· m s+1 i 6 ∅

can 0 if s =0, 1 (3) m (x1,...,xs)= . s,0 x1 x2 + x1 (( 1)| || | | |x1 x2 if s =2 ′ − ′ ∪ x + + xi can can (4) ( 1)| 1| ··· | | m (x ,...,m (x ,... ),...,x )=0. − s1,β1 1 s2,β2 i+1 s s1+s2=s+1 β1+Xβ2=β (5) mcan(..., 1 ,... )=0 if β =0 or s 3. s,β L 6 ≥ ′ ′ ′ can x0 ( x1 + + xs ) can (6) m (x1,...,xs), x0 cyc = ( 1)| | · | | ··· | | m (x0, x1,...,xs 1), xs cyc. h s,β i − h s,β − i can s We deﬁne m : H•(L; Λ)⊗ H•(L;Λ) by s,ρ → ω(β) can 2π can ms,ρ := ρ(∂β)T ms,β . β H (X,L;Z) ∈ 2X Here H•(L; Λ) is considered as a Z/2Z-graded vector space. By the above conditions, we have the following theorem: can Theorem 3.2 (see, e.g., [20, Section 2.2]). H•(L; Λ),ms,ρ , 1L, , cyc deﬁnes an n-cyclic curved A algebra, i.e., an n-cyclic curved A category with one object.h i ∞ ∞ 3.2. Maurer-Cartan elements. To construct (non-curved) A algebras, we introduce (weak) odd ∞ Maurer-Cartan elements. Let b+ H (L;Λ+) be an odd degree element. The pair (ρ,b+) is denoted by b. Set ∈

l0 l1 ls can can ms,b (x1,...,xs) := ms+l + +ls,ρ(b+,...,b+, x1, b+,...,b+, x2,b+,...,b+, xs, b+,...,b+). 0 ··· 0 l ,l ,...,ls ≤ 0X1 z }| { z }| { z }| { can Remark 3.3. Since b H•(L;Λ ), m converge with respect to the energy ﬁltration. + ∈ + s,b We easily see that operators mcan also satisfy the A relations and the curvature mcan(1) is equal s,b ∞ 0,b can to ms,ρ (b+,...,b+). 0 s X≤ COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 11

Definition 3.4. b = (ρ,b+) is called a (weak) Maurer-Cartan element if b satisfies the Maurer- Cartan equation, i.e., there exists W Λ such that L,b ∈ mcan(b ,...,b )= W 1 . s,ρ + + L,b · L 0 s X≤ For a Maurer-Cartan element b, operators mcan (s 1) define an A algebra. Moreover, using s,b ≥ ∞ Condition 3.1, we easily see the following:

can Theorem 3.5. Let b = (ρ,b ) be a Maurer-Cartan element. Then H•(L; Λ),m , 1 , , + s,b L h icyc deﬁnes an n-cyclic A algebra, here we set mcan =0. ∞ 0,b can The cohomology of H•(L;Λ) with respect to m1,b is denoted by HF •(L,b). This is naturally considered as a Z/2Z-graded ring with the unit 1L. Deﬁnition 3.6. Let L be a compact oriented Lagrangian submanifold with a spin structure. L is called weakly unobstructed if there exists a Maurer-Cartan element. We will use the next lemma to state the divisor axiom Assumption 4.3 (2) .

2 Lemma 3.7. Let η Hc (X L; C) be a compact support cohomology class and b = (ρ,b+) be a Maurer-Cartan element.∈ Set \ ω(β) can (3.1) i∗ (η) := η,β ρ(∂β)T 2π m (b ,...,b ) L,b h i s,β + + β H2(X,L;Z) ∈ X0 s ≤ can Then m1,b iL,b∗ (η) =0. Proof. By Condition 3.1 (4), we have

can ω(β) can m i∗ (η) + η,β ρ(∂β)T 2π m (b ,...,b , W 1 ,b ,...,b )=0 1,b L,b h i s,β + + L,b · L + + X can Combined with Condition 3.1 (5), we see m1,b iL,b∗ (η) =0. 3.3. Fukaya categories with one Lagrangian submanifold. For a compact oriented Lagrangian submanifold with a spin structure L and a group homomorphism ρ, we deﬁne an n-cyclic A L,ρ L,ρ ∞ category Fuk(X) . Ob(Fuk(X) ) is the set of Maurer-Cartan elements (ρ,b+) of L. For objects b1 = (ρ,b1,+) and b2 = (ρ,b2,+), the morphism space is deﬁned by

H•(L; Λ) if WL,b1 = WL,b2 hom•(b1,b2) := . (0 otherwise

For Maurer-Cartan elements bi = (ρ,bi,+) and xi hom•(bi 1,bi) (1 i s), the A structure L,ρ ∈ − ≤ ≤ ∞ maps ms are deﬁned by

l0 ls L,ρ can ms (x1,...,xs) := ms+l + +ls,ρ(b0,+,...,b0,+, x1,b1,+,...,bs 1,+, xs, bs,+,...,bs,+) 0 ··· − 0 l ,l ,...,ls ≤ 0X1 z }| { z }| { L,ρ if WL,b0 = = WL,bs . If otherwise, then ms (x1, x2,...,xs) is deﬁned to be 0. Then operators L,ρ ··· ms (1 s) satisfy the A relations (see, e.g., [15, Proposition 1.20]). Moreover we easily see the following:≤ ∞ Theorem 3.8. Fuk(X)L,ρ is an n-cyclic A category, here units and pairings are the same as Theorem 3.5. ∞ COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 12

3.4. Quantum cohomology. In 3.4, we introduce some notations about quantum cohomology. See [22] for more details. § Let GW (α; x , x ,...,x ) C l 1 2 l ∈ be the Gromov-Witten invariant, here α H2(X; Z) and xi (i =1, 2,...,l) are elements of H•(X; C). This is defined by using the moduli space∈ of genus zero stable holomorphic maps with l marked points and of homology class α. Set ω(α) GW (x , x ,...,x ) := T 2π GW(α; x , x ,...,x ). l 1 2 l · 1 2 l α X l We extend GWl linearly and GWl gives a morphism from H•(X; Λ)⊗ to Λ. We define the quantum product x x by the following equation: 1 ∗ 2 x x , x := GW (x , x , x ). h 1 ∗ 2 3iX 3 1 2 3 The quantum product makes H•(X; Λ) into a Z/2Z-graded commutative ring with a unit 1X . H•(X; Λ) equipped with this ring structure is called the quantum cohomology and denoted by QH•(X). 4. Fukaya categories and open-closed maps 4.1. Assumptions on Fukaya categories and open closed maps. In 4.1, we recall some expected properties of Fukaya categories. The construction of Fukaya categories§ with these properties is announced by [1] (see also [26]). Let be a finite set of weakly unobstructed compact oriented L Lagrangian submanifolds with spin structures. We assume that L ⋔ L′ for L = L′ . 6 ∈ L Assumption 4.1. There exists an n-cyclic A category Fuk(X)L which satisfies the following: ∞ (1) ObFuk(X)L = (L,b) L , b is a Maurer-Cartan element of L . { | ∈ L }

H•(L ; H om(ρ ,ρ )) Λ if L = L , W = W . 1 1 2 ⊗ 1 2 L1,b1 L2,b2  Λ p if L1 = L2, WL1,b1 = WL2,b2 . (2) hom•((L1,b1), (L2,b2)) = h i 6 p L L  ∈M1∩ 2 0 if otherwise, H here om(ρ1,ρ2) is the local system on L1 corresponding to the group homomorphism 1  ρ ρ− . Note that p L  L is equipped with a Z/2Z-grading . 2 · 1 ∈ 1 ∩ 2

(3) For L and a group homomorphism ρ, Fuk(X)L induces an n-cyclic A structure on the ∈ L ∞ set of Maurer-Cartan elements (ρ,b+) of L. We assume this n-cyclic A category is equal to Fuk(X)L,ρ introduced in Theorem 3.8. ∞ Remark 4.2. In [38], Sheridan constructs Fukaya categories for compact monotone symplectic manifolds. Sheridan also shows that the monotone Fukaya category is naturally equipped with a weak proper Calabi-Yau structure. But this monotone Fukaya category is not cyclic. For cyclicity, see also [25].

The cohomology H•hom•((L1,b1), (L2,b2)) is denoted by HF • (L1,b1), (L2,b2) . Moreover, we assume the following: Assumption 4.3 (see also [17], [19], [21]). There exists a closed open map

q : QH•(X) HH•(Fuk(X)L). → We define +n p : HH (Fuk(X)L) QH• (X) • → COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 13 by the equation n X (4.1) x, p(X) = ( 1) ·| | q(x), X , h iX − h icyc here x QH•(X) and X HH (Fuk(X)L). They satisfy the following conditions: ∈ ∈ • (1) q is a unital ring homomorphism. (2) For η H2(X L; C), we assume ∈ c \ [r ] q(η)= i∗ (η), (L,b) ◦ L,b here η is naturally considered as an element of QH•(X). (3) For X, X′ HH (Fuk(X)L), we assume ∈ • n(n+1) +? p(X), p(X′) = ( 1) 2 X, X′ , h iX − h iMuk here the sign ? only depends on parity of n and X . (4) Let [L] Hn(X; Z) be the Poincarédual of the homology| | class of L, i.e., [L] is defined by the ∈ equation L x L = X [L] x. We consider 1L HF •(L,b) as an element of HH (Fuk(X)L). We assume | ∪ ∈ • R R p(1L) = [L]. Remark 4.4. There is many results closely related to this assumption. See, e.g., [24, Appendix B], [38, Propositions 2.1, 2.2, 2.6, Lemmas 2.3, 2.7, 2.14] for monotone cases (see also [32]). For toric cases, see, e.g., [21, 4.7]. § By Equation (2.18), we see that p is a QH•(X)-module map, i.e., (4.2) p q(x) X = x p(X) ∩ ∗ for x QH•(X) and X HH (Fuk(X)L). To simplify notation, [Z] is also denoted by Z. As a direct∈ consequence of Assumption∈ • 4.3 (3), we see that ′ n(n+1) +? (4.3) q p(X) = ( 1) 2 Z(X), ◦ − here ?′ =?+ n X . · | | Lemma 4.5. Let A Fuk(X)L be a subcategory. Suppose that H•(A ) is not equivalent to the zero ⊂ category. Then p [cA ] =0. ◦ 6 Proof. Assume p [cA ]=0. By Equation (2.10) and the definition of p, we have ◦ n X 1 , p [cA ](X) = ( 1) ·| | [rA ] q(1 ), X h X ◦ iX − h ◦ X icyc for all X HH (A ). From this we have 1Hoch = [rA ] q(1X )=0, which implies 1L HF •(L,b) are ∈ • ◦ ∈ equal to zero for all (L,b) A . Thus (L,b) = 0, which contradicts the assumption H•(A ) = 0. ∈ ∼ 6∼ 4.2. Decomposition of Fukaya categories with respect to idempotents. In 4.2, we consider decomposition of Fukaya categories by using open closed maps (see [1], [6], [14], [36]).§ For A Fuk(X)L and (L,b) Fuk(X)L, set ⊂ ∈ pA := p [cA ], qA := [rA ] q, p := p [c ], q := [r ] q. ◦ ◦ L,b ◦ (L,b) L,b (L,b) ◦ We note that qL,b is also a ring homomorphism and pL,b is a QH•(X)-module map. Let e be an 2 idempotent of QH•(X), i.e., e = e. Note that e is even degree. Let Fuk(X; c)eL be the n-cyclic A subcategory with ∞ Ob Fuk(X; c)L := (L,b) W = c, q (e)=1 . e { | L,b L,b L} Set Fuk(X; c)L := Fuk(X; c)1LX , Fuk(X)eL := Fuk(X; c)eL. c Λ [∈ COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 14

Note that Fuk(X)1LX = Fuk(X)L. The restriction of p to HH (Fuk(X)eL) is also denoted by p. Using Theorem 2.12, we have the following theorem which is due to• Abouzaid-Fukaya-Oh-Ohta-Ono [1].

Theorem 4.6 ([1]). Suppose that A Fuk(X)L and e Im(pA ). Then A split generates Fuk(X)L. ⊂ e ∈ e Proof. By assumption, we have 1 Im([r ] q p [cA ]) for (L,b) Fuk(X)L. Combined with L ∈ (L,b) ◦ ◦ ◦ ∈ e Equation (4.3), we have 1 Im(Z [cA ]). The theorem follows from Theorem 2.12. L ∈ (L,b) ◦

We show some basic properties of Fuk(X)eL.

Lemma 4.7. Let e ,e be idempotents of QH•(X) and (L ,b ) Fuk(X)L (i =1, 2). Suppose that 1 2 i i ∈ ei e e =0. Then HF • (L ,b ), (L ,b ) =0. 1 ∗ 2 1 1 2 2

Proof. For x HF • (L 1,b1), (L2,b2) , we have x = 1L1 x 1L2 = qL1,b1 (e1) x qL2,b2 (e2). By Equation (2.5),∈ we see that · · · · q (e ) x q (e )= x q (e ) q (e )= x q (e e )=0. L1,b1 1 · · L2,b2 2 · L2,b2 2 · L2,b2 1 · L2,b2 2 ∗ 1 Hence we have x =0.

Lemma 4.8. p HH (Fuk(X)eL) QH•(X)e • ⊂ Proof. Since p HH (Fuk(X)eL) is finite dimensional, we can take X1,..., Xk HH (Fuk(X)eL) such • ∈ • that p(X1),..., p(Xk) span p(HH (Fuk(X)eL). By definition, we have qL,b(1 e)= 0 for (L,b) • N 1 N− N ∈ Fuk(X)eL. By Lemma 2.3, there exists ϕ CC•(Fuk(X)eL) such that M (ϕ )=0, [ϕ ]= q(1 e) N ∈ N − and ϕs =0 for s N. By taking enough large N, we can assume [ϕ ] Xi = 0 for i =1, 2,...,k. Hence we have q(e≤) X = X . Since p is a module map, we see e p(X∩) = p q(e) X = p(X ), ∩ i i ∗ i ∩ i i which implies p(X ) QH•(X)e. i ∈ The next lemma is used in 5 to construct orthogonal idempotents of QH•(X). § Lemma 4.9. Let A1, A2 Fuk(X)L. Suppose that HF • (L1,b1), (L2,b2) = 0 for each (Li,bi) A ⊂ A ∈ i. Then p(X1) p(X2)=0 for Xi HH ( i), here we identify Xi with [cAi ](Xi). ∗ ∈ • Proof. We have ′ n(n+1) +? p(X ) p(X )= p q p(X ) X = ( 1) 2 p Z(X ) X . 1 ∗ 2 ◦ 1 ∩ 2 − 1 ∩ 2 Hence it is sufficient to show that Z(X ) X =0. By Equation (2.10), we see that 1 ∩ 2 Z(X ) X = [cA ] [rA ] Z(X ) X , 1 ∩ 2 2 2 ◦ 1 ∩ 2 which implies Z(X ), X = [rA ] Z(X ), X . h 1 2icyc h 2 ◦ 1 2icyc A By Lemma 2.7, we have X1, X2′ Muk = 0 for all X2′ HH ( 2), which implies h i ∈ • [rA ] Z(X ), X′ = Z(X ), X′ = X , X′ =0. h 2 ◦ 1 2icyc h 1 2icyc h 1 2iMuk From this, it follows that [rA ] Z(X )=0. Hence we have Z(X ) X =0. 2 ◦ 1 1 ∩ 2 4.3. Automatic split-generation. The next statement is a main theorem of this paper. We note that a similar statement has independently been obtained by Ganatra [24].

Theorem 4.10 (cf. [24]). Let A Fuk(X)L be a subcategory. Suppose that Z of A is an isomor- ⊂ phism. Then there exists an idempotent e QH•(X) such that A Fuk(X)L and A split generates ∈ ⊂ e Fuk(X)eL. Moreover, +n pA : HH (A ) QH• (X)e • → is an isomorphism and qA induces an isomorphism between QH•(X)e and HH•(A ) by restriction. COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 15

Proof. Since Z is an isomorphism, there exists X HH (A ) such that qA pA (X)=1Hoch. Set ∈ • ◦ e := pA (X). Since pA is a module map, we have

e e = pA qA pA (X) X = pA 1 X = pA (X)= e. ∗ ◦ ∩ Hoch ∩ A By construction, we have Fuk(X)eL and e Im(pA ). Combined with Theorem 4.6, it follows A ⊂ ∈ that split generates Fuk(X)eL. ′ +n n(n+1) +? By Lemma 4.8, we see that pA HH (A ) QH• (X)e. Since Z = ( 1) 2 qA pA is an • ⊂ − ◦ isomorphism, pA is injective. Since e pA HH (A ) and pA is a module map, it follows that pA is +n ∈ • a surjection to QH• (X)e. Hence pA is an isomorphism. Injectivity of the restriction qA • |QH (X)e follows from Equation (4.1). Since the dimensions of QH•(X)e, HH (A ), and HH•(A ) are the • same, it follows that qA • is an isomorphism. |QH (X)e Remark 4.11. A By Lemma 4.5, we see that e =0 if H•( ) ∼= 0. • If A is smooth, then the assumption6 of Theorem6 4.10 is satisfied. • 4.4. Eigenvalues of c1. Let c1 be the first Chern class of the tangent bundle TX. Crit(X∨) denotes the set of eigenvalues of the operator c . For c Crit(X∨), the generalized eigenspace of 1∗ ∈ c with the eigenvalue c is denoted by QH•(X) and the idempotent corresponding to QH•(X) 1∗ c c is denoted by ec (i.e., ec is the QH•(X)c summand of the unit 1X ). By definition, ec satisfies QH•(X)c = QH•(X)ec.

Proposition 4.12 (cf. [4, Theorem 6.1], [19, Theorem 23.12]). Let (L,b) Fuk(X)L. If b+ H1(L;Λ ), then q (c )= W 1 . ∈ ∈ + L,b 1 L,b · L 2 R 2 R Proof. We note that the Maslov class µL H (X,L; ) ∼= Hc (X L; ). By Assumption 4.3 (2), we have ∈ \ ω(β) can q (c )= i∗ (µ /2) = k ρ(∂β)T 2π m (b ,...,b ). L,b 1 L,b L · s,β + + k Z µ (β)=2k X∈ L X Since b H1(L;Λ ), the degree of mcan(b ,...,b ) is 2 µ (β). From this it follows that + ∈ + s,β + + − L ω(β) 0 if k =1 2π can ρ(∂β)T ms,β (b+,...,b+)= 6 (WL,b 1L if k =1, µL(Xβ)=2k · which proves the proposition. The next lemma gives a relationship between eigenvalues of c and W . 1∗ L,b Lemma 4.13. If q (c )= W 1 , then c p (x)= W p (x) for x HF •(L,b). L,b 1 L,b · L 1 ∗ L,b L,b · L,b ∈ Proof. Since pL,b is a module map, we have c p (x)= p (q (c ) x)= W p (x). 1 ∗ L,b L,b L,b 1 ∩ L,b · L,b

Combined with Assumption 4.3 (4), we have (4.4) c [L]= W [L] 1 ∗ L,b · 1 if b H (L;Λ ). The categories Fuk(X; c)L and Fuk(X)L are related as follows: + ∈ + ec Lemma 4.14. Suppose that (L,b) Fuk(X; c)L satisﬁes q (c )= W 1 . Then ∈ L,b 1 L,b · L (L,b) Fuk(X)L . ∈ ec COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 16

Proof. By the deﬁnition of e , there exists α QH•(X)(1 e ) such that (c c) α =1 e . c ∈ X − c 1 − ∗ X − c By assumption, we have q (1 e )= q (c c) q (α) = 0 for all (L,b) Fuk(X; c)L. Hence L,b X − c L,b 1 − · L,b ∈ we have qL,b(ec)= qL,b(1X )=1L, which implies the lemma.

4.5. Potential functions. In 4.5, we assume that H1(L; Z) of L is torsion free. Moreover we assume the following condition§ ([20, Condition 6.1]): ∈ L

Condition 4.15. Let L be a compact oriented Lagrangian submanifold. If β H2(X,L; Z) satisfies M (L; β) = and β =0, then µ (β) 2. ∈ 1 6 ∅ 6 L ≥ We note that this condition is satisfied if 2 L is monotone, i.e., µL = c ω H (X,L; Z) for some c R>0. • L is a torus fiber of a compact· ∈ symplectic toric Fano manifold.∈ • Note that if L satisfies Condition 4.15, then mcan(1) Q 1 and 0,β ∈ · L ∂β,b s (4.5) mcan(b,b,...,b)= h i mcan(1) s,β s! · 0,β for b H1(L; C) (see [20, Appendix 1]). Define n Q by n 1 = mcan(1). For b H1(L;Λ ), ∈ β ∈ β · L 0,β + ∈ + we see that ω(β) W = n T 2π ρ(∂β) exp( ∂β,b ) L,b β · · · h +i µLX(β)=2 1 by using Equation (4.5) (see [20, Theorem A.2]). Thus all elements of Hom π1(L), C∗ H (L;Λ+) are Maurer-Cartan elements. Let Λ[H (L; Z)] be the Laurent polynomial ring with the× coefficient 1 Λ corresponding to the monoid H1(L; Z). The monomial corresponding to a H1(L; Z) is de- a ∈ noted by y . The completion of the ring Λ[H1(L; Z)] with respect to the Gauss norm is denoted by Λ H1(L; Z) (see [8], [21]). Then Λ H1(L; Z) is considered as the ring of Λ-valued functions hh 1 ii hh ii defined on H (L;Λ=0) := Hom H1(L; Z), Λ0∗ .

Definition 4.16. We define a potential function WL of L by ω(β) ∂β W := n T 2π y . L β · · µLX(β)=2 By the Gromov compactness, this is an element of Λ H (L; Z) . hh 1 ii By the isomorphism Hom π (L), C∗ = Hom H (L; Z), C∗ and the natural inclusion C∗ Λ∗, 1 ∼ 1 ⊂ 0 we consider Hom π (L), C as a subset of H1(L;Λ ). Then Hom π (L), C H1(L;Λ ) is isomor- 1 ∗ =0 1 ∗ + 1 b+ × 1 phic to H (L;Λ=0) by sending (ρ,b+) to ρ e . Then we see that WL(b)= WL,b for b H (L;Λ=0). m · ei i 1 ∈ Take a basis ei i=1 of H1(L; Z) and set yi := y . Let e H (L; Z) be the dual basis of ei . By definition, we{ see} { }⊂ { }

can i ∂WL (4.6) m1,b (e )= yi (b) 1(L,b), ∂yi · 2 can i j can j i ∂ WL (4.7) m2,b (e ,e )+ m2,b (e ,e )= yiyj (b) 1(L,b). ∂yi∂yj · 1 for b H (L;Λ=0). The next proposition is not used later, but it may be useful to understand Landau-Ginzburg∈ mirror of Fano manifolds. Proposition 4.17 (cf. [23, Remark 3.1.3]). Suppose that H (X; Z)=0 and c /r H2(X; Z) for 1 1 ∈ r Z 1. Let β1,...,βm be elements of H2(X,L; Z) such that ∂βi is a Z-basis of H1(L; Z) and let ∈ ≥ { } ζ be an rth root of unity. Set k := µ (β )/2 Z and y := y∂βi . Then we have i L i ∈ i k1 km WL(ζ y1,...,ζ ym)= ζWL(y1,...,ym). COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 17

Proof. By deﬁnition we have ω(β) a a W = n T 2π y 1 y m . L β · · 1 ··· m µL(β)=2 ∂β=a ∂β X+ +am∂βm 1 1 ··· Since µL(β) = 2, we see that k a + + k a + c (β a β a β )=1. 1 1 ··· m m 1 − 1 1 −···− m m We note that β a β a β H (X; Z). By assumption, we have − 1 1 −···− m m ∈ 2 c (β a β a β ) rZ, 1 − 1 1 −···− m m ∈ which implies the proposition. 2 1 Let η Hc (X L; C) and b′ := (ρ′,b+′ ) Hom π1(L), C∗ H (L;Λ+). By Equation∈ (4.5),\ we see that ∈ × ω(β) i∗ (η)= n η,β T 2π ρ(∂β) exp( ∂β,b ) 1 . L,b β · h i · · · h +i · (L,b) µLX(β)=2 Proposition 4.18. If i∗ (η) = i∗ ′ (η), then HF • (L,b ), (L,b′ ) =0. L,b 6 L,b + + Proof. This easily follows from Equation (2.5) and Assumption 4.3 (2).

5. Applications 5.1. Tori. In 5.1, we assume L L ,L satisfies Condition 4.15, and L is homeomorphic to the § n ∈ n-dimensional torus T . Let Cℓn be the Clifford algebra of an n dimensional Λ-vector space with a non-degenerate quadratic form. 1 Lemma 5.1. If b = (ρ,b ) H (L;Λ ) is a non-degenerate critical point of W , then HF •(L,b) + ∈ =0 L is isomorphic to the Clifford algebra Cℓn as a Z/2Z-graded ring. Proof. The proof is the same as [21, Theorem 3.6.2] (see also [10]).

Since Cℓn is formal ([38, Corollary 6.4]) and its Hochschild homology is one-dimensional (see the proof of [28, Proposition 1]), the Hochschild homology of the A algebra hom•((L,b), (L,b)) is also ∞ one-dimensional. Hence the trace map gives a canonical isomorphism HH hom•((L,b), (L,b)) = • ∼ Λ. Let p be the element of Hn(L; Z) such that p = 1. Then p gives an element of the Hochschild L homology corresponding to 1 Λ. Since Cℓn is smooth (see [13]), we have p,p Muk = 0 by Theorem 2.5. ∈ R h i 6 n(n+1)/2 Remark 5.2. By easy computation, we see that ( 1) p,p Muk is equal to the determinant 2 i,j=n − h i ∂ WL of the matrix yiyj ∂y ∂y (b) (see [21, Section 3.7]), which also implies p,p Muk =0. i j i,j=1 h i 6 h i Set p (p) e := L,b . (L,b) p,p h iMuk By Proposition 4.12 and Lemma 4.13, it follows that c1 e(L,b) = WL,b e(L,b). Combined with (the proof of) Theorem 4.10, we have the following statement:∗ ·

Proposition 5.3. e(L,b) is an idempotent which gives an ﬁeld factor, i.e., QH•(X)e(L,b) ∼= Λ. Moreover, e is an eigenvector of c with the eigenvalue W . Finally, (L,b) split generates (L,b) 1∗ L,b Fuk(X) . eL(L,b) To obtain orthogonal idempotents, we show the following lemma: COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 18

1 Lemma 5.4. Let bi = (ρi,bi,+) (i =1, 2) be elements of H (L;Λ=0) with WL(b1)= WL(b2). Assume that b are non-degenerate critical points of W and b = b . Then HF • (L,b ), (L,b ) =0. i L 1 6 2 1 2 Proof. We ﬁrst show that H•(L; H om(ρ1,ρ2)) = 0 if ρ1 = ρ2. If n =1, then this statement follows by easy computation. The case n > 1 is follows by the6 K¨unneth formula. Hence we can assume ρ = ρ . Then hom•((L,b ), (L,b )) = H•(L; Λ) and m (1 )= b b =0. Thus it follows that 1 2 1 2 1 L 2 − 1 6 n dimHF • (L,b1), (L,b2) < 2 .

Since HF • (L,b1), (L,b2) is a Z/2Z-graded HF •(L,b1) HF •(L,b2) bimodule and HF •(L,bi) − op are isomorphic to Cℓ , we see that HF • (L,b ), (L,b ) is a left Cℓ Cℓ module, here n 1 2 n ⊗ n ⊗ is the tensor product of Z/2Z-graded rings. Hence we see HF (L,b ), (L,b ) is a left Cℓ • 1 2 2n module. Since dimensions of non-trivial irreducible representations of Cℓ are 2n, it follows that 2n HF • (L,b1), (L,b2) =0. Combined with Lemma 4.9, we see the following: Proposition 5.5. Under the same assumptions of Lemma 5.4, we have e e =0. (L,b1) ∗ (L,b2) 5.2. Even-dimensional spheres. In 5.2, we assume that n is even and L is an oriented spin rational homology sphere equipped with§ the rank one trivial local system. We ﬁrst recall the following result: Theorem 5.6 ([17, Corollary 3.8.18]). L is weakly unobstructed. odd can Since H (L; Λ) = 0 , b = 0 is a unique Maurer-Cartan element. Since m1,0 is odd degree, we can Z Z see that m1,0 = 0 and HF •(L, 0) ∼= H•(L;Λ) as a /2 -graded vector space. Let pL be the element n of H (L; Z) such that L pL =1. Then pL pL = αL pL + βL 1L for some αL,βL Λ. Using cyclic symmetry, we have · · · ∈ R α = p p = 1 ,mcan(p ,p ) = p ,p =0. L L · L h L 2,0 L L icyc h L Licyc ZL The monotone cases of the next proposition is obtained by Biran-Membrez ([7, Theorem 3.3]). Proposition 5.7. [L]3 =4β [L] L · Proof. By Equation (2.11), we see that 1 , 1 =2, 1 ,p = p , 1 =0, p ,p =2β , h L LiMuk h L LiMuk h L LiMuk h L LiMuk L which implies Z(L,0)(1L)=2 pL and Z(L,0)(pL)=2βL 1L. Hence it follows that · ′ · ′ [L]2 = p (1 ) p (1 ) = ( 1)n/2+? p (Z (1 )) = ( 1)n/2+? 2 p (p ) L,0 L ∗ L,0 L − L,0 (L,0) L − · L,0 L and ′ [L]3 = ( 1)n/2+? 2 p (p ) p (1 )=2 p (Z (p ))=4β p (1 )=4β [L], − · L,0 L ∗ L,0 L · L,0 (L,0) L L · L,0 L L · here we use Assumption 4.3 (4) and Equation (4.2). If β = 0, then we easily see that L 6 1 1 2 (5.1) eL± := [L]+ [L] ±4√βL · 8βL · + Z Z are idempotents and eL eL− = 0. Moreover, we have HF •(L, 0) ∼= Λ Λ as a /2 -graded ring. Since Λ Λ is formal and∗ smooth, we have the following: × × + Proposition 5.8. If β = 0, then e± give ﬁeld factors and e e− = 0. Eigenvalues of these L 6 L L ∗ L idempotents with respect to the operator c1 are WL,0. Moreover (L, 0) split generates Fuk(X)L+ − . ∗ eL +eL COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 19

If there exist two Lagrangian submanifolds, we have the following:

Proposition 5.9. Let L1,L2 be oriented spin Lagrangian rational homology spheres equipped with the rank one trivial local systems such that L1 ⋔ L2.

(1) If [L1] [L2]=0, then [L1] [L2]=0. (2) If [L ] ∪ [L ] =0, then β =∗ β and W = W . 1 ∪ 2 6 L1 L2 L1,0 L2,0 Proof. (1) Set q ([L ]) = a 1 + b p (a,b Λ). Note that L2,0 1 · L2 · L2 ∈ n x n x p (x)= 1 , p (x) = ( 1) ·| | 1 , x = ( 1) ·| | x Li,0 h X Li,0 iX − h Li icyc − ZX ZLi for x HF •((Li, 0)). Since pL2,0 is a module map, we have [L1] [L2]= pL2,0 qL2,0([L1]). Hence it follows∈ that ∗ ◦ [L ] [L ]= p q ([L ]) = ( 1)n q ([L ]) = ( 1)nb. 1 ∗ 2 L2,0 ◦ L2,0 1 − L2,0 1 − ZX ZX ZL2 By assumption, we have [L ] [L ]= [L ] [L ]=0, which implies b = 0 and [L ] [L ] is X 1 ∗ 2 X 1 ∪ 2 1 ∗ 2 proportional to [L ]. Similarly, we see that [L ] [L ] is proportional to [L ]. Since [L ] [L ]= 2 R R 1 ∗ 2 1 X i ∪ i ( 1)n/2 2 and [L ] [L ]=0, [L ] is not proportional to [L ]. Thus we have [L ] [L ]=0. − · 1 ∪ 2 1 2 1 ∗ R 2 ± (2) We ﬁrst consider the case βL1 =0,βL2 =0. In this case, we have four idempotents eLi (i =1, 2). 6 6 + + By assumption, we see [L1] [L2] =0. Hence we can assume e = e . Since ∗ 6 L1 L2 1 e+ = ( 1)n/2 , Li − · 4β ZX Li we have βL1 = βL2 . + We next consider the case βL =0. If βL = 0, then we can assume [L1] e = 0 since [L1] [L2] = 1 2 6 · L2 6 ∗ 6 0. Since e+ is a ﬁeld factor, there exists a non-zero constant a Λ such that [L ] e+ = a e+ . L2 1 L2 L2 3 + 3 + 3 ∈ · · Hence we have [L1] e = a e . This contradicts [L1] =4βL [L1]=0. Hence we have βL =0. · L2 · L2 1 · 2 The proof for the case βL2 = 0 is similar.

Finally, we show WL1,0 = WL2,0. By Equation (4.4), we see that [Li] is an eigenvector of c1 with the eigenvalue W . Since [L ] [L ] = 0, we have W = W . ∗ Li,0 1 ∗ 2 6 L1,0 L2,0 6. Examples 6.1. Toric Fano manifolds. In 6.1, we give a simple proof of the toric generation criterion for toric Fano manifolds with Morse potential§ functions which is due to Ritter [33]. Let N = Zn be a finitely generated free lattice and M be the dual lattice. Set MR := M R. Let ∼ ⊗ (X,ω) be a compact symplectic toric Fano manifold corresponding to a moment polytope P MR. ⊂ The normal fan of P is denoted by Σ. Let v1, v2,...,vm N be the set of primitive generators of rays of Σ. Then { }⊂ P = u MR v ,u + λ 0 (i =1, 2,...,m) { ∈ | h i i i ≥ } for some λ1, λ2,...,λm R. Set ℓi(u) := vi,u + λi. Since X is smooth, we can assume that v , v ,...,v is an integral∈ basis of N. Eachh v isi uniquely written as a v + a v + + a v for { 1 2 n} i 1 1 2 2 ··· n n some a1,a2,...an Z and set ωi := λi a1λ1 a2λ2 anλn. We define a potential function WX by ∈ − − − · · · m W := T ωi ya1 ya2 yan . X 1 2 ··· n i=1 X 1 1 This is an element of the Laurent polynomial ring Λ[N] ∼= Λ[y1,y1− ,...,yn,yn− ]. The Lagrangian torus fiber corresponding to u intP is denoted by L , here intP is the set of interior points of P. ∈ u Lu are equipped with the standard spin structures ([9], [11]). We note that Lu satisfy Condition COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 20

4.15. By the torus action, H1(Lu; Z) is naturally identiﬁed with N and hence Λ[N] is naturally considered as a subset of Λ H (L ; Z) . By [18, Theorem 4.5] (see also [11]), we have hh 1 u ii m W (y ,y ,...,y )= T ℓi(u)ya1 ya2 yan . Lu 1 2 n 1 2 ··· n i=1 X Therefore it follows that

ℓ1(u) ℓn(u) (6.1) WLu (y1,...,yn)= WX (T y1,...,T yn).

Let c = (c ,...,c ) Λn be a critical point of W . We deﬁne u(c) MR by the condition 1 n ∈ X ∈ ℓi u(c) = val(ci) (i = 1, 2,...,n). By [18, Theorem 7.8], it follows that u(c) intP. Set b(c) := val(c ) val(c ) val(cn) 1 ∈ (T − 1 c ,T − 2 c ,...,T − c ). Under the identiﬁcation H L ; Z = M, b(c) gives an 1 2 n u(c) ∼ element of H1 L ;Λ . By Equation (6.1), b(c) is a critical point of W . Moreover, b(c) is u(c) =0 Lu(c) non-degenerate if and only if c is non-degenerate. Theorem 6.1 ([33, Theorem 6.17], see also [14]). Suppose that contains all L such that c is L u(c) a critical point of WX . If WX is Morse, i.e., all critical points are non-degenerate, then L ,b(c) c is a critical point of W { u(c) | X } split generates Fuk(X)L.

Proof. By [18, Theorem 6.1], the number of non-degenerate critical points of WX is dimQH•(X). Hence the theorem follows from Theorem 4.6, Proposition 5.3, and Proposition 5.5. 6.2. Blow-up of CP 2 at four points. Let X be the blow-up of CP 2 at 4 points of general position equipped with the standard complex structure. Then X has a unique symplectic structure such that 2 the cohomology class of the symplectic form ω is equal to c1. Let H H (X; Z) be the pullback 2 2 ∈ of the hyperplane class of CP and E1, E2, E3, E4 H (X; Z) be the Poincar´edual classes of the exceptional divisors. Then we know that c =3H ∈E E E E . 1 − 1 − 2 − 3 − 4 By [30, Theorem 1.4], there exist oriented Lagrangian spheres S1,S2,S3,S4 such that [Si] = E E (i =1, 2, 3), [S ]= H E E E (see also [7, Section 5.1]). Each S is equipped with i − i+1 4 − 1 − 2 − 3 i a unique spin structure. To simplify notation, Si with the Maurer-Cartan element 0 is also denoted by Si. By [31, Theorem 4.25], there exists a monotone Lagrangian torus with a spin structure L such that the potential function WL is given by 1 1 (1 + y + y )(1 + )(1 + ) 3 T 1 2 y y − · 1 2 with respect to an appropriate basis of H1(L; Z). By direct computation, it follows that WL has three critical points b1,b2,b3 with critical values 5+5√5 5 5√5 T, − T, 3 T respectively. Moreover these critical points are non-degenerate. By 2 · 2 · − · Hamiltonian perturbation, we can assume S1,S2,S3,S4,L intersect each other transversely. We assume that contains these ﬁve Lagrangian submanifolds. L Theorem 6.2. Objects S1,S2,S3,S4, (L,b1), (L,b2) split generate Fuk(X)L. Moreover, QH•(X) is semi-simple.

Proof. Semi-simplicity of QH•(X) has already been known ([5], [12]), but we give another proof. By

Proposition 5.3 and Proposition 5.5, it follows that there exist three idempotents e(L,b1),e(L,b2),e(L,b3) such that they are orthogonal to each other and QH (X)e Λ. Since [S ] [S ] = 0, we see • (L,bi) ∼= X 1 1 that [S ], [S ], [S ], [S ], [S ] [S ] are linearly independent. By Equation (4.4) these∗ are eigenvector6 s 1 2 3 4 1 1 R of the operator c . We note∗ that [S ] and [S ] [S ] have the same eigenvalue W . We easily 1 ∗ 1 1 ∗ 1 S1,0 COMPUTATION OF QUANTUM COHOMOLOGY FROM FUKAYA CATEGORIES 21 see that [S ] [S ] = 0 if and only if i j 1. By Proposition 5.9, this implies β = β and i ∪ j 6 | − | ≤ Si Sj WSi,0 = WSj ,0 for each i, j. If βSi =0, then [S1], [S2], [S3], [S4], [S1] [S1] are nilpotent. Since e(L,bi) are linearly independent idempotents, e ,e ,e , [S ], [S ∗], [S ], [S ], [S ] [S ] are linearly (L,b1) (L,b2) (L,b3) 1 2 3 4 1 ∗ 1 independent. This contradicts dimQH•(X) = 7. Thus, we have β = 0. By Proposition 5.8, we Si 6 have two idempotents e± for each S . Since W and W are irrational and W are rational, Si i L,b1 L,b2 Si,0 we see that e is orthogonal to e± for i =1, 2, j =1, 2, 3, 4. By Proposition 5.9 and [Si] [S4]=0 (L,bi) Sj ∪ for i =1, 2, we have [S ] [S ] = [S ] [S ]=0. Combined with Equation (5.1), it follows that e± ,e± 1 4 2 4 S1 S2 ∗ ∗ + + are orthogonal to e± . Since [S1] = [S2], we can assume e = e± . Thus e(L,b ),e(L,b ),e± ,e ,e± S4 6 S2 6 S1 1 2 S1 S2 S4 are idempotents orthogonal to each other. Since dimQH•(X)=7, this implies the semi-simplicity. Let A be the subcategory of Fuk(X)L with objects (L,b1), (L,b2),S1,S2,S4. Then idempotents + e ,e ,e± ,e ,e± are contained in p(HH (A )), which implies 1 p(HH (A )). By The- (L,b1) (L,b2) S1 S2 S4 X A • ∈ • orem 4.6, it follows that split generates Fuk(X)1LX = Fuk(X)L.

References

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Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake- cho, Sakyo-ku, Japan E-mail address: [email protected]