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Fukaya categories in Koszul duality theory SatoshiSugiyama∗ 7 1 3rdJanuary2017 0 2 n a Abstract J 2 Inthispaper,wedefineA∞-KoszuldualsfordirectedA∞-categoriesintermsof twistsintheirA -derivedcategories.Then,wecomputeaconcreteformulaofA - ∞ ∞ ] KoszuldualsforpathalgebraswithdirectedA -typeGabrielquivers. Tocompute G n an A -Koszuldualofsuchanalgebra A,weconstructadirectedsubcategoryof ∞ S aFukayacategorywhichareA -derivedequivalenttothecategoryofA-modules ∞ . and compute Dehn twists as twists. The formula unveils all the ext groups of h t simplemodulesoftheparhalgebrasandtheirhighercompositionstructures. a m Contents [ 1 1 Introduction 2 v 9 2 2 Algebraicpreliminaries 4 4 2.1 BasicdefinitionsandpropertiesofA -categories . . . . . . . . . . . 4 ∞ 0 2.2 DirectedA -categoriesandA -Koszulduals . . . . . . . . . . . . . 5 ∞ ∞ 0 2.3 A -Koszuldualsandtwists . . . . . . . . . . . . . . . . . . . . . . . 7 . ∞ 1 0 3 Geometricpreliminaries 9 7 3.1 DefinitionoftheFukayacategories . . . . . . . . . . . . . . . . . . . 9 1 3.2 Algebraictwistsversusgeometrictwists . . . . . . . . . . . . . . . . 12 : v i 4 Mainresults 12 X 4.1 ComputationofA -Koszulduals . . . . . . . . . . . . . . . . . . . . 12 ∞ r a 4.2 Combinatorialsetup . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 A -Koszuldualsofpathalgebras. . . . . . . . . . . . . . . . . . . . 14 ∞ 4.4 Combinatoriallemmas . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 ConstructionofRiemannsurfacesandLagrangianbranes 17 5.1 Lemmasforconstruction . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Construction(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.3 Construction(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ∗[email protected] 1 6 DirectedFukayacategoriesforRiemanndiagrams 26 7 ComputationofDehntwists 26 7.1 Choiceofrepresentataives . . . . . . . . . . . . . . . . . . . . . . . 27 7.2 Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (cid:122) 7.3 CoreofS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.4 Determinationofdegree . . . . . . . . . . . . . . . . . . . . . . . . 40 7.5 Countingdiscs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7.6 Someexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1 Introduction ThepurposeofthispaperistogiveanewexpressionofA -Koszuldualsofcertainpath ∞ algebraswithrelations(Theorem4.5). WeusethetechniqueoftheFukayacategories andDehntwiststocomputeA -Koszulduals. Ourapproachdoesnotcontainanything ∞ new in the standpoint of the abstract theory of Koszul duality. However, we show thatthetechniqueoftheFukayacategoriescanbeusedforaconcretecomputationof analgebraicproblem. Moreover,ourdescriptioncomputedviatheFukayacategories provides a new way of understanding of Koszul duality as a duality between higher productsandrelations. TheFukayacategoriesareA -categoriesassociatedtosymplecticmanifoldsdefined ∞ byusingthetechniqueofFloertheory[FOOO10],[Se08]. TheFukayacategoriesare mainly studied in the context of homological mirror symmetry [Ko94]. The concept of Fukaya categories emerges in the context of Koszul duality in the paper of A. J. Blumberg, R. L. Cohen, and C. Teleman [BCT09] and the paper of T. Etgu¨ and Y. Lekili[EL16]. ThesepapersstatethatEndA -algebrasoftwocertainobjectsinsome ∞ Fukaya categories are Koszul dual to each other. Therefore, they say that the Koszul dualitypatternsemergeinthecontextofFukayacategories. Inourpaper,thedirection isopposite. WeusetheFukayacategoriestocomputeA -Koszuldualsofpathalgeb- ∞ raswithrelations. ThereforewecansaythatFukayacategoriesemergeinthecontext ofKoszuldualitytheory. Beforeweseethemaintheoremofthispaper,letusreviewthefundamentalresults about Koszul duality in [Lo¨86]. (The results presented here is a simplified version.) Let A = k be a field, A be a finite dimensional vector space and I be a subspace 0 1 of A ⊗A . Define A (cid:66) T(A )/I as the quotient algebra of the tensor algebra of A 1 1 1 1 over A = k. Then, we have E (cid:66) Ext (k,k) (cid:27) T(A∗)/I⊥, where (−)∗ is the linear 0 A 1 dual over k and I⊥ ⊂ A∗ ⊗A∗ is the annihilating submodule of I ⊂ A ⊗A (we use 1 1 1 1 thenaturalisomorphismbetweenA∗ ⊗A∗ and(A ⊗A )∗). Letusfixanisomorphism 1 1 1 1 betweenA andA∗.Then,IandI⊥aremutuallycomplemental.Hence,wecansaythat 1 1 the products and relations interchange between A and E. By the above computation, Ext (k,k)isnaturallyisomorphictoA. ThisiswhatwecallKoszuldualityandwecan E saythatKoszuldualityisadualitybetweenproductsandrelationsrepresentedbythe Yoneda Ext algebra. Moreover, certain derived categories of A and E are equivalent [BGS96]. (In that paper, the setting above is generalized to the case of that A is a 0 finitedimensionalsemi-simplealgebra.) 2 Nowadays,manyphenomenarelatedtotheKoszuldualityarewidelyobserved,for example, the Koszul duality for Koszul algebras [Pr70], [Lo¨86], [BGS96], its generalisation to augumented-A algebras [LPWZ04], a generalisation to Koszul operads ∞ [GK94], [Va07], [LV12], anditsrelationtothestudyofsymplecticgeometry[EL16] andmirrorsymmetry[AKO08]. Inthispaper,weareinterestedinthecasethatthereexisthigherdegree(homogen- (cid:76) ous)relation,i.e. forthealgebraA = T(A )/I withI((cid:49) A⊗2) ⊂ A⊗d. Ingeneral, 1 1 d≥2 1 thereisnoeasydescriptionofE. Moreover,theextalgebraExt (A ,A )andAareno E 0 0 longerisomorphic.However,wecanovercomethisdifficultybyreferringtheresultsin [LPWZ04].TheygeneralisetheconceptofKoszuldualtotheaugmentedA -algebras. ∞ Afterthat,theyprovethatthetwicedualisquasi-isomorphictotheoriginalaugmented A -algebra and their derived categories are equivalent (under some finiteness condi- ∞ tion). TheabovealgebraAisanexampleofanaugmentedA -algebra,sowehaveits ∞ dual. ButthedescriptionistoocomplicatedandwecannotinterprettheKoszuldual asthedualitybetweenproductsandrelations. In this paper, we define the notion of A -Koszul dual for directed A -categories ∞ ∞ (Definition2.4)andpresentanexplicitdescriptionof A -Koszuldualofcertainclass ∞ ofpathalgebraswithrelations(Theorem4.5)whichenableustounderstandtheKoszul duality as a duality between higher products and relations. The notion of A -Koszul ∞ dualisanaturalgeneralisation. Thisissupportedbythefollowingtwocorollaries: the A -KoszuldualCofBisnaturallyquasi-isomorphictoA(Corollary4.3); Aandits ∞ KoszuldualBareA -derivedequivalent,i.e. TwA(cid:27)TwB(hence,inparticular,they ∞ arederivedequivalent,i.e. DA(cid:27) DB)(Corollary4.2). The computation of the A -Koszul dual takes place in the Fukaya categories of ∞ exactRiemannsurfaces.Theroughsketchofthecomputationisasfollows.Ingeneral, theKoszuldualcanbecomputedbytheoperationinthederivedcategorycalledtwist. Firstwe“embed”ourdirected A -categoryA = A(R)intotheFukayacategoryF = ∞ Fuk(M)ofanexactRiemannsurfaceMconstructedbyusingthedataofrelationsofR. Seidelprovedin[Se08]thatthetwistsare“quasi-isomorphic”totheDehntwistsinthe Fukayacategory. Thus,wecomputetheDehntwistsoftheobjectswhicharelyingin theimageofthe“embedding”A(cid:44)→ Fuk(M).Finally,weinvestigatehowtheresulting curvesintersectandencirclepolygonstocomputethemorphismspacesandtheirhigher compositions. Afterthat, wefind thatthere isa (d+1)-gon in M corresponding toa degreedrelation,andthe(d+1)-gongeneratesthed-thhighercompositionµd. Thisis ourgeometricexplanationofthedualitybetweenhigherproductsandrelations. Some typicalexampleispresentedinCorollary4.7andSubsection7.6. Here, we fix some notations we often use. In this paper, k is a fixed field; all categories are of over k; all graded vector spaces are assumed to have the property that theie total dimensions are finite; for a graded vector space V = (Vd)d∈Z, V[r] (cid:66) (Vd+r)d∈Z is the r-th shift of V; all modules are always right modules; all manifolds are oriented; all the additional structures on manifolds are assumed to be compatible withtheirorientations;thecharacterF alwaysstandsfortheFukayacategoryFuk(M) of M where M is “the” exact symplectic manifold we consider in each paragraph; if M hassomesubscriptslike M thenF standsfortheFukayacategoryof M , unless 1 1 1 otherwisestated. The structure of this paper is as follows. In section 2, we prepare the algebraic 3 notionsanddefinetheA -Koszuldual. Insection3,wepreparethegeometricnotions, ∞ e.g. exact symplectic manifolds and their Fukaya categories. At the last part of the section,wepresentthekeytheoremprovedbySeidelwhichstatestheequivalenceof algebraictwistsandDehntwists. Insection4, westatethemaintheorem. Insection 5and6,weconstructexactRiemannsurfaceswhoseFukayacategoriesarethetargets ofthe“embedding”fromdirectedA -categories. Insection7,wedothecomputation ∞ of A -Koszul duals, i.e. the computation of Dehn twists. The computation and the ∞ formulaofA -Koszuldualsarethemainingredientsofthispaper. ∞ Acknowledgement I would like to thank my supervisor Toshitake Kohno for giving me great advice andnavigatingmeandthisstudytoanappropriatedirection. IalsowanttothankA. Ikeda for teaching me about Koszul duality theory and to F. Sanda, M. Kawasaki, T. Kuwagaki,J.Yoshida,andR.Satoforfruitfuldiscussion. Finally,Iamdeeplygreatful tomyfriendI.Hoshimiya,A.Kiriya,R.Shibuki,andY.Todoforsupportingmewhen Iwasindifficultsituations. This work was supported by the Program for Leading Graduate Schools, MEXT, Japan. 2 Algebraic preliminaries In this section, we review the definitions of algebraic objects we use in this paper, and define the A -Koszul dual, the key concept in this paper. For the notation of ∞ signs,wefollowSeidel’snotationin[Se08]. ThedefinitionoftheKoszuldualforA - ∞ algebraswithsomepropertiesalreadyexists[EL16],[LPWZ04]. Ourconstructionisa generalisationtodirectedA -categories. ∞ 2.1 Basicdefinitionsandpropertiesof A -categories ∞ Definition2.1(A -category) AnA -categoryA,consistsofthefollowingdata: ∞ ∞ 1. asetOb(A), (cid:77) 2. aZ-gradedvectorspacehom (X,Y)= homi (X,Y)foreachX,Y ∈Ob(A), A A i∈Z 3. mapscalledhighercompositionmaps µd: hom (X ,X )⊗hom (X ,X )⊗···⊗hom (X ,X ) A d−1 d A d−2 d−1 A 0 1 →hom (X ,X )[2−d], A 0 d ford ≥1andX ,X ,...,X ∈Ob(A). 0 1 d Weimposethattheµ’ssatisfytheA -associativityrelation: ∞ (cid:88) (−1)(cid:70)iµl(ad,...,ai+j+1,µj(ai+j,...,ai+1),ai,...a1)=0 i,j,l (cid:88) ford ≥1,where(cid:70) = (|a|−1), (|a|=deg(a)). i l i i 1≤l≤i 4 Let us see the first few A -relations. The A -relation of d = 1 is µ1(µ1(a )) = 0 ∞ ∞ 1 and deg(µ1) = 2−1 = 1. Hence, (hom (X ,X ),µ1) forms a cochain complex. The A 0 1 secondcase,therelationisµ1(µ2(a2,a1))+µ2(a2,µ1(a1))−(−1)|a1|µ2(µ1(a2),a1) = 0. Whenwewriteda=(−1)|a|µ1(a)anda2◦a1 =(−1)|a1|µ2(a2,a1),therelationiswritten by d(a2 ◦a1) = da2 ◦a1 +(−1)|a2|a2 ◦da1. Thus, the second relation expresses the gradedLeibniz’rule. Ifallthehighercompositionmapsarezero,i.e. µd =0ford ≥3, thentheA -categoryisnothingbutadgcategorybytheabovedand−◦−. Therefore, ∞ thenotionofA -categoriesisageneralisationofdgcategories. ∞ Thethirdrelationissomewhatcomplicated: a ◦(a ◦a )−(a ◦a )◦a =±d(µ3(a ,a ,a ))±µ3(da ,a ,a )+(othertwoterms). 3 2 1 3 2 1 3 2 1 3 2 1 Ingeneral,therighthandsidedoesnotvanish,sothecompositiondefinedbyµ2isnot associative.However,µ3formsahomotopybetweena ◦(a ◦a )and(a ◦a )◦a ,hence 3 2 1 3 2 1 µ2definesanassociativecompositiononcohomologylevel.Wedefinethecohomology category H(A) by Ob(H(A)) (cid:66) Ob(A), hom (X ,X ) (cid:66) H(hom (X ,X ),µ1), H(A) 0 1 A 0 1 and [a2]◦[a1] (cid:66) (−1)|a1|[a2 ◦a1]. The resulting category H(A) has an associative composition. Thus,wesaythatµ2 ishomotopyassociative. Wealsodefine H0(A)in theobviousway. We don’t assume that the A -category admits identity morphisms, so H(A) and ∞ H0(A)maynothaveidentitymorphisms. IfH(A)admitsidentitymorphismsforeach object,thenwesaythatAiscohomologicallyunitalorc-unital. Inthispaper,allthe A -categoriesareofc-unitalunlessotherwisestated. Wesaythattwoobjects X and ∞ 0 X inanA -categoryarequasi-isomorphiciftheyareisomorphicinH0(A). 1 ∞ We do not present the definitions of A -functors, quasi-equivalences, and quasi- ∞ isomorphismsof A -categorieshere. Thesearegeneralisationsinthecaseofdgcat- ∞ egories. Forprecisedefinitionandproperties,pleasereferSection1and2in[Se08]. 2.2 Directed A -categoriesand A -Koszulduals ∞ ∞ Inthispaper,wemainlyconsiderthedirectedA -categories. ∞ Definition2.2 AnA -categoryAissaidtobedirectedwhen ∞ 1. thesetOb(A)isfinite, 2. hom (X,X)=k·1 ,and A X 3. there exists a total order on Ob(A) such that the hom space hom (X,Y) (cid:44) 0 A onlywhenX ≤Y. Foratotallyorderedfiniteset A,wehaveacanonicalisomorphism A (cid:27) {0 < 2 < ···<n}. ThereforewewritetheobjectsofandirectedA -categoryas0<1<···<n, ∞ X < X <···< X ,andsoon. 0 1 n 5 Definition2.3 Let A be an A -category and Y = (Y ,Y ,...,Y ) be a collection of ∞ 1 2 n objects in A. Then, we define the associated directed subcategory A→(Y) of A by settingOb(A)={Y ,Y ,...,Y }, 1 2 n  homA→(Y)(Yi,Yj)=0hko·meiA(Yi,Yj) (((iii>=< jjj))), andµ’sofA→(Y)arecanonicallyinducedfromthoseofA. Now, we begin the definition of A -Koszul duals. For an A -category A, we ∞ ∞ call an A -functor M from Aop to C(k) (cid:66) Cb (k) a (right) A -module, where Aop ∞ dg ∞ is the opposite A -category of A and Cb (k) is the dg category of bounded cochain ∞ dg complexes of finite dimensional k vector spaces considered as an A -category. It is ∞ known that such A -modules form a triangulated dg category Q (cid:66) mod(A). (Note ∞ that all the hom spaces of this category are finite dimensional iff #Ob(A) < ∞.) Let A be a directed A -category with its object set {0 < 1 < ··· < n}. We define an ∞ A -moduleS(j)for j∈Ob(A)determinedbythedata ∞  S(j)(i)=k ifi= j, 0 ifi(cid:44) j andcallitasimpleA-modulecorrespondsto j,whereweconsiderkasaone-dimensional cochain complex concentrated in the degree zero part. Then, it is known that the full sub A -category A! of Q with object set {S(n) < S(n − 1) < ··· < S(0)} ∞ ∞ forms a directed A -category. Hence, A! is naturally isomorphic to Q→(S), where ∞ ∞ S=(S(n),S(n−1),...,S(0))isacollectionofobjectsinQ. Thedetailscanbefound in(5j)and(5o)in[Se08]. Definition2.4 LetAbeadirected A -categorywithobjectset{0 < 1 < ··· < n}. A ∞ directedA -categoryBquasi-isomorphictoA! iscalledanA -KoszuldualofA. ∞ ∞ ∞ Remark2.5 Theabovedefinitionisananalogyoracategoryversionofthedefinition in[BGS96]. Inthatpaper,theydealwithKoszulringAandgiveadifferentdefinition ofitsKoszuldual A!. However, Theorem2.10.1inthatpaperstatesthatExt•(k,k) (cid:27) A (A!)opp canonically. Eventhoughthereexistmanydifferentnotations,wecantranslate fromonetotheother. Also, our definition is an analogy of the definition in [LPWZ04]. In that paper, they define Koszul dual E(A) (in their notation) for Adams connected A -algebra A ∞ by E(A) (cid:66) RHom (k,k). The right hand side of the definition is a straightforward A◦ generalisationofthedefinitionin[BGS96], hencethedefinitionsinthatpaperandin ourpapersharesthecommonorigin. In this paper, we treat with A -categories, not A -algebras and we focus on the ∞ ∞ veryspecialcase,directedA -categories. ∞ 6 →− Example2.6 Let R = k(∆,ρ) be a path algebra with relations over a finite direc- →− ted quiver ∆. Here, a finite quiver is a quiver with a finite set of vertices (which we write ∆ ) and a finite set of arrows (which we write ∆ ); a directed quiver is a 0 1 quiverwithoutorientedcycles. WecanseeRasanA -categoryA = A(R)bysetting ∞ Ob(A) = ∆ , hom0 (i, j) = e Ae, homd (i, j) = 0 for d (cid:44) 0, µ2 is induced from the 0 A j i A product structure of A, and µd = 0 for d (cid:44) 2. (We write the product of two paths α fromito jandβfrom jtolasβαlateron.) Now,thedimensiondim Rasakvector k →− space is finite since its quiver ∆ has no oriented cycles. Thus, we can deduce that mod(A) and C(R) (cid:66) Cb (R) are naturally isomorphic as triangulated dg categories, dg whereCb (R)isthedgcategoryoffinitelygeneratedR-modules. (Recallthatafunctor dg fromA(R)op (consideredask-linearcategory)tothecategoryoffinitedimensionalk vector spaces vect(k) can be naturally considered as a right R-module.) The natural isomorphismmapsS(j)inmod(A)for j ∈ ∆ intothesimplemoduleS(j)inCb (R) 0 (cid:16)(cid:76) (cid:76) dg (cid:17) corresponds to j ∈ ∆ . Set a graded algebra R! (cid:66) hom∗ S(cid:101)•(j), S(cid:101)•(j) , 0 dg Cb(R) dg whereS(cid:101)•(j)isaprojectiveresolutionofS(j)andthedirectsumistakenover∆ . We 0 call it the dg Koszul dual of R. Then we can compute A(R)! by Ob(A(R)! ) = ∆ , (cid:16)(cid:76) (cid:76) (cid:17) ∞ ∞ 0 homd (i, j)=homd S(cid:101)•(i), S(cid:101)•(j) ,andµ’sareinducedfromthedifferen- A(R)!∞ Cdbg(R) tialdandtheproductstructure−·−ofR! . dg If our algebra R is Koszul, equivalently, the relations are of quadratic, then the cohomology algebra H(R! ) is nothing but the Koszul dual R! of R. Hence, the dg dg Koszul dual R! is a generalisation of the Koszul dual to general path algebras over dg finite directed quivers. The dg Koszul dual R! can be reconstructed from A(R)! by (cid:77) dg (cid:16)(cid:77) (cid:77) (cid:17) ∞ R! = hom (S(i),S(j)) = hom S(i), S(j) , wherethelast dg A(R)!dg mod(A(R)!∞) i,j∈∆0 twodirectsumsaretakenover∆ . 0 Wefinishthissubsectionbycollectingsomeusefullemmasfrom[Se08]. Lemma2.7((5n)in[Se08]) Let F: A → B be a cohomologically full and faithful (c-full and faithful in short) A -functor and Y = (Y ,Y ,...,Y ) be a collection of ∞ 1 2 n objectsin A. Then, there existsa canonicalquasi-isomorphism betweenA→(Y) and B→(FY),whereFY =(FY ,FY ,...FY ). 1 2 n Lemma2.8(Lemma5.21in[Se08]) LetY andY(cid:48) becollectionsofobjectsinAand theseobjectsarepairwisequasi-isomorphic,i.e. Y (cid:27) Y(cid:48) inH0(A)forevery j. Then, j j theassociateddirectedsubcategoriesA→(Y)andA→(Y(cid:48))arequasi-isomorphic. 2.3 A -Koszuldualsandtwists ∞ In this subsection, we develop the method to compute an A -Koszul dual of a given ∞ directedA -categoryA.AllthedetailsandprecisedefinitionscanbefoundinChapter ∞ Iof[Se08]. First, we fix some notations. For an A -category A, we define the category of ∞ A-modules Q (cid:66) mod(A) = fun(Aop,C(k)). For such categories, we can define the 7 Yoneda embedding functor ι: A → Q, by setting (ιX)(Y) = hom (Y,X). We set A the triangulated A -category TwA by the full subcategory generated as triangulated ∞ A -category by the objects which are lying in the image of the Yoneda embedding ∞ ι(Ob(A)). Now, we have three embeddings of A -categories, A (cid:44)→ TwA (cid:44)→ Q = ∞ mod(A). Thesethreeembeddingsareknowntobec-fullandfaithful. ForX ∈ Ob(A)andM ∈ Ob(Q),wecandefinethetwistof MalongX ,whichis wtrittenbyT M,bythemappingconeoftheevaluationmorphismιX⊗hom (ιX,M)→ X Q M. This is a generalisation of the case when A = A(R) as in Example 2.6. If there existsZ ∈Ob(A)suchthatιZandT (ιY)arequasi-isomorphic,wewriteZ =T Yand X X call it a twist of Y along X. This is a fact that TwA is closed under twist. There are tworemarksonthenotionoftwists. ThefirstoneisthatsuchaZ maynotbeunique. ThereforewheneverwewriteT Y,wechooseoneofsuchobjects. Thesecondoneis X thatwhenwewriteT Y,wealwaysassumetheexistenceoftherepresentativeofT ιY. X X Finally,thefollowingholds: Lemma2.9(Lemma5.24. in[Se08]) Let A be a directed A -category with object ∞ set{X < X < ··· < X },andsetS(cid:48)(j) (cid:66) T T ···T X ∈ Ob(TwA) (cid:44)→ Ob(Q). 0 1 n X0 X1 Xj−1 j ThentheresultingobjectS(cid:48)(j)isquasi-isomorphic(inQ)tothesimplemoduleS(X ). j This lemma is a generalisation of the case that the category A is a directed A - ∞ category A(R) associated with a path algebra with relations R over a finite directed quiver. By this lemma, we can compute an A -Koszul dual by iteration of twists. ∞ We abbreviate S(cid:48)(j) into S(j). Together with the definition of A! for directed A - ∞ ∞ categoryA,onehasanaturalisomorphismbetweenA! and(TwA)→(S),whereS= ∞ (S(n),S(n−1),...,S(0)). Wefinishthissectionbyrecallingusefullemmas. Lemma2.10(Lemma5.6in[Se08]) SupposeF: A→Bbeac-fullandfaithfulA - ∞ functorandtheseY andY beobjectsinA. Then,thereexistsacanonicalisomorph- 0 1 isminH0(B)betweenF(T Y )andT F(Y ). Y0 1 F(Y0) 1 Lemma2.11(Lemma5.11in[Se08]) Suppose that Y is a spherical object in A. 0 Then,T isaquasi-equivalencefromAtoitself. Y0 Corollary2.12 Let Y and Y be objects in A and Y is spherical. Then, for any 0 1 0 object Z ∈ Ob(A), there exists a natural quasi-isomorphism between T Z and TY0Y1 T T T−1Z. Y0 Y1 Y0 Here,thedefinitionofasphericalobjectscanbefoundin(5h)in[Se08]. For a collection of objects Y = (Y ,Y ,...,Y ) in an A -category A, we define 0 1 n ∞ a new collection LjY (cid:66) (Y0,...,,Yj−1,TYjYj+1,Yj,Yj+2,...,Yn) in A and call it a mutationofY. Lemma2.13(Lemma5.23in[Se08]) LetY =(Y ,Y ,...,Y )beacollectionofspher- 0 1 n ical objects in an A -category A and define U (cid:66) L Y. Then there is a quasi- ∞ j equivalence between Tw(cid:0)A→(Y)(cid:1) and Tw(cid:0)A→(U)(cid:1). In particular, there is a equival- enceofderivedcategoriesbetweenD(A→(Y))andD(A→(U))astriangulatedcategor- ies. 8 Lemma2.14 Let Y = (Y ,Y ,...,Y ) be a collection of spherical objects in an A - 0 1 n ∞ categoryAanddefineU (cid:66)L Y. LetuswriteY asanobjectinA→ (cid:66)A→(Y)asY(cid:101) , j i i thecollectionofthemas(cid:101)Y =(Y(cid:101),Y(cid:101),...,Y(cid:101)),andthemutationasU(cid:101)(cid:66)L (cid:101)Y(thetwist 1 2 n j takesplaceinA→,notinA). Then,thereexistsaquasi-isomorphismbetweenA→(U) and(A→)→(U(cid:101)). Conceptually,thislemmasaysthatforsphericalobjectsthetwistinAandA→are equivalent in the above sense. This lemma is proved in the proof of Lemma 5.23 in [Se08]. 3 Geometric preliminaries In this section, we prepare the notation of the Fukaya categories of exact Riemann surfacesanddiscussthetwistsintheFukayacategories. 3.1 DefinitionoftheFukayacategories Thedefinitionitselfcanbefoundin[Se08]anditscombinatorialdescriptionwhichwe mainlyusecanalsobefoundin[Su16]. However,werepeattherelevantpartsofthose papersforthesakeofcompleteness. An exact symplectic manifold M = (M,ω,θ,J) consists of a symplectic manifold with non-empty boundary (M,ω), a primitive θ of ω, i.e. θ is an 1-form satisfying dθ=ω,andanω-compatiblealmostcomplexstructureJ. Weimposethatthenegative Liouville vector field X , defined by ω(−,X ) = θ(−), points strictly inward on the θ θ boundary∂M. Now, wesee the definition of theFukaya category F = Fuk(M) of agiven exact Riemannsurface M. Infact, weonlyusetheFukayacategoryF oftheformF→(L) for some collection of objects L in this paper, hence what we really need to define is as follows: the set of objects Ob(F), the hom spaces hom (L#,L#) for two dis- F 0 1 tinctobjectsL#,L# ∈Ob(F),andthehighercompositionmapsµd: hom (L# ,L#)⊗ 0 1 F d−1 d hom (L# ,L# )⊗···⊗hom (L#,L#)→hom (L#,L#)formutuallydistinctobjects. F d−2 d−1 F 0 1 F 0 d To define the objects of the Fukaya category F = Fuk(M) of an exact Riemann surface M, we fix a trivialization of TM as a complex line bundle (this is possible since M possessesnon-emptyboundary). Thankstothecomplexstructure J, wecan identify the trivialization with a non-vanishing vector field X. Let L (cid:27) S1 (cid:44)→ M˚ be (cid:82) a Lagrangian submanifold. We say that L is exact when θ = 0. Let η: [0,1] → L M be a compositon [0,1] (cid:44)→ R → R/Z = S1 (cid:44)→ M representing L. We choose dη a function (cid:101)α: [0,1] → R such that dt(t) ∈ R>0 · (eπi(cid:101)α(t)Xη(t)) ⊂ Tη(t)M holds. Set w(L) (cid:66) (cid:101)α(1) −(cid:101)α(0) and call it the writhe of L. We say that L is unobstructed if w(L)=0. ForanexactunobstructedLagrangiansubmanifoldL,wedefineitsgrading α: L → R by α(η(t)) = (cid:101)α(t). We call a triple L# = (L,α,p) of an unobstructed Lagrangian submanifold L, its grading α, and arbitrary point p ∈ L a Lagrangian brane. Here,wecallthethirdcomponentoftheLagrangianbrane paswitchingpoint. Finally, we define the set of objects Ob(F) of F by the set of all Lagrangian branes. 9 Note that a grading α of a Lagrangian brane L# defines a new orientation of L by (p (cid:55)→ eπiα(p)X ) ∈ Γ(TL). Wecallitthebraneorientation. Wecanseethatafunction p α[n](p)(cid:66)α(p)−nforn∈ZisanothergradingofL. Wecallitthen-foldshiftofα Remark3.1 TherearefewdifferencesinthedefinitionofobjectsofFukayacategor- ies between Seidel [Se08] and this paper. In Seidel’s definition, one uses a quadratic volumeformη2 whichisasectionof(∧topT∗M)⊗2 (wherethetensorproductistaken M overC)whileweuseanon-vanishingvectorfieldX. Therelationofthesetwoisgiven byη2 (X⊗X) = 1. Then,ourgradingαisnothingbutagradingα# ofSeidel’ssense. M TherelevantconstructionsareverysimplifiedfromSeidel’snotationinthispapersince weonlytreatwithexactRiemannsurfaces(whileSeidelconsideredexactsymplectic manifoldsofanydimension). A Lagrangian brane in Seidel’s sense is a triple L# = (L,α#,P#), here L and α# arethesamewithourdefinitionbut P# isaPinstructureof L. Inthedefinitionofthe Fukayacategories,thePinstructuresareusedforthedeterminationoftheorientationof themodulispaces. Hencetheyareusedforthedeterminationofthesignofthehigher compositionmapsµ’s. Todeterminetheorientationofthemodulispaces,Seideluses a real line bundle β associated to the Pin structure P#. In the case of exact Riemann surfaces, the Pin structure P# must be non-trivial in order to achieve Theorem 3.5 so weassumethat. Thereforethereallinebundleβinourpaperisalwaysthenon-trivial one. Corresponding to that, a fixed point p ∈ L is used as follows. Since our real line bundleβisnottrivial,wecannottrivializeβonwholeLbutwecanonL\{p}.Withthis trivialization, we can consider that the orientation of β changes when we go through the point p. This is the meaning of p. The choice of a point p does not cause the difference of objects, i.e. two Lagrangian branes L# = (L,α,p ) and L# = (L,α,p ) 0 0 1 1 are quasi-isomorphic in F. In the comparison with the definition by Seidel, we fix a trivializationofareallinebundleβinsteadoffixingofthePinstructure,sowehaveas “S1timesmany”objectsasSeidel’sdefinition. Next,wedefinethehomsetfromL# =(L ,α ,p )toL# =(L ,α ,p ). Fromnow 0 0 0 0 1 1 1 1 on, we assume that any collection of Lagrangian branes is in general position unless otherwise stated, i.e. any two submanifolds intersect transversally, there is no triple point, and the switching point of one Lagrangian brane never contained in the other Lagrangian branes. In this assumption, the hom set is defined by hom (L#,L#) (cid:66) (cid:77) F 0 1 k·[p] as a vector space. Here, [p] is a formal symbol corresponds to p. We p∈L0∩L1 sometimesabbreviate[p]into p. Foranintersectionpoint p ∈ L ∩L asamorphism 0 1 fromL#toL#,wedefineitsindexbyi(p)=[α (p)−α (p)]+1andsethomd(L#,L#)(cid:66) (cid:77) 0 1 1 0 F 0 1 k·[p]. p∈L0∩L1, i(p)=d Finally,wedefinetheA -structureµ’s. Thisisjustarepetitionbutweonlydefine ∞ themapsµd: hom (L# ,L#)⊗hom (L# ,L# )⊗···⊗hom (L#,L#)→hom (L#,L#) F d−1 d F d−2 d−1 F 0 1 F 0 d undertheconditionthatL (cid:116) L and p (cid:60) L fori(cid:44) j. i j i j Let∆d+1bea(d+1)-gon. Wenameitsverticesv ,v ,...,v counterclockwise,the 0 1 d verticesconnectingvj−1 andvj by[vj,vj+1](0 ≤ j < d),andthevertexconnectingv0 10