of special isogenous tori Equivariant split generation and mirror symmetry Advances in Mathematics

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Advances in Mathematics

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Equivariant split generation and mirror symmetry of special isogenous tori

Weiwei Wu

DepartmentofMathematics,UniversityofGeorgia,Athens,GA30602,USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received29January2015

Receivedinrevisedform23October 2017

Accepted24October2017 Availableonline7November2017 CommunicatedbyL.Katzarkov

Keywords:

Fukayacategory

Homologicalmirrorsymmetry Symplectictori

WeproveaversionofequivariantsplitgenerationofFukaya categorywhenasymplecticmanifold admitsafreeactionof afinitegroupG.Combiningthiswithsomegeneralizationsof Seidel’salgebraicframeworksfrom[35],weobtainnewcases ofhomologicalmirrorsymmetryforsomesymplectictoriwith non-split symplectic forms, which we call special isogenous tori.Thisextends theworkof Abouzaid–Smith[2].Wealso showthatderivedFukayacategoriesarecompleteinvariants ofspecialisogenoustori.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

InthispaperweinvestigatetherelationbetweentheFukayacategoryofasymplectic manifoldand itsfinite coverings.This aspect ofFukaya categorieshas been considered inmanydifferentperspectives.Forexample,in[32]SeidelrelatedanequivariantFukaya categoryof abranchedcoverover aquartic surfacetotheonedownstairs,whichled to thefirstproofofhomologicalmirrorsymmetryofcomplexdimensionhigherthan1.This ideawasexploitedfurtherin[36][39][12][11][38]formanyotherinstancesofhomological mirrorsymmetry.Anothercloselyrelatedresultwasgivenin[30],whereAlexRitterand

E-mailaddress:[email protected].

https://doi.org/10.1016/j.aim.2017.10.036 0001-8708/©2017ElsevierInc.Allrightsreserved.

Ivan Smithshowedthatforafinite coveringπ:X →X withdecktransformation G, if BisacollectionofLagrangianbranesthatsplitgeneratestheFukayacategoryFuk(X), then theirliftsalsosplit generatesFuk(X).

We would like to addressingeneral theother direction ofRitter–Smith’s result: let π:X →X be afinitecovering, ifB splitgeneratesFuk(X),dotheirimagesunderthe projectionπalsosplitgenerateFuk(X)?Theanswerisbothyesandno:oneimmediately notices thatnot all images inπ(B) are guaranteed to be embedded thus are noteven objectsofFukayacategoryofXinthecommondefinition.However,itisconceivablethat, ifoneiswillingtoincludeimmersedobjectsintheformulationoftheFukayacategory,the result shouldstillhold byestablishinganappropriate versionofAbouzaid’sgeneration result[1].

Theapproachweadoptinthisarticleistechnicallysimpler.Wefirstdefineaversion of equivariant Fukaya category Fuk(X)^G. The particular formulation we use here is very close to that in [12], to which we will compare in Section 4. This category, as promised,atechnicalreplacementoftheimmersedFukayacategory ofX, inwhich the extraimmersedobjectsarereplacedbyG-orbitsoftheirliftsinX,hencesimplifyingthe situation.

The main tool of our study of the equivariant Fukaya categories is the following G-equivariant version of Abouzaid’s generation criterion. Throughout,we work over a groundfieldK,suchthatchar(K) iscoprimewith |G|.

Theorem1.1(Corollary 4.7).LetBbeasubcategoryofFuk(X).Iftheopen–closedstring map OC : HH_∗(B) → HF^∗(X) hits the identity, then G· B split generates Fuk(X)^G over agroundfield K,suchthat char(K)iscoprimewith |G|.

Following[1],wesaythesubcategoryBresolvesthediagonalifitsatisfiesthecondition inTheorem 1.1.To relatethisequivariantversion ofFukayacategory tothe usualone, weconsiderthefollowingfullyfaithfulfunctorfromFuk(X) totheG-invariantcategory Fuk(X)^G.

Theorem1.2(Theorem 4.6).ThereisatransferfunctorT :Fuk(X)→ Fuk(X)^G which is fulland faithful.

ThefunctorT isnaturallydefinedusingtheclassicaltransfermapforcoveringspaces in the Floer context. By definition, it is easy to see that T is an equivalence in the following circumstance:

Corollary 1.3 (Corollary 4.9). If B ∈ Ob(Fuk(X)) resolves the diagonal, and π(L) is embeddedforanyL∈ B,thenthecollectionπ(B)splitgenerates Fuk(X)overaground field K,suchthat char(K)iscoprimewith |G|.

Ournexttaskisto understandsomeapplicationsof thisequivariantsplitgeneration mechanism. We arefirst interestedin ageneralalgebraic reductionscheme of amirror

functorwhenthesymplecticside isequippedwithafreefiniteG-action. Wesetupthe probleminaratheralgebraic mannerasfollows.

Givenanequivalence functorF :C → D betweentriangulated categories,where C is endowedwithastrictG-actionfor afinite groupG. Ddoesnotinherit anaturalstrict G-action from F ingeneral, but only acoherent G-action. We showedthat, when the G-action can be lifted up to A_∞-level in an appropriate sense, then one may reduce F to a fully faithful functor F^{f ix} from C^{f ix} to D^{f ix}, while the two fixed (non-full) subcategories and F^{f ix} are again shown to be triangulated. Also, to obtain a more natural set up to study D^{f ix} when D is only endowed with a coherent G-action, we proposeto considerastrictification of G-actionD^strict of DinSection2.3. Weproved the existenceof a strictificationmodel for any finitely generated groups (although this willnotbeusedintherestofourpaper).TheseareagainreminiscentofanideaofSeidel in[35, (14b)], where heconsiders the casewhen G=Z/2.Combining theresultsfrom thefirst part, givena mirrorfunctor m :D^πFuk(X)→D^bCoh(X^∨) with afinite free G-actiononX, wehavefully faithfulfunctorsm^{f ix}:D^πFuk(X/G)→D^b(X^∨)^{f ix} and m¯^{f ix}:D^πFuk(X/G)→(D^b(X^∨)^strict)^{f ix}.

Wethenturntoanewcaseofhomologicalmirrorsymmetryfollowingasuggestionof PaulSeidel.Wecall asymplecticform ω_lin ofR²ⁿ linear,ifitscoefficientsareconstant everywhere.Forany linearsymplecticforms andafulllatticeΓ<R²ⁿ, (R²ⁿ,ωlin)/Γ is asmoothT²ⁿ endowedwithaquotientsymplecticform,whichbyabuseofnotationwill bedenotedasω_lin again.Suchsymplecticforms onT²ⁿ willalsobecalled linear.

Notethatthe symplectomorphismtypeoflinearsymplecticforms onT²ⁿ aredeter- minedcompletely by linearalgebra.Namely, asymplectomorphismbetweentwo linear symplecticformsinducesalinearmaponH¹(T²ⁿ,R).Fixinganintegralbasisinthefirst cohomologygroups,suchamapbelongstoGL(2n,Z),whichshowsthatthecohomology classesof thetwo linearsymplecticforms are congruent(asan anti-symmetric bilinear form)bysuchamatrix.Sincelinearsymplecticforms arecompletelydeterminedbythe cohomologyclasses,onemayindeedchooseasymplectomorphismwhichliftstoalinear transformationontheuniversalcover.

Themaincomputationconcernsthemirrorsymmetryofaspecialtypeoflinearsym- plectictori,whichwecall thespecialisogenoussymplectictori,denotedT( ¯α)¯l.Roughly speaking, these are tori finitely covered by split symplectic tori in aspecific way. On theB-side, we consider abelian varietiesA( ¯α)¯l overthe Novikovfield, whichare again isogenous inaspecific way to split analytic tori. We referthe reader to Definition 5.1 and Example 5.6 for the precise definitions. Our main homological mirror symmetry statementreads:

Theorem1.4. D^π(Fuk(T( ¯α)¯l))isequivalent toD^b(A( ¯α)¯l).

The reconstruction theorem due to Bondal and Orlov [9] shows that the derived category ofanalgebraic varietywithample (anti-)canonicalline bundlecompletely de- termines the variety. In other words, the derived category is a complete invariant of

varieties of this sort. In contrast, this is not the case for abelian varieties [23]. More- over, Polishchuk [28] and Orlov[25] gave anexplicit criterion for two abelian varieties to bederivedequivalentover afieldchar(k)= 0.Asthemirrorofderivedcategoriesin algebraic geometry, the reconstruction theorem of a Fukaya category still seems to be out of reachcurrently, however, it is still curioushow farthe Fukaya category is from a completeinvariant. Withthe mirrorsymmetry Theorem 1.4, oneverifiesthe derived Fukayacategory isacomplete invariantofspecialisogenoustori.

Theorem 1.5. Two special isogenous toriare symplectomorphicif and only if theyhave equivalent derived Fukaya categories.

TheproofcruciallyreliesonOrlov’sresult,however,theverificationofOrlov’scondi- tionisfarfromstraightforward:wewillneedtoinvolveflavorsofrigidanalyticgeometry, whichwewillrecallinSection5.1.2and5.4.Wefurtherproposethefollowing question:

Question 1.6. Is thederived Fukayacategory a complete invariantof linear symplectic tori?

Anaffirmativeanswertothis questionshouldbe usefulfordistinguishingsymplectic manifolds of shapesT( ¯α)¯l×M, where theelementarylinearalgebramethodwould no longer work.This willbethetopicofaforthcomingwork.

Notation:ThroughoutGwillbeafinitegroupactingfreelyonasymplecticmanifoldM unless otherwisespecified.WhenL⊂M is Lagrangiansubmanifold,

• M=M/G,andL=L/GifGpreserves theLagrangiansubmanifoldL⊂M;

• GL={g∈G:g(L)=L}istheisotropygroupofL;

• GL=G·L=

g∈Gg·L;

• whenx∈CF^∗(L₀,L₁),wedenote G·x=

g∈G

g·x∈

g∈G

CF^∗(gL₀, gL₁);

• givenastrict/coherent G-actiononacategoryC, wedenote C^{f ix} or C^G as thesub- categoryconsistingofinvariantobjectsandmorphisms.Thesameappliestoanaive G-actiononanA_∞-category;

• theuniversalNovikovringis

ΛR={^∞

i=0

aiT^λi, ai ∈R, λi→+∞, λi∈R} (1.1) foranycommutativeringR.

Standing assumption:To simplifythe technicality ofthe paper,we assumethroughout that

(1) All Lagrangian submanifolds we consider are spin. We also require that they ei- therboundsno holomorphicdisks forgeneric choiceofcompatiblealmost complex structures;ormonotone, i.e.

[ω] =β·[c1(M, L)], β >0 (1.2) and

w1(L) = 0 =w2(L). (1.3)

Seemorediscussionsonthemonotonicityconditionfrom 4.1.2.

(2) gcd(ord(G),char(R))= 1.WhenaZ/N-gradingis consideredforasymplectic/La- grangianmanifold, gcd(ord(G),N)= 1 (seeSection4fordefinitionand discussions ongradings).

Acknowledgments:TheauthorisparticularlygratefultoMohammedAbouzaidforvery informative discussions on [2] and many other aspects of homological mirror symme- try,which was crucial for the author to initiate this projectduring the “Workshopon ModuliSpaces ofPseudo-holomorphicCurves I”inSimons Center;andto PaulSeidel, whosuggestedconsideringthespecialisogenoustori as anapplicationofourreduction method. Discussions with Cheol-Hyun Cho, Octav Cornea, Luis Haug, Richard Hind, HeatherLee,YankiLekili,Tian-JunLi,CheukYuMak,DusaMcDuff,EgorShelukhin, JingyuZhaohavegreatlyinfluencedthispaper.Partofthisworkwascompletedduring theauthor’sstayinMichiganStateUniversityandsupportedbySelmanAkbulutunder NSFFocusedResearchGrantsDMS-0244663;OctavCorneaandFrancoisLalondehave generouslysupportedtripsrelatedtothisworkduringmystayinUniversitedeMontreal asaCRMpostdoc;LingyanXiao helpedtypesettingpartofthefirstdraftofthiswork.

Mycordial thanksareduetoallofthem.

2. Algebraicpreliminaries

Thepurpose of this section is two-fold:first we wouldlike to recallbasic notionsof A_∞-categoriesand fixnotationsfortherest ofthepaper.Thenwe discusssomepurely algebraicresultsrelevanttoreducingamirrorfunctorm:D^π(Fuk(M))→ Dbyafinite groupaction.HereDcanbeanytriangulatedcategory,inactionitisusuallythederived categoryofcoherentsheavesormatrixfactorizationsofthemirrorvariety/singularity.

The reader will note thatwe almost always focus on the cohomological level hence willmostlyonlydealwithordinary(triangulated)categories. Thisismostlydue tothe attemptofmakingourdiscussionassuccinctaspossible.Infact,oncethecohomological

level is clear, there is amachinery of obstruction theory developedby PaulSeidel [31, Proposition 14.5]and[37] ofupgradingequivariant objectsfrom cohomologicallevelto chainlevel(weaklyequivarianttocoherentequivariantinSeidel’sterminology),therefore, we willsaveourselvesfromreplicatinghis work.

2.1. Reminderon A_∞-category

Wecollectnecessary notionofA_∞-categorieswe willneed,mostlyfrom [35] without proofs. Interestedreadersarereferredthereofforasystematictreatmentonthetopic.

Definition 2.1.Fixanarbitraryfieldk.AnA_∞-category consistsofthefollowingdata:

(1) a setofobjectsOb(A),

(2) a gradedk-vectorspacehom_A(X0,X1) foreachX0,X1∈Ob(A), (3) a k-linearcompositionmaps, foreachd≥1,

μ^d_A:hom_A(X_d−1, X_d)⊗ · · · ⊗hom_A(X₀, X₁)−→hom_A(X₀, X_d)[2−d], whichsatisfiesquadraticequations

m,n

(−1)ⁿμ^d−m+1_A (ad, . . . , an+m+1, μ^m_A(an+m, . . . , an+1), an. . . , a1) = 0 with n = n

j=1|a_j|−n and where the sumruns over all possible compositions:

1≤m≤d,0≤n≤d−m.

Inparticular,hom_A(X₀,X₁) isacochaincomplexwithdifferentialμ¹_A;thecohomolog- icalcategoryH(A) hasthesameobjectsasAbutmorphismgroupsarethecohomologies of these cochain complexes. Inthis case,the naturalcomposition maps inherited from μ²areassociative.

Definition 2.2.AnA_∞-functorF:A→ BbetweenA_∞-categoriesAandB comprises (1) a mapF:Ob(A)→Ob(B),

(2) a sequenceofmultilinearmaps ford≥1

F^d:hom_A(X_d−1, X_d)⊗ · · · ⊗hom_A(X₀, X₁)→hom_B(FX₀,FX₁)[1−d]

satisfyingthepolynomialequations

r

s1+···+sr=d

μ^r_B(F^sr(a_d, . . . , a_d−sr+1), . . . ,F^s1(a_s1, . . . , a₁))

=

m,n

(−1)ⁿF^d−m+1(ad, . . . , an+m+1, μ^m_A(an+m, . . . , an+1), an, . . . a1).

Such anA_∞-functordefines anordinaryfunctor H(F):H(A)→H(B) whichtakes [a] →[F¹(a)].

Definition 2.3. If H(F) isan isomorphism (resp.full or faithful),we say F is aquasi- isomorphism(resp.cohomologicallyfullor faithful).

ForanyA_∞-categoryA, onemayconsidertheA_∞-modulesoverA.

Definition2.4. A left A-module Massociates toeach objectX ∈ObAagraded vector spaceM(X),together withmaps

μ^k :hom(Xk−1, Xk)⊗. . .⊗hom(X0, X1)⊗ M(X0)→ M(Xk) (2.1) whichsatisfiesA_∞relationsfromDefinition 2.2sothatMbecomesanA_∞-functorfrom AtoCh,thedg-category formedbychaincomplexes overk.

TheleftA-modulesform adg-category,whichwill bedenotedA-mod.Similarlyone definesthecategory ofrightA-modules,denotedasmod-A.There isaA_∞analogueof Yonedaembedding:

Definition2.5.GivenanobjectK∈ A,wedefineitsYonedaembedding,aleftmoduleYK^l , with

YK^l (L) :=hom_A(K, L) (2.2)

μ^k_Yl

K(L):=μ^k+1_A . (2.3)

In this way Yoneda embedding extends to a cohomologically fully faithful A_∞- embedding Y^l : A → A-mod and the same holds true for right mod-A. (For explicit formulaeofnaturaltransformationsbetweenmodulessee[35, Section2g].)

Oneofthebasicmerits A-moduleenjoysisthenaturaltriangulatedstructure.More concretely,ifc∈hom⁰_A(Y0,Y1) isacocycle,Cone(c) isanA_∞-moduledefinedby

Cone(c)(X) =hom_A(X, Y₀)[1]⊕hom_A(X, Y₁) (2.4) andwithoperationsμ^d_Cone(c)((b0,b1),a_d−1,. . . ,a1) givenbythepairofterms

μ^d_A(b0, ad−1, . . . , a1), μ^d_A(b1, ad−1, . . . , a1) +μ^d+1_A (c, b0, ad−1, . . . , a1) .

Inparticular onemay applyYoneda embeddingand obtainnaturally atriangulated envelop ofAfrom A-mod,whichis thesmallestfullsubcategory whichcontainsY^l(A) andareclosedundertakingconesandapplyingshiftfunctors.AnA_∞-categorywhichis closedunderthesetwooperationsarecalledatriangulated A_∞ category.Atriangulated envelopcanalsobe recastbyaconcrete constructioncalledthetwisted complex.

Definition 2.6.Atwisted complex overAisapairofdata (X,δX),sothat (1) X isaformaldirectsumoverfiniteindex setI

X=⊕i∈IVⁱ⊗Xⁱ

with{Xⁱ}∈Ob(A) andVⁱ finite-dimensionalgraded k-vectorspaces.

(2) δ_X isamatrixofdifferentials

δX = (δ^ji_X); δ_X^ji=

k

φ^jik⊗x^jik

with φ^jik ∈ Hom_k(Vⁱ,V^j), x^jik ∈hom_A(Xⁱ,X^j) and having totaldegree |φ^jik|+

|x^jik|= 1.ThedifferentialδX shouldsatisfythetwoproperties

• δ_X isstrictly lower-triangularwithrespect tosomefiltrationofX;

• _∞

r=1μ^r_ΣA(δX,. . . ,δX)= 0.

OneobservesthattwistedcomplexesformanA_∞-categoryT w(A) (see[35]),whichis closedundermappingconesandhasanaturalautomorphismbythedegreeshiftfunctor

⊗k[1].OnemayshowthatT w(A) isnaturallyquasi-isomorphicto thetriangulateden- velopoftheYonedamodules.Forconcreteness,wewillmostlysticktotwistedcomplexes inthispaper. FromthetriangulatedstructureofT w(A),onemayprove

Lemma 2.7([35,3.29]).H⁰(T w(A))isan(ordinary) triangulated category.

Nextwe willdiscussidempotents. Given anadditivecategory C andX ∈ C,suppose we have an idempotent endomorphism p ∈ hom(X,X). A triple (Y,i,r) is called an image of pifthefollowingholds:

i∈hom(Y, X), r∈hom(X, Y), ri=idY, ir=p.

C is called idempotent complete or Karoubi complete if every idempotent endomor- phism hasan image inC. An idempotent completionΠC of C can be constructed ina fairly formalway.

Definition 2.8.TheidempotentcompletionΠC ofanadditivecategory Cisdefinedas:

• Ob(ΠC)={(X,p_X):X ∈ C, p²_X =p_X ∈hom(X,X)},

• hom((X0,pX0),(X1,pX1))=pX1hom(X0,X1)pX0.

Idempotentcompletionsbehavewellwithrespect tothetriangulatedstructure:

Theorem2.9([4]).IfCistriangulated,thenΠChasanaturalinducedtriangulatedstruc- ture.

Thereis alsoanotionof idempotents intheA_∞ sense.LetA bean A_∞-category as usual.

Definition 2.10. An idempotent up to homotopy for an object Y is a non-unital A_∞-functor P:K→ Asuchthat P(∗)=Y.

ForeachidempotentuptohomotopyP,onemayassociateanabstract imageof P whichisanA-module(see[35, (4b)]).WethensayAissplitclosed ifanysuchabstract imageisquasi-isomorphictoaYonedaimage ofanobjectofA.

However, as we mentioned in the beginning of the section, we will only consider idempotentsafterpassingtothehomotopylevel.Thisdoesnotlosetoomuchinformation dueto followingobservationbySeidel.

Lemma 2.11 ([35,4c]).Given A_∞-categories A, B and a cohomologically fully faithful functor F : A → B. Then (B,F) forms an A_∞ split-closure of A iff (H^∗(B),H^∗(F)) formsasplit-closureof H^∗(A).

Inparticular, H⁰(ΠT wA)isequivalent toΠ(H⁰T wA).

WethendefinethederivedcategoryD^πAofanA_∞-categoryAas D^πA:=H⁰(ΠT wA)∼= Π(H⁰T wA)

Whilethefirstmodelisadoptedbymostoftheexistingliterature,wewillalsomake use of the latter. For situationswe will consider, this ambiguity does not incur extra complications.

2.2. Groupactions onacategory

Recallthedefinitionof strictandcoherentgroupactiononacategory from[35].

Definition2.12.LetGbeadiscretegroup,Abeanadditivecategory,and{T_g}g∈Gaset ofautoequivalencesofAsuchthatT_e=id_A.Then{T_g}g∈G forms

• astrict G-actionifTg1◦Tg0 =Tg1g0,

• a coherent G-action if there is a system of isomorphism of functors ϕ_g1,g0 : T_g1 ◦ Tg0

−→∼ Tg1g0,suchthatϕg1,g0 istheidentitywheng0=eor g1=e, andthat:

Tg3◦Tg2◦Tg1

LTg3(ϕg1,g2)

RT1(ϕg3,g2)

Tg3g2◦Tg1

ϕg3g2,g1

Tg3◦Tg2g1

ϕg3,g2g1

Tg3g2g1

(2.5)

• We also consider the casewhen A is an A_∞ category with anaive G-action. This means{Tg}g∈G isasystemofA_∞-autoequivalencessothat

(1) theyform aG-actiononOb(A),

(2) themaps T_g¹:hom(X,Y)→hom(T_g(X),T_g(Y)) formsaG-action, (3) T_gⁱ ≡0 wheni>1.

Thereasonweneedboth notionsofG-actionsisthefollowing:

Lemma 2.13.Given anequivalence between categoriesF:C → D andastrict G-action on C.Thenforany choiceofquasi-inverseof G ofF,{F ◦Tg◦ G}g∈G formsacoherent G-actionon D.

Proof. See[35, 10c]. 2

Definition2.14.LetCbeanadditive(A_∞,resp.)categorywitheitherastrictorcoherent (naive,resp.)G-action.ItsfixedpartC^{f ix}denotesthesubcategoryconsistingbothobjects and morphismsfixedbytheG-action.

OnenotesthatC^{f ix}is equivalenttothenotationC^invin[35].

It goes without saying thatthe situation of Lemma 2.13 models the caseof mirror symmetry, where we have constrained our data so that there is astrict action on the A-side,andinduceacoherentactionontheB-side.

The proof of Lemma 2.13 involves a choice of G. Much of the theory of G-action carriesoutwithouttheexplicitmentionofG,however,itturnsouttheinvariantpartof theG-action(theequivariantcategory)issensitivetothisparticularchoice.Thismeans thatifwechoosethequasi-inverseinanarbitraryway,F doesnotdescendtoafunctor between the invariant part naturally(even when the coherent action is strictified, see Section2.3).Theauthor doesnot knowifthere isanaturalway toinduceaninvariant functor F^{f ix} insuchcases.Instead,weimposethefollowingextracondition,whichcan alwaysbe achieved.

Definition 2.15. In the situation of Lemma 2.13, a quasi-inverse G : D → C is called admissible ifthefollowing holds:

(1) IfY ∈ F(C^{f ix}),thenG(Y)∈ C^{f ix} andF ◦ G(Y)=Y.

(2) If Y0,Y1 ∈ F(C^{f ix}), the pair F : hom(G(Y0),G(Y1)) → hom(Y0,Y1) and G : hom(Y0,Y1)→hom(G(Y0),G(Y1)) areinversesofeachother.

Lemma2.16.InthesituationofLemma 2.13,givenan admissiblequasi-inverseGof F, F induces afunctorF|_C^{f ix}:C^{f ix}→ D^{f ix},whichisfully faithful.

Proof. Let{Tg}g∈G denotethefunctorsinG-actionasinDefinition 2.12.Forany X ∈ C^{f ix},sinceGF(X)∈ C^{f ix},wehaveF ◦T_g◦ G(FX)=F ◦ G(FX)=F(X).Thelefthand side isprecisely the inducedcoherent actionon F(X).This proves F(X)∈ D^{f ix}. The propertyofbeingfullyfaithfulfollowsdirectlyfromthedefinitionofadmissibility (2). 2 Animportant feature of mirror functors is their triangulated structures, hence this property of F^{f ix} isour nexttopic ofstudy. To begin with, we assume throughout the restofthissection that:

• In the situation of Lemma 2.13, C, D and F are all triangulated, and the quasi- inverseG ofF isalwaysadmissible. (The readershouldbe remindedthatG is also triangulatedinthiscase,see[20, p.4].)

• WhenGisanaiveactiononanA_∞-category,theG-actionsendscohomologicalunits to cohomologicalunits(theG-actionisc-unital).

We say that C^{f ix} inherits a triangulated structure from C if it has a triangulated structureso thatallexacttrianglesinC^{f ix} aretrianglesinC.

Lemma2.17.IfC^{f ix}inheritsatriangulatedstructurefromC,thenD^{f ix}andF^{f ix}inherits atriangulated structurefromD andF.

Proof. Given Y₀,Y₁∈ D^{f ix},there isatriangleinC foreveryf ∈hom(Y₀,Y₁)^{f ix} GY0 Gf

−−→ GY1 i

−→Cone(Gf)−→ G^j Y0[1]. (2.6) SinceGYi∈ C^{f ix} andGf ∈hom(GY0,GY1)^{f ix} (thelatterfollowsfrom admissibility), we havea choiceof Cone(Gf)∈ C^{f ix} from assumption,and i, j are also inthe corre- spondingfixedpartsofmorphismgroups.ApplyingF to(2.6),thefollowingisatriangle inDfrom theadmissibility:

Y₀−→^f Y₁−−→ F^Fi (Cone(Gf))−−→^Fj Y₀[1]. (2.7) Butwe haveseenthatF|_C^{f ix} :C^{f ix} → D^{f ix} is awell-defined functor,which implies (2.7)is indeedatriangleinDwith allobjects andmorphisms belongingto D^{f ix}.This verifiesthatD^{f ix}hasaninheritedtriangulatedstructurefromD.Verificationofaxioms fortriangulatedstructureonD^{f ix} isstraightforwardandsimilartotheaboveproof.

ThetriangulatedstructureofF^{f ix} thenfollows fromthatofF. 2

ThetriangulatedstructureonC^{f ix}comesforfreeoncewebringin abitmoreinforma- tionfromthechainlevel.Inparticular,thisalwaysholdswhenCisthederivedcategory of anA_∞-category withanaiveG-actionas thefollowinglemmashows.

Lemma 2.18. For a triangulated A_∞-category A with a naive G-action, D^πA is a tri- angulated category with a strict G-action. Moreover, its fixed part (D^πA)^{f ix} is also triangulated.

This is a mild generalization of a result in [14] where Elagin proved the result for dg-categories. Of course, the derived category model we have adopted here is much moreelementary,inparticular, wedidnotexplicitlyinvolveA_∞-moduleshiddeninthe constructionof twistedcomplexes.

Proof. RecallthatusingDefinition 2.8,anobjectinD^πAcanberepresentedas Z = (Z = (

i∈I

V_i⊗X_i, δ), p_Z),

where pZ ∈Hom⁰(Z,Z) isanidempotent. UsingthenotationfromDefinition 2.12, the strict action is given by {H⁰(Tg)}. Hence, an object of (D^πA)^{f ix} is pair consisting a twisted complex and an idempotent, which are both fixed by G. Given a G-invariant morphism between two objects f : (Z₁,p₁)→ (Z₂,p₂) in D^πA, ifp_i =id_Zi, then the usual constructionof twisted complex shows themapping cone Z₀ is also aninvariant twisted complex.Forthegeneralcaseweargueasintheproofof[4,Lemma1.13].

Considerthefollowingdiagram oftriangles:

Z₁

p1

Z₂

p2

Z₀

t

Z₁[1]

p1[1]

Z₁ Z₂ Z₀ Z₁[1]

(2.8)

From the triangulation structure of H⁰T wA, there is at which makes the diagram commute.ByaveragingonemayalsoassumettobeG-invariant.Then[4,Lemma1.13]

shows

p₀=t+ (t²−t)−2t(t²−t) = 3t²−2t³

is an idempotent, by which onemay replace t in diagram (2.8). Hence(Z0,p0) is the mappingconeoff.

Now (TR2) and (TR3) follows from definition and the above averaging trick. This appliestotheoctahedronaxiomaswell,butweprovidealittlemoredetailshere.Given a lower capof the octahedron, one firstcompletes allobjects to a G-invarianttwisted complexbyaddinganotherdirectsummand.Thisisalwayspossiblesinceweassumedthe

cohomologicalunit andall projectionsinvolved to be G-invariant. Onethen completes theoctahedron bythe canonicalconstruction of mappingcones for twistedcomplexes, which again consists of G-invariant twisted complexes. The averaging trick makes all maps on the upper cap G-invariant again. Now the rest of the proof follows literally from thatof [4,1.15],bynoting thatallmaps and conesinvolvedarenow G-invariant, hencethispropertycarriesoverthewholeproof. 2

WeconcludethegeneraldiscussionsoncategoricalG-actionsbybridgingtheG-fixed derivedcategories andthederivedcategory of aG-equivariantA_∞-category, as willbe neededinSection4.

Lemma2.19.LetA beanA_∞-categorywithanaiveG-action. Thenthere isan isomor- phism ofcategoriesS: ΠH⁰((T wA)^{f ix})→(ΠH⁰T wA)^{f ix}= (D^πA)^{f ix}.

Proof. Theproof istautologicalbutwe includeittodispel possible doubts.Define the functorS : ΠH⁰((T wA)^{f ix})→(ΠH⁰T wA)^{f ix} bythenaturalinclusion.Given (Zi,pZi) with Z_i ∈ (T wA)^{f ix}, i = 0,1, where p_Zi ∈ Hom⁰(Z_i,Z_i)^{f ix} are idempotents. The full andfaithfulnessofS isequivalenttotheclaim

(pZ1Hom(Z0, Z1)pZ0)^{f ix}=pZ1Hom(Z0, Z1)^{f ix}pZ0.

Butany morphism on theleft has theform ofa G-orbit G·(p_Z1◦f ◦p_Z1)=p_Z1◦ (Gf)◦pZ0. On the other hand, for an object (Z,p) ∈ ΠH⁰(T wA) to be fixed by the G-action,Z andpmustbeboth fixedbydefinition. 2

2.3. Strictificationof acoherent G-action

InthissectionwewillexplainhowtoobtainacanonicalstrictG-actionmodeloutof acoherentone. Theadvantageofthis modelisthatthediscussion ofinducedcoherent actionontheB-sidebecomesmorecanonical.However,forreducingthemirrorfunctor we still need the admissibility condition on the quasi-inverse G. Such a strictification modelwasinvestigatedby PaulSeidel in[35, 14b]forthe caseofG=Z/2.Weextend this result to allfinitely generated groupsusing Cayley graph,which is actually much morethan whatwe need. Onemayeasily see from our argumentthat this evenworks forabroadgeneralityofinfinitelygeneratedG,as longasonestaysincasesthatanap- propriateversionofaxiomofchoicecanbesetup,sothatthecorrespondinggeneralized Cayleygraphhasamaximalsubtree.

Definition2.20.LetCbeacategorywithacoherentG-action{Tg,φg0,g1}g,g0,g1∈G.Then thecanonicalstrictification model C^strict isacategory equippedwith astrict G-action, definedbythefollowing data:

(1) Objects: X ∈ Ob(C^strict) has the form X = ((Xg)_g∈G,F), where Xg ∈ Ob(C), F = {Fg}g∈G. Here Fg ∈

h∈Ghom(Xgh,gXh) areisomorphisms. Moreover, we requirethecompatibilitycondition

ϕ⁻_g1¹_,g0◦ Fg1g0 =T_g1(Fg0)◦ Fg1

Fe=id^⊗G

(2.9) Suchasystemofisomorphismswill becalled acompatiblesystemof isomorphisms.

(2) Morphisms: hom_Cstrict(_X,_Y) = hom_C(_Xe,_Ye) for _X = ((_Xg)_g∈G,F^X), _Y = ((Yg)_g∈G,F^Y).

(3) G-action:

• On the object level, for s ∈ G, X ·s = ((_Xg)g∈G,F). Here _Xg = Xgs and Fg∈

h∈Ghom_C(_Xgh,g_Xh)=

h∈Ghom_C(_Xghs,g_Xhs) isashiftedcopyofFg.

• Onthemorphismlevel, ifΦ∈hom(_X,_Y),thenΦ·s= (Fs^Y)⁻¹◦T_s(Φ)◦ Fs^X. Remark 2.21. It is straightforward to check that the G-action defined above is strict.

The equivalence of C^strict and C is equallytransparent once we construct X for each X ∈ObC sothat_Xe=X inProposition 2.22. Oneshouldnoticethattheisomorphism type ofan objectX is completely determined byXe.The inheritanceoftriangulation structure shouldalso betransparent.

However, one also notes thatour model has apartially unsatisfactory feature, that we are forced to trade the left coherentG-action for aright strict G-action on C^strict, unless Gisabelian.Thisofcoursecouldberemediedinanaiveway:wemayapply the process againtoturn theright actionbacktoaleftonewhenabsolutely necessary.

Ourmain propositionofthissectionis:

Proposition2.22.ForanyGfinitelygenerated,ifCisequippedwithacoherentG-action, C^strict isequivalent toC.

A note onnotations: Inthe proof we willnot specifywhich componentof Fg is under investigation whenitisclearfromthecontext(forexample,whenitssourceandtarget are specified).Wewilluset·toreplacethenotationofgroupactionT_tfort∈G.Using thisnotation,ϕ_t0,t1 issimplyafunctorisomorphismfromt₀·t₁·(−) to(t₀t₁)·(−).Hence we willalsosuppressthesubscriptsof ϕwhenthecontext causesnoconfusions.

Proof. From discussions in Remark 2.21, what we need is to construct an X ∈ Ob(C^strict) for any X ∈ ObC, so that Xe = X. We will show a stronger statement thatforanyset ofobjects{Xg}g∈G satisfying Xg ∼=gXe,thereisacompatiblesystem of isomorphismsbetweenthese objects.

The first step is to takea set of generatorsof G, {t₁,· · ·,t_k}, and we consider the CayleygraphΓ forthisgenerating set.Recallthat

Fig. 1.CayleygraphforanobjectX.AllsinglearrowsarepartoftheCayleygraph,whilethesolidarrows belongtoT r,andtherestaredotted.DoubledottedarrowsarenotpartoftheCayleygraph,butelements ofisomorphismsystemsdeterminedby(2.9).Thetopleftarrowsinthecenterdepicts(2.10)intheactual Cayleygraph.Extrabulletsareaddedfordemonstrationpurposes.

• vert(Γ)=Gasaset,

• thereisanorientededgeewith sources(e)=v1 andtarget t(e)=v2 iffv1v⁻₂¹=tl

forsomel.Inthiscaseweendoweanextralabel tl.

NotethatourdirectionofarrowsisoppositefromtheusualnotationinCayleygraph.

Replace vertices marked as g by the object Xg. Our goal is to assign each edge marked by ti acomponentof Fti, so thatthe compositions arecompatible with (2.9).

More concretely, if two consecutive edges marked as t_i and t_j are each assigned an isomorphismcomponents of Fti and Ftj, then (2.9) uniquely determines acomponent ofFtitj. Ourpropositionis equivalentto theassertionthat,there isachoiceof Fti for edgesoftheCayleygraph,sothatthesecompositionsdependonly onthestartandend points.Wedemonstratepartof theCayleygraphinFig. 1.

The naive strategy is to start by assigning arbitrary isomorphisms for some edges, thenusetheaboveobservationtocompletethewholecompatiblesystem.Therearetwo potentialconflicts incompletingthisprocess:

(i) (Compatibility along a path.) For triple compositions, Ftitjtk determined by {Ftitj,Ftk},and{Fti,Ftjtk}mustcoincide.

(ii) (Compatibility alongaloop.)Whentherearemorethanoneorientedpathconnect- ing _Xg0 and _Xg1, the isomorphism component of Fg⁻¹₀ g1 determined by the two pathsmustcoincide.

Weclaimthat(i),thecompatibilityalongapathdoesnotimposeextraobstructions.

Xt1t2t3g Ft1

t₁·Xt2t3g t1Ft2

t₁·t₂·Xt3g t1·t2Ft3

ϕ

t₁·t₂·t₃·Xg ϕ

(t₁t₂)·Xt3g

(t1t2)·Ft3

(t₁t₂)·t₃·Xg ϕ

t1·(t2t3)·X _ϕ

ϕ

(t1t2t3)·Xg

(2.10) In diagram (2.10), we have included all possible compositions of relevant isomor- phisms, corresponding to three consecutive arrows inthe Cayleygraph. Arrows inthe first rowaregiven arbitraryisomorphismcomponentsof F,and this willdetermine all the restofthedata inthediagram.Tosee thisyields acommutativediagram,wenote first thatall thesolid arrows form acommutative sub-diagram: there aretwo loops in question,where thetop-rightsquareiscommutativesinceϕis anisomorphismof func- tors; thebottom-rightdiagram issimplythecoherenceconditionoftheaction.Now by definition, thethree dottedarrows aredetermined bythe sub-diagramformed by solid arrows, thecommutativityhencefollows.

For(ii), wetakeaconnectedmaximalsubtreeT rofΓ withthefollowingproperty:

• Xe∈T r

• thereisanorientedpathinT rstartingfromXe andendsatXg forallg∈G.

NotethatT rcontainsallverticesinΓ:otherwise,takeanymissingvertex Xm.mcan be decomposedas

m=ti1· · ·tik.

Letl = max{r: 1≤r≤k, t_i1·t_i2· · · ·t_ir ∈T r}.Addingvertices{Π_1≤ν≤pt_iν}l≤p≤l

and correspondingedgesyieldsastrictlylargersubtreethanT r.

Nowassignarbitraryisomorphismcomponentscorrespondingtoeachedgebelonging to T r. Then the compatibilityonpaths proved aboveyieldsa systemof isomorphisms through(2.9)onallcompositionsalongT r.

ForedgesthatarenotinT r,bydefinition,addinganyofthemtoT ryieldsprecisely one loop to the new graph. Hence the prescribed data on T r and (2.9) automatically assignsanisomorphismtotheseedges,andbydefinition(orrepeatingtheproofofcom- patibility along paths),this way of assigning isomorphismsyields acompatible system of isomorphismsas desired.

Lastly, notice that we indeedfixed a bit moredata than in the proof by requiring Fe≡id^⊗G. Butaquick reflectionshows this onlycause potential problem when there isaloopstartingfromavertexandback,whichconsistsaloopthatwecoulddealwith asin(ii)intheproof. 2

Note that although the embedding C → C^strict is canonical only up to a choice of compatibilitysystem ofisomorphism foreach object, itis canonicalon C^{f ix} where the systemofisomorphismscanbechosenasidentity(strictlyspeakingoneneedstorequire ϕtobe identityonC^{f ix},too,butthisisonlycosmetic).Hencebycombining2.19,2.16 fromprevioussectionsweobtainthefollowing:

Proposition 2.23. If A isa triangulated A_∞-category with a naive G-action, and there is an triangulated equivalence m : D^πA → B, then the equivalence can be reduced to triangulatedfully faithfulembeddings

m^{f ix}: (D^πA)^{f ix}→ B^{f ix},

m¯^{f ix}: (D^πA)^{f ix}→(B^strict)^{f ix}. (2.11) Question:Whenare m^{f ix} andm¯^{f ix} exact equivalences?

Abstractly, this question is related to the equivariant theory recently developed by Paul Seidel in [37][31]. Using terminologies from his work, the potential failure for (D^πA)^{f ix} → B^{f ix} to be essentially surjective lies in that whether one could find a weakly equivariant object with image Y for all Y ∈ B^{f ix}. If this is the case, [31, Lemma 14.10]providesawaytoproducethenecessaryobjectin(D^πA)^{f ix}.Thenextstep forB^{f ix}→(B^strict)^{f ix}wouldprobablyrequireadditionalconsiderationsonobstruction theoriesforcoherentactions.

Inthis perspective,whatwehaveshownisquitepreliminary:givenachainlevel(for example, anaive one) G-action onA and aquasi-equivalence to A, the strictification model provided an approach so that one could discuss weakly equivariant objects on both sides as astart. However, problems about upgradingthe G-action or equivariant objectsonA tothechainlevel,aswellasthecomparisonoftheseobjectsonAandA, remain mysterious. However, we will see later that, in some geometric situations, one mayshowtheequivalencefromextractinganisomorphicpiecefromeachside,wherethe inducedG-actiononB isstillstrict.

3. EquivariantFukayacategory andsplitgeneration

3.1. Modulispacesof bordered holomorphiccurves

Westartbyconsideringvariousmodulispacesofpseudo-holomorphiccurvesinvolved in our discussions. Since a comprehensive account on these moduli spaces in rather

generalcontextshasbeencarefullywrittendownin[1][18],weonlyrecallrelevantnotions involvedinourlaterdiscussionsandreferinterestedreaderstotheirdetailedexpositions.

LetD²={z∈C:|z|≤1}.Denotethemoduli space

Di,j ={D²\Σ⁺∪Σ⁻: Σ⁺∪Σ⁻⊂∂D²;|Σ⁺|=i,|Σ⁻|=j}, and

Di,j^± ={D²\Σ⁺∪Σ⁻∪p^± : Σ⁺∪Σ⁻ ⊂∂D²;|Σ⁺|=i,|Σ⁻|=j;p^±∈int(D²)}, where D² is always equipped with the standard complex structure. In both cases, we orderΣ⁺andΣ⁻counterclockwiselyas{z₁⁺,· · ·,z⁺_i }and{z₁⁻,· · ·,z_j⁻}.Tobemoreper- tinenttoourapplications,wefurtherrestrictourattentiontothefollowingcomponents of theabovemodulispaces:

Definition3.1.Di,j ⊂Di,j,D^±i,j⊂D^±i,jarecomponentssuchthatthefollowingadditional restrictionsaresatisfied:

(1) j= 0,1 or2;

(2) Σ⁻ liesonthesameconnectedcomponentof∂D²\Σ⁺.

We will denote theDeligne–Mumford compactificationof these two typesof moduli spaces as Di,j and D^±i,j,where thelower strata arestratified bystable diskswith more than onecomponent.

Next we consider the moduli space for annuli, denote A_r = {1≤ |z|≤ r}, and the moduli spaceCd⁻={(Ar,z0,z1,...zd)|z0= 1,z1=r,|zk|=r,∀k≥2}.

TheDeligne–Mumford compactifiedmoduli spaceC⁻d hassimilarboundarystrataas Di,j,i.e. bubblingupdisksattheboundary|z|=r.However,twotypesofnewboundary strata of codimension 1appears where the annulusbreaks into two disks D₁ and D₂, and either

(1) D1∈ D_d,0⁻ andD2∈ D⁺0,1

(2) D1∈ Dd1,2,D2∈ Dd−d1+2,1.

Inmanyoccasionsbelow,wewilldealwithD^±i,jandCd⁻uniformly.Insuchcaseswewill simply use R and R to denote anyof thesemoduli spacesand theircompactifications, respectively.

3.2. Floerdataandconsistent choices

Definition 3.2. A G-invariant Floer datum on a disk S ∈ R consists of the following choiceoneachcomponent: