Meet homological mirror symmetry

MEET HOMOLOGICAL MIRROR SYMMETR Y MA TTHEW ROBER T BALLARD Abstract. In this paper, we introduce the intereste d reader to homological mirror symmetry . After recalling a little back ground knowledge, we tack le the simplest cases of homological mi rror symm etry: curves of genus zero and one. W e close b y outlining the curren t state of the ﬁeld and men tioning what homo- logical mi rror symmetry has to say ab out other asp ects of mir ror symmetry . Contents 1. Int ro duction 2 2. Building s ome c ommon k nowledge 3 2.1. Homological algebra 3 2.2. Algebraic geometry 6 2.3. Symplectic geometry ba ckground 6 2.4. Floer co homology and F uk aya categor ies 8 3. Homologica l mirror sy mmetr y for the pro jective line 11 3.1. B-branes o n the pro jective line 11 3.2. A-branes on the mirror to the pro jective line 12 3.3. B-branes o n the mirr or to the pro jectiv e line 14 3.4. A-branes on the pro jective line, a .k .a. the sphere 17 4. Homologica l mirror sy mmetr y for elliptic curves 21 4.1. A-branes on an elliptic curve 21 4.2. B-branes o n an elliptic curve 23 4.3. An o utline of the equiv ale nce 25 5. F urther results 27 5.1. A-branes on (near) F ano manifolds versus B-br anes on LG-mo dels 27 5.2. A-branes on F ano manifolds v ersus B-br anes o n LG-mo dels 28 5.3. A-branes v ersus B-br anes on Calabi-Y a u manifolds 29 6. How is homo logical mirror symmetry related to mir ror symmetry? 30 6.1. SYZ 30 6.2. Mirror ma ps 31 6.3. Instant on num ber s 31 References 32 2000 Mathematics Subje ct Classiﬁc ation. Primary: 14J32; Secondly: 18E30, 53D40, 53D35. Key wor ds and phr ases. homological mirror symmetry , A and B -branes, A ∞ -algebra, F uk ay a category , autoequiv alence, elliptic curves, Calabi-Y au manifold. 1 2 MA TTHEW ROBER T BALLARD 1. Introduction As it s tands to day , mirror symmetry is a well-established ﬁeld of ma thematics. Seven teen y e a rs ha ve passed since physicists correctly predicted the n umber of ra- tional cur ves of given degr ees o n the F ermat quintic [9]. In the interv ening years, the ph ysical inspir ation has energized the mathematics communit y . In ma ny cases , in- cluding a nti-canonical hypersurfa ces and Calabi- Y au co mplete intersections in toric v arieties, mathematical mirr or co ns tructions hav e b een realis ed a llowing one to see the mirror r eﬂection in the Ho dge diamond a nd establish the relation b etw een curve counts and perio d integrals. Unfortunately , rigorous formulations o f the ph y s ical arguments yielding these predictions hav e not aris en. Su ch rig orous formulations would allow mathematicians to tap into the deep physical understanding of the situation, but mo dern physical to o ls , such as the path in tegr al, lie just b eyond the current scop e o f mathematics . Consequently , to fully understa nd the phenome- non of mirror symmetry , mathematicians must fo llow the physical inspiration and develop their o wn ov era rching appro aches. A t the International Co ngress of Mathematicians in 1994, M. Kontsevic h pro- po sed such an appro ach [34]. As is it commonly known now, homolog ic al mirror symmetry , or the homo lo gical mir ror conjecture, reformulates mirror symmetry as an equiv alence of triangulated categor ies built from diﬀerent asp ects of the Calabi- Y au geometry of tw o ma nifolds, c a lled mirror manifolds. The C a labi-Y au metric is determined by tw o pieces of data — the complex str ucture and the K¨ ahler form. Each piece lands in a diﬀerent bra nch of mathematics . If we o nly care abo ut the complex structure, we are natur ally led into alg e braic geo metry . If we instead re- mem be r the K¨ ahler form, w e study s ymplectic geometry . Up on pas sage to the mirror manifold, we should see a n exchange of these t w o geometr ies. Kon tse v ich sought to capture this exchange. T o day , homologica l mirror symmetry r emains an in tr iguing and daun ting challenge to mathematics. The scop e has expanded to encompass o ther ma nifo lds beyond Ca labi-Y au manifolds. It has b eco me a p ow- erful source of inspiration motiv ating insights in algebraic geometry , symplectic geometry , homological alg ebra, no ncommutativ e g eometry , and b eyond. Mixing all these ﬁelds, ho mological mirror symmetry is a very attrac tive co njec- ture, but it rema ins o utside the c ommon k nowledge o f the working ma thematician. Wh y? Alo ng with its relative youth, its pr o p er formulation requir es a n impo sing amount of unfamiliar technology . This article oﬀers a introduction to homolo gical mirror symmetr y thro ugh tw o ex plic it exa mples. The cases of P 1 and elliptic curves are very conc r ete. W e give a ho listic appro ach that treats b oth sides of mirro r sym- metry for P 1 , instead of sp ea king to one side without reference to the other as is often the case in the litera ture. Since homologica l mirr or sy mmetry manifests itself in noticeably diﬀeren t w ays whether one consider s F a no o r Cala bi-Y au v a rieties, w e also review the case o f elliptic curves. See ing b oth cases side by side will hop efully give the reader a deep er apprec ia tion of the duality . T o mov e into examples we need to review a little of the necessar y for malism. This is done in sectio n 2. With this ov er, we mov e on to the examples in s ections 3 and 4. Algebr aic curves o f genus zer o and one provide s imple case studies. Within these, the reader can meet co ncrete incarnations of the relev a nt categor ies and a ppreciate the unexpected equiv alences that homolog ical mirror symmetry pr edicts. After the examples have b een covered, we o utline the current state of k nowledge in the ﬁeld in sec tio n 5 and then men tion MEET HOMOLOGICAL MIRROR SYMMETR Y 3 how ho mologica l mir ror symmetry relates to other a sp ects o f mirro r symmetry in section 6. The a utho r hea rtily thanks his a dviser, Cha rles Dor a n, for encouraging the cre- ation of this survey , the referee for his careful reading , a nd Ursula Whitc her for her time a nd numerous suggestions to improv e the article. All erro rs be long s olely to the author. 2. Building some co mmon knowledge Part of the diﬃculty in dealing with homolog ical mirror symmetry is the breadth of knowledge requir ed for a prop er for mu lation. Before we can dive int o the promised simple examples, we re call so me terminolog y a nd r esults from homolog ical algebra, a lg ebraic g eometry , and symplectic geometr y . 2.1. H omolog i cal algebra. I a ssume the r e ader has s ome fa miliarity with derived categorie s, at lea st in the case o f mo dules ov er a n a sso ciative algebra, Hochschild cohomolog y , and dg-alge br as. A go o d reference for homolo gical algebra is [20], a go o d refer ence for ho mo logical alge bra and Ho chschild cohomology is [60], and a go o d reference for dg-algebra s is [33]. Our goal is here is to deﬁne a tr ia ngulated ca teg ory which app ears on each side of mirror symmetry . The main algebraic to ol is the A ∞ -algebra . A ∞ -algebra s are av oida ble in algebr aic geo metr y but no t in s ymplectic geometry . (How ever, the A ∞ - algebras that app ear in homo logical mirror symmetry for P 1 are honest asso ciativ e algebras ). Deﬁnition 2.1. An A ∞ -algebr a over a b ase ﬁeld k is a gr ade d k -mo dule A with k -line ar maps m n : A ⊗ n → A of de gr e e 2 − n for e ach n > 0 , satisfying the fol lowing quadr atic r elations for al l n > 0 : X r,s ( − 1) r s +( n − r − s ) m l (id ⊗ r ⊗ m s ⊗ id ⊗ n − r − s ) = 0 Let us loo k a t the ﬁrst thre e of these relations. The ﬁrst relation says m 2 1 = 0 so ( A, m 1 ) is a chain co mplex . The second says m 2 is a chain map when we use the diﬀerential 1 ⊗ m 1 + m 1 ⊗ 1 on A ⊗ 2 . The third s ays that m 2 is asso ciative up to a homotopy m 3 . Thus, w e can pass to the cohomolog y with resp ect to m 1 and g et an a sso ciative algebra H ( A ). F amiliar examples o f A ∞ -algebra s are a sso ciative alge br as, where m n = 0 for n 6 = 2, and dg - algebra s, wher e m n = 0 for n 6 = 1 , 2. An A ∞ -algebra with m 1 = 0 is called minimal. W e can (a nd often have to) do something troublesome and a dd a degree 0 mul- tiplication m 0 : k → A a nd con tin ue to require the qua dratic relations to hold. F or example, the ﬁrst t w o relations b ecome m 1 ( m 0 ) = 0 m 1 ( m 1 ) + m 2 ( m 0 , id) + m 2 (id , m 0 ) = 0 This in ge neral destroys the p ossibility of taking cohomo logy . An A ∞ -algebra po ssessing such an m 0 is called curved or o bstructed. If we as sume, that m 0 is central (with resp ect to m 2 ) a nd m n (id ⊗ r ⊗ m 0 ⊗ id ⊗ n − r − 1 ) = 0 for a ll n > 2 we can once again take coho mology H ( A ). In this ca se, A is called weakly obstructed. Below our A ∞ -algebra s will be unobstructed unless explicitly indicated. 4 MA TTHEW ROBER T BALLARD Deﬁnition 2.2. A morphism f : A → B of A ∞ -algebr as over k is a c ol le ction of k -line ar maps f n : A ⊗ n → B of de gr e e 1 − n satisfying X ( − 1) ♥ m k ( f i 1 ⊗ · · · ⊗ f i k ) = X ( − 1) sr +( n − s − r ) f l (id ⊗ s ⊗ m r ⊗ id ⊗ n − r − s ) wher e ♥ = ( k − 1 )( i 1 − 1) + ( k − 2)( i 2 − 1) + · · · + ( i k − 1 − 1) . The ﬁrst r elation says that f 1 commutes with the diﬀerential. The se c ond says f 1 resp ects m 2 up to f 2 . f is a called a quasi- is omorphism if f 1 : H ( A ) → H ( B ) is an is omorphism. Remark 2.3 . These deﬁnitions b e c ome mor e c omp act and p erhaps cle ar er when one p asses fr om A to the b ar c omplex B ( A ) on A . The A ∞ -algebr a structu r e on A is e quivalent to a c o derivation on B ( A ) . A ∞ -algebr a morphisms ar e then c o algebr a morphisms c ommut ing with the c o derivations. F or a r efer en c e, se e [21] . Now we recall a result that a llows us to pa ss to cohomology of a n A ∞ -algebra without losing information. Theorem 2 .4. [31, 40, 37] Given an A ∞ -algebr a A over a ﬁeld k , cho ose a splitting A = H ⊕ B ⊕ D over k , wher e H is the c ohomolo gy and m 1 : D → B is an iso- morphism. Then, t her e exists an A ∞ -algebr a structu r e on H ( A ) with zer o ﬁrst or- der c omp osition, se c ond or der c omp osition induc e d by m 2 , and quasi-isomorphisms i : H ( A ) → A, π : A → H ( A ) . Mor e over, π 1 is the asso ciate d r estriction map A → H and i 1 is the asso ciate d inclusion map H → A . Consider for a moment a dg-algebra A . Applying this pro cedure yields a mini- mal A ∞ -algebra struc tur e on H ( A ). Th us, the chain-level data lost when we just consider H ( A ) has b een tr ansmogr iﬁed, re turning as the higher comp os itio ns in the A ∞ -structure. How co mplex can A ∞ -algebra s b e? A partial a nswer to this question is given by the follo wing obser v ation. F rom theorem 2 .4, there is an A ∞ -structure on H ( A ) which makes it quasi- isomorphic to the A ∞ -algebra A . Given a minimal A ∞ -algebra ( A, m n ), take m k to b e the ﬁrs t non-zero o pe r ation with k > 2. Then, the ﬁrst A ∞ -relation in volving m k is X r ( − 1) r m k (id ⊗ r ⊗ m 2 ⊗ id ⊗ k − 1 − r ) + m 2 ( m k , id) − m 2 (id , m k ) = 0 This equation states that m k is a Hochschild co chain for the algebr a ( A, m 2 ). Sup- po se we wan ted to ﬁnd ano ther minimal A ∞ -structure m ′ n on ( A, m 2 ) which is isomorphic to our or iginal o ne . Then, we hav e our collection f k : A ⊗ k → A [1 − k ]. W e need f 1 to b e an automorphis m of the alg ebra ( A, m 2 ), co nsequently we can inductively solve X ( − 1) ♥ m k ( f i 1 ⊗ · · · ⊗ f i k ) = X ( − 1) s + r ( n − s − r ) f l (id ⊗ s ⊗ m r ⊗ id ⊗ n − r − 1 ) for m ′ n . Thus, the f n and m n uniquely determine m ′ n . So , if we wan t to ﬁnd an isomorphic ( A, m ′ n ) with m ′ l = 0 for 2 < l ≤ k , we just need to solve X s f l (id ⊗ s ⊗ m 2 id ⊗ n − r − 1 ) + m 2 ( f l , id) + m 2 (id , f l ) = m k ( f ⊗ k 1 ) and then take f l = 0 for 1 < l < k and f l arbitrar y for l > k . Applying an automorphism, we can set f 1 = id. The previo us equa tion says that m k is a Ho chsc hild cobounda ry for ( A, m 2 ). Thus, we get the following res ult. MEET HOMOLOGICAL MIRROR SYMMETR Y 5 Lemma 2. 5. If t he r elevant pie c es of t he Ho chschild c ohomolo gy of ( A, m 2 ) ar e zer o, we c an trivialise up to any arbitr ary or der. F r om the c onstru ction, the limiting c omp osition of t hese maps ex ists, giving a trivialisatio n for the A ∞ -structu r e. As in the cas e o f an a s so ciative algebr a, we w is h to for m a categ ory o f modules ov er a n A ∞ -algebra A . W e do so in a manner strongly analo gous to forming chain complexes o f modules ov er a n asso ciative a lgebra. Deﬁnition 2.6. A (right) mo dule M over an A ∞ -algebr a A is a gr ade d mo dule over k e quipp e d with k -line ar maps m M n : A ⊗ n − 1 ⊗ M → M of de gr e e 2 − n for e ach n > 0 : satisfying the fol lowing quadr atic r elations for al l n > 0 : X r,s,n − r − s> 0 ( − 1) r s +( n − r − s ) m M l (id ⊗ r ⊗ m s ⊗ id ⊗ n − r − s )+ X u,v ( − 1) uv m M l (id ⊗ u ⊗ m M v ) = 0 A morphism g : M → N of A -mo dules is a c ol le ct ion of k -line ar maps g n : A ⊗ n − 1 ⊗ M → N satisfying quadr atic r elations similar t o the c ase of morphisms of A ∞ - algebr as. F or mor e details, se e [21] . Let us repa ck age this deﬁnition. Instea d of thinking of A as a n A ∞ -algebra , we shall think o f it as a n A ∞ -categor y with one ob ject ∗ . The morphisms in this category are only the endomorphisms of ∗ and Hom( ∗ , ∗ ) := A . T o give a categ ory A an A ∞ -structure, w e need to deﬁne m ulti-co mpo sitions of morphisms m n : Hom( X 0 , X 1 ) ⊗ · · · ⊗ Hom( X n − 1 , X n ) → Hom( X 0 , X n ) satisfying the A ∞ -relations. Here X i are o b jects of A . In our case, w e just use the op erations m n : A ⊗ n → A coming fro m the A ∞ -algebra structure o f A . F or an A ∞ -categor y A , each ob ject X of A furnishes a functor Hom A ( X, · ) from A to chain complexes ov er k , C h ( k ). These are naturally A ∞ -functors (for more, see [32]). Deﬁnition 2.7 . Given two A ∞ -c ate gories A and B . An A ∞ -functor F : A → B is an assignment of obje cts X 7→ F ( X ) and a c ol le ction of maps F n : Hom A ( X 0 , X 1 ) ⊗ · · · ⊗ Hom B ( X n − 1 , X n ) → Hom B ( F ( X 0 ) , F ( X n )) which satisfy quadr atic r elations similar to those given in the deﬁnition of a mor- phism of A ∞ -algebr as. The A ∞ -version of the Y oneda embedding is given by sending X to Hom A ( X, − ). This gives an A ∞ -functor Y fr om A to the categor y of A ∞ -functors from A op to C h ( k ). Denote this categ ory b y Mo d A . The standard Y one da embedding is full and fa ithful. After taking H 0 , Y r educes to the standard Y oneda embedding. Since H 0 ( Y ) is an eq uiv alence onto its image, we say that Y is a qua si-equiv alence onto its imag e. If we consider the A ∞ -categor y with a single o b ject and morphisms algebra A , this gives another deﬁnition of A -mo dule. It is a go o d e x ercise for the reader to translate b etw een the tw o deﬁnitions of A -mo dules for an A ∞ -algebra . Since an y A -mo dule is, in particular, a c hain complex , we have a notion of quasi-isomo rphism in the ca tegory of A -mo dules. In analog y with the case of an asso ciative algebr a, we wis h to in vert these. When we do , we get D ( A ), the derived category of A -mo dules . As in the case of a sso ciative algebr as, the r esulting categor y 6 MA TTHEW ROBER T BALLARD is triangula ted. W e let D π ( A ) b e the smalle st tria ngulated subc ategory of D ( A ) containing A and closed under taking triangle s, dir ect sums, and direct summands. T o ﬁnish, we reca ll the following useful res ult. F or a pro of, see [49], where the reader can ﬁnd more details ab out the construction of D ( A ). Prop ositi o n 2.8 . If A and B ar e quasi-isomorphic A ∞ -algebr as, then D ( A ) is triangle e quivalent t o D ( B ) and D π ( A ) is triangle e quivalent to D π ( B ) . 2.2. Al gebraic ge o metry. Algebraic geo metr y is the most na tural geometric ﬁeld for the application o f homologic a l algebra. Indeed, homological alg ebra p ermeated algebraic g eometry long ag o. W e shall only recall a sma ll amo unt . F or more deta ils, see [27]. Let us recall some standard ab elia n ca teg ories asso ciated to an algebra ic v ariety X . Any a lgebraic v ariety X comes with its sheaf of functions O X . A sheaf E on X is ca lled a quasi-co herent s heaf if there is a n ac tion o f O X on E and, loca lly in the Zariski to p o logy , E is the cokernel of a morphism betw een free O X -mo dules. E is called co herent if lo cally , in the Zariski top o lo gy , E is the cokernel of a morphism betw een free, ﬁnite-rank O X -mo dules. Let us restr ict ours e lves to the case that X is quasi-pro jective over a ﬁeld k . In this case, the catego r y of coherent sheaves on X , Co h( X ), is an ab elia n category . The catego ry o f qua si-coherent sheav es, QCoh( X ), is also ab elian and it po ssesses enough injective ob jects. Ther efore, we can r esolve any quasi-cohe r ent sheaf by a b ounded b elow co mplex of injective sheav es . Th us, we can form the derived catego ry of Co h( X ), or QCoh( X ), by taking the ho motopy categ o ry of the sub catego ry of all b o unded b elow complexes of injectiv es with b ounded c o herent coho mology , or by taking the homotopy categor y of the sub categor y of all injectiv es with quasi-coher ent cohomolo gy . An imp or tant class of cohere nt sheav es is the lo cally free co herent sheav es, i.e. ones which a re lo cally is omorphic to a ﬁnite-rank free O X -mo dule. Given a loca lly free cohere n t sheaf E , we can asso cia te to it an alg ebraic vector bundle E whose sheaf of sections is the dua l sheaf of E , i.e. the sheaf whose sections over U a r e given by Hom U ( E ( U ) , O X ( U )). This gives a con trav ar ia nt equiv alence b etw een the categorie s of loca lly free coherent sheav es and algebr a ic vector bundles. 2.3. Sym plectic geometry bac kground. In this sectio n, we review some of the underlying notio ns of symplectic geo metry . With this knowlege in ha nd, we pro- ceed, in this section, to deﬁne La grangia n intersection Flo er homology . An excellent reference for all things symplecto - top ologica l is [38] and, cor r esp ondingly , an ex- cellent intro duction to the uses of J -holomo rphic curves in symplectic top olog y is [39] Let M b e a smooth manifold and ω an an ti- symmetric t wo-form. Deﬁnition 2 .9. ω is a symple ctic form if dω = 0 and ω is non-de gener ate, i.e. t he p airing on ve ct ors in T x M at al l p oints x ∈ M is non-de gener ate. Note that a symplectic manifold must have ev en dimensio n. Example 2.10. (1) The c anonic al example of a symple ctic manifold is the c otangent sp ac e T ∗ X to any smo oth manifold X . L et us lo ok at an op en chart U and denote the c o or dinates on U by p and denote the c o or dinates in the ﬁb er dir e ction by q . T ∗ U ∼ = U × R n . We let σ can = q dp . One c an che ck that even though we have deﬁne d this lo c al ly, it is glob al ly wel l- deﬁne d. We set ω can = dσ can = dq ∧ dp . ω can is cle arly non-de gener ate as ve ctors in MEET HOMOLOGICAL MIRROR SYMMETR Y 7 the b ase p air with ve ctors in the ﬁb er. ( T ∗ X , ω can ) is also an imp ortant example of an exact symple ctic manifold, that is, a symple ctic manifold for which ω = dσ for some one-form σ . (2) Any smo oth pr oje ct ive c omplex algebr aic variety is a symple ctic manifold. The F ubini-Stu dy K¨ ahle r form on P n C r estricts to a symple ctic form on any c omplex s ubmanifold. The follo wing result shows that symplectic manifolds ar e all locally isomorphic. Prop ositi o n 2.11. (Darb oux) Given a symple ct ic manifold ( M 2 n , ω ) and a p oint x ∈ M , then ther e is a neighb orho o d U of x and a set of c o or dinates ( x 1 , . . . , x n , y 1 , . . . , y n ) on U such that ω | U = P i dx i ∧ dy i . If ω is a n anti-symmetric tw o-form on a vector space T , for any subspace S we can deﬁne the s ymplectic ortho gonal S ⊥ = { v ∈ T : ω ( s, v ) = 0 ∀ s ∈ S } . If ω is non-deg enerate, then S ⊕ S ⊥ = T . A subspace S is iso tr opic if S ⊂ S ⊥ and coisotro pic if S ⊥ ⊂ S . It is called Lag rangian if it is b oth isotropic and coisotropic and symplectic if S ∩ S ⊥ = { 0 } . Given an sy mplec tic manifold ( M , ω ), w e say that an e mbedded submanifold S is isotro pic (resp. coisotro pic, Lagr angian, s ymplectic) if all the tangent spaces are isotro pic (re s p. co isotropic, Lagrangian, symplectic). F or T ∗ X with ω can , the zero section X is a Lagra ng ian submanifold, a s is each ﬁber . The general case is similar. Prop ositi o n 2.12. (Weinstein) L et L b e a L agr angian su bmanifold of a symple ctic manifold ( M , ω ) . Then ther e is a t u bular neighb orho o d U of L which is symple cto- morphic to a n eighb orho o d of L in T ∗ L with ω can . Since a symplectic form is non-degener a te, it aﬀords a w ay to turn vectors into one-forms and vice versa. If η is a one-form, we can deﬁne η ∨ to be the vector ﬁeld such that ω ( η ∨ , − ) = η . W e o ften start with the diﬀerential of a s mo oth function f : M → R , or of a time-v arying function f : M × R → R . Giv en such a function, we deﬁne the Hamilto nia n vector ﬁeld X f t to be d f ∨ t . Given a vector ﬁeld, w e can often in tegrate it out to a diﬀeomorphism called a ﬂo w φ t : M → M . The ﬂo w asso ciated to X f t is called a Ha miltonian ﬂow. O ne can c heck that since d f t is closed, Hamiltonian ﬂows preserve the symplectic form. Note that given a submanifold N of M , φ t ( N ) deﬁnes an isoto py betw een N and φ 1 ( N ). This is called a Hamiltonian isotopy . Deﬁnition 2.13. A bund le endomorphi sm J : T X → T X is c al le d an almost c omplex stru ct ur e if J 2 = − 1 . The simplest ex a mple of an almost complex structure is m ultiplica tion b y ı on C ∼ = R 2 . An a lmost complex s tructure on a vector space is the same thing as a complex s tructure. Complex manifolds therefore hav e natural a lmost complex structures, namely multiplication by ı . An almost complex structure J that makes X into a complex manifold is called a complex structure. How ever, not all almost complex structures ar e co mplex s tr uctures. The ı -eigenspaces of J on T X ⊗ C must be inv o lutive. See [42]. Deﬁnition 2 . 14. If ( M , ω ) is a symple ctic manifold, then an almost c omplex struc- tur e J is c al le d c omp atible if ω ( J − , − ) is a Riemannian metric, i.e. s ymm et ric, p ositive deﬁnite, and non-de gener ate. 8 MA TTHEW ROBER T BALLARD Given a symplectic manifold ( M , ω ), we ca n a lwa ys lo cally ﬁnd a compatible almost complex str ucture using Darb oux’s theo rem. Since Riemannian metrics form a conv ex spac e (we c a n add them), we can ﬁnd a partition of unit y and patch together our loca l compatible almost complex structures in to a g lobal almo st complex structure J . Given a complex ma nifold Y with complex struc tur e j , we say that a map φ : Y → M is J - holomorphic if dφ ◦ j = J ◦ dφ . If Y = M = C and j = J = ı , then this e q uation is equiv alent to the set of Cauch y -Riemann equatio ns. Solutions of the Cauc h y-Riemann equations satisfy nice prop e rties (all of complex algebraic geo metry stems fro m this.) Gromov realised [22] that J -holomorphic maps hav e ma ny o f the same nice prop erties. He intro duced the study of J -ho lomorphic curves into symplectic manifolds and gav e a useful new to ol to symplectic geo metry . 2.4. Fl o er cohom ology and F uk ay a categorie s. Now we ca n outline Lagra ng ian int ersection Flo er co homology . Let L and L ′ be tw o closed Lagra ng ian subma ni- folds o f a compact symplectic manifo ld. If they a r e not transverse, r e pla ce L by a Hamiltonian isotopic Lag rangian. W e can then ass ume that L and L ′ int ersect transversely and thus L ∩ L ′ is a ﬁnite set. Given tw o intersection p oints p and q , consider the s et of J -ho lo morphic maps φ : D → M from the unit disc with tw o marked p oints a t − 1 and 1 such that φ ( ∂ D ∩ H ) ⊂ L, φ ( ∂ D ∩ ( C − H )) ⊂ L ′ , φ (1) = q , and φ ( − 1) = p . The exp ected dimension of the space of solutions can be deter - mined as follo ws. The pullback tang ent bundle φ ∗ T M is trivial since we are work- ing with a disc. Ther e is a real sub-bundle along the b oundar y determined by the tangent spaces to the Lagr angians. W e ca n change o ur tr iv ialisation s o that φ ∗ ( T p L ′ ) = ı φ ∗ ( T p L ) and φ ∗ ( T q L ) = ıφ ∗ ( T q L ′ ). If we rotate a t p and q , we get a lo op of La grang ia n s ubs pa ces in R 2 n . Let Λ n denote the Lagra ngian Gra ssmannian of R 2 n . H 1 (Λ n ) ∼ = Z a nd there is a distinguished generato r µ called the Maslov class. The index of the ope rator is given by a pplying µ to the lo op o f Lagrang ian subspaces. Let us denote this index by µ φ ( p, q ), or b y µ ( p, q ). Note that this a relative gra ding in the sense that w e only know the diﬀerence betw een p and q . No absolute grading on the critical p oints is sp eciﬁed. The space of s uch J -holo mo rphic discs is at leas t one-dimensiona l since we hav e a free action by conformal auto mo rphisms of the unit dis c P S L (2 , R ) on the unit disc. If the dimension is one, taking the quo tient by the free action we e x pe c t to get a zero-dimensio nal ma nifold, M ( p, q ), as the mo duli space of so lutions. A wonderful fact is that there ar e natur a l compactiﬁcations of spaces o f J -holomor phic maps called Gro mov compactiﬁcations ¯ M ( p, q ) [39]. The co dimensio n one co mpo nent of the boundar y of thes e compa c tiﬁca tions, in the best situations, is ∂ ¯ M ( p, q ) = a r ∈ L ∩ L ′ M ( p, r ) × M ( r , q ) This is analo g ous to ﬁnite-dimensional Mor se theory where, to compactify the space of gr adient tr a jectories, one adds in tra jectories which a re br oken at an intermediate critical points. The co dimension one piece is where w e only ha ve o ne intermediate critical point. I n the case wher e µ ( p, q ) is one, there is no r w e can squeeze in and hence M ( p, q ) is compact. Thus, M ( p, q ) is a ﬁnite set of p oints which we ca n count. Deﬁne the chain complex C F ( L, L ′ ) as the free Z / 2 Z -graded vector space with basis L ∩ L ′ and s e t m 1 ( p ) = X q : µ ( p,q )=1 n pq q MEET HOMOLOGICAL MIRROR SYMMETR Y 9 where n pq is the num b er of points in M ( p, q ). Prop ositi o n 2.1 5 . m 2 1 = 0 . T o s ee this, let µ ( p, q ) = 2 and consider the co e ﬃcient of the q -term in m 2 1 ( p ). It is exactly P r n pr n r q , whic h is the n um ber o f points in ` r ∈ L ∩ L ′ M ( p, r ) × M ( r, q ). ` r ∈ L ∩ L ′ M ( p, r ) × M ( r, q ) is the b oundar y of ¯ M ( p, q ), whic h is one-dimensiona l, hence m ust b e a n ev e n num b er. W e ca n impro ve the counts to lie in Z , instead of Z / 2 Z , when w e can coher ent ly orient the mo duli spaces of solutio ns . This requires a Spin structure [15]. The cohomolo gy o f C F ( L, L ′ ) is called the (La grangia n intersection) Floer co - homology of L and L ′ and is denoted by H F ( L, L ′ ). What could go wrong with this situation? The r e are a few problems re la ted to the Gr o mov compactiﬁcation. The compactness relies on b ounds on the energy R φ ∗ ω of a J -ho lomorphic dis c φ . How ever, in general, w e could hav e energies of a sequence of J -ho lomorphic discs tending to inﬁnit y . T o r emedy this, w e can include a formal parameter which keeps track of the symplectic ar ea of these discs. This int ro duces Novik ov ring s into the discussion [29]. If we are lucky , as we will be la ter in this pap er, the formal series actually con verges if we sp ecia liz e it. Another, mo re ser ious, pro blem is that if the symplectic form do es not v anish on the seco nd homotopy gr oup of M , π 2 ( M ), we would have to inc lude sphere bubbles in the Gromov co mpactiﬁcation. Perturbing the almo s t complex structure a nd adding a zer o th order term are no long er enoug h to guara ntee that the co mpactiﬁed mo duli spaces have the pro p e r structure. This issue was ov er c ome by int ro ducing a new, mo r e genera l metho d o f p er turbation — virtual per turbation theor y . F or the details see [19, 48, 55, 36]. On the other hand, if ω do es not v anish on π 2 ( M , L ), we could have disc bubbles which g e nerally o ccupy a (r eal) c o dimension one subset of the compa c tiﬁed moduli spaces. Thus, if µ ( p, q ) = 2, the bo unda ry of ¯ M ( p, q ) may now also include other comp onents. Consequently , m 2 1 may no longe r be zero. While this problem cannot alwa ys b e ﬁxed, there is a nice gener al fra mework developed in [1 8] which a llows one to addres s the issue. F or a simple and illuminating ex ample of what could go wrong in this case, see [30]. Assume now that our Lag rangia n L is sitting inside a Calabi-Y au manifold X . Since X is Cala bi-Y au, it p o s sesses a no n- v anishing holomor phic volume fo r m Ω. Given a Lagr angian submanifold L of X , w e take a frame { v 1 , · · · , v n } for it at a po int x ∈ X . This gives a c omplex n umber θ ( x ) = Ω( v 1 , · · · , v n ) 2 | Ω( v 1 , · · · , v n ) | 2 and consequently , a map θ : L → S 1 called the phase squar ed ma p. A g raded Lagra ng ian subma nifo ld is a La grangia n L and a lifting of the phase map to α : L → R . This induces a grading on the intersection po int s of graded Lagrang ia n submanifolds. One can chec k that this grading is compatible with the r elative Maslov grading giv en in the Flo er co mplex; se e [51]. A choice o f Spin structur e , if it exists, and the gr ading allow one to work, in principle, with Z -co eﬃcients and Z -gra ding s in Flo er cohomology . There is no r e ason to stop at J -holomor phic disc s with t w o marked p oints. One can als o consider J -holomor phic discs with n > 2 marked po ints. W e outline the construction a nd r efer the inquisitive reader to the refere nc e s . First we consider the 10 MA TTHEW ROBER T BALLARD universal fa mily of n -p ointed discs, A n . By an n -p o in ted disc, we mean a disc with n distinguished po ints on the b oundar y consider ed up to equiv alence given by M¨ obius transformatio ns. The ma rked p oints ξ 0 , · · · , ξ n − 1 are or dered cyclically and we la b el the b o undary segment b et ween ξ i and ξ i +1 as C i . Given Lag rangians L 0 , · · · , L n and intersection po ints p ii +1 ∈ L i ∩ L i +1 where i is taken mo dulo n + 1, we ca n lo ok for J -ho lomorphic maps φ fr o m an ( n + 1)-p ointed disc into our symplectic manifold M so that φ ( C i ) ⊂ L i and φ ( ξ i ) = p ii +1 . When ˜ p n 0 = 2 − n + P ˜ p ii +1 , the space of such ma ps (when considering all p ossible ( n + 1)-p ointed discs, a ssuming transversality) is zer o-dimensional. As b efor e, there are natur al co mpactiﬁcations of these mo duli spaces which add pieces of co dimension larger o r equal to one. Therefore, in the case where we hav e chosen the corr e ct intersection p o in ts, we ca n count the num b er of such J -holomor phic maps. Denote this count by n p 01 ··· p n − 1 n p n 0 and deﬁne m i ( p 01 , · · · , p n − 1 n ) = X q ∈ L 0 ∩ L n : ˜ q =2 − n + P ˜ p ii +1 n p 01 ··· p n − 1 n q q Recall tha t m 1 satisﬁed m 2 1 = 0 . The total co llection o f the m i satisfy the A ∞ - relations. Let us sketc h why this is true. W e ﬁrs t consider the case that M is simply a p oint and lo ok at the natur al co mpactiﬁcations of these mo duli spaces . An y n - po int ed disc provides a J -ho lomorphic ma p. Thus, our moduli space of such maps is all of A n . W e need to compactify this s pace. T o do this w e add deg enerations of the form / / /o /o /o • • • • • • ξ 0 ξ 1 ξ 2 ξ 3 ξ 4 • • • • • ξ 3 ξ 2 ξ 0 ξ 1 ξ 4 as the codimension one b oundary strata and work our w ay down inductiv ely . The co dimension one b oundary strata of A n are given by Y n − 1 >j > 2 A j × A n +1 − j corres p o nding to the splitting o f a single marked disc into tw o p ointed discs, the ﬁrst w ith j marked p oints and the second with n + 1 − j ma rked points, joined at a marked p oint. If we view these a s counting discs, we se e that the relations in the co dimension one b oundary strata are exactly the A ∞ -relations with v anishing m 1 . The compac tiﬁca tion in the genera l case works in a v ery s imilar way . When we genera lise from a p oint, the c ompactiﬁcations include J -holomor phic maps with t wo marked p oints cor r esp onding to the non-v a nishing m 1 . Thus, in the b est o f circumstances, we can form an A ∞ -categor y called the F uk aya category of M , F uk ( M ). Its ob jects a re Lag rangia n submanifolds (with some extra struc tur e) and its morphis ms are intersection p oints b etw een tw o given Lagrangia ns . The m ulti- comp ositions a re those outlined ab ov e. W e shall see a few v ar iations o n the theme of the F uk ay a categ o ry a s we pro ceed. Most v ariations corr esp ond to r e stricting one’s attent ion to pieces of a lar g er F uk aya category . Other s corr esp ond to slight mo diﬁcations if we do not actually exp erience the best of cir cumstances. MEET HOMOLOGICAL MIRROR SYMMETR Y 11 Remark 2. 16. If one wishes to examine a ful ler tr e atm ent of the c onstruction of the F ukaya c ate gory in the c ase of an exact symple ctic manifold, se e [49] and t he r efer enc es ther ein. F or the r e ader inter este d in the most gener al r esu lts, se e [18] . 3. Homological mirror symmetr y for the projective l ine 3.1. B - branes on the pro jectiv e line. W e b egin by tackling B-branes b efore A-branes. The name “bra nes ” is sho rt for membranes a nd is a r eﬂection of the sub ject’s physical origin. In s tring theo r y , branes ar e bo unda ry conditions for an op en s tr ing propr ogating thro ugh s pacetime. The r elative simplicity of the algebro- geometric side, or B-side, of mirror symmetry spurs this alphab etic reb ellion. After deﬁning the categor y of B-br anes, we seek a nd ﬁnd a co ncrete descr iptio n of the category in terms of a directed graph, i.e. a quiver. Let k be a ﬁeld. Deﬁnition 3.1. The c ate gory of B-br anes on P 1 k is t he b ounde d derive d c ate gory of c oher ent s he aves on P 1 k . If we are seeking a c a tegory that r eﬂects the a lg ebraic geometry of P 1 k , there is per haps no b etter choice. Evidence is given by the following result of Bondal a nd Orlov [8]. In passing from P 1 k to D b (Coh( P 1 k )), w e lo s e no thing . Theorem 3 . 2. L et X and Y b e smo oth pr oje ctive varieties and let ω X or ω − 1 X b e ample. If D b (Coh( X )) is e quivalent t o D b (Coh( Y )) , then X is isomorphi c to Y . In order to get a better handle on D b (Coh( P 1 k )) we seek an a lter native descrip- tion. Consider the structure sheaf O and the t wisting sheaf O (1). Let E = O ⊕ O (1). The co herent sheaf E “touches everything” in D b (Coh( P 1 k )). More precisely , Lemma 3.3. The smal lest triangulate d sub c ate gory of D b (Coh( P 1 k )) c ontaining E and close d u nder taking dir e ct summands is D b (Coh( P 1 k )) . Pr o of. The se t of ob jects {O ( n ) } n ∈ Z generates D b (Coh( P 1 k )), since any coher ent sheaf on P 1 k has a ﬁnite reso lution b y members of this collection ([28]). Consider the follo wing exact sequence 0 → O ( n − 2 ) ( y , − x ) → O ( n − 1) ⊕ 2 ( x,y ) t → O ( n ) → 0 Setting n = 2 shows that we can r esolve O (2) b y O (1) and O and consequently b y E . Letting n = 0 shows that O ( − 1 ) can also b e r esolved (in the o ther direction) by copies of E . Th us, O (2) , O ( − 1 ) lie in the smallest tria ngulated s ubca tegory o f D b (Coh( P 1 k )) gener a ted b y E and close d under taking idemp otents. Iterating the argument shows inductively tha t O ( n ) also lies in the sub categ ory generated by E , for all n ∈ Z . Hence, the smallest triang ulated ca tegory in D b (Coh( P 1 k )) containing E is D b (Coh( P 1 k )) itself.  The next lemma is w ell known. Lemma 3.4. Ext i ( E , E ) = 0 for i > 0 . The o nly interesting Ex t- group is then Ext 0 ( E , E ) = Hom( E , E ). A simple gra ph- ical means of enco ding the data of this algebr a is the following q uiver Q . • •   D D 12 MA TTHEW ROBER T BALLARD The path algebr a k Q of Q is the alge bra gene r ated by oriented paths in Q with m ultiplication given b y concatenation of paths. Note that the trivial paths a t each of the vertices ar e idemp otents. Lemma 3.5. Hom( E , E ) ∼ = k Q . Pr o of. O corres po nds to the ﬁrst no de a nd O (1) to the s econd. The tw o arrows are the morphisms x, y : O → O (1).  F r om what we hav e just seen, the following s hould no t be to o s urprising. Prop ositi o n 3.6 . D b (Coh( P 1 k )) is triangle e quivalent to D b ( k Q - mo d) . The pro of of this statement requir e s a little so phistication. Recall that, to g et D b (Coh( P 1 k )), we replaced all our bo unded c o mplexes o f coher e nt sheav es with choices of injective resolutions and then to ok the homoto p y ca tegory . Let us ta ke an injectiv e r esolution o f E and deno te it by I E and co nsider the category I ( P 1 k ) formed by injectiv e complexe s with b o unded co herent cohomology . W e then consider the functor Hom I ( P 1 k ) ( I E , − ) from I ( P 1 k ) to C h ( k ). In fac t, the image of an ob ject J in C h ( k ) has the s tructure of a left dg-mo dule (a nd hence A ∞ -mo dule) ov er the dg-algebr a A = Hom I ( P 1 k ) ( I E , I E ) given by prec omp osition. Thus, we hav e a functor from I ( P 1 k ) to A -Mo d. This induces a functor from I ( P 1 k ) to D ( A ). The question then b ecomes: do e s it descend to a functor from D b (Coh( P 1 k )) to D ( A op )? In other words, a re all quasi-iso morphisms in I ( P 1 k ) in verted by the functor to D ( A op )? The answer is yes. Since any qua si-isomor phism has an inv ers e up to homotopy in I ( P 1 k ), we just nee d to c heck tha t our functor kills a ll n ull-homotopic maps. This is equiv- alent to checking that the functor kills a ll null-homotopic co mplexes, i.e. those complexes for whic h the identit y map is null-homotopic. But the co nt racting ho- motopy for the identit y map of the complex, N , provides the co nt racting homotopy for any chain ma p in Hom I ( P 1 k ) ( I E , N ). Th us, the functor descends. Let us deno te the res ulting functor by RHom( E , − ). The dg-algebr a A is quasi-iso morphic to k Q . Applying Pr op osition 2.8, we ca n ass ume that RHom( E , − ) ma ps fro m D b (Coh( P 1 k )) to D ( k Q op ). RHom( E , − ) is full and fa ithful on the smallest tria ngulated categ ory containing E and closed under dire c t summands and he nce is fully faithful. It is also es sentially surjective o nt o D π ( k Q op ) since b oth hav e to b e the smallest tr ia n- gulated c ategories co nt aining k Q and closed under dir ect summands. Now, as is well known, k Q ha s ﬁnite global dimensio n th us, D π ( k Q op ) ∼ = D b ( k Q -mo d); see [47]. Thu s, the structure of the seemingly complicated D b (Coh( P 1 k )) is controlled by the relatively s imple algebra k Q . W e shall see the quiver Q again shortly . 3.2. A-b ranes on the mi rror to the pro jectiv e line. The mirr or par tner for P 1 k is something a little mor e exo tic than a v ariety . W e consider the function W : C × → C g iven b y W ( z ) = z + q / z with q ∈ C × . Such a pair ( C × , W ) is called a La nda u-Ginzburg mo del, o r LG-mo de l. Since the v alue of q do es not aﬀect the category o f A-branes, we set q = 1. When dealing with LG-mo dels, a useful moral to keep in mind is that we should study the geometry of the critical locus and ho w it rela tes to the am bien t spac e . Now, on the A-s ide, we ar e interested in symplectic geometry , in particula r Lagrangians. First, we need a symplectic form. W e take it to b e ı dz ∧ d ¯ z z ¯ z . MEET HOMOLOGICAL MIRROR SYMMETR Y 13 W e could just take Lagr angian submanifolds of C × as o ur A-branes but that would not r eﬂect the prese nc e of W . Instead, we will lo ok at non-closed La grangia n submanifolds of C × whose b oundar ies lie o n W − 1 ( z ) for some g eneric choice z . W e could pro ceed as in [1] and inv es tigate a ll suc h Lagr angians with some additional requirements. This appro ach se e ms muc h more na tur al, but for s implicity’s sa ke we shall follow the ide a s of Seidel in [5 3] o r [4] and only consider a few sp ecial Lagra ng ians. W has non-deg e nerate singular ities, i.e., for ea ch p such that W ′ ( p ) = 0 we have W ′′ ( p ) 6 = 0. E ach such p is called a critica l p oint; W ( p ) is called the critical v alue. Ther e is a na tural connection on the complement of the singular ﬁber s coming in this situation. Given a vector ﬁeld X on C we can ﬁnd a horizo nt al lift ˜ X by using the s ymplectic form to split the tange nt spa ce of the domain of W int o the tangent space of the ﬁb er and its s ymplectic or thogonal. W e take ˜ X to lie in this orthog onal and pro ject down to X . With a connection, we can para llel transp ort vectors. Thank s to our choice of connection, parallel transp ort pr eserves the symplectic form. This gives our Lag r angians. Fix a non-cr itical point q . If we take a path γ : [0 , 1] → C so that γ (0) = p and γ (1 ) = q , we can lo ok at the set { x ∈ W − 1 ( q ) : lim t → 0 T γ | [ t, 1] ( x ) = q } ∪ { q } where T γ | [ t, 1] ( x ) is the symplectomo rphism coming from pa rallel transp ort along γ | [ t, 1] . The resulting set is just an in terv al (or a one- dimensional disc). It is a Lagra ng ian submanifold with bo undary on W − 1 ( q ). Stated in this language, the fact that the stable manifold is an n -dimensiona l Lagrangia n disc holds in grea ter generality; see [53]. This Lagr angian is called the v anishing thimble asso ciated with p . The b oundar y , which is an ( n − 1)-dimensio nal sphere, is ca lled the v anishing cycle ass o ciated with p . Since our example is low-dimensional, w e can a c tually draw the v a nishing thim bles asso cia ted to the tw o critical p oints for z + 1 /z . ◦ × × • • L 0 L 1 W e can now deﬁne our categ ory o f A-bra nes a sso ciated with the LG-mo del. Our tw o critical po ints give us tw o Lag rangia n thim bles which in tersect in tw o po int s. Order the La grang ian thim bles L 0 and L 1 . Deﬁnition 3.7. The c ate gory of vanishing cycles, F S ( W ) for the L andau-Ginzbur g mo del W is an A ∞ -c ate gory whose obje ct s ar e c ol le ction of vanishing thimbles L i , and whose morphisms ar e Hom( L i , L j ) =      C F ( ∂ L i , ∂ L j ) if i < j k id L i if i = j 0 otherwise We use the usu al A ∞ -structu r e on t he vanishing cycles of the re gular ﬁb er W = 0 . The c ate gory of A-br anes is the b ounde d derive d idemp otent c ompletion of the c ate gory of vanishing cycles, D π F S ( W ) . 14 MA TTHEW ROBER T BALLARD One should ﬁrst notice that the A ∞ -relations re sp e ct the (str ict) ordering , in the sense that an y deg eneration of an or dered disc splits into t wo ordered discs . Thus, the A ∞ -structure on the F uk ay a category g ives F S ( W ) its A ∞ structure. One should also note that, a priori, D π F S ( W ) dep ends o n the c hoic e of paths. Diﬀerent choices yield diﬀerent Lagr angians. But, in fact, D π F S ( W ) is indep endent up to equiv alence of the choice of paths [5 2]. Deﬁnition 3.8. The c ate gory of A-br anes for the LG-mo del W i s the c ate gory D π F S ( W ) . Now, let us take a closer lo o k at the catego ry of A-bra ne s for W ( z ) = z + 1 /z . W e hav e tw o ob jects, t wo morphisms from the ﬁrs t ob ject to the second, and the ident it y mo r phisms. Th us, it a lmost a tautology that the morphism alg e bra of F S ( W ) is the path algebr a o f our quiver Q . D π ( F S ( W )) is simply the b ounded derived ca teg ory of ﬁnite-dimensional repr esentations of Q . Indeed, the categor y o f B-branes on P 1 k is equiv a lent to the categ ory of A-branes o n W . Remark 3.9. The simplicity of this example hides sever al imp ortant details. One of these details is that O , · · · , O ( n ) is the wro ng exc eptional c ol le ction to lo ok at. One should inste ad c onsider Ω n ( n ) , · · · , Ω(1) , O . The c ase of t he mirr or to P 2 k pr ovides another il luminating ex ample of homolo gic al mirr or symmetr y; se e [4] . 3.3. B - branes on the mirror to the pro jectiv e line. With half of the homo lo g- ical mirror symmetry c orresp o ndenc e done, we now turn our attention to the other half. The naive guess for B-branes on the Landau-Ginzburg theory W : C × → C would simply b e co he r ent sheav es on C × or equiv alently mo dules ov er C [ z , z − 1 ]. What we should remem b er here is that, as in the case of A-branes on the Landa u- Ginzburg mo del, we wan t to take the potential W into account. W e must measure the singular ities of W using complex geo metry . This reﬂects the similar co nsid- erations on the A-side where the singula r ﬁb er s of W ga ve r ise to the v a nishing thim bles. How do we measure singularities? By a classical r esult of Serre, a No e - therian commutativ e a lgebra A is r egular if and only if ev ery mo dule ov er A has a pro jective resolution of uniformly b ounded leng th. F o r a general A , mo dules with such b ounded reso lutio ns form a tr iangulated subca tegory of D b (mo d- A ). Let us denote this sub catego ry by Perf ( A ). Then a mea s ure of the sing ularity of A is the quotient D b (mo d- A ) / Perf ( A ). This is often called the stable category o f A o r the category o f singula rities of A a nd it is indeed the desired deﬁnition of B-branes on W . Deﬁnition 3.10. The c ate gory of B-br anes of W : Y → C is D sing ( W ) = Y λ ∈ C D b (Coh( Y λ )) / Perf ( Y λ ) wher e Perf ( Y λ ) is the ful l triangulate d sub c ate gory of c omplexes of c oher en t she aves admitting a b oun de d lo c al ly fr e e r esolution. Note that, since we are working ov er C , ther e a re only a ﬁnite num ber o f sing ular v alues and hence the pro duct is ﬁnite. Of co urse, for this deﬁnition to b e us e ful, one needs to k now how to take the quotient o f D b (Coh( Y λ )) b y P erf ( Y λ ). W e will not cov er this, b eca use there is a more computationally acce s sible version o f the category of B- br anes o n W — the catego ry of matrix factoriza tions of W , M F ( W ). MEET HOMOLOGICAL MIRROR SYMMETR Y 15 Let A b e an alg ebra and f ∈ A . A matrix factorizatio n of f is a p er io dic c omplex · · · → P 0 d 0 P → P 1 d 1 P → P 0 d 0 P → P 1 → · · · with P 0 , P 1 pro jective mo dules over A and d 0 P d 1 P = d 1 P d 0 P = f id. Notice that we could repack a ge the data b y consider ing P = P 0 ⊕ P 1 with d P =  0 d 0 P d 1 P 0  so that d 2 = f id. Complexes o f this for m will be the ob jects in the ca tegory of ma trix factorizations of f . Given t wo matrix factorizations P • and Q • , we deﬁne the set of morphisms Hom C F ( f ) ( P • , Q • ) to b e Hom A ( P, Q ). It is Z / 2 Z -gr aded, with the degree zero piece being g iven by Hom A ( P 0 , Q 0 ) ⊕ Hom A ( P 1 , Q 1 ) a nd the degree one piece b eing given b y Hom A ( P 0 , Q 1 ) ⊕ Hom A ( P 1 , Q 0 ). O ne can view this a s a per io dic version of the o rdinary Ho m-complex. One nice fea ture is that while P • and Q • are not chain complexes, Hom( P • , Q • ) is. The action of the diﬀere ntial d P,Q is s imply sup er - commutation, namely d P,Q φ = d Q φ − ( − 1 ) ˜ φ φd P . The pro o f of the follo wing result is left as a s imple exerc ise to the reader. Lemma 3. 11. The c omp osition d 2 P,Q is 0 . Thus it makes sen s e to sp e ak of t he c ohomolo gy of Hom( P • , Q • ) . W e also have a shift functor [1] which sends P i to P i +1 mo d 2 and d i to − d i +1 mo d 2 and which acts o n mor phisms in the s tandard way . Notice that [2] ∼ = Id. W e can form cones ov er morphisms φ : P • → Q • by taking C ( φ ) = P [1] ⊕ Q with factorizations d C ( φ ) =  d P [1] φ 0 d Q  . Deﬁnition 3.12. The c ate gory of matrix factorizations of f , M F ( f ) , is the c ate- gory whose obje cts ar e matrix factorizations and whose morphisms ar e Hom M F ( f ) ( P • , Q • ) = H 0 (Hom C F ( f ) ( P • , Q • )) Prop ositi o n 3.13 . M F ( f ) is a triangulate d c ate gory wher e the shift is as ab ove and triangles ar e of the form A B C ( φ ) φ / /          [1] Z Z 4 4 4 4 4 4 4 This prop osition is pr ov ed in [4 4], where we ﬁnd the following v er y useful theo- rem. Theorem 3 .14. Consider a L andau-Ginzbur g mo del W : Y → C with Y aﬃne. Then D sing ( W ) = Q λ ∈ C M F ( W − λ ) = : M F ( W ) as t riangulate d c ate gories. Remark 3. 15. This is an extension of the pr o of by Eisenbud that the c ate gory of matrix factorizations is e quivalent to the c ate gory of Cohen-Mac aulay mo dules [16] . Here is an elucidating example. Example 3.16. L et A = k [ x ] / ( x 2 ) . Then the minimal fr e e r esolution of A/ ( x ) is · · · A x → A x → A → A/ ( x ) → 0 Lifting this r esolut ion to k [ x ] we get a matr ix factorization of x 2 . 16 MA TTHEW ROBER T BALLARD In fact, our situatio n is quite similar to the previous example. In the ca se o f the mirror to P 1 k , W = z + q /z . The critical p oints of W are ± √ q . The critical v alues a re z = ± 2 √ q and W ± 2 √ q = (1 / z )( z ± √ q ) 2 . Let us take the matrix factorizatio n, F , for W − 2 √ q , given by P 0 = P 1 = C [ z , z − 1 ] and d 0 P = (1 /z )( z − √ q ) , d 1 P = ( z − √ q ). First, le t us compute the cohomo logy a lgebra Hom M F ( W − 2 √ q ) ( F, F ). Prop ositi o n 3.17. Hom M F ( W − 2 √ q ) ( F, F ) is isomorp hic to t he Cliﬀor d algebr a on a single gener ator φ with φ · φ = − ( ∂ 2 z W )( √ q ) . Pr o of. A general mor phism ψ ∈ Hom C F ( W − 2 √ q ) ( F, F ) is of the form ψ =  A B C D  and d F, F ψ =  0 (1 /z )( z − √ q ) ( z − √ q ) 0   A B C D  −  A − B − C D   0 (1 /z )( z − √ q ) ( z − √ q ) 0  =  B / z + C D − A B / z + C A − D   z − √ q 0 0 z − √ q  Thu s fo r ψ to lie in the kernel we m ust ha ve A = D and B = − C z . Over C [ z , z − 1 ], the cohomolog y is genera ted by the identit y and by φ =  0 1 /z − 1 0  . Note that φ 2 =  − 1 /z 0 0 − 1 /z  . Notice that d 2 F = W − λ implies tha t d F ( ∂ z d F )+ ( ∂ z d F ) d F = ∂ z W . Hence ∂ z W id is triv ia l in cohomolo gy . Th us, the co homology a lg ebra b e- comes a mo dule over the Jac o bian ring C [ z , z − 1 ] / ( ∂ z W ), which in particular is ﬁnite dimensional ov er C . Now φ 2 = ( − 1 /z ) id. The image of 1 /z in the Jacobia n ring is 1 / √ q a nd hence φ 2 = − ( ∂ 2 z W )( √ q ).  A similar re s ult holds at z = − √ q . Usually , the cohomolo gy alg e br a is not enough data. W e nee d to know so mething ab out the underlying A ∞ -structures. How ever, in this cas e, we actually have all the data we need. Ho chsc hild coho mology of a gra ded asso c ia tive alge bra A ca n b e computed a s E xt ∗ A e ( A, A ). Prop ositi o n 3.18. A Cliﬀor d algebr a C , c onsider e d as a Z / 2 Z - gra de d algebr a, is pr oje ctive over C e . W e shall no t rec a ll the pro o f here, but o nly r efer the r eader to [6] for satisfactio n. Appea ling to the previous result and lemma 2.5, we can conclude that the dg- algebra Hom C F ( W − 2 √ q ) ( F, F ) is quasi-isomor phic to its cohomolog y . Again a sim- ilar r e s ult holds at z = − √ q . Finally , we ca n a ppe al to Orlov’s result, theorem 3.14, to conclude that the zero ob ject and F gener ate M F ( W − 2 √ q ), since the ﬁber ov er the singular p oints is isomorphic to k [ x ] / ( x 2 ). W e can conclude that M F ( W − 2 √ q ) is equiv alent to the bo unded derived ca teg ory o f mo dules over the Cliﬀord algebra Ho m M F ( W − 2 √ q ( F, F ) via a n argument analogo us to tha t in subsection 3.1. Thus, the c a tegory of B - branes for z + q /z is equiv alent to tw o copies of the b ounded derived ca tegory of modules ov er the Cliﬀord algebra s k h x i /  x 2 = ± 1 / √ q  . MEET HOMOLOGICAL MIRROR SYMMETR Y 17 3.4. A-b ranes on the pro jectiv e line , a.k.a. the sphere. Equip P 1 C with the F ubini-Study metric and its asso cia ted K¨ ahler form ω . In a Riemann sur fa ce, any curve is a La g rangia n. W e shall choo se to deal only with embedded and closed Lagra ng ians, i.e. simple close d curves in P 1 C . Any such cur ve divides the sphere into t wo, p e rhaps unequa l (with r esp ect to the F ubini- Study ar ea form), halves. W e ﬁrs t repro duce an illustrative ca lculation fr om [30]. Assume w e ha ve t wo Lagrang ians γ and γ ′ that intersect in exactly tw o po in ts. Let us denote the regions as in the image b elow. A B C D • q • p γ γ ′ Let u b e a forma l parameter that we will use to keep track of the symplectic area of the discs of int erest. W e shall use the standard complex structur e on the sphere. Be fore embarking on the computatio n, w e must make sure tha t w e have the signs asso cia ted to our Flo er diﬀerential correct and w e must make sure that we have the pro p e r gra dings on the intersection p oints. A thorough discussion of a coherent sign conven tion would take us o ﬀ track, so we shall ca rry out our computations with Z / 2 Z co eﬃcients. This will initially eliminate co nsiderations of the line bundles supp orted alo ng the Lagr angians. The issue o f gra ding, how ever, cannot b e a voided. T o simplify the situation, he r e we shall assume that one of o ur Lagra ng ians is a small Hamiltonian deformation as so ciated to a Mor se function. In this case, the intersection points are iden tiﬁed with the critica l p oints of the Morse function and we can use the asso ciated Morse index to gra de them. Now w e compute the Flo er diﬀeren tial. m 1 ( q ) = ( u ω ( C ) + u ω ( D ) ) p m 1 ( p ) = ( u ω ( A ) + u ω ( B ) ) q Then, m 2 1 ( p ) = ( u ω ( C ) + u ω ( D ) )( u ω ( A ) + u ω ( B ) ) p = u ω ( C )+ ω ( B ) ( u ω (Int( γ )) + u ω (Int ( γ ′ )) )( u ω (Int( γ ′ )) + u ω (Out( γ ) ) p Since γ ′ is a Hamiltonian defor mation of γ the areas o f the in ter iors of γ and γ ′ are equal. Hence, m 2 1 ( p ) = 0 and similarly m 2 1 ( q ) = 0 . Thus, the Flo er coho mology betw een γ and γ ′ is well-deﬁned. Note that the eq uality of the interior ar eas implies that ω ( A ) = ω ( B ). Thus, m 1 ( p ) = 0. Now consider m 1 ( q ). m 1 ( q ) = ( u ω ( C ) + u ω ( D ) ) p = u ω ( C ) (1 + u ω ( D ) − ω ( C ) ) p = u ω ( C ) (1 + u ω (Out( γ )) − ω (Int( γ ′ )) ) p = u ω ( C ) (1 + u ω (Out( γ )) − ω (Int( γ )) ) p 18 MA TTHEW ROBER T BALLARD Thu s, unless γ cuts the sphere in half, m 1 ( q ) 6 = 0 and the Floer cohomolog y of γ with itself v anis hes. In the cohomolo gical category of the F uk aya categ ory of P 1 C , γ is isomorphic to the zer o o b ject if and only if γ separates the spher e into t wo unequal halv e s . The mos t obvious choice o f a γ which is co homologica lly non-ze r o is a grea t circle. It is a classical result of Poincar´ e tha t any le ng th minimizing area -bisecting curve in the spher e must b e a grea t circle. The go od news is that any area -bisecting curve is Hamiltonian isotopic to a great c ir cle [43]. Th us, up to isomor phis m the only no ntrivial ob jects in the F uk aya c ategory for P 1 C are the gr eat circles. Given t wo great circles, we can take the p o les of the sphere to be their intersection po in ts and take the heig ht function h on the sphere. The asso cia ted Hamiltonian vector ﬁeld is simply J ∇ h wher e J is the complex structure on P 1 C . Thus, the asso ciated Hamiltonian ﬂow will simply rotate the sphere alo ng the given axis and even tually take one great circle in to a nother. F r om our brusque analy sis, we hav e concluded that with Z / 2 Z -co eﬃcients there is only o ne (cohomolo gically) non-zero ob ject in the F uk ay a catego ry — a choice of a grea t circle. If we k eep trivial ﬂat line bundles, this contin ues to hold with Z co eﬃcients. W e shall switch to C co eﬃcients and outline the computation showing that the ho lonomy of our ﬂat line bundle must b e ± 1 in o rder for the A-brane to be non-zer o. The computation sho wing that m 2 1 = 0 remains essentially the sa me. In the presence of a non-triv ia l ﬂat line bundle, we get the fo llowing equation m 1 ( q , v ) = u ω ( C ) ( v − e R γ ′ A ′ v e R γ A u ω (Out( γ )) − ω (Int( γ )) ) p = u ω ( C ) ( v − (Hol γ ( A )) 2 v u ω (Out( γ )) − ω (Int( γ )) p Thu s, the holonomy of the ﬂat line bundle must be ± 1 and as before γ m ust bisect P 1 C . Consequently , we hav e tw o non-tr ivial ob jects in the F uk ay a catego r y — a great circle with either trivial line bundle or a ﬂat line bundle with holonomy − 1. T o determine the full structure of the catego ry we m ust co mpute the morphism spaces. A c o mputation s imilar to the abov e shows that Hom(( γ , L 1 ) , ( γ , L − 1 )) has zero coho mo logy . Th us, there a re tw o non-zero o b jects which are o r thogonal to each other, at lea st co homologic ally . The structure o f the categ ory o f A-branes o n P 1 C now b eg ins to resemble the structure of the category of B-br anes on the mirror to P 1 C . It is imp orta nt to keep in mind that the a bove discussion sidestepp ed many of the crucial p oints in the constr uction of the F uk ay a category , such as whether it honestly factors through Hamiltonian isotopy , o rientations of the relev ant mo duli spaces, tra nsversality , ch ecking the A ∞ -structure, etc. Many of these issues are resolved. F or instance, the issues of Ha miltonian isotopy a nd transversality a re taken care o f in this case; s ee [1 2]. Now that we have seen that there are really only tw o non-zero ob jects of the F uk aya categ ory of P 1 C and that the o nly non-zer o morphism se ts are their endomor - phisms, we will present another wa y of computing H F ( L, L ) which was o riginally conceived in [1 8]. It is a natur al adaptation of Bott’s g eneralisatio n of Mo r se the- ory to allow non-isola ted cr itica l p oints. Naturally , it is called Bott-Mo rse-Flo er cohomolog y . As men tioned ab ov e, bubbling holomor phic discs can o c cur in co di- mension one strata of the mo duli spa ce of holo morphic discs. The res olution of this problem was the ce n tral mo tiv a tion of [1 8]. The presence of no nt rivial discs with bo undary on a given Lag rangian should defor m the singular coho mology alg e bra MEET HOMOLOGICAL MIRROR SYMMETR Y 19 structure in mu ch the same wa y that the presence of nontrivial holomorphic spher e s deforms the s ing ular cohomolog y in to the (small) quant um cohomo logy . Consider the s ingular c hain complex of our great circle w ith its usua l intersection pro duct. The intersection pr o duct is not deﬁned on a ll singular chains s o we must r estrict the allow e d class of co mpo sable ob jects to tho se that are trans versal. As the reader might appreciate now, the issue of transversality , or more precisely the lack thereo f, crops up often in these cases. One mea ns to sidestep the is sue is the deﬁnition o f a pre-catego ry [35, 1]. While we shall not be so formal, it is imp ortant to keep this po int in mind. W e now outline the computation o f Bott-Mor se-Flo er cohomolo gy obtained in [10, 11]. Let M k ( β ) b e the mo duli space of stable maps f from ( k + 1)-marked discs to P 1 C with f ∗ [ ∂ D ] = β ∈ π 2 ( P 1 C , L ). Here we hav e cyclically ordere d the marked p oints. The mar kings provide us with ev aluation morphisms ev i : M ( β ) → L . No w, we deform the intersection pr o duct as follows: m 2 ( S, T ) = S ∩ T + X β ∈ π 2 ( P ,L ) ev 0 ( ev ∗ 1 S ∩ ev ∗ 2 T ) u ω ( β ) . As the reader mig ht hav e gues sed by now, the defo rmed comp os ition is no longer asso ciative. W e must add in compo sitions of all order s to o btain an A ∞ -algebra . In fact, in this case, we must add in an m 0 also. W e set m 0 = X β ∈ π 2 ( P ,L ) ev 0 ( M 0 ( β )) u ω ( β ) and m k ( S 1 , · · · , S k ) = X β ∈ π 2 ( P 1 C ,L ) ev 0 ( ∩ ev ∗ i S i ) u ω ( β ) In fact we only get a Z / 2 Z -g rading on the resulting A ∞ -algebra . If we added in another for mal parameter counting the Maslov index of β , w e co uld rectify this. The prop er wa y to view expr essions such as ev 0 ( ∩ ev ∗ i S i ) is as currents, i.e. distribution-v alued diﬀere ntial forms. Th us, the image, e v 0 ( ∩ ev ∗ i S i ), will repre- sent the s ame curren t as its clo sure. Prop ositi o n 3.19. m k ( x 1 , · · · , [ L ] , · · · , x k − 1 ) = 0 for k 6 = 2 and m 2 ([ L ] , x ) = ( − 1) 1 − ˜ x m 2 ( x, [ L ]) = x . Pr o of. It is cle ar that the fundamen ta l cla s s is closed since any quantum cor rection gives a chain with to o large a degree . Since any class meets L , the class ev 0 ( ev ∗ 1 L ∩ ev ∗ i S i ) lies inside ev 0 ( ∩ ev ∗ i S i ). If k > 2, the dimension o f M k − 1 ( β ) is one les s than the dimension of M k ( β ) since we have lost a marked po int and consequently a degre e of freedom. Th us, the actua l dimensio n of the curren t is smaller than the exp ected dimension and the current v anishes. The same reasoning tells us that in the case k = 2 any quantum corrections v anish leaving only the standard cap pro duct. The sign arises from the choice of conv e ntion in [11].  Notice that m 0 is a m ultiple of the fundamental class. As we remarked previously , we still have a n honest A ∞ -algebra structure on co homology . Since L is a strict unit, the only comp ositio ns that matter are those in volving chains in the clas s of po int s. Now we chec k the previo us calculatio n concerning the (Flo er-co homologic a l) non-triviality of a given Lagr angian. 20 MA TTHEW ROBER T BALLARD Prop ositi o n 3. 20. U nless t he L agr angian divides P 1 C in half, the Flo er c ohomolo gy vanishes. Pr o of. m 1 ( p ) = [ L ]( u ω ( β 1 ) − u ω ( β 2 ) ). The reversal in the sig n comes fro m the diﬀerent o rientations induced on the b ounda r y from each attaching disc. With a single marked p oint we trace out the whole o f L . Now, if the symplectic ar ea asso ciated to β 1 is not equa l to that β 2 , we see that m 1 ( p ) is a nonzero multiple of the [ L ]. N ote that, for a ny one-chain S , m 1 ( S ) contains a piece cor r esp onding to the b oundary of S a s a chain. The r est is a singular chain o f dimensio n one. Hence, for a singular chain of dimension o ne to b e in the kernel of m 1 , the chain needs to be the fundamen tal class or some m ultiple o f it. Thus, if the Lagrangia n do es not cut P 1 C in half, the Flo er cohomology is zero.  The arg ument in the previo us pr op osition a lso shows that, if w e ta ke the equator as Lag rangian, the Flo e r cohomology is isomorphic to the standard co homology as a mo dule ov er o ur r ing C [ u ]. If we equip the Lag rangia n with a ﬂat line bundle L , we incorp orate the holo nomy into the A ∞ -structure as follows m k ( S 1 , · · · , S k ) = X β ∈ π 2 ( P 1 C ,L ) ev 0 ( ∩ ev ∗ i S i ) u ω ( β ) Hol ∂ β ( L ) As be fo re, we have reduced to the tw o A-branes of interest: the equator equipp ed with the trivia l line bundle and the equato r equipp ed with the non-trivial ﬂat line bundle. In these tw o cases, the Flo er cohomology is non-v anishing and iso morphic as a module to the usua l coho mology . The richness lies in the deformation o f the standard algebr a structure. In our case, the usual exterio r algebra str ucture of the cohomolog y of a circle is defo r med to a Cliﬀord a lgebra k h x i / D x 2 = ± u ω ( P 1 C ) / 2 E . Prop ositi o n 3.21. Le t p, q b e distinct p oints thought of as tr ansversal chains. Then m 2 ( p, q ) = [ L ] e ω ( β ) Hol ∂ β ( L ) Pr o of. When we consider a disc with three marked p oints t wo o f whic h m ust lie on p and q , we can either or ient the disc so that the zero th marked po int is b efor e or after p . The tw o orientations are realized b y the tw o discs attaching to L . Th us, we tr ace out all of L . The subtlety is in the signs . The mo duli spa c e s of the discs all hav e natural orientations from a uxiliary c hoices. In this c ase the induced orientations match up and we ge t all of the fundamental class .  Stopping and taking stock o f the situa tio n, w e see that on the A-side and B-side we hav e tw o sp ecial ob jects whose Z / 2 Z -graded endomorphism alg e br as ar e Cliﬀord algebras . If w e specia lize our for mal parameter u to e , as is ph y sically motiv ated, w e end up in the same situation as b efore. An y A ∞ -structure on H F ( L, L ) is necessa r- ily isomor phic to the trivia l one. When we pass to D π (F uk ( P 1 C )) we w ill get a sum of the bo unded derived category of mo dules ov er the tw o Cliﬀord a lgebras k h x i / D x 2 = ± e ω ( P 1 C ) / 2 E . “Prop ositi o n” 3.22. The c ate gory of A-br anes on P 1 C is e qu ivalent to the c ate gory of B-br anes on W : C × → C , W ( z ) = z + q /z wher e q = e − ω ( P 1 C ) . Mathematic al ly, the derive d idemp otent-c omplete d F ukaya c ate gory of P 1 k is e quivalent to the trian- gulate d c ate gory of matrix factorizations of its mirr or. MEET HOMOLOGICAL MIRROR SYMMETR Y 21 Remark 3.2 3. We c an interpr et this c orr esp ondenc e of A-br anes on P 1 and B- br anes on W as an instanc e of T-duality. T her e is for the torus. P 1 admits a torus ﬁbr ation over the interval given by the m omentum m ap of the U (1) - r otation along an axis. In T-duality, we r eplac e these tori with their dual tori, i.e. Hom( T , U (1)) . Often, we think ab out the torus as having some metric structu r e which is a choic e of length R . The T-dual torus wil l then have length 1 /R . Thus, if we T-dualize the moment u m map torus ﬁbr ation for P 1 , the two en ds op en u p and we get the algebr aic torus C × . The two n on-trivial A-br anes map to 1 and − 1 as determine d by their holonomies. These ar e exactly the critic al p oints of the sup erp otential on C × . Remark 3.24. As mentione d b efor e, ther e ar e a few details missing fr om a c om- plete pr o of of pr op osition 3.22, such as the c omplete deﬁnition of F uk( P 1 C ) ! A rig- or ous r eformulation of the pr evious pr op osition would just r e quir e an assumption that a F uk( P 1 C ) exists su ch that the c omputations c arrie d out ab ove ar e valid. 4. Homological mirror symmetr y for elliptic cur ves The name “ mirror symmetry” implies that t wo ob jects r elated by the cor re- sp ondence should b e similar enoug h for one to b e a reﬂectio n of the other. In the previous section we saw that P 1 k had a s mirr or not another algebra ic v a riety , not even another top olog ical s pace, but a function. T o the un trained eye, these tw o ob jects appea r quite dissimilar. F o r something a little mor e symmetric, we move to the next case of homolog ical mirr or symmetr y : elliptic curves. F o r the purp oses of mir r or symmetry , an e lliptic cur ve is a smo o th Ca labi-Y au v ariety o f dimensio n one. W e shall stic k to C as a gr o und ﬁeld. A s mo oth v ar iety X is Calabi- Y a u if its canonical bundle is trivia l. F or most Ca la bi-Y au v arieties, mirror symmetry is a little mor e s y mmetric. Unlik e the pre v ious section where we had four diﬀerent deﬁnitions of categ ories of branes, for Calabi-Y au’s w e only have t wo. Given a Ca labi-Y au X , the categor y of B-branes o n X is D b (Coh( X )) and the category of A-branes is D π (F uk ( X )). Homo logical mirror symmetry should simply exchange A-branes a nd B-branes of each ellipitic curv e. T esting this conjecture on the case of elliptic curves allows us to explo it so me additional structure. W e can uniformize elliptic curves. Namely , we ca n pres e n t any elliptic curve as C / ( Z + τ Z ) for some τ in the upp er half of the co mplex plane. This determines its complex structure, as we simply take the complex s tructure provided by C . This a lso determines the symplectic structure, since dz ∧ d ¯ z is inv ariant under translations . W e can of cours e scale the volume of our elliptic curve without changing the complex struc tur e. This makes sense for any p ositive real num b er, but we will wan t to co mplexify and allow scalings by complex num b er s with p ositive imaginar y co mpo nents. The r esulting complex- v a lued symplectic form is us ua lly called the co mplexiﬁed K¨ ahler for m. As such, we ca n desc rib e our elliptic curve E with complex iﬁed K¨ ahler form by tw o num b ers τ and ρ which lie in the upper ha lf plane. Let us incorp or ate this observ a tion by denoting our curve by E ρ τ . Then, mirr or s y mmetry is simply the exchange E ρ τ ↔ E τ ρ . Below, we generally follo w the notation from [46]. 4.1. A-b ranes on an ell iptic curv e . Another useful pr op erty of elliptic c ur ves is that mean curv ature ﬂow b ehav es well on them. One can a lso note that mean 22 MA TTHEW ROBER T BALLARD curv ature ﬂow is a Hamiltonian defor mation [57]. Thus, w e can ﬂow our (homo- logically non-trivial) La grang ian (curve) into a geo desic and be none the worse oﬀ in terms of the F uk ay a ca tegory . The result is that we only need to consider sp e- cial Lagrangians when working with the F uk ay a categor y . Of course, the sta nda rd means of computing C F ( L, L ) is b y using a slight Hamiltonia n deformatio n so w e need either to mo dify our deﬁnitions or forget abo ut these types of computations. While one could attempt generalisations o f F uk ay a -type comp ositio ns using singu- lar ho mology and incidence r equirements [18], such a discussion would certainly take us b eyond the scope of o ur in tro ductory pa per . W e shall therefore restrict our attent ion to a sub c la ss of morphisms when deﬁning co mpo sitions. W e r educe to the following. The ob jects o f our categ ory will b e sp ecial La- grangia n submanifolds o f E ρ 1 . Our spe c ia l Lagra ngians ar e simply lines with ra- tional (o r inﬁnite) slop e. W e will of co ur se need to choos e a g rading a nd a spin structure as b efore. The choice of spin structure we shall s uppress. Let us re- call how a grading of a Lagrang ian L is chosen. W e need to lift the phase ma p L → S 1 . O ur holomo rphic volume form is simply dz . Restricting that volume for m to the line mx = ny , we get ( n + ım ) √ m 2 + n 2 dl wher e l is some co o rdinate and we have l → (1 / √ m 2 + n 2 )( nl, ml ). With the par ametrization, the induced volume for m on our line is simply dl . The phase map is co nstant and equal to e 2 π ıα for some choice of α so that e π ıα is equal to the ar gument of n + ım . Thus, to choose a gr ading for our Lagrangian L we simply need to make a choice of α . There are Z -fold o ptio ns and ea ch ch oice giv es a diﬀeren t ob ject in our categ ory o f A-bra nes. Now, the grading (which we tak e from [46]) on a n in tersection p oint p ∈ L i ∩ L j is then µ ( p ) = − [ α j − α i ] where [ r ] is the greatest int eger less than o r equal to r . If we exclude v ertical lines, we ca n choose a gr ading that sa tisﬁes µ ( p ) = 0 if s i < s j and µ ( p ) = 1 if s i > s j for all lines simultaneously . W e also include ﬂat line bundles E on o ur sp ecial Lagrangians with mono dro my of unit mo dulus (or ﬂa t U (1)-bundles). Then an A-br ane o n o ur e lliptic curve is a triple ( L, α, E ) with L a sp ecia l Lagra ngian, α a lift of the arg ument of the slop e, and E a ﬂat line bundle on L . W e shall denote the collec tion o f this data a s L . Let us s e t Hom( L i , L j ) = C F ( L i , L j ; Hom( E i , E j )). As discuss ed ab ov e, we restr ict the cla ss of allowed comp ositions. In our A ∞ - category , w e are only allow ed to form compo sitions m n : Hom( L 1 , L 2 ) ⊗ · · · ⊗ Hom( L n − 1 , L n ) → Hom( L 1 , L n ) where all Lagra ngians inv olved are distinct. This of course forces us to abandon ident it y mo rphisms. W e hav e the comp ositions. m n : Hom( L 1 , L 2 ) ⊗ · · · ⊗ Hom( L n − 1 , L n ) → Hom( L 1 , L n ) m n ( p 1 , · · · , p n ) = X p 0 ∈ L 1 ∩ L n X φ ∈M ( p 0 , ··· ,p n ) ± exp(2 π ı Z D φ ∗ ω ) P exp( φ ∗ β ) when deg( p 0 ) = P n i =1 deg( p i ) + 2 − n , M ( p 0 , · · · , p n ) is the s pace of ho lomorphic maps fr om an ( n + 1)-pointed disc (mo dulo a utomorphisms when n = 1 ) with φ ( ξ i ) = p i and φ ( S i ) ⊂ L i , and P exp( φ ∗ β ) is the path order ed exp one ntial P exp( φ ∗ β ) = P exp( Z S n β n ) t n P exp( Z S n − 1 β n ) t n − 1 · · · t 1 P exp( Z S 1 β n ) MEET HOMOLOGICAL MIRROR SYMMETR Y 23 Here β is a connec tio n for m for the ﬂat bundle E . Consider tw o A-bra nes L and L ′ . T hen as a set the mo r phisms betw een L and L ′ are the same — simply L ∩ L ′ . Ass ume we have equipp ed these A-bra nes with a g rading with phase within ( − 1 / 2 , 1 / 2). Then, all the mo rphisms betw een L and L ′ are either of degree zero or of degree one. Let us as sume they ar e of degree zero. Assume that L has s lop e m and L ′ has slop e n . Then there are n − m in terse ction po int s o n the torus given b y  k n − m , mk n − m  for k ∈ Z / ( n − m ) Z . Assume now that the x -a xis int ercept of L is ( α 1 , 0) and the x -axis int ercept for L ′ is α 2 for 0 ≤ α i ≤ 1. Then the equa tion of L beco mes ( α 1 + t, ( n − 1 ) α 1 + nt ) and the equation of L ′ is ( α 2 + t, ( m − 1) α 2 + mt ). Thus the in ters ection p oints a re  k + α 2 − α 1 n − m , mk + nα 2 − nα 1 n − m  for k ∈ Z / ( n − m ) Z . There is a natural non-degenerate pairing of degree o ne Hom( L , L ′ ) × Hom( L ′ , L ) → C where w e declare p ∈ Hom( L , L ′ ) dual to p ∈ Hom( L ′ , L ). W e then take the trace on the bundle comp o nents. Let us deno te this by ( · , · ). Then ( m 2 ( a, b ) , c ) = ( a, m 2 ( b, c )) If we set L = L ′ , then the pair ing ( · , · ) is the Flo er manifestation of Poincar´ e duality . The cyclic symmetry and nondegenera cy o f the pairing allows us to re construct tw o- fold comp ositions inv olv ing degree one mor phisms from the t wo-fold co mp os itions only inv o lving degree zer o morphis ms . Th us, a t the level o f g raded ca tegories, if we for mally add in iden tit y morphisms to H 0 (F uk( E τ )) then w e can determine the prop er tw o-fold compos itio n rule s for the degr ee one piece of Hom( L , L ). 4.2. B - branes on an elliptic curve. W e now mov e onto the other side. If w e quotient C out by z 7→ z + 1 ﬁrst, our elliptic curve E τ can b e viewed as C × / Z where the action by Z is given by multiplication by e 2 π ıτ . W e can pull back any line bundle from E τ to C × . Any line bundle on C × is necessar ily trivial (and consequently so is the pullback of any vector bundle). Since E τ is pro jective and smo oth, the collection o f all line bundles on E τ generates D b (Coh( E τ )). All line bundles, since they are trivia l when pulled back to C × , can b e describ ed a s the quotient ( C × × C ) / Z wher e Z acts by ( u, v ) 7→ ( q u, φ ( u ) v ) for some holo mo rphic map φ : C × → C × . The map φ is the choice o f holomor phic tr iv ialisation over C × of the pulled- back bundle. Let L ( φ ) denote the line bundle determined in this w ay . With the similar ity of the F uk aya-compos itions to theta functions kept in mind, there a re ob vious c hoice s for φ . Let φ 0 ( u ) = exp( − π ıτ ) u − 1 . The sections of L ( φ 0 ) are the classica l theta functions. A general theta function is giv en as follows θ [ c ′ , c ′′ ]( τ , z ) = X m ∈ Z exp(2 π ı ( τ ( m + c ′ ) 2 / 2 + ( m + c ′ )( z + c ′′ ))) The classical theta function is simply θ [0 , 0 ]( τ , z ). T o chec k that this is actually a section o f L , we simply need to check that it behaves proper ly when we translate 24 MA TTHEW ROBER T BALLARD by 1 a nd τ . θ [0 , 0]( τ , z ) is clearly in v ariant under the shift z 7→ z + 1. And θ [0 , 0]( τ , z + τ ) = ∞ X m = −∞ exp(2 π ı (( m 2 / 2) τ + mz + mτ )) Letting l = m + 1 , we hav e ( m 2 / 2) τ + mz + mτ = ( l 2 / 2) τ + l z − (1 / 2) τ − z . Thus, θ [0 , 0]( τ , z + τ ) = e − π ıτ e − 2 π ız θ [0 , 0]( τ , z ) , so it is indeed a section of L . One can similarly chec k that θ [ a/ n, 0]( nτ , nz ) for a ∈ Z /n Z are s ections o f L n . By pulling ba ck L n via tr a nslation, w e g et sections for all p ossible line bundles. The degree of L ( φ 0 ) = L is 1. Any line bundle on an elliptic curve is the tensor pro duct of a degree z ero line bundle and L n . It is well known that Pic 0 ( E τ ) is isomorphic to E τ . Thus, an y line bundle is simply t ∗ x L ⊗ L n − 1 where x ∈ E a nd n is the degree . If the degree o f a line bundle is < 0, the corresp onding divisor is not equiv ale nt to a n eﬀective diviso r. Hence, we can hav e no g lobal s ections. F or n > 0, the s ections of L n are also given by theta functions: θ [ a/ n, 0]( nτ , nz ) for a ∈ Z /n Z . The sections of t ∗ α + ıβ L n are then t ∗ α + ıβ θ [ a /n, 0]( nτ , nz ) = θ [ a/n , 0]( nτ , n ( z + α + ıβ )) Note that Hom( t ∗ x L ⊗ L n − 1 , t ∗ y L ⊗ L m − 1 ) ∼ = Hom( t ∗ x L − 1 ⊗ t ∗ y L ⊗ L m − n ) Similarly , t ∗ x L − 1 ⊗ t ∗ y L − 1 can hav e no global sections if x 6 = y . Thus, there are no morphisms from t ∗ x L ⊗ L n − 1 to t ∗ y L ⊗ L m − 1 unless m > n or m = n and x = y . If we let x = α 1 + ıβ 1 and y = α 2 + ıb 2 , then we can also rewr ite t ∗ x L − 1 ⊗ t ∗ y L − 1 as ( t ∗ α 21 + ıβ 21 L ) m − n where α 21 = α 2 − α 1 m − n , β 21 = β 2 − β 1 m − n The sec tions o f ( t ∗ α 21 + ıβ 21 L ) m − n are g iven b y θ [ a / ( m − n ) , 0](( m − n ) τ , ( m − n )( z + α 21 + ıβ 21 )) Since E τ is a Cala bi-Y au v ariety , the canonical bundle ω E τ is tr iv ial. F or a general smo o th pro jective X , we have Ext ∗ ( F, G ) ∼ = Ext n −∗ ( G, F ⊗ ω X ) ∨ [28]. The pairing is given b y comp osing a ∈ Ex t ∗ ( F, G ) and b ∈ Ext n −∗ ( G, F ⊗ ω X ), taking the tra ce on F giving a n element of H n ( ω X ) ∼ = C . The functor S : D b (Coh( X )) → D b (Coh( X )) g iven by F 7→ F ⊗ ω X [ − dim X ] is called the Serre functor [7]. F or a Calabi-Y au, we g e t something ra ther sp ecial as S ∼ = [ − dim X ]. W e also ha ve ( a · b, c ) = ( − 1) ˜ a ( ˜ b +˜ c ) ( b · c, a ) Thu s, as in the cas e of A-branes, one ca n then use Ser r e dua lit y and the degre e zero morphisms to compute the Ex t-groups of these line bundles and r econstruct D b (Coh( E τ )) as a graded category . In [46], Polishc h uk and Zaslow tr eat all p ossible vector bundles on E τ and torsio n sheav es . MEET HOMOLOGICAL MIRROR SYMMETR Y 25 4.3. An o utline of the equiv alence. The ﬁrst step in ch ecking homolog ical mir- ror symmetry for elliptic curves is to g ive a functor fro m the categ o ry of line bundles (with only honest mor phisms) to the sub-categ ory of the F uk aya categ ory given by lines of no n-vertical slop e with ﬂat U (1)-bundles on them. F r om the discuss ion ab ov e, o ne can see that the co rresp ondence b etw een theta functions and intersec- tion p oints is pretty str ong, so we make the ob vious deﬁnition. Le t Φ : L ( E τ ) → H 0 (F uk( E τ )) Φ( t ∗ ατ + β L ⊗ L n − 1 ) = (Λ , A ) where Λ = ( α + t, ( n − 1) α + nt ) and A = ( − 2 π ı β ) dx . Recall that morphis ms betw een t ∗ α 1 τ + β 1 L ⊗ L n 1 − 1 and t ∗ α 2 τ + β 2 L ⊗ L n 2 − 1 are the s ame thing a s sections of t ∗ α 21 τ + β 21 L n 2 − n 1 where α 21 = α 2 − α 1 n 2 − n 1 , β 21 = β 2 − β 1 n 2 − n 1 . t ∗ α 21 τ + β 21 L n 2 − n 1 has sections given by f k θ [ k / ( n 2 − n 1 ) , 0](( n 2 − n 1 ) τ , ( z + α 21 τ + β 21 )) for k ∈ Z / ( n 2 − n 1 ) Z . Corresp o ndingly the in tersection points of Φ( t ∗ α 1 τ + β 1 L ⊗ L n 1 − 1 ) and Φ( t ∗ α 2 τ + β 2 L ⊗ L n 1 − 1 ) are g iven b y e k =  k + α 2 − α 1 n 2 − n 1 , n 1 k + n 1 α 2 − n 1 α 1 n 2 − n 1  for k ∈ Z / ( n 2 − n 1 ) Z . W e then set Φ( f k ) = exp( − π ıτ α 2 21 ( n 2 − n 1 )) · e k . Theorem 4.1. [46 ] Φ is ful l and faithful. It gives an e quivalenc e with the ful l sub- c ate gory of H 0 (F uk ( E τ )) c onsisting of sp e cial L agr angians with non-vertic al slop e and t he chosen sp e cial gr adings p air e d with ﬂat U (1 ) -bund les. Mor e over, Φ natu- r al ly extends to an e quivalenc e of the ab elian c ate gory of c oher ent she aves on E τ and the sub c ate gory H 0 (F uk ( E τ )) c onsisting of al l L agr angians with ﬂat U ( N ) -bund les and the chosen sp e cial gr adings. In lie u of re viewing the pr o of, we shall treat an illuminating example. The example is taken directly from [46], a s the autho r feels he canno t improve up on the choice. Under Φ we ta ke the structure s heaf of E τ to the x -axis with zero c o nnection. L n is ma pp ed to a line pas sing through the origin with slop e n and no connection. Let us consider the composition Hom( O , L ) × Hom( L, L 2 ) → Hom( O , L 2 ). The vector space Hom( O , L ) is o ne-dimensional and spa nned b y the cla ssical theta function θ [0 , 0]( τ , z ). As Hom( L, L 2 ) ∼ = Hom( O , L ), this space is also spanned by the clas sical theta function. On the other side, we hav e Λ 1 ∩ Λ 2 = { e 1 } , Λ 2 ∩ Λ 3 = { e 1 } , and Λ 1 ∩ Λ 3 = { e 2 , e 3 } where e 1 and e 2 are the orig in and e 3 = (1 / 2 , 0 ). Under Φ, we identify e 1 with θ [0 , 0]( τ , z ) and e 2 with θ [0 , 0](2 τ , 2 z ) and e 3 with θ [1 / 2 , 0](2 τ , 2 z ). 26 MA TTHEW ROBER T BALLARD                                                        Λ 1 Λ 2 Λ 3 e 1 e 2 • • e 3 T o compute the comp osition in the F uk ay a ca tegory , we lift our diag ram to R 2 in all p o ssible ways and c o unt the num ber of tr iangles with appropr iate weights determined by the K¨ ahler form. These will give the structure co eﬃcien ts . Of course, triangle s which a re translates of ano ther tria ngle are equiv a le nt. Therefore, we need to pin down one of the vertices that we will take to b e the origin, i.e . e 1 . W e need to compute the coeﬃcients of e 2 and e 3 . The coeﬃcient of e 2 comes fr o m counting tr iangles with vertices o n integer tr anslates of the or igin. Since we ha ve pinned down the ﬁr st v ertex we are lo oking at tria ngles in R 2 given by (0 , 0) , ( n, 0 ) , (2 n, 2 n ). Each suc h triangle ha s a rea τ n 2 . Summing ov er them w e get ∞ X n = −∞ exp(2 π ıτ n 2 ) = θ [0 , 0](2 τ , 0) The co e ﬃcient of e 3 comes from triang les with vertices at (0 , 0) , (0 , n + 1 / 2) , (2 n + 1 , 2 n + 1) . W e get ∞ X n = −∞ exp(2 π ıτ ( n + 1 / 2 ) 2 ) = θ [1 / 2 , 0 ](2 τ , 0) . Thu s, we have m 2 ( e 1 , e 1 ) = θ [0 , 0](2 τ , 0) e 2 + θ [1 / 2 , 0](2 τ , 0) e 3 . Let us now inv estig ate the other side. W e are lo oking at the square o f the classica l theta function. Since Φ is a functor, we nee d to show that θ ( τ , z ) 2 = θ [0 , 0](2 τ , 0) θ [0 , 0](2 τ , 2 z ) + θ [1 / 2 , 0](2 τ , 0 ) θ [1 / 2 , 0](2 τ , 2 z ) This is simply an application o f the addition formula for theta functions [41]. This is a basic example but it nicely illus trates the cor resp ondence. Using the eq uiv alence b etw een Co h( E τ ) and the sub category of H 0 (F uk ( E τ )) with ob jects whose gra ding is in the interv a l ( − 1 / 2 , 1 / 2), one can then e x tend the functor to Φ : D b (Coh( E τ )) → H 0 (F uk( E τ )) uniquely by requiring that it b e compatible with Serr e a nd Poincar´ e duality . Th us, w e get Theorem 4.2. D b (Coh( E τ )) and H 0 (F uk( E τ )) ar e e quivalent as gr ade d c ate gories. The next step is to determine if the tria ngulated structures coincide. The trian- gulated structure des cends fro m a cone co ns truction at the chain lev el. Thus, the natural way to pro ceed is to s how that as A ∞ -categor ies D b ( E τ ) and H 0 (F uk ( E τ )) are quasi-isomo r phic. But r e call that the deﬁnition o f the F uk ay a category is miss- ing Hom( L , L ). W e must r estrict our c la ss of comp os able morphisms to Hom-spaces betw een diﬀerent ob jects. With this limitation in mind, one c a n then a tta ck the problem. MEET HOMOLOGICAL MIRROR SYMMETR Y 27 In [45], the p os sible minimal A ∞ -structures on D b (Coh( E τ )) compa tible with Serre duality (i.e. cyclic) and the tensor structure and with a suitably la rge class of allow ed comp ositions ar e classiﬁed up to homotopy . The pr o blem then reduces to computing a single triple pro duct as the test of homotopy equiv a lence. The triple pro ducts on D b (Coh( E τ )) and H 0 (F uk ( E τ )) co incide. Thus, w e get the following result. Theorem 4.3. Consider D b (Coh( E τ )) with the admissible class of morphisms de- termine d by t he e quivalenc e with H 0 (F uk( E τ )) . The A ∞ -structu r es induc e d on this c ate gory by t he dg-c ate gory of line bund les and t he F ukaya c ate gory c oincide. In fact, the result is a little s tronger. If one can extend the deﬁnition of the F uk aya ca teg ory to include endomor phism sets while preser ving the A ∞ -structure already present on comp osable mor phisms, then one knows that the resulting A ∞ - category will b e homotopy equiv alent to D b ( E τ ) with the minimal A ∞ -structure determined by the dg-catego ry of line bundles. O f course , o ne can e x tend the A ∞ -structure in such a wa y since the A ∞ -structure inherited from the dg-c ategory of line bundles allows all po ssible comp os itions. So such an extensio n certainly exists; it is just not of a symplectic nature yet. Mo dulo this small deta il we have homologica l mirror symmetry for elliptic curves. 5. Fur ther resul ts In this section, we give a r apid review of the established ca ses o f homologica l mirror symmetry b eyond dimensio n one. By no means is this mea n t to b e a compre- hensive treatment. The in terested r eader is enco ur aged to consult the refer e nces. 5.1. A-b ranes on (near) F ano manifolds v ersus B-branes on LG-mo dels. 5.1.1. F ano surfac es. In the previous examples, w e were for tunate to b e able to compute comp ositions in F uk aya categor ies. The Landau-Ginzburg mirror to P 1 k was an extremely simple situation since the ﬁbers o f the p o ten tial were zer o- dimensional manifolds as were the Lag rangia ns inv o lved. F or P 1 C , any choice of almost co mplex structure was automatically integrable and regula r. Th us, we could apply the Riemann mapping theor em and reduce our counts to pure ly topo logical consideratio ns. If we consider A-branes on the mirro r of P 2 , the ﬁb er s are no longer zer o- dimensional but are now non-compact Riemann surfac e s. Thus, the co unt ing ca n again b e r educed to top ologica l co nsiderations. This at least gives o ne the hop e of applying previous methods to this pr oblem. Mor eov er , if we replace P 2 by an y other tw o-dimensio nal smo o th v ariety (or stack), the Landau-Ginzbur g mirro r (if it exists) should b e approachable via the same methods. In fact, this is done in a pair of pape rs [4, 5], see also [58]. In [4], Auro ux, Katzarkov, and Orlov tackle weighted pro jective surfaces a s stacks and demonstrate the follo wing. Theorem 5.1. (Aur oux -Katzarkov-Orlov) L et a, b, c b e mut ual ly prime p ositive inte gers and c onsider the stack CP ( a, b, c ) . Then D b (Coh( CP ( a, b, c )) is triangle e quivalent to D π ( F S ( W )) wher e W : { x a y b z c = 1 | x, y , z ∈ C } → C is given by W ( x, y , z ) = x + y + z . 28 MA TTHEW ROBER T BALLARD Moreov er, they also addr e ss how ho mological mirror symmetry be haves with resp ect to non-commutativ e deformations of the homoge neo us co or dinate r ing of the weigh ted pro jective plane. In addition, using the results for weighted pro jective planes, they prove a similar result for Hirzebr uch surfaces. In [5], Auroux, Katzarkov, and Orlov contin ue the study of this side o f mir r or symmetry for surfaces . They in vestigate the eﬀect of blowing up CP 2 at ≤ 9 p oints. T o pr op erly a ddress this, they m ust compactify the mirr o r to CP 2 and defo r m it. Theorem 5.2 . ( Aur oux -Katzarkov-Orlov) L et K b e a set of k p oints in CP 2 . Con- sider the del Pezzo surfac e X K given by blowing up CP 2 at those p oints. Then D b (Coh( X K )) is triangle e quivalent to D π ( F S ( W K )) wher e W K : M K → C is an el liptic ﬁbr ation obtaine d by deforming a c omp actiﬁe d mirr or of CP 2 . F o r more details, see [4, 5]. In [58], Ueda studies toric blow-ups of CP 2 . 5.1.2. Pr oje ctive, s m o oth toric varieties. In [1], Ab ouzaid takes a slight ly diﬀer e nt approach to the study o f homologic a l mirror symmetry for smo oth pro jective toric v arieties. Motiv ated by the results on F a no surfaces a nd the underlying ph ysics, one can sp eculate that the mir ror of a such a toric v ar iety X ∆ determined by the po lytop e ∆ is the La nda u-Ginzburg mo del W ∆ : ( C × ) dim X ∆ → C where W is the Laurent polynomia l whose Newton p olytop e is ∆. In fact, symplectically , one has a lot of freedom in cho osing the co eﬃcients of W ∆ . This a llows one to ta ke a limit that cor resp onds to a tropical degeneration. In this limit, one can iden tify A and B br anes on either side with cer tain Morse chain complexes on the moment po lytop e o f X ∆ . Morally , one r educes mir ror symmetry in this setting to data on the shared moment p olytop e. O f cour se, a bit of work go es into rela ting these “tropica l Lagra ng ians” back to the usual Lag rangians . In the end, one gets the fo llowing result. Let F uk( W ∆ ) denote the A ∞ -(pre)-catego ry of a dmissible La grangia ns in ( C × ) dim X ∆ . Essentially , these are Lagra ngians with b ounda ry o n a r egular ﬁb er ov er a p oint p of W ∆ which pro ject under W ∆ , near p , to curves emana ting from p . Theorem 5.3. (Ab ouzaid) Ther e is a ful l and faithful triangle fun ctor i : D b (Coh( X ∆ )) → D π (F uk( W ∆ )) . Mor e over, if X ∆ is ample, the functor is essent ial ly surje ctive. F o r more details, see [2]. 5.2. A-b ranes on F ano m anifolds v ersus B-branes on LG-m o dels . Although there are few results on the F uk aya categ ory of a toric v ariety , similar to subsection 3.4, one ca n deﬁne a symplectic categ ory that works. In [12], Cho and O h deﬁne a version of Flo er cohomolog y called adapted Flo er cohomolo gy which uses the standard complex s tr ucture on the toric v ar iety instea d of a g eneric almost co m- plex structur e. If the tor ic v ar iety is co nv ex, then this reduces to the usua l Flo er cohomolog y . They als o restr ict themselves to the Lagra ng ian torus ﬁb ers (with ﬂa t line bundles) of the momen t map ﬁbration. As in subsection 3.4, there are o nly a ﬁnite nu mber of tor us ﬁber s that hav e non- v anishing Flo er cohomo lo gy . These are called balanced tor us ﬁb ers. Cho and Oh show that bala nced torus ﬁbe rs are in o ne-to-one corresp o ndenc e with the cr itical po int s of the mirror LG s uper p otential. There is no Flo er cohomolo g y b etw een t wo distinct balanced tor us ﬁb er s . And, given a balanced tor us ﬁber L , the Z / 2 Z - graded Flo er cohomolo gy is H F ( L, L ) is a Cliﬀord alg ebra on dim L gener ators with MEET HOMOLOGICAL MIRROR SYMMETR Y 29 non-degenera te pa ir ing given by the holomorphic Hes sian of W at the critica l p oint p corresp onding to L . As in subsectio n 3 .4, the intrinsic formality of the Cliﬀord algebra uniquely determines any underlying A ∞ -structure on H F ( L, L ). F o r B-branes o n the mirr o r, a simple computation shows that the skys crap er sheav es a t the sing ular p oints are mu tually orthogo nal, and tha t the endomor phism algebra of a ny of these ob jects is a Cliﬀord algebra on dim L generator s with the bilinear form given by the holo morphic Hess ian at the corr esp onding p oint. As befo re, to match up completely , we need to c hange v a riables from C × to C using the exponential map. Nevertheless, w e get the following. Theorem 5.4. F or a c onvex sm o oth pr oje ctive toric variety X ∆ with mirr or W ∆ , the sm al lest t r iangulate d thick su b c ate gory c ont aning t he skyscr ap er she aves at the singular p oints of W ∆ in D sing ( W ∆ ) is triangle e quivalent to the smal lest triangu- late d thick sub c ate gory c ontaining the b alanc e d torus ﬁ b ers in D π (F uk( X ∆ )) . 5.3. A-b ranes v ersus B-branes on Calabi-Y au manifolds. 5.3.1. Ab elian varieties. F ollowing the re sults o f [46], the natural inclina tion was to consider highe r dimensional ab e lia n v a rieties A using similar metho ds. And, indeed, in [17], this is what is done. F uk ay a is able to demonstrate that the corr e sp onding Hom-spaces match up a nd, in the case wher e all three La grang ians a re mutually transverse, the m 3 -comp ositions match also. Another appr oach was undertaken in [3 5]. K ontsevic h a nd Soib elman use non- Archimedean analysis to skirt convergence issues and, similar to [1], use a deg en- eration to the bas e of a to rus ﬁbration to show that D b (Coh( A )) embeds a s a triangulated subc a tegory of D π (F uk ( A ∨ )) where A ∨ is the mirro r ab elia n v a riety to A . Sp eciﬁcally , it is eq uiv alent to the categ ory o f Lagrangians tra nsverse to the base. 5.3.2. Quartic su rfac e. In [50], Seidel implements his plan fro m [5 4] which is to use deforma tion theo ry a nd r esults ab out directed categ o ries of v anishing cycles to deduce homological mirr or symmetry for a Ca labi-Y au v ar iety . The idea is a s follows. The computations on the algebr o-geometr ic side are not so bad. On the symplecto-geo metric side, all the rele v a nt Lagrangia ns sit in an aﬃne patch of the Calabi-Y au, in this ca se a quartic surface in P 3 , a s v a nis hing cycles of a Picard- Lefschetz ﬁbration. Here they can be viewed as matching paths co rresp onding to low er- dimensional Picard-Lefschetz ﬁbrations. This viewpo int allo ws one to auto- mate an inductive pro cedur e that computes Flo er ho mology (F uk ay a categ ories) asso ciated to these Lag rangia ns from the directed categor y o f v a nishing cycles as- so ciated to the Picar d-Lefschetz ﬁbratio n. Then, to pass fr om the Lagrang ia ns in the aﬃne patch to the L a grang ians in the total s pace one has to include pseudo- holomorphic discs tha t hit the divisor at inﬁnit y , i.e. the complement of the aﬃne patch. Co unting these discs can be viewed a s a formal defor mation of the A ∞ - structure asso ciated endomorphism algebra of the v anishing cycles in the aﬃne patch. Then, the endomorphism algebra o f the v anishing cycles in the co mpa ct Calabi-Y au is obtained b y spe cialising this for mal deforma tion to a no n-zero v alue of the parameter. The key to ma king this result fea s ible is that the deformation spaces in volv ed are small, i.e . o ne-dimensional, and carr y a k × action. Thus, ther e are really o nly tw o diﬀere nt deformatio ns p ossible, the tr ivial o ne and the non-trivia l one. The is sue o f triviality of the deforma tion ca n be r educed to a computation in- volving Ho chschild c ohomolog y of the underlying A ∞ -algebra which ca n b e check ed 30 MA TTHEW ROBER T BALLARD on o ne side via explicit computations and o n the other b y the non- de g eneration of a certain spe ctral sequence. O f cour s e, Seidel still needs to avoid the conv ergence problem, so he w o rks with Novik ov ring s. Theorem 5.5. (Seidel) L et X b e a smo oth quartic su r fac e in P 3 C and let Λ Q b e t he r ational Novikov ﬁeld over C . T ake t he quartic surfac e in P 3 Λ Q deﬁne d by x 4 + y 4 + z 4 + w 4 + q ( xy z w ) = 0 , quotient by the standar d action of ( Z / 4 Z ) 2 , t ake the minimal cr ep ant r esolution, and denote it by Y . Then ther e is an e quivalenc e of triangulate d c ate gories D π (F uk ( X )) ∼ = ˆ ψ ∗ D b (Coh( Y )) wher e ψ is a c ontinu ous automorphism of Λ N . F o r more details, see [50]. 6. How is homological mirror symmetr y rela ted to mirror symmetr y? Mirror sy mmetr y is a wide and v a ried sub ject. Homolo gical mirro r symmetry is a single attempt to unify a ma jority o f the v arious topics. As such, it must prov e its utility in other mir ror s y mmetric mathematica l inv es tigations. In this section, we pr esent the relations b etw een homologica l mir ror symmetry and three other prominent mathematical aspects of mirror s y mmetry . 6.1. SYZ. Perhaps the most tantalising appro ach to mirro r symmetry was outlined by Strominger , Y au, and Zaslow in [56]. Here, mirr or symmetry of tw o Calabi-Y au manifolds co uld b e rea lis ed geometrically , a s opp osed to homolog ically . Namely , given a Cala bi-Y au ma nifold X over C there should exist a ﬁbr ation by sp ecial La- grangia n tor i. Dualis ing this tor us ﬁbration (and accoun ting for the singular ﬁber s appropria tely) should realise the mirro r Cala bi-Y au Y . Such an explicit geometric op eration s ho uld provide an explicit functor be tw een the categorie s of branes on each side of the mirr or corres p o ndence. Unfor tunately , sp ecial Lagrang ian torus ﬁbrations ar e har d to come b y on Calabi- Y a u ma nifolds. Of course , many exp ect mirror symmetry to hold not necessarily throughout the whole mo duli spac es of theories on eac h side, but in a small neig hborho o d of some s pe c ial singularities on each side. Th us, one can try to implement SYZ in conjunction with a degenera- tion of the structures . Indeed, motiv ated b y the SYZ co njecture, M. Gr oss and B. Siebe r t hav e b egun a pro gram [26, 24, 25] to ex plain mirro r s y mmetry as a Leg endre transform of dual aﬃne structures obta ine d fro m tor ic degenerations. F or a n in tr o- duction se e [23]. W e hav e already s een such log ic employ ed in the previo us sections in inv estigating ho mological mirr or sy mmetry b oth in the case of Ca labi-Y au v ar i- eties, sp eciﬁcally ab elian v arie ties, and a lso in the case o f F a no v a rieties. Indeed, per haps the most sp ectacular a pplication of SYZ-inspired idea s to homo logical mir- ror symmetry is found in [1], wher e one can actually see how the branes on each side cor resp ond by repack aging them as data o n the ba se moment p olytop e. And SYZ informs our under standing of the other s ide of homolog ical mirror symmetry for F ano v a r ieties. It is under SYZ tha t the balanced torus ﬁb ers of [12] corres po nd to the critical po in ts of the mirror superp otential. This is explo red in [30]. MEET HOMOLOGICAL MIRROR SYMMETR Y 31 6.2. M irror maps. Another common a r ea of inv estig ation in mirror sy mmetry is the determination of the mirror map. The mirror map is an isomorphism betw een neighborho o ds of tw o sp ecia l singula r p oints in mo duli on each side of the mirr or corres p o ndence. Aside from its evident imp ortance, mirr or maps turn out to p os- sess rich arithmetic prop erties. Mirr or maps are interesting and elusive. If indeed homologica l mirr or symmetry provides the fullest explanation for the mirr or sym- metry phenomenon, then we sho uld somehow b e a ble to extra ct the mirror ma p. Indeed, in some cases, one can do exactly that. In [3 ], Aldi a nd Zaslow use the following idea of Seidel. A pr o jective v ariety X is determined by its homog eneous c o ordinate ring. The data of the ho moge- neous co or dinate ring sits inside D b (Coh( X )) and can be extrac ted assuming w e know the very ample line bundle L corresp o nding to the embedding a nd the auto- equiv alence co rresp onding to − ⊗ L . Then, the homogeneo us c o ordinate ring is ⊕ n> 0 Hom D b (Coh( X )) ( L, L ⊗ n ). Giv en a triangulated categ ory , a n o b ject, a nd an auto equiv alence we c an alwa ys for m such a ring but there is no reas on to e xp e ct commutativit y . But, if we know that homolo gical mirro r symmetry holds, then we know w ha t ob ject and functor to lo ok a t — the mirror to L and the mirror to − ⊗ L . In gener al, it is exp ected that the mirro r auto equiv ale nce to tensoring b y a line bundle L is g iven by the Dehn twist τ S ab out a v anishing cycle S mirror to L . Assuming homologica l mirr or symmetry , we ca n extract the homo geneous co ordinate ring of the mirro r a s ⊕ n> 0 Hom D π (F uk( Y )) ( S, τ n ( S )). The str ucture of this ring dep ends only on the symplectic structure of Y . Thus, as we v ar y the symplectic structure of Y , we should see the v ariation of the complex structure of the mirro r X manifested in the change in the homog eneous co ordinate r ing of X . In this wa y , Aldi and Za slow of [3] extract the mirror map in a num ber of examples and check that the log ic is correc t. The upshot is that if one can chec k ho mologica l mir ror symmetry in a g iven case b y e x hibiting an explicit equiv alence, one should b e able to compute the cor - resp onding mirr o r map and extr act the wealth of arithmetic informatio n that is exp ected to lie within. 6.3. Ins tan ton n umbe rs. Mir r or sy mmetry br o ke o n to the mathematica l scene by pr op erly co unt ing curves [9] on C a labi-Y au three fo lds, and curve, or instan- ton, count s have been a cen tral fea tur e of mirro r symmetry s inc e . In deed, mirror symmetry was originally view ed as the study o f these counts and their relation to solutions of cer tain diﬀerential eq uations. If ho mological mirror s ymmetry provides the fullest ex planation of the mirro r symmetry phenomenon, then o ne sho uld b e able to extract these instanton co unts so lely fr o m the categorica l data. Physically , the instanton counts come from the c lo sed string sector of our topo logical ﬁeld theory , wher eas what has b een discussed earlier conce rns the op en string secto r . The o b jects in each o f our categories a re appr opriate b oundary conditions for these op en strings in diﬀer ent theories. One needs a mathematical co nstruction that re- lates the tw o. In [13, 14], Costello considers the no tion o f a op en-clo sed topolo gical conformal ﬁeld theory inside which sit op en and closed TCFTs. He shows that an op en TCFT is (mora lly) the s ame as an A ∞ -categor y with a cyc lically-inv ar iant inner pa ir ing a nd demonstr a tes a means to construct, fro m an op en TCFT, an ope n- closed TCFT whose spa ce of closed states is the Ho chschild chain complex C ∗ ( A, A ) of the A ∞ -categor y ass o ciated to the initial op en TCFT. He a lso demonstrates how to build a Gromov-Witten po ten tial from a TCFT a nd thus gives instanton counts 32 MA TTHEW ROBER T BALLARD asso ciated to an A ∞ -categor y with a cyclically-inv ar ia nt inner pairing. In the case of the F uk aya ca tegory , the passag e to Hochschild ho mology has a mor e natural ex- planation. F o r an A ∞ -categor y with a cyclically inv ar iant inner pr o duct, one has a quasi-isomo rphism b etw een C ∗ ( A, A ) a nd C ∗ ( A, A ), the Hochschild chain complex. 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Meet homological mirror symmetry

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