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18.969 Topics in Geometry: Mirror Symmetry 
Spring 2009 

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0.1. Lagrangian Floer Homology (contd). Let {M,cu) be a symplectic man- 
ifold, Lq, Li compact Lagrangian submanifolds intersecting transversely. Recall 
that the complexes CF{Lo, Li) = A^^°'~^^^^ carry a differential mi, product m2, 
and higher operations 

(1) CF*(Lo, Li) • • • ® CF*{Lk-i, Lk) ^ CF*{Lo, Lfe)[2 - k] 

We looked at J-holomorphic maps u from disks with marked boundary 
points to disks in the manifold between Lo,...,Lk with u{zo) = q E Lq Cl 
Lk,u{zi) = Pi E Li-i n Li. We find that the expected dimension of our man- 
ifold A^(pi, . . . ,pfe, Q', [w], J) is deg q - (deg pi H deg pk) -\-k -2. Assuming 


(2) qeLoHLk 

md{[u\) =0 

By looking at the d (1-dimensional moduli space), we obtained the relations: 
Proposition 1. Assuming no bubbling of disks and spheres, Wm > l,{pi, . . . ,Pm), 

Pi e Li_i n Li, 

^ {-iyme{pm, . . . ,Pj+k+i,mk{pj+k, ■ ■ ■ ,Pj+i),Pj, . . . ,pi) = 

(3) k,i>l 

k + £^m + l 

where * — deg (pi) + • • • + deg (pj) + j. 

This implies that mi is a differential, m2 satisfies the Leibniz rule, and m2 is 
associative up to homotopy given by ms (i.e. it is associative in HF*). 

Definition 1. An A^ category is a linear "category" where rnorphism spaces 
are equipped with algebraic operations {mk)k>i satisfying the A^o-relations (those 
defined above). 

In our case, we have the following categories: 



• A Fukaya category is any of a number of categories whose objects 
are Lagrangian submanifolds (with extra data) , the morphisms are Floer 
complexes, and the algebraic operations are as above. 

• So far we only have an 'Aoo-precategory" because the homomorphisms 
have only been defined for transverse pairs of objects. 

• At the homology level, we can also define the Donaldson- (Fukaya cate- 
gory) whose homomorphisms arc the cohomologies HF, so that compo- 
sition is automatically associative. This is technically easier, but we lose 
some information that we need for mirror symmetry. 

• We eventually want to define our Fukaya category to be over C, rather 
than over the Novikov ring. So far, we have counted disks with weights 
j'a;(u) g g^j^j Gromov compactness tells us that there are only finitely 
many contributions below a certain area. That is, the sums may be 
infinite, but they converge in the Novikov ring. Physicists usually write 
the terms as e"^'^'^^") e R instead of T'^("), and hope for convergence. 
Changing the value of T is equivalent to rescaling the symplectic form, 
i.e. working over A is equivalent to working with a family M, {ut = too), 
with T = e~'^^^. We thus work near the large volume limit t ^ oo 
and compute Floer homologies for all t simultaneously. We call this the 
"convergent power series" Floer homology: even when defined, this is 
often not Hamiltonian isotopy invariant. 

• For Lagrangians Li equipped with (Eij'Vi) Li complex vector bundles 
with flat (unitary) connections. We think of these as local systems of 
coefficients on our Lagrangians. We define an associated complex with 
twisted coefficients: 

(4) CF((Lo,£;o,Vo),(Li,£;i,Vi)) = 

for Lq, Li transverse. Then given pi, . . . ,Pk,Pi G Li^iHLi, Wi, . . . , Wk, Wi G 
Hom((£;j_i)p., (^j)pj, we let 


mk{wk,...,wi)= (#A<(pi,...,pfc,g, [m], J))T'^^"^P[a„](wfc,...,wi) 

q e LoHLk 
md([u]) = 

where P[a«](wfc, . . . ,Wi) e Hom((£;o)g, {Ek)q) is defined by 
(6) Vidu] {wk, . . . , wi) = 7fc o o 7fc_i o • • • o 7i o wi o 7o 

where parallel transport along du from q ^ pi gives 70 G Hom((£'o)(?) (-E'o)pi), 
and similarly parallel transport from pj — > pj+i using Vj gives 7^ G 
Hom((£'j)p., (£'j)p.^J. For Vj flat, this only depends on [du]. In particu- 
lar, if Ei is the topologically trivial line bundle C x Lj and Vj is a flat C/(l) 



connection (up to gauge equivalence), Vj = d + iAi for Ai a closed 1-form, 
this encodes the data of holonomies 7ri(Li) — > U{1). Then, using trivial- 
izations, we get CF — A^°'~''^^' with generators p,w — id : Eq^ — > Ei^ 
and nik counts disks with weight T''^(") • hol(9w), where 

(7) hol{du) = exp Aj 

is the holonomy of parallel transport. 
We can now construct our first iteration of the Fukaya category: 

• The objects are C — {L,E,V), where L is a compact spin Lagrangian 
(Z-graded: /jLl — with grading data) and {E, V) a flat hermitian vector 

• The morphisms for transverse Co,^i is given by hom^Co, Ci) = CF*. 

(1) What if Lo is not transverse to Li (in particular, if Lq = I/i)? 

(2) What if L bounds disks? 

For the first problem, sec Seidel's book: the idea is to use a Hamiltonian perturba- 
tion (pH to get Li to be transverse to Lq, and define CF*{Lq, Li) to be generated 
by Lon0i^(Li) (the vector bundles carry without change). We perturb all the d- 
equations by suitable terms: in the strip-like ends, we have |^ + J{^+Xh{u)) — 
for = if(Lj_i, Li). We need a procedure to associate to (L, L') a Hamltonian 
H[L, L'), and to a sequence Lq, . . . ,Lk some compatible perturbation data, and 
further to show that different choices give equivalent Aoo-categories. Note that 
this will not be strictly unital, and will only get a homology unit. 

Alternatively, one can use "Morse-Bott" Floer theory (e.g. FOOO). We define 
CF*(L,L) = C=k(L;A) to be the space of singular chains on L: when defining 
the operations rrik, instead of strip-like ends, we have a marked point z on the 
boundary such that when evaluating at z, and require u{z) to be in the chain. For 
instance, in the product m2, one considers disks with boundary points zq,zi,Z2 
with three evaluation maps evj : Ato,3(-^, L; J, P) — ^ L, 

(8) m2(C2,Ci) = Yl T''^''\evo).{[Mo,3{M,L;J,(3)]nevlCinev;C2) 

For the class /3 = 0, we find that the contribution of constant disks gives the 
intersection product on C*(L). The higher rrik follow similarly, though mi does 
not allow (3 = and adds the classical dC instead. More generally, if Lq fl Li 
have a "clean intersection" (i.e. Lq n Li is smooth), then we set CF*{Lq, Li) =