A8 FUNCTORS FOR LAGRANGIAN CORRESPONDENCES
S. MAU, K. WEHRHEIM, AND C. WOODWARD
Abstract. We describeaconstructionofA88functorsassociatedtomonotone
Lagrangian correspondences, and a proof that the composition of A
functors is homotopic to the functor for the composition, in the case that the
composition is smooth and embedded.
1. Introduction In the paper [16] the second two
authors defined a Weinstein-Floer 2category whose objects are monotone symplectic manifolds, 1-morphisms
are Lagrangiancorrespondences, and2-morphismsareFloerhomologyclasses,
and showed thatcompositioninthiscategoryagrees, upto2-isomorphism,
withthe geometric composition in the case that it smooth and embedded. This
paper describes anextension ofthese resultstothe(co)chainlevel,
usingFukaya’sA8
#categories. For any monotone symplectic manifold M there is an associated
Fukaya category Fuk(M), whose objects aregeneralized Lagrangianbranes in
M. For a Lagrangian correspondence L01⊂ M0× M1we define an Afunctor
F(L01#) : Fuk(M0#) → Fuk(M1). For a Floer cocycle a ∈ CF(L801,L' 01) we define
a natural transformation Taof the corresponding Afunctors. The two
constructions give an A8functor Fuk(M0× M1) → Func(Fuk#(M0#),Fuk8(M))
Given Lagrangian correspondences L01⊂ M0× M1,L121⊂ M1× M, we show that
if L01◦ L12is smooth and embedded in M0× M22then there exists a homotopy of
A8functors F(L01)◦ F(L12) ~ F(L01◦ L12). In particular, the associated derived
functors are isomorphic. Applying these
constructions to moduli spaces of flat bundles yields A8-functor-valued
invariants of three-dimensional cobordisms.
A complete chain-level version of the earlier work is still missing. Namely,
one would like to construct a Weinstein-Fukaya A82-category, whose objects
are symplectic manifolds, 1-morphisms are Lagrangian correspondences,
and 2-morphisms are Floer cochains.
The results of this paper depend on a gluing result proved by the first
author in [10] and on computations of signs by the second two authors in
[15], both
1
2 S. MAU, K. WEHRHEIM, AND C. WOODWARD
of which are still undergoing revision. Therefore, for the moment this paper
should be considered a research announcement.
2. Acategories Our conventions for A88categories attempt to follow those of
Seidel [13]. Other references for this material are Fukaya [3],
Lef´evre-Hasegawa [7], and Lyubashenko [8]. Let N > 0 be an even integer. A
ZN-graded non-unital A8
category C , consists of the following data:
(a) A set of objects Obj(C );
(b) For each pair (C1,C2) ∈ Obj(C ), a ZN-graded abelian group of morphisms
HomC(C1,C2) = i∈ZNHomi C(C1,C). (c) For each d = 1 and (d + 1)-tuple
C02,...,Cd∈ Obj(C ), a multilinear composition map
: HomC(Cd- 1,Cd)× ...× HomC(C0,C1) → HomC(C0,Cd
µd C8
(-1)ℵµe C(ad,...,an+m+1,µj C(an+m,...,an+1),an,...,a1).
1
where 1,...,a
d|
a
and
)[2- d] satisfying the A-associativity equations (1) 0 =
| ,...,| a
n+m<d are homogeneous elements of
d
degree | a
n | a i|
(2) ℵ = n+
.
i=1
To any A8 category C is associated an ordinary homological category H(C ), with the
same objects, morphisms HomH(C )(C1,C2) = H(HomC(C1,C2),µ1) and
composition given by
8 | a] = (-1)1|[µ2 C(a1,a2
◦ a1
to
1,C2
functor
F
from
C
C
1
1
2
(3) [a2
)]. Let Cbe A8categories. An A
) → Obj(C2
∈
0,...,Cd
Obj(C
(a) A map Obj(F ) :
Obj(C
(4)
d- 1
(a
i+j+1
1
j+i
i+1
1
),a ,...,a
r
e
→ Hom(Obj(F )(C0),Obj(F )(
1
d
ℵ(-1)F
consists of the following data:
); (b) For any d = 1 and d+1-tuple C
,C d,...,a
,µj C (a ,...,a
: Hom(C
i d)×...Hom(C0,Cd)
d- i
µ r C2(Fr(ad
), a map F))[1-d] such that the following holds:
d
1
),...,Fi1(a
1))
i,...,a)
=
i
,...,a
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 3
where the first sum is over integers i,j with i + j = d, the second is over
partitions d = i1+ ... + ir, and ℵ is given by (2). The composition of A8
functors is defined by composition of maps on the level of objects, and (5)
(F1◦ F2)d(ad,...,a1) =
F1,d(F2,ir (ad,...,ad- ir ),...,F2,ir (ai1 ,...,a1))
i1+...+ir=d+1- r
functor F : C1 → C2
on the level of morphisms. Any A8
1,C0
1(C0),F2(Cd
1
2
defines an ordinary functor H(F ) : C1→ C2| a|, acting in the same way on
objects and on morphisms by H(F )([a]) = (-1)[F(a)]. Let F1,F2: C1→ C21be
Afunctors. A pre-natural transformation T from F1to F28consists of the
following data: For each d = 0 and d + 1-tuple of objects C0,...,Cd∈ Obj(C) a
multilinear map Td(Cd,...,C0) : Hom(Cd1,Cd- 1)× ...× Hom(C) → Hom(F)). Let
Hom(F1,F2) denote the space of pre-natural transformations from Fto F2.
Define a differential on Hom(F1,F) by (6)
(µ1 Hom(F1,F2)T )d =
(-1)†µr (F2(ad,...,ad- ir+1),F2(ad- ir
,...),...,
k,r
i1,...,ir C2
,...,),...,F1(ai1,...,a1
(ai+e- 1,...,ai+1),ai,...,a
,...,ai1+...+ik- 1+1),F1(ai1+...+ik- 1
ℵ+| T| - 1(-1)Te(ad,...,ai+e,µe C
)
: F0 (-1) → F1, T2 : F1
d
di
→ F2, define µ2(T2,T
k+1
r
2
pre-natural transformations T1
2
2
(µ2(T2,T1))d(ad,...,a1
r,k,l i1,...,ir
i1+...+ik- 1
(| T2| - 1)(| ai| - 1)+
i1+...+ij- 1
i=1
‡=
i=1
(| T1| - 1)(| ai| - 1).
(...),
Ti(ai1+...+ik )) - 11
k- 1
| +...+| a 1
where † = (| T| - 1)(| a
by (7)
)=
1
k- 1
). A natural transformation is a closed pre-natural transformation. Given
two
1)
k
‡µr
C
i
r
(F i (a ,...,a
k- 1
),...,F
1
1
i
i1 (ai1
0
,...,a
i
i
(ai1+..
.+i
1
,...,a
1+..
.+i
+
1
il- 1(...),...,F
)
,
F
il+1 ik- 1(...),...,F
T2 l(...), Tlil- 10
+1),F
)) where
Higher compositions are defined similarly. Any A8natural transformation T :
F1→ F2induces a natural transformation of the corresponding homological
1+...+i
,...,a
1+...+i
(aik
4 S. MAU, K. WEHRHEIM, AND C. WOODWARD
functors H(F1) → H(F2) in the obvious way. A8structures, resp. morphisms
resp. natural transformations have natural interpretations as dg-structures
resp. maps resp. deformations of maps on the shifted tensor algebra [6].
8
Let Hom(C1,C2) denote the space of A8 functors from C1 to
8
C2
1,C2) the structure of an A8
d, d = 3. Given an A8
8 1,F2
category C∨
: C1 → C2
2
∨. Suppose thatF1,F2
to F
1
, with
morphisms given by pre-natural transformations. The higher compositions give
Hom(Ccategory [3, 10.17], [7, 8.1], [13, Section 1d]. In particular, let Ch denote
the dg-category of cochain complexes of finite dimensional vector spaces,
considered as an Acategory with vanishing higher differentials µcategory C ,
the Yoneda dual is the A
:= Hom(C ,Ch). The map C → Hom(C,· ) defines an Afunctor from C to C
arefunctorsthatactthe same way onobjects. A homotopy from Fis a natural
transformation T ∈ Hom(F) of degree -1 such that
- F2 = µ1
to F1, and
d.
T1
Each
1
facet
8
of Kd 1
d
0
1
T1
2
0
In this section we construct A
2
8
1
d
m
d
n
from F1
to
F2
3
d
d
m,n
1:
Kd- m+1 × Km → Kd
...xn+m)xn+m+1
n
(
x
...xd.
n
+
1
(8) F1
n
(T ) where µ(T ) is defined by (6). Note that the assumption on de
substantially simplifies the signs. Homotopy of Afunctors is an equ
relation [13, p.12-13]. Given homotopies Tfrom F, the sum
+µ
2
(T
,T ) ∈ Hom(F0,F2
+T2
=
(
K
. The associahedron K4
×K
) is a natural
transformation from Fto F. 3. Fukaya categories for monotone symplectic
manifolds
categories for monotone symplectic
manifolds, adapted from Fukaya [5] and Seidel [13].
3.1. The associahedra. Let d > 2 be an integer. The d-th associahedron K
,...,xis a CW-complex of dimension d- 2 whose vertices correspond to the
possible ways of parenthesizing d variables xis the pentagon shown in Figure
1.
The associahedra can be defined inductively as follows. Let Kbe the closed
unitinterval. Letd > 3andsuppose thatwehave constructed theassociahedra
Kfor n = d- 1. Define first the boundary (9) ∂K)/ ~ where the union is over the
facets of K, and the equivalence relation ~ is the obvious identification of
facets along codimension two faces. Then Kis the cone on ∂Kis the image of
an embedding (10) ∂...x, corresponding to the expression x
A8
(x1(x2x3))x4
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 5
x1((x2x3)x4)
x1(x2(x3x4))
(x1x2)x3)x4
x4)
The associahedra can be realized as the moduli space of stable mark
disks. A nodal disk D is obtained from a union of disks (called the
components of D) by identifying points on the boundary. Any singular
(x1x2)(x3 is assumed to belong to exactly two disk components. The combinato
type of the nodal disk is the planar graph (by assumption, a tree) obta
by replacing each disk with a vertex, and each singularity with an edg
set of markings is a set { z0,...,zd} of the boundary ∂D in clockwise ord
distinct from the singularities. A nodal disk with markings is semistable
each disk component contains at least three singularities or markings
F
combinatorial type of a nodaldisks
withmarkings
i
isobtainedfromthegraphabovebyaddingsemiinfinite edges associated
g graph has a distinguished vertex defined
the markings. The resulting
u the zeroth marking z. Thus the combinatori
the component containing
type of a nodal disk with rmarkings is a rooted planar tree. A morphism
between nodal disks is aecollection of holomorphic isomorphisms betw
the disk components, preserving the singularities and markings. Let M
1
denote the set of isomorphism
classes of semistable nodal marked di
.
of combinatorial type G, and
K
Md
=
G
4
Md,G.
is shown below in Figure 2. The topology on Mdis the smallest one for which
The moduli space M4
the cross-ratios for any choice of four marked points are continuous.
3.2. Fukaya category of submanifolds. By a monotone symplectic manifold,
we mean a pair (M,ω) consisting of a smooth manifold M and a closed
non-degenerate two-form ω such that
(a) for some t = 0, we have [ω] = tc1(TM), and (b) if t > 0 then M is
compact. If t = 0 then M is (necessarily) noncompact but satisfies ’bounded geometry’ assumptions as in [13].
6 S. MAU, K. WEHRHEIM, AND C. WOODWARD
Figure 2. M4
Let M be a monotone symplectic manifold. A Lagrangian submanifold L ⊂ M is
admissible if
(a) L is compact and oriented; (b) L is monotone, that is, the symplectic
action and index are related by
2A(u) = tI(u) ∀u ∈ p2(M,L) where the t = 0 is (necessarily) the
monotonicity constant of M.
(c) L has minimal Maslov number at least 3, or minimal Maslov number 2 and
disk invariant F= 0; (d) the image of p1L(L) in p1(M) is torsion. We denote by
Lag(M) → M the bundle whose fibers are Lagrangian subspaces
Nof the tangent bundle. Suppose that N > 0 is an even integer and M is
equipped with an N-fold Maslov cover Lag2(M) → Lag(M) in the sense of
Kontsevich and Seidel [12], see also [16, Section 3.3]. We suppose that the
induced cover Lag(M) → M is the bundle of oriented Lagrangian subspaces. A
gradingof a Lagrangian submanifold L ⊂ M is a lift of the canonical section sL:
L → Lag(M),l → TlL to a section sN L: L → LagN(M). If L is oriented, we assume
that the induced lift s2 L2is the canonical lift. A brane structure on an admissible
L consists of a grading and a relative pin structure with background class b ∈
H(M,Z2). An admissible Lagrangian equipped with a brane structure will be
called a Lagrangian brane. Define the objects of Fuk(M,ω,b) to be Lagrangian
branes in M with background class b.
01,LThe morphism spaces in the Fukaya category are spaces of Floer
cochains. Let L⊂ M be Lagrangian branes. Let H ∈ C8([0,1]× M) be a smooth
function, Xt∈ Vect(M) the Hamiltonian vector field for Ht,t ∈ [0,1], and
A8
ft
1,L) =
(L d∈ZN
Define CF
Id 0(L 1,L), Id
0I
0(L 1,L 0)
d 0(L,L1)
1
0
, 1,L)
) =
L
=
=
Id 0(L
d∈ZN
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 7
0
= { x ∈ I (L 1,L), | x| = d} .
Zx
1,L)
(L 01,L
CF
t
d
C
F
(
L
±
(
a
)
l
i
m
0
). The differentials on the morphism spaces
count pseudoholomorphic
and Hom(L strips,
also known as Floer trajectories. We denote the space of time-dependent
almost complex structures
s→ ±
8
(t)
;
(b
)
u(
s,
k)
: M → M its flow. Let Ham(L
Jt(M,ω) = Map([0,1],J (M,ω)). Forany such (J,H)we say that amap u : R×
[0,1] → M is (J,H)-holomorphic
if (11) ∂su(s,t)+Jt,u(s,t)(∂tu(s,t)- X0(u(s,t))) = 0. For any x∈ I (L1,L) let M(x+,x-)
denote the moduli space of finite Henergy (J,H)-holomorphic maps u : R×
[0,1] → M satisfying the conditions
0
1
u(s,t) = x±
1;H) ⊂ Jt
k
reg t(L
1,L)
be the s
L01,Ltransve
0,1} the set
partition
∈
L,
k
=
0,
1
u
p
to
tr
a
n
sl
at
io
n
in
s.
B
y
w
or
k
of
Fl
o
er
[2
],
O
h
[1
1]
a
n
d
F
u
k
a
y
aO
hO
ht
aO
n
o
0,L
[4], there exists a subset
J
01,L
+,x-)0
(L
± ,L
,L
⊂ M(x+,x(12) ∂M(x+
-)1
0x∈I
(L
1,L)
-)1,x±
M(x +,x)0 × M(x,x-)0
0∈ I (L 1,L
1,L
+,x-)0,M(x+,x
)
of
B
ai
re
s
e
c
o
n
d
c
at
e
g
or
y,
s
u
c
h
th
at
) is a smooth manifold for all x
⊂ M(x+,x
0
-)1
-
1,L
+,x
). (b) Suppose that Lhave minimal Maslov numbers at least 2. Then the 1∈ I (L
zero-dimensional component M(x0) is finite. (c) Suppose that Lhave
minimum Maslov numbers at least 3. Then the one-dimensional
component M(x) has a compactification as a one-dimensional manifold
with boundary
0(d) If (L ) is relatively pin, then there exist a coherent set of orientations on
M(x), that is, orientations compatible with (12).
(a) M(x+,x
-
,x
=
8 S. MAU, K. WEHRHEIM, AND C. WOODWARD
0,L
reg(L
1,L),J
+,x-)0
0
, )→
L CF
re 0(L 1,L
g
∈J
1),
∂( x
)=
s(u) x
-)
.
1point. Define
µ: CFd(L
0The
pair (H,J),H ∈ Ham(L
1
d+1(L
1;H)
is called a perturbation datum for (L01). The set of all regular
perturbation data is denoted P,L). From (d) we obtain a map s : M(x
±1} obtained by comparing the given orientation to the canonical
orientation of a
+
,L
0
0
u∈M(x+,x
It follows from Theorem 3.2.1 that (µ1)2= 0.
ThehighercompositionmapsaredefinedfrommodulispacesofJholomorphic
d+1-marked disks, as follows. Let S be a d+1-marked disk.
A set of strip-like ends for S is a set of proper holomorphic
embeddings
lim: R= 0 (s,t) = z
x
q L
oS,k
L
r
2
o
z
3
z
.
z2
p
d,G
z
4
z
d
d,G
d,G
d,G
y
× [0,1] → S, k = 0,...,d such that
S,k
k
1
0
s→ 8
ML
1
uL
0
3
d
4
L
, ∀t ∈ [0,1]. Suppose
that S is a nodal disks with d+1 markings on the boundary.
A set
Figure 3. Apseudoholomorphic map
fromasurface withstriplike ends
of strip-like ends for S means a choice of strip-like end for
each marked and singular point, for each component of S.
We wish to make a choice of striplike ends for all surfaces
S ∈ M, varying smoothly in S in the coordinates given by
the cross-ratios. Note that strip-like ends for a nodal disk S
induce strip-like ends for any resolution of singularities, for
sufficiently small gluing parameters. Suppose we have
constructed strip-like ends for all surfaces in a
neighborhood of the boundary of M. Since the space of
strip-like ends is convex, one can extend this choice to M,
and then use gluing to extend this to a choice of strip-like
ends for all surfaces in a neighborhood of M. Proceeding
recursively by dimension of the stratum M, one constructs
striplike ends for all surfaces in M
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 9
For any surface S without nodes we denote by j the complex structure on S
and Ikthe component of the boundary of S between zkand z0d. Let L,...,Lk⊂ M
be Lagrangian submanifolds. For each pair L,Lk+1k+1we assume we have
chosen a perturbation datum (Hk,Jk) ∈ Pregk(Lk+1,L0d,...,L). A perturbation
datum for (S;L) consists of (a) KS1∈ Ω(S,C8(M,R)) (b) JS∈ Map(S,J (M,ω))
such that YS,JSagree with the given perturbation datum on each end in the
sense that
d
1
JS,oS,k(s,t) = Jk
S
* S,kKS
k
0
S,JS
0,...,L
S
k
d
k
∈
and u : S → M
satisfies
u(I
I
0
(
L
(t) o= H(t)dt for k = 0,...,d. A
perturbation datum for (L
) for each S ∈ M
d
,...,L
∈Ω
d
) is a choice of
perturbation datum (K, varying smoothly in S. Let P (L)denote the space
ofperturbation data. Let Y(S,Vect(M)) denote the Hamiltonian vector
field-valued one-form on S determined by Kk+1,L. Let x) for k = 0,...,d and
M(x,...,x) the moduli space of pairs (S,u), where S ∈ Md
kk) ⊂ L
+JS,z,u(z) ◦ (du(z)- YS,z,u(z)
(13) du(z)- YS,z,u(z)
))◦ j and
lim u(os,k(s,t)) = xk
s→ 8
8(M)
(L ,...,L
0,...,xd
k
8 λ-norm
λ=
H8
λk
(t)
) of perturbation data of second category so th
for k = 0,...,d.
are smooth of the expected dimension. Name
By the Sard-Smale theorem there exists a subset of almost complex structures the smooth topo
P
subspace of smooth functions with finite C
sup|
D
H|
k . Let K lie in Kd(M) := Map(Md1,Ω(S,C88(M))). For some functio
for some sequence of real numbers Md) with compact support, let Kd(M)λdenote the space of onefinite
λk
(14) K8 λ
=
o - 1sup| ρDkK|
k
k
considering K as a function on Md× TS × M. These functions form a
sufficiently large set that the universal moduli space is smooth, and the
Sard-Smale
10 S. MAU, K. WEHRHEIM, AND C. WOODWARD
theorem applied to the projection onto Kd(M)λshows that for each J, there is a
set of K of Baire second category such that the moduli space M(x0,...,x0d,...,L)
is smooth of expected dimension. We denote by P (L)regdthe subset of
perturbations (J,K) with this property.
0d,...,LSuppose that S is a semistable nodal disk with d+1 markings on the
boundary. A perturbation datum for S is a choice of perturbation datum for
each component, and is regular if the perturbation datum for each
component is regular, with the additional requirement that on each
component with exactly two singular or marked points the perturbation datum
is of Floer type, i.e. translationally invariant. A perturbation datum for Lof
combinatorial type G is a choice of perturbation datum for each nodal curve
S of type G, varying smoothly in S. Let PG0(Ld,...,L) denote the set of such
perturbation data. For each edge e of G, let Le,L+ edenote the Lagrangians
adjacent to the corresponding point in S, and xe∈ I (Le,L+ e) a set of
trajectories for the perturbation data associated to Le,L+ e. Letting x = (xe)we
denote by MGe∈E(x ) the moduli space of pairs (S,u) of combinatorial type G
such that the restriction of u to each component Sis a pseudoholomorphic
map with the given boundary conditions and limits. Let I(u) denote the
generalized Maslov-Viterbo index of u, as in [4]. The same argument shows
that there exists a subset PG,reg0(Ld,...,L) in PGa0(Ld,...,L), second-category
with respect to the second factor, such that MG(x0,...,xd) is a smooth manifold
of expected dimension
dim(M)+
I(ui)+di +si - 2
i are the numbers of markings, singularities on the i-th compone
where di,si
for the moduli spaces MG(x0,...,x) to fit together to a partial com
of M(x0,...,xdd'), the perturbation data must be chosen compatib
a nodal curve with markings on the boundary and S'a nodal deg
of it. Let G,Gdenote the combinatorial type of S, resp. S'. The
manifold-with-corners structure on Mdtranslates into the existen
maps
'
: MG',0,d+1 ×
o,G( → MG,0,d+1
inPG 0(L d,...,L
m
#S S'
S
S
'
[0,o)
L
G,o0(L,...,L
(15) #G G'
0
G
. These can be given
recursively by gluing in the boundary of the bubbled disks
dof depth i into an interval of width given by the corresponding gluing
parameter into the boundary of the unique adjacent disk of depth i - 1. Let
P)denotetheinverse imageoftheimageof#). Define a map
0 d,...,L)
(P ) : P ' (L d,...,L) → P
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 11
by gluing together the perturbation data on each component; the perturbation
data on the disks with two markings are those for the pairs Le,L+ e0d,...,L. Let P
(L) ⊂ PG0(Ld,...,L)
G semistable
be the subset of perturbation data such that the gluing maps are compatible
in the sense that #S S'G(P )◦ #'G''(P ) agrees with #G G(P ) on the intersection of
their domains, for each stable combinatorial type G. By induction on the
dimension of MG0, P (Ld,...,L) is non-empty. Let u ∈ M' (xG0,...,xd+m) and take
on MG''(x0,...,x) perturbation datum induced by #G G'd(P ). Floer’s gluing
construction gives rise to a map #G Gfrom a neighborhood of (u,0) in M'
(xG0,...,xd+m)× [0,o)mto MG(x0,...,xd'), that is a local diffeomorphism onto its
image U(u). Define M(x0,...,xd) = sMG(x0,...,xd).
G semistable
d,...,L ,...,L) such that
0
admissibleLagrangianbranes. Thereexistsasubset P0Theorem 3.2.1. Let
M be a monotone symplectic manifold, and Lreg0(Ld,...,L) ⊂ P (L
0
of M(x0,...,xd
)0
d)1
(b) Theclosure M(x0,...,xd)1
has the structure of a manifold with boundary
∂M(x0,...,xd)1= MG(x0,...,xd
G
(a) The zero-dimensional component M(x0,...,xd)0 ) is finite;
oftheone-dimensionalcomponentM(x ,...,x
where G ranges over semistable combinatorial types with 2
components. The proof in the exact case is given in Seidel [13]. To achieve
transversality,
one takes the Hamiltonian perturbations to lie in a space Kd(M)λ, defined as
before in (14) but with a function ρ compactly supported and non-zero on the
complement of the images of the gluing maps (15). By assumption, the
perturbation data are already regular on the complement of the support of ρ,
and an application of Sard-Smale shows that generic Hamiltonian
perturbations make the moduli space transverse on the complement of the
gluing maps as well. By Gromov compactness and compatibility of the
perturbation data, any sequence of solutions with constant index has a
solution converging up to sphere and disk bubbling, and degeneration of the
disk to a semistable nodal disk. The assumption on the Maslov numbers
implies that sphere and disk bubbles capture at least index two, so that there
is no sphere or disk bubbling on the components of index zero or one.
12 S. MAU, K. WEHRHEIM, AND C. WOODWARD
The moduli spaces can be oriented as follows. At any element (S,u) ∈
Md(x0,...,x) the tangent space is the kernel of a parametrized operator ˜D du:
TSMd0⊕ Ω(S,u*0TM;Ld,...,L0,1) → Ω(S,u*˜ DTM). The operatoruis canonically
homotopic to the operator (0,Du), and so its determinant line admits a natural
isomorphism
Λtop(T(S,u)Md(x0,...,xd)) → det(Du)⊗ Λtop(TSMd) whereDuistheCauchy-Riemann
operatorobtainedbylinearizingtheequation (13). Define an orientation on
Mdd+1as follows. Orient ∂D by the counterclockwise direction. The product
(∂D)inherits an orientation from the factors, and restricts to an orientation on
the subset (∂D)d+1 +of cyclically ordered points. Fixing the first three points
gives a global slice for the SL(2,R) action, and this induces an orientation on
the quotient M. The determinant line for Dudinherits an orientation from the
construction of [4],[13], [15]. Together these induce an orientation on
Md(x0,...,x0d,...,Ld) at (S,u). Given objects Ldd- 1define the higher composition
maps µ: Hom(Ld,L0)× ...× Hom(L1,L0) → Hom(Ld,L) by
♥(-1)s(u) x0
(16) µd( x1 ,..., xd ) =
u∈M(x0,...,xd)0
d
where ♥ =
i=1
→ Md(x0
-graded A8
d)0
m(y,xn+1,...,xn+m)0
i| xi|
.
d)1
#S ,u1#u2
u2)
((S1,u1),(S2,u2)) → (S
→ det(Du1#u2
u
1
m
♦
By Theorem 3.2.1, these data define a ZN category Fuk(M,ω), called the Fukaya category of (M,ω
signs which are discussed indetailin[4], [13].
AcohomologicalunitforanobjectLisdefined bycounting p
pseudoholomorphic once-marked disks with boundary
We give a brief discussion of the signs. Consider the gluing
,...,x
map M× Md- m+1(x0,x1,...,y,...,x
2
1
) ⊗ det(D
uM
m
u2,
and a contribution m
i>n+m
(i+1)| xi|.
). The gluing map for determinant lines det(D× M) is orientation preserving,
d i=1
by [4], [13], [15]. One easily checks that the sign of the gluing map Md- m+1→
Mdis m(d- n)+m+n. The sign of the gluing map has contributions ♥, a
contribution m(d- m) from permuting T| xwith kerDi| from permuting the ends
into their correct order, all of which sum to give ℵ times an overall sign (-1),
where ♦ =
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 13
Figure 4. Attaching strips
3.3. Fukaya category of generalized submanifolds. We wish to enlarge this
category by allowing generalized Lagrangian submanifolds. Recall that a
generalized Lagrangian correspondence L from M-to Mconsists of (a) a
sequence N0,...,Nr+of any length r = 0 of symplectic manifolds with N0= M-and
Nr= M, (b) asequence L01,...,L+ofcompactLagrangiancorrespondences with L(j1)j⊂ Nj- 1× Nj(r- 1)rfor j = 1,...,r. A generalized Lagrangian submanifold in M is
generalized Lagrangian correspondence from a point to M. A generalized Lagrangian brane is a generalized
Lagrangian submanifold equipped with a grading and relative pin structure.
Let (M,ω) be a compact symplectic manifold satisfying (M1-2). Define an
A8category Fuk##(M) as follows. The objects of Fuk(M) are generalized
Lagrangian branes satisfying (L1-3), together with a positive real width ofor
each symplectic manifold Niiin the sequence. The space of morphisms is the
Floer cochain group
0(L 1,L ) = CF(L0 (- r0)(- r0+1)
,...,L1 (- r1)(- r1+1)
HomFuk#(M)
(M)
i jd,
integers
d Fuk
ji
n0
,
i
ji
0
,L
=
0
,
.
.
.
,
d
,
j
d
=
1
,
.
.
.
,
n
i
#
). The composition maps µare defined as follows. For any element S ∈ M,...,n,
and positive real numbers oj i
let S denote the quilted surface whose components are S together with
infinite strips Sof width o. See Figure 4. The quilted surface S depends on
not only the choice of widths o, but also the parametrization of the boundary
components Iused in the attaching maps. The space of such
parameterizations is convex, since a map from R to itself is an orientation
preserving diffeomorphism if and only if it has positive first derivative
everywhere. As in the construction of strip-like ends, we construct S
inductively on the strata of M0,...,L(M) and xso that S depends smoothly on
S. Given objects Ljj- 1∈ I (Lj,L) for j = 0,...,d (mod d) suppose we have chosen
a perturbation datum (Jj i,Hj i) for each pair Li j- 1) for each strip, as well as a
perturbation datum (J,K) for the component S. Let M(x,...,xd) denote the set
of collections of maps ui j: Si j→ Mi j,
14 S. MAU, K. WEHRHEIM, AND C. WOODWARD
is the i-th symplectic manifold appearing
where Mi j
(a) each map uj iis perturbed J-holomorphic in the sense of (11) (b) the map
u = u0 iis perturbed J-holomorphic in the sense of (13); (c) The restriction of ui
j× ui j- 1to the intersection (seam) of the surfaces S i jn Si j- 1maps to Li (j- 1)j; (d)
the limit of u along the j-th strip-like end is x. The same argument as in the
definition of µd Fukjshows that for generic choices of perturbation data
∂ Md(x
0(x,...,xd
d,...,x)0
G
d)1
d(x0
d)0
d0(x,...,x
(
a
)
M
d
) is a smooth manifold of expected dimension (b) the zero-dimensional
component Mis finite, (c) the one-dimensional component Mhas a
compactification as a one-manifold with boundary the union
0d,...,x)1
=
M d,
0(x,...,x
G
where G ranges over semistable combinatorial types with two components,
and
(d) there exist a set of orientations on the zero and one-dimensional
components, so that the gluing sign is (2).
satisfy the A8
8
#(M)
#
d- 1,L
→ Fuk(M)∨
8
∨
(- 1)0
(- r)(- r+1)
d
-relations. Cohomological
units are constructed in [16]. The unit for
T
hus the composition maps µd L is defined by counting perturbed pseudoholomorphic once-punctu
with boundary in L ; that is, the single boundary component of the
once-punctured disk has been attached to a sequence of strips, the
boundaries of which lie in the Lagrangian correspondences in L .
3.4. Embedding of the generalized category in the Yoneda dual. In t
section we define an Afunctor (17) Fuk. To any object L of Fuk(M) w
assign the A-functor L: Fuk(M) → Ch defined by
L → CF(L,L ) = CF(L,L
,...,L
CF(L
1) →
0,L ),CF(L
)⊗ ...⊗ CF(L
Hom
) and
d
0,L
Ch(CF(L
,L)) defined
by the signed count of moduli spaces of J-holomorphic maps of the
associated quilted surfaces with strip-like ends obtained by attaching a
number of infinite strips to a d + 2-marked disk, shown in Figure 5. For any
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 15
d i=1
,z
let M
d+1(y,x1,...,xd,z)
4L
y ,x1,...,xd
- 3(- 2)
d+1(y,x1,...,xd
L- 2(- 1)
L(- 1)0
L
s(u) z
0
d
| +(d+1)| z | .
∨L(x1
2
3L
,z) denote the resulting moduli space. As before, for generic perturbation
data the zero-dimensional component is finite and the one-dimensional
component is compact. We define
,...,x )(y ) =
♣(-1)
0
u∈ M
L
1
w
hi| xi
e
r
e L
L
♣
=
Figure 5. The dual of a generalized Lagrangian submanifold
is an A8 functor from Fuk(M) to Fuk(M)∨
M
× Fi2
∨
2 Fuk(M)∨
j Fuk(M)
e(1× i
× 1× k
0
d
8
Proposition 3.4.1. L
∨
. Proof. We freely use the gluing results discussed
in [10]. The boundary o
d+1(y,x1,...,xd,z) consists of semistable surfaces with two nodes. The
degenerations in whichhigher compositions in Fuk(M)vanish. The remaining terms are of the form
the left and right ends F× µ) on the left-hand side of (4). The gluing signs are the same as for the
are separated
higher compositions in the Fukaya category. This means
correspond to the termsthat if the left and right ends are separated then there is no sign, whereas
µ(Fi1) on the right-hand the remaining signs are as in (2).
side of (4); note that the
NextwedefinetheA functor(17)onthelevelofmorphisms. LetL
,...,L
0 1
e,...,L
→ L∨ d
,
e,Ld
#Fuk(M)(Lj- 1,Lj
j
.
.
Fuk(M)(Lj- 1,Lj
k
.
1,...,ad) : L∨ 0
(
,
T
b
e
(
a
1
,L0
,
.
.
.
,
a
d
1,...,be
j
w
is the number of symplectic
manifolds in the
h
sequence
e
r
e
n
j
j
) → CF(L objects of Fuk(M), a∈ Hom),j =
pre-natural transformation T (a
) by counting
J-holomorphic maps of the associated quilted surfaces with striplike ends shown in Figure 6, with a1,...,adinserted at the lower ends and
binserted at the upper ends. That is, given a d+e+2-marked disk let S
denotethequiltedsurfaceobtainedbyattachingn- 1stripstotheboundary
component I
be objects of
Fuk
(M), L
#
))(
b
0)
: CF(L
16 S. MAU, K. WEHRHEIM, AND C. WOODWARD
∈ I (Lj- 1 j,L),yk
d
d+e+2
k- 1∈
d+e+2
; y1
I (L k,L),w ∈ I (L0
(x1,...,x
ji
ji
0,L),z
∈ I (Ld
e,L
ji
→M
,y1,...,ye
e
T ( x1 ,..., x
Lj. Given xj
,..., y
)( w ) =
(x,y,w,z) = M), let Md,w,z) denote the moduli s
a stable d + e + 2-marked disk and u is a
collection of maps from the components of S
such that the map u : S → M satisfies (13) an
We define
♣(-1)s(u) z .
z ,u∈Md+1(y,x1,...,xd,y1,...,ye,z ,w)0
L2
LL
1L
0L
L0
L3
L3
2
L1
L4
Figure 6. Dual functor on morphisms
Proposition 3.4.2. The data L → L∨, (a1,...,ad) → T (a1,...,ad)
form an A8functor from Fuk#(M) to Fuk(M)∨. Proof. The
boundary of the one-dimensional component consists of
nodal surfaces of three types: stable disks in which the left and right
ends are separated, stable disks in which the left and right
ends lie on the same component as the top ends, and
stable disks in which the left and right ends have been
grouped with the bottom ends. The second corresponds to
the
⊗ µj Fuk#(M)
2 Fuk(M)∨
j) and µ
Kd,0
d,0
1
3
,
0
⊗
is
1
⊗
k
∨
(
T
d
∨
2
8
,
.
.
.
,
x
d
d,0
erms
Te(1⊗ i
). The remaining types
correspond to terms µ(T i,T1 Fuk(M)) in (4), using the
t
definitions (6), (7) and vanishing of all higher compositions in Fuk(M). The
signs for gluing are the same as those for the Fukaya category. The
additional signs arise from the Koszul convention and the sign in (3) which
defines µfor the dg-category of cochain complexes.
4. Quilted disks and Afunctors for correspondences 4.1. Themultiplihedron.
In[14]StasheffintroducedafamilyofCW-complexes
, which play the same role for maps between loop spaces as the
associahedra play in the recognition principle for loop spaces. The d-th
multiplihedron Kis a complex of dimension d - 1 whose vertices correspond to
ways of bracketing d variables xand applying an operation, say f. The
multiplihedron Kis the hexagon shown in Figure 7. The definition of K
A8
f((x1x2)x3
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 17
f(x1(x2x3))
1x2)f(x3)f(x)
f(x1)f(x2x3
(f(x1)f(x2))f(x3) f(x1)(f(x2)f(x3)) Figure 7.
Vertices of K3,0
inductive, as in (9). In a previous paper [9] we gave a realization of the
multiplihedron as the moduli space of quilted disks. A quilted disk consists of a
disk D together with a circle C ⊂ D (the seam of the quilt) passing through a
unique point in the boundary. Thus C divides the interior of D into two
components. Given quilted disks (D0,C0) and (D1,C), a morphism from (D0,C0)
to (D1,C1) is a holomorphic isomorphism D01→ D1mapping Cto C10. Any quilted
disk is isomorphic to the pair (D,C) where D is the unit disk in the complex
plane and C the circle of radius 1/2 passing through 1 and 0. Consider the map
D → H given by z → -i/(z - 1). The image of C is the horizontal line L through i.
Thus the automorphism group of (H,L) is the group T ⊂ SL(2,R) of translations
by real numbers.
Let d = 2. A quilted disk with d+1 markings on the boundary consists of a
disk D, distinct points z0,...,zd∈ ∂D and a circle C through zof radius small than
that of D. A morphism (D0,C0;z0,...,zd) → (D1,C00;w0,...,w) is a holomorphic
isomorphic D0→ D1mapping C0to C1and zjto wjdfor j = 0,...,d. Let Md,0be the set
of isomorphism classes of d + 1-marked quilted disks. We compactify Md,0as
follows. A nodal d+1-quilted disk S is a collection of quilted and unquilted
marked disks, identified at pairs of points on the boundary. We require that
(a) The combinatorial type of S is a (planar) tree. (b) Each quilted disk
component is attached to only unquilted components; (c) The unique
non-self-crossing path from the semi-infinite edge marked
z to the semi-infinite edge zj
0
crosses exactly one quilted vertex, for
each j = 1,...,d.
A nodal quilted disk is called stable if (a) Each quilted disk component contains
at least 2 singular or marked
points; (b) Each non-quilted disk component contains at least 3 singular
or marked
points.
18 S. MAU, K. WEHRHEIM, AND C. WOODWARD
Thus the automorphism group of any disk component is trivial. Let
Md,0denote the set of isomorphism classes of semistable d+1-marked nodal
quilted disks. For example M3,0is the hexagon shown in Figure 8.
z3
z0
z1
z2
Figure 8. M
3,
0
,...,L d
In [9] we showed that Md,0 is CW-isomorphic to the multiplihedron K. For each S, let S denote a
quilted surface with strip-like ends obtained by attaching njinfinite strips
each of the boundary components Ijd,0. As before, we may construct S
inductively over the strata, so that S depends smoothly on S. That is, a
choice of strip-like ends and attaching maps for any stratum induces
achoiceinaneighborhoodofthatstratum, andsincethesetofpossible choic
is convex we may extend the choices inductively until a choice of S is
given for all S ∈ Md,0. See Figure 9. 4.2.
Moduliofpseudoholomorphicquilteddisks. GivenobjectsL0
of Fuk#(M) let P (L0 d,...,L
j d,0(x0,...,xd
∈ I (Lj- 1 j,L ),j = 0,...,d let M
and u = (uj i
j- 1).
j
j
j
ji
) denote the fiber bundle whose fiber at S consists of perturbation data for
each disk component, which match with the given perturbation data on the
strip-like ends. Since the set of perturbation data for a given S is convex, we
may construct the perturbation data recursively so that near any stratum, the
perturbation data is induced by the gluing construction. Given x) denote the
set of pairs (S,u) where S ∈ Md,0) is a collection of perturbed
pseudoholomorphic maps from the components Sof S to the symplectic
manifolds Min L, and the asymptotic limit along the j-th strip-like end is x∈ I
(L,L
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 19
Figure 9. Gluing quilted surfaces
ForasecondcategorysubsetofsectionsoftheperturbationbundleP (L0 d,...,L
)1 =
M d,0,G(x0,...,x )0
d0(L d,...,L
∂ Md,0(x ,...,x
), the moduli space M) has finite zero-dimensional component, and the
one-dimensional component has a compactification as a one-manifold with
boundary
0
d
G
d
where G ranges over semistable combinatorial types with two components.
The stable types correspond to the faces of Md,0, while the remaining types
correspond to “bubbling off a trajectory” along the strip-like end.
Define an orientation on Md,0(x0,...,xd) as follows. At any element (S ,u) ∈
Md,0(x0,...,x) the tangent space is the kernel of a parametrized operator ˜Du:
TdSMd,00⊕ Ω(S ,u*0TM ;Ld,...,L,L010,1) → Ω*˜ DTM ). As in the unquilted case,
the operatoru(S,uis canonically homotopic to the operator (0,Dtopu), and so its
determinant line admits a natural isomorphism Λ(T(S,u )Md,0(x0,...,xd)) →
det(Du)⊗ Λtop(TSM) → 0. On Md,0d,0we choose the orientation induced by the
slice given by fixing the first two points and the radius of the interior circle. An
orientation on det(D) is given by choosing orientations for the operators on
the once-punctured disks for each end, see [4], [13], [15]. This induces an
orientation on Md,0(x0u,...,xd).
20 S. MAU, K. WEHRHEIM, AND C. WOODWARD
L01 functors
⊂ M0 × M1
for correspondences. Let L01
01
4.3. A8
#(M0)
→ Fuk#
1
,...,L
be an admissible Lagrangian brane equipped with a width
o: Fuk0(M> 0. Define F) on objects by
FL01(L(- r)(- r+1),...,L(- 1)0) = (L(- r)(- r+1)
(- 1)0,L
Fhas
width
o.
On
morphisms
define
0
,L
)
→
Hom(L0 d
0
Fd L010: Hom(Ld- 1d,L
d)0
s(S,u ) x
( x1 ,..., xd ) =
) where M
)× ...× Hom(L
1
♥(-1)
d L01
(S,u )∈Md,0(x0,...,x
m(d- n)+m+n+1
1,L
0
d,0
#(M
) is an A
(x0,...,xd)1
j,j
∈ Ij
8
∪ ... ∪ Ir
)
by
where s(S,u ) is the difference between the given
orientation of (S,u) and the canonical orientation of a point.
8 #functor from Fuk(M0) to Fuk
1
01
d,0
n+1,...,zn+m
wherel
j(| Ij| - 1).
♠
♠=
j=1
). Proof. The boundary of Mconsists of two combinatorial type
single unquilted bubble, a collection of quilted bubbles, or a b
trajectory [10]; these correspond to the terms in the definition
We give a brief discussion of the signs. Recall Mhas two kind
those corresponding a single bubble with no interior circle co
marked points z, and those given by a partition I= { 1,...d} , w
markings zon the j-th bubble, each containing an interior circl
facets of the first type, the sign of the gluing map is (-1). For f
second type the gluing map acts on orientations by a sign (-1
Proposition 4.3.1. F(L
The isomorphism of determinants of the Cauchy-Riemann
operators det(Du0)⊗ det(Dui) → det(Du) induced by gluing is
orientation preserving, by [15]. Combining the signs from
the Koszul convention, the occurrences of ♥ in the
definitions of µd Fukand F(L01#(M)), and the gluing sign ♠
gives the signs in (4). More generally, a sequence L of
Lagrangians
L01 ⊂ M0 × M1,...,...Lk- 1,k
⊂ Mk- 1 × Mk
defines
an
A
8
#(M0
#(Mk)
together with a sequence of widths o0,...,ok- 1 functor F(L ) : Fuk) → Fuk
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 21
in a similar way, by replacing the seam in the quilted disk by a sequence of
infinite strips. We denote the sequence L = L01#o1...#okL(k- 1)k.
5. Twice-quilted disks and composition of A8functors for
correspondences
In order to study the composition of A8functors for Lagrangian
correspondences, one needs generalizations of the multiplihedra which
allow multiple interior circles. These spaces also appear in a recent preprint
of Batanin [1] under the name of Getzler-Jones polytope. To study
composition of A8functors, one needs only the special case of twice-quilted
disks with no interior markings.
5.1. Twice quilted disks. Let Kd,0,0denote the CW-complex whose vertices
correspond to expressions in variables x1,...,xdand operations f1,f. For
example, the space K2,0,02is the hexagon shown in Figure 10. The definition
of Kd,0,0is inductive, as in (9). The space Khas a realization as the moduli
f1(f2(x1)f2(xd,0,02)) f1(f2(x1))f1(f2(x2))
(f1f2)(x1)(f1f2)(x) f1(f2(x1x2))2
(f1f2)(x1x2)
Figure 10. The space K2,0,0
space of twice-quilted disks with markings on the boundary. A twice-quilted
disk is a disk D with a pair of interior circles C1,C2⊂ D, such that Cis
contained in and not equal to C1, and C1,C22,D has a unique intersection
point denoted z0. A nodal twice-quilted disk is a collection of unquilted,
quilted, and twice-quilted disks joined at boundary points. A d + 1 marked
nodal twicequilted disk is a nodal twice-quilted disk together with d points
z1,...,zdin cyclic order on the boundary, disjoint from the singularities. We say
that such a disk is stable if each unquilted (resp. quilted, resp. twice-quilted
component)
22 S. MAU, K. WEHRHEIM, AND C. WOODWARD
contains at least 3 resp. 2 resp. 2 singular or marked points. Let Md,0,0denote
the set of isomorphism classes of semistable d+1 marked nodal disks.
The topology on Md,0,0is the smallest one for which the product of forgetful
morphisms to the one-dimensional moduli spaces M3,M2,0,M1,0,0is continuous.
For each i,j,k,l distinct we have a forgetful morphism Md,0,0→ M3
obtained by forgetting both circles, all but four of the markings and collapsing
all unstable components. For each pair i,j distinct and non-zero we have two
forgetful morphisms Md,0,0→ M2,0obtained by forgetting one of the circles and
all but three of the markings, and collapsing all unstable components. Finally,
for each non-zero i we have a forgetful morphism Md,0,0→ M1,0,0
obtained by forgetting all but two of the markings and collapsing all unstable
components. The topology on Mis by definition the minimal topology such that
the product of these forgetful morphisms is continuous. Since the product of
the one-dimensional moduli spaces is compact and Hausdorff, so is Md,0,0. The
charts for Md,0,0d,0,0are similar to those for Md,0. (a) There are two types of
gluing parameters: Firstly there is a gluing parameter for each node of the disk. Secondly, if the S contains no region
ofmiddle shaded type thenthere isasingle aparameter d describing the
width of the middle shaded region. There are no relations involving the
parameter d, and d = 0 is the image of the embedding Md,0→ Md,0,0
mentioned above. (b) There are two sets of relations corresponding to
the two sets of circles.
Namely, the interior circles in any quilted disk divide it into regions of three
possible shadings, which we call lightly, medium, and darkly shaded. For any
pair of quilted disk components containing a lightly (resp. darkly) shaded piece,
the product of gluing parameters over the two paths to the component with
z0should be equal. The construction of the homeomorphism to Md,0,0is similar
to that for M, and left to the reader. The moduli space M1,0,0resp M2,0,0d,0is
shown below in Figure 11 resp. (12). The edges of Md,0,0are of three types:
Figure 11. The twice-quilted moduli space M1,0,0
(a) exchange of f1(f2(...)) for (f1f2)(...), (b) exchange of an expression
...(w1(w2w3))... for ...(w1(w2w)) (c) exchangeoff1(w1w2)(resp.
f2(w1w2))forf1(w1)f1(w23)resp. f2(w1)f2(w2).
d; (b) the image of the embedding× ...× Mik,1)× Mk,1
→ Md,0,0
d,0,0
Md,1
d1
d2,0,0
R
+d2,0,0 → Md ,0,
1
0
×R
Mi
i
A8
are of four types: (a) the image of an
1
,
embedding (Mi1,1
0
+...+i
,
1k=
...×
2,0,0
F
j
0
d,0
T
he
facets
of M
× M. (c) the image of an embedding M
, with i
→M
+ ... + ij
1
; (d) the image of an embedding M
d,0,
0
for some partition i= d; the fiber product
ratios of the radii of the interior circles m
A combinatorial type G corresponds to a facet if all gluing parameters are
equivalent by relations. The possibilities are:
(a) one lightly shaded unquilted disk and one twice-quilted disk, and
no relations;
(b) a darkly shaded unquilted disk and a collection of twice-quilted
disks; (c) a collection of quilted disks with lightly and medium shaded
regions,
and a disk with a medium and darkly shaded region; (d) one lightly
shaded unquilted disk and one lightly and darkly shaded
quilted disk. The facets of M2,0,0are shown in Figure 13. 5.2.
Pseudoholomorphic twice-quilted disks. In order to study the relation between the functors F(L01)◦ F(L12) and F(L01#oL12) let Mo d,0,0 ⊂ M d,0,
0
denote the subset obtained by removing a tubular neighborhood of the
face isomorphic to Md,1where the two circles are equal. Any element of
Mo d,0,0
o d,0,0
d,0,0
i,
has the same facets as
M
d,0,
0
we have a forgetful map fi
of markings i,j, the maps fi,f
contains a region of middle shading, and Mo . An alternative description of Mis as follows. For an
d,0,0
M→ M1,0,0~ = [0,8 ]. For any two choices
j are equal. In fact, the coordinate on
1,0, i
0
M
s
24 S. MAU, K. WEHRHEIM, AND C. WOODWARD
Figure 13. Facets of M2,0,0
Figure 14. Desingularization of a twice-quilted nodal disk
the ratio of the radii of the two inner circles minus 1, which is independent of
the position of the marking. Then Mo d,0,0~ = f- 1 i[o,8 ]. For any sequence of
number n1,...,n, we define for each element S ∈ Md,0,0d, a nodal quilted
surface with strip-like ends S by attaching nj,n+1 or nj+2 infinite strips to the
boundary component Ijj, depending on whether a segment bounds a lightly
shaded (resp. middle shaded, darkly shaded) region. See Figure 14. As
usual, we construct the mapS → S inductively on thestrata of Md,0,0, so that S
depends smoothly on S and near the strata is induced by gluing.
0Let L d,...,L
#be objects of Fuk(M), of length n0,...,nd, and L01,L12
Lagrangian correspondences, Let xj∈ I (Lj- 1,Lj), j = 1,...,d and
x0 ∈ I (L12#oL01#L0,L12#oL01#Ld
j o
' → Mj i
o
: Sj i
f
d,0,0(x0,...,xd)
i f
'
to the corresponding
j
M M
S
component
o
'
d
,
0
,
0
). A stable
pseudoholomorphic twice-quilted disk is a pair (S,u ) of an element S
, a nodal twice-quilted disk with strip-like ends Swith stabilization S , together
with perturbed J-holomorphic maps ufrom each component Sj iintothe
corresponding Lagrangiancorrespondence. Let M, mapping each seam ofS
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 25
denote the moduli space of stable pseudoholomorphic twice-quilted disks
with asymptotic limits x0,...,x. Let P (x0,...,xddreg) be the set of choices of
perturbation data constructed inductively on the strata. Using the
Sard-Smale theorem one shows that there exists a subset P(x0,...,xd) ⊂ P
(x0,...,xd) of Baire second category such that
ofMd,0,0(x0,...,xd
(b) The one-dimensional component Md,0,0(x0,...,xd)
d,0,0(x0,...,x) with boundary ∂
Md,0,0(x0d,...,xd)1= Md,0,0,G(x0
1 0
d
G
a) Thezero-dimensionalcomponentMd,0,0(x0,...,xd)0
,...,x )
) is finite;
(
has compact closure in M
where G ranges over semistable combinatorial types with two
components.
The construction of orientations for Mo d,0,0(x0,...,xd) is similar to the previous
cases. It depends on a choice of orientation on Md,0,0, which itself depends on
a choice of slice for the SL(2,R) action on the set of twice-quilted disks with
marking. Our slice is the set of disks with first two marked points fixed and the
first inner circle fixed at radius 1/2.
5.3. Composition of A8functors for Lagrangian correspondences. Recall that
if L01⊂ M0× M1,L12⊂ M1× M2are Lagrangian correspondences, then their
geometric composition is
0 L01 ◦ L12 L12 = p02(L01 ×M1 L L12
Theorem 5.3.1. F
oL
01
12
◦F
2
L01#0
01
0d
:= L12 ◦ L01 0)×...Hom(L
d
u∈Md,0,0(x0,...,xd
1,L)
→ Hom(F(L01#oL
)Ld,F(L
0 .
d- 1,L
d
,..., x1
H0 d
◦ L). If the fiber product is smooth, then L× Mis an immersed Lagrangian
correspondence in M. It will be convenient to denote the composition by
# L
0
:= L01 ◦ L12
12
o 12)L
01#
L12
02
0
1
)=
12
♥s(u)
x
Proof. Define maps
H: Hom(L
(x
(-1)
)o 0
.
The following is the main result of this paper:
Lis
homotopic to F, for any o > 0, and if Lis a smooth embedded Lagrangian
correspondence, also for o = 0.
) by
26 S. MAU, K. WEHRHEIM, AND C. WOODWARD
d,0,0(x0,...,xd)1
Considertheboundaryoftheone-dimensionalmodulispace M
8
L12#oL01
d,0,0(x0,...,xd)1
are of the
r
0
Fuk
2)(H1,o(xIr
form M
G,d,0,0(x0,...,xd)0
0
G,d,0, 8 1
0
),..
u∈(Md,0,0(xi,...,x
o,I1,...,Ir,i
1
where
d)0
. The boundary points correspond to boundary facets in Md,0,0, or bubbled off
trajectories. The first type of facet, see Section 5.1, corresponds to the terms
in the definition of the composition of Afunctors (5). The second type of facet
corresponds to the terms in F. The remaining boundary components of
Mwhere G is a combinatorial type consisting of an unquilted disk mapping to
Mand a twicequilted disk, or a collection of twice-quilted disks attached to a
unquilted disk mapping to M, Facets of the third type correspond to the first
terms in the definition of homotopy of Afunctors, see (6). It remains to show
that facets of the fourth type correspond to the last set of terms. On the r
doubly-quilted disks we have r - 1 relations, requiring that the inner/outer
ratios be equal.
On the other hand, by assumption the moduli space
(x0,...,x
M
G,d,0,0(x0,...,x
)'
0
I
I
µ
(x i
has dimension zero. Thus the moduli space Mdof disks without the
requirement of equal ratios, has dimension r- 1. Since this is a product of the
moduli spaces for the various components, and each is of expected
dimension, we see that for r - 1 of the bubbles, the unconstrained moduli
space is dimension 1, and exactly forone ofthe bubbles, say thei-th, the
unconstrained moduli space is dimension 0. Thus the contribution of this
type of facet is (18)
)=
(-1)♥s(u) x0
#(M
)
H1,o(xi,...,x
0)
counts over the moduli space of expected dimension one, of fixed ratio o.
(Note that this is not the moduli space Md,0,0(xi,...,x0)o 0which is non-empty only
for discrete set of values of o, used in the definition of H .
In order to define a homotopy between F(L01)◦ F(L12) and F(L01#oL), we have
to “integrate over o”. Divide (0,8 )into intervals (oi,oi+1121) such that there is at
most one contribution to Hin each interval. Then Moid,0,0(xi,...,x0)1
[oi,oi+1 (xi,...,x
0
Moi+1d,0,0(xi,...,x0)1 are components in the boundary of M ] d,0,0
). The other boundary components correspond to bubbling off unquilted
disks or
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 27
bubbling off a number of quilted disks. Thus for o ∈ (oi,o1,o) non-singular (19)
H(xd,...,x1) = H1,oi(xd,...,x)+ ± µr Fuk#(M2)1(Hnr,d(xIr),...,Hn1i+1,d(xI1))
d∈[oi,o],I1,...,Ir
±
H0,d(xd,...,µd Fuk#(M0)(xn+m,...,xn+1),xn,...,x1).
∈ { 0,1} and r i=1(ni - 1) = -1. By assumption on [oi,oi+1]
d∈[oi,o],n,m
0,o
where each ni
1
1,o(xi,...,x1)
µr Fuk#(M2)(Hnr,d(xIr
= H1,oi+1(xi
),...,Hn1,d(xI1))
d∈[o,oi+1],I1,...,Ir
i H0,d(xd,...,µd Fuk#(M0)(xn+m,...,xn+1),xn,...,x1)
±
d∈[o,oi+1],n,m
= H1,oi+1(xi,...,x1
8
0,o,
we obtain the terms in the definition of A8
and those for ratio oi+1
the second and third terms vanish. Substituting this expression in (18) for the
terms before H,...,x, and the opposite expression (20) H) ±
) for
the terms after Hhomotopy between the functors with ratio o. Since homotopy
of Afunctors is an equivalence relation, this proves the theorem for o > 0, up to
sign.
We give a brief discussion of the signs. First, consider the signs of the
inclusions of strata into Md,0,0. (a) An embedding Me,0,0× Mf→ Md,0,0i+e(de+1)corresponding to an unquilted bubble containing the markings i+1,...,i+f
has sign (-1). (b) For the facets induced by embeddings
Me,0 × (M| I1| ,0 × ...× M| Ir| ,0) → Md,0,0
gluing acts on signs by (-1) Pr j=1(r- j)(| Ij| - 1)
Mr × (M| I1| ,0,0 ×R ...×R M| Ir| ,0,0) → Md,0,0
. (c) For the facets induced by embeddings
Pr j=1(r- j)(| Ij| - 1)(wheretherealnumberistheratiooftheradiiofthetwointeriorcircles)
the gluing map has sign (-1). (d) for the facet given by the embedding
Md,0→ Md,0,0the gluing map is orientation preserving.
The gluing map for the Cauchy-Riemann operators induces an
orientationpreserving isomorphism of determinant lines det(Dui) → det(D#ui),
in each of the cases above. To obtain the sign of each term in the formulas
(8) (6)
28 S. MAU, K. WEHRHEIM, AND C. WOODWARD
we combine the two occurrences of ♥ for each facet, the Koszul signs, and
the gluing signs for the inclusions of the facets of M. For the facets induced
by embeddings Me,0× (M| I1| ,0× ...× M| Ir| ,0d,0,0) → Md,0,0and Mr× (M| I1| ,0,0×R
e
×
M
,
)1, where So
d,0,0(x0,...,x
0
,
0
1
|
M
I
f
r
|
,
0
,
0
)
.
.
.
×
R
u,J o,uo )
∈M
(x → Md,0,0
→
M
d
,
0
,
0
u,J
02
1)1
,
.
.
.
,
x
ℵ,
the signs combine to
d,0,0(x0
d
)
,...,xd
. these signs cancel, as in the proof of (4.3.1). For the
facets M, as in the proof at the end of Section 3.2.
For the case o = 0, we require that for elements S ∈ Mwith no middle region,
the seam between the darkly and lightly shaded region maps to L. As before,
we construct perturbation data inductively on the combinatorial type G. As
discussed in [16], regular perturbation data for types with no middle region
gives regular perturbation data for surfaces with sufficiently small size of the
middle region. Suppose (S,u) is an element over M, where S is a surface with
no middle region. We claim that there exists a family (Sd,0,00is a surface
obtained by replacing the seam by a strip of width o. The proof is the same as
that of [16] (except that it
is the parametrized linear operator ˜ Drather than the linear operator itself
Dthat is surjective.) The argument of [16] also shows that, due to the
monotonicity assumptions, disk, sphere, and figure eight bubbles cannot occur
in the limit o → 0. Hence Mis compact and has the required boundary terms.
6. Natural transformations for Floer cocycles The main result of this section
is the following.
→ F(L ) extends to an A8 to M1
8
#(M1
0
0
01,L
01
01,L
' 01
a
∈ Hom(F(L01 ),F(L'
01)),
d,..
with T
.,z
1
(
a
)
= µ1(Ta
d,e
0
Theorem 6.0.2. The assignment L × Mfunctor Fuk(M1#) → Func(Fuk(M0),Fuk)). In particular, let
Lagrangian correspondences from M, equipped with brane st
F(L),F(L) the corresponding A
'
functors. Then any Floer cochain a ∈
0
CF(L
1
) defines a prenatural
transformation Tµ). Thus any Floer cocycle defines a natural transformation.
The corresponding moduli space of disks are the moduli spaces Mwith d+1
markings zon the boundary, an interior circle passing through z0
,...,we
0. M
d,e
i
is homeomorphic to a CW-complex Kd,e
1,...,xd
0,...,fe, and 2-morphism symbols t1,...,te
j(...) and tk
i,fj,tk’s must appear in order.
a
on the interior
circle, different from z2
nd no other point of the boundary, and additional markings wthat is defined purely combinatorially
variables x, 1-morphism symbols f. E
where ... represents some expression
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 29
Figure 15. M
2,
1
is the interval with vertices f0(x1)t1 and t1f1(x1). K2,1
0(x1x2)t1,(f0(x1)f0(x2))t1,f0(x
(x 1)(f0(x2)t1),f0(x1)(t1f
1 1
1 2),t (f (x )f (x )), t f (x x
0(x1)t1)f1(x2), (t1
j
j+1
Example 6.0.3.
K1,1
1
f (x ))f (x
j
f
is theoctagonwithverticesf
1
1
1
2
1 1
1
1 2
)
, (f). Two vertices are connected by an edge if the corresponding expressions
are
related by (a) a change in parenthesizing (· (· · )) → ((· · )· );
(· · ) → f(· )f
j+1
j(· )tj+1
(b) a distribution fj → t(· ) or vice-versa; (c) conjugation f(· ) or vice-versa. The faces
correspond to weakening of these expressions, with partial parenrelative to fk
: having defined Kf,g,Kh
d,e
d,e
for (f,g) < (d,e) and all h, one defines
a space L
1
d,e to be the cone on Ld,1
d,1
For any S ∈ Md,e
j
d,e
0d,...,L
1
0
0
1
be a collection of admissible Lagrangian branes in
M0
e
0 of L0,...,L
0 to M1
1
0
1
0
1
,F(Le 01)L
i
i- 1 01,Li 01
1
d
and u is a
d
,
,...,ze e
(· ). The definition
t
of Kis by induction, as for Kdan
hesizing and partial ordering of the expressions t
various facets, and defines K. Note in particular t
Kcorrespond to the terms in the definition of µfor
, let S denote a quilted surface with strip-like end
j
ior markings, obtained by replacing each
boundary component with a collection of
ninfinite strips. We construct S → S inductively
on the strata of M, using convexity of the space
of choices. Let L, and L,...,La sequence of
correspondences from M,...,x. Given
intersection points xdda sequence of elements
z∈ I (L0), and an element y ∈ I (F(L)Ld), let
Md,1(y,z,x,...,x) denote the moduli space of pairs
(S,u ), where S is an element of M
30 S. MAU, K. WEHRHEIM, AND C. WOODWARD
collection of perturbed pseudoholomorphic maps from the components of S
to the symplectic manifolds M0,M1j, and those appearing in L0d,...,L, mapping
the outersegmentsoftheboundarytoL,thesegments oftheinteriorcircle to L 0
01,...,Le 01,, and with the given limits at the strip-like and cylindrical ends
corresponding to the markings. Given a sequence of cochains a 1,...,ae
aj =
nj (z) z ∈ CF(Lj- 1 01 ,Lj
01)
z∈I (Lj- 1
,Lj 01)
01
1
d,1(y,z1,...,z
e(a1 ,...,ae
d
) ∈ Hom(F(L0 01),F(Le 01
0
(T
d
define T
)) by
e(a1,...,ae))d(x1,...,x
z,y,u∈
M
)=
,...,x ♥ + n(z)(-1)s(u)
)
e,x
e
where
=
y
i| ai|
.
i=1
e(a1,...,ae
)1
(y,z1,...,ze,x1,...,x
We claim that the maps L01
→ F(L01),(a1,...,ae)
→T
) define Afunctor. ConsiderthecomponentofdimensiononeMd,e
,...,zi+e
0 × M1
1 (x ,...,x
(-1)ℵT
),a
j
e- 1
+1(...)T
8
d
)
0 01
8
1,...,ir,j1,...,je
)
)
i
j
functor axiom up to sign.
1
1
. Its boundary components are of three combinatorial types:
on the interior
circle have bubbled off onto a disk
(
Facets for which the radius of the interior circle h
a) Facets where some subset of the
ating a number of bubbles, each containing
markings z
markings.
(c) Bubbling off trajectories at the interior or exte
Figure 16 for the case of two interior markings; the
resents the limit when the two interior marked point
s of the first type gives an expression
i+1
e
0
1
))ij e +1 (...)T
je(...)
i,e
while the second contributes a sum of terms of the
form (21) ±µe Fuk#(M1)(F(Le 01))ir(...)...(F(L
i
d- e(ad,...,µe Fuk(M0× M1)(an+m,...,an+1
n,...,a
)d
1
d
e
(F(Le- 1 01))ij e - (...)...(F(Le- 1 01)ij e- 1
1
j(...)
...T11(...)(F(L0 01e- 1(...)...(F(L)(...)) where each ... is an expres
the ai’s. By (7),(6) this proves the A
A8
FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 31
Figure 16. The moduli space M1,2
Mdi,ei× Mf,0→ Md,eWe give a brief discussion of the signs. The sign for the
inclusion of a facet
i
Pis
e
(-1)r i=1(di- 1+ei)i+ Pi<j(di- 1)(ej). The degree of T (ae,...,a1) is
(| ai| +1).
deg(Te(ae,...,a1) = 1-
i=1
(| xj i| - 1)(| yl k| - 1).Hence the signs appearing in (7) and higher compositions
are given by sums
i<k,j,l
The terms of the form | xj i| | yl k| are accounted for by Koszul signs.
Combining with the two occurrences of ♥, and comparing with the orientation
on
modified by the sign (-1)Pd i=0(i+1)| xi | + Pe y=1(j+1)| yj|
M
d,e(x,y)
1
0,M
#
8
# (M0 #),Fuk(M1
8
8
gives the signs claimed in (4).
We leave it to the reader to extend this to a functor from
Fuk
(M
1
)→
Func(Fuk)), by replacing the segments of the inner circle by collections of
infinite strips.
It would be interesting to know whether one can make a stronger statement
on the existence of an A2-functor from a Fukaya-Weinstein category to an
A2-category of Acategories, by using moduli spaces involving arbitrary
numbers of interior circles with markings.
32 S. MAU, K. WEHRHEIM, AND C. WOODWARD
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