text: 3   (cid:104) |±(cid:105) | T(cid:105)he spin components of the matrix above are in the (cid:126)r   σ± basis. They can be converted to the original spin   − − − ∓1 basis of Eq. (3) th†ro(cid:88)ugh the unitary transformation ∝ exp(λ(+)(r R)) exp(λ(−)(r R)) ± i . H(2B);0 = UH˜(2B);0U with 2 i exp(iφ) | (cid:105)(cid:104) | (cid:105)(cid:104) | exp(iφ) U = σ σ σ˜ σ˜ . ↑ ↓ σ=( / ),σ˜=σ± To obtain the normalization constant, we evaluate the (cid:29) follo(cid:90)wing integral in the limit of λ(±)R 1, This yields (cid:18) (cid:19) R − − − 1 I − dr 2πr(exp(λ(+)(r R)) exp(λ(−)(r R)))2 H(2B);0 = A pφσz + 2R σ Bkzσφ, (8) 0 − ≈ πR(λ − λ )2 ( ) (+) ≡ − . λ − λ (λ + λ − ) where pφ ( i∂φ)/R. Eq. (8) coincides with the ( ) (+) (+) ( ) usual Rashba form of the Dirac fermion Hamiltonian × · (cid:28) v((cid:126)σ p(cid:126)) nˆ on the surface of the cylinder expanded (Ref. [13] writes of the λ ± R limit but we ﬁnd that (cid:29) ( ) out in cylindrical coordinates with the identiﬁcation of λ ± R 1 is a better description of what we end up oper- (cid:126) ( ) A = B/ = v except for the presence of the additional ationally doing.) The normalized zero-energy eigenstates A term. The latter has been interpreted as a manifesta- thus read as   2R tion of the spin Berry phase [13] but can be ignored for   most transport calculations as a constant shift in energy.  1  Finally, we consider the remaining terms in Eq. (3) ∓ (cid:104) |±(cid:105) i that have not yet been dealt with, and which correspond (cid:126)r = ρ(r) ± (7) i exp(iφ) to the warping terms, i.e. (cid:113) exp(iφ) − − H (cid:107) = R Γ (k3 3k k2) + R Γ (3k2k k3). (4B), ;R 1 3 x x y 2 4 x y y − − where ρ(r−) = λ(−π)Rλ((λ+()−(λ)−(+λ)(++λ))(2−))[exp(λ(+)(r R)) These terms require a bit more subtlety to handle as exp(λ − (r R))]. they contain r and its derivatives. We often see the mo- ( ) × (cid:39) − B. Projection of H (cid:107) into zero-energy (2 2) mentum in the x direction being written as pˆ i∂ , (4B), x x subspace of H(4B),⊥ so t(cid:16)hat the e(cid:17)xpectation value∗of−the momentum at posi- tion x gets written as ψ(x) ( i∂ ψ(x)). We should, ∗ −x We now proceed to derive the eﬀective Hamiltonian on however, recognize that ψ(x) ( i∂xψ(x)) is actually (cid:104) | | (cid:105)(cid:104) | | (cid:105) the surface state of the Bi2Te3 cylinder. We ﬁrst conside−r ψ x x px ψ where we are taking the expectation H = (M + M k2 + M k(cid:107)2)Γ + BΓ k + A(Γ k | (cid:105)(cid:104) | (4B),0 0 1 z 2 5 4 z 1 y value of the non-Hermitian operator x x p with re- | (cid:105) x Γ k ) [where the “0”’ in the subscript denotes the ab- 2 x spect to the state ψ . This does not lead to any prob- (cid:104) | (cid:105) sence of the R and R hexagonal warping terms in Eq. 1 2 lems when ψ(x) = x ψ = exp(ikx) where k is real, but (3)]. From this, we extract the Hamiltonian H (cid:107) (4B), ;0 yields a complex value where k is complex. This con- that corresponds to motion parallel to the surface of the travenes the fact that an expectation value which cor- cylinder [i.e., the terms which are not already included responds to a physical measurement should yield a real in H ⊥ in Eq. (4)]: (cid:16) (cid:17) (4B), value. The most natural way around this is to modify ∗ − the expression for the momentum to Re(ψ ( i∂ ψ)) = (cid:104)(cid:126)r|H(4B),(cid:107);0 = (M1kz2)Γ5 + BΓ4kz − iAr σr ⊗ τ1∂φ. 12((i∂xψ∗)ψ + ψ∗(−i∂xψ)) = 12(cid:104)ψ| {|x(cid:105)(cid:104)x|, px} |ψx(cid:105). The preceding discussion is relevant for the case of a TI cylin- We then project H (cid:107) into the subspace spanned by |±(cid:105) |±(cid:105) (cid:82) (4B), ;0 der because the r dependence of the states take the , the zero-eigenstates of H ⊥. In other words, we − (4B), form of exp( rλ) which can be interpreted as a wave treat H (cid:107) as a perturbation to H ⊥. An explicit (4B), ;0 (cid:104) ±(cid:48)| | ±(4(cid:105)B), with a complex wavevector. Following× our discussion, calculation of dr r (cid:126)r,±(cid:48) H(4B±),(cid:107);0 (cid:126)r, for the four pos- the matrix elements of the projected (2 2) Hamiltonian sible combinatio(cid:18)ns of and yields (cid:19) H which represents the warping terms acting on (2B);R the surface states, are given by (cid:2) (cid:3) (cid:104) | | (cid:105) (cid:104) | |−(cid:105) ≡ (cid:18) + H (cid:107) (cid:19)+ + H (cid:107) H˜ (cid:104)−| (4B), ;0| (cid:105) (cid:104)−| (4B), ;0|−(cid:105) (2B);0 BkH(4ABp),(cid:107);0 + H(4B),(cid:107);0 H(2B);R ±(cid:48),± = 12(cid:104)±(cid:48)|{R1Γ3(p3x − 3pxp2y) = z − φ . − | (cid:105)(cid:104) |}|±(cid:105) Ap Bk +R Γ (3p2p p3), (cid:126)r (cid:126)r . φ z 2 4 x y y ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: spin

text: components

text: of

text: the

text: matrix

text: above

text: are

text: in

text: the

text: σ±

text: basis.

text: They

text: can

text: be

text: converted

text: to

text: the

text: original

text: spin

text: ∓1

text: basis

text: of

text: Eq.

text: the

text: unitary

text: transformation

text: exp(λ(+)(r

text: R))

text: exp(λ(−)(r

text: R))

text: i

text: with

text: i

text: exp(iφ)

text: exp(iφ)

text: U

text: σ

text: σ

text: σ˜

text: σ˜

text: σ=(

text: ),σ˜=σ±

text: To

text: obtain

text: the

text: normalization

text: constant,

text: we

text: evaluate

text: the

text: integral

text: in

text: the

text: limit

text: of

text: λ(±)R

text: This

text: yields

text: R

text: I

text: dr

text: R))

text: exp(λ(−)(r

text: A

text: pφσz

text: σ

text: Bkzσφ,

text: πR(λ

text: λ

text: λ

text: λ

text: (λ

text: λ

text: where

text: pφ

text: i∂φ)/R.

text: Eq.

text: coincides

text: with

text: the

text: usual

text: Rashba

text: form

text: of

text: the

text: Dirac

text: fermion

text: Hamiltonian

text: nˆ

text: on

text: the

text: surface

text: of

text: the

text: cylinder

text: expanded

text: (Ref.

text: writes

text: of

text: the

text: λ

text: R

text: limit

text: but

text: we

text: ﬁnd

text: that

text: out

text: in

text: cylindrical

text: coordinates

text: with

text: the

text: identiﬁcation

text: of

text: λ

text: R

text: is

text: a

text: better

text: description

text: of

text: what

text: we

text: end

text: up

text: oper-

text: A

text: B/

text: v

text: except

text: for

text: the

text: presence

text: of

text: the

text: additional

text: ationally

text: doing.)

text: The

text: normalized

text: zero-energy

text: eigenstates

text: A

text: term.

text: The

text: latter

text: has

text: been

text: interpreted

text: as

text: a

text: manifesta-

text: thus

text: read

text: as

text: tion

text: of

text: the

text: spin

text: Berry

text: phase

text: but

text: can

text: be

text: ignored

text: for

text: most

text: transport

text: calculations

text: as

text: a

text: constant

text: shift

text: in

text: energy.

text: Finally,

text: we

text: consider

text: the

text: remaining

text: terms

text: in

text: Eq.

text: i

text: that

text: have

text: not

text: yet

text: been

text: dealt

text: with,

text: and

text: which

text: correspond

text: ρ(r)

text: i

text: exp(iφ)

text: to

text: the

text: warping

text: terms,

text: i.e.

text: exp(iφ)

text: H

text: R

text: Γ

text: R

text: Γ

text: ;R

text: x

text: x

text: y

text: x

text: y

text: y

text: where

text: ρ(r−)

text: R))

text: These

text: terms

text: require

text: a

text: bit

text: more

text: subtlety

text: to

text: handle

text: as

text: exp(λ

text: (r

text: R))].

text: they

text: contain

text: r

text: and

text: its

text: derivatives.

text: We

text: often

text: see

text: the

text: mo-

text: B.

text: Projection

text: of

text: H

text: into

text: zero-energy

text: mentum

text: in

text: the

text: x

text: direction

text: being

text: written

text: as

text: pˆ

text: i∂

text: x

text: x

text: subspace

text: of

text: so

text: the

text: value∗of−the

text: momentum

text: at

text: posi-

text: tion

text: x

text: gets

text: written

text: as

text: ψ(x)

text: i∂

text: ψ(x)).

text: We

text: should,

text: −x

text: We

text: now

text: proceed

text: to

text: derive

text: the

text: eﬀective

text: Hamiltonian

text: on

text: however,

text: recognize

text: that

text: ψ(x)

text: i∂xψ(x))

text: is

text: actually

text: the

text: surface

text: state

text: of

text: the

text: cylinder.

text: We

text: ﬁrst

text: conside−r

text: ψ

text: x

text: x

text: px

text: ψ

text: where

text: we

text: are

text: taking

text: the

text: expectation

text: H

text: (M

text: M

text: M

text: BΓ

text: k

text: A(Γ

text: k

text: z

text: z

text: y

text: value

text: of

text: the

text: non-Hermitian

text: operator

text: x

text: x

text: p

text: with

text: re-

text: x

text: Γ

text: k

text: [where

text: the

text: “0”’

text: in

text: the

text: subscript

text: denotes

text: the

text: ab-

text: x

text: spect

text: to

text: the

text: state

text: ψ

text: This

text: does

text: not

text: lead

text: to

text: any

text: prob-

text: sence

text: of

text: the

text: R

text: and

text: R

text: hexagonal

text: warping

text: terms

text: in

text: Eq.

text: lems

text: when

text: ψ(x)

text: x

text: ψ

text: exp(ikx)

text: where

text: k

text: is

text: real,

text: but

text: From

text: this,

text: we

text: extract

text: the

text: Hamiltonian

text: H

text: yields

text: a

text: complex

text: value

text: where

text: k

text: is

text: complex.

text: This

text: con-

text: that

text: corresponds

text: to

text: motion

text: parallel

text: to

text: the

text: surface

text: of

text: the

text: travenes

text: the

text: fact

text: that

text: an

text: expectation

text: value

text: which

text: cor-

text: cylinder

text: [i.e.,

text: the

text: terms

text: which

text: are

text: not

text: already

text: included

text: responds

text: to

text: a

text: physical

text: measurement

text: should

text: yield

text: a

text: real

text: in

text: H

text: in

text: Eq.

text: value.

text: The

text: most

text: natural

text: way

text: around

text: this

text: is

text: to

text: modify

text: the

text: expression

text: for

text: the

text: momentum

text: to

text: Re(ψ

text: i∂

text: ψ))

text: iAr

text: σr

text: τ1∂φ.

text: ψ∗(−i∂xψ))

text: px}

text: The

text: preceding

text: discussion

text: is

text: relevant

text: for

text: the

text: case

text: of

text: a

text: TI

text: cylin-

text: We

text: then

text: project

text: H

text: into

text: the

text: subspace

text: spanned

text: by

text: der

text: because

text: the

text: r

text: dependence

text: of

text: the

text: states

text: take

text: the

text: the

text: zero-eigenstates

text: of

text: H

text: In

text: other

text: words,

text: we

text: form

text: of

text: exp(

text: rλ)

text: which

text: can

text: be

text: interpreted

text: as

text: a

text: wave

text: treat

text: H

text: as

text: a

text: perturbation

text: to

text: H

text: An

text: explicit

text: with

text: a

text: complex

text: wavevector.

text: Following×

text: our

text: discussion,

text: calculation

text: of

text: dr

text: r

text: for

text: the

text: four

text: pos-

text: the

text: matrix

text: elements

text: of

text: the

text: projected

text: Hamiltonian

text: sible

text: of

text: and

text: yields

text: H

text: which

text: represents

text: the

text: warping

text: terms

text: acting

text: on

text: the

text: surface

text: states,

text: are

text: given

text: by

text: H

text: H

text: H˜

text: z

text: φ

text: Ap

text: Bk

text: +R

text: Γ

text: φ

text: z

text: x

text: y

text: y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 18 S. MA’U, K. WEHRHEIM, AND C. WOODWARD Figure 2. A quilted surface with strip-like ends and a destabiliza- tion of it ∈ where j (r, z) is the complex structure on the quilt S at z S , and p∈ r,p r,p Y (r, z) Vect(M ) is the Hamiltonian vector ﬁeld associated to the Hamil- p p tonian perturbation K (r, z). p (cid:48) (cid:48) (cid:48) (c) (Isomorphism) Two pseudoholomorphic quilts (r, Sds, u), (r , Sds, , u ) are iso- (cid:48) r r morphic if r = r and there exists an isomorphism of destabilizations φ : → (cid:48) (cid:48) ◦ Sds Sds, inducing the identity on S such that u φ = u. r r r (d) (Regular pseudoholomorphic quilts) Associated to any pseudoholomorphic S → S quilt u : ds M with destabilization is a Fredholm linearized operator for any integer p > 2 R × ∗ ∗ → ∗ (11) DS : T Ω0(Sds, u T M, ∂u T L) Ω0,1(Sds, u T M) ,r,u r r 1,p r 0,p (cid:55)→ 1 ◦ ◦ (δr, ξ) (J du Dj (δ)) + D (ξ). 2 u Sdrs Sdrs,u Here D (ξ) is the usual linearized Cauchy-Riemann operator of e.g. [22, Sds,u r ∗ ∗ ∗ Chapter 3], acting on the space Ω0(Sds, u T M, ∂u T L) of sections of u T M r 1,p of Sobolev class W 1,p with Lagrangian boundary and seam conditions, and Dj (δr) is the inﬁnitesimal variation of the complex structure on Sds deter- Sds r r mined by δr. A pseudoholomorphic quilt is regular if the associated linearized operator is surjective. We introduce the following notation for moduli spaces. Denote the moduli space of isomorphism class of pseudoholomorphic quilts with varying domain M { → } = (r, u : Sds M) /isomorphism. r ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: Figure

text: A

label: caption

text: quilted

text: surface

text: with

text: strip-like

text: ends

text: and

text: a

text: destabiliza-

text: tion

text: of

text: it

label: list

text: where

text: j

text: (r,

text: z)

text: is

text: the

text: complex

text: structure

text: on

text: the

text: quilt

text: S

text: at

text: z

text: S

text: and

text: p∈

text: r,p

text: r,p

text: Y

text: (r,

text: z)

text: Vect(M

text: is

text: the

text: Hamiltonian

text: vector

text: ﬁeld

text: associated

text: to

text: the

text: Hamil-

text: p

text: p

text: tonian

text: perturbation

text: K

text: (r,

text: z).

text: p

text: (c)

text: (Isomorphism)

text: Two

text: pseudoholomorphic

text: quilts

text: (r,

text: Sds,

text: u),

text: (r

text: Sds,

text: u

text: are

text: iso-

text: r

text: r

text: morphic

text: if

text: r

text: r

text: and

text: there

text: exists

text: an

text: isomorphism

text: of

text: destabilizations

text: φ

text: Sds

text: Sds,

text: inducing

text: the

text: identity

text: on

text: S

text: such

text: that

text: u

text: φ

text: u.

text: r

text: r

text: r

text: (d)

text: (Regular

text: pseudoholomorphic

text: quilts)

text: Associated

text: to

text: any

text: pseudoholomorphic

text: S

text: S

text: quilt

text: u

text: ds

text: M

text: with

text: destabilization

text: is

text: a

text: Fredholm

text: linearized

text: operator

text: for

text: any

text: integer

text: p

text: R

text: DS

text: T

text: u

text: T

text: M,

text: ∂u

text: T

text: L)

text: u

text: T

text: M)

text: ,r,u

text: r

text: r

text: r

text: (δr,

text: ξ)

text: (J

text: du

text: Dj

text: (δ))

text: D

text: (ξ).

text: u

text: Sdrs

text: Sdrs,u

text: Here

text: D

text: (ξ)

text: is

text: the

text: usual

text: linearized

text: Cauchy-Riemann

text: operator

text: of

text: e.g.

text: Sds,u

text: r

text: Chapter

text: acting

text: on

text: the

text: space

text: u

text: T

text: M,

text: ∂u

text: T

text: L)

text: of

text: sections

text: of

text: u

text: T

text: M

text: r

text: of

text: Sobolev

text: class

text: W

text: with

text: Lagrangian

text: boundary

text: and

text: seam

text: conditions,

text: and

text: Dj

text: (δr)

text: is

text: the

text: inﬁnitesimal

text: variation

text: of

text: the

text: complex

text: structure

text: on

text: Sds

text: deter-

text: Sds

text: r

text: r

text: mined

text: by

text: δr.

text: A

text: pseudoholomorphic

text: quilt

text: is

text: regular

text: if

text: the

text: associated

text: linearized

text: operator

text: is

text: surjective.

text: We

text: introduce

text: the

text: following

text: notation

text: for

text: moduli

text: spaces.

text: Denote

text: the

text: moduli

text: space

text: of

text: isomorphism

text: class

text: of

text: pseudoholomorphic

text: quilts

text: with

text: varying

text: domain

text: M

text: (r,

text: u

text: Sds

text: M)

text: /isomorphism.

text: r

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text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 3 recall that the geometric composition of Lagrangian correspondences ⊂ − × ⊂ − × L M M , L M M 01 0 1 12 1 2 is ◦ × (1) L L := π (L L ) 01 12 02 01 M1 12 × × → × where π : M M2 M M M is the projection onto the product of the 02 0 1 2 0 2 ﬁrst and last factors. If the ﬁber product is transverse and embedded by π then ◦ 02 L L is a smooth Lagrangian correspondence. 01 12 Theorem 1.2. (Geometric composition theorem) Suppose that M , M , M are 0 1 2 ⊂ monotone symplectic manifolds with the same monotonicity constant. Let L − × ⊂ − × 01 M M , L M M be admissible Lagrangian correspondences with spin 0 1 12 1 2 ◦ − × structures and gradings such that L L is smooth, embedded by π in M M , 01 12 02 0 2 and admissible. Then there exists a homotopy of A∞ functors ◦ (cid:39) ◦ Φ(L ) Φ(L ) Φ(L L ). 12 01 01 12 There is a slightly more complicated statement in the case that the correspondences are only relatively spin, which involves a shift in the background class. In particular, the theorem implies that the associated derived functors ∼ ◦ ◦ → D(Φ(L )) D(Φ(L )) = D(Φ(L L )) : DFuk (M ) DFuk (M ) 12 01 01 12 0 2 are canonically isomorphic. The result extends to generalized Lagrangian correspon- dences, in particular the empty correspondence. In the last case the result shows that the Fukaya categories constructed using two diﬀerent systems of perturbation data are homotopy equivalent. A complete chain-level version of the earlier work is still missing. Namely, one would like to construct a Weinstein-Fukaya A∞ 2-category whose objects are sym- plectic manifolds and morphism categories are the extended Fukaya categories of correspondences. Furthermore one would like an A∞ categoriﬁcation functor given by the extended Fukaya categories on objects and the functor of Theorem 1.1 on morphisms. This theory would be the chain level version of the Weinstein-Floer 2-category and categoriﬁcation functor constructed in [42]. A possibly-related no- tion of A∞ 2-category is given by a construction of Batanin [2]. Some steps in this direction have been taken by Bottman [5], [6]. The A∞ structures, functors, and natural transformations are deﬁned using a general theory of family quilt invariants that count pseudoholomorphic quilts with varying domain. This theory includes families of quilts associated to the associa- hedron, multiplihedron, and other polytopes underlying the various A∞ structures. Unfortunately these families of quilted surfaces come with the rather inconvenient (for analysis) property that degeneration is not given by “neck stretching” but rather by “nodal degeneration”. Our ﬁrst step is to replace these families by ones that are more analytically convenient, see Section for the precise deﬁnitions. We say that ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: recall

text: that

text: the

text: geometric

text: composition

text: of

text: Lagrangian

text: correspondences

text: L

text: M

text: M

text: L

text: M

text: M

text: is

label: equation

text: L

text: L

text: π

text: (L

text: L

text: where

text: π

text: M

text: M

text: M

text: M

text: is

text: the

text: projection

text: onto

text: the

text: product

text: of

text: the

text: ﬁrst

text: and

text: last

text: factors.

text: If

text: the

text: ﬁber

text: product

text: is

text: transverse

text: and

text: embedded

text: by

text: π

text: then

text: L

text: L

text: is

text: a

text: smooth

text: Lagrangian

text: correspondence.

text: Theorem

text: (Geometric

text: composition

text: theorem)

text: Suppose

text: that

text: M

text: M

text: M

text: are

text: monotone

text: symplectic

text: manifolds

text: with

text: the

text: same

text: monotonicity

text: constant.

text: Let

text: L

text: M

text: M

text: L

text: M

text: M

text: be

text: admissible

text: Lagrangian

text: correspondences

text: with

text: spin

text: structures

text: and

text: gradings

text: such

text: that

text: L

text: L

text: is

text: smooth,

text: embedded

text: by

text: π

text: in

text: M

text: M

text: and

text: admissible.

text: Then

text: there

text: exists

text: a

text: homotopy

text: of

text: A∞

text: functors

text: Φ(L

text: Φ(L

text: Φ(L

text: L

text: There

text: is

text: a

text: slightly

text: more

text: complicated

text: statement

text: in

text: the

text: case

text: that

text: the

text: correspondences

text: are

text: only

text: relatively

text: spin,

text: which

text: involves

text: a

text: shift

text: in

text: the

text: background

text: class.

text: In

text: particular,

text: the

text: theorem

text: implies

text: that

text: the

text: associated

text: derived

text: functors

text: D(Φ(L

text: D(Φ(L

text: D(Φ(L

text: L

text: DFuk

text: (M

text: DFuk

text: (M

text: are

text: canonically

text: isomorphic.

text: The

text: result

text: extends

text: to

text: generalized

text: Lagrangian

text: correspon-

text: dences,

text: in

text: particular

text: the

text: empty

text: correspondence.

text: In

text: the

text: last

text: case

text: the

text: result

text: shows

text: that

text: the

text: Fukaya

text: categories

text: constructed

text: using

text: two

text: diﬀerent

text: systems

text: of

text: perturbation

text: data

text: are

text: homotopy

text: equivalent.

text: A

text: complete

text: chain-level

text: version

text: of

text: the

text: earlier

text: work

text: is

text: still

text: missing.

text: Namely,

text: one

text: would

text: like

text: to

text: construct

text: a

text: Weinstein-Fukaya

text: A∞

text: whose

text: objects

text: are

text: sym-

text: plectic

text: manifolds

text: and

text: morphism

text: categories

text: are

text: the

text: extended

text: Fukaya

text: categories

text: of

text: correspondences.

text: Furthermore

text: one

text: would

text: like

text: an

text: A∞

text: categoriﬁcation

text: functor

text: given

text: by

text: the

text: extended

text: Fukaya

text: categories

text: on

text: objects

text: and

text: the

text: functor

text: of

text: Theorem

text: on

text: morphisms.

text: This

text: theory

text: would

text: be

text: the

text: chain

text: level

text: version

text: of

text: the

text: Weinstein-Floer

text: and

text: categoriﬁcation

text: functor

text: constructed

text: in

text: A

text: possibly-related

text: no-

text: tion

text: of

text: A∞

text: is

text: given

text: by

text: a

text: construction

text: of

text: Batanin

text: Some

text: steps

text: in

text: this

text: direction

text: have

text: been

text: taken

text: by

text: Bottman

text: The

text: A∞

text: structures,

text: functors,

text: and

text: natural

text: transformations

text: are

text: deﬁned

text: using

text: a

text: general

text: theory

text: of

text: family

text: quilt

text: invariants

text: that

text: count

text: pseudoholomorphic

text: quilts

text: with

text: varying

text: domain.

text: This

text: theory

text: includes

text: families

text: of

text: quilts

text: associated

text: to

text: the

text: associa-

text: hedron,

text: multiplihedron,

text: and

text: other

text: polytopes

text: underlying

text: the

text: various

text: A∞

text: structures.

text: Unfortunately

text: these

text: families

text: of

text: quilted

text: surfaces

text: come

text: with

text: the

text: rather

text: inconvenient

text: (for

text: analysis)

text: property

text: that

text: degeneration

text: is

text: not

text: given

text: by

text: “neck

text: stretching”

text: but

text: rather

text: by

text: “nodal

text: degeneration”.

text: Our

text: ﬁrst

text: step

text: is

text: to

text: replace

text: these

text: families

text: by

text: ones

text: that

text: are

text: more

text: analytically

text: convenient,

text: see

text: Section

text: for

text: the

text: precise

text: deﬁnitions.

text: We

text: say

text: that

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 22 S. MA’U, K. WEHRHEIM, AND C. WOODWARD that is non-zero an(cid:90)d that vanishes on the(cid:90) image of the linearized operator. In particular η vanishes on the image of D , i.e. S ,u r (cid:104) (cid:105) (cid:104) ∗ (cid:105) D (ξ), η = 0 = ξ, D (η) Sr,u Sr,u S S ∈ r∗ r ∗ for all ξ W 1,p(S , u T M). This argument implies D (η) = 0, and elliptic r S ,u r regularity ensures that η is of class Cl at least in the interiors of the patches of the quilt S . To prove that η is zero, it suﬃces to show that it vanishes on an open subset r ∗ of each patch of the quilt S , since by unique continuation for solutions of D η = 0 r S ,u r it follows that it vanishes on all of S . We may assume that the complement of the r (cid:90) images of the strip-like ends contains such an open subset. Considering the image of (0, 0, 0, δK) shows that (cid:104) (cid:105) (δY )0,1, η = 0 S r for all δY , where δY is the Hamiltonian vector ﬁeld associated to δK. But now, for each z in the complement of the inverse image of U and the complement of the T images of the strip-like ends, there exists a sequence of functions δK in l that n K are supported on successively smalle(cid:90)r neighborhoods of z and such that (δY )0,1 ⊗ n converges to the delta function δ η(z). It follows that there exists a limit z | | (cid:104) (cid:105) η(z) 2 = l→im∞ (δY n)0,1, η = 0. n S r So η(z) = 0 on an open subset of each patch of the quilt. By unique continuation it must vanish everywhere. Thus, η = 0, which is a contradiction. Hence the linearized − operator is surjective, so by the implicit function theorem for Cl 1 maps of Banach M ∈ E − spaces, univ(y , e ) is also a Banach manifold of class Cl 1. One can now e consider the projection M ∈ E → J × K (cid:55)→ Π : univ(y , e ) l l, (r, u, J, K) (J, K) d e M ∈ E on the subset univ(y , e ) of parametrized index d, which is a Fredholm map d e between Banach manifolds of index d. By the Sard-Smale theorem, for l > max(d, 0) the subset of regular values P { ∈ J × K | ∀ ∈ M } l = (J, K) l l coker(D Π) = 0, (r, u, J, K) univ(y ) reg r,u,J,K e is comeager, hence dense. Now the regular values of the projection correspond M ∈ E precisely to regular perturbation data for the moduli spaces (y , e ), thus the J × Ke subset of regular Cl-smooth perturbation data comeager in l l. ∞ The ﬁnal step is to pass from Cl-smooth regular perturbation data, to C regular perturbation data. This is a standard argument due to Taubes, see Floer-Hofer- (cid:92) (cid:92) Salamon [8] and McDuﬀ-Salamon [22] for its use in pseudoholomorphic curves, which we explain in the unquilted case for simplicity. Let us write J J K K P J × K := l, := l, := ≥ ≥ l 0 l 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: that

text: is

text: non-zero

text: that

text: vanishes

text: on

text: image

text: of

text: the

text: linearized

text: operator.

text: In

text: particular

text: η

text: vanishes

text: on

text: the

text: image

text: of

text: D

text: i.e.

text: S

text: ,u

text: r

text: D

text: (ξ),

text: η

text: ξ,

text: D

text: (η)

text: Sr,u

text: Sr,u

text: S

text: S

text: r∗

text: r

text: for

text: all

text: ξ

text: W

text: u

text: T

text: M).

text: This

text: argument

text: implies

text: D

text: (η)

text: and

text: elliptic

text: r

text: S

text: ,u

text: r

text: regularity

text: ensures

text: that

text: η

text: is

text: of

text: class

text: Cl

text: at

text: least

text: in

text: the

text: interiors

text: of

text: the

text: patches

text: of

text: the

text: quilt

text: S

text: To

text: prove

text: that

text: η

text: is

text: zero,

text: it

text: suﬃces

text: to

text: show

text: that

text: it

text: vanishes

text: on

text: an

text: open

text: subset

text: r

text: of

text: each

text: patch

text: of

text: the

text: quilt

text: S

text: since

text: by

text: unique

text: continuation

text: for

text: solutions

text: of

text: D

text: η

text: r

text: S

text: ,u

text: r

text: it

text: follows

text: that

text: it

text: vanishes

text: on

text: all

text: of

text: S

text: We

text: may

text: assume

text: that

text: the

text: complement

text: of

text: the

text: r

text: images

text: of

text: the

text: strip-like

text: ends

text: contains

text: such

text: an

text: open

text: subset.

text: Considering

text: the

text: image

text: of

text: δK)

text: shows

text: that

text: (δY

text: η

text: S

text: r

text: for

text: all

text: δY

text: where

text: δY

text: is

text: the

text: Hamiltonian

text: vector

text: ﬁeld

text: associated

text: to

text: δK.

text: But

text: now,

text: for

text: each

text: z

text: in

text: the

text: complement

text: of

text: the

text: inverse

text: image

text: of

text: U

text: and

text: the

text: complement

text: of

text: the

text: T

text: images

text: of

text: the

text: strip-like

text: ends,

text: there

text: exists

text: a

text: sequence

text: of

text: functions

text: δK

text: in

text: l

text: that

text: n

text: K

text: are

text: supported

text: on

text: successively

text: neighborhoods

text: of

text: z

text: and

text: such

text: that

text: (δY

text: n

text: converges

text: to

text: the

text: delta

text: function

text: δ

text: η(z).

text: It

text: follows

text: that

text: there

text: exists

text: a

text: limit

text: z

text: η(z)

text: l→im∞

text: (δY

text: η

text: n

text: S

text: r

text: So

text: η(z)

text: on

text: an

text: open

text: subset

text: of

text: each

text: patch

text: of

text: the

text: quilt.

text: By

text: unique

text: continuation

text: it

text: must

text: vanish

text: everywhere.

text: Thus,

text: η

text: which

text: is

text: a

text: contradiction.

text: Hence

text: the

text: linearized

text: operator

text: is

text: surjective,

text: so

text: by

text: the

text: implicit

text: function

text: theorem

text: for

text: Cl

text: maps

text: of

text: Banach

text: M

text: E

text: spaces,

text: univ(y

text: e

text: is

text: also

text: a

text: Banach

text: manifold

text: of

text: class

text: Cl

text: One

text: can

text: now

text: e

text: consider

text: the

text: projection

text: M

text: E

text: J

text: K

text: Π

text: univ(y

text: e

text: l

text: l,

text: (r,

text: u,

text: J,

text: K)

text: (J,

text: K)

text: d

text: e

text: M

text: E

text: on

text: the

text: subset

text: univ(y

text: e

text: of

text: parametrized

text: index

text: d,

text: which

text: is

text: a

text: Fredholm

text: map

text: d

text: e

text: between

text: Banach

text: manifolds

text: of

text: index

text: d.

text: By

text: the

text: Sard-Smale

text: theorem,

text: for

text: l

text: max(d,

text: the

text: subset

text: of

text: regular

text: values

text: P

text: J

text: K

text: M

text: l

text: (J,

text: K)

text: l

text: l

text: coker(D

text: Π)

text: (r,

text: u,

text: J,

text: K)

text: univ(y

text: reg

text: r,u,J,K

text: e

text: is

text: comeager,

text: hence

text: dense.

text: Now

text: the

text: regular

text: values

text: of

text: the

text: projection

text: correspond

text: M

text: E

text: precisely

text: to

text: regular

text: perturbation

text: data

text: for

text: the

text: moduli

text: spaces

text: (y

text: e

text: thus

text: the

text: J

text: Ke

text: subset

text: of

text: regular

text: Cl-smooth

text: perturbation

text: data

text: comeager

text: in

text: l

text: l.

text: The

text: ﬁnal

text: step

text: is

text: to

text: pass

text: from

text: Cl-smooth

text: regular

text: perturbation

text: data,

text: to

text: C

text: regular

text: perturbation

text: data.

text: This

text: is

text: a

text: standard

text: argument

text: due

text: to

text: Taubes,

text: see

text: Floer-Hofer-

text: Salamon

text: and

text: McDuﬀ-Salamon

text: for

text: its

text: use

text: in

text: pseudoholomorphic

text: curves,

text: which

text: we

text: explain

text: in

text: the

text: unquilted

text: case

text: for

text: simplicity.

text: Let

text: us

text: write

text: J

text: J

text: K

text: K

text: P

text: J

text: K

text: l,

text: l,

text: l

text: l

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 30 S. MA’U, K. WEHRHEIM, AND C. WOODWARD (cid:8) (cid:0) (cid:1) (cid:12) (cid:9) (cid:12) are given by the orientation of the moduli space M R × → − R (x−, x+)0 := u = uj : [0(cid:0), δj] Nj j=0(cid:1),...,r (19) (22), Ind(Du) = 1 / of tuples of pseudoholomorphic maps − ∀ (19) ∂ u = ∂ u + J ∂ u Y (u ) = 0 j = 0, . . . r, Jj,Hj j s j j t j j j satisfying the seam conditions ∈ ∀ ∈ R (20) (u (s, δ ), u (s, 0)) L j = 0, . . . r, s , j j j+(cid:90)1 j(j+1) (cid:88) with ﬁnite energy r ∗ ∞ (21) u ω + d(H (u )dt) < , R× j j j j j=0 [0,δj] and prescribed limits · ± ∀ (22) →lim±∞ uj(s, ) = xj j = 0, . . . , r. s The Floer coboundary operator is the ﬁrst in a sequence of operators associated to pseudoholomorphic quilts with varying domain. In [37] we showed that ∂2 = 0, and hence the quilted Floer cohomology • • HF (L) = H (CF (L, ∂)) is well deﬁned. Here we work on chain level, and in case M = pt interpret ∂ =: µ1 s as the ﬁrst of the A∞ composition maps on Fuk (M), • − → • − µ1 : CF (L+, L ) CF +1(L+, L ). The objects in the extended Fukaya category are generalized Lagrangian branes. The morphism spaces in the extended Fukaya category are the quilted Floer chain complex associated to the cyclic generalized La(cid:16)g(cid:88)rangian correspo(cid:88)ndence L of l(cid:17)ength − r = r+ + r shifted in degree − − 1 − Hom(L+, L ) := CF (L+, L )[d], d = 2 dim(Nk+) + dim(Nk−) − k+ k where − − CF (L+, L ) := CF (L+ (L )T ) = CF (L). (cid:15)=1 The higher composition maps in Fukaya categories are deﬁned by counting pseu- doholomorphic polygons with Lagrangian boundary condition. The domain of each polygon corresponds to a point in a Stasheﬀ associahedra as follows. The domain of each polygon corresponds to a point in a Stasheﬀ associahedra as ≥ K follows. Let d 2 be an integer. The d-th associahedron d is a cell complex of − dimension d 2 deﬁned recursively as the cone over a union of lower-dimensional associahedra, whose vertices correspond to the possible ways of parenthesizing d variables a , . . . , a , see Stasheﬀ [33]. More precisely, any such expression corre- 1 d sponds to a tree Γ describing the parenthisization, which is stable in the sense that | | the valence v of any vertex Vert(Γ) is at least 3. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: are

text: given

text: by

text: the

text: orientation

text: of

text: the

text: moduli

text: space

text: M

text: R

text: R

text: (x−,

text: u

text: uj

text: δj]

text: Nj

text: Ind(Du)

text: of

text: tuples

text: of

text: pseudoholomorphic

text: maps

text: u

text: u

text: J

text: u

text: Y

text: (u

text: j

text: r,

text: Jj,Hj

text: j

text: s

text: j

text: j

text: t

text: j

text: j

text: j

text: satisfying

text: the

text: seam

text: conditions

text: R

text: (u

text: (s,

text: δ

text: u

text: (s,

text: L

text: j

text: r,

text: s

text: j

text: j

text: with

text: ﬁnite

text: energy

text: r

text: u

text: ω

text: d(H

text: (u

text: )dt)

text: R×

text: j

text: j

text: j

text: j

text: and

text: prescribed

text: limits

text: →lim±∞

text: uj(s,

text: xj

text: j

text: r.

text: s

text: The

text: Floer

text: coboundary

text: operator

text: is

text: the

text: ﬁrst

text: in

text: a

text: sequence

text: of

text: operators

text: associated

text: to

text: pseudoholomorphic

text: quilts

text: with

text: varying

text: domain.

text: In

text: we

text: showed

text: that

text: ∂2

text: and

text: hence

text: the

text: quilted

text: Floer

text: cohomology

text: HF

text: (L)

text: H

text: (CF

text: (L,

text: is

text: well

text: deﬁned.

text: Here

text: we

text: work

text: on

text: chain

text: level,

text: and

text: in

text: case

text: M

text: pt

text: interpret

text: µ1

text: s

text: as

text: the

text: ﬁrst

text: of

text: the

text: A∞

text: composition

text: maps

text: on

text: Fuk

text: (M),

text: µ1

text: CF

text: (L+,

text: L

text: CF

text: L

text: The

text: objects

text: in

text: the

text: extended

text: Fukaya

text: category

text: are

text: generalized

text: Lagrangian

text: branes.

text: The

text: morphism

text: spaces

text: in

text: the

text: extended

text: Fukaya

text: category

text: are

text: the

text: quilted

text: Floer

text: chain

text: complex

text: associated

text: to

text: the

text: cyclic

text: generalized

text: L

text: of

text: r

text: r+

text: r

text: shifted

text: in

text: degree

text: Hom(L+,

text: L

text: CF

text: (L+,

text: L

text: )[d],

text: d

text: dim(Nk+)

text: dim(Nk−)

text: k+

text: k

text: where

text: CF

text: (L+,

text: L

text: CF

text: (L+

text: (L

text: )T

text: CF

text: (L).

text: The

text: higher

text: composition

text: maps

text: in

text: Fukaya

text: categories

text: are

text: deﬁned

text: by

text: counting

text: pseu-

text: doholomorphic

text: polygons

text: with

text: Lagrangian

text: boundary

text: condition.

text: The

text: domain

text: of

text: each

text: polygon

text: corresponds

text: to

text: a

text: point

text: in

text: a

text: Stasheﬀ

text: associahedra

text: as

text: follows.

text: The

text: domain

text: of

text: each

text: polygon

text: corresponds

text: to

text: a

text: point

text: in

text: a

text: Stasheﬀ

text: associahedra

text: as

text: K

text: follows.

text: Let

text: d

text: be

text: an

text: integer.

text: The

text: d-th

text: associahedron

text: d

text: is

text: a

text: cell

text: complex

text: of

text: dimension

text: d

text: deﬁned

text: recursively

text: as

text: the

text: cone

text: over

text: a

text: union

text: of

text: lower-dimensional

text: associahedra,

text: whose

text: vertices

text: correspond

text: to

text: the

text: possible

text: ways

text: of

text: parenthesizing

text: d

text: variables

text: a

text: a

text: see

text: Stasheﬀ

text: More

text: precisely,

text: any

text: such

text: expression

text: corre-

text: d

text: sponds

text: to

text: a

text: tree

text: Γ

text: describing

text: the

text: parenthisization,

text: which

text: is

text: stable

text: in

text: the

text: sense

text: that

text: the

text: valence

text: v

text: of

text: any

text: vertex

text: Vert(Γ)

text: is

text: at

text: least

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 35 z5 z4 z6 z2 z7 z1 z 0 Figure 5. The part of the boundary between two markings The forgetful maps induce the structure of ﬁber bundles on the moduli spaces with contractible ﬁbers. Indeed the gluing construction identiﬁes all nearby ﬁbers G D δ with the interval obtained by removing small disks around the nodes and identifying (cid:16) (cid:17) the endpoints: − ∪ ∼ (26) (∂G D) = (∂D) m B (w ) / . δ i i k=1 δ1/2 k k On the other hand the intervals on the right-hand side of (26) are homeomorphic to (∂D) itself, by extending a homeomorphism of the complement i − ∪ → − ∪ { } (∂D) m B (w ) (∂D) m w i k=1 δ1/2 k i k=1 k k R d to the boundary. It follows that f induces on the structure of a ﬁber bundle R − i d 1 over with interval ﬁbers. The discussion above shows by induction that the R d moduli space is a topological disk. (And it follows from the isomorphism with the associahedron that they are isomorphic, as decomposed spaces, to convex polytopes.) In this section we construct the higher composition maps on the Fukaya category of generalized Lagrangian branes. These are deﬁned by family quilt invariants for families of surfaces with strip-like ends over the associahedra constructed in the following proposition: Proposition 4.7. (Existence of families of strip-like ends over the associahedra) For ≥ S d each d 2 there exists a collection of families of quilted surfaces with strip-like R S R ends over d with the property that the restriction d of the family to a stratum d R Γ Γ that is isomorphic to a product of ij, j = 1, . . . , k is a product of the corresponding S S families ij, and collar neighborhoods of d are given by gluing along strip-like ends. Γ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: z

text: Figure

text: The

text: part

text: of

text: the

text: boundary

text: between

text: two

text: markings

text: The

text: forgetful

text: maps

text: induce

text: the

text: structure

text: of

text: ﬁber

text: bundles

text: on

text: the

text: moduli

text: spaces

text: with

text: contractible

text: ﬁbers.

text: Indeed

text: the

text: gluing

text: construction

text: identiﬁes

text: all

text: nearby

text: ﬁbers

text: G

text: D

text: δ

text: with

text: the

text: interval

text: obtained

text: by

text: removing

text: small

text: disks

text: around

text: the

text: nodes

text: and

text: identifying

text: the

text: endpoints:

text: (∂G

text: D)

text: (∂D)

text: m

text: B

text: (w

text: δ

text: i

text: i

text: k

text: k

text: On

text: the

text: other

text: hand

text: the

text: intervals

text: on

text: the

text: right-hand

text: side

text: of

text: are

text: homeomorphic

text: to

text: (∂D)

text: itself,

text: by

text: extending

text: a

text: homeomorphism

text: of

text: the

text: complement

text: i

text: (∂D)

text: m

text: B

text: (w

text: (∂D)

text: m

text: w

text: i

text: k

text: i

text: k

text: k

text: R

text: d

text: to

text: the

text: boundary.

text: It

text: follows

text: that

text: f

text: induces

text: on

text: the

text: structure

text: of

text: a

text: ﬁber

text: bundle

text: R

text: i

text: d

text: over

text: with

text: interval

text: ﬁbers.

text: The

text: discussion

text: above

text: shows

text: by

text: induction

text: that

text: the

text: R

text: d

text: moduli

text: space

text: is

text: a

text: topological

text: disk.

text: (And

text: it

text: follows

text: from

text: the

text: isomorphism

text: with

text: the

text: associahedron

text: that

text: they

text: are

text: isomorphic,

text: as

text: decomposed

text: spaces,

text: to

text: convex

text: polytopes.)

text: In

text: this

text: section

text: we

text: construct

text: the

text: higher

text: composition

text: maps

text: on

text: the

text: Fukaya

text: category

text: of

text: generalized

text: Lagrangian

text: branes.

text: These

text: are

text: deﬁned

text: by

text: family

text: quilt

text: invariants

text: for

text: families

text: of

text: surfaces

text: with

text: strip-like

text: ends

text: over

text: the

text: associahedra

text: constructed

text: in

text: the

text: following

text: proposition:

text: Proposition

text: (Existence

text: of

text: families

text: of

text: strip-like

text: ends

text: over

text: the

text: associahedra)

text: For

text: S

text: d

text: each

text: d

text: there

text: exists

text: a

text: collection

text: of

text: families

text: of

text: quilted

text: surfaces

text: with

text: strip-like

text: R

text: S

text: R

text: ends

text: over

text: d

text: with

text: the

text: property

text: that

text: the

text: restriction

text: d

text: of

text: the

text: family

text: to

text: a

text: stratum

text: d

text: R

text: Γ

text: Γ

text: that

text: is

text: isomorphic

text: to

text: a

text: product

text: of

text: ij,

text: j

text: k

text: is

text: a

text: product

text: of

text: the

text: corresponding

text: S

text: S

text: families

text: ij,

text: and

text: collar

text: neighborhoods

text: of

text: d

text: are

text: given

text: by

text: gluing

text: along

text: strip-like

text: ends.

text: Γ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 38 S. MA’U, K. WEHRHEIM, AND C. WOODWARD After this identiﬁcation the gluing map (29) takes the form (omitting tensor prod- ucts from the notation to save space) R R D D− D− R − D D− D− D− (31) det( ) det(T m) + . . . det(T d m+1) + . . . . . . . y xn+1 xn+m x0 x1 y xd To determine the sign of this map, ﬁrst note that the gluing map × R × R − → R (0, (cid:15)) m d m+1 d on the associahedra is given in coordinates (using the automorphisms to ﬁx the R R − location of the ﬁrst and second point in m to equal 0 resp. 1 and d m+1) by (32) (δ, (z , . . . , z ), (w , . . . , w − )) 3 m 3 d m+1 → (w , . . . , w , w + δ, w + δz , . . . , w + δz , w , . . . , w − ). 3 n+1 n+1 n+1 3 n+1 m n+2 d m − This map acts on orientations by a sign of 1 to the power − − (33) (m 1)(n 1). These signs combine with the contributions (cid:88) (cid:88) (cid:88) n d m | | | | − | | − | | (34) k x + (n + 1) y + (k m + 1) x + (k n) x k k k k=1 k=n+m+1 k=n+1 in the deﬁnition of the structure maps, and a contribution (cid:88)   − | | | | (35) (d m + 1)m + m y + x i ≥ i n D− D R − from permuting the determinant lines , j = n+1, . . . , n+m, + with det(T d m+1) xj D− ≤ D− y and permuting these determinant lines with the , i n, . On the other hand, xi y the sign in the A∞ axiom contributes (cid:88) n | | (36) ( x + 1) k k=1

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: After

text: this

text: identiﬁcation

text: the

text: gluing

text: map

text: takes

text: the

text: form

text: (omitting

text: tensor

text: prod-

text: ucts

text: from

text: the

text: notation

text: to

text: save

text: space)

text: R

text: R

text: D

text: D−

text: D−

text: R

text: D

text: D−

text: D−

text: D−

text: det(

text: det(T

text: m)

text: det(T

text: d

text: y

text: xn+m

text: y

text: xd

text: To

text: determine

text: the

text: sign

text: of

text: this

text: map,

text: ﬁrst

text: note

text: that

text: the

text: gluing

text: map

text: R

text: R

text: R

text: m

text: d

text: d

text: on

text: the

text: associahedra

text: is

text: given

text: in

text: coordinates

text: (using

text: the

text: automorphisms

text: to

text: ﬁx

text: the

text: R

text: R

text: location

text: of

text: the

text: ﬁrst

text: and

text: second

text: point

text: in

text: m

text: to

text: equal

text: resp.

text: and

text: d

text: by

text: (δ,

text: (z

text: z

text: (w

text: w

text: m

text: d

text: (w

text: w

text: w

text: δ,

text: w

text: δz

text: w

text: δz

text: w

text: w

text: m

text: d

text: m

text: This

text: map

text: acts

text: on

text: orientations

text: by

text: a

text: sign

text: of

text: to

text: the

text: power

text: (m

text: These

text: signs

text: combine

text: with

text: the

text: contributions

text: n

text: d

text: m

text: k

text: x

text: (n

text: y

text: (k

text: m

text: x

text: (k

text: n)

text: x

text: k

text: k

text: k

text: in

text: the

text: deﬁnition

text: of

text: the

text: structure

text: maps,

text: and

text: a

text: contribution

text: (d

text: m

text: m

text: y

text: x

text: i

text: i

text: n

text: D−

text: D

text: R

text: from

text: permuting

text: the

text: determinant

text: lines

text: j

text: n+m,

text: with

text: det(T

text: d

text: xj

text: D−

text: D−

text: y

text: and

text: permuting

text: these

text: determinant

text: lines

text: with

text: the

text: i

text: n,

text: On

text: the

text: other

text: hand,

text: xi

text: y

text: the

text: sign

text: in

text: the

text: A∞

text: axiom

text: contributes

text: n

text: x

text: k


text: 42 S. MA’U, K. WEHRHEIM, AND C. WOODWARD homotopic to the gluing map. Now the action on orientations is given by the ac- tion on local homology groups, and homotopic maps induce the same action. So by replacing the gluing map with this parametrization we may assume that the M { } gluing map is an embedding. The moduli space 2(L , J , (x , x ) ) 1 j(j+1) j(j+1) j j+1 has orientation at [u] induced by determinaRnt×line det∼(DRu) and the determinant of the inﬁnitesimal automorphism group aut( [0, 1]) = . On the other hand, the M { } orientation on 2(L , J , (x , x ) ) is induced by the orientation of the 1 j(j+1) j(j+1) j j+1 determinant lin∼e det∞(Dv×j(j+R1)) and the determinant line of the automorphism group Aut(D, ∂D, 1) = (0, ) . The gluing map induces an orientation-preserving iso- → morphism of determinant lines det(D ) det(D ) by [41]. Under gluing the u vj(j+1) R × second factor in Aut(D, ∂D, 1) agrees with the translation action on [0, 1]. On the other hand, the ﬁrst factor agrees approximately with the gluing parameter, in the sense that gluing in a dilation of v by a small constant ρ > 0 using gluing j(j+1) parameter τ is approximately the same as gluing in v with gluing parameter (cid:80) Mj(j+1) τρ. Thus v represents a boundary point of (x, x) with the opposite of j(j+1) 1 the induced orientation from the interior. Summing over the boundary of the one- M − dimensional manifold ∂ (x, x) proves that ∂2 r w(L ) Id = 0. The proof 1 j=0 j(j+1) (cid:3) that two choices lead to cochain-homotopic pairs (CF (L), ∂) is given in [37]. Remark 4.14. (Floer theory of a pair of Lagrangians) In the special case L = (L , L ) ⊂ 0 1 of a cyclic correspondence consisting of two Lagrangian submanifolds L , L M we − − 0 1 have w(L) = w(L ) w(L ). Indeed the J -holomorphic discs with boundary on ⊂ − ×{ } 0 1 1 ⊂ L M pt are identiﬁed with J -holomorphic discs with boundary on L M 1 1 1 via a reﬂection of the domain. This reﬂection is orientation reversing for the moduli spaces. In particular, the diﬀerential for a monotone pair L = (L, ψ(L)) with any ∈ symplectomorphism ψ Symp(M) always squares to zero, since w(L) = w(ψ(L)). ∈ Z Following Sheridan [30], for each integer w deﬁne the generalized Fukaya category Fuk (M, w) whose objects are generalized Lagrangians L whose total disk invariant is w(L) = w, and whose morphisms are the quilted Floer cochain groups deﬁned above. ∈ Z Theorem 4.15. Let M be a symplectic background, w an integer and µd for ≥ d 1 the maps deﬁned in (27) for some choice of family of surfaces of strip-like ≥ ends and perturbation data. Then the maps µd, d 1 deﬁne an A∞ structure on Fuk (M, w). Proof. The possibility of disk bubbling occurs only the deﬁnition of µ2, since it is 1 only in this case that there exist holomorphic quilts in a moduli space that is not of expected dimension (the constant trajectories). The proof is therefore the same as in the case of Maslov number at least three, with the added requirement that the perturbation data on the ends makes the disks in the deﬁnition of the disk invariant (cid:3) regular. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

text: homotopic

text: to

text: the

text: gluing

text: map.

text: Now

text: the

text: action

text: on

text: orientations

text: is

text: given

text: by

text: the

text: ac-

text: tion

text: on

text: local

text: homology

text: groups,

text: and

text: homotopic

text: maps

text: induce

text: the

text: same

text: action.

text: So

text: by

text: replacing

text: the

text: gluing

text: map

text: with

text: this

text: parametrization

text: we

text: may

text: assume

text: that

text: the

text: M

text: gluing

text: map

text: is

text: an

text: embedding.

text: The

text: moduli

text: space

text: J

text: (x

text: x

text: j

text: has

text: orientation

text: at

text: [u]

text: induced

text: by

text: determinaRnt×line

text: det∼(DRu)

text: and

text: the

text: determinant

text: of

text: the

text: inﬁnitesimal

text: automorphism

text: group

text: aut(

text: On

text: the

text: other

text: hand,

text: the

text: M

text: orientation

text: on

text: J

text: (x

text: x

text: is

text: induced

text: by

text: the

text: orientation

text: of

text: the

text: j

text: determinant

text: lin∼e

text: and

text: the

text: determinant

text: line

text: of

text: the

text: automorphism

text: group

text: Aut(D,

text: ∂D,

text: The

text: gluing

text: map

text: induces

text: an

text: orientation-preserving

text: iso-

text: morphism

text: of

text: determinant

text: lines

text: det(D

text: det(D

text: by

text: Under

text: gluing

text: the

text: u

text: R

text: second

text: factor

text: in

text: Aut(D,

text: ∂D,

text: agrees

text: with

text: the

text: translation

text: action

text: on

text: On

text: the

text: other

text: hand,

text: the

text: ﬁrst

text: factor

text: agrees

text: approximately

text: with

text: the

text: gluing

text: parameter,

text: in

text: the

text: sense

text: that

text: gluing

text: in

text: a

text: dilation

text: of

text: v

text: by

text: a

text: small

text: constant

text: ρ

text: using

text: gluing

text: parameter

text: τ

text: is

text: approximately

text: the

text: same

text: as

text: gluing

text: in

text: v

text: with

text: gluing

text: parameter

text: τρ.

text: Thus

text: v

text: represents

text: a

text: boundary

text: point

text: of

text: (x,

text: x)

text: with

text: the

text: opposite

text: of

text: the

text: induced

text: orientation

text: from

text: the

text: interior.

text: Summing

text: over

text: the

text: boundary

text: of

text: the

text: one-

text: M

text: dimensional

text: manifold

text: (x,

text: x)

text: proves

text: that

text: ∂2

text: r

text: w(L

text: Id

text: The

text: proof

text: that

text: two

text: choices

text: lead

text: to

text: cochain-homotopic

text: pairs

text: (CF

text: (L),

text: is

text: given

text: in

text: Remark

text: (Floer

text: theory

text: of

text: a

text: pair

text: of

text: Lagrangians)

text: In

text: the

text: special

text: case

text: L

text: (L

text: L

text: of

text: a

text: cyclic

text: correspondence

text: consisting

text: of

text: two

text: Lagrangian

text: submanifolds

text: L

text: L

text: M

text: we

text: have

text: w(L)

text: w(L

text: w(L

text: Indeed

text: the

text: J

text: -holomorphic

text: discs

text: with

text: boundary

text: on

text: L

text: M

text: pt

text: are

text: identiﬁed

text: with

text: J

text: -holomorphic

text: discs

text: with

text: boundary

text: on

text: L

text: M

text: via

text: a

text: reﬂection

text: of

text: the

text: domain.

text: This

text: reﬂection

text: is

text: orientation

text: reversing

text: for

text: the

text: moduli

text: spaces.

text: In

text: particular,

text: the

text: diﬀerential

text: for

text: a

text: monotone

text: pair

text: L

text: (L,

text: ψ(L))

text: with

text: any

text: symplectomorphism

text: ψ

text: Symp(M)

text: always

text: squares

text: to

text: zero,

text: since

text: w(L)

text: w(ψ(L)).

text: Z

text: Following

text: Sheridan

text: for

text: each

text: integer

text: w

text: deﬁne

text: the

text: generalized

text: Fukaya

text: category

text: Fuk

text: (M,

text: w)

text: whose

text: objects

text: are

text: generalized

text: Lagrangians

text: L

text: whose

text: total

text: disk

text: invariant

text: is

text: w(L)

text: w,

text: and

text: whose

text: morphisms

text: are

text: the

text: quilted

text: Floer

text: cochain

text: groups

text: deﬁned

text: above.

text: Z

text: Theorem

text: Let

text: M

text: be

text: a

text: symplectic

text: background,

text: w

text: an

text: integer

text: and

text: µd

text: for

text: d

text: the

text: maps

text: deﬁned

text: in

text: for

text: some

text: choice

text: of

text: family

text: of

text: surfaces

text: of

text: strip-like

text: ends

text: and

text: perturbation

text: data.

text: Then

text: the

text: maps

text: µd,

text: d

text: deﬁne

text: an

text: A∞

text: structure

text: on

text: Fuk

text: (M,

text: w).

text: Proof.

text: The

text: possibility

text: of

text: disk

text: bubbling

text: occurs

text: only

text: the

text: deﬁnition

text: of

text: since

text: it

text: is

text: only

text: in

text: this

text: case

text: that

text: there

text: exist

text: holomorphic

text: quilts

text: in

text: a

text: moduli

text: space

text: that

text: is

text: not

text: of

text: expected

text: dimension

text: (the

text: constant

text: trajectories).

text: The

text: proof

text: is

text: therefore

text: the

text: same

text: as

text: in

text: the

text: case

text: of

text: Maslov

text: number

text: at

text: least

text: three,

text: with

text: the

text: added

text: requirement

text: that

text: the

text: perturbation

text: data

text: on

text: the

text: ends

text: makes

text: the

text: disks

text: in

text: the

text: deﬁnition

text: of

text: the

text: disk

text: invariant

text: regular.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 45 We introduce the following notations for moduli spaces of quilted disks. Denote R by d,0 for the set of isomorphism classes of combinatorial type Γ. The moduli space R Γ R d,0 is the union of d,0 over combinatorial types Γ. As before in (24), forgetting Γ R → R − a marking and stabilizing produces a ﬁber bundle f : d,0 d 1,0 with interval i − ﬁbers isomorphic to the part of the boundary between the i 1 and i+1-st markings. R d,0 This implies by induction that is a topological disk. There is also a forgetful map R → R d,0 d (41) f : R forgetting a seam which is, restricted to any stratum d,0, a ﬁber bundle with open Γ R interval ﬁbers. For example, the top-dimensional stratum d,0 may be identiﬁed with the space of tuples (wH2∼< . .−. <{ w}d) by ﬁxing w1 = 0 and taking the seam to be given by Im(z) = 1 in = D 1 . The forgetful morphism for forgetting the seam is then in coordinates → (w < . . . < w ) (w /w < . . . w /w ). 2 d 3 2 d 2 This map is smooth and surjective with ﬁber the space of quilted disks represented by tuples [0, λ, λw /w , . . . , λw /w ]. 3 2 d 2 R 3,0 Example 5.5. (The third multiplihedron) The moduli space is the hexagon shown in Figure 7. The picture shows how the interior circle on the open stratum can “bubble oﬀ” into the bubble disks on the boundary. The local structure of the moduli space of quilted disks is described as follows, see Ma’u-Woodward [20]. Deﬁnition 5.6. (Balanced gluing parameters) A collection of gluing parameters → ∞ δ : Edge ∞(Γ) [0, ) < ∈ is balanced if the following condition holds: for each pair v , v Vert(1)(Γ) of colored 1 2 (cid:89) vertices, let γ denote the shortest path in Γ from v to v . Then the relation 12 1 2 ± ∀ ∈ (42) δ(e) 1 = 1, v , v Vert(1)(Γ) 1 2 ∈ e γ12 − holds where the sign is +1 resp. 1 if the edge points towards resp. away from the root edge. The following theorem describes the local structure of the moduli space of quilted ⊂ R disks near any stratum. Let Z Map(Edge ∞(Γ), ≥ ) be the set of balanced Γ < 0 gluing parameters from Deﬁnition 5.6. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: We

text: introduce

text: the

text: following

text: notations

text: for

text: moduli

text: spaces

text: of

text: quilted

text: disks.

text: Denote

text: R

text: by

text: for

text: the

text: set

text: of

text: isomorphism

text: classes

text: of

text: combinatorial

text: type

text: Γ.

text: The

text: moduli

text: space

text: R

text: Γ

text: R

text: is

text: the

text: union

text: of

text: over

text: combinatorial

text: types

text: Γ.

text: As

text: before

text: in

text: forgetting

text: Γ

text: R

text: R

text: a

text: marking

text: and

text: stabilizing

text: produces

text: a

text: ﬁber

text: bundle

text: f

text: d

text: with

text: interval

text: i

text: ﬁbers

text: isomorphic

text: to

text: the

text: part

text: of

text: the

text: boundary

text: between

text: the

text: i

text: and

text: markings.

text: R

text: This

text: implies

text: by

text: induction

text: that

text: is

text: a

text: topological

text: disk.

text: There

text: is

text: also

text: a

text: forgetful

text: map

text: R

text: R

text: d

text: f

text: R

text: forgetting

text: a

text: seam

text: which

text: is,

text: restricted

text: to

text: any

text: stratum

text: a

text: ﬁber

text: bundle

text: with

text: open

text: Γ

text: R

text: interval

text: ﬁbers.

text: For

text: example,

text: the

text: top-dimensional

text: stratum

text: may

text: be

text: identiﬁed

text: with

text: the

text: space

text: of

text: tuples

text: w}d)

text: by

text: ﬁxing

text: and

text: taking

text: the

text: seam

text: to

text: be

text: given

text: by

text: Im(z)

text: in

text: D

text: The

text: forgetful

text: morphism

text: for

text: forgetting

text: the

text: seam

text: is

text: then

text: in

text: coordinates

text: (w

text: w

text: (w

text: /w

text: w

text: /w

text: d

text: d

text: This

text: map

text: is

text: smooth

text: and

text: surjective

text: with

text: ﬁber

text: the

text: space

text: of

text: quilted

text: disks

text: represented

text: by

text: tuples

text: λ,

text: λw

text: /w

text: λw

text: /w

text: d

text: R

text: Example

text: (The

text: third

text: multiplihedron)

text: The

text: moduli

text: space

text: is

text: the

text: hexagon

text: shown

text: in

text: Figure

text: The

text: picture

text: shows

text: how

text: the

text: interior

text: circle

text: on

text: the

text: open

text: stratum

text: can

text: “bubble

text: oﬀ”

text: into

text: the

text: bubble

text: disks

text: on

text: the

text: boundary.

text: The

text: local

text: structure

text: of

text: the

text: moduli

text: space

text: of

text: quilted

text: disks

text: is

text: described

text: as

text: follows,

text: see

text: Ma’u-Woodward

text: Deﬁnition

text: (Balanced

text: gluing

text: parameters)

text: A

text: collection

text: of

text: gluing

text: parameters

text: δ

text: Edge

text: ∞(Γ)

text: is

text: balanced

text: if

text: the

text: following

text: condition

text: holds:

text: for

text: each

text: pair

text: v

text: v

text: of

text: colored

text: vertices,

text: let

text: γ

text: denote

text: the

text: shortest

text: path

text: in

text: Γ

text: from

text: v

text: to

text: v

text: Then

text: the

text: relation

text: δ(e)

text: v

text: v

text: e

text: holds

text: where

text: the

text: sign

text: is

text: resp.

text: if

text: the

text: edge

text: points

text: towards

text: resp.

text: away

text: from

text: the

text: root

text: edge.

text: The

text: following

text: theorem

text: describes

text: the

text: local

text: structure

text: of

text: the

text: moduli

text: space

text: of

text: quilted

text: R

text: disks

text: near

text: any

text: stratum.

text: Let

text: Z

text: Map(Edge

text: ∞(Γ),

text: be

text: the

text: set

text: of

text: balanced

text: Γ

text: gluing

text: parameters

text: from

text: Deﬁnition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 46 S. MA’U, K. WEHRHEIM, AND C. WOODWARD z0 z1 z3 z2 R Figure 7. 3,0 Theorem 5.7. (Existence of compatible tubular neighborhoods) For any integer ≥ d 1 there exists a collection of open neighborhoods U of 0 in Z and collar Γ Γ neighborhoods R × → R G : d,0 U d,0 Γ Γ Γ R satisfying the following compatibility property: If d,(cid:48)0 is contained in the closure of R R Γ d,0 and the local coordinates on d,0 are induced via the gluing construction from Γ R Γ those on d,(cid:48)0 then the diagram Γ R × (cid:45) R × dΓ,(cid:48)0 UΓ(cid:48) dΓ,0 UΓ (cid:115) (cid:43) R d,0 commutes. Sketch of proof. The proof uses a version of the gluing construction for nodal disks ∈ to the quilted case. Let D be a stable nodal quilted disk and δ Z . The glued disk Γ G (D) is deﬁned as follows. For each component D , let w be the node connecting δ j j the disk with component containing the root marking z , or z if D is that com- 0 − {0 } j ponent. We assume that for each D an identiﬁcation of D w with half-space H { ≥ } j j j = Im(z) 0 has been ﬁxed. Such an identiﬁcation is given by ﬁxing two addi- tional markings or nodes for an unquilted component, or one additional marking or { } node and the seam of the quilt as the line C = Im(z) = 1 . Note that the space of such coordinates is convex. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

font: default

text: R

text: Figure

text: Theorem

text: (Existence

text: of

text: compatible

text: tubular

text: neighborhoods)

text: For

text: any

text: integer

text: d

text: there

text: exists

text: a

text: collection

text: of

text: open

text: neighborhoods

text: U

text: of

text: in

text: Z

text: and

text: collar

text: Γ

text: Γ

text: neighborhoods

text: R

text: R

text: G

text: U

text: Γ

text: Γ

text: Γ

text: R

text: satisfying

text: the

text: following

text: compatibility

text: property:

text: If

text: is

text: contained

text: in

text: the

text: closure

text: of

text: R

text: R

text: Γ

text: and

text: the

text: local

text: coordinates

text: on

text: are

text: induced

text: via

text: the

text: gluing

text: construction

text: from

text: Γ

text: R

text: Γ

text: those

text: on

text: then

text: the

text: diagram

text: Γ

text: R

text: R

text: UΓ

text: R

text: commutes.

text: Sketch

text: of

text: proof.

text: The

text: proof

text: uses

text: a

text: version

text: of

text: the

text: gluing

text: construction

text: for

text: nodal

text: disks

text: to

text: the

text: quilted

text: case.

text: Let

text: D

text: be

text: a

text: stable

text: nodal

text: quilted

text: disk

text: and

text: δ

text: Z

text: The

text: glued

text: disk

text: Γ

text: G

text: (D)

text: is

text: deﬁned

text: as

text: follows.

text: For

text: each

text: component

text: D

text: let

text: w

text: be

text: the

text: node

text: connecting

text: δ

text: j

text: j

text: the

text: disk

text: with

text: component

text: containing

text: the

text: root

text: marking

text: z

text: or

text: z

text: if

text: D

text: is

text: that

text: com-

text: j

text: ponent.

text: We

text: assume

text: that

text: for

text: each

text: D

text: an

text: identiﬁcation

text: of

text: D

text: w

text: with

text: half-space

text: H

text: j

text: j

text: j

text: Im(z)

text: has

text: been

text: ﬁxed.

text: Such

text: an

text: identiﬁcation

text: is

text: given

text: by

text: ﬁxing

text: two

text: addi-

text: tional

text: markings

text: or

text: nodes

text: for

text: an

text: unquilted

text: component,

text: or

text: one

text: additional

text: marking

text: or

text: node

text: and

text: the

text: seam

text: of

text: the

text: quilt

text: as

text: the

text: line

text: C

text: Im(z)

text: Note

text: that

text: the

text: space

text: of

text: such

text: coordinates

text: is

text: convex.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 54 S. MA’U, K. WEHRHEIM, AND C. WOODWARD 6. Natural transformations for Floer cocycles In this section we associate to any Floer cocycle for a pair of correspondences a natural transformation between the functors constructed in the previous section. We then complete the proof of Theorem 1.1. The natural transformations are deﬁned by counts of quilted disks with markings on the interior circle. In the manner of Stasheﬀ [33] we deﬁne for any pair d, e of K K positive integers a cell complex d,e inductively. Each face of d,e corresponds to an expression in formal variables a , . . . , a , formal 1-morphism symbols h, and formal 1 d 2-morphism symbols t , . . . , t . Each(cid:18)type of sym(cid:19)bol must appear in the given order 1 e with parentheses in an expression of the form t . . . t 1 e h . a . . . a 1 d Each such expression corresponds to a pair of colored trees Γ = (Γ , Γ ) equipped − → − 1 2 with an isomorphism of subtrees Γ Γ of subtrees above the colored vertices. 1 2 The expression is built from the tree by using the parts of the tree below the colored vertices to build the parenthesized expressions in the variables t and a , and the i j part of the tree above the colored vertices to build the parenthesized expressions · · in the expressions h( ), where represents the expression below the colored vertex. Thus in the ﬁgure on the right Figu(cid:18)re 8, where(cid:19)the(cid:18)red colo(cid:19)r is used for the second (cid:0) (cid:1) tree below the colored vertices, the resulting expression is t h (t t ) h h 3 . 1 2 a (a a ) a a 1 2 3 4 5 We deﬁne (cid:116) ∼ Vert(Γ) = (Vert(Γ ) Vert(Γ ))/ 1 2 the set of vertices modulo th−e n∼atu−ral identiﬁcation of the subtrees and similarly for the edge set. The subtrees Γ = Γ describe the parenthesizations of the symbols of · 1 2 the form h( ), while the trees below the colored vertices describe the parenthesization of symbols t resp. a . There is a natural partial order on such tree pairs where (cid:22) (cid:48) i j → (cid:48) Γ Γ if there is a morph−ism∼ of−trees Γk Γk, k = 1, 2 contracting edges compatible with the identiﬁcations Γ = Γ . For any combinatorial type of bicolored tree Γ and 1 2 each colored resp. uncolored vertex v of Γ, let t(v) = (d(v), e(v)) resp. t(v) = d(v) denote the number of edges of each type incident to v. K (cid:48) (cid:48) The deﬁnition of the Stasheﬀ-style complexes is inductive. Suppose d ,e have (cid:48) (cid:48) K (cid:48) (cid:48) been deﬁned for d < d or e <(cid:91)e in such a way th(cid:89)at each d ,e is a stratiﬁed space with decomposition into combinatorial types K (cid:48) (cid:48) K K K d ,e = Γ, Γ := t(v). ∈ Γ v Vert(Γ) → (cid:48) If π : Γ Γ then there is a natural inclusion (cid:48) K → K (cid:48) ιΓ : Γ Γ Γ

text: S.

text: MA’U,

text: K.

text: WEHRHEIM,

text: AND

text: C.

text: WOODWARD

label: section

text: Natural

text: transformations

text: for

text: Floer

text: cocycles

text: In

text: this

text: section

text: we

text: associate

text: to

text: any

text: Floer

text: cocycle

text: for

text: a

text: pair

text: of

text: correspondences

text: a

text: natural

text: transformation

text: between

text: the

text: functors

text: constructed

text: in

text: the

text: previous

text: section.

text: We

text: then

text: complete

text: the

text: proof

text: of

text: Theorem

text: The

text: natural

text: transformations

text: are

text: deﬁned

text: by

text: counts

text: of

text: quilted

text: disks

text: with

text: markings

text: on

text: the

text: interior

text: circle.

text: In

text: the

text: manner

text: of

text: Stasheﬀ

text: we

text: deﬁne

text: for

text: any

text: pair

text: d,

text: e

text: of

text: K

text: K

text: positive

text: integers

text: a

text: cell

text: complex

text: d,e

text: inductively.

text: Each

text: face

text: of

text: d,e

text: corresponds

text: to

text: an

text: expression

text: in

text: formal

text: variables

text: a

text: a

text: formal

text: symbols

text: h,

text: and

text: formal

text: d

text: symbols

text: t

text: t

text: of

text: must

text: appear

text: in

text: the

text: given

text: order

text: e

text: with

text: parentheses

text: in

text: an

text: expression

text: of

text: the

text: form

text: t

text: t

text: e

text: h

text: a

text: a

text: d

text: Each

text: such

text: expression

text: corresponds

text: to

text: a

text: pair

text: of

text: colored

text: trees

text: Γ

text: (Γ

text: Γ

text: equipped

text: with

text: an

text: isomorphism

text: of

text: subtrees

text: Γ

text: Γ

text: of

text: subtrees

text: above

text: the

text: colored

text: vertices.

text: The

text: expression

text: is

text: built

text: from

text: the

text: tree

text: by

text: using

text: the

text: parts

text: of

text: the

text: tree

text: below

text: the

text: colored

text: vertices

text: to

text: build

text: the

text: parenthesized

text: expressions

text: in

text: the

text: variables

text: t

text: and

text: a

text: and

text: the

text: i

text: j

text: part

text: of

text: the

text: tree

text: above

text: the

text: colored

text: vertices

text: to

text: build

text: the

text: parenthesized

text: expressions

text: in

text: the

text: expressions

text: h(

text: where

text: represents

text: the

text: expression

text: below

text: the

text: colored

text: vertex.

text: Thus

text: in

text: the

text: ﬁgure

text: on

text: the

text: right

text: is

text: used

text: for

text: the

text: second

text: tree

text: below

text: the

text: colored

text: vertices,

text: the

text: resulting

text: expression

text: is

text: t

text: h

text: (t

text: t

text: h

text: h

text: a

text: (a

text: a

text: a

text: a

text: We

text: deﬁne

text: Vert(Γ)

text: (Vert(Γ

text: Vert(Γ

text: the

text: set

text: of

text: vertices

text: modulo

text: th−e

text: n∼atu−ral

text: identiﬁcation

text: of

text: the

text: subtrees

text: and

text: similarly

text: for

text: the

text: edge

text: set.

text: The

text: subtrees

text: Γ

text: Γ

text: describe

text: the

text: parenthesizations

text: of

text: the

text: symbols

text: of

text: the

text: form

text: h(

text: while

text: the

text: trees

text: below

text: the

text: colored

text: vertices

text: describe

text: the

text: parenthesization

text: of

text: symbols

text: t

text: resp.

text: a

text: There

text: is

text: a

text: natural

text: partial

text: order

text: on

text: such

text: tree

text: pairs

text: where

text: i

text: j

text: Γ

text: Γ

text: if

text: there

text: is

text: a

text: morph−ism∼

text: of−trees

text: Γk

text: Γk,

text: k

text: contracting

text: edges

text: compatible

text: with

text: the

text: identiﬁcations

text: Γ

text: Γ

text: For

text: any

text: combinatorial

text: type

text: of

text: bicolored

text: tree

text: Γ

text: and

text: each

text: colored

text: resp.

text: uncolored

text: vertex

text: v

text: of

text: Γ,

text: let

text: t(v)

text: (d(v),

text: e(v))

text: resp.

text: t(v)

text: d(v)

text: denote

text: the

text: number

text: of

text: edges

text: of

text: each

text: type

text: incident

text: to

text: v.

text: K

text: The

text: deﬁnition

text: of

text: the

text: Stasheﬀ-style

text: complexes

text: is

text: inductive.

text: Suppose

text: d

text: ,e

text: have

text: K

text: been

text: deﬁned

text: for

text: d

text: d

text: or

text: e

text: in

text: such

text: a

text: way

text: each

text: d

text: ,e

text: is

text: a

text: stratiﬁed

text: space

text: with

text: decomposition

text: into

text: combinatorial

text: types

text: K

text: K

text: K

text: K

text: d

text: ,e

text: Γ,

text: Γ

text: t(v).

text: Γ

text: v

text: Vert(Γ)

text: If

text: π

text: Γ

text: Γ

text: then

text: there

text: is

text: a

text: natural

text: inclusion

text: K

text: K

text: ιΓ

text: Γ

text: Γ

text: Γ


text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 55 K − → K (cid:91) − ⊂ induced by the maps π 1(v) t(v), where π 1(v) Γ is the subgroup that maps to v. Deﬁne K K ∼ ∂ d,e = Γ/ Γ ∼ where the union is over all combinatorial types with at least two vertices, and is (cid:48) K the equivalence relation induced by the maps ιΓ . Then ∂ d,e is naturally a stratiﬁed Γ K space whose links are the disjoint unions of the links in the stratiﬁed spaces Γ. Let K K d,e = Cone(∂ d,e) K K K − K K denote the cone on ∂ d,e. Deﬁne int( Γ) = Γ ∂ Γ. We call the sets int( Γ) K faces of d,e. K Example 6.1. (One or(cid:18)two(cid:19)seam m(cid:18)ark(cid:19)ing(cid:18)s) (cid:19)1,1 is t(cid:18)he i(cid:19)nter(cid:18)val w(cid:19)ith faces correspond- ing to the expressions t t t (cid:18) (cid:19) 1 1 1 h , h h , h h . a a a 1 1 1 K t 2,1 is th(cid:18)e octago(cid:19)n wi(cid:18)th o(cid:19)pen(cid:18)fac(cid:19)e co(cid:18)rres(cid:19)pon(cid:18)ding(cid:19)to (cid:18)the(cid:19)exp(cid:18)ress(cid:19)ion (cid:18)h (cid:19) 1 and a a 1 2 facets corresponding to the expressions (cid:18) (cid:19) (cid:18) (cid:19)t (cid:18) (cid:19) t (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) t (cid:18) (cid:19) t(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) 1 1 1 1 h , h h h h , h h h , (a a ) a a a a a a 1 2 1 2 1 2 1 2 t t t t 1 1 1 1 h h h , h h h , h h , h h . a a a a a a a a 1 2 1 2 1 2 1 2 Remark 6.2. (Facets) For any positive integers d, e, there is a bijection between K facets of d,e and the following expressions: (a) insertion of parenthesis around a sub-expression a . . . a − ; i+1 i+j 1 (b) insertion of parentheses around a sub-expression t . . . t − t ; · · · i i+j 1 i+j (c) a product of expressions h( )h( ) . . . h( ), corresponding to a partition of the symbols t , . . . , t , a , . . . , a , such that each element of the partition contains 1 e 1 d at least one symbol. K In particular, the facets of d,1 correspond to the terms in the deﬁnition (81) of µ1 for pre-natural transformations. The spaces above may be identiﬁed with moduli spaces of stable quilted disks with markings on the boundary or seam: Deﬁnition 6.3. (Quilted disks with inner and outer markings) Let d, e be positive integers. A quilted disk with d outer markings and e inner markings is a tuple (D, C, ζ, z . . . , z , w , . . . , w ) where 1 d 1 e (a) D is a holomorphic disk; ∩ { } (b) C is a circle in D with unique intersection point C ∂D = ζ ;

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: K

text: K

text: induced

text: by

text: the

text: maps

text: π

text: t(v),

text: where

text: π

text: Γ

text: is

text: the

text: subgroup

text: that

text: maps

text: to

text: v.

text: Deﬁne

text: K

text: K

text: d,e

text: Γ/

text: Γ

text: where

text: the

text: union

text: is

text: over

text: all

text: combinatorial

text: types

text: with

text: at

text: least

text: two

text: vertices,

text: and

text: is

text: K

text: the

text: equivalence

text: relation

text: induced

text: by

text: the

text: maps

text: ιΓ

text: Then

text: d,e

text: is

text: naturally

text: a

text: stratiﬁed

text: Γ

text: K

text: space

text: whose

text: links

text: are

text: the

text: disjoint

text: unions

text: of

text: the

text: links

text: in

text: the

text: stratiﬁed

text: spaces

text: Γ.

text: Let

text: K

text: K

text: d,e

text: Cone(∂

text: d,e)

text: K

text: K

text: K

text: K

text: K

text: denote

text: the

text: cone

text: on

text: d,e.

text: Deﬁne

text: int(

text: Γ)

text: Γ

text: Γ.

text: We

text: call

text: the

text: sets

text: int(

text: Γ)

text: K

text: faces

text: of

text: d,e.

text: K

text: Example

text: (One

text: is

text: faces

text: correspond-

text: ing

text: to

text: the

text: expressions

text: t

text: t

text: t

text: h

text: h

text: h

text: h

text: h

text: a

text: a

text: a

text: K

text: t

text: is

text: and

text: a

text: a

text: facets

text: corresponding

text: to

text: the

text: expressions

text: t

text: t

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: (a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: t

text: t

text: t

text: t

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: h

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: a

text: Remark

text: (Facets)

text: For

text: any

text: positive

text: integers

text: d,

text: e,

text: there

text: is

text: a

text: bijection

text: between

text: K

text: facets

text: of

text: d,e

text: and

text: the

text: following

text: expressions:

text: (a)

text: insertion

text: of

text: parenthesis

text: around

text: a

text: sub-expression

text: a

text: a

text: i+j

text: (b)

text: insertion

text: of

text: parentheses

text: around

text: a

text: sub-expression

text: t

text: t

text: t

text: i

text: i+j

text: i+j

text: (c)

text: a

text: product

text: of

text: expressions

text: h(

text: )h(

text: h(

text: corresponding

text: to

text: a

text: partition

text: of

text: the

text: symbols

text: t

text: t

text: a

text: a

text: such

text: that

text: each

text: element

text: of

text: the

text: partition

text: contains

text: e

text: d

text: at

text: least

text: one

text: symbol.

text: K

text: In

text: particular,

text: the

text: facets

text: of

text: correspond

text: to

text: the

text: terms

text: in

text: the

text: deﬁnition

text: of

text: µ1

text: for

text: pre-natural

text: transformations.

text: The

text: spaces

text: above

text: may

text: be

text: identiﬁed

text: with

text: moduli

text: spaces

text: of

text: stable

text: quilted

text: disks

text: with

text: markings

text: on

text: the

text: boundary

text: or

text: seam:

text: Deﬁnition

text: (Quilted

text: disks

text: with

text: inner

text: and

text: outer

text: markings)

text: Let

text: d,

text: e

text: be

text: positive

text: integers.

text: A

text: quilted

text: disk

text: with

text: d

text: outer

text: markings

text: and

text: e

text: inner

text: markings

text: is

text: a

text: tuple

text: (D,

text: C,

text: ζ,

text: z

text: z

text: w

text: w

text: where

text: d

text: e

text: (a)

text: D

text: is

text: a

text: holomorphic

text: disk;

text: (b)

text: C

text: is

text: a

text: circle

text: in

text: D

text: with

text: unique

text: intersection

text: point

text: C

text: ∂D

text: ζ


text: A∞ FUNCTORS FOR LAGRANGIAN CORRESPONDENCES 57 Example 6.5. An example of a nodal (3, 5)-marked disk is given in Figure 8. In the ﬁgure, we use black leaves for outer circle markings, and red (in the online version) leaves for inner circle markings. The black leaves are required to respect the order of the outer marked points, likewise the red leaves respect the order of the inner marked points. Figure 8. A nodal (3,5)-marked disk A topology on the moduli space of marked quilted disks can be deﬁned by in- troducing a suitable notion of convergence. Consider a sequence (D , C , z , w ) of ν ν ν ν marked quil∼tedPdi∼skCs w∪it{h∞sm} ooth domain. That is, we assume that Dν has no nodes so that D = 1 = . For simplicity we may assume that the seam is given ν by Im(z) = ρ and the boundary given by Im(z) = (cid:15) . Let (D, C, z, w) be another ν ν marked quilted disk. For (D , C , z , w ) to converge to (D, C, z, w) we require the ν ν ν ν → following. For each irreducible component D of D let π : D D denote the i i i unique map that is the identity on D and constant on all other components. We → i require a family φ : D D of holomorphic embeddings (preserving the seams, ν i ν − if D is quilted) such that for each k, either φ 1(z ) converges to z , if z lies on i ν,i k,ν k k− D , or otherwise to a node π (z ) of D ; and similarly for each j either φ 1(w ) i i k i ν,i j,ν converges to w or to a node π (w ). The deﬁnition of convergence for a ﬁxed combi- j i j natorial type is similar, treating each irreducible component separately. In the case of sequences with varying combinatorial type, a subsequence of each ﬁxed type is R d,e required to converge to the same limit. We equip with the topology given by declaring a subset to be closed if it is closed under limits. An argument analogous to [22, 5.6.5] shows that the convergence in the resulting topology is in fact the R d,e above notion of convergence. As before in (25), comes with a universal curve U U d,e containing a ﬁberwise boundary ∂ d,e as well as a universal seam consisting of ∈ isomorphisms classes of tuples (D, C, z, w, y) where y C. The local structure of these moduli spaces of quilted disks with seam and boundary markings can be described in terms of balanced gluing parameters. Suppose that Γ ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A∞

text: FUNCTORS

text: FOR

text: LAGRANGIAN

text: CORRESPONDENCES

text: Example

text: An

text: example

text: of

text: a

text: nodal

text: disk

text: is

text: given

text: in

text: Figure

text: In

text: the

text: ﬁgure,

text: we

text: use

text: black

text: leaves

text: for

text: outer

text: circle

text: markings,

text: and

text: red

text: (in

text: the

text: online

text: version)

text: leaves

text: for

text: inner

text: circle

text: markings.

text: The

text: black

text: leaves

text: are

text: required

text: to

text: respect

text: the

text: order

text: of

text: the

text: outer

text: marked

text: points,

text: likewise

text: the

text: red

text: leaves

text: respect

text: the

text: order

text: of

text: the

text: inner

text: marked

text: points.

text: Figure

text: A

text: nodal

text: disk

text: A

text: topology

text: on

text: the

text: moduli

text: space

text: of

text: marked

text: quilted

text: disks

text: can

text: be

text: deﬁned

text: by

text: in-

text: troducing

text: a

text: suitable

text: notion

text: of

text: convergence.

text: Consider

text: a

text: sequence

text: (D

text: C

text: z

text: w

text: of

text: ν

text: ν

text: ν

text: ν

text: marked

text: quil∼tedPdi∼skCs

text: w∪it{h∞sm}

text: ooth

text: domain.

text: That

text: is,

text: we

text: assume

text: that

text: Dν

text: has

text: no

text: nodes

text: so

text: that

text: D

text: For

text: simplicity

text: we

text: may

text: assume

text: that

text: the

text: seam

text: is

text: given

text: ν

text: by

text: Im(z)

text: ρ

text: and

text: the

text: boundary

text: given

text: by

text: Im(z)

text: Let

text: (D,

text: C,

text: z,

text: w)

text: be

text: another

text: ν

text: ν

text: marked

text: quilted

text: disk.

text: For

text: (D

text: C

text: z

text: w

text: to

text: converge

text: to

text: (D,

text: C,

text: z,

text: w)

text: we

text: require

text: the

text: ν

text: ν

text: ν

text: ν

text: following.

text: For

text: each

text: irreducible

text: component

text: D

text: of

text: D

text: let

text: π

text: D

text: D

text: denote

text: the

text: i

text: i

text: i

text: unique

text: map

text: that

text: is

text: the

text: identity

text: on

text: D

text: and

text: constant

text: on

text: all

text: other

text: components.

text: We

text: i

text: require

text: a

text: family

text: φ

text: D

text: D

text: of

text: holomorphic

text: embeddings

text: (preserving

text: the

text: seams,

text: ν

text: i

text: ν

text: if

text: D

text: is

text: quilted)

text: such

text: that

text: for

text: each

text: k,

text: either

text: φ

text: converges

text: to

text: z

text: if

text: z

text: lies

text: on

text: i

text: ν,i

text: k,ν

text: k

text: k−

text: D

text: or

text: otherwise

text: to

text: a

text: node

text: π

text: (z

text: of

text: D

text: and

text: similarly

text: for

text: each

text: j

text: either

text: φ

text: i

text: i

text: k

text: i

text: ν,i

text: j,ν

text: converges

text: to

text: w

text: or

text: to

text: a

text: node

text: π

text: (w

text: The

text: deﬁnition

text: of

text: convergence

text: for

text: a

text: ﬁxed

text: combi-

text: j

text: i

text: j

text: natorial

text: type

text: is

text: similar,

text: treating

text: each

text: irreducible

text: component

text: separately.

text: In

text: the

text: case

text: of

text: sequences

text: with

text: varying

text: combinatorial

text: type,

text: a

text: subsequence

text: of

text: each

text: ﬁxed

text: type

text: is

text: R

text: d,e

text: required

text: to

text: converge

text: to

text: the

text: same

text: limit.

text: We

text: equip

text: with

text: the

text: topology

text: given

text: by

text: declaring

text: a

text: subset

text: to

text: be

text: closed

text: if

text: it

text: is

text: closed

text: under

text: limits.

text: An

text: argument

text: analogous

text: to

text: shows

text: that

text: the

text: convergence

text: in

text: the

text: resulting

text: topology

text: is

text: in

text: fact

text: the

text: R

text: d,e

text: above

text: notion

text: of

text: convergence.

text: As

text: before

text: in

text: comes

text: with

text: a

text: universal

text: curve

text: U

text: U

text: d,e

text: containing

text: a

text: ﬁberwise

text: boundary

text: d,e

text: as

text: well

text: as

text: a

text: universal

text: seam

text: consisting

text: of

text: isomorphisms

text: classes

text: of

text: tuples

text: (D,

text: C,

text: z,

text: w,

text: y)

text: where

text: y

text: C.

text: The

text: local

text: structure

text: of

text: these

text: moduli

text: spaces

text: of

text: quilted

text: disks

text: with

text: seam

text: and

text: boundary

text: markings

text: can

text: be

text: described

text: in

text: terms

text: of

text: balanced

text: gluing

text: parameters.

text: Suppose

text: that

text: Γ

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##