Contents Introduction - math.berkeley.edukatrin/papers/fieldb.pdfContents Introduction -...

FLOER FIELD THEORY FOR TANGLES

KATRIN WEHRHEIM AND CHRIS WOODWARD

Abstract. We construct functor-valued invariants invariants for cobordisms pos-sibly containing tangles and certain trivalent graphs. The latter gives gauge-theoretic invariants similar to Khovanov-Rozansky homology [12].

Contents

1. Introduction 12. Flat bundles with fixed holonomy 23. Derived Floer theory 214. Category-valued field theory 29References 33

Preliminary version.

1. Introduction

Let X be a compact connected oriented surface. This paper constructs whatmight be called a Floer field theory in 2 + 1 dimensions for tangles in X × [−1, 1]:a functor from the category of (finite subsets of X, tangles in X × [−1, 1]) to thecategory of (categories, isomorphism classes of functors). The categories associatedto 2-manifolds with markings are Donaldson-Fukaya category of the correspondingmoduli spaces of parabolic bundles. The invariants should be related to singularinstanton invariants as considered by Collin-Steer [4] and Kronheimer-Mrowka [15],by some version of the Atiyah-Floer conjecture. The relevant symplectic geometry isdiscussed further in Jacobsson-Rubinsztein [10]. As an application, we use a resultof Seidel [28] to prove the non-triviality of the symplectic mapping class groups ofcertain representation varieties.

As in [33] we first construct a functor associated to a Cerf decomposition of thetangle, and then prove that the functor is independent up to isomorphism of thechoice of Cerf decomposition. We use freely the terminology and notation from [33],in particular the notion of Cerf decomposition, partial functor, and weak topologicalfield theory. Let Cobd+1 denote the category of d+ 1 dimensional cobordisms up toequivalence. Let C be a category. The following is explained in [33, Corollary 2.2.2]:

Corollary 1.0.1. Any partial functor Cobd+1 → C, which associates

(a) to each compact, connected, oriented d-manifold X, an object C(X) ∈ Obj(C),1

2 KATRIN WEHRHEIM AND CHRIS WOODWARD

(b) to each equivalence class of compact, connected, oriented simple cobordism Yfrom X− to X+, a morphism Φ(Y ) from C(X−) to C(X+),

(c) to the trivial cobordism [0, 1] ×X the identity morphism 1C(X) of C(X);

and satisfies the Cerf relations

(a) If Y1 from X0 to X1 and Y2 from X1 to X2 are simple cobordisms such thatY1 ∪X1

Y2 is a cylindrical cobordism via critical point cancellation, then

Φ(Y1) Φ(Y2) = Φ(Y1 ∪X1Y2);

(b) If Y1, Y2 and Y ′1 , Y

′2 are simple cobordisms related by critical point reversal,

then

Φ(Y1) Φ(Y2) = Φ(Y ′1) Φ(Y ′

2);

(c) If Y1, Y2 are simple cobordisms, one of which is cylindrical, then

Φ(Y1) Φ(Y2) = Φ(Y1 ∪X1Y2);

extends to a unique weak d+1-dimensional C-valued topological field theory Cobd+1 →C.

In other words, to define a topological field theory it suffices to define the mor-phisms for simple cobordisms and prove the Cerf relations. In the first stage, our

target category is the category Symp#1/2c of (monotone symplectic manifolds with

monotonicity constant 1/2c, generalized Lagrangian correspondences with relativespin structures and minimal Maslov numbers at least two). In the second stage, ourtarget category will be the category FCat of categories enriched in matrix factoriza-tions, defined in Section 3.

We thank Paul Seidel for encouragement in the project.

2. Flat bundles with fixed holonomy

In this section we construct a symplectic-valued field theory for cobordisms possi-bly containing trivalent graphs. The corresponding moduli spaces are moduli spacesof bundles with holonomy around each strand in a fixed conjugacy class.

2.1. Conjugacy classes. Throughout, G is a compact, connected, simply-connectedLie group with Lie algebra g. We denote its maximal torus by T ⊂ G, its Cartansubalgebra by t ⊂ g, and its rank by r = rank(G) := dim t. Let N(T ) denote thenormalizer of T , then W := N(T )/T is the Weyl group of G. The action of W onT by conjugation induces an action on t∗ denoted µ 7→ wµ for w ∈ W,µ ∈ t∗. TheLie algebra admits a splitting

g = t ⊕l⊕

i=1

gαi

where gαiare the two-dimensional root spaces, labelled by the roots ±α1, . . . ,±αl ∈

t∗, which are defined up to sign. In our convention for the roots, the elementexp(ξ) ∈ T acts on gαi

∼= R2 by

Ad(exp(ξ))ζ = R±αi(ξ)(ζ)

FLOER FIELD THEORY FOR TANGLES 3

where Rαi(ξ) denotes rotation by angle αi(ξ). The equations αi(ξ) = 0 decomposethe Cartan subalgebra t into chambers. Let t+ be a choice of positive chamber; thisfixes a choice of positive roots α1, . . . , αl by requiring αj(t+) ⊂ R≥0. Let α1, . . . , αr

be the simple roots, i.e. the minimal set such that

t+ = ξ|αj(ξ) ≥ 0, j = 1, . . . , r.

Let α0 ∈ α1, . . . , αl denote the highest root, defined as the unique positive rootsuch that α0(ξ) ≥ αj(ξ) for all ξ ∈ t+ and j = 1, . . . , l.

Now conjugacy classes in G are parametrized by the Weyl alcove. We denote

Cµ = g exp(µ)g−1 | g ∈ G for µ ∈ A = ξ ∈ t+ | α0(ξ) ≤ 1.

Since G is 1-connected, the conjugacy classes of G are simply-connected. To seethis, let A(S1) = Ω1(S1, g) denote the space of connections on the trivial bundle onthe circle. The holonomy fibration

ΩG→ Ω1(S1, g) → G

restricts to a fibration

(1) ΩG→ LG · µdθ → Cµ

where LG · µ is the orbit under the loop group LG = Map(S1, G) acting affinely onµdθ ∈ Ω1(S1, g). Since ΩG is connected and LG ·µ is simply-connected [27, Section8.6], it follows that Cµ is simply-connected as well.

We denote by ∗ the (possibly trivial) involution of the alcove defined by

(2) ∗ : A → A, C∗µ = C−1µ .

Let βi denote the unique vertex of A such that αj(βi) = 0 for all i ∈ 1, . . . , r\j.Let Λ = exp−1(1) ⊂ t denote the coweight lattice and let ωi ∈ Λ be the i-thfundamental coweight, that is, ωi is the positive multiple of βi that generates Λ∩Rβi.

Furthermore, we denote the half-sum of positive roots and the dual Coxeter num-ber of G [11, 6.1] by

(3) ρ = 12

∑li=1αi ∈ t∗, c = 〈ρ, α0〉 + 1

using the basic inner product [27, p. 49]. We can also use this to identify t with t∗.We remark that the image of ρ/c in t is independent of the inner product used, andinvariant under the involution (2).

Definition 2.1.1. We say that an element µ ∈ A is

(a) monotone1 if µ = projσ(c−1ρ) for some face σ of A, where projσ denotesorthogonal projection onto σ.

(b) standard if µ is the projection of ρ/c onto an edge of A containing 0, that is,µ = 1

2βi for some i = 1, . . . , r.(c) spin if the corresponding conjugacy class Cµ admits a G-equivariant spin

structure. Such a structure is necessarily unique, since Cµ is simply-connected.

1We will see later in Theorem 2.3.1 the relationship with monotonicity.


Example 2.1.2. If G = SU(r + 1) then c = r + 1. Under the standard identification

t∗ ∼= t ∼= (ξ1, . . . , ξr+1) ∈ Rr+1 |∑r+1

i=1 ξi = 0 and with the standard unit vectorsei ∈ Rr+1 we have simple roots αi = ei+1 − ei for i = 1, . . . , r, highest root α0 =e1 − er+1, ρ = ((r + 2)/2, r/2, . . . ,−r/2, (−2 − r)/2), and the Weyl alcove

(4) A ∼=

(λ1, . . . , λr+1) ∈ R

r+1

∣∣∣∣∣∣λ1 ≥ λ2 ≥ . . . ≥ λr+1 ≥ λ1 − 1,

r+1∑

j=1

λj = 0

.

The vertices of A are βi = ir+1(e1 + . . . + ei) + i−r−1

r+1 (ei+1 + . . . + er+1), and the

fundamental coweights are ωi = (r + 1)βi. We will use this enumeration of simpleroots, vertices, and fundamental coweights whenever working with SU(r + 1).

(a) If G = SU(2), then c = 2 and we will identify A ∼= [0, 12 ] by the map

(λ,−λ) 7→ λ so that Cµ consists of matrices with eigenvalues exp(±2πiµ).We have α0 = 1 and ρ = 1

2 . If µ = 14 , Cµ consists of all traceless matrices.

The monotone elements are 0, 12 , 1

4 . The unique standard element is 14 .

(b) If G = SU(3), we have ρ = α0 = (1, 0,−1). The monotone elements are(0, 0, 0), (1

3 ,−16 ,−

16), (2

3 ,−13 ,−

13), (1

2 , 0,−12 ), (1

3 ,13 ,−

23), (1

6 ,16 ,−

13), (1

3 , 0,−13 ).

The standard elements are (13 ,−

16 ,−

16), (1

6 ,16 ,−

13).

Figure 1. Standard conjugacy classes for SU(3)

2.2. Holonomy description of moduli spaces. We choose to describe the modulispaces via representations of the fundamental group, rather than gauge theory as in[33]. LetX be a compact, connected, oriented manifold, possibly with boundary, andlet K ⊂ X be an oriented, embedded submanifold of codimension 2. Let K1, . . . ,Kn

denote the connected components of K. For each j = 1, . . . n let γj : S1 → X \Kbe a small loop around Kj , so that the induced orientation on the normal bundleof Kj agrees with that induced by the orientations of Kj and X. Each γj definesa conjugacy class [γj] ⊂ π1(X \K). (Here we always implicitly fix a base point inthe definition of the fundamental group π1(X \K).) For µ = (µ1, . . . , µn) ∈ An letM(X,K,µ) denote the moduli space of flat G-bundles on XrK whose holonomyaround γj lies in the conjugacy class Cµj

. We call the element µj the label of thecomponent Kj. This moduli space has a description in terms of representations ofthe fundamental group, which we take as a definition:

(5) M(X,K,µ) :=ϕ ∈ Hom(π1(XrK), G))

∣∣ϕ([γj ]) ⊂ Cµj∀j

/G.

Here G acts by conjugation. In case X is not connected, this definition is replacedby the product of moduli spaces for the connected components of X. Changing the


orientation of a component Kj (i.e. of γj) corresponds to changing the label µj by the

involution ∗ of the alcove A, given by (2). That is, if K denotes the tangle obtainedby changing the orientation on Kj and µ is the set of labels obtained by replacing

µj with ∗µj then there is a canonical homeomorphism M(X,K,µ) →M(X, K, µ).

Remark 2.2.1. If the conjugacy classes Cµ = (Cµ1, . . . , Cµn

) each have finite order (as

in all our examples), then one can identify the moduli space M(X,K,µ) with themoduli space of flat connections of an orbibundle over X, see [20, 14, 21] for two- andfour-dimensional cases. If the orbifold cohomology groups H0(dα) = H2(dα) = 0vanish, then M(X,K,µ) has only orbifold singularities, as in [33]. This providesM(X,K,µ) with a smooth structure. In all our cases, this is compatible with thesmooth structure induced by the holonomy description (5) and an appropriate choiceof finite presentation of the fundamental group, in which the relations cut out trans-versely a smooth manifold on whichG acts with constant central stabilizer. However,we will avoid using the equivariant descripiton and instead check explicitly, in specificpresentations, the smoothness of those moduli spaces that enter our constructions.

2.3. Moduli spaces for marked surfaces. Let X be a compact, connected, ori-ented surface of genus g, and let x = x1, . . . , xn be a finite set of distinct, orientedmarkings xj ∈ X. Define ǫj = ±1 depending on whether the orientation of xj agreeswith the standard orientation of a point. The fundamental group π1(Xrx) hasstandard presentation

π1(Xrx) ∼= 〈 α1, . . . , α2g, γ1, . . . , γn | Πgj=1[α2j , α2j+1]Π

nj=1γ

ǫj

j = 1 〉,

where γj is a loop around xj, oriented corresponding to ǫj. Let µ ∈ An be a setof labels for x. The moduli space of flat G-bundles with fixed holonomy can bedescribed in terms of a standard presentation of π1(X \ x) by

M(X,x, µ) =ϕ ∈ Hom(π1(Xrx), G))

∣∣ϕ([γj ]) ⊂ Cµj∀j

/G

∼=(a, b) ∈ G2g × Cµ | Φ(a, b) = 1

/G,(6)

where G acts on G2g × Cµ diagonally by conjugation and

(7) Φ((a1, . . . , a2g), (b1, . . . , bn)) =

g∏

j=1

[a2j , a2j+1]n∏

j=1

bǫj

j .

Any other presentation of standard form induces an equivariant diffeomorphism ofG2g+n which preserves Φ−1(1), and hence induces the same smooth structure onM(X,x, µ). In case X is not connected, we define M(X,x, µ) to be the productof moduli spaces for its connected components. Alternatively, M(X,x, µ) can berealized as the moduli space of flat connections on the trivial G-bundle over Xrxwith fixed holonomies around x (see [20, 21]) and as such has a symplectic form.The form can be described explicitly in the holonomy description [1] as follows.

First suppose that X is a surface of genus g, without markings. Let θ, θ ∈Ω1(G, g)G be the left and right-invariant Maurer-Cartan forms. Define a form ω1 on


G2 by

ω1 ∈ Ω2(G2), ω1 =1

2〈l∗θ ∧ r∗θ〉 +

1

2〈l∗θ ∧ r∗θ〉

where l, r : G2 → G are the projections on the first and second factor. Definetwo-forms ωg ∈ Ω2(G2g) inductively by

(8) ωg = ωg1+ ωg2

+1

2〈Φ∗

g1θ ∧ Φ∗

g2θ〉

where g = g1 + g2 is any splitting with g1, g2 ≥ 1, Φgj: (G2)gj → G the product of

commutators, and we omit pull-backs to the factors of G2g ∼= G2g1 ×G2g2 from thenotation. A theorem of Alekseev-Malkin-Meinrenken [1, Theorem 9.3], extendingearlier work of Weinstein, Jeffrey, and Karshon, states that the restriction of ωg tothe identity level set of Φg descends to the symplectic form on the locus of irreducibleconnections in M(X).

More generally, suppose that (X,x, µ) is a marked, labelled surface. A symplecticstructure on M(X,x, µ) can be defined as follows. For any label µ ∈ A, define a2-form ωµ on the conjugacy class Cµ by

ωµ(vξ(g), vη(g)) = 〈θ(vη(g)) + θ(vη(g)), ξ〉, g ∈ Cµ, ξ, η ∈ g

where vξ, vη ∈ Vect(Cµ) are the generating vector fields for ξ, η. Define two-formsωg,µ ∈ Ω2(G2g×

∏µ∈µ Cµ) inductively by ωg,∅ = ωg, ω0,µ = ωµ, and for any splitting

g = g1 + g2, µ = µ1∪ µ

2, where for each j = 1, 2 either gj > 0 or µ

jis non-empty,

setting

(9) ωg,µ = ωg1,µ1+ ωg2,µ

2+

1

2〈Φ∗

g1,µ1

θ ∧ Φ∗g2,µ

2

θ〉

where

Φgj,µj: G2gj ×

∏

µ∈µj

Cµ → G, (h, (cµ)µ∈µj) 7→ Φgj

(h)∏

µ∈µj

cµ.

By [1, page 27], the restriction of ωg,µ to the identity level set of Φg,µ descends to

the symplectic form on the locus of irreducible connections in M(X,x, µ).Sufficient conditions for monotonicity of the moduli spaceM(X,x, µ) are provided

by [22, Theorem 4.2]:

Theorem 2.3.1 (Monotonicity). If each label µj is monotone and M(X,x, µ) is

smooth then M(X,x, µ) is monotone with monotonicity constant τ−1 = 2c, where cis the dual Coxeter number of (3).

We remark that the symplectic form is not in general integral; as we will see later,the minimal Chern number of M(X,x, µ) may be as small as 2.

Proposition 2.3.2 (Smoothness). Each of the following is a sufficient condition forM(X,x, µ) to be a smooth orbifold. Conditions (d-f) in fact imply that M(X,x, µ)is a manifold.

(a) Every flat G-connection on X \ x with holonomies around x in Cµ has finiteautomorphism group.


(b) Every solution (a, b) ∈ G2g × Cµ to Φ(a, b) = 1 has finite stabilizer.

(c) For each tuple w1, . . . , wn ∈W we have

(10) 〈∑n

j=1wjµj, ωi〉 /∈ 〈Λ, ωi〉 ∀i = 1, . . . , r.

(d) G = SU(2), all labels are µi = 14 , and the number n of markings is odd.

(e) G = SU(r), there are n = r + 1 markings and an integer j coprime to r(with respect to our fixed enumeration of vertices) such that µi = 1

2βj fori = 1, . . . , r + 1.

(f) G = SU(r), all labels µi are standard, and 2∑n

i=1 µi = βd mod Λ for somed coprime to r.

Proof. By [1], the space M(X,x, µ) can be realized as a symplectic quotient of

the Hamiltonian G-manifold M(X,x, µ) := G2g × Cµ with group-valued moment

map Φ : M(X,x, µ) → G given by the product (6). (a) and (b) are immediateconsequences, since the identity level set of the moment map is cut out transversallyif and only if all stabilizers are discrete, by [1, Definition 2.2,Condition B3]. (c)

describes the wall structure of the moment image of M(X,x, µ). For each tuplew = (w1, . . . , wn) ∈ W n and subset I ⊂ 1, . . . , l such that the span of (αi)i∈I isnot all of t∗ ∼= t, define the wall corresponding to w by

Hw,I := exp(∑n

j=1wjµj + span(αi)i∈I

).

Let TX,sing denote the singular values of Φ contained in T . We claim

TX,sing ⊆ ∪w,IHw,I .

From (1) follows that any orbit stratum in Cµjwith infinite stabilizer group contains

a T -fixed point in its closure, since the same is true for coadjoint orbits by equivari-ant formality of Hamiltonian actions [7, Appendix C]. Similarly, the closure of anyorbit stratum in G2g with infinite stabilizer is equal to H2g, for some subgroup Hcontaining T 2g. Now T 2g maps to the identity under the product of commutatorsΦ. Putting everything together, any orbit-type stratum Y in M (X,x, µ) containsa T -fixed point y in its closure, and such a fixed point has image contained inexp(

∑nj=1wjµj) for some w ∈W n. The tangent space TyY is a sum of root spaces,

and the assumption that the stabilizer of Y is infinite implies that the span of theroots appearing in the sum is not all of t∗. The claim follows.

To prove part (c), suppose that some orbit-type stratum with infinite stabilizerintersects Φ−1(1). Taking the roots that lie in the span of αi, i ∈ I as above givesa root subsystem of the root system for g, generating a Lie subalgebra h ⊂ g.After acting by some Weyl group element, we may assume that the simple roots of g

contain the simple roots of h. Any fundamental coweight ωi for a simple root of g thatis not a simple root of h is perpendicular to h, and satisfies

∑(wiµi, ωj) ∈ (Λ, ωj).

The cases (d-e) are special cases of (f). (f) is a special case of (c), since∑

wiµi −∑

µi ∈ Λ/2

and (βd/2, ωj) = jd/r mod Z is never an integer for r coprime to d and j = 1, . . . , r−1.


Definition 2.3.3. Let Diff+(X,x, µ) be the subgroup of orientation preserving dif-feomorphisms ϕ ∈ Diff+(X) that preserve the marked points, orientations, andlabels,

Diff+(X,x, µ) =ϕ ∈ Diff+(X)

∣∣ϕ(x) = x, ϕ∗ǫ = ǫ, ϕ∗µ = µ.

Here we denote by ǫ and µ the maps x → ±1, xi 7→ ǫi and x → A, xi 7→ µi. Sofor ϕ ∈ Diff+(X) with ϕ(xi) = xj the conditions are ǫi = ǫj and µi = µj . LetMap+(X,x, µ) be the quotient of Diff+(X,x, µ) by isotopy.

The action of Map+(X,x, µ) on π1(Xrx) induces an action on M(X,x, µ) bysymplectomorphisms on the smooth stratum, see [1, Section 9.4]. In particular, ifX is a sphere and all labels are equal, µ = (µ, . . . , µ), then the spherical braid group

Map+(S2, x) acts on M(X,x, µ). Explicitly if βi ∈ Map+(S2, x) is the half-twist ofxi and xi+1 then for suitable choice of presentation of π1(Xrx) we have

βi[b1, . . . , bn] = [b1, . . . , bi−1, bi+1, b−1i+1bibi+1, bi+2, . . . , bn].

2.4. Moduli spaces for cobordisms with tangles. Let Y be a compact, con-nected, oriented cobordism between connected, oriented surfaces X− and X+. Atangle in Y is an oriented, embedded one-manifold K ⊂ Y with boundary ∂K ⊂∂Y . The pair (Y,K) will also be called a cobordism-with-tangle. We denote byx± = (x±,1, . . . , x±,n±

) the intersections x± = K ∩ (∂Y )±.

Definition 2.4.1. A Morse datum for (Y,K) consists of a pair (f, b) of a Morsefunction f : Y → R that restricts to a Morse function f |K : K → R, and an orderedtuple b = (b0 < b1 < . . . < bm) ⊂ R

m+1 such that

(a) X− = f−1(b0) and X+ = f−1(bm) are the sets of minima resp. maxima of f ,(b) each level set f−1(b) for b ∈ R is connected, that is, f has no critical points

of index 0 or d+ 1,(c) f has distinct values at the critical points of f and f |K , i.e. it induces a

bijection Crit(f)∪Crit(f |K) → f(Crit(f)∪Crit(f |K)) between critical pointsand critical values,

(d) b1, . . . , bm−1 ∈ R\f(Crit(f)∪Crit(f |K)) are regular values of f and f |K suchthat each interval (bi−1, bi) contains at most one critical value of either f orf |K .

In the special case Y = X × [b−, b+], we say that (f, b) is a cylindrical Morse datumfor (Y,K) if ∂tf(x, t) > 0 for all (x, t) ∈ Y , and hence each level set is diffeomorphicto X.

Definition 2.4.2. The Cerf decomposition of (Y,K) induced by a Morse datum(f, b) is the sequence (Yi := f−1([bi−1, bi],Kj := Yj∩K)i=1,...m of simple cobordisms-with-tangles between the connected level sets Xi := Yi ∩ Yi+1 = f−1(bi), xi = Ki ∩Ki+1 = f−1(bi) ∩K. Here we have X0 = X−, Xm = X+, ∂Yi = Xi−1 ⊔Xi, and thesequence (Yi,Ki)i=1,...m corresponds to the decomposition

(11) Y = Y1 ∪X1Y2 ∪X2

. . . ∪Xm−1Ym, K = K1 ∪x1

K2 ∪x2. . . ∪xm−1

Km.

In the special case Y = X × [b−, b+], a cylindrical Cerf decomposition of the tangleK is a Cerf decomposition induced by a cylindrical Morse datum.


Definition 2.4.3. A cobordism-with-tangle (Y,K) is a

(a) simple cobordism-with-tangle if (Y,K) admits a Cerf decomposition with asingle piece,

(b) a compression body-with-tangle if it admits a compression body decomposi-tion with a single piece

(c) a simple tangle if (Y,K) admits a Cerf decomposition with a single piece andno critical points on Y . That is, Y is a cylindrical cobordism and fK has atmost one critical point on K.

Thus a cylindrical Cerf decomposition is a decomposition of the trivial cobordismY = [b−, b+]×X into cylindrical cobordisms Y1∪X1

. . .∪Xm−1Ym which decomposes

the tangle K = K1 ∪x1. . . ∪xm−1

Km into simple tangles (Yj ,Kj).

Remark 2.4.4. The decomposition (11) is a compression body decomposition if (a) and(c) in the definition of Morse datum are satisfied but in (b) each piece f−1((bk−1, bk))can contain several critical points of f or fK , but not both, and the indices of allcritical points are equal.

Figure 2. Cyindrical Cerf decomposition of a tangle

The following is a special case of Cerf theory, for the special case of cylindricalCerf decompositions.

Theorem 2.4.5. Let K be a tangle in Y = X × [b−, b+]. Then any two cylindricalCerf decompositions of (Y,K) are related by a finite sequence of the following movesand diffeomorphism equivalences:


(a) min-max cancellation in which two simple tangles (Yi,Ki), (Yi+1,Ki+1), whichcarry a local minimum resp. local maximum, both of which lie on the samestrand of K∩Yi∪Yi+1, are replaced by the simple tangle Yi∪Xi

Yi+1,Ki∪xiKi+1

which admits a Morse function with no critical point;(b) critical point reversal in which two simple tangles (Yi,Ki), (Yi+1,Ki+1) which

carry critical points of index k and l on strands whose intersection with(Xi, xi) is disjoint,are replaced by two simple tangles which carry criticalpoints of index l and k such that Yi ∪Xi

Yi+1 = Y ′i ∪X′

iY ′

i+1;

(c) cylinder gluing in which two simple cobordisms Yi, Yi+1, one of which is cylin-drical, are replaced by the simple cobordism Yi ∪Xi

Yi+1.

See Figures 3 and 4 for depictions of the first two moves.

Figure 3. Critical point cancellation

Proof. Let (fj , bj), j = 0, 1 be cylindrical Morse data for (Y,K). Let fs = (1−s)f0+sf1 be the linear interpolation between f0 and f1. Then ∂tfs > 0 for all s ∈ [0, 1].Now consider the family fs|K . We claim that there exist tubular neighbourhoods ofX × b± on which none of the fs|K has a critical point.

Indeed, recall that K ⊂ Y = X × [b−, b+] is a submanifold with boundary. Thismeans that there exist tubular neighbourhoods τ± : X× [0, δ) → Y of the boundaryτ±(X×0) = X×[b±] such that τ−1

± (K) = x±×[0, δ). Moreover, each fj is minimalresp. maximal on X×b± but not critical (since critical points are nondegenerate),so by choosing δ > 0 small we can ensure that τ∗±fs has positive resp. negative

derivative in the [0, δ)-coordinate (in particular along τ−1± (K)) for s = 0, 1 and

hence for all s ∈ [0, 1].So the homotopy (fs)s∈[0,1] is already of the desired form on a tubular neigh-

bourhood of ∂Y . In particular, (fs|K) has singularities or critical points only on acompact set in the interior of K. By Cerf theory for the interior int(K), there existsa smooth perturbation (hs : K → R)s∈[0,1] supported in int(K) × (0, 1) such that(fs|K+hs)s∈[0,1] is a good homotopy of functions in the sense that fs|K+hs is a Morsefunction injective on its critical set for all but finitely many values of s ∈ [0, 1], and atthose values at most one of the following happen: two critical values become coincideor a fold singularity occurs. Since K is a submanifold of Y , any such perturbationhas an extension to a smooth family of functions (hs : Y → R)s∈[0,1], supported inthe interior of [0, 1] × Y , whose derivatives are bounded by a constant times those

of (hs). We may assume that (hs) is sufficiently small so that ∂t(fs + hs)(x, t) > 0for all (x, t) ∈ Y and s ∈ [0, 1].


So this homotopy has only finite many values c1 < . . . < cn for which fcj+ hcj

does not satisfy (a-c) in Definition 2.4.5. Each of these has either a fold singularity(corresponding to critical point cancellation) or two critical values coinciding (cor-responding to reversing order of critical points) on K. The theorem then followsfrom a similar decomposition as in [33].

We expect that a similar statement holds for general Cerf decompositions ofcobordisms-with-tangles. This would lead to a construction of invariants for gen-eral cobordisms-with-tangles (Y,K). We will give the construction of Lagrangiancorrespondences in this case but will not prove invariance.

Figure 4. Critical point reversal

Given a cobordism-with-tangle (Y,K) as above we construct Lagrangian corre-spondences as follows. Let K1, . . . ,Kp be the connected components of K and fixlabels ν = (ν1, . . . , νp) ∈ Ap for K. In Section 2.2 we defined the moduli spaceM(Y,K, ν) of flat G-bundles on Y rK with holonomy around Kj in Cνj

. On the

boundary ∂Y = (X−)− ∪X+ the labels ν induce labels µ±∈ An± , given by νj for

∂Kj . Restriction to the boundary defines a map

(12) M(Y,K, ν) →M(X−, x−, µ−)− ×M(X+, x+, µ+).

More precisely, the inclusion of the boundary and a choice of paths between basepoints induces a map of fundamental groups that is well-defined up to conjugacy.This induces a dual map from the representation variety of the cobordism to theproduct of repesentation varieties of its boundary components, which is independentof the choice of path. For any cobordism-with-tangle (Y,K) and labels ν we denotethe image of (12) by

L(Y,K, ν) ⊂M(X−, x−, µ−)− ×M(X+, x+, µ+).

Lemma 2.4.6. Let (Xi, xi) be marked surfaces for i = 0, 1, 2, and let (Y01,K01) resp.(Y12,K12) be cobordisms-with-tangle from (X0, x0) to (X1, x1) resp. from (X1, x1) to(X2, x2). Let ν01 and ν12 be labels for the cobordisms with tangles such that theyinduce the same label µ

1for (X1, x1). Then gluing provides a cobordism with tangle

(Y01#Y12,K01#K12) from (X0, x0) to (X2, x2) with labels ν01#ν12. The inducedlabels µ

0for x0 and µ

2for x2 are the same as the ones induced from ν01 and ν12,

and we have the equality of subsets of M(X0, x0, µ0) ×M(X2, x2, µ2

)

L(Y01#Y12,K01#K12, ν01#ν12) = L(Y01,K01, µ01) L(Y12,K12, µ12

).


Proof. By Seifert-van Kampen we have an isomorphism (using a base point on X1)

π1(Y01#Y12 \K01#K12) ∼= π1(Y01 \K01) ⋆π1(X1\x1)π1(Y12 \K12).

In particular, any representation on Y01#Y12 \K01#K12 induces representations onboth sides, whose restriction to X1\x1 agree. Conversely, any pair of representationson the two sides, whose restrictions to X1 \ x1 are conjugate, we can conjugate oneof the sides so that the restrictions agree. This induces a representation on the gluedspace.

Proposition 2.4.7. If (Y,K) is a simple cobordism-with-tangle as in Definition2.4.1, then the moduli space L(Y,K, ν) is a smooth Lagrangian correspondence.

The proof will be based on the following special cases.

Lemma 2.4.8 (Correspondences for simple cobordisms-with-tangles).

(a) Suppose that (Y,K) admits a Morse function f with no critical points on Yor K. Then f : (Y,K) → [min f,max f ] is a ”marked” fiber bundle witha trivialization Φ : (X− × [min f,max f ], x− × [min f,max f ]) → (Y,K),where Φ|f−1(min f) = IdX−

and φ := Φ|f−1(max f) : (X−, x−) → (X+, x+)

is an isomorphism of marked surfaces. L(Y,K, µ) is the graph of (φ−1)∗ :M(X−, x−, µ−) →M(X+, x+, µ+

).

(b) Let X be a compact oriented surface, Y = X × [−1, 1] and suppose that Kcontains a single critical point that is a maximum, and so consists of n − 2strands meeting both the incoming and outgoing boundary, and one strandthat connects two incoming markings xi, xj , as in Figure 5. The projectionπ+ from L(Y,K, µ) to M(X+, x+, µ+

) is a coisotropic embedding, and the

projection π− from L(Y,K, µ) to M(X−, x−, µ−) is a fiber bundle with fiber

Cµi∼= Cµj

.(c) Suppose that Y contains a single critical point of f of index 1. The projection

π+ from L(Y,K, µ) to M(X+, x+, µ+) is a coisotropic embedding, and the

projection from L(Y,K, µ) to M(X−, x−, µ−) is a fiber bundle with fiber G.

Furthermore, in each case L(Y,K, µ) is a smooth Lagrangian correspondence in

M(X−, x−, µ−)− ×M(X+, x+, µ+).

Figure 5. A cap

Proof. The trivialization in case (a) is constructed as the flow ψt of the gradientvector field ∇f ∈ Vect(Y ) for a special metric. Since K is transverse to ker df we


can choose the metric such that ∇f is parallel to K, and then scale it such that|∇f | ≡ 1. Then Φ(x−,min f + t) = ψt(x−) gives the claimed trivialization.

Now any element of L(Y,K, µ) arises from a representation of Y \ K, which is

the pullback (Φ−1)∗ρ of a representation ρ of the trivial cylinder over X− \ x−.Since the restrictions of ρ to the two boundary components are clearly conjugate,the boundary restriction of (Φ−1)∗ρ will be the graph of pullback under φ−1 =Φ−1|X+

: X+ → X−. Moreover, φ−1 preserves the orientations and labels and henceinduces a symplectomorphism of moduli spaces M(X−, x−, µ−) → M(X+, x+, µ+

).

So L(Y,K, µ) is the graph of a symplectomorphism and hence a smooth Lagrangiancorrespondence.

In case (b), to prove that L(Y,K, µ) is a smooth, Lagrangian fiber bundle wechoose a system of generators of the punctured surfaces such that in the holonomydescription

L(Y,K, µ) =(

[a1, . . . , a2g, c1, . . . , cn+2],

[a1, . . . , a2g, c1, . . . , ci, . . . , cj , . . . , cn+2]) ∣∣ cicj = e

.

Such a system of generators can be found as follows. We pick

b+ := f(X+) > c+ > f(k0) > c− > f(X−) =: b−

such that we obtain the following normal form on f−1([c−, c+]). The gradient ∇f isparallel to all strands of K except for the one through k0, which remains the uniqueminimum of fK . Now the gradient flow induces a diffeomorphism f−1([c−, c+]) ∼=X− × [b−, b] which maps K to the union of trivial strands x− × [b−, b] with a caplinking two marked points in X− × b. Now f−1(c+) is obtained from f−1(c−) byreplacing a twice punctured disk D− by a disk D+ without punctures. We choosea system of generators for f−1(c−) that except for the generators around the i-thand j-th markings do not meet D−, and the corresponding system of generators forf−1(c+).

Since the i-th and j-st strands are connected by a cap, the holonomies around thepunctures xi, xj ∈ X+ are inverse, up to conjugacy, and hence µi = ∗µj . In thesecoordintes π− : L(Y,K, µ) → M(X−, x−, µ−) clearly is a smooth fibration. We can

identify the fiber with the anti-diagonal ∆i := (ci, c−1i ) | ci ∈ Cµi

in Cµi× C∗µi

=Cµi

× Cµjor with either of the conjugacy classes Cµi

or Cµjby projection. The

symplectic form on M(X+, x+, µ+) is given by reduction from the 2-form (9) on

G2g × Cµ+

∼= G2g × Cµ−× Cµi

× C∗µi. In the latter splitting the 2-form is

(13) ωg,µ−

+ ω0,µi,∗µi +1

2〈Φ∗

g,µ−

θ ∧ Φ∗0,µi,∗µi

θ〉,

and one can check that the second and third term vanish on G2g × Cµ−× ∆i . The

same holds after taking quotients, hence L(Y,K, µ) is isotropic and half-dimensional,thus Lagrangian. This also implies that π+ : L(Y,K, µ) → M(X+, x+, µ+

) is a

coisotropic embedding.Case (c) is similar.


Proof of Proposition 2.4.7. There are three cases to consider, depending on whethera critical point occurs in the tangle, in the ambient cobordism, or not at all. Thethird case is case (a) of the subsequent Lemma. In the first case, the critical pointmust be a maximum or minimum. Up to symmetry, this is exactly the setting ofLemma 2.4.8 (b), so the claim follows. For a critical point in the ambient cobordismthe index of the critical point must be either one or two; up to symmetry this isLemma 2.4.8 (c).

Lemma 2.4.9. Let (Y,K) be an oriented simple tangle. Then L(Y,K, µ) is simply-connected and canonically oriented. Furthermore, if the labels in µ are spin thenL(Y,K, µ) has a unique relative spin structure with background class (w2(M(X,x−, µ−)), 0)

and also a unique relative spin structure with background class (0, w2(M(X,x+, µ+))).

Proof. Suppose first that K contains a critical point of index 1, connecting thestrands marked i and j. Suppose that the orientation of xj resp. xi is the sameresp. opposite of the standard orientation of a point. We show that L(Y,K, µ)is simply-connected. By Lemma 2.4.8, L(Y,K, µ) is diffeomorphic to a Cµj

-bundleover M(X,x−, µ−). The base M(X,x−, µ−) is simply-connected by the Atiyah-Bott

method for bundles with fixed holonomy [23]. The conjugacy classes of G are simply-connected, by (1). Hence so is L(Y,K, µ).

Next we show that L(Y,K, µ) is canonically oriented. Since the base is simply-connected and the structure group of the bundle is connected, an orientation L(Y,K, µ)is induced by the symplectic orientation on the base M(X,x−, µ−) and the orienta-

tion on the fiber Cµjgiven by the volume form constructed in [2, Section 3.5].

Finally, suppose that Cµjis equivariantly spinnable. Since it is simply-connected

by (1), it has a unique spin structure. The identity mapM(X,x−, µ−) →M(X,x−, µ−)

has a canonical relative spin structure with background class w2(M(X,x−, µ−)).

The relative spin structures on the base and equivariant spin structure on thefiber induce on L(Y,K, µ) → M(X,x−, µ−) a relative spin structure with back-

ground class w2(M(X,x−, µ−)). Hence the inclusion L(Y,K, µ) →M(X,x−, µ−)−×

M(X,x+, µ+) has a relative spin structure with background class (w2(M(X,x−, µ−)), 0).

Since it is simply connected, this relative spin structure is unique.The other cases (critical point of index 0, or no critical point) are similar.

Given a Cerf decomposition (f, b) of (Y,K), let xk denote the intersection f−1K (bk)∩

K with labels µj

induced from those on K. Denote by Kj := f−1K [bj−1, bj ] the tangle

between the levels bj−1 and bj.

Definition 2.4.10 (Generalized Lagrangian correspondence for a cobordism-with–tangle). Let (f, b) be a cylindrical Cerf decomposition of the cobordism-with-tangle(Y = X × [−1, 1],K), and (Yj = f−1([bj−1, bj ]),Kj) the simple cobordisms ap-pearing in the decomposition. Let µ be a set of monotone, spin labels for the


components of K. Let L(Yj,Kj) ⊂ M(Xj−1, xj−1, µj−1)− ×M(Xj , xj , µj

) the La-

grangian submanifold of representations that extend over Kj, equipped with ori-entations and relative spin structures as in Lemma 2.4.9. Define L(Y,K, f, b) :=L(Y1,K1)# . . .#L(Ys,Ks).

The generalized correspondence L(Y,K, f, b) is a morphism from M(X,x−, µ−)

to M(X,x+, µ+) in Symp#, that is, a generalized Lagrangian correspondence with

brane structure. The relative spin structures have background classes that alternate

(. . . , 0, w2(M−1), 0), or (. . . , w2(M−2)), 0, w2(M0)).

Remark 2.4.11. One could allow compression body decompositions in the aboveformula; the Lagrangian correspondences for compression bodies are also smooth,oriented, relatively spin.

2.5. Invariance.

Proposition 2.5.1 (Invariance under Cerf moves). Let K be a tangle in Y =X × [b−, b+] and (f0, b0) and (f1, b1) cylindrical Morse data. Then L(Y,K, f0, b0)and L(Y,K, f1, b1) are equivalent as generalized Lagrangian correspondences fromM(X,x−, µ−) to M(X,x+, µ+

).

Proof. The cylindrical Cerf decompositions for f0, f1 are related by a sequence ofCerf moves described in Theorem 2.4.5. We claim that in each case the movesinvolve replacing pairs L(Yi,Ki), L(Yi+1,Ki+1) with the composition L(Yi,Ki) L(Yi+1,Ki+1). This follows from suitable equivariant versions of the results of [33].However, we prefer to give an explicit computation. Consider first the case of min-max cancellation. We may suppose that Ki is a cup connecting the strands i,i− 1,and Ki+1 is a cup connecting the strands i, i+1. In terms of the holonomies aroundthe strands a1, . . . , a2g−2 for xi−1, b1, . . . , b2g for xi and c1, . . . , c2g−2 for xi+1 wehave

L(Yi,Ki) = bi−1 = b−1i , bj = aj for j < i− 1, bj = aj−2 for j > i,

L(Yi+1,Ki+1) = bi = b−1i+1, bj = cj for j < i, bj = cj−2 for j > i+ 1.

Their composition L(Yi,Ki) L(Yi+1,Ki+1) is the diagonal in M(Xi−1, xi−1, µi−1).

The tangent space

T (L(Yi,Ki) × L(Yi+1,Ki+1) = ξi−1 = −ξi, ξ′i = ξ′i+1

intersects TM(Xj−1)×∆M(Xj)×TM(Xj+1) transversally. Invariance under criticalpoint switches is similar.

Definition 2.5.2. (Decorated Tangle Category) Let Tan2+1(d, r,X) denote the cat-egory of whose objects are collections of distinct points of a compact, oriented surfaceX with standard labels satisfying (f), see Definition 2.1.1, and whose morphisms aremarked cobordisms-with-tangles (X× [−1, 1],K) modulo diffeomorphisms fixing theboundary.


Theorem 2.5.3 (Symplectic-valued field theory for tangles). For G = SU(r), themap (X,x, µ) 7→M(X,x, µ), (Y = X× [b−, b+],K, ν) → L(Y,K, ν) defines a functor

Tan2+1(r, d,X) → Symp#1/2c.

Proof. By Proposition 2.5.1 and Lemma 2.4.9.

Remark 2.5.4. It seems likely that, by a more detailed examination of Cerf theory,one can allow simultaneously tangles and non-trivial cobordisms, but we have notchecked the details.

2.6. Extension to graphs. In this section we sketch an extension to invariants ofgraphs in trivial cobordisms. Let X−,X+ be closed, connected, oriented 2-manifoldsand let Y be an oriented compact connected cobordism from X− to X+.

Definition 2.6.1. A graph in Y consists of a graph Γ and an injection |Γ| → Y ofthe underlying topological space that restricts to embeddings on all edges, maps thevalence one vertices of Γ to the boundary of Y , the valence greater-than-one verticesof Γ to the interior of Y , and the interior of each edge to the interior of Y .

For simplicity we omit the embedding from the notation and denote by Γ ⊂ Ythe image of the given embedding. An oriented graph is a graph equipped withorientations of its edges.

In the following we will study pairs (Y,Γ) consisting of a trivial cobordism Y =X × [b−, b+] of a closed, connected, oriented 2-manifold X and an oriented graph Γin Y .

Definition 2.6.2. A cylindrical Morse datum for (Y,Γ) consists of a pair (f, b) suchthat

(a) f : Y → R is a smooth function such that(i) X × b+ resp. X × b− is the set of maxima resp. minima of f ;(ii) ∂tf(x, t) > 0 for all (x, t) ∈ Y ;(iii) f restricts to a Morse function on each edge of Γ;(iv) the restriction of f to any edge has critical points only on the interior

of the edge; and(v) f |Γ is injective on the union of the critical set of f |Γ and the set of

valence-greater-than-one vertices of Γ.(b) b = (b− = b0 < . . . < bs = b+) is an ordered subset such that each f−1(bj)

contains no critical points of f |Γ or vertices and each f−1(bk−1, bk) containsat most one critical point of f |Γ or vertex of Γ.

Definition 2.6.3. Any cylindrical Morse datum (f, b) of (Y,Γ) gives rise to acylindrical Cerf decomposition of (Y,Γ) into simple cobordisms-with-graphs (Yj :=f−1([bj−1, bj ]),Γj := Yj ∩ Γ). That is, each Yj is a cylindrical cobordism and Γj hasat most one critical point or vertex.

Theorem 2.6.4. Any two cylindrical Cerf decompositions are related by a finitesequence of

(a) min-max cancellations


Figure 6. Cerf decomposition of a graph

(b) critical point order reversals(c) critical point/vertex order reversals,(d) vertex/vertex order reversals,(e) vertex/critical point cancellations, and(f) gluing simple graphs with no critical points or vertices to adjacent simple

graphs.

See Figures 7, 8, 9.

Figure 7. Critical point/vertex switch

Figure 8. Vertex/vertex switch

Proof. The proof is similar to that of Theorem 2.4.5. The new feature is thatbecause the edges are non-compact, a critical point may run into a vertex; themove depicted in Figure 9 results. Let (fj, bj) be two Morse data for (Y,Γ). We


Figure 9. Vertex/critical point cancellation

construct a homotopy f : Y × [0, 1] → R with the properties that (i) ∂tf(y, s) > 0

for all (y, s) ∈ Y × [0, 1] and (ii) the restriction f |e of f to each edge e gives a good

homotopy in the sense that f |e is a Morse function injective on its critical set exceptfor a finite number of times s1, . . . sn ∈ [0, 1] where at most one of the followingoccur: (a) min-max cancellation occurs in the interior (b) a critical point occurs atan endpoint or (b) two critical points or endpoints have the same value. First note,that for no min-max cancellation to occur at the endpoints it suffices that for eachendpoint p and time s, either dfs|e(p) or d2fs|e(p) is non-zero. Now the edge e hasa well-defined tangent direction Tpe at p, and the set of edges meeting p is finite.It follows that therre exists a function h : Y → R supported near p with dh|e(p)non-zero on every edge e meeting p. Similarly, we can find g : Y → R supported nearp with d2g|e(p) non-zero on every edge e meeting p. Let f : X× [0, 1] → [0, 1] be the

height function f(x, t) = t. A generic time-dependent combination fpre of f, g, h hasonly non-degenerate critical points at p. Now as Cerf theory a generic perturbationf of fpre will have only fold singularities on the interiors of the edges, and distinctvalues on the critical points and vertices, which completes the proof.

Suppose that Γ is equipped with a set of labels ν. We wish to associate toit a generalized Lagrangian correspondence. For this, a certain condition on thelabels at each vertex must be satisfied. For each j, let νj denote the labels ofthe edges of Γj , induced by those of Γ. If Γj has no vertex, then (Yj ,Γj) is asimple tangle, and we associate to it the Lagrangian correspondence L(Yj ,Γj, νj)as in Section 2.4. Furthermore, we associate to any simple graph (Y,Γ) with asingle vertex v a Lagrangian correspondence, as follows. Denote the boundary ofY by ∂Y = X− ∪X+. Let B(v) ⊂ Y be a small open ball containing v, and S(v)the boundary of B(v), oriented as an outgoing boundary component of B(v). Thecomplement YrB(v) of B(v) can be viewed as a three-dimensional cobordism fromX− ∪ S(v) to X+, containing a tangle

K := Γ\(B(v) ∩ Γ).

Let µ±

denote the labels for x± = K ∩X±. Let

x(v) := S(v) ∩ Γ

denote the intersection of S(v) with the graph Γ. Let µ(v) denote the set of labelsfor x(v), given by the labels of the edges meeting the vertex (or their image under(2) if the induced orientation differs from the standard orientation of a point).


Definition 2.6.5. We say that µ(v) is a vertex-admissible collection if the modulispace of flat bundles on the punctured sphere M(S(v), x(v), µ(v)) is either empty ora point.

Recall that βj denotes the j-th fundamental coweight of SU(r). We denote byj = βj/2.

Lemma 2.6.6. Suppose that G = SU(r+ 1). The labels (1,1, ∗2) and (r, r, ∗r − 1)are vertex-admissible.

Proof. The product C21

has symplectic quotient at λ isomorphic to the moduli spaceof flat bundles with labels (1,1, ∗λ), see [1, Section 9]. The moment polytope con-tains all points of the form w11+w21 ∈ A, w1, w2 ∈ A, and in particular, β1 = 1+1

and 2 = (β1 + s12β1)/2. A dimension count shows that the moment polytope is theconvex hull of β1 and 2, and that the symplectic quotients are points, which provesthe claim.

The triple above is analogous to Khovanov-Rozansky’s 1,2 (or thin, thick) mark-ings [12].

Definition 2.6.7. Let M(Y,Γ, ν) denote the moduli space of flat bundles on thecomplement of Γ in Y with holonomies around the edges of Γ given by ν. Restrictionto the boundary defines a map

(14) M(Y,Γ, ν) →M(X−, x−, µ−)− ×M(X,x+, µ+).

Here M(S(v), x(v), µ(v))− is a point by assumption. We denote the image of (14)by L(Y,Γ, ν).

Lemma 2.6.8. Let (Y,Γ) be a simple graph containing a single vertex and µ(v) avertex-admissible labelling of the edges of Γ. Then L(Y,Γ, ν) is a smooth Lagrangiancorrespondence from M(X,x−, µ−) to M(X,x+, µ+

).

Proof. This follows from equivariant versions of the results of [33], since the inclusionof the boundary component with the most labels induces a surjection of fundamentalgroups. We check it explicitly in the holonomy description. We write µ

±rµ(v) resp.

µ±∩ µ(v) for the labels of those markings in x± that are not resp. are connected to

v by an edge. By (9), the symplectic forms on the two ends are those obtained byreduction from

(15) ωg,µ±rµ(v) + ω0,µ

±∩µ(v) +

1

2〈Φ∗

g,µ±rµ(v)θ ∧ Φ∗

0,µ±∩µ(v)θ〉.

Let d be the value of f at the vertex and ǫ a small number. The level sets f−1(d−ǫ), f−1(−1) are isomorphic, by the flow used in the proof of Lemma 2.4.8. Choosea presentation for the fundamental group of f−1(d− ǫ); then a presentation for thefundamental group of f−1(d − ǫ) is obtained by replacing the generators for thestrands incoming to the vertex with those outgoing. With respect to this set ofgenerators, the correspondence defined by the cobordism is

(16)∏

µ∈µ−∩µ(v)

cµ =∏

µ∈µ+∩µ(v)

cµ


and descending to the quotient. Now this equation defines an isotropic submanifoldof C−

µ−∩µ(v) × Cµ

+∩µ(v) since the moduli space for the sphere around the vertex is

a point. It follows from (15) that the (16) defines an isotropic, hence Lagrangiansubmanifold of the product M(X−, x−, µ−)− ×M(X+, x+, µ+

).

Lemma 2.6.9. Let Γ be a simple graph containing a single vertex with incomingmarkings 1,1 and outgoing markings 2. Then L(Γ) is a P1-bundle overM(X−, x−, µ−).

In particular, L(Γ) is simply-connected and admits unique relative spin structureswith background classes (w2(M(X−, x−, µ−)), 0) and (0, w2(M(X+, x+, µ+

))).

Proof. This will be proved in [32].

Definition 2.6.10. An admissible labelling of Γ is a labelling of the edges of Γby admissible labels, such that at each vertex the collection of labels is vertex-admissible. An admissible graph is a graph equipped with an admissible labelling.A standard labelling of Γ is a labelling of each edge by 1 or 2, so that each vertex istrivalent with labels 1,1,2.

The following associates a generalized Lagrangian correspondence to any graphwith admissible labelling:

Definition 2.6.11 (Generalized Lagrangian correspondence for a decorated graph).Let (f, b) be a cylindrical Cerf decomposition of Γ equipped with vertex-admissible,monotone, spin labels µ. Let L(Yj ,Γj , νj) ⊂ M(Xj−1, xj−1, µj−1

)− ×M(Xj , xj, µj)

denote the Lagrangian submanifold of representations that extend over Γj. Define

L(Y,Γ, f, b, ν) := L(Yr,Γr, νr)# . . .#L(Y1,Γ1, ν1).

Proposition 2.6.12 (Invariance under Cerf moves). Let (Γ, ν) be an admissibledecorated graph in Y = X × [−1, 1] and (f0, b0) and (f1, b1) cylindrical Morse data.Then L(Y,Γ, f0, b0) and L(Y,Γ, f1, b1) are equivalent as generalized Lagrangian cor-respondences from M(X,x−, µ−) to M(X,x+, µ+

).

Definition 2.6.13 (Category of decorated graphs). Let Graph2+1(d, r,X) denotethe category of whose objects are collections of distinct points of a compact, orientedsurface X with standard labels satisfying (f), see Definition 2.1.1, and whose mor-phisms are equivalence classes of cylindrical cobordisms-with-graphs (X× [−1, 1],Γ)equipped with labellings by 1,2, such that the labels at each vertex are 1,1,2. Andiffeomorphism of marked cobordisms with graphs is a diffeomorphism of cobordisms-with-graphs acting trivially on the boundary.

The following extends Theorem 2.5.3 to graphs.

Theorem 2.6.14 (Symplectic-valued field theory for graphs). For G = SU(r), the

map (X,x, µ) 7→M(X,x, µ) extends to a functor Graph2+1(r, d,Σ) → Symp#1/2c.

Proof. By Proposition 2.5.1 and Lemma 2.6.9.


3. Derived Floer theory

In this section we describe a framework for Floer theory for Lagrangians withminimal Maslov number two. Namely, even if the Floer differential ∂ for a cyclic gen-eralized Lagrangian correspondence L does not square to zero, the object DF (L) :=(CF (L), ∂) in the derived category of matrix factorizations 2 is independent of allchoices up to isomorphism. In our application to SU(r) knot Floer cohomology in[33], the invariant associated to a trivalent graph will be an object in such a derivedcategory, and the language is chosen to make it match up with that in Khovanov-Rozansky [12]. Even in the case that the differentials have vanishing square, workingin the derived category has certain advantages. For example, it makes duals andtensor products work the way they should. We emphasize that the version of thederived category needed here is not something deep, but essentially only a questionof language. We also emphasize that the derived category construction discussedhere is separate from the derived category construction applied by Kontsevich toFukaya’s A∞ category.

3.1. Matrix factorizations. We define categories of matrix factorizations as fol-lows, see e.g. [25, p.17].3

Definition 3.1.1. For any w ∈ Z, let Fact(w) denote the category of factorizationsof w Id.

(a) The objects of Fact(w) consist of pairs (C, ∂), where(i) C is a Z2-graded free abelian group C = C0 ⊕ C1;(ii) ∂ is a group homomorphism ∂ : C• → C•+1, satisfying ∂2 = w Id.

(b) For objects (C, ∂) and (C ′, ∂′), the space of morphisms HomFact((C, ∂), (C ′, ∂′))is the space of grading preserving maps f : C• → (C ′)• such that f∂ = ∂′f .

Given an object (C, ∂) ∈ Obj(Fact(w)), there exists a dual object (C, ∂)∨ =(C∨, ∂∨), where C∨ = Hom(C0,Z)⊕Hom(C1,Z) and ∂∨ is the dual of ∂. Similarlyfor a morphism f : (C, ∂) → (C ′, ∂′) we obtain a dual morphism f∨ : (C ′, ∂′)∨ →(C, ∂)∨. Thus we obtain a contravariant dualization functor

Fact(w) → Fact(w), (C, ∂) 7→ (C, ∂)∨.

Similarly, there is a covariant tensor product functor

Fact(w1)×Fact(w2) → Fact(w1+w2),((C1, ∂1), (C2, ∂1)

)7→ (C1⊗C2, ∂1⊗Id+Id⊗∂2).

Here ⊗ is the graded tensor product, so that (Id⊗ ∂2)(x1 ⊗ x2) = (−1)|x1|x1 ⊗ ∂2x2

for homogeneous x1 ∈ C|x1|1 .

For any matrix factorization (C, ∂) let H((C, ∂) ⊗Z Zw) denote the cohomologyof the differential ∂ ⊗Z Id : C ⊗Z Zw → C ⊗Z Zw obtained from ∂ by tensoringwith Zw. Any morphism in Fact(w) defines a homomorphism of the corresponding

2More standard notation would simply be CF (L). However, we find this notation very confusingin this setting.

3The assumption that C0, C1 are free avoids the more complicated localization procedure usedin e.g. Hartshorne [9].


cohomology groups, and so we have a cohomology with coefficients functor to thecategory Ab of Z2-graded abelian groups,

(17) Fact(w) → Ab, (C, ∂) 7→ H((C, ∂) ⊗Z Zw).

Definition 3.1.2. (a) A morphism f : C → C is called null-homotopic if thereexists a map h : C• → C•−1 such that f = h∂ + ∂h.

(b) The derived category of matrix factorizations DFact(w) is the category withthe same objects as Fact(w), and morphisms given by the quotient of Hom(Fact(w))by null-homotopic morphisms.

(c) The trivial object inDFact(w) is the trivial complex C0 = C1 = 0 equippedwith the trivial differential ∂. (Note that ∂2 = w Id, for any w.)

Remark 3.1.3. (a) DFact(w) is naturally a triangulated category, with distin-guished “exact” triangles given by the mapping cone construction: Given amorphism of matrix factorizations f : (C1, ∂1) → (C2, ∂2), its mapping coneis the factorization

Cone(f) :=

(C1[1] ⊕ C2,

(−∂1 f0 ∂2

)).

The exact triangles in DFact(w) are by definition those isomorphic to trian-gles

. . .→ C1 → C2 → Cone(f) → C1[1] → . . . .

In particular, if C1f→ C2 → C3 → C1[1] is an exact triangle then C3 is

(non-canonically) isomorphic to the mapping cone on f . The proofs are thesame as for the case w = 0 of complexes, see e.g. [6].

(b) The cohomology with coefficients functor (17) factors through the derivedcategory to give a functor DFact(w) → Ab. Any exact triangle in DFact(w)gives rise to a long exact sequence of cohomology groups with coefficients inZw.

3.2. Derived Floer theory for a pair of Lagrangians. The following is astraightforward extension of results of Oh [24] to cyclic generalized Lagrangian cor-respondences. Let D ⊂ C be the unit disk and fix the base point 1 ∈ ∂D. Let (M,ω)be a compact monotone symplectic manifold and L ⊂M an oriented monotone La-grangian submanifold. That is, we assume (M1-2) and (L1-2) with τ ≥ 0 but not(L3). (Note that, by convention, (L3) is always satisfied in the exact case τ = 0.)

For any J ∈ J (M,ω) and submanifold X ⊂ L, let M21(L, J,X) denote the moduli

space of J-holomorphic disks u : (D,∂D) → (M,L) with Maslov number 2 and onemarked point satisfying u(1) ∈ X, modulo automorphisms of the disk fixing 1 ∈ ∂D.

Proposition 3.2.1. For any ℓ ∈ L there exists a subset J reg(ℓ) ⊂ J (M,ω) of Bairesecond category such that M2

1(L, J, ℓ) is a finite set. Any relative spin structureon L induces an orientation on M2

1(L, J, ℓ). Letting ǫ : M21(L, J, ℓ) → ±1

denote the map comparing the given orientation to the canonical orientation of apoint, the disk number of L,

w(L) :=∑

u∈M21(L,J,ℓ)

ǫ(u),


is independent of J ∈ J reg(ℓ) and ℓ ∈ L.

Proof. First, we prove that for generic J and a generic point m ∈ M there areno J-holomorphic spheres with Chern number one passing through m. For anysubmanifold X ⊂ M and almost complex structure J ∈ J (M,ω) let M1

1(M,J,X)denote the moduli space of J-holomorphic maps u : P1 → X with Chern number oneand u(0) ∈ X, modulo holomorphic automorphisms of P

1 fixing 0 ∈ P1. Standard

arguments using the Sard-Smale theorem for the universal moduli space show thatfor J in a subset J reg

sphere(X) ⊂ J (M,ω) of Baire second category, the moduli space

M11(M,J,X) is a smooth manifold of dimension dim(X) − 2. (Because the Chern

number is minimal by monotonicity, multiple covers are impossible, so every mapu in the universal moduli space is somewhere injective.) Similarly a parametrized

version shows that for a subset J regsphere(X) of homotopies Jtt∈[0,1] of Baire second

category, the parametrized moduli space M11(M, Jtt∈(0,1),X) is a smooth manifold

of dimension dim(X) − 1. In particular, if X is a finite subset of M then both theordinary and parametrized moduli spaces are empty.

Next, consider an oriented submanifold X ⊂ L. Another standard argumentshows that for J in a subset J reg

disk(L,X) ⊂ J (M,ω) of Baire second category themoduli space M2

1(L, J,X) is a smooth manifold of dimension dim(X). (The resultsof [17, 16] produce from a J-holomorphic disk that is not somewhere injective asomewhere injective disk of lower energy. By monotonicity and minimality of theMaslov number, this is impossible and so every map u in the universal moduli spaceis somewhere injective.)

Given a relative spin structure on L one obtains an orientation on M21(L, J,X) as

follows. First, let M21(L, J,X) be the moduli space of parametrized J-holomorphic

maps u : (D,∂D) → (M,L) with Maslov index 2 and u(1) ∈ X. As explainedin [5, 34], the orientation on X and relative spin structure on L induce an ori-

entation on M21(L, J,X). To obtain an orientation on the quotient M2

1(L, J,X) =

M21(L, J,X)/Aut(D,∂D, 1) it suffices to define an orientation on the automorphism

group. For that purpose we identify D\1 with the half-space H = z ∈ C, Im(z) ≥0. Then we have Aut(D,∂D, 1) ∼= (0,∞) × R, where (0,∞) acts by dilations onH and R acts by translations on H. The standard orientations on the two factorsinduce the orientation on Aut(H).

Now suppose that X = ℓ is a point. Then for J in

J reg(ℓ) := J regsphere(ℓ) ∩ J

regdisk(L, ℓ)

the moduli space M21(L, J, ℓ) is compact. (For if a sphere or disk bubbled off,

then the remaining principal component of the disk would be constant and carrythe marked point, therefore taking values in ℓ. In the case of a sphere bubble thisis impossible since M1

1(M,J, ℓ) is empty. The case of a disk bubble is excludedsince this configuration is not stable. ) This moduli space is moreover oriented andzero dimensional, hence the sum in the definition of w(L) is well defined. To seethat w(L) is independent of ℓ we consider the moduli space M2

1(L, J, γ((0, 1))) foran embedded path γ : [0, 1] → L from γ(0) = ℓ0 to γ(1) = ℓ1. This is a smooth, ori-ented 1-dimensional manifold and a compactness argument similar to the one above


shows that M21(L, J, γ((0, 1))) gives an oriented cobordism from M2

1(M,L, ℓ0) toM2

1(M,L, ℓ1), which shows that w(L) is the same for either choice. Similarly, theindependence of w(L) from J follows from an oriented cobordism that is providedby the parametrized moduli space M1

1(L, Jtt∈(0,1), ℓ) associated to a generichomotopy from J0 to J1.

We will now extend the definition of quilted Floer cohomology, using the setup of[35] but dropping the assumption (L3).

Theorem 3.2.2. Let L = (Lj(j+1))j=0,...,r be a cyclic generalized Lagrangian corre-spondence. Suppose that the underlying symplectic manifolds Mj satisfy (M1-2) withthe same value of the monotonicity constant τ ≥ 0, the Lagrangian correspondencesLj(j+1) satisfy (L1-2), and L satisfies the monotonicity assumption of [35] and isrelatively spin. Then, for any H ∈ Ham(L), widths δ = (δj > 0)j=0,...,r, and forJ in a subset J reg

t (L,H) ⊂ Jt(L) of Baire second category, the Floer differential∂ : CF (L) → CF (L) satisfies

∂2 = w(L) Id, w(L) =

r∑

j=0

w(Lj(j+1)).

The image DF (L) of (CF (L), ∂) in DFact(w(L)) is independent of the choice of Hand J , up to isomorphism.

Proof. We sketch the proof, following Oh in the case of Z2 coefficients. For anyx± ∈ I(L), the zero dimensional component M(x−, x+)0 of Floer trajectories is afinite set. From part (a) of that theorem we also know that the one-dimensionalcomponent M(x−, x+)1 is smooth, but the “compactness modulo breaking” in part(c) does not hold in general: Apart from the breaking of trajectories, a sequence ofFloer trajectories of Maslov index 2 could in the Gromov compactification convergeto a constant trajectory and either a sphere bubble of Chern number one or adisk bubble of Maslov number two. All other bubbling effects are excluded bymonotonicity. Thus failure of “compactness modulo breaking” can occur only whenx− = x+.

The subset J regt (L;H) consists of those time-dependent almost complex struc-

tures Jj : [0, δj ] → J (Mj , ωj) for which all M(x−, x+) are smooth and the uni-versal moduli spaces of spheres M1

1(Mj , Jj(t)t∈[0,δj ], xj) are empty for all x =

(xj)j=0,...,r ∈ I(L). This excludes the Gromov convergence to a constant trajec-tory and a sphere bubble. We can now restrict to those J ∈ J reg

t (L;H) such thatJj(j+1) := (−Jj(δj)) ⊕ Jj+1(0) ∈ J reg(Lj(j+1), (xj , xj+1)) for all x ∈ I(L) andj = 0, . . . , r. This still defines a subset in Jt(L) of Baire second category. We claimthat now each one-dimensional moduli space M(x, x)1 of self-connecting trajectorieshas a compactification as a one-dimensional manifold with boundary

∂M(x, x)1 ∼=⋃

y∈I(L)

(M(x, y)0×M(y, x)0

)∪

⋃

j=0,...,r

M21(Lj(j+1), Jj(j+1), (xj , xj+1))

−

and that furthermore the orientations on these moduli spaces induced by the relativespin structures are compatible with the inclusion of the boundary. Here M2

1(. . .)−


denotes the moduli space M21(. . .) with orientation reversed. The proof of the claim

uses a gluing theorem of non-transverse type for pseudoholomorphic maps withLagrangian boundary conditions, which can be adapted from [19, Chapter 10] asfollows: We replace L with its translates under the Hamiltonian flows of H, thenthe Floer trajectories are unperturbed Jj-holomorphic strips (where the Jj havesuffered some Hamiltonian transformation, too).

Pick vj(j+1) ∈ M21(Lj(j+1), Jj(j+1), (xj , xj+1)), then the gluing construction

gives maps

(18) (T,∞) × (−T−1, T−1) −→ M2(x, x)

to the moduli space of parametrized Floer trajectories of index 2. This constructionidentifies vj(j+1) with a map vj(j+1) : H → M−

j × Mj+1 on the half space H ∼=

D \ 1; then for (τ, σ) ∈ (T,∞) × (−T−1, T−1) it shifts this map by σ and outside

of a half disk of radius 12τ

1/2 around 0, interpolates it to the constant solution

(xj , xj+1) outside of the half disk of radius τ1/2 (using a slowly varying cutoff functionin submanifold coordinates of Lj(j+1) ⊂ M−

j × Mj+1 near (xj , xj+1)). Then it

rescales this map by τ to a half-disk of radius τ−1/2 centered around 0 in the stripR × [0, τ−1/2), again extended constantly. This gives rise to two maps (u′j , uj+1) :

R× [0, τ−1/2) →M−j ×Mj+1. Here uj+1 is approximately Jj+1-holomorphic, and we

reflect uj(s, t) := u′j(s, δj − t) to obtain an approximately Jj-holomorphic map uj :

R × (δj − τ−1/2, δj ] →Mj . For T ≥ maxδ−2j , δ−2

j+1 these strips can be extended to

width δj resp. δj+1 and together with the constant solutions uℓ ≡ xℓ for ℓ 6∈ j, j+1form a tuple of maps u = (uℓ : R× [0, δℓ] →Mℓ)ℓ=0,...,r that is an approximate Floertrajectory. An application of the implicit function theorem gives an exact solutionfor T sufficiently large.

It remains to examine the effect of the gluing on orientations. The gluing con-struction can equivalently be described by viewing the domain (T,∞)×(−T−1, T−1)of the gluing map as a subset of the automorphism group Aut(D,∂D, 1) ∼= (0,∞)×R

and identifying it with its image

U :=((T,∞) × (−T−1, T−1)

)· vj(j+1) ⊂ M2

1(Lj(j+1), Jj(j+1), (xj , xj+1))

on vj(j+1). The resulting map U → M2(x, x) is simply the parametrized gluing map(with fixed gluing parameter) on pairs of disks (after capping off the strip-like endsof the constant solution) and so orientation preserving by the definitions of [34]. Nowthe infinitesimal translation action of (−T−1, T−1) on U approximately agrees under

the gluing with the inverse of the infinitesimal translation action on M2(x, x). So,after quotienting by the translations the gluing map induces an embedding (T,∞) →

M(x, x)1 = M2(x, x)/R for sufficiently large T , and we see that this embedding isorientation reversing. Summing over the boundary of the one-dimensional manifold∂M(x, x)1 thus proves ∂2 −

∑rj=0w(Lj(j+1)) Id = 0.

The proof that the image of (CF (L), ∂) in the derived category of matrix fac-torizations is independent of all choices up to isomorphism is essentially the same


as that in [35], which produces a pair of chain maps whose compositions are nullhomotopic.

Remark 3.2.3. In the special case L = (L0, L1) of a cyclic correspondence consistingof two Lagrangian submanifolds L0, L1 ⊂ M we have w(L) = w(L0) − w(L1). Thisis since the −J1-holomorphic discs with boundary on L1 ⊂M−×pt. are identifiedwith J1-holomorphic discs with boundary on L1 ⊂M via a reflection of the domain,which is orientation reversing for the moduli spaces.

Definition 3.2.4. Let L be a cyclic generalized Lagrangian correspondence as inTheorem 3.2.2.

(a) We define the derived Floer factorization DF (L) to be the image of the Floermatrix factorization (CF (L), ∂) in Obj(DFact(w(L)).

(b) We define the Floer cohomology with coefficients in Zw, w := w(L) to be theimage

HF (L; Zw) := H((CF (L), ∂) ⊗Z Zw

),

of (CF (L), ∂) under the cohomology with coefficients functor (17).

Remark 3.2.5. (a) In the case w = w(L) = 0 this definition coincides with theusual definition of Floer cohomology with Z coefficients in the sense that thefunctor taking cohomology with Z coefficients from D Fact(0) to the categoryof finitely generated Z2-graded abelian groups induces a bijection betweenisomorphism classes of objects.

(b) Theorem 3.2.2 and Remark 3.1.3 show that the Floer cohomology HF (L; Zw)is independent of all choices up to isomorphism in the category of abeliangroups.

(c) The differential for a monotone pair L = (L,ψ(L)) with any symplecto-morphism ψ ∈ Symp(M) always squares to zero, since w(L) = w(ψ(L)) byProposition 3.2.1.

Remark 3.2.6. Duals and tensor products behave as expected for the derived Floerinvariants. (Note that even in the case of Maslov number at least three, the derivedinvariants have better properties than the Floer homology in this respect.)

(a) Suppose that (L0, L1) is a monotone, relatively spin pair of compact, orientedLagrangian submanifolds. Then DF (L0, L1) is canonically isomorphic toDF (L1, L0)

∨. To see this we use the canonical identification of the generatorsof CF (L0, L1) and CF (L1, L0). Then ∂L0,L1

is given by the transpose ∂TL1,L0

of ∂L1,L0. The bijection between trajectories is given by rotating the strip

with boundary L0, L1 by 180 degrees.(b) Suppose that L0

0, L01 ⊂ M0 and L1

0, L11 ⊂ M1 are monotone, relatively spin

pairs of compact, oriented Lagrangian submanifolds in compact. ThenDF (L00×

L10, L

01×L

11) is canonically isomorphic to DF (L0

0, L01)⊗DF (L1

0, L11). The right

hand side is the matrix factorization whose differential is the graded tensorproduct ∂L0

0,L0

1⊗ Id + Id ⊗ ∂L1

0,L1

1.


In our main Theorem in [35] the assumption (L3) on the minimal Maslov numberwas needed only for the definition of the Floer cohomologies. We thus obtain thefollowing generalization.

Theorem 3.2.7. Let L = (L01, . . . , Lr(r+1)) be a cyclic generalized Lagrangiancorrespondence satisfying (M1-2) and (L1-2) with the same monotonicity constantτ ≥ 0, and such that L is monotone, relatively spin, and graded. Suppose that forsome 1 ≤ j ≤ r the composition L(j−1)j Lj(j+1) is embedded and is monotone

in the sense of (L1) and such that the modified sequence L′ := (L01, . . . , L(j−1)j Lj(j+1), . . . , Lr(r+1)) is monotone.

Then, with respect to the induced relative spin structure, orientation, we havew(L) = w(L′) =: w and there exists a canonical isomorphism in DFact(w) betweenthe derived objects DF (L) and DF (L′), induced by the canonical identification ofintersection points.

Proof. The bijection between the trajectory spaces for small widths and for thecomposed Lagrangian correspondence in [35] only requires that the minimal Maslovnumber of the Lagrangians is at least two (which is automatic in the monotone ori-entable case). The comparison of orientations in [34] is also independent of Maslovindices, hence the morphism f : CF (L) → CF (L′) given by the canonical identifi-cation of intersection points satisfies f ∂ = ∂′ f , where ∂ and ∂′ are the Floerdifferentials on CF (L) resp. CF (L′). Now, for any x ∈ I(L) we obtain

w(L)f(〈x〉) = f(∂∂〈x〉) = ∂′∂′f(〈x〉) = w(L′)f(〈x〉).

Unless the generalized intersection is empty (and hence both complexes are trivial)this implies w(L) = w(L′), and hence f is a morphism in Fact(w) for w := w(L).Similarly, the inverse f−1 : CF (L′) → CF (L) satisfies ∂′f−1 = f−1∂ and hence isanother morphism in Fact(w), inverse to f . So f defines an isomorphism betweenthe derived Floer factorizations DF (L) and DF (L′).

3.3. Derived relative invariants. Given a quilted surface S, a collection M ofcompact, monotone symplectic manifolds (satisfying (M1-2) with a fixed constantτ ≥ 0) and a collection L of compact, oriented, monotone Lagrangian boundaryconditions that satisfy (L1-2) and are monotone and relatively spin in the sense of[35] one obtains a derived invariant

DΦS :⊗

e∈E+(S)

DF (Le) →⊗

e∈E−(S)

DF (Le),

where the derived objects are the images of the corresponding chain groups in thederived category of matrix factorizations. The proof is exactly the same as in theFloer cohomology case, since in fact we used the assumption on the minimal Maslovnumber only to make the Floer cohomology groups well-defined. Note in particularthat the tensor products of derived matrix factorizations are elements of DFact(w)with the same w for incoming and outgoing ends. This is since each noncompactseam σ of S connects either an incoming and an outgoing end, contributing w(Lσ) toboth sides, or it connects two ends of the same kind, contributing w(Lσ)+w(Lt

σ) = 0to one side.


Example 3.3.1. (a) Given an admissible Lagrangian L ⊂M one obtains an iden-tity morphism IL : Z → DF (L,L), where Z is the trivial complex in degree 0.The differential for CF (L,L) automatically squares to zero by Remark 3.2.5,so after passing to cohomology the identity morphism IL induces the identityobject 1L ∈ HF (L,L).

(b) Given an admissible triple L0, L1, L2 ⊂ M of Lagrangians one obtains aderived composition morphism

Dµ2 : DF (L0, L1) ⊗DF (L1, L2) → DF (L0, L2)

in DFact(w) for w = w(L0)−w(L1) +w(L1)−w(L2) = w(L0)−w(L2) Thederived composition morphism is also associative.

3.4. Donaldson-Fukaya category of Lagrangians. Let (M,ω) be a compact,monotone symplectic manifold, equipped with a Maslov cover LagN (M) → Lag(M)and background class b ∈ H2(M,Z2). One can define a category-like structureDon(M) by taking as objects the set of Lagrangian branes as in [35], but withoutthe assumption on the minimal Maslov number. To any pair of objects (L0, L1),the morphism object Hom(L0, L1) := DF (L0, L1) is an object in the disjoint unioncategory

DFact =∐

w∈Z

DFact(w).

That is, Obj(DFact) is the disjoint union of the objects of DFact(w), and there areno morphisms between objects with different values of w. Composition is given byDµ2.

Don(M) might be called a category enriched in the derived category of matrixfactorizations, except that the morphism object is not a set; or a category object inthe derived category of matrix factorizations, except that only the morphisms areobjects in this category. The results of [35] hold with appropriate modifications ofcategories to categories enriched in DFact. In particular, one can define an enriched2-category whose objects are compact monotone symplectic manifolds, morphismsare Lagrangian correspondences, and for each pair of 1-morphisms we have as 2-morphisms an object of DFact. The standard representation becomes a 2-functorto the 2-category whose objects are categories enriched in DFact, 1-morphisms arefunctors, and 2-morphisms are natural transformations.

The matrix factorizations here appear in [5] in the following guise. A weak A∞

category consists of a collection of objects and composition maps µd, d ≥ 0, satisfyingthe A∞ associativity relations modified to include an operation µ0 which assigns toany object X an element µ0(X) ∈ Hom(X,X). The first A∞ relation is (µ1)

2 =µ2(µ0 ⊗ 1 − 1 ⊗ µ0). Special features of the Fukaya category in the monotone caseare that µ2(µ0 ⊗ 1 − 1 ⊗ µ0) is a multiple of the identity morphism for any objectX, and a homotopy of perturbation data produces a chain homotopy for µ1, due tolack of disk bubbling.


4. Category-valued field theory

In this section we construct a field theory for tangles, which associates to anymarked surface with odd number of markings a category enriched in the derivedcategory of matrix factorizations FCat, and apply it to generalize a result of Seidel onDehn twists in moduli spaces of flat bundles. The construction itself is an immediateconsequence of Theorem 2.6.14 and the main result of [35]:

Theorem 4.0.1. For r, d coprime, the maps

(X,x, µ) 7→ Don#(M(X,x, µ)), (Y,K, µ) → Φ(L(Y,Γ, µ))

define functors

Tan2+1(r, d,X) → FCat, Graph2+1(r, d,X) → FCat .

As in [33] there is a version of Floer’s excision property:

Proposition 4.0.2. Let K be a morphism from x− to x+ in Graph2+1(r, d,X). Letc ∈ [−1, 1], x = K ∩ (X ×c), K− = K ∩ (X × [−1, c]), and K+ = K ∩ (X × [c, 1]).Then

(19) Φ(K) = Φ(K−) Φ(K+).

If x is order one, then the functor Φ(K) is trivial. If x is order three, then

(20) Hom(Φ(K)L−, L+) ∼= Hom(Φ(K−)L−,pt) ⊗ Hom(Φ(K+) pt, L+).

Here the Donaldson-Fukaya category of the empty set has a single object, theempty Lagrangian, and the Hom spaces are trivial. By the trivial functor, we meanthe functor that associates to any object the empty set, and to any morphism thetrivial element of the trivial group. The proof is the same as that of the excisionproperty in [33], and will be omitted.

The theory above does not apply to links, which have even numbers of intersectionpoints with the level sets X × c for generic c. We define invariants of links in S3

by the following trick suggested to us by Seidel. Suppose K is a link in S3 withcomponents labelled by standard conjugacy classes in SU(r). Let Ir+1 ⊂ S2×[−1, 1]to be the tangle with r+1 trivial strands labelled by 1 := β1/2. Let (S3,K)#(S2 ×[−1, 1], Ir+1) denote the connect sum of (S3,K) with (S2×[−1, 1], Ir+1), as in Figure10 for the case r = 2.

Definition 4.0.3. For any link K ⊂ S3, define the sphere compactified Floer matrixfactorization (CF sc(K), ∂K) to be the matrix factorization (CF (Φ((S3,K)#(S2 ×[−1, 1], Ir+1)) pt,pt), ∂(Φ((S3 ,K)#(S2×[−1,1],Ir+1)) pt,pt)).

Consider the case that all components of the link have the same marking ν.Let M2n+r+1(S

2, x, νn, (∗ν)n,1r+1) denote the moduli space of flat bundles withfixed holonomies, where superscript n denote n-fold repetition etc. Consider theLagrangian submanifolds

L = [g1, . . . , g2n+r+1] ∈M(S2, x, νn, (∗ν)n,1r+1), gjg2n+1−j = 1, j = 1, . . . , n


Figure 10. Adding three trivial strands

Figure 11. A distinguished crossingless matching

Proposition 4.0.4. Suppose that K is the braid closure of an element β ∈ B2n.Then there exists an isomorphism of matrix factorizations

(CF sc(K), ∂K) ∼= (CF sc(L, (β × 1n)(L)), ∂(L,(β×1n)(L))).

Proof. Take a cylindrical Cerf decomposition of K given by a Morse function withthe property that all index 0 critical points occur before the index 1 critical points.Suppose that c1, . . . , cn are the index 0 critical values. Since the composition Ln . . . L1 is smooth and embedded, we can use it to compute the Floer homology,by [35, Theorem 7.2.6]. Applying the same result to the index one critical pointscompletes the proof.

Corollary 4.0.5. The coboundary ∂K for CF sc(K) has ∂2K = 0.

Proof. The disk invariant w(L) for L is equal to that for β(L), see Section 7 of [35].Hence ∂2

(L,β(L)) = 0. It follows that ∂2K = 0, by Proposition 4.0.4.

Let HF sc(K) denote its homology, the SU(r) sphere-compactified Floer homologyof K.

Example 4.0.6 (The unknot). Take the cylindrical Cerf decomposition of the unknotO consisting of a cup and cap. We have L = g1g2 = e. Since L is diffeomorphic toCP r−1, which admits a Morse function with only even indices, we have as explainedin Pozniak [26] HF sc(K) = HF (L,L) = H(CP r−1,Z).

Kronheimer-Mrowka [15] investigate the coincidence with Khovanov-Rozanskyhomology in greater detail.


4.1. Application to symplectic mapping class groups. Recall from (2.3.3)that the action of the orientation-preserving mapping class group of X induces ahomomorphism ϕX : Map+(X) → Map(M(X), ω) where Map+(X) denotes thegroup of orientation-preserving diffeomorphisms up to isotopy. Specializing to thecase that X is a sphere with n marked points, Map+(X) is known as the sphericalbraid group and denoted Bn.

We denote by Mn(µ) the moduli space M(X,x, µ) of flat bundles on the sphereX with a set of markings x of order n, G = SU(2), and all labels equal to µ. Recallthat any Fano surface is isomorphic to one of the del Pezzo surfaces dPn, obtainedby blowing up P2 at n < 9 points.

Proposition 4.1.1. M5(µ) is isomorphic, as a smooth variety, to dP4 for µ < 15 ,

and dP5, for 15 < µ < 2

5 . For µ > 25 , M5(µ) is empty.

Proof. The Betti numbers of Mn(µ) for µ < 1/n can be computed by the methodof Kirwan [13], since in this case the moduli space is the geometric invariant theoryquotient of (P1)n by the diagonal action of SL(2,C). Let

Pn(µ, t) =∞∑

j=0

rank(Hj(Mn(µ)))tj

denote the Poincare polynomial of Mn(µ), µ < 1/n. By [13, p.193]

Pn(µ, t) = (1 + t2)n(1 − t4)−1 −∑

n2<r≤n

(nr

)t2(r−1)(1 − t2)−1.

In particular, P5(µ, t) = 1 + 5t2 + t4 for µ < 1/5. The Poincare polynomial Pµ,n(t)for arbitrary µ can be computed by two techniques, the original approach of Atiyah-Bott [3] and the recursive approach of Thaddeus [31]. In the special case µ = 1

4 , thefirst approach gives

Pn(µ, t) = (1 + t2)n(1 − t2)−1(1 − t4)−1 − 2n−1(1 − t2)−2

while the second gives

Pn(µ, t) = Pn−2(µ, t)(1 + t2)(1 + t4) + 2n−3tn−3.

Both approaches give for n = 5,

P5(µ, t) = 1 + 6t2 + t4

for 1/5 < µ < 2/5.As the labels vary, the moduli space of parabolic bundles undergoes a series of

flops [8], [30]. Furthermore, if the moduli space is empty on one side then the modulispace on the other side is a projective bundle over the final geometric invariant theoryquotient. To study the topology we consider the family of moduli spaces M5(µ

4, ν)obtained by varying the last parameter, for µ < 1

5 and ν. For ν > 4µ the modulispace is empty. The critical values for ν are 4µ, 2µ, 0, corresponding to the modulispaces that contain reducible bundles. The moduli space M5(µ

4, 3µ) is thereforeisomorphic to P2. The moduli space M5(µ

4, 2µ) has discrete singular set of order 4.It follows that M5(µ

5) is obtained by M5(µ4, 3µ) by 4 blow-ups or blow-downs at


points; from the description of the Poincare polynomial we see that all are blow-ups.

Finally M5(15

5) has connected, isolated singular set, so M5(µ) for µ > 1

5 is obtained

from M5(ν) for ν < 15 by a single blow-up or blow-down. From the description of

the Poincare polynomial M5(µ) is a blow-up of M5(ν). (Alternatively, one could usethe fact that M5(µ

5) is Fano, and the classification of Fano surfaces together withthe Betti number computation above.)

Let τ2 ∈ B5 be the full twist around the first two strands, Γ(τ2) ⊂M5(14)−×M5(

14 )

and the graph of its action on M5(14 ). Let ∆ ⊂M5(

14 )− ×M5(

14 ) be the diagonal.

Theorem 4.1.2. ∆ is not isomorphic to Γ(τ2) in Don(M5(1/4),M5(1/4)).

Proof. This is essentially a result of Seidel [28, Example 1.13]. Let Γ(τ) denote theLagrangian associated to the half-twist, and Γ(τ−1) the Lagrangian associated tothe half-twist inverse. If f ∈ Hom(∆, L) were an isomorphism, the composition withthe identity would induce an isomorphism

∆ Γ(τ−1) → Γ(τ2) Γ(τ−1) = Γ(τ).

Any such isomorphism is automatically compatible with the module structure overHom(∆,∆) = QH(M5(1/4)). The rest of Seidel’s argument is the same.

Theorem 4.1.3. For n ≥ 0 let τn ∈ Diff(M2n+3(1/4)) denote the symplectomor-phism associated to a half-twist of strands 1, 2, Γ(τn) ⊂M2n+3(1/4)

− ×M2n+3(1/4)the corresponding Lagrangian, and ∆n ⊂ M2n+3(1/4)

− ×M2n+3(1/4) the diagonal.Γ(τ2

n) is not isomorphic to ∆n in Don(M2n+3(1/4),M2n+3(1/4)).

Proof. By induction on n. This is Seidel’s result for n = 1. Suppose n > 1 and τ2 issymplectically isotopic to the identity. In this case the relative invariant would givean isomorphism

f ∈ Hom(∆2n+3Γτ2)

where Γτ2 is the graph of the corresponding symplectomorphism. Let Y+, Y− denotea cup, cap at the 3, 4 strands resp. 4, 5 strands. Then (Y+, τ

2n, Y−) is a Cerf decom-

position for τ2n−1, as in Figure 12. Any isomorphism Hom(∆, τ2

n) would therefore

=

Figure 12. The half twist

induce an isomorphism

Φ(Y+) Φ(τ2n) Φ(Y−) → Φ(Y+) Φ(∆n) Φ(Y−)

and therefore an isomorphism Φ(τ2n−1) → Φ(∆n−1) which is impossible by the in-

ductive hypothesis.


Corollary 4.1.4. τ2n is not symplectically isotopic to the identity, for any n ≥ 1.

Hence the action of the braid group on M2n+3(1/4) does not factor through the actionof the symmetric group.

Remark 4.1.5. It would be interesting to know for which labels the braid groupaction on Mn(µ) factors through the symmetric group and to identify the kerneland image of the map

Bn → Map(Mn(µ), ω).

In the case without labels, M. Callahan (unpublished) announced a similar resultfor a separating Dehn twist of a genus two surface, in the moduli space of fixed-determinant bundles of rank two and degree one. Callahan’s result together with theresults of this paper would imply that a separating Dehn twist is not symplecticallyisotopic to the identity in any genus.

The surgery exact triangles for the theory (in the rank two are a consequenceof a generalization of a triangle for Dehn twists by Seidel, proved in [32]. Theinvariants constructed here should be related to the invariants of Collin-Steer [4]and Kronheimer-Mrowka [15], by a suitable version of the Atiyah-Floer conjecture.

We remark that the nilpotent slices used in the definition of the Seidel-Smith[29] and Manolescu [18] invariants embed in the moduli spaces of parabolic bun-dles, for sufficiently small weights; the map is given by assigning to a given matrixin the nilpotent slice the quasiparabolic structure given by the eigenspaces, whichsit insidea canonical rank 2 sub-bundle of the trivial bundle of rank 2n. Assumingthat these are in fact symplectic versions of the Khovanov and Khovanov-Rozanskyinvariants, it would be interesting to know whether the inclusion induces a spec-tral sequence relating the instanton know homologies with the Khovanov-Rozanskyinvariants, and how much information the spectral sequence contains. Kronheimerhas shown that the instanton knot invariants associated to these moduli spaces arestrictly smaller, in some cases, than Khovanov’s.

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