Exact triangle for fibered Dehn twists

We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply it to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.


Introduction
Seidel's long exact sequence [42,Theorem 1] describes the effect of a symplectic Dehn twist on Lagrangian Floer homology, and is one of the main tools used to describe Fukaya categories of Lefschetz fibrations. However in many examples (moduli spaces of bundles, nilpotent slices, etc.) the relevant fibrations have Morse-Bott rather than Morse singularities and the associated monodromy maps are fibered Dehn twists. Many years ago Seidel suggested that this sequence should generalize to the fibered case. In this paper we show how to carry this out using quilted Floer theory developed in [51,45,46], which gives an expression of the third term in the exact triangle as a push-pull functor, similar to the situation in algebraic geometry.
To state the main result suppose that M is an exact or monotone symplectic manifold. A Lagrangian brane in M is a Lagrangian submanifold L equipped with a grading and relative spin structure. We say that L is admissible if it is monotone or exact, compact, and has minimal Maslov number at least three, or minimal Maslov number two and disk invariant zero and torsion fundamental group. 1 Suppose that L 0 , L 1 are admissible Lagrangian branes in M . The Lagrangian Floer Partially supported by NSF grants CAREER 0844188 and DMS 0904358. 1 For convenience; alternatively one can assume further monotonicity conditions. homology group HF (L 0 , L 1 ) has underlying complex generated by perturbed intersection points of L 0 , L 1 and differential that counts finite energy holomorphic strips with boundary in L 0 , L 1 ; taking these are morphism spaces one obtains the cohomology of the Fukaya category of Lagrangian branes in M . For the moment, we work with Z 2 coefficients, although the main result will be stated with Z coefficients. Let C ⊂ M be a Lagrangian sphere equipped with a brane structure. Associated to C is a symplectic Dehn twist τ C : M → M , equal to the antipodal map on C and supported in a neighborhood of it. In the case of a Lefschetz fibration, if C is a vanishing cycle then τ C is the monodromy around the corresponding critical value. Theorem 1.1. (Seidel exact triangle, [42]) Let M be a compact monotone or exact symplectic manifold, and C, L 0 , L 1 a monotone triple of admissible Lagrangian branes such that C is a sphere of codimension at least two. There is a long exact sequence in Lagrangian Floer homology HF (L 0 , L 1 ) HF (L 0 , C) ⊗ HF (C, L 1 ).
HF (L 0 , τ C L 1 ) By taking the Euler characteristics one essentially recovers the Picard-Lefschetz formula as in Arnold [4], for which reason Seidel's result is often referred to as a categorification of Picard-Lefschetz. The Fukaya version (conjectured by Kontsevich) is developed in Seidel's book [43]. The codimension one case also holds, in theory, but requires further assumptions or Novikov rings since monotonicity does not usually hold for the Lefschetz fibrations involved in the arguments.
Many interesting fibrations that arise in representation theory or gauge theory (such as nilpotent slices or moduli spaces of bundles over a family of curves) have not just Morse singularities but rather Morse-Bott singularities. Extensions of the Picard-Lefschetz formula to fibrations π : E → S with more general singularities are considered by Clemens [9], Landman [22], and many subsequent authors; the Morse-Bott situation is a particularly easy case. Let s 0 ∈ S be a critical value, and B ⊂ E s 0 the critical locus in its fiber, which need no longer be isolated. Let s ∈ S be a generic nearby point. The analog of the vanishing cycle in this case is a manifold C ⊂ E s , consisting of points that converge to B under parallel transport, called by Clemens the vanishing bundle. In the symplectic setting, C is a coisotropic manifold and the map to B is smooth with maximally isotropic fibers. The monodromy τ C is a fibered Dehn twist, that is, a Dehn twist in each of the fibers of C → B. Let C → B be the vanishing bundle in the generic fiber M = E s of a Lefschetz-Bott fibration E for a path to a critical value s 0 and c the codimension of C. A special case of Clemens [9,Theorem 4.4] gives that the monodromy τ C acts on α ∈ H(M ) by the formula . The main result of this paper is a categorification of the fibered Picard-Lefschetz formula (1) to the setting of Floer-Fukaya theory, that is, a generalization of Seidel's triangle to the fibered case. As before, for technical reasons we need to assume suitable monotonicity conditions, although presumably a version of the triangle holds in general.
Definition 1.2. (Spherically fibered coisotropics) A spherically fibered coisotropic submanifold of a symplectic manifold M is a coisotropic submanifold C ⊂ M of codimension c ≥ 1 such that (a) (Fibrating) the null-foliation of C is fibrating over a symplectic base B with fiber S c a sphere of dimension c and (b) (Orthogonal structure group) the structure group of p : C → B reduces to SO(c + 1), that is there exists a principal SO(c + 1)-bundle P → B and a bundle isomorphism P × SO(c+1) S c ∼ = C.
Any spherically fibered coisotropic gives rise to a fibered Dehn twist τ C ∈ Aut(M, ω), see Section 2. We identify C with its Lagrangian image in M − × B, where M − denotes M witih symplectic structure reversed, and denote by C t its transpose in B − × M . Thus C defines Lagrangian correspondences from M to B and vice-versa, which fit into the framework of quilted Floer theory developed in [50], [51], [45], [46]. We assume the reader is familiar with the notion of monotonicity of a tuple of Lagrangian correspondences in [45]; monotonicity implies an energy-index relation for pseudoholomorphic quilts. Assuming monotonicity, the correspondence C defines functors from the Fukaya category of B to that of M and vice versa. On the level of homology, the latter functor gives rise to a map of Lagrangian Floer homology groups HF (L 0 , graph(τ C ), L 1 ) .
The proof is similar to that given by Seidel in [42], except that the pair of pants map in [42] is replaced by a "quilted pants" map from [45]. A similar triangle was developed by T. Perutz, as part of the program described in [33]. As in Seidel's case, the codimension one case also holds in theory, but requires further assumptions to rule out bubbling. In Section 7.1, we extend the triangle above to an exact triangle in the derived Fukaya category: where C t C L is a generalized Lagrangian brane in the sense of [46].
One can also write the bottom object in the exact triangle where Φ(C t ), Φ(C) are the A ∞ functors associated to Lagrangian correspondences constructed in [26]. The formulation of the third term as a push-pull functor makes clear that the exact triangle is the mirror partner of Horja's exact triangle in [16]. We remark that Perutz [35] proves a related exact triangle describing a symplectic version of the Gysin sequence; roughly speaking this describes the composition of the functors for C t , C in the opposite order as a mapping cone for the map given by multiplication of the Euler class. We briefly outline the contents of the paper. Section 2 contains background results on fibered Dehn twists and Lefschetz-Bott fibrations. Section 3 describes various situations in which surface Dehn twists induce generalized Dehn twists on moduli spaces of flat bundles; these are mostly minor improvements of results of Seidel and Callahan. Sections 4 and 5 contain the proof of the exact triangle. Section 6 applies the triangle to moduli spaces of flat bundles to obtain generalizations of Floer's exact triangle for surgery along a knot, as well as surgery exact triangles for crossing changes in knots which have the same form as the surgery exact triangles as Khovanov [18] and Khovanov-Rozansky [20]. Finally Section 7 describes generalizations to the A ∞ setting, which are limited to the case of minimal Maslov number greater than two.
We thank Mohammed Abouzaid, Tim Perutz, and Reza Rezazadegan for helpful comments, and especially Paul Seidel who got us involved in this area.
The purpose of this paper is to provide more details on a draft that we have circulated since 2007. While the first author wishes to note that readers should not expect her specific level of rigour in the present version, she has agreed to publication of this version in the hope that its availability will be useful to the field.

Lefschetz-Bott fibrations and fibered Dehn twists
This section covers the generalization of the theory of symplectic Lefschetz fibrations to the Lefschetz-Bott case, that is, to the case that the singularities of the fibration are not isolated but still non-degenerate in the normal directions. Most of this material is covered in an unpublished manuscript of Seidel [40] and in the works of Perutz [33], [34]. For more recent appearance of fibered Dehn twists, see Chiang et al [8].
2.1. Symplectic Lefschetz-Bott fibrations. Lefschetz-Bott fibrations have a natural definition in the setting of holomorphic geometry: one requires the projection to be proper and Morse-Bott. In the setting of symplectic geometry, there are several analogous definitions which we discuss below. We begin with the holomorphic setting: Let S be a complex curve. A Lefschetz fibration over S is a complex manifold E equipped with a proper holomorphic map π : E → S such that π only has critical points of Morse type. A Lefschetz-Bott fibration over S is a complex manifold equipped with a proper holomorphic map π : E → S that has only Morse-Bott singularities. That is, the critical set is a smooth (necessarily holomorphic) submanifold and the Hessian of π is nondegenerate along the normal bundle of E crit . By the parametric Morse lemma [3, p.12] near any critical point e ∈ E crit there exists coordinates (z 1 , . . . , z n ) : E → C n such that where n is the dimension of E and c + 1 is the codimension of E crit at e.
In our examples we will not have global complex structures on E and S (at least no canonical ones). Instead, we work with symplectic versions of Lefschetz-Bott fibrations.
(a) (Symplectic fibrations) A symplectic fibration is a manifold E equipped with a closed two-form ω E and a fibration π : E → S over a smooth surface S, such that the restriction of ω E to any fiber of π is a symplectic manifold.

(b) (Symplectic Lefschetz-Bott fibrations) A symplectic Lefschetz-Bott fibration
consists of (i) a smooth manifold E equipped with a closed two-form ω E ; (ii) a smooth, oriented surface S; (iii) a smooth proper map π : E → S with critical set E crit := {e ∈ E | rank(D e π) < 2} and critical values S crit = π(E crit ) ⊂ S ; (iv) a positively oriented complex structure j 0 ∈ End(T S| U ) defined in a neighborhood U ⊂ S of the critical values S crit ; (v) an almost complex structure J 0 ∈ End(T E| V ) defined in a neighborhood V ⊂ E of the critical set E crit satisfying the following conditions: (i) E crit ⊂ E is a smooth submanifold with finitely many components; (ii) the Hessian D 2 π at any critical point restricts to a non-degenerate complex bilinear form on the normal bundle of E crit ⊂ E; (c) (Normally Kähler symplectic Lefschetz-Bott fibration) We do not make any assumption about integrality of J 0 near the critical set, while Perutz [33] adds the condition that J 0 be normally Kähler.
Symplectic Lefschetz-Bott fibrations are somewhat better behaved than Lefschetz fibrations in that they form a closed category under products, under the following genericity assumption: Suppose that π k : E k → S, k = 1, 2 are (symplectic) Lefschetz-Bott fibrations such that critval(π 1 ) ∩ critval(π 2 ) is empty, where critval denotes the set of critical values. Then is a (symplectic) Lefschetz-Bott fibration. In particular, the fiber product of a fibration with a Lefschetz fibration is a Lefschetz-Bott fibration.
Associated to any Lefschetz-Bott fibration we have a natural notion of parallel transport in the fibers, leading to coisotropic vanishing cycles. The usual notion of parallel transport monodromy of Lefschetz fibrations extends to the symplectic Lefschetz-Bott situation. First, suppose that π : E → S, ω E ∈ Ω 2 (E) is a symplectic fibration with connected base S, for simplicity. The canonical symplectic connection on E is the connection defined by Here the superscript denotes the symplectic complement with respect to ω E , which has dimension 2 due to the nondegeneracy of ω E | ker(Deπ) . Moreover, for any horizontal vector field v ∈ Γ(T h E) and fiber E s we have Hence, given any smooth path γ : [0, 1] → S − S crit the parallel transport for any t, τ ∈ [0, 1] is a symplectomorphism. (Also see [15,Section 1.2].) This gives a reduction of structure group of the fibration to the symplectomorphism group of any fiber. Now suppose that E is a Lefschetz-Bott fibration. The smooth locus E − E crit is a fibration over S − S crit with vertical tangent spaces T v e E = ker(D e π) and a canonical symplectic connection T h E ⊂ T (E − E crit ) defined by (3). Given an embedded path γ ending on the critical locus γ(1) ∈ S crit parallel transport extends to a continuous map ρ t,1 : E γ(t) → E γ(1) , which is pointwise defined by Indeed, choose a tubular neighborhood of γ. After rescaling, parallel transport becomes the gradient flow of the function f • π for some coordinate function f : S → R with respect to the metric ω E (·, J 0 ·), see [42,Lemma 1.13]. Since the critical points of π are Morse-Bott, the gradient flow is hyperbolic and the limit is welldefined [44]. Vanishing thimbles and cycles for Lefschetz-Bott fibrations are defined as follows. Let γ : [0, 1] → S be a smooth embedded path with γ(1) ∈ S crit such that γ([0, 1)) ⊂ S −S crit . Fix a connected component B ⊂ E crit ∩E γ(1) of the critical set in the endpoint fiber. Then the vanishing thimble for the path γ and component B is The vanishing thimble T γ,B ⊂ E is a smooth submanifold with boundary, since it is the stable manifold of B, see [44]. The intersections with the smooth fibers of π for t ∈ [0, 1) are called the vanishing cycles for the path γ.
Proposition 2.2. Each vanishing cycle C t from (5) is a coisotropic submanifold of the fiber E γ(t) . The map ρ t,1 : C t → B is smooth and gives C t the structure of a spherically fibered coisotropic submanifold in the sense of Definition 1.2.
Proof. The parallel transport map ρ t,1 of (4) is a smooth fibration with fibers cdimensional spheres and structure group SO(c + 1) by the stable manifold theorem, see for example [44]. The dimension c is the dimension of the fiber as well as the codimension of C t ⊂ E γ(t) , by the normal form of π. The parallel transport can also be written as a rescaled Hamiltonian flow of g • π for some coordinate function g : S → R, as in the unfibered case described in [42]. Since the Hamiltonian flow preserves the symplectic form, the fibers of C t are parallel to the null foliation of C t . Since π is J 0 -holomorphic on B, the tangent space T B (which is the null space of the Hessian of π) is J 0 -invariant. Hence nondegeneracy of ω E in a neighborhood of B implies that the restriction of ω E to B is symplectic. This proves that the total space C t is coisotropic and the projection ρ t,1 : C t → B is the null foliation as claimed.
2.2. Fibered Dehn twists. The symplectic Dehn twist along a Lagrangian sphere in [42] can be generalized to spherically fibered coisotropics using the associated symplectic fiber bundle construction. Recall that a connection one-form on a principal G-bundle P is a one-form α ∈ Ω 1 (P, g) G satisfying the following two conditions: (a) α(ξ P ) = ξ for any ξ ∈ g, where ξ P ∈ Γ(T P ) is the corresponding vector field, and (b) g * α = Ad(g) −1 α for any g ∈ G.
Any connection one-form α determines a connection (invariant subspace complementary to the vertical subspace) on the principal bundle P given by ker(α).
Given a principal bundle with connection and a Hamiltonian action, a symplectic fibration may be formed as follows. Suppose the following are given: (a) a base symplectic manifold (B, ω B ); (b) a compact structure group G with Lie algebra g; (c) a principal bundle π : P → B with structure group G; (d) a fiber symplectic manifold (F, ω F ) equipped with a Hamiltonian G-action with moment map Φ F : F → g ∨ := Hom(g, R); and (e) a connection one-form α ∈ Ω 1 (P, g) G .
Denote by π 1 , π 2 the projections to the factors of P × F . The minimally coupled form on P × F is Theorem 2.3. (Symplectic associated fiber bundles) (see e.g. [14]) Let P be a principal G-bundle, F a Hamiltonian G-manifold and α a connection one-form as above.
(a) The minimally coupled form ω P ×F,α descends to a closed two-form ω P (F ),α on the quotient which is non-degenerate on the fibers in a neighborhood Here |Φ F | denotes the norm of Φ F with respect to an invariant metric on g ∼ = g ∨ . Hence P (F ) ǫ is a symplectic fiber bundle over B for sufficiently small ǫ > 0.
Example 2.4. (Associated bundles with cotangent-sphere fibers) For any integer c ≥ 1, let S c denote a sphere of dimension c and T ∨ S c its cotangent bundle. Consider T ∨ S c with canonical symplectic form ω T ∨ S c and the canonical SO(c + 1)action, which is Hamiltonian with a moment map Φ T ∨ S c whose zero level set is S c . Thus for any principal SO(c + 1)-bundle π : P → B the associated fiber bundle construction yields a symplectic fiber bundle P (T ∨ S c ) ǫ over B. By functoriality, any automorphism τ of ( , which is an isomorphism of symplectic fiber bundles on P (T ∨ S c ) ǫ . For any sphere S c , a model Dehn twist τ S c along the zero section S c in the cotangent bundle T ∨ S c is a compactly supported symplectomorphism equal to the antipodal map on S c , given as follows. Let ζ ∈ C ∞ (R) be a function satisfying In particular, ζ is compactly supported and ζ ′ (0) = 1/2. Fix the standard Riemannian structure on S c and let denote the Riemannian norm, which is smooth on the complement of the zero section.
The time 2π flow of v → ζ(|v|) extends to a smooth symplectomorphism τ S c of T ∨ S c which acts on the zero section by the antipodal map, that is, Furthermore τ S c is SO(c + 1)-equivariant and preserves the moment map for the SO(c + 1)-action. Any two model Dehn twists given by different choices of ζ are equivalent up to symplectomorphism generated by a compactly supported Hamiltonian.
We construct fibered Dehn twists along spherically fibered coisotropics as follows.
Definition 2.6. Let C be a spherically fibered coisotropic submanifold of a symplectic manifold M . (a) (Coisotropic embedding) Let (M, ω) be a symplectic manifold and C ⊂ M a spherically fibered coisotropic submanifold of codimension c ≥ 1, fibering C → B over a symplectic manifold (B, ω B ), as in Definition 1.2. Recall that C is diffeomorphic to an associated fiber bundle P (S c ) := P × SO(c+1) S c , for some principal SO(c + 1)-bundle π : P → B. By the coisotropic embedding theorem [14, p.315], a neighborhood of C in M is symplectomorphic to a neighborhood of the zero section in P (T ∨ S c ) ǫ as in Theorem 2.3. (b) (Model fibered Dehn twists) Given a symplectomorphism of a neighborhood of C ⊂ M with P (T ∨ S c ) ǫ we define a symplectomorphism τ C : M → M by τ T ∨ P (S c ) on the neighborhood of C and the identity outside. We call τ C a model fibered Dehn twist along C. (c) (Fibered Dehn twists) A symplectomorphism of M is called a fibered Dehn twist along C if it is Hamiltonian isotopic to a model Dehn twist.

Equivariant fibered Dehn twists.
In this section we discuss the interaction of equivariant fibered Dehn twists with symplectic reduction. In this section G is a compact Lie group and (M, ω) be a symplectic G-manifold. A spherically fibered coisotropic G-submanifold is an invariant coisotropic submanifold C ⊂ M of codimension c ≥ 1 such that there exists a principal SO(c + 1)-bundle P → B equipped with an action of G by bundle automorphisms (i.e. SO(c + 1)-equivariant diffeomorphism) and a G-equivariant bundle isomorphism P × SO(c+1) S c ∼ = C, where the G-action is induced by the action on the first factor. Given such a C, one obtains a G-equivariant Dehn twist on P (T ∨ S c ) by the G-equivariant version of the associated symplectic fiber bundle construction, and a G-equivariant model fibered Dehn twist on M via the G-equivariant coisotropic embedding theorem. A G-equivariant symplectomorphism τ ∈ Diff(M, ω) is a G-equivariant fibered Dehn twist if it is equivariantly Hamiltonian isotopic to a model Dehn twist, that is, Hamiltonian isotopic via a symplectomorphism generated by a family of G-invariant Hamiltonians. We now show that equivariant fibered Dehn twists give rise to fibered Dehn twists in symplectic quotients. Let (M, ω) be a Hamiltonian G-manifold with moment map Φ : M → g ∨ . The symplectic quotient of M by G is Assuming that G acts freely on Φ −1 (0), M/ /G is a symplectic manifold with unique symplectic form that lifts to the restriction of the symplectic form to Φ −1 (0). Lemma 2.7. Let (M, ω) be a Hamiltonian G-manifold with moment map Φ. Let C ⊂ M be a G-invariant coisotropic submanifold. Suppose that 0 is a regular value of Φ, the action of G on Φ −1 (0) is free, and C intersects Φ −1 (0) transversally. Then Now (ker DΦ) ω ∼ = g is the tangent space to the G-orbits and contained in T C, while Lemma 2.8. Suppose that C ⊂ M is a spherically fibered G-coisotropic over a base B where M is a Hamiltonian G-manifold with moment map Φ. Then the induced action of G on B is Hamiltonian with moment map denoted Φ B : B → g ∨ induced by the moment map Φ, which is constant on the fibers of C.
Proof. By assumption, the action of G on P is SO(c + 1)-equivariant and so induces an action on B. For any ξ ∈ g the infinitesimal action ξ M ∈ Γ(T M ) is tangent to C, hence L v Φ, ξ = ω(ξ M , v) = 0 for all fiber tangent vectors v ∈ T vert C = T C ω .
In this setting every G-equivariant fibered Dehn twist along C descends to a fibered Dehn twist of M/ /G along C/ /G: Theorem 2.10. Let (M, ω) be a Hamiltonian G-manifold with moment map Φ such that 0 is a regular value of Φ and the action of G on Φ −1 (0) is free. Let C ⊂ M be a spherically fibered G-coisotropic over a base B. Suppose that C intersects Φ −1 (0) transversally, and that the induced action of G on the base Φ −1 B (0) ⊂ B is free. Let τ C ∈ Diff(M, ω) be a G-equivariant fibered Dehn twist along C. Then the symplectomorphism is a fibered Dehn twist [τ C ] =: τ C/ /G along C/ /G.

Proof.
By definition τ C is Hamiltonian isotopic to a model Dehn twist on P (T ∨ S c ) given by τ 0 . The latter is G-equivariant since the G-action commutes with the SO(c+ 1) action. Any G-equivariant local model P (T ∨ S c ) → M induces a local modelP (T ∨ S c ) → M/ /G and one obtains a Dehn twist onP (T ∨ S c ) along C/ /G. The equivariant Hamiltonian isotopy of τ C to τ 0 C induces a Hamiltonian isotopy of τ C/ /G to τ 0 C/ /G , which completes the proof. 2.4. Lefschetz-Bott fibrations associated to fibered Dehn twists. In this section we explain that any fibered Dehn twist appears as the monodromy of a symplectic Lefschetz-Bott fibration. Conversely, the monodromy of a symplectic Lefschetz-Bott fibration is given by a fibered Dehn twist, at least up to isotopy and under the "normally Kähler assumption" by a theorem of Perutz [33] below. (The theorem is not used in this paper; we mention it only for its conceptual importance linking Lefschetz-Bott fibrations and fibered Dehn twists.) Proposition 2.11. Let M be a symplectic manifold, C ⊂ M a spherically fibered coisotropic, and τ C : M → M the corresponding fibered Dehn twist. There is a standard Lefschetz-Bott fibration E C with generic fiber M and symplectic monodromy τ C .
Proof. Let p : C → B denote the fibration, and P → B the associated SO(c + 1)bundle. Recall from [42] the symplectic Lefschetz fibration associated to a model Dehn twist. Given the standard representation of SO(c + 1) on V = C c+1 we have a vector bundle P (V ) := (P × V )/SO(c + 1) → B.
Let ω V ∈ Ω 2 (V ), Φ V : V → so(c + 1) ∨ denote the symplectic form and moment map for the SO(c + 1)-action induced from the identification V → T ∨ R c+1 . The associated symplectic fiber bundle construction above produces a closed form ω P (V ) on P (V ), equal to ω B on B and ω V on the fiber V . The projection (8) π is SO(c + 1)-invariant and so induces the structure of a Lefschetz-Bott fibration on P (V ) over C. Let S c ⊂ R c+1 ⊂ V denote the unit sphere and so that T z is a submanifold of the fiber P (V ) z . By [42, p. 14] the symplectic form on V can be changed slightly so that the symplectic monodromy around 0 is a Dehn twist along S c . By [42, 1.13] there exists an isomorphism of Hamiltonian . By the coisotropic embedding theorem, a neighborhood of C in M is symplectomorphic to the fiber bundle P (U ), where U is a neighborhood of the zero section in T ∨ S c . It follows that P (V ) − P (T ) is symplectomorphic to P (C × T ∨ S c − S c ) in a neighborhood of P (T ) resp. P (C × S c ). By replacing a neighborhood of C × C in C × M with a neighborhood of P (T ) in P (V ), one obtains a Lefschetz-Bott fibration E C → C with monodromy τ C .
Theorem 2.12. [33,Theorem 2.19] Suppose that π : E → S is a normally Kähler symplectic Lefschetz-Bott fibration, and s 0 ∈ S crit . Then the symplectic monodromy around s 0 is a fibered Dehn twist.  Proof. The flow of ψ • Φ is rotation by the angle ψ ′ . The rest is immediate from the definitions.
Remark 2.14. (Standard Dehn twists as symplectic Dehn twists) Standard Dehn twists of complex curves arise from the construction in 2.13 as follows. Suppose that M is a complex curve and C → M an embedded circle C ∼ = S 1 . Choose an area form on M , which is automatically symplectic with C as a Lagrangian submanifold. By the Lagrangian embedding theorem, there exists a tubular neighborhood M 0 = C × (c − , c + ) on which the symplectic form is standard. Then the U (1) action by rotation on the left factor of M 0 is free, and the projection Φ on the second factor is a moment map. For any ψ with the properties in Proposition 2.13, the flow of ψ • Φ is a standard Dehn twist.
Next we consider SU (2)-actions. We fix a metric on the Lie algebra su(2), so that non-zero elements ξ with exp(ξ) = 1 have minimal length 1.
Proof. By the coisotropic embedding theorem, there exists an equivariant symplectomorphism of a neighborhood of Φ −1 (0) in M with a neighborhood of the zero section in Φ −1 (0) × SU (2) S c , where c = 3 resp. c = 2, in the trivial stabilizer resp. U (1)-stabilizer case, such that the moment map is the moment map for the action of SU (2) on T ∨ S c . In the local model |Φ| becomes the norm on the fibers T ∨ S c and the flow of ψ • |Φ| becomes rotation by angle 2πψ ′ . The claim follows.

Fibered Dehn twists on moduli spaces of flat bundles
This section describes a natural collection of fibered Dehn twists on moduli spaces of flat bundles, which are our motivating examples.
3.1. Moduli spaces of flat bundles. Moduli spaces of bundles play a role in many recent topological invariants. Here we focus on the case of moduli spaces of flat bundles on surfaces with markings, that is, flat bundles on the complement of the markings with holonomies around them in fixed conjugacy classes. (a) (Conjugacy classes in compact 1-connected Lie groups) Let G be a simple compact, 1-connected Lie group, with maximal torus T and Weyl group W = N (T )/T . Let g, t denote the Lie algebras of G and T . We choose a highest root α 0 ∈ t ∨ and positive Weyl chamber t + ⊂ t. Conjugacy classes in G are parametrized by the Weyl alcove see [36]. For any µ ∈ A, we denote by the corresponding conjugacy class. (b) (Marked surfaces) By a marked surface we mean a compact oriented connected surface X equipped with a collection of distinct points x ∈ X n and a collection of labels µ ∈ A n . For simplicity, we denote such a surface (X, µ).
(a) (Action of central bundles) Let Z(G) denote the center of G and M Z (X) denote the moduli space of Z(G)-bundles on X with trivial holonomy around the markings. The group multiplication induces a group structure on M Z (X), isomorphic to Z 2g where g is the genus of X. The action of Z on G induces a symplectic action of M Z (X) on M (X, µ).
In the case G = SU (r), the central markings are exactly the vertices of the alcove A. Several central markings may be combined into a single central marking as follows: Suppose that λ 1 ∈ A resp. λ 2 ∈ A are labels corresponding to z 1 , z 2 ∈ Z, λ 12 ∈ A corresponds to z 1 z 2 ∈ Z. Then there is a symplectomorphism This follows immediately from the description of the moduli space as representations of the fundamental group. The moduli space of flat bundles M (X, µ) may be realized as the symplectic quotient of the moduli space of framed bundles on a cut surface described in [30]. Let Y ⊂ X be an embedded circle, that is, a compact, oriented, connected one-manifold, disjoint from the markings x. Let X cut denote the surface obtaind from X by cutting along Y as in Figure 1, with boundary components (∂X cut ) j ∼ = S 1 , j = 1, 2. Let X cap denote the surface obtained from X cut capping off with a pair of disks, with an additional marked point on each disk, as in Figure 1. We summarize some discussion from [30]. Let M (X cut , µ) be the moduli space of flat bundles with framings (trivializations) on the boundary of X cut . This moduli space is a symplectic Banach manifold, with symplectic form given by the usual pairing of one forms and integration. Let LG = Map(S 1 , G) denote the loop group of G, so that LG 2 acts on M (X cut , µ) by changing the framing. The moment map for the action of LG 2 is restriction to the boundary Noting that the orientations on the two extra boundary circles are opposite, one obtains that the diagonal action of LG has moment map Φ given by the difference of the moment maps for each boundary component: By [30] M (X, µ) is naturally symplectomorphic (on the smooth locus) to the symplectic quotient of M (X cut , µ) by the diagonal action of LG, that is, In particular, the symplectic structure on M (X, µ) descends from a symplectic structure on M (X cut , µ). We use this description below to analyze the effect of a Dehn twist, which is easier to understand for the moduli space of bundles on the cut surface. Locally M (X, µ) can be written as a finite-dimensional symplectic quotient of a subset of M (X cut , µ). In particular, the subset consisting of connections of "standard form" ξdθ, ξ ∈ A • in a neighborhood U of the circle Y can be written as a finite dimensional symplectic quotient of the subset by the diagonal action of the maximal torus T .

Symplectomorphisms induced by Dehn twists.
Any Dehn twist on a marked surface induces a symplectomorphism of the moduli space of flat bundles. One can write down explicitly this symplectomorphism as the Hamiltonian flow of a non-smooth function. In good cases, this description shows that the symplectomorphism is a symplectic Dehn twist. Let (X, µ) be a marked surface. Let Map + (X, µ) the group of isotopy classes of orientation preserving diffeomorphisms φ of X preserving the markings The following is elementary and left to the reader: In particular, a Dehn twist on the surface induces a symplectomorphism of the moduli space of bundles. Our aim is to describe this symplectomorphism as a Hamiltonian flow of a function relating to the holonomy. Denote parallel transport from t 0 to t 1 in R, the universal cover of S 1 Given the embedded oriented circle Y in X disjoint from the markings given as the image of a map ι : [0, 1] → X with ι(0) = ι(1), let ρ 0,1 denote parallel transport around Y and define by mapping an equivalence class of flat connections [A] to the conjugacy class of the holonomy of any representative A around Y . This is independent of the choice of base point on Y . Let Proof. We assume for simplicity of notation that there are no markings. The idea of the proof is to show the claim first "upstairs" on the moduli space of the cut surface, where it follows from generalities involving the moment map for the loop group, and then descend to the glued surface. Consider the action of a Dehn twist τ ∈ Diff(X) along a small translation of the boundary component (∂X cut ) 2 , so that the twist is the identity on the boundary and so induces a Dehn twist on X cut , also denoted τ . Note that the generators of the fundamental group of X cut may be represented by loops not meeting the support of the Dehn twist. Given a connection A, the twisted connection (τ −1 ) * A has the same holonomies as A, since the generators of π 1 (X) have image in a neighborhood of the complement of the boundary. Therefore, (τ −1 ) * A is gauge equivalent to A, via a gauge transformation g : Σ → G given by the difference in parallel transports of A and (τ −1 ) * A. The difference between parallel transports from a base point to a point in the boundary (∂X cut ) 2 is the holonomy ρ(A|(∂X cut ) 2 )) along the boundary (∂X cut ) 2 . We denote by g 0 ∈ G parallel transport from the base point in X to a base point on the boundary (∂X cut ) 2 . Then the parallel transports of A resp. (τ −1 ) * A to a point t ∈ (∂X cut ) 2 are ρ 0,t g 0 resp. ρ 0,t ρ 0,1 g 0 where ρ 0,t := ρ 0,t (A|(∂X cut ) 2 ). Therefore the gauge transformation g relating A and (τ −1 ) * is given by After gauge transformation we may assume that ρ 0,t form a one-parameter subgroup in which case ρ 0,t and ρ 0,1 commute, hence g|(∂X cut ) 2 = ρ 0,1 is constant on the boundary. Similarly, the restriction of the gauge transformation g to the first boundary component (∂X cut ) 1 is trivial. Thus the twist τ Y acts by changes the framing of A by the holonomy along the second boundary component: The restriction ofτ Y to M (X cut ) • of (10) has the form This map is the time-one Hamiltonian flow of the functioñ For later use, we recall the following facts about level sets of holonomy maps. Proofs can be found in for example Meinrenken-Woodward [30].

Full twists for rank two bundles. Dehn twists on a surface induce fibered
Dehn twists of the moduli space of flat bundles for G = SU (2) because the group and its maximal torus are both spheres. The following is a slight generalization of a result of M. Callahan (unpublished) resp. P. Seidel [40,Section 1.7] in the case of a separating resp. non-separating curve on a surface. (a) (Separating case gives a codimension one Dehn twist) If Y is separating then τ Y acts on M (X, µ) by a fibered Dehn twist along C λ := ρ −1 Y (λ) for any λ ∈ (0, 1 2 ) such that M (X, µ, λ, * λ) contains no reducibles. (b) (Non-separating case gives a codimension three Dehn twist) If Y is nonseparating and M (X cap , µ) contains no reducibles, then τ Y acts by a fibered Dehn twist along the coisotropic Proof. To prove the theorem we realize M (X, µ) as a symplectic quotient as in Remark 3.3 and use the results of Section 2.5. For simplicity, we consider the case that µ is empty. By the irreducible-free assumption, the generic stabilizer of the LG 2 action on M (X cut ) is Z(G) #π 0 (Xcut) , where Z(G) is the center of G, and this is also the stabilizer for the action of T 2 on the space M (X cut ) • of (10). Denote the subset with generic holonomy In the non-separating resp. separating case, exp(1/2) = −I (resp. exp(1/4)) acts as a non-trivial square root of the identity. In particular in the exp(1/2) resp. exp(1/4) acts as the antipode map on any generic U (1)-orbit. Suppose first that Y is separating. The group element exp(1/2) acts by the identity on M (X cut ), so τ * Y is Hamiltonian isotopic to a symplectomorphism supported on M (X) • , given by Proposition 3.5 as the Hamiltonian flow of ρ 2 Y /2. By Remark 3.3 and the results of Section 2.5, τ * Y is a Dehn twist along any ρ −1 Y (λ) for λ generic. Explicitly, by the coisotropic embedding theorem a neighborhood of Half twists for rank two bundles with fixed holonomies. A half-twist on a marked surface (X, µ) induces a fibered Dehn twist on the moduli space M (X, µ) of flat G = SU (2) bundles with fixed holonomy. Definition 3.8. (Half-twist and corresponding coisotropic) Given a pair of points x i , x j with the same label µ i = µ j , let Y denote a circle around x i , x j and σ Y a halftwist along Y , which interchanges x i , x j . Denote the coisotropic of bundles with trivial holonomy along Y Example 3.9. (Moduli spaces of flat bundles on the sphere punctured five times) Let X = S 2 with five markings x 1 , . . . , x 5 all with labels 1/4 ∈ A so that /G is real dimension four, and all markings µ 1 , . . . , µ 5 = 1/4. It is well-known that M (X, µ) is the real manifold underlying the fifth del Pezzo surface. This can be seen, for example, by computing its homology by any number of standard techniques and noting its rationality, or by the more detailed discussion in [24]. The submanifold C Y given by a loop Y around the i-th and j-th marking is a Lagrangian sphere described as bundles whose holonomy along a loop containing the i-th and j-th markings is the identity: . The intersection diagram of these Lagrangians reproduces the root system A 4 corresponding to the fifth del Pezzo, discussed in general in Manin [25]. After choosing suitable generators for the fundamental group we may assume that i, j are adjacent.
The submanifold C Y consists of closed spherical polygons of side lengths π such that the i and j-th edges are coincident, that is, the polygon consists of a bigon and a triangle as in Figure 2. The intersection points for loops   The following is slight generalization of a result of Seidel [40].
Theorem 3.11. (For G = SU (2) half-twists around pairs of markings act by codimension two fibered Dehn twists on the moduli space) Let G = SU (2). Let (X, µ) be a marked surface such that M (X, µ) contains no reducibles and µ i = µ j = 1/4. Let Y be an embedded circle disjoint from the markings which is the boundary of a disk containing x i , x j . Then C Y is a spherically fibered coisotropic submanifold of codimension 2, and the action of σ Y on M (X, µ) a fibered Dehn twist along C Y .
Proof. First we show that the half-twist is a Hamiltonian flow of a non-smooth function. Let X cut denote the surface obtained by cutting X along Y , so that (10). By restriction an element of M (X cut , µ) gives a flat connection on the twice-punctured disk, of the form ξdθ near the boundary of the disk, with holonomies g i , g j around the punctures. We choose convenient representatives for g i , g j as follows. Let n be a representative of the non-trivial element w of the Weyl group W . Note n −1 = −n ∈ C 1/4 . Furthermore, the stabilizer G n acts on C 1/4 by rotation fixing n and n −1 . The set nT is diffeomorphic to a circle passing through n. Thus after conjugating g i , g j by some g ∈ G, we may assume g i = n, g j = tn for some t ∈ T .
We compute the action of the half-twist on the holonomies: for the i-th holonomy 2 )g i (the square root is unique up to an element of the center, which acts trivially) and for the j-th holonomy This shows that σ Y acts on g i , g j by conjugation by Ad((−g i g j ) 1 2 ). Restricted to M (X cut , µ) • , Φ 2 /2 gives a real-valued function whose Hamiltonian flow is the action of −I. Hence Φ 2 /4 has Hamiltonian flow (−I) 1/2 = exp(1/4). Similarly Φ 2 2 /4 has Hamiltonian flow given by the action of (g i g j ) 1/2 . Combining these remarks shows that the action of σ Y on M (X cut ) • is the time-one Hamiltonian flow of Φ 2 (Φ 2 + 1)/4, hence the action on M (X) • is the time-one Hamiltonian flow of ρ Y (ρ Y + 1)/4.
Next we show that the Hamiltonian flow is a Dehn twist along a smooth coisotropic. Since M (X, µ) contains no reducibles, M (X, µ−{µ i , µ j }) also contains no reducibles. Indeed, if some point in M (X, µ − {µ i , µ j }) has a one-parameter group of automorphisms then there is also a point in M (X, µ) with a one-parameter group of automorphisms, since the stabilizers of (g i , g j ), g i g j = 1, g i , g j ∈ C 1/4 range over all one-parameter subgroups of SU (2). Thus every point in Φ −1 2 (0) has stabilizer isomorphic to U (1), see Lemma 3.6, and so Φ −1 2 (0) ⊂ M (X cut , µ) is a spherically fibered coisotropic of codimension 2. The flow on M (X, µ) • extends over ρ −1 Y (0) as the action of (−I) 1/2 on the holonomies, which is the antipodal map on the fibers of . The flow at ρ Y = 1/2 is the action of (−I) 2/2 = −I, which is trivial. Hence the action of σ Y is a Dehn twist along ρ −1 Y (0), as claimed. 3.5. Half-twists for higher rank bundles. For special markings, a half twist on the marked surface X induces a fibered Dehn twist on the moduli space of flat SU (r) bundles with fixed holonomy. Definition 3.12. (Khovanov-Rozansky modification on moduli spaces) Let r ≥ 2 and let ω k , k = 1, . . . , r − 1 denote the fundamental weights of G = SU (r), which via the identification of g with g ∨ are identified with vertices of the fundamental alcove A. Let µ = (µ 1 , . . . , µ n ) be a collection of markings with k is the midpoint between the vertices ω k , ω k+2 of the alcove A, . . , µ n , ν 2 k ) the moduli space of flat bundles on X with markings obtained by removing µ i , µ j = ν 1 k and adding ν 2 k . Theorem 3.13. (Codimension two fibered Dehn twists via half-twists) Suppose that G = SU (r) and µ i , µ j , ν 1 k , ν 2 k as in Definition 3.12 such that the moduli spaces M (X, µ) and M (X, µ, (µ i , µ j ) → ν 2 k ) contain no reducibles. Let Y ⊂ X denote an embedded circle enclosing only the i-th and j-th markings. Then the subset C Y from (11) is a spherically fibered coisotropic submanifold of codimension 2, fibered over Proof. First we identify the action of the half-twist on the holonomies as a conjugation. Let g i resp. g j denote the holonomies around the i-th resp. j-th marking. The half-twist replaces g i , g j with g i g j g −1 i , g i . Let C 1 denote the conjugacy class of exp(ω 1 /2). It suffices to consider the case that g i , g j ∈ C 1 , that is, ν 1 k = µ i = µ j = ω 1 /2, ν 2 k = ω 2 /2. We begin by choosing convenient representatives of g i , g j . First we take = diag(− exp(πi/r), exp(πi/r), . . . , exp(πi/r).
The centralizer of g i is therefore Let O ⊂ G denote the one-parameter subgroup generated by rotation in the first two coordinates in C r . Since g i is the product of diag(−1, 1 . . . , 1) with a central element in U (r), the adjoint action of Now C i is a symmetric space of rank one, and in particular Z g i acts transitively on the unit sphere in T g i C 1 which implies that the map Also note that since O is conjugate to the one-parameter subgroup generated by α 1 (or rather its dual coweight) the square of C 1 in G C ω 1 +ǫα 1 the union of conjugacy classes of exp(ω 1 + ǫα 1 ) where ǫ ∈ [0, − 1 2 ]. In particular, the conjugacy class C 2 of exp(ω 2 /2) appears in C 2 1 .
We compute the action of the half-twist on the holonomies: In particular if g j = exp(s 12 ω 1 /2) so that g i g j = exp(ω 2 )/2 then o 1/2 rotation by 180 degrees in the first two coordinates, that is, o = diag(−1, −1, 1, . . . , 1). Now The fact that the action is trivial when g i = g j fixes the square root. On the subset M • (X cut , µ) of M (X cut , µ) which takes values in the interior of the alcove on the two boundary components, the action of the half-twist is conjugate to that of the constant loop with values o 1/2 = exp(ω 1 /2 − Φ 2 /2). By (12), the image of M (X cut , µ • under Φ 2 is the interval with endpoints ω 1 , ω 2 /2. It follows that the action of o 1/2 is the Hamiltonian flow of the shifted moment map component (Φ 2 , ω 1 /2) − (Φ 2 , α 1 ) 2 /2. This descends to the non-smooth function on the moduli space of the closed surface Thus we have written the half-twist as the Hamiltonian flow of a non-smooth function.
The subset (11) with fixed holonomy ρ −1 Y (ω 2 /2) is a spherically fibered coisotropic: . Furthermore, the action of exp(ω 1 /2) is equal to the antipodal map on the fibers. This shows that the Hamiltonian flow of (13) is a fibered Dehn twist along C Y , and completes the proof of Theorem 3.13.

Pseudoholomorphic sections of Lefschetz-Bott fibrations
In this section we describe the relative invariants associated to Lefschetz-Bott fibrations, which are maps between Lagrangian Floer cohomology groups obtained by counting pseudoholomorphic sections. 4.1. Monotone Lagrangian Floer homology over Novikov rings. Novikov rings are not needed to define Lagrangian Floer homology for a monotone pair of Lagrangian submanifolds. However, we will need our cochain complexes to admit "filtrations by energy" and for this we find it convenient to use the Novikov version, as in the construction of the spectral sequence in Fukaya-Oh-Ono-Ohta [11]. (One could alternatively use the index as in Oh [32]). Since we also want to include the case that the Floer operator is a matrix factorization, we review the constructions. Let M be a symplectic manifold with symplectic form ω. (ii) (Monotonicity) ω is a symplectic form on M which is monotone, i.e.
[ω] = τ c 1 (T M ); (iii) (Background class) b ∈ H 2 (M, Z 2 ) is a background class, which will be used for the construction of orientations; and (iv) (Maslov cover) Lag N (M ) → Lag(M ) is an N -fold Maslov cover such that the induced 2-fold Maslov covering Lag 2 (M ) is the oriented double cover. We often refer to a symplectic background (M, ω, b, Lag N (M )) as M .
area-index relation for disks with boundary in L, that is, in H 2 (M, L)) is again the symplectic area and I(u) is the Maslov index of u as in [29,Appendix].
monotone if the following holds: Let Σ be any compact surface with boundary where I(u) is the sum of the Maslov indices of the totally real subbundles (u| Ce ) * T L e in some fixed trivialization of u * T M . There is a similar definition of monotonicity for tuples of Lagrangian correspondences [46]. Note that monotonicity of any tuple implies monotonicity of any subset, in particular, of any pair which implies that the Floer homology groups are well-defined. (d) (Lagrangian branes) A grading of a Lagrangian submanifold L ⊂ M is a lift of the canonical section L → Lag(M ) to Lag N (M ), as in Seidel [41]. A brane structure on L consists of a relative spin structure for the embedding L → M (equivalent to a trivialization of w 2 (M ) in the relative chain complex for (M, L), see for example [49]) and a grading. A Lagrangian brane is an oriented Lagrangian submanifold equipped with a brane structure.
Let (L 0 , L 1 ) be a compact, monotone pair of Lagrangian branes in M . Choose a Hamiltonian H ∈ C ∞ ([0, 1] × M ) whose time-one flow φ 1 ∈ Diff(M, ω) satisfies the condition that φ 1 (L 0 ) intersects L 1 transversally. Define the set of perturbed intersection points The gradings on L 0 , L 1 induce a degree map I(L 0 , L 1 ) → Z N given by the Maslov index of a path from the lifts in the Maslov cover [41]; by the assumption on the Maslov cover, the mod 2 degree is determine purely by the orientations, that is, is +1 resp. −1 if the two Lagrangians meet positively resp. negatively after perturbation.
We denote by I d (L 0 , L 1 ) ⊂ I(L 0 , L 1 ) the subset of Maslov index d so that denote the space of time-dependent ω-compatible almost complex structures. For any x ± ∈ I(L 0 , L 1 ) we denote by M(x − , x + ) the space of finite energy (J, H)holomorphic maps modulo translation in s ∈ R, and M(x − , x + ) 0 the subset of formal dimension 0, that is, index 1. The relative spin structures on L 0 , L 1 induce a map measuring the difference between the orientation on each element u and the canonical orientation of a point [43,49]. The Floer cochain complex is the direct sum The Floer coboundary operator ∂ : For the following, see [31], [11]. We wish for our Floer cochain complexes to admit action filtrations. To obtain we pass to versions over Floer cochains using Novikov rings.
of powers of a variable q where n j ∈ Z, ν j ∈ R.
The Floer cochain complex over the coefficient ring Λ is the direct sum CF (L 0 , L 1 ; Λ) = x∈I(L 0 ,L 1 ) Λ x with coboundary incorporating the energies of trajectories: Let H # t ∈ Vect(M ), t ∈ [0, 1] be the Hamiltonian vector field of H and (s, t)))dsdt the perturbed energy. Then

4.2.
Relative invariants for Lefschetz-Bott fibrations. We may now associate to Lefschetz-Bott fibrations over surfaces with strip-like ends relative invariants that are morphisms of Floer cohomology groups associated to the ends, given by counting pseudoholomorphic sections. The following material can also be found, in a slightly different form, in Perutz [34]. (a) Given u ∈ Γ(E) with image disjoint from the critical set define its index and symplectic area Note that the form ω E is only fiber-wise symplectic, so that A(u) may be negative.
Proposition 4.6. Let π : E → S be a symplectic Lefschetz-Bott fibration with S, E compact. If all vanishing cycles in E have codimension at least 2 and the generic fiber M of E is monotone, then E is monotone.
Proof. First note that the homology classes of any two sections differ by homology classes in the fibers. For each critical value s i ∈ S crit let ρ i : M → E s i denote the map given by symplectic parallel transport which collapses the null foliation of the vanishing cycles. We claim that for any two sections u 0 , u 1 ∈ Γ(E), the push-forwards u j, * [S] differ by an element in the span of the homology of the fibers To see this, apply the spectral sequence for the map π : E → S, see for example Segal [37], and the cover consisting of S −{s 1 , . . . , s n } and a collection of balls around the critical values. The first differential in the Leray spectral sequence vanishes and E 2 -term is The degree 2 part of the kernel of π * under the projection map is generated by H 2 (M ) (embedded as any fiber) and H 2 (E s i ), as claimed. Since monotonicity holds on H 2 (M ), it suffices to check that the inclusion maps H 2 (M ) → H 2 (E s ) are surjective for all s ∈ S. Consider the long exact sequence where Cone(ρ i ) is the mapping cone on ρ i . Since ρ i is a diffeomorphism away from the vanishing cycle, Cone(ρ i ) admits a deformation retraction to Cone(p i ) where where ∼ is the equivalence relation which collapses the sections given by fixed points (0, . . . , 0, ±1) ∈ S c+1 . For c ≥ 2, the spectral sequence for the fibration P i × SO(c+1) S c+1 → B i implies that any degree two homology class arises from the base. Since the base is contracted in the mapping cone, H 2 (Cone(p i )) = 0. Hence H 2 (M ) surjects onto H 2 (E s i ).
) is a local holomorphic coordinate on the closure of the image of ǫ j ; the end is called incoming resp. outgoing if the sign is negative resp. positive. A collection of strip-like ends E is a set of strip-like ends, one for each j = 1, . . . , n. We write E = E − ∪ E + the union of the incoming and outgoing ends. (ii) each fiber F z ⊂ E z , z ∈ ∂S is a Lagrangian submanifold; (iii) for each e ∈ E there exist Lagrangian submanifolds L 0,e , L 1,e ⊂ M such that F is constant sufficiently close to z e that is, ϕ S,e (F ǫ S,e (s,j) ) = L j,e , ±s ≫ 0; and (iv) for each e ∈ E, the intersection L 0,e ∩ L 1,e is transversal.  Proof. In the case without ends, as in Proposition 4.6, any two sections differs by a homology sphere in the fiber, and these are generated by classes in the generic fiber by the codimension assumption, for which monotonicity holds. In the case with ends, as in [45], one may use gluing to reduce to the case that S has no strip-like ends as in the statement of the Proposition.
We now turn to the construction of relative invariants associated to symplectic Lefschetz-Bott fibrations. (a) (Compatible almost complex structures) Let π : E → S be a Lefschetz-Bott fibration over a surface with strip-like ends S. A complex structure j on S is compatible with E if j = j 0 in an neighborhood of S crit . An almost complex structure J on E is compatible with π, j iff (i) J = J 0 in a neighborhood of E crit ; (ii) π is (J, j)-holomorphic in a neighborhood of E crit , that is, J • dπ = dπ • j; and (iii) ω E (·, J·) is symmetric and positive definite on T E v x , for any x ∈ E. Let J (E) denote the set of (π, j)-compatible almost complex structures. For sufficiently j-positive symplectic forms ω S on S, the sum ω E + π * ω S is tamed but not necessarily compatible with J.
is given by differentiating the Cauchy-Riemann operator along a path exp u (tξ) of geodesic exponentials, and using parallel transport Π −1 tξ back to u. The operator D u is Fredholm since the boundary conditions at infinity are assumed transversal.
consisting of pairs of a section with bubbled-off trajectory; and The proof is similar to that of [42] in the exact case. Bubbling for sections can occur only in the fiber, and so sphere and disk bubbling on the zero and onedimensional moduli spaces is ruled out by the monotonicity condition. The construction of coherent orientations is given in [49]. if the disk invariants of the Lagrangians cancel. More generally (not requiring disk invariant or ignoring torsion) one obtains a morphism in the derived category of matrix factorizations in Definition 5.10 below.
The relative invariants of fibrations with "non-negative curvature", in the following sense, have particularly nice properties. Recall the symplectic connection (3); the spaces T E h e have canonical complex structures, induced from the complex structure j on the base S. We say that a Lefschetz-Bott fibration E with two-form ω E has non-negative curvature if ω E (v, jv) ≥ 0 for all v in the horizontal subspace T E h .
Non-negative curvature has the following consequence: Recall that the total space of any Lefschetz-Bott fibration π : E → S admits a canonical isotopy class of symplectic structures given as follows. If ω S ∈ Ω 2 (S) is a sufficiently positive twoform then ω E + π * ω S is a symplectic form on T E e for any e ∈ E. If E is compact, then ω E + π * ω S is symplectic on E for ω S sufficiently positive. If E is non-negative, then ω E + π * ω S is symplectic for any positive form ω S ∈ Ω 2 (S).

Proposition 4.13. (Non-negative curvature of standard Lefschetz-Bott fibrations)
If C ⊂ M is a spherically fibered coisotropic, then the standard Lefschetz-Bott fibration E C of 2.11 has non-negative curvature.
Proof. Let v ∈ V , the standard representation of SO(c + 1), and (p, v) ∈ P × V . The horizontal subspace H v ⊂ T v V pairs trivially with ker(α) × T (π −1 V (π V (v))) under the pairing given by the two-form (6), where π V is the projection (8). It follows that the image [H v ] of H v in P (V ) is the horizontal subspace at [p, v]. Let J V denote the standard complex structure on V , and J 0 the induced complex structure on E.
Proposition 4.14. Let E be a Lefschetz-Bott fibration with Lagrangian boundary condition F and relative spin structure. If E has non-negative curvature, then the coefficients of q in the formula (18) are all non-negative.
Proof. Since the form ω E (·, J·) is non-negative for any J ∈ J (E), any pseudoholomorphic section has non-negative area.
We do not give formula for the degree of the relative invariant, see [45] for a formula in the case without singularities.

4.3.
Invariants for quilted Lefschetz-Bott fibrations. The main difference between the triangle for the fibered case and the original Seidel triangle [42] is the appearance of invariants associated to quilted surfaces. The following definitions are taken from [45]. (a) (Quilted surfaces with strip-like ends) A quilted surface with strip-like ends S consists of the following data: (i) a collection S = (S k ) k=1,...,m of surfaces with strip-like ends, see [42], [45]. Each S k carries a complex structure j k and has strip-like ends (ǫ k,e ) e∈E(S k ) near marked points lim s→±∞ ǫ k,e (s, t) = z k,e ∈ ∂S k with the boundary components ∂S k = (I k,b ) b∈B(S k ) ; (ii) a collection S of seams: pairwise disjoint 2-element subsets σ ⊂ m k=1 b∈B(S k ) and for each σ = {I k,b , I k ′ ,b ′ } a real analytic isomorphism where the isomorphisms ϕ σ should be compatible with the strip-like ends, in the sense that on each end ϕ σ should be a translation; where I k,b ranges over true boundary components resp. I k 0 ,b 0 , I k 1 ,b 1 range over identified boundary components, such that (i) each fiber F (k 0 ,b 0 ),(k 1 ,b 1 ),z ⊂ E k 0 ,z × (ϕ * (k 0 ,b 0 ),(k 1 ,b 1 ) E k 1 ) z over z ∈ S is a Lagrangian submanifold; and (ii) over the strip-like ends the submanifolds the fibers F (k 0 ,b 0 ),(k 1 ,b 1 ),z over z ∈ S are given by fixed Lagrangians L ke,be on the strip-like ends, with the property that the composition   (More generally, for arbitrary coefficients not requiring vanishing disk invariant, one obtains a morphism in the derived category of matrix factorizations.) These invariants satisfy a composition relation for gluing along strip-like ends [45].

4.4.
Vanishing theorem. In this section we using gluing along a seam to obtain a vanishing theorem analogous to [42, Section 2.3] for the invariants associated to standard fibrations associated to a fibered Dehn twist.
Remark 4.17. (Gluing along a seam for quilted Lefschetz-Bott fibrations) (a) (Glued surface) For k = 0, 1 let S k be quilted surfaces with d k + 1 strip-like ends, and z k a seam point in S k . Let ρ > 0 be a gluing parameter, and S ρ the quilted surface with d 0 + d 1 + 2 strip-like ends formed by gluing together quilted disks D 0 , D 1 around z 0 , z 1 using the map z → ρ/z. (b) (Glued bundles) Let (E k , F k ) be Lefschetz-Bott fibrations over S k , equipped with a trivialization of E k , F k in a neighborhood of z k , and a symplectomorphism of (E k,z k , F k,z k ) for k = 0, 1. The seam connect sum E ρ → S ρ is formed by patching E 0 and E 1 , and similarly for the boundary and seam conditions F ρ . (c) (Glued complex structures) Given an admissible almost complex structure J k for E k → S k that is constant in a neighborhood of z k (with respect to the given trivialization) for k = 0, 1 and such that J 0 agrees with J 1 on the glued fiber, we can patch together to obtain an admissible almost complex structure J for E → S.
Suppose that (E ρ , F ρ ) is obtained from gluing (E 0 , F 0 ) and (E 1 , F 1 ) and all three are monotone symplectic Lefschetz-Bott fibrations. Let ev k : M(E k , F k ) → F k,z k denote the evaluation maps at the nodal points z k , k = 0, 1.
(a) the evaluation map ev 0 × ev 1 is transverse to the diagonal; (b) for any ρ ≫ 0 and any pair (u 0 , u 1 ) there exists a gluing map on a neighborhood U (u 0 , u 1 ) of (u 0 , u 1 ) given by (c) As ρ varies and (u 0 , u 1 ) varies over points in the zero-dimensional component of the left-hand-side, Θ ρ is surjective onto the zero-dimensional component of M(E ρ , F ρ ); and (d) for any u ∈ M(E 0 , F 0 ) × ev 0 ,ev 1 M(E 1 , F 1 ), the sequence Θ ρ (u) Gromov converges to u as ρ → 0, that is, converges up to sphere bubbling, disk bubbling, and bubbling off of Floer trajectories on the strip-like ends.
See McDuff-Salamon [29, Chapter 10] for the case of gluing at an interior point, and Abouzaid [1] for the details of gluing along a point in the boundary. The case of seam gluing is similar; however, we will only use the dimension count in the above argument. Standard technique, as explained in McDuff-Salamon [29], show the existence of a Gromov limit.
The vanishing formula depends on a formula for the dimension for the pseudoholomorphic sections of the standard fibration studied in 2.11, 4.13. . Equip E C with the Lagrangian boundary condition given by the fiber product F C := P (T ) | ∂D where T is the union of vanishing cycles in the local model P (V ) as in (9). As in [42] this problem fits into a family of problems E C,r , F C,r , the standard fibrations of Section 2.4 over a disk D r of radius r. Each member of the family is formed by patching together C × SO(c+1) V with (M × B) − (i(C) × p(C)), and boundary condition F C,r with fiber over z ∈ ∂D r given by P  Proof. We may explicitly describe a family of pseudoholomorphic sections as follows. Suppose that the almost complex structure on E C,r is induced from an almost complex structure on B and the standard almost complex structure on V . For each a ∈ V, b ∈ B fix a local trivialization of P and define w r,a,b : D r → E C,r , z → r −1/2 az + r 1/2 a.
The conditions for w r,a,b to be a section and have boundary values in F C,r are (20) 2a · a = 1, a · a = a · a = 0.
We claim that the Maslov index satisfies the following formula: Indeed, by definition the Maslov index of w := w r,a,b is the index of the pair (w * T vert E C , (∂w) * T vert F C ), which fits into the exact sequence Now T w(z) E = (T b B) 2 ⊕ C c+1 is trivial, and the boundary condition has vertical part as pairs over (D, ∂D). Using (23) and (22) we have and so which proves the claim (21). We claim that w r,a,b is a section of lowest index for r sufficiently small. Indeed, the area A(w r,a,b ) goes to zero as r → 0, for all a. Choose r sufficiently small so that A(w r,a,b ) ≤ 1/2λ. The area-index monotonicity relation and non-negativity of the curvature (4.13) implies that any other section u has positive area and so at least the index of w r,a,b .
Remark 4.22. (Double cover in the codimension one case) In the case that c = 1, r = 1, the evaluation map on the moduli space M(E C,r , F C,r ) 0 of zero-index pseudoholomorphic sections of (E C,r , F C,r ) at z = 1 induces a double cover ev 1 : M(E c,r , F C,r ) 0 → C, w c,a,b → w c,a,b (1) = a + a of the fiber of the vanishing cycle C. The relations (20) become a 2 = ±ia 1 , a 1 a 1 = 1/2. Hence w c,a,b (1) = (Re(a 1 ), ± Im(a 1 )). We note for later use that the double cover is orientation preserving resp. reversing on the component with a 2 = ia 1 resp. a 2 = −ia 1 . Corollary 4.23. (Vanishing of the relative invariant associated to a standard fibration) Suppose that E → S is a Lefschetz fibration over a surface with strip-like ends S, Lagrangian boundary condition F obtained by a seam connect sum from a Lefschetz-Bott fibration (E C , F C ) over the disk corresponding to a spherically fibered coisotropic submanifold C, with an arbitrary quilted Lefschetz-Bott fibration E 0 → S 0 with boundary condition F 0 . Suppose that all these fibrations with boundary conditions are monotone and equipped with relative spin structures, so that in particular the relative invariant Φ(E, F ) is defined. Then Φ(E, F ) = 0.
Proof. Suppose that E, F are as in the statement of the Corollary. If Φ(E, F ) is nonzero then by Gromov compactness for the family of surfaces obtained by stretching the neck, there exist (u 0 , u 1 ) in the zero-dimensional component (ev 0 , ev 1 ) −1 (∆), where ∆ is the diagonal in the gluing fiber F 0,z 0 ∼ = F C,z 1 . The gluing Theorem 4.18 for k = 0, 1. In the codimension two case Corollary 4.23 now follows from Lemma 4.21 and Remark 4.20, which gives a contradiction. The codimension one depends on the remark on orientations in 4.22: in this case the limiting moduli space is not empty, but has two components corresponding to the choices of sign in 4.22. As explained in [49], the orientations on the moduli spaces of disks of Maslov index zero are induced via the orientations on the Lagrangians defined by evaluation at a point. Since the evaluation map is orientation preserving resp. reversing for the first resp. second component, the contributions from the two components cancel.
4.5. Horizontal invariants. As in Seidel [42], the computation of the relative invariants in the special cases needed for the exact triangle uses only horizontal sections, defined as follows. (a) (Horizontal sections) A section u : S → E (that is, a collection of sections  Proof. As in [42,Lemma 2.11], any non-horizontal pseudoholomorphic section has positive symplectic area and so does not lie in the same component as a horizontal section, which has zero symplectic area. We give a criterion for the zero-dimensional component M(E, F ) 0 to consist entirely of horizontal sections in the monotone case; the exact case was discussed in Seidel [42]. Let E → S be a monotone Lefschetz-Bott fibration with Lagrangian boundary/seam conditions F .  Proof. If u is a pseudoholomorphic section with formal index 0, then it has nonnegative symplectic area equal to c((x e ) e∈E )). If this constant vanishes, then all such sections must be horizontal.

Floer versions of the exact triangle
The proofs of the exact triangles described in the introduction follow the lines of the proof of Floer's exact triangle [7], in Seidel [42] and Perutz [35]. Namely, one first constructs a short sequence of cochain groups that is exact up to leading order, and then uses a spectral sequence argument to deduce the existence of a long exact sequence of cohomology groups. In this section we also describe various extensions, to the case of minimal Maslov number two and the case of periodic Floer cohomology.

Fibered Picard-Lefschetz formula.
In this section we prove the exact triangle on the level of vector spaces; this is essentially equivalent to the fibered Picard-Lefschetz formula in Theorem 1.
Definition 5.1. (Angle functions) Recall from Section 2.2 that a Dehn twist in a local model T ∨ S c is defined by rotating a vector in T ∨ S c with norm t by an angle function θ(t) with θ(0) = π and θ(t) = 0, t ≫ 0. The angle function θ(t) is related to the choice of ζ(t) of (7) by θ(t) = ζ ′ (t). For the rest of the paper we assume that the function θ(t) is decreasing with t. We wish to use angle functions which go to 0 sufficiently quickly. In particular, given an angle function θ we consider the family of angle functions defined by (24) θ λ (t) := θ(λt).  ) be a fibered Dehn twist using a given local model and the angle function θ λ (t). There exists a constant λ 0 such that if λ > λ 0 then there exists a bijection Proof. By taking λ 0 sufficiently large we may suppose that τ C,λ is supported in a neighborhood U C disjoint from L 0 ∩ L 1 , so there is an inclusion C,λ L 1 . We wish to identify the remaining intersections points with (L 0 × C t ) ∩ (C × L 1 ).
For the case B trivial, this is Seidel [42,Lemma 3.2]: The Dehn twist τ −1 C acts at [v] ∈ P (T ∨ S c ) by normalized geodesic flow by time θ λ (|v|). There exists a unique v ∈ L 1,F of norm less than π such that its time π − |v| normalized geodesic flow lies in L 0,F , and the unique point w ∈ R >0 v with θ λ (|w|) = π − |v| gives the desired intersection point.
We reduce to the B trivial case by the use of suitable local models as follows. Let l 0 ∈ L 0 ∩C, l 1 ∈ L 1 ∩C be points with the same projection b ∈ B. Consider the action of scalar multiplication on the vector bundle P (π T ∨ S c ) : P (T ∨ S c ) → P (S c ), which induces a map of a neighborhood of the identity in R × on M via the coisotropic embedding. Note that τ C,λ = λτ C λ −1 wherever the right-hand-side is defined. For k = 0, 1, the submanifolds (λL k ∩ U C ) λ>0 fit together with P (π T ∨ S c ) −1 (L k ∩ C) in the limit λ → ∞ to a smooth family at λ = ∞. Indeed, if L k is given locally by {f (b, x, y) = 0} in local coordinates b on B and cotangent coordinates (x, y) on T ∨ S c then λL k = {f (b, x, λ −1 y) = 0}; transversality with the zero section implies that this cuts out a smooth family including at λ = ∞. By the case B trivial, the intersection at λ = ∞ given by By the implicit function theorem, the set of intersection points L 0 ∩ τ −1 C,λ L 1 forms a smooth manifold parametrized by λ ≫ 0 and intersection points (L 0 × C t ) ∩ (C × L 1 ) as desired.

Lagrangian Floer version.
We are now read to prove the exact triangle Theorem 1.3. Since τ C is a symplectomorphism, it suffices to prove the theorem with L 1 replaced by τ −1 C L 1 , that is, that there is a long exact sequence . . . HF (L 0 , C t , C, τ −1 C L 1 ) → HF (L 0 , τ −1 C L 1 ) → HF (L 0 , L 1 ) . . . . 5.2.1. Definition of the maps. Let M be a symplectic background and C ⊂ M a spherically fibered coisotropic submanifold of codimension c. Let L 0 , L 1 ⊂ M be monotone Lagrangian branes of minimal Maslov number at least 3. We also assume that the image of C in B × M is monotone with minimal Maslov number at least two, all Lefschetz-Bott fibrations discussed below are monotone, and the triple (C, L 0 , L 1 ) is monotone. These monotonicity conditions hold, for example, if L 0 , L 1 are simply connected, M has minimal Chern number at least two, and c ≥ 2 by Lemma 4.9. We may assume, after Hamiltonian perturbation, that C, L 0 , L 1 all intersect transversally.
(a) (Chaps map) The first map in the exact sequence is defined as the relative invariant associated to a "quilted pair of pants", or more accurately, "quilted chaps" in American dialect. Let S 1 denote the quilted surface shown in Figure 4: Let E = (M, B), where B is the base of the fibration p : C → B. We identify C with its image in M × B. Let F denote the Lagrangian seam/boundary condition for E given by L 0 , L 1 , C and consider the relative invariant defined in Remark 4.16 by counting points in the zero-dimensional component of the moduli space M 1 of pseudoholomorphic quilts on S 1 . (b) (Lefschetz-Bott map) The second map in the exact sequence is a relative invariant associated to a Lefschetz-Bott fibration with monodromy given by the Dehn twist. Namely let E C → D denote the standard Lefschetz-Bott fibration with monodromy τ C from Lemma 2.11. By gluing in E C with the trivial fibration over a strip (using the identity as transition map to the left of the disk, and τ C as transition map to the right) we obtain a Lefschetz-Bott fibration (E 2 , F 2 ) over the infinite strip shown in Figure 5. Let The first step in the proof of the exact sequence is to show that the composition of the chaps and Lefschetz-Bott maps vanishes: Lemma 5.4. (Exactness at the middle term) The composition Φ = Φ 1 • Φ 2 (the relative invariant associated to picture on the left in Figure 6) vanishes.
Proof. The composition of the two relative invariants is the relative invariant associated to a Lefschetz-Bott fibration over the glued surface S = S 1 S 2 by [45,Theorem 2.7]. Consider the deformation S t of S obtained by moving the critical value of the Lefschetz-Bott fibration towards the boundary marked C and pinching off a disk in M × B with boundary values in C shown in the second two pictures in Figure 6. The bundles E and Lagrangian boundary/seam conditions F naturally extend to families E t , F t , which are obtained from gluing for t ≫ 0. It follows from Proposition 4.23 that the relative invariant Φ vanishes. Figure 6. Pinching off a disk at the seam 5.2.2. Exactness to leading order. The proof that the maps Φ 1 , Φ 2 of (26), (27) fit into a long exact sequence follows a standard argument [7] in which one first proves that the "leading order terms" in the cochain-level map define a short exact sequence. Recall that L 0 ∩ τ −1 C L 1 is the disjoint union of the images of i 1 and i 2 of the map in Lemma 5.2.
Theorem 5.5. (Exactness of the short sequence to leading order) Let C, L 0 , L 1 be as in Theorem 1.3. There exists ǫ > 0 such that (a) (Horizontal sections as leading order contributions to CΦ 2 ) there exists δ > 0 such that any fibered Dehn twist τ C defined using a function ζ(t) with supp ζ ′ ⊂ (−∞, δ] and |ζ(0)| < δ has the following property: for any x ∈ L 0 ∩ L 1 , (i) if y = i 1 (x), the coefficient of x in CΦ 2 ( y ) is equal to 1 plus terms of the form q ν x with ν > ǫ. (ii) if y = i 1 (x), then the coefficient of x in CΦ 2 ( y ) is a sum of terms with coefficients q ν , for ν > ǫ; (b) (Small triangles as leading order contributions to CΦ 1 ) There exists δ > 0 such that if τ C is a Dehn twist with 2π|ζ(0)| < δ and x ∈ ( is equal to q ν , for some ν < ǫ/3, plus terms with coefficients q µ with µ > ǫ. (ii) if y = i 2 (x) then the coefficient of y in CΦ 1 ( x ) is a sum of terms with coefficients q µ with µ > ǫ.
Proof. (a) First we show that the terms with no q powers arise from horizontal sections. Let u be the horizontal section of E 2 on the infinite strip with value x. Then Ind(D u,J ) = 0, since the boundary conditions are constant. Hence u is regular for horizontal J, and the count for y = i 1 (x) follows by Proposition 4.26. The map Φ 2 has degree 0, since the horizontal sections have zero index.
In the case of limits x, y with y = i 1 (x), by the mean value inequality, for any neighborhood U ⊂ M of x, there exists an ǫ ′ > 0 such that any holomorphic strip with boundary (L 0 , L 1 ) with energy less than ǫ ′ is contained in U . (To see, this, paths to the boundary of the strip S 2 and integrate the inequality on the energy density given by the mean value inequality.) Taking U sufficiently small, this implies that any holomorphic quilt in the definition of Φ 2 with energy at most ǫ ′ cannot have limits x, y with y = i 1 (x). By taking ǫ to be the minimum of all such ǫ ′ , the claim follows.
(b) We aim to reduce to the exact, unfibered case considered by Seidel [42]. The simplest case to deal is the case in which the points on the spherical fiber are antipodes. That is, let x = (m, b, m ′ ) be as in the statement of the Theorem, so Suppose that m, m ′ are antipodes, so that L 0 , τ −1 C L 1 intersect at m. Let u denote the constant section with value m and boundary conditions L 0 , L 1 , C. Let J any horizontal almost complex structure, for which u is a pseudoholomorphic section with zero area. We claim that u is J-regular. By non-negativity of the curvature and Proposition 4.25, it suffices to show that the index of the linearized operator D u,J for u the constant section is zero. The unfibered case that C is Lagrangian is covered in [42]. If C is equal to M , then the index problem is equivalent to the index problem for a section of T M with constant values in T L 0 , T L 1 . Since the kernel and cokernel in this case is also trivial, the cokernel of the linearized operator for u is trivial. Hence J is regular for horizontal u, which implies we may use J to compute the coefficient of To reduce to the antipodal case, suppose that (m, b, m ′ ) give an intersection point of (L 0 , C, C t , L 1 ). Choose a family of fiber-wise metrics on C depending on t ∈ [0, 1] such that for t = 1 the points m, m ′ are antipodes in C b . The family of metrics defines a reduction of the structure group depending on t, that is, a family of principal SO(c + 1) bundles P t together with a family of diffeomorphisms Let τ t C denote the resulting family of fibered Dehn twists, and consider the family of boundary conditions given by τ t C L 0 , L 1 . The intersection points I(τ t C L 0 , C t , C, L 1 ) fit into smooth families depending on t as in the proof of Lemma 5.2. Consider the parametrized moduli space of pseudoholomorphic curves for this deformation. Standard arguments show that there is a parametrized regular family of almost complex structures for the deformation, in the sense that the moduli space (28) is a smooth, finite dimensional manifold with boundary.
The sections contributing to the coefficient of i 1 (x) in CΦ 1 ( x ) are of small area: We claim for δ sufficiently small, all such u of index 0 (parametrized index 1) connecting i 1 (x) with x have E(u) is at most ǫ/3. To compute the constant in the area-index relation, suppose u is a small triangle in a neighborhood of C b with Lagrangian boundary conditions L 0 , τ −1 C L 1 , C, vertices close to m, m ′ , b. Near C b all Lagrangians are exact and path connected and so the area computation reduces to the computation of action differences in [42], which shows that the area is at most ǫ/3 for δ sufficiently small. As a quilt the triangle has index zero, since it deforms to the constant index problem at t = 1. By monotonicity, any holomorphic quilt of index zero with boundary/seam conditions L 0 , τ −1 C L 1 , C has the same area. The count of holomorphic triangles is invariant under the deformation (28). Indeed, for δ sufficiently small, bubbling off trajectories in the deformation (28) is impossible by Remark 5.6 below, while bubbling off holomorphic disks and spheres is impossible because of the monotonicity conditions. Hence the component of the moduli spaceM 1 (i 1 (x), x) of formal dimension one is compact and so (29) where o(u) = ±1 are the orientations. The second sum in (29) is equal to 1 if the almost complex structure is horizontal. For the unique contribution then comes from the horizontal section with value i 1 (x) = (m, b, m ′ ).
It remains to deal with the pseudoholomorphic sections connecting other limits. Suppose that u = (u 0 , u 1 ) is a quilt from S 1 connecting x = (m, b, m ′ ) with y with energy at most ǫ with u 1 resp. u 0 mapping to M resp. B. For any neighborhood U of C b , there exists ǫ > 0 sufficiently small so that the image of u 1 is contained in U . To see this, let (s 0 , t 0 ) denote the coordinate of the top-most point in the seam and write the quilted surface as a union of "halves" In the top half S + 1 , integrating the mean value inequality over paths to the boundary shows proximity to L 0 and τ −1 C L 1 , hence proximity to L 0 ∩ τ −1 C L 1 . The bottom half S − 1 gives rise, by "folding", to a strip with values in M ×B ×M with boundary values L 0 × C t and C × τ −1 C L 1 . Integrating the mean value inequality over paths to the boundary shows proximity to each boundary condition, hence to a fiber of C meeting both L 0 and L 1 , which necessarily must be C b . Since the top half is contained in a small neighborhood of L 0 ∩ τ −1 C L 1 and the quilt is connected, the quilt lies entirely in U . We suppose that U is sufficiently small so that the components of L k ∩ U are in one-to-one correspondence with the intersection points L k ∩ C b for k = 0, 1. Then i 1 (x) is the unique intersection point of L 0 ∩ τ −1 C L 1 connected to L 0 ∩ C b by a path from m by path in L 0 ∩ U , and a path from m ′ by a path in τ −1 C L 1 ∩ U , see the proof of 5.2. Since the image of u 0 is contained in U , we must have y = i 1 (x).
Remark 5.6. (Energy gap for Floer trajectories) Let C, L 0 , L 1 be as in the previous two Lemmas. We claim that there exists ǫ sufficiently small such that for all model fibered Dehn twists τ C with 2π|ζ(0)| < ǫ/3, any non-constant Floer trajectory for (L 0 , C t , C, τ −1 C L 1 ) or (L 0 , τ −1 C L 1 ) has symplectic area at least ǫ. First, consider the trajectories for a fixed fibered Dehn twist τ 1 along C. By a suitable Hamiltonian perturbation chosen near each intersection point, one can achieve that all (not necessarily holomorphic) trajectories of index one necessarily connecting distinct intersection points have area at least 2ǫ; this is a local computation with action functionals. On the other hand, the computation in Seidel [42,Section 3] shows that in the exact case, for any Dehn twist τ C with |ζ(0)| < ǫ/3, the areas of trajectories of index one are within 2π|ζ(0)| < ǫ/3 of those of τ 1 .

5.2.3.
Isomorphism with the mapping cone. Every mapping cone of cochain complexes gives rise to an exact triangle, so to construct an exact triangle it suffices to prove an isomorphism of a third complex with a mapping cone. So to prove Theorem 1.3 it suffices to show the following: Theorem 5.7. (Isomorphism with the mapping cone) Let L 0 , L 1 , δ, ǫ, τ C as in Theorem 5.5. Then the map CΦ 2 induces an isomorphism of CF (L 0 , τ −1 C L 1 ) with the mapping cone on CΦ 1 .
Before we give the proof of the theorem we recall some basic facts of homological algebra, explained for example in [13].
(a) (Mapping cone) If C j = (C j , ∂ j ), j = 0, 1 are cochain complexes and f : C 0 → C 1 is a cochain map then the mapping cone on f is the complex (b) (Quasiisomorphisms from mapping cones) A cochain map from Cone(f ) to a complex C 2 is equivalent to pair (k, h) consisting of a cochain map k :

Such a map induces a quasi-isomorphism if and only if Cone(k[1] ⊕ h)
is acyclic.
We will need the following criterion, which is a slightly modified version of Perutz [35,Lemma 5.4] for a cochain map (k, h) as in the second item above to induce a quasi-isomorphism: Lemma 5.9. (Double mapping cone lemma) Fix ǫ > 0. Suppose that C 0 , C 1 and C 2 are free, finitely-generatedΛ-cochain complexes with differentials δ 0 , δ 1 , δ 2 . Suppose is a sequence of cochain maps (not necessarily exact at the middle term) and h : Then the induced map (h, k) : Cone(f )C 2 is a quasiisomorphism.
Proof. The proof is exactly the same in Perutz [35] with the exception that the lowest order term of f 0 is not required to be greater than 2ǫ. However, the leading order differential in Cone((Cone(f ), C 2 ) is still acyclic by standard homological algebra: Proof of Theorem 5.7. We apply Lemma 5.9 to the Floer complexes . Let k denote the cochain level map CΦ 1,Λ defined by the quilted surface in Figure 4. For any x ∈ I(L 0 × C t , C × τ −1 C L 1 ), z ∈ I(L 0 , L 1 ) consider the parametrized moduli spaceM(x, z) for the one-parameter family of deformations (E ρ , F ρ ) (with ρ representing the length of the neck) connecting the pair obtained by gluing (E 1 , F 1 ) and (E 2 , F 2 ) along a strip-like end to the nodal surface equipped with bundles (E, F ), (E C,r , F C,r ), as shown in Figure 6. An element ofM(x, z) consists of a quilted surface S ρ in the family, together with a pseudoholomorphic quilt u ρ with the given boundary and seam conditions. By Corollary 4.23, for r sufficiently small and ρ sufficiently large the parametrized moduli space is empty. The boundary of M(x, z) 1 is therefore where the first union consists of pairs of pseudoholomorphic sections of E 1 and where o(u) = ±1 are the orientations constructed in [49]. The description of the boundary components ofM(x, z) 1 in (30) gives the relation ∂h + h∂ = k • f, so by Remark 5.8 the pair (k, h) define a morphism We claim that the mapping cone is acyclic. Theorem 5.5 show that for Dehn twists satisfying the conditions in the Theorem, the differential ∂ splits into the sum of an operator ∂ 0 whose q-coefficients lie in [0, ǫ/3) and a term ∂ − ∂ 0 whose q-coefficients lie in (ǫ, ∞). The result for Λ coefficients follows from Lemma 5.9. The monotonicity relations as in Remark 4.4 show that all modules involved in the argument are actually free Λ-modules, so acyclicity overΛ implies acyclicity over Λ, and the result for q = 1 follows by specialization. The quilted version in Theorem 1.3 is proved similarly, but replacing the boundary labelled L 0 , L 1 with collections of strips corresponding to the symplectic manifolds appearing in the generalized Lagrangian branes L 0 , L 1 .

5.3.
Minimal Maslov two case. In general, Lagrangian Floer cohomology is defined only the case that certain holomorphic disk counts vanish. In the case that one of the Lagrangians has minimal Maslov number two, the relevant disk count is that of Maslov index two holomorphic disks. First we recall some basics of the derived category of matrix factorizations from [48]. is the space of grading preserving maps f : the cohomology of the differential ∂ ⊗ Z Id : C ⊗ Z Z w → C ⊗ Z Z w obtained from ∂ by tensoring with Z w . Any morphism in Fact(w) defines a homomorphism of the corresponding cohomology groups. The cohomology with coefficients functor has target the category Ab of Z 2 -graded abelian groups, (c) (Derived category of matrix factorizations) (i) A morphism f : C → C is called null-homotopic if there exists a map h : The derived category of matrix factorizations D Fact(w) is the category with the same objects as Fact(w), and morphisms given by the quotient of Hom(Fact(w)) by null-homotopic morphisms.
equipped with the trivial differential ∂. (Note that ∂ 2 = w Id, for any w.) D Fact(w) is naturally a triangulated category, with distinguished "exact" triangles given by the mapping cone construction: Given a morphism of matrix factorizations f : (C 1 , ∂ 1 ) → (C 2 , ∂ 2 ), its mapping cone is the factorization The exact triangles in D Fact(w) are by definition those isomorphic to triangles is an exact triangle then C 3 is (non-canonically) isomorphic to the mapping cone on f .
We recall the definition of the Maslov index two disk count studied in Oh [31]. Let M be a symplectic background and L ⊂ M be a compact Lagrangian submanifold with minimal Maslov number equal to 2. For any ℓ ∈ L, consider the moduli space M 1 2 (L, J, ℓ) of J-holomorphic disks u : (D, ∂D) → (M, L) with Maslov number 2, mapping a point 1 ∈ ∂D to ℓ, modulo automorphisms preserving 1. By results of Kwon-Oh [21] and Lazzarini [23], for J in a comeager subset J reg (M, ω, L) ⊂ J (M, ω) the moduli space M 1 2 (L, J, ℓ) is a finite set. Suppose L is equipped with a relative spin structure on L, which by [11], [49] induces an orientation on the moduli space The element w(L) is independent of J ∈ J reg (M, ω, L) and ℓ ∈ L.
Denote by J t (M, ω, L 0 , L 1 ) ⊂ J t (M, ω) the subset of t-dependent almost complex structures whose restriction to a neighborhood of t = 0 resp. t = 1 lies in J reg (M, L 0 ) resp. J reg (M, L 1 ).
gives rise to a (non-canonical) isomorphism Cone(f ) → C 2 : Since C 2 is free, there exists a lift l : C 2 → C 1 , and exactness implies that [l, ∂] = f h for some h : If the quotients ⊕ n C n /C n−1 are quasiisomorphic to zero in the derived category then the short exact sequences (32) 0 → ⊕ n C n /C n−j → ⊕ n C n+1 /C n−j → ⊕ n C n+1 /C n−j+1 → 0 and induction on j imply that the quotients ⊕ n C n+1 /C n−j are also quasiisomorphic to 0, for all j, by the triangulated category axioms for the derived category of matrix factorizations. This implies under suitable "collapsing" assumptions that C is also equivalent to zero.
The exact triangles (32) and the same arguments as in the Maslov number at least three case imply the following: .

Periodic Floer version.
One can also formulate a version of the exact triangle for symplectomorphisms, that is, in periodic Floer theory. In this formulation, the exact triangle relates the symplectic Floer cohomology of the Dehn twist with the Lagrangian Floer homology of the vanishing cycle and the identity: HF (τ C ) The ideas are very similar and we only sketch the proof. The map HF (id) → HF (τ C ) is the relative invariant corresponding to a Lefschetz-Bott fibration over the cylinder in Figure 7 below. In the Figure the outer and inner boundary represent cylindrical ends, the former with a twisted boundary condition with monodromy τ C . The map HF (C t , C) → HF (id) is the relative invariant associated to the quilted τ C Figure 7. Lefschetz-Bott fibration defining HF (id) → HF (τ C ) cylinder in Figure 8. In the Figure, the outer boundary represents a cylindrical end while the inner boundary represents a quilted cylindrical end with seams labelled C t , C. The proof is similar to that of Theorem 1.3. Namely, HF (id) is canonically isomorphic to HF (L 0 , L 1 ) where L 0 = L 1 = ∆ is the diagonal in M 2 , at least with Z 2 coefficients. Similarly, HF (τ C ) = HF (L 0 , (τ C × 1)L 1 ). There is a natural bijection I(L 0 , (τ C × 1)L 1 ) → I(L 0 , L 1 ) ∪ I(L 0 , C, C t , L 1 ); this amounts to repeating the argument of Lemma 5.2. The same filtration arguments as before are used to construct the exact triangle. With integer coefficients, there is a slight confusion caused by the fact that ∆ does not have a canonical relative spin structure, so the argument works only after treating the diagonal as the "empty correspondence", see Section 7.2 below.

Applications to surgery exact triangles
In this section we apply the exact triangle to obtain versions of the Floer [10], Khovanov [18], and Khovanov-Rozansky [20] exact triangles. Note that we have already established in Section 3 that Dehn twists of surfaces induce fibered Dehn twists of moduli spaces.
6.1. Exact triangle for three-cobordisms. In [47], we introduced a categoryfield theory associated to certain connected, decorated surfaces and cobordisms. We use freely the notation and terminology from [47]: (a) (Decorated surface) A decorated surface is a compact connected oriented surface X equipped with a line bundle with connection D → Y . The degree of D is the integer d = (c 1 (D), X). (b) (Decorated cobordisms) A decorated cobordism of degree d is a compact connected oriented cobordism Y between decorated surfaces (X ± , D ± ) equipped with a line-bundle-with connection D → Y such that D|X ± = D ± . Note that given a decorated cobordism (Y, D) between (X ± , D ± ) one can obtain another decorated cobordism by tensoring D with any line bundle that is trivial on the boundary of Y . In [47] we describe how to equip L(Y, D) with relative spin structures and gradings, so that L(Y, D) has the structure of an admissible Lagrangian brane.
For the purposes of the exact triangle, we will need an alternative description as flat bundles with fixed holonomy around an additional marking.
(a) (Marked surfaces) For any integer n ≥ 0, an n marked surface is a compact oriented surface X equipped with a tuple x = (x 1 , . . . , x n ) of distinct points on X and a labelling µ = (µ 1 , . . . , µ n ) of labels. In this section we take markings all equal to 1/2 corresponding to the central element −I of A. where D → Y is a line bundle whose first Chern class is dual to the tangle K. Thus, in particular, the addition to K of a circle component K ′ corresponds to twisting the determinant line bundle D by a line bundle whose first Chern class is dual to the homology class of K ′ .
The following proposition relates the moduli spaces of bundles with fixed holonomy around an embedded circle with the Lagrangian correspondences associated to elementary cobordisms. (a) If Y is a cobordism from X − to X + containing a single critical point of index 1 and a trivial tangle K (that is, a union of intervals connection x − to x + ) and C ⊂ X + is the attaching cycle then L(Y, K) is diffeomorphic via the projection to M (X + , µ + ) to the subset of connections on X + − x + with holonomy along C equal to I. (b) if Y is a decorated cobordism from X − to X + containing a single critical point of index 1, C ⊂ X + is the attaching cycle, K 0 is a trivial cobordism connecting x − to x + and K 1 ⊂ Y the unstable manifold of the critical point, then L(Y, K 0 ∪ K 1 ) is diffeomorphic to the subset of flat bundles on X + − x + with holonomy along C equal to −I.
Proof. By Seifert-van Kampen, π 1 (Y − K) is the quotient of π 1 (X + − K) by the ideal generated by the element [C] obtained from C by joining by a path to the base point. Hence in the first case, L(Y, K) is diffeomorphic to the submanifold of M (X, µ + ) obtained by setting the holonomy along C equal to the identity. For the second case, the gradient flow the Morse function defines a deformation retract of Since C is a loop around K 1 , the holonomy around K 1 is equal to the holonomy along C, hence the claim.
In order to obtain smooth Lagrangian correspondences, we break the given correspondence into elementary cobordisms.
(a) (Cerf decompositions) A Cerf decomposition of a cobordism Y is a decomposition of Y into elementary cobordisms Y 1 , . . . , Y K , that is, cobordisms admitting a Morse function with at most one critical point. Associated to any Cerf decomposition where the second equality is by definition of the Donaldson-Fukaya category. Since we have ignored absolute gradings, this a relatively Z 4 -graded group depending only on the equivalence class of the decorated cobordism Y .
We prove the following surgery exact triangle for the invariants HF (Y ; L − , L + ). Definition 6.5.
(a) (Knots) A knot in a cobordism Y is an embedded, connected 1-manifold K ⊂ Y disjoint from the boundary. (b) (Knot framings) A framing of a knot K ⊂ Y is a non-vanishing section of its normal bundle. Given a framed knot, the other framings are obtained by twisting by representations of π 1 (K) ∼ = Z into SO (2), and so are indexed by Z.
(c) (Knot surgeries) For λ ∈ Z the λ-surgery Y λ,K of Y along K is obtained by removing a tubular neighborhood of K and gluing in a solid torus D 2 × S 1 so that the meridian ∂D 2 × {pt} is glued along the curve given by the framing of the knot corresponding to λ. We denote by K λ the knot in Y λ,K is the contraction of the solid torus glued in.
Remark 6.6. (Knot surgeries in terms of decompositions into elementary cobordisms) The three-manifolds Y 0,K , Y −1,K obtained by a 0 resp. −1-knot surgery have decompositions into elementary cobordisms given as follows. Suppose that Y is decomposed into elementary cobordisms Y 1 , . . . , Y l , so that K is contained in the boundary (∂Y i ) + = (∂Y i+1 ) − and the framing is the direction normal to the boundary. Gluing in D 2 × S 1 produces two new critical points, the first of which has stable manifold with unit sphere equivalent to K and the second has unstable manifold with unit sphere equivalent to K. Thus (a) The zero-surgery Y 0,K has a decomposition into elementary cobordisms with two additional pieces, Proof. Choose a Morse function f : Y → R such that f is constant on K and the framing is given by the gradient of f at K. The level set f −1 (λ) containing K can be made connected by adding 1-handles in Y , so that f −1 (λ) becomes a connected surface containing K separating the boundary components of Y . The results of Gay-Kirby [12] imply that f can be deformed away from f −1 (λ) so that f is Morse and has connected fibers.
Theorem 6.8. (Exact triangle for knot surgery) Let Y be a cobordism from X − to X + and K ⊂ Y a framed knot contained in the interior of Y , and Y −1,K , Y 0,K the −1 and 0-surgeries on K. Let L − , L + be admissible Lagrangian branes in M (X ± ). There is a long exact sequence of (relatively graded) Floer homology groups where the determinant bundle on Y 0,K has been shifted by the dual class of the knot K 0 ⊂ Y 0,K , or equivalently, Y 0,K is considered as a marked cobordism with knot K 0 .
Proof. By combining Remark 6.6, Lemma 6.7, Theorem 3.7, the statement becomes a special case of Theorem 1.3: If Y = Y − ∪ X Y + is a decomposition into compression bodies with K ⊂ X a non-separating knot and τ the corresponding Dehn twist then we have an exact sequence (33) .
The addition of the knot K 0 is equivalent to a shift in the determinant line bundle, as explained in Remark 6.6; we thank Guillem Cazassus for pointing out the missing shift in an earlier version of the paper.
6.2. Exact triangles for tangles. We obtain Floer-theoretic invariants of tangles constructed in [48] exact triangles that are the same as those obtained by Khovanov [19,18] and Khovanov-Rozansky [20]. We assume freely the terminology from [48], in particular moduli spaces of bundles with fixed holonomy for marked surfaces. in the case with tangles the Donaldson-Fukaya categories are enriched in the category of matrix factorizations, see [48]. In [48] we proved that Φ(K) is independent, up to isomorphism, of the choice of decomposition K 1 ∪. . .∪K r . (c) (Group valued invariants) Given objects L ± ∈ Obj(Don (M (X ± , µ ± ))), the complex is a (Z 2 -relatively graded) invariant of K in the derived category of matrix factorizations. In the case that K is a knot, the matrix factorization is of w Id = 0 and so defines a cohomology group HF (K; L − , L + ).
We prove the following surgery exact triangle for these invariants. Let K , K be the tangles obtained by modifying K by a half-twist, respectively adding a cup and cap as in Figure 9.
. Figure 9. Exact triangle for a crossing change, SU (2) case Theorem 6.10. (Exact triangle for changing a crossing) Let K, K , K be tangles in a cylindrical cobordism Y = X × [−1, 1] between X ± = X × {±1} as above and L ± admissible Lagrangian branes in M (X ± , µ ± ). The matrix factorizations for K , K, K are related by an exact triangle Proof. The crossing change depicted in Figure 9 has the following effect on the decomposition of elementary tangles. Suppose that K = K 1 ∪ . . . ∪ K r is such a decomposition. We denote by the corresponding Lagrangian correspondences. The tangle K is obtained by inserting a half-twist of the j and j + 1-st markings after some elementary tangle K i . We may assume that K i is a cylindrical cobordism (that is, admits a Morse function with no critical points) so that the cylindrical Cerf decomposition of K is obtained from that of K by replacing K i with a half-twist. Similarly for K , let K ∩ j , K ∪ j denote the tangles corresponding to the cap and cup of the j-th and j + 1-st strands, so that . . ∪ K r is a cylindrical Cerf decomposition of K . The Lagrangian correspondence for K ∪ j is that associated to the coisotropic submanifold where g i is the holonomy around the j-th marking. Similarly the correspondence for K ∩ j is C ∩ j = (C ∪ j ) t . These correspondences are simply-connected, hence automatically monotone. Let τ C j ∈ Diff(M (X i+1 , µ i+1 ) denote the corresponding fibered Dehn twist. We have CF (K; L − , L + ) = CF (L − , L 1 , . . . , L r , L + ) . . , L r , L + ). Theorem 6.10 now follows from Theorems 1.3 and 3.11.
More generally, in higher rank invariants we obtain an exact triangle for the Khovanov-Rozansky modification.
(a) (Admissible labels) An admissible label is a projection of the barycenter of A onto some face. Under suitable conditions given a marked surface (X, x) with admissible labels µ the moduli space M (X, µ) of bundles with fixed holonomy around the markings x in the conjugacy classes associated to µ is a smooth, compact, monotone symplectic manifold. obtained by composing the functors Φ(L(K i )) is independent of the choice of decomposition into elementary graphs [48]. (c) (Khovanov-Rozansky modification of a graph) Suppose K ⊂ X × [−1, 1] is a trivalent graph with admissible labels. If (K 1 , . . . , K e ) is a cylindrical Cerf decomposition and K ∩ (X × {b i }) a slice such that two points have the same label ν 1 k from 3.12, we obtain a new trivalent graphs K , K by inserting a half-twist, respectively inserting two new vertices as shown in Figure 10 where the intermediate edge represented by a squiggle is labelled ν 2 k from 3.12.
By Theorems 1.3 and 3.13, Theorem 6.12. (Exact triangle for a Khovanov-Rozansky modification) Suppose that G = SU (r), (X, µ) is a marked surface with µ i = µ j = ν 1 k , with ν 1 k = 1 2 (ω k + ω k+1 ). Let ν 2 k = 1 2 (ω k+2 + ω k ) and suppose that K, K , K , L ± are as in Definition 6.11. There exists an exact triangle in the derived category of matrix factorizations   (a) (Necessity of matrix factorizations?) If K is a tangle then by choosing L ± with the same disk number, two of the terms in the above exact sequence may be taken to be cochain complexes (matrix factorizations of zero.) But we know of no reason why the matrix factorization associated to the graph could be taken to be a cochain complex. (b) (More general surgery formulas?) The exact triangle of Khovanov-Rozansky [20] has a similar form. The theories for the other standard markings will not in general have surgery exact triangles of this form, since the corresponding symplectomorphisms are not generalized Dehn twists. It would be interesting to understand whether there is a replacement for the surgery exact triangle in these more general cases.

Fukaya versions of the exact triangle
In [26] the authors constructed A ∞ functors for Lagrangian correspondences between Fukaya categories. The gluing results necessary for the construction of the A ∞ functors for Lagrangian correspondences are proved in [27]. Applied to the Lagrangian correspondences arising from moduli spaces of flat bundles one obtains a (partial) A ∞ -category-valued field theory. We now explain the Fukaya versions of the exact triangle for fibered Dehn twist. 7.1. Open Fukaya version. Let M be a symplectic background. Recall that Fuk(M ) is the A ∞ category whose objects are Lagrangian branes satisfying suitable monotonicity conditions. For the remainder of the paper we take an admissible branes to be monotone with torsion fundamental group and minimal Maslov number at least three, that is, we disallow the minimal Maslov number two case. Morphisms are Floer cochain groups and composition maps count (perturbed) holomorphic polygons with boundary on the given Lagrangians. In [26] we defined a similar category Fuk (M ) whose objects are generalized Lagrangian submanifolds. Let D ♭ Fuk (M ) denote its bounded derived category, as defined by Kontsevich, see [43]. Given any Lagrangian correspondence L 01 ⊂ M − 0 × M 1 with admissible brane structure, [26] constructs  Figure 13. We consider a family of deformations of this surface depending on a parameter ρ as follows. As ρ → 1, we deform the glued surface so that a disk with values in (B, E C,r ) bubbles off. By the proof of Proposition 4.23, the relative invariant for the picture on the right corresponding to ρ = 1 is zero on the cochain level for r sufficiently large. As ρ → 0, we pinch off a pair of pants as in the left side of Figure 13. Let Here M(x, y, z), M(x, y) are the moduli spaces for the 2 resp. 3 marked disk, counted by the compositions µ 1 , µ 2 . The first part consists of the ρ = 0 boundary ofM(x) 1 , and corresponds to µ 2 (f, k). The other boundary components at ρ ∈ (0, 1) are formed by splitting off Floer trajectories v ∈ M(x, y) for L, C t , C, L. as claimed. Now let L 1 be another object in D ♭ Fuk (M ), for simplicity unquilted. Acyclicity of the differential (38) is shown as follows. It suffices to prove acyclicity with L replaced with τ −1 C L. The terms of lowest order in q are µ 2 (f, a) and µ 2 (k, b). As in Section 5.2, the leading term of µ 2 (f, a) corresponds to the canonical injection I(L 1 , C t , C, τ −1 C L) → I(L 1 , τ −1 C L) while the leading term of µ 2 (k, b) corresponds to the canonical injection I(L 1 , τ −1 C L 1 ) → I(L 1 , L). As before, the lowest order terms in complex are acyclic, and filtering the complex by energy shows that entire complex is acyclic. An application of Lemma 7.3 completes the proof of Proposition 7.2.

graph(τ C )
Sketch of proof. The proof is the similar to that of Theorem 7.2, replacing the striplike ends with cylindrical ends. The morphism from C t C to ∅ is obtained from Figure 14. Quilt defining the morphism from (C t , C)[dim(B)] to ∅ the quilted cap in Figure 14, where the outer circle represents a quilted cylindrical end with seams C t , C, the lightly shaded patch maps to M and the darkly shaded patch maps to B. The morphism from ∅ to graph(τ C ) is defined by the Lefschetz-Bott fibration over the cap shown in Figure 15, where the outer circle represents Figure 15. Quilt defining the morphisms from ∅ to graph(τ C ) a cylindrical end with monodromy around the end given by τ C . The composition of the two maps is defined by the surface shown in Figure 16. By deforming the singularity on the surface onto a disk with boundary condition in C, one obtains a null-homotopy of the composition, and so an isomorphism of the mapping cone Cone(C t C[dim(B)] → ∅) with τ C . Applying the functor Applying this exact triangle to any object L of DFuk (M ) leads to the exact triangle given in Theorem 7.1.
Remark 7.5. (A ∞ results for minimal Maslov two?) Note that the A ∞ results do not apply to minimal Maslov number two. These results give exact triangles for the A ∞ -valued field theory using Fukaya categories for moduli spaces of bundles with fixed determinant, but not the version with tangles. It would be interesting to carry out an extension to this case.