Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$

We realise Stroppel's extended arc algebra in the Fukaya-Seidel category of a natural Lefschetz fibration on the generic fiber of the adjoint quotient map on a type $A$ nilpotent slice with two Jordan blocks, and hence obtain a symplectic interpretation of certain parabolic two-block versions of Bernstein-Gelfan'd-Gelfan'd category $\mathcal{O}$. As an application, we give a new geometric construction of the spectral sequence from annular to ordinary Khovanov homology. The heart of the paper is the development of a cylindrical model to compute Fukaya categories of (affine open subsets of) Hilbert schemes of quasi-projective surfaces, which may be of independent interest.


1.2.
Context. Fix a semisimple complex Lie algebra g. The Berstein-Gelfan'd-Gelfan'd category O is an abelian (Noetherian and Artinian) category of finitely generated g-modules, which plays a central role in many parts of representation theory. It contains all the highest-weight modules, is closed under the operations of taking submodules, quotient modules and under tensoring with finite-dimensional modules, and is "minimal" with respect to those properties. Concretely for sl m , it comprises the finitely generated U (sl m )-modules which are h-semisimple and which are locally finite for the nilpotent subalgebra n ⊂ sl m . Given a parabolic subalgebra p ⊂ g, there is a parabolic subcategory O p of O consisting of those modules on which p acts locally finitely. Again for sl m , a partition m = m 1 +· · ·+m k determines a parabolic p; the extreme cases m = 1+· · ·+1 and m = m respectively define O p = O and O p = O f d , the semisimple category of finite-dimensional sl m -modules, so the parabolic subcategories can be viewed as intepolating between these. The action of h decomposes O p into blocks and the block containing the trivial modules is called the principal block O p 0 . It is known that for any p ⊂ g, the category O p 0 is equivalent to the category of modules over a finitedimensional associative algebra; there are algorithmic descriptions of these algebras via quivers with relations [46], but nonetheless working with them concretely is often rather non-trivial. For parabolics p ⊂ sl m associated to two-block partitions m = n + (m − n), the corresponding category O p 0 = O n,m has several other descriptions: it is Morita equivalent to the category of perverse sheaves on the Grassmannian Gr(n, m) constructible with respect to the Schubert stratification [10], and the underlying associative algebra 1 K alg n,m has a diagrammatic description due to Stroppel [44] and Brundan-Stroppel [13,12], in which context it is known as the 'extended arc algebra'. This paper gives a new interpretation of these algebras in terms of Fukaya-Seidel categories associated to natural Lefschetz fibrations on the generic fiber of the adjoint quotient map of the nilpotent (Jacobson-Morozov type) slices associated to nilpotent matrices with two Jordan blocks, and hence a symplectic-geometric construction of these particular principal blocks of parabolic category O. This fits into the general dictionary between symplectic geometry and aspects of representation theory related to categorification, cf. [22,11], and simultaneously extends the symplectic viewpoint on Khovanov homology [41,1,2] to a corresponding viewpoint on annular Khovanov homology, complementing recent work of [18,8].
1.3. Main result. Let π : A m−1 → C be the standard Lefschetz fibration on the Milnor fibre of the A m−1 -surface singularity, so π is an affine conic fibration with m Lefschetz critical points; the symplectic topology of this fibration has been extensively studied, see for instance [22,30]. The Hilbert scheme Hilb n (A m−1 ) has a distinguished divisor D r of subschemes whose projection under π has length < n, the complement of which is an affine variety Y n,m . (When 2n ≤ m, this space is isomorphic to the generic fibre of the restriction of the adjoint quotient map sl m → h/W to a transverse slice meeting the nilpotent cone at a matrix with Jordan blocks of size {n, m−n}, as studied in classical Springer theory [43].) The map π induces a map π n,m : Y n,m → C which is a Lefschetz fibration in the weak sense that its only critical points are of Lefschetz type (it may, however, have critical points at infinity when n > 1). Nonetheless, there is a well-defined Fukaya-Seidel A ∞ -category FS(π n,m ), governing the Floer theory of the Lefschetz thimbles associated to any collection of vanishing paths. We write D π (C) for the derived category of C (split-closure of twisted complexes), working over a field K.
Theorem 1.1. If n ≤ m and K has characteristic zero, D π FS(π n,m ) is quasi-equivalent to the dgcategory of perfect modules perf-K alg n,m over K alg n,m , hence Morita equivalent to parabolic category O n,m .
Remark 1.2. For the extreme case n = m, we have D π FS(π n,m ) = D b (K) (see Corollary 5.17) and K alg n,m := K. When n > m, the map π n,m has no critical point and FS(π n,m ) is an empty category; while K alg n,m is not well-defined. When n = 0, our convention is FS(π n,m ) := K and K alg n,m := K. Under this convention, Theorem 1.1 is true for all non-negative integers n, m. Remark 1.3. The hypothesis on the ground field arises from a formality theorem for the A ∞ -structure on the symplectic side, which we prove, following [1], via methods of non-commutative algebra which rely on inverting all primes. We work over a characteristic zero field unless stated otherwise. Remark 1.4. It is reasonable to expect that symplectic models for the categories O p , for parabolics p corresponding to partitions with more parts (or indeed in other simple g), can also be obtained from the symplectic geometry of nilpotent slices. In general, however, such slices are not known to be reinterpretable in terms of Hilbert schemes of surfaces, so different techniques would be needed to analyse them. Note that the map π n,m : Y n,m → C is itself obtained from the embedding of Y n,m into a Hilbert scheme; it would be interesting to characterise π n,m intrinsically Lie-theoretically.
There is a well-known 'tautological correspondence' between a holomorphic disc in the symmetric product of a complex manifold X and a map of a branched cover of the disc to X itself, cf. [15,14,25,4] amongst others. Following [25] such correspondences are referred to as 'cylindrical models' for computing Floer cohomology. Theorem 1.1 uses the embedding Y n,m → Hilb n (A m−1 ) [28], and the development of a cylindrical model for computing Fukaya-Seidel categories of Hilbert schemes. Whilst the tautological correspondence has been widely exploited before, the proof nonetheless requires numerous technical innovations: we require a model for the whole Fukaya category, our Lagrangian submanifolds are not compact and not cylindrical at infinity, and the Hilbert scheme is related to the symmetric product by a non-trivial crepant resolution. A brief summary of the relevant issues is given in Section 2.1 (our treatment requires only classical transversality theory, but an appeal to virtual perturbation theory would not bypass many of the difficulties).
1.4. Consequences. As commented previously, K alg n,m is Morita equivalent to a category of perverse sheaves on the Grassmannian Gr(m, n). From the (Schubert-compatible) isomorphism Gr(n, m) = Gr(m − n, m), one obtains a geometrically non-trivial: Corollary 1.5. D π FS(π n,m ) is quasi-equivalent to D π FS(π m−n,m ).
Note that the categories appearing in Corollary 1.5 are associated to Hilbert schemes of n respectively m − n points, so take place in symplectic manifolds of different dimension. For another non-trivial symmetry, the identity K alg n,m ∼ = (K alg n,m ) op of the arc algebra with its opposite -which has an easy diagrammatic proof -is non-trivial on the geometric side, because of the non-trivial boundary conditions and 'wrapping' involved in setting up a Fukaya-Seidel category (cf. Section 9.2).
The Fukaya-Seidel category of any Lefschetz fibration has a full exceptional collection. Additivity of Hochschild homology under semi-orthogonal decomposition [24] immediately yields: Corollary 1.6. The Hochschild homology of the extended arc algebra HH * (K alg n,m ) vanishes in degree * = 0, and in degree zero is free of rank m n . The Corollary was previously established by Beliakova et al [8] by an ingenious and involved algebraic argument involving a 'quantum deformation' of the Hochschild complex.
Underlying and extending Theorem 1.1, and the previous Corollaries, there is a rich dictionary between objects in the representation theory and in the symplectic geometry, which is useful in both directions. The blocks of parabolic category O are renowned instances of highest-weight categories, i.e. ones with a full exceptional collection where the exceptional objects generate rather than just splitgenerate. A key point in the proof of Theorem 1.1 is that the endomorphism algebra of the collection of Lagrangian submanifolds associated to projective modules is formal. Crucially, the 'projective' Lagrangians are predicted by the diagrammatic combinatorics on the algebra side; away from Milnor fibres of surface singularities, they have no obvious counterpart for Hilbert schemes of general Lefschetz fibrations on quasi-projective surfaces.
Remark 1. 7. In contrast to a number of recent formality results which hold (for instance) for degree reasons, the formality of the symplectic extended arc algebra holds for non-trivial geometric reasons, and is established, following [1], by building a non-commutative vector field b ∈ HH 1 (FS(π n,m )) counting certain holomorphic curves with prescribed behaviour at infinity. The endomorphism algebra of the Lefschetz thimbles is in general not formal (cf. Appendix A), and it seems hard to prove Theorem 1.1 by directly comparing the A ∞ -algebras associated to distinguished bases of exceptional objects.
In the other direction, the geometry gives rise rather directly to the following: Theorem 1.8. There is a semi-orthogonal decomposition perf-K alg n,m = A n , . . . , A 0 where A j is quasiequivalent to perf-(K alg j,n ⊗ K alg n−j,m−n ) for all j = 0, . . . , n.
The equivalence of Theorem 1.8 does not send tensor products of projectives to projectives, which may make it less transparent from the viewpoint of the extended arc algebras themselves. 1.5. Invariants of braids. The braid group Br m acts by symplectomorphisms on Y n,m ; a braid β defines a symplectomorphism φ This is by definition an invariant of the braid. Recall from [41] that there is a distinguished closed exact Lagrangian submanifold L ℘ ⊂ Y n,2n with the property that Kh symp (κ(β)) := HF * (L ℘ , φ is an invariant of the link closure κ(β) of β ∈ Br n , known as 'symplectic Khovanov cohomology'. This was introduced in [41] as a singly graded sibling to combinatorial Khovanov homology; working over a characteristic zero field K, the isomorphism Kh(κ(β)) ∼ = Kh symp (κ(β)) of Z-graded 3 vector spaces was established in [1,2]. Theorem 1.9. There is a spectral sequence AKh symp (β) ⇒ Kh symp (κ(β)) from the symplectic annular Khovanov homology of β to the symplectic Khovanov cohomology of the link closure κ(β) of β.
Annular Khovanov homology was introduced by Asaeda, Przytycki and Sikora in [3], via a diagrammatic calculus for links in a solid torus. Roberts [34] showed there is a spectral sequence AKh(β) ⇒ Kh(κ(β)), but the fact that AKh(β) splits into summands which can be identified with Hochschild homology groups is non-trivial; this was conjectured by Auroux, Grigsby and Wehrli [6] and proven very recently by Beliakova, Putyra and Wehrli [8]. By contrast, if one had defined the annular Khovanov invariant as such a direct sum of Hochschild homologies, the existence of the spectral sequence would seem rather mysterious: the bimodules P (j) β over the categories FS(π j,m ) do not in themselves have enough information to determine the differentials in the spectral sequence (which do not preserve the decomposition by j). Theorem 1.9 gives a geometric explanation for the existence of a spectral sequence from a direct sum of Hochschild homologies to (symplectic) Khovanov cohomology; the crucial input is the semi-orthogonal decomposition from Theorem 1.8. We remark that there is an analogous spectral sequence from knot Floer homology, viewed as Hochschild homology of a suitable bimodule via [27], to Heegaard Floer homology; it would be interesting to see if that can be derived following the methods of this paper. 2 There are numerous results relating fixed-point Floer cohomology and Hochshild homology, see e.g. [38], which for instance show that the two invariants co-incide when n = 1. When n > 1, since πn,m has critical points at infinity, the established results do not apply in our case; for simplicity we take our basic invariant of a braid to be the bimodule. 3 In this paper we will not discuss the grading; the methods of [1] give rise to a second grading by elements of K, which is conjecturally integral, Markov-invariant and lifts the Z-graded equivalence to a bigraded equivalence.
1.6. Outline of the paper. Let π E : E → C be a Lefschetz fibration on a quasi-projective surface. The map π E defines a map π [n] E : Hilb n (E) → C, by taking the map induced from the sum of copies of π E on the product E n . The map π [n] E is a Lefschetz fibration when restricted to the affine open subvariety of the Hilbert scheme Hilb n (E) comprising subschemes whose projection to C has length n. Section 2 develops a cylindrical model FS cyl for the Fukaya(-Seidel) category of this associated Lefschetz fibration; section 3 establishes some basic properties of FS cyl analogous to the usual Fukaya(-Seidel) category; section 4 relates the cylindrical model to the usual model FS, yielding in general an embedding FS → FS cyl ; and section 5 proves that this embedding is essentially surjective for type A Milnor fibres. Sections 6-7 construct a symplectic version of the extended arc algebra, which is a priori an A ∞ -algebra, and prove that its cohomology agrees with its combinatorial sibling; sections 8-9 then prove the A ∞ -structure is actually formal in characteristic zero. Section 10 illustrates the dictionary between natural objects in the representation theory of the extended arc algebras and their geometric counterparts. Sections 11 and 12 introduce symplectic annular Khovanov homology, and relate this to ordinary Khovanov homology via a particular semiorthogonal decomposition of the Fukaya-Seidel category associated to the (n, 2n)-nilpotent slice (associated to a nilpotent with Jordan blocks of size (n, n)). The Appendix shows that the A ∞ -endomorphism algebra of the thimbles in the Fukaya-Seidel category is in general not formal; this underscores the important contribution of the relationship to representation theory, which picks out a different set of generators for the category (as studied in sections [5][6][7][8][9] which do have formal endomorphism algebra.
Acknowledgements. The authors are grateful to Mohammed Abouzaid, Robert Lipshitz, Jacob Rasmussen, Catharina Stroppel and Filip Zivanovic for many helpful conversations.
2. The cylindrical Fukaya-Seidel category 2.1. Overview. The aim of this section is to define, given a Lefschetz fibration π E on a complex surface E equipped with an exact symplectic structure (that satisfies some mild additional hypotheses, see Section 2.2.4) and a positive integer n, an A ∞ category FS cyl,n (π E ), which we call the n-fold cylindrical Fukaya-Seidel category. Objects in this category are, roughly speaking, unordered n-tuples of pairwise disjoint exact Lagrangians L = {L 1 , . . . , L n }, with L i ⊂ E. The product of the L i defines a Lagrangian (which descends to one) in Sym n (E), which can be lifted to a Lagrangian Sym(L) in Hilb n (E).
Let π [n] E : Hilb n (E) → C be the map induced from the sum of n copies of π E . There is a divisor D r of Hilb n (E) such that Y E := Hilb n (E) \ D r admits an exact symplectic structure, and such that is a Lefschetz fibration in the weak sense that all the critical points of π Y E are of Lefschetz type 4 . Every Lefschetz thimble of π Y E is given by Sym(L) for some L. If E is the A m−1 -Milnor fiber and π E is the conic fibration with m critical points, then Y E and π Y E coincide with Y n,m and π n,m , respectively.
The A ∞ structure of FS cyl,n (π E ) is carefully defined so that it has the property that the category of Lefschetz thimbles of π Y E embeds into FS cyl,n (π E ) as a full subcategory, where on the object level, it is given by Sym(L) → L. This reduces calculations in the Fukaya-Seidel category of π Y E to more accessible calculations in FS cyl,n (π E ), which involve holomorphic curve counts in E itself.
Ideally, we would like to have a choice of perturbation scheme such that there is a bijective correspondence between (perturbed)-holomorphic polygons u : S → Y E contributing to the A ∞ structure of FS(π Y E ) and pairs (π Σ , v) such that π Σ : Σ → S is an n-fold branched covering and v : Σ → E is a solution to a (perturbed)-holomorphic equation. In this case, if we define the A ∞ structure of FS cyl,n (π E ) as a signed rigid count of the pairs (π Σ , v), then it would be tautological that the A ∞ categories FS(π Y E ) and FS cyl,n (π E ) would be equivalent. Problems arise when one implements this idea in practice: • The domain-dependent almost complex structure on E has to be complex (and hence induce an almost complex structure on Hilb n (E)) to have any hope of a bijective correspondence between u and (π Σ , v), which puts some restrictions on the perturbation scheme; • Hamiltonian perturbations in Hilb n (E) that are simultaneously induced from Hamiltonian perturbations in E and which preserve D r are not general enough to achieve transversality; strictly speaking, there is no perturbation scheme that can allow us to define the A ∞ structure of FS cyl,n (π E ) by merely counting (π Σ , v); • We need to include some L for which Sym(L) is not cylindrical in Y E (e.g. to obtain formal A ∞ Floer cochain algebras in the setting of Theorem 1.1), so compactness of the moduli involved has to be carefully addressed. Our main contribution to the construction of FS cyl,n (π E ) is to overcome the above difficulties, roughly as follows: • Instead of using moduli of polygons R d+1 , we use moduli of polygons with unordered interior marked points R d+1,h to define the A ∞ structure. The interior marked points keep track of the branch points of π Σ : Σ → S, and the domain-dependent almost complex structure is only required to be integrable near the interior marked points. This gives us more flexibility for the perturbation scheme and at the same time partially recovers the bijective correspondence between u and (π Σ , v). • Whilst the A ∞ structure of FS cyl,n (π E ) is defined by counting certain solutions u mapping to Hilb n (E) \ D r , rather than pairs (π Σ , v), our set up is sufficiently flexible that the Floer differential and product can be computed by counting solutions (π Σ , v). • To achieve compactness, we need to avoid that solutions in Hilb n (E)\ D r escape to the vertical boundary, horizontal boundary or D r . No escape along the vertical boundary is achieved by modifying Seidel's ingenious set-up in [40]; one takes the base of the Lefschetz fibration to be the upper half-plane, and uses hyperbolic isometries to gauge back perturbed holomorphic curves to actual holomorphic curves. No escape along the horizontal boundary and D r are achieved by positivity of intersection, which relies on a delicate choice of Hamiltonian perturbation scheme and some particular geometric features of D r . Familiarity with [40] may be helpful.

2.2.
Definitions and the setup.
2.2.1. Domain moduli. Let R d+1,h be the moduli space of discs with d + 1 punctures on the boundary and h (unordered, not necessarily pairwise distinct) interior marked points (counted with multiplicities). Let S d+1,h be the universal family of R d+1,h . We fix a distinguished puncture ξ 0 for the elements in R d+1,h consistently, and order the remaining punctures ξ 1 , . . . , ξ d counter-clockwise along the boundary. For S ∈ R d+1,h , we use ∂ j S to denote the boundary component of S between ξ j and ξ j+1 for j = 0, . . . , d (ξ d+1 is understood as ξ 0 ). We use mk(S) to denote the set of interior marked points of S and mult(z) to denote the multiplicity of z ∈ mk(S). Consider a ribbon tree T with d + 1 semi-infinite edges. We choose one of the semi-infinite edges to be distinguished (called the root). We associate to each vertex v of T a non-negative integer label h v . The integer label of (T, indexed by the vertices of T , where each component S v is an element in R |v|,hv and |v| is the valency of v. From now on, we assume the stability condition holds: |v| + 2h v ≥ 3 for every vertex v of T . In particular, we allow a one-valent vertex if the associated integer is positive. Let R T,{hv} be the moduli space of broken discs modelled on (T, {h v } v∈T ). The disjoint union of moduli of broken discs over all stable trees with d + 1 semi-infinite edges and integer label h is denoted by Let Σ be a compact (not necessarily connected) Riemann surface with n(d + 1) pairwise distinct marked points on the boundary. Let Σ be the Riemann surface obtained from Σ by deleting the boundary marked points. Let S ∈ R d+1,h . If there is an n to 1 branched covering π Σ : Σ → S such that such that the branched locus is contained in mk(S) (counted with multiplicities), then (Σ, π Σ ) is called an n-fold lift of S. In other words, the branch locus and mk(S) define two divisors D bl and D mk(S) on S, respectively, and an n-fold lift of S is an n-fold covering such that D mk(S) − D bl is an effective (or zero) divisor. The punctures of Σ that go to ξ j under π Σ are called type-ξ j punctures. Two n-fold lifts (Σ 1 , π Σ 1 ) and (Σ 2 , π Σ 2 ) of S are equivalent if there is a biholomorphism φ : Σ 1 → Σ 2 such that π Σ 2 • φ = π Σ 1 .

2.2.2.
Strip-like ends and marked-points neighborhoods. For each R d+1,h , we make a choice of striplike ends = { 0 , . . . , d } for elements in R d+1,h such that ξ 0 is an output and ξ j is an input for j = 1, . . . , d. Thus, for each S ∈ R d+1,h , we have holomorphic embeddings varying smoothly respect Furthermore, we choose a 'marked-points neighborhood' ν(mk(S)) for each S ∈ R d+1,h , which is a (possibly disconnected) open subset of S containing mk(S), such that Since π E is symplectically locally trivial outside π −1 E (C H ), for H ∈ H γ (E), X H is uniquely determined by the property that for all x ∈ E. If we extend X H to a smooth vector field on E, we also have Next, we consider the class L of properly embedded Lagrangian submanifolds L ⊂ E such that (2.20) and φ H is the associated Hamiltonian diffeomorphism (which is well-defined everywhere because where g A ∈ G aff is the parallel transport from 0 to 1 with respect to A (see Figure 2.1). Finally, we consider the set L cyl,n of (unordered) n-tuples of Lagrangians L = {L 1 , . . . , L n } such that L k ∈ L for k = 1, . . . , n (2.21) there are pairwise disjoint contractible open sets U L k in H such that (2.22) π E (L k ) ⊂ U L k and U L k ∩ ∂H is either empty or contractible In particular, (2.22) implies that the {L k } are pairwise disjoint. We define Sym(L) := q Sn (L 1 × · · · × L n ) ⊂ Conf n (E) ⊂ Sym n (E) (2.23) where q Sn : E n → Sym n (E) and q Sn,H • : (H • ) n → Sym n (H • ) are the quotient maps by the symmetric group. When λ L = ∅ (i.e. when π E (L k ) is compact for all k = 1, . . . , n), we define λ L := {0} so λ L is a non-empty closed interval for all L ∈ L cyl,n .

2.2.5.
Hilbert scheme of points. Let Hilb n (E) be the Hilbert scheme of n points on E, defined with respect to the complex structure J E on E. We denote the Hilbert-Chow divisor by D HC and the relative Hilbert scheme with respect to π E by D r , i.e. this is the divisor of subschemes whose projection under π E has length < n. (We will sometimes write Y E for the complement Hilb n (E)\D r when the particular value of n is implicit or plays no role.) Let π HC : Hilb n (E) → Sym n (E) be the contraction of D HC and Sym n (π E ) : Sym n (E) → Sym n (H • ) be the natural map induced by π E .
For H ∈ C ∞ (E, R), we define H [n] ∈ C ∞ (Conf n (E), R) to be then φ H is a well-defined Hamiltonian diffeomorphism of Conf n (E) and the Hamiltonian vector field X Ht satisfies 1] and H t ∈ H n at (E) for all t, the associated Hamiltonian vector field satisfies (2.31) and φ H is also a well-defined Hamiltonian diffeomorphism of Conf n (E). Moreover, φ −1 H (Sym(L)) = Sym(L ) outside a compact set of Conf n (E) \ D • r for some L ∈ L cyl,n . Therefore, we can define λ φ −1 H (Sym(L)) = λ L . Remark 2.6. We introduce both H n,pre γ (E) and H n γ (E) because, on the one hand, H n γ (E) gives us more freedom to achieve transversality, but on the other, working with H n,pre γ (E) simplifies explicit computations for cases that transversality can be achieved within that more restricted class.
For L 0 = {L 0,k } n k=1 and L 1 = {L 1,k } n k=1 in L cyl,n , a perturbation pair is Lemma 2.8. For any L 0 , L 1 ∈ L cyl,n , the set of perturbation pair is non-empty.
Proof. For any choice of (A L 0 ,L 1 , H L 0 ,L 1 ) satisfying (2.38), by (2.32) and (2.36), we have so the transversality condition (2.39) is satisfied outside a compact subset. By definition (2.29), we are free to perturb H L 0 ,L 1 inside a compact subset so the resullt follows (cf. Remark 2.6).
Given L 0 , L 1 ∈ L cyl,n , a perturbation pair (A L 0 ,L 1 , H L 0 ,L 1 ) and a smooth family J = (J t ) t∈[0,1] , J t ∈ J n (E), we define the Floer cochains (as a vector space over a chosen coefficient field 6 by where o x is the orientation line at x. In this case, we call (A L 0 ,L 1 , H L 0 ,L 1 , J) a Floer cochain datum for L 0 , L 1 ∈ L cyl,n .
If all Lagrangians in L 0 and L 1 are equipped with a grading function, Lemma 2.3 shows the product of grading functions on Sym(L i ) is a grading function, and we use that to grade X (H L 0 ,L 1 , L 0 , L 1 ). Example 2.10. In the situation of Example 2.9, the grading of x = (x 1 , . . . , x n ) is where |x j | is the Floer grading of x j in E.
2.2.7. Floer data. We define the following group of gauge transformations: Proof. It sits inside the weak fibration and recalling that N d+1,h mk is contractible with smooth boundary.
Similarly to (2.33), we define Consider a trivial G aff -bundle over S d+1,h with fiber R and equip it with the trivial connection 0 aff .
(2.52) Now, for S ∈ R d+1,h , we define P aff (S, {(A j , λ j,0 , λ j,1 )} j ) by A cylindrical Lagrangian label is a choice of an element L j ∈ L cyl,n associated to ∂ j S for all j. We choose a cylindrical Lagrangian label and Floer cochain data (A 0 , H 0 , J 0 ) and (A j , H j , J j ) for (L 0 , L d ) and (L j−1 , L j ) for j = 1, . . . , d, respectively.
, where for j = 0, it should be understood as (A j , λ L j−1 , λ L j ) := (A 0 , λ L 0 , λ L d ). Recall that we have chosen J j = (J j,t ) t∈[0,1] , J j,t ∈ J (E) for j = 0, . . . , d. We equip S with the following additional data a smooth family J = (J z ) z∈S , J z ∈ J n (E) such that J z = J [n] E in ν(mk(S)) (2.54) and J j (s,t) = J j,t for all j K ∈ Ω 1 (S, C ∞ (Conf n (E), R)) such that for each w ∈ T S, K(w) ∈ H n A S (w) (E) (2.55) and * j K converges to H j dt exponentially fast in any C r topology for all j. Moreover, we require K to vanish in ν(mk(S)).
Let X K ∈ Ω 1 (S, C ∞ (Conf n (E), T E)) be the associated one-form with values in Hamiltonian vector fields Remark 2.12. We require * j K to converge to H j dt exponentially fast, instead of co-inciding with it, because it makes it easier to achieve regularity of the moduli whilst maintaining compatibility with gluing (see [40,Remark 4.7]).
Remark 2.13. Note that J can be extended smoothly to a family of tamed almost complex structures in Conf n (E ) and K can be extended smoothly to an element in Ω 1 (S, C ∞ (Conf n (E ), R)).
We choose a smooth family of (A S , J, K) for S varying in R d+1,h . Given x 0 ∈ X (H 0 , L 0 , L d ) and for z ∈ mk(S), the multiplicity of intersection with D HC equals mult(z) u(z) ∈ Sym(L j ) for z ∈ ∂ j S lim s→±∞ u( j (s, ·)) = x j (·) uniformly (2.56) Note that the conditions: X K vanishes near mk(S), J = J

[n]
E near mk(S), and (Du| z − X K | z ) 0,1 = 0, together imply that u is a J

[n]
E -holomorphic map near mk(S). Therefore, there is a well-defined multiplicity of intersection between u and D HC at z ∈ mk(S). Moreover, u −1 (D HC ) = mk(S) guarantees that (Du| z − X K | z ) 0,1 = 0 is a well-defined equation for all z ∈ S, by identifying Conf n (E) as a subset of Hilb n (E). Lemma 2.14. For generic (J, K) such that (2.54) and (2.55) are satisfied, every solution u ∈ R d+1,h (x 0 ; x d , . . . , x 1 ) is regular.
Proof. Without loss of generality, we fix the domain S ∈ R d+1,h . Let B be the space of smooth maps u : S → Hilb n (E) such that u −1 (D HC ) ⊂ ν(mk(S)). Note that B is an open subset of C ∞ (S, Hilb n (E)).
We want to run the Fredholm theory for appropriate Sobolev completions of B k for each k.
For each k, using Lemma 2.4, we pick a symplectic form ω k on Hilb n (E) which agrees with ω Conf n (E) outside V k and tames the complex structure on Hilb n (E). This induces a family of Riemannian metrics g k = (g k,z ) z∈S on Hilb n (E) which agree with the metric induced from ω Conf n (E) and J z outside V k . We use g k to form an appropriate Sobolev completion of B k (or the corresponding function space of the graphs of the maps).
The boundary conditions ensure that every solution u of (2.56) has non-empty intersection with a compact subset of Conf n (E) \ D • r outside mk(S). For u ∈ B k , Gromov's graph trick applies, because on the one hand, X K | S\ν(mk(S)) is the Hamiltonian field with respect to ω k , and on the other, X K | ν(mk(S)) = 0, so there it is tautologically the Hamiltonian field with respect to ω k . Hence regularity of u can be achieved by choosing K generically amongst functions satisfying (2.55), i.e. although we require that K| ν(mk(S)) = 0, the freedom of K outside ν(mk(S)) is sufficient to achieve regularity.
The outcome is a sequence of residual sets S k in the space of all K satisfying (2.55) such that for every K ∈ S k , every u ∈ B k satisfying (2.56) is regular. Therefore, we can take S := ∩ k S k , which is still dense and every u ∈ B satisfying (2.56) is regular for every K ∈ S.
There is a correspondence of maps as follows. An n-fold branched covering π Σ : Σ → S and a continuous map v : Σ → E together uniquely determine a continuous map u : S → Sym n (E), given by u(z) = v(π −1 Σ (z)), counted with multiplicity. Conversely, if u : S → Sym n (E) is complex analytic near the big diagonal ∆ E and Im(u) is not contained in ∆ E , then u uniquely determines an n-fold branched covering π Σ : Σ → S and a continuous map v : We call this the 'tautological correspondence'. For a proof in the case that dim C E = 1, see for example, [31,Lemma 3.6] or [26,Section 13]. In particular, it implies that Lemma 2.15 (Tautological correspondence). Every solution u of (2.56) determines uniquely an nfold lift (Σ, π Σ ) of S and a map v : Proof. Every solution u of (2.56) is complex analytic near D HC and Im(u) is not contained in D HC . Therefore, by the tautological correspondence, we get an n-fold branched covering π Σ : Σ → S and a continuous map v : Σ → E such that π HC • u(z) = v(π −1 Σ (z)). The last thing to check is that the branch locus of π Σ is contained in mk(S) (counted with multiplicities), which in turn follows from (2.57) and the third condition in (2.56).
Remark 2.16. Suppose that we are in the situation of Example 2.9 for all the pairs (L 0 , L d ) and (L j−1 , L j ). Suppose also that J z = (J z ) [n] ∈ J n (E) and K(w) = (K (w)) [n] ∈ H n,pre A S (v) (E) in (2.54) and (2.55), respectively. Then we can formulate (2.56) in terms of a set of constraints to maps v : Σ → E and (Σ, π Σ ). In particular, we would have (Dv| z − X π * Σ K | z ) 0,1 = 0 with respect to (J π Σ (z) ) v(z) for all z ∈ Σ (2.58) Remark 2.17. Suppose we are in the situation of Remark 2.16. The corresponding Fredholm problem has a virtual dimension and we can calculate it as follows: The Euler characteristic of Σ is n − h. The sum of Maslov indices at the inputs and outputs are |x 0 | − n dim C E and d j=1 |x j |, respectively. The dimension of the moduli space of domains in which Σ varies is 2h + (d − 2). Therefore, the virtual dimenison is Since dim C E = 2, the virtual dimension is given by which coincides with the virtual dimension of the solutions to (2.56), as expected. In other words, we replace the punctures of S by closed intervals, which can be identified with { j (±∞, t)|t ∈ [0, 1]}. For u ∈ R d+1,h (x 0 ; x d , . . . , x 1 ), we define G(u) := (Sym n (π E ) • π HC • u, id S ) : S → Sym n (H • ) × S, which can be extended continuously to by sending j (±∞, t) to (Sym n (π E ) • π HC • x j (t), j (±∞, t)) for all j. Note that, G(u)(∂B(S)) lies inside Proof. Since Sym(U L j ) is a contractible open set, we have Map(x 0 ; x d , . . . , x 1 ) = π 2 (C n+1 , S 1 ) = π 1 (S 1 ) = Z. In this case, the intersection pairing is a multiple of the winding number along the boundary and the winding number of [G(u)] is 1, by definition. Proof. Let u := Sym n (π E ) • π HC • u : S → Sym n (H • ). There are two kinds of intersections between u and ∆ H • , namely, at or away from mk(S). If z ∈ mk(S), then (2.54), (2.55) and (2.56) imply that u is (j Sym n (H • ) , j S )-holomorphic near z. It implies that the contribution of the algebraic intersection at z is mult(z) and z is the only intersection with ∆ H • in a small neighborhood of z. Summing over all z ∈ mk(S), the contribution to the algebraic intersection is h. We need to show that the contribution from other intersections with ∆ H • is positive. Let z 0 ∈ S \ mk(S) be such that u(z 0 ) ∈ ∆ H • . Let B(z 0 ) ⊂ S be a small disc centred at z 0 . By Lemma 2.2, we must have u(z 0 ) ∈ D r . To show that the contribution at u(z 0 ) is positive, it suffices to show that the algebraic intersection number between u(B(z 0 )) and D r at u(z 0 ) is positive. This is in turn achieved by Gromov's graph trick, and the fact that our choice of X K is tangential to D r at u(z 0 ).
We give a detailed explanation of the last sentence. By the definitions (2.26) and (2.29), near D r , we have J z 0 = (J ) [n] and K(w) = (π * E H w ) [n] for some H w ∈ H A S (w) (H • ) for w ∈ T B(z 0 ). It means that, by (2.10) and by shrinking B(z 0 ) if necessary, we have (2.68) (X K (η)) τ ∈ T τ D r for all η ∈ T B(z 0 ) and τ ∈ D r , where T τ D r is understood to be the tangent space of the smallest strata of D r that contains τ .
We consider the graph trick. Let F = (u, id) : B(z 0 ) → Conf n (E) × B(z 0 ). Let J F be the almost complex structure of Conf n (E) × B(z 0 ) characterised by is the projection. In holomorphic coordinates (s, t), we have where the last line uses (Du − X K ) 0,1 = 0. Similarly, we also have (DF + J F • DF • j S )(∂ t ) = 0 and hence (DF ) 0,1 = 0 and F is (J F , j S )-holomorphic.
Notice that all strata of D • r × B(z 0 ) are J F -holomorphic due to (2.68), (2.69) and the fact that ∆ H • is a J z -holomorphic subvariety. As a result, the intersection F (z 0 ) between F (B(z 0 )) and D • r × B(z 0 ) is positive. This completes the proof. Proof. One can choose (A L , H L ) such that there exists a total ordering with respect to intersection, in the following sense: one can order the tuple L = {L 1 , . . . , L n } such that, if the Lagrangians are moved by the positive isotopy to {L + 1 , . . . , L + n }, then L i ∩ L + j = ∅ only if i ≥ j. In that case, the only possible chords are tuples with x i ∈ L i ∩ L + i for each i, which then ensures that Sym n (π E )(π HC (x(t))) ∈ Sym(U L ) for all t. The result follows, because the boundary conditions also project to Sym(U L ), which is contractible.
2.2.9. Energy. The product symplectic form on Conf n (E) cannot be smoothly extended to Hilb n (E), and nor can the induced metric. Therefore, we will define energy and discuss compactness with the help of the corresponding maps v : Σ → E obtained from Lemma 2.15.
By (2.54) and (2.55), there exists a compact subset C of Conf n (E) \ D • r and a smooth family J = (J z ) z∈S , J z ∈ J (E), and (2.73) where U is the closure of U . By adjusting U , we assume that u(S) is transversal to ∂U so that u −1 (∂U ) ⊂ S is a smooth manifold with boundary. Note that, by definition, mk(S) ⊂ S out and S in contains small neighborhoods of the punctures. We also define Σ in = π −1 Σ (S in ) and Σ out = π −1 Σ (S out ).
Definition 2.23 (Energy). The energy of u is defined to be where g is the metric induced by J and ω Conf n (E) and g is the metric induced by J and ω E .
The energy is defined this way, instead of as 1 2 S Du − X K 2 g dvol, because u * X K is not defined at u −1 (D HC ) ⊂ mk(S), so it is not a priori clear that the latter expression is related to the action of the asymptotes of u. However, we show the following (see also (2.81)).
This is in turn clear because by (2.75) and (2.76), both J and X K split as products. Since ω Conf n (E) is also a product, the metric g is the product metric.
More precisely, let G ⊂ S \ mk(S) be a small open set such that u(G) ∩ C = ∅ and π −1 Σ (G) is a disjoint union of open sets G 1 , . . . , G n ⊂ Σ. For z ∈ G and z j := G j ∩ π −1 Σ (z) for j = 1, . . . , n, we have canonical identifications By (2.75) and (2.76), both J z and X K (and ω Conf n (E) ) respect this product decomposition and every summand is given by J z and X π Σ * K , respectively. Therefore, (2.79) is true for G. Now, the result follows by summing over these small open subsets G ⊂ S \ mk(S).
By taking a sequence of larger and larger U and applying Lemma 2.24, we have Our next task is to derive a uniform upper bound for E(u) that depends only on (A S , J, K) and the Lagrangian boundary condition.
Consider again the graph construction. Letv := (v, id) : Σ → E × Σ and define on E × Σ the following 2-forms: so, by tameness of J , we have There is a constant T > 0 such that for any choice of solution u of (2.56) (and hence the corresponding v) and U , we have | Σ outv * R π * Σ K | < T .
Proof. By (2.13), there exists K ∈ Ω 1 (S, C ∞ (H • )) such that K = π * E K and K (w) ∈ H A S (w) (H • ) for all w ∈ T S. Recall also that K = 0 near mk(S). It implies that where R − is the corresponding curvature defined by − using (2.84). Note that by (2.38) and (2.55), K converges exponentially fast in any C r topology with respect to s over strip-like ends of S so there is a constant T > 0 such that for every j and any section f : S → H • × S, we have | Im( j ) f * R K | < T . Moreover, (2.12), (2.38) and the flatness of A S implies that R K takes values in functions of H • that are supported in a compact subset of H • . Therefore, by (2.87), R π * Σ K takes values in functions of E that are uniformly bounded (the bound only depends on K but not u), and there is a constant T > 0 such that for every j and any section f : Let Σ e ⊂ Σ be the closure of the complement of the strip-like ends. The discussion in the previous paragraph implies that | Σ outv * R π * Σ K | is bounded above by (d + 1)T plus the integral of a bounded function over Σ e , where the bound on that function depends only on K . The result follows.
Similarly, Letû := (u, id) : S → Conf n (E) × S and define on Conf n (E) × S the following 2-forms We have the parallel lemma.
Lemma 2.26. There is a constant T > 0 such that for any choice of solution u of (2.56) and U , we have | S inû * R K | < T .
Proof. As before, there exists T > 0 such that for every section S → Conf n (E) × S and every j, we have | Im( j ) f * R K | < T . Ouside U , R K takes values in bounded functions on Hilb n (E) \ U with bound determined by K . Since U is relatively compact, there is also a bound for the function-values of R K inside U that is independent of u. Over all, if we let S e be the closure of the complement of strip-like ends in S, then | S inû * R K | is bounded above by (d + 1)T plus an integration of a bounded function over S e , so the result follows.
The primitive one form θ E for ω E induces a primitive one form θ Conf n (E) for ω Conf n (E) . It is clear that are primitives of ω top π * Σ K and ω top K , respectively.
so there is a constant T > 0 such that for all u satisfying (2.56), we have E(u) < T .
Proof. Apply Stoke's theorem to the terms on the left hand side of (2.92). Note that, because the Floer data splits into a product outside U and the orientation of the curves in the two summands are opposite to one other. Similarly, Therefore, we get (2.92).
Moreover, there is a constant T > 0 (independent of u) such that ∂Sû * θ top K < T (see [ The right hand sided is in turn bounded above by a constant independent of u, by Lemma 2.27.
everywhere, so the symmetric product of the asymptotes of v corresponding to a fixed puncture of S is not necessarily a X K -Hamiltonian chord. This implies that the integral over the strip-like ends might diverge.
Lemma 2.30. Fix S ∈ R d+1,h . Let u k : S k → Hilb n (E) be a sequence in R d+1,h (x 0 ; x d , . . . , x 1 ) such that S k converges to S ∈ R d+1,h . If there exists z k ∈ S k \ ν(mk(S k )) such that Du k (z k ) − X K (z k ) g diverges to infinity, then for any compact subset C S in the universal family over R d+1,h and N > 0, there exists k > N such that z k / ∈ C S .
Proof. For simplicity, we assume S k = S for all k; the reasoning below adapts directly to the general case. Suppose the lemma were false, then there is a subsequence of z k which converges to z ∞ ∈ S. First assume that z ∞ is an interior point of S. Let B ⊂ S be a small ball centered at z ∞ and conformally identify B with a 3 -ball in C centered at the origin. Under this identification, we can assume that for all k, z k lies in the -ball centered at z ∞ .
We apply the rescaling trick to the -ball B k centered at z k . The outcome is a sequence of maps u k • φ k : r k B k → Conf n (E), for appropriate r k > 1 and φ k , such that there exists T > 0 with sup r k B k D(u k • φ k ) < T for all k. Since π −1 Σ (B) is a disjoint union of n discs in Σ, the u k • φ k induce corresponding maps V k,j : r k B k → E for j = 1, . . . , n as in Lemma 2.15. By the same reasoning in Lemma 2.28, we know that there exists T > 0 such that sup r k B k DV k,j g < T for all k.
Since the metric on E extends to a metric on E and V k,j ∈ W 1,p (for p > 2) with uniformly bounded W 1,p -norm, by applying compactness of W 1,p in C 0 to V k,j for each j in turn, we get a subsequence which converges uniformly to continuous functions V ∞,j : C → E for all j = 1, . . . , n. Moreover, V ∞,j satisfies the (j C , J z∞ )-holomorphic equation for all j, because of the stipulation that J z = (J z ) [n] for all z. Elliptic bootstrapping shows that V ∞,j is smooth. Moreover, since Du k (z k ) − X K (z k ) g diverges to infinity, at least one of V ∞,j is not a constant map. By Lemma 2.28, DV ∞,j has finite L 2 -norm so we can apply removal of singularities to conclude that every V ∞,j extends to a J z∞ -holomorphic map CP 1 → E (and at least one of them is not a constant).
By assumption (2.11), every non-constant J z∞ -holomorphic map CP 1 → E has stricly positive algebraic intersection with D E , which in turn implies that this is true for v k for large k (because all other sphere bubbles, if any, also contribute positively to the algebraic intersection). However, Im(v k ) is contained in E, giving a contradiction. Now, if instead z ∞ ∈ ∂S, then we can apply the same rescaling trick and the outcome is a J z∞holomorphic disc with appropriate Lagrangian boundary for each j, and at least one of them is nonconstant. By exactness of the Lagrangian boundary, the J z∞ -holomorphic disc has strictly positive algebraic intersection with D E , which in turn implies that that is true for v k for large k, yielding the same contradiction.
Next, we consider the case that we have a uniform bound on sup S k \ν(mk(S k )) Du k (z k ) − X K (z k ) g .
for all k, then there exists u ∞ ∈ R d+1,h (x 0 ; x d , . . . , x 1 ) such that a subsequence of u k converges (uniformly on compact subsets) to u ∞ .
Proof. Without loss of generality, assume S k = S for all k. Let v k : Σ k → E be the corresponding maps. Since there are only finitely many n-fold lifts of S, by passing to a subsequence, we can assume Σ k = Σ for all k.
In particular, for every relatively compact open subset G of S, . Therefore, so does v ∞ , and elliptic bootstrapping implies that v ∞ is smooth outside C E and satisfies (2.85). Let u ∞ : G → Sym n (E ) be the corresponding continuous map induced from v ∞ . We can apply elliptic bootstrapping away from ∆ E so u ∞ is actually smooth away from ∆ E , so v ∞ is smooth everywhere.
We will prove that Given these, u ∞ uniquely lifts to a map u ∞ : G → Hilb n (E) such that u −1 ∞ (D HC ) ⊂ G ∩ ν(mk(S)). Note that this will be true for for every relatively compact open subset G of S.
On the other hand, we will also prove that, by possibly passing to a subsequence, there is a compact subset C of Conf n (E) such that u k (Im( j )) ⊂ C for all k and j (2.102) so we can apply compactness of u k over strip-like ends inside C. Combining all this information, and by a diagonal subsequence argument, we obtain u ∞ ∈ R d+1,h (x 0 ; x d , . . . , x 1 ) such that u k has a subsequence that is uniformly converging to u ∞ over S \ ν(mk(S)). Note that we cannot guarantee the convergence is also uniform over ν(mk(S)), because u ∞ is lifted from u ∞ .
However, we can argue as follows. If there is a subsequence of u k which converges uniformly over ν(mk(S)), then it converges pointwise to u ∞ over ν(mk(S)) \ mk(S). By continuity, the limit must be u ∞ and we are done. If there is no such subsequence, then for all T > 0 and N > 0, there exists k > N such that sup ν(mk(S)) Du k g > T , where g is the induced metric from a choice of a symplectic form on Hilb n (E) that tames J E -holomorphic sphere intersect D r strictly positively (see Lemma 2.5) so Im(u k ) ∩ D r = ∅ for large k. This contradicts our assumption that h = I x 0 ;x d ,...,x 1 , by Corollary 2.20. Now, we need to justify the claims in the previous proof. Proof. We can make (2.85) more akin to the situation in [40], namely, if we define A Σ = π * Σ A S , K Σ = π * Σ K and (J Σ ) z = J π Σ (z) for z ∈ Σ, then outside a compact set of E, (2.85) becomes Note also that our choice of Floer data (A S , K, J) (see (2.25) and (2.36)) and the induced Floer data (A Σ , K Σ , J Σ ) guarantee that over each strip-like end, we have for all a i and b i and all j = 1, . . . , d. For j = 0, we have Proof. Suppose not. Then there is a point z 0 ∈ G \ mk(S) such that u ∞ (z 0 ) ∈ ∆ E (here, we have applied Lemma 2.32 and Proposition 2.33 to replace ∆ E by ∆ E ).
Then, by the same reasoning as in Lemma 2.19, Sym n (π E ) • u ∞ (G) and ∆ H intersect positively at z 0 . It violates the assumption that h = I x 0 ;x d ,...,x 1 .
Proof. Suppose not, then there exists j ∈ {0, . . . , d} and a sequence z k = j (s k , t k ) such that u k (z k ) goes to the infinite end of Conf n (E). Moreover, we know that t k has a subsequence converging to some t ∞ ∈ [0, 1] and s k goes to infinity. Pick a small neighborhood D ⊂ C of (0, t ∞ ) and consider the mapsû k : D → Conf n (E) given byû k (s, t) = u k ( j (s + s k , t)). Letv k,a : D → E be the corresponding maps for a = 1, . . . , n.
Since there is no energy concentration, there is a subsequence of k such that for each a = 1, . . . , n, v k,a converges uniformly to a continuous mapv ∞,a : D → E ; these in turn induce a continuous map u ∞ : D → Sym n (E ). By applying Lemma 2.32, 2.34 and Proposition 2.33, we get the corresponding results (2.99), (2.100) and (2.101) forv ∞,a andû ∞ . This contradicts the assumption that u k (z k ) goes to infinity in Conf n (E).
2.2.11. The definition. An object of FS cyl,n (π E ) is an element L ∈ L cyl,n . Given two objects under the additional assumption that if L 0 = L 1 , the perturbation pair chosen is given by Lemma 2.22. The corresponding morphism space is Then we pick a consistent choice of Floer data as follows: For a generic consistent choice of Floer data, all elements in R d+1,h (x 0 ; x d , . . . , x 1 ) are regular.
The A ∞ structure is now defined by where, when h = 0 and d = 1, #R d+1,h (x 0 ; x d , . . . , x 1 ) is understood as the signed count after dividing out by the R-symmetry (and where we suppress the discussion of signs).

Properties of the cylindrical Fukaya-Seidel category
At this point, given a four-dimensional exact Lefschetz fibration π E : E → H, we have constructed for each n ≥ 1 an A ∞ category FS cyl,n (π E ) which captures certain Floer-theoretic computations in the Hilbert scheme of E, or rather its affine subset Y E . Unsurprisingly, the resulting A ∞ category satisfies typical properties of Fukaya-type categories, with proofs which are minor modifications of those which pertain in more familiar settings. This short section briefly elaborates upon some of these; to keep the exposition of manageable length, the proofs are only sketched.
Sketch of proof. To give a cochain representative of the cohomological unit of HF (L, L), we need to consider S ∈ R 0+1,0 . We put the Floer cochain datum (A L,L , H L,L , J L,L ) on the outgoing strip-like end and stablize the disc S by picking (A S , K, J) satisfying (2.52), (2.54) and (2.55). The Lagrangian boundary condition is a moving one from φ H L,L (Sym(L)) to Sym(L).
We then form the moduli space of maps u : S → Hilb n (E) \ (D r ∪ D HC ) such that u is asymptotic to x 0 ∈ X (H L,L , L, L) near the strip-like end. The signed rigid count of this moduli gives a cochain level representative of the cohomological unit.
Note that we only need to consider R 0+1,h for h = 0 because, by Lemma 2.22, the corresponding moduli space for h > 0 is necessarily empty.
Lemma 3.2 (Hamiltonian invariance). If L 0 , L 1 ∈ L cyl,n can be connected by a smooth family of elements Sketch of proof. Without loss of generality, we can assume that L 1 is C 1 -close to L 0 . In particular, we can assume that For each integer h ≥ 0 and S ∈ R 0+1,h , we equip S with the moving Lagrangian boundary label L t on ∂S. Then, we pick (A S , K, J) such that (2.52), (2.54) and (2.55) are satisfied and compatible with gluing with elements in R 1+1,k for all k. The rigid count of the corresponding solutions of (2.56) give us an element α in CF 0 (L 0 , L 1 ).
We claim that α is a cocycle. Indeed, by the same argument as in Lemma 2.22, when the Floer cochain datum is carefully chosen, the rigid count of solutions for h > 0 vanishes. In this case, α is a sum of continuation elements Similarly, we can construct a cocycle β in CF 0 (L 1 , L 0 ). By gluing, one checks that µ 2 (α, β) and µ 2 (β, α) are cohomological units (for suitable choices of Floer cochain datum as above, these cohomological units agree with the classical units).

Canonical embeddings. Let W be a contractible (hence connected) open subset of H such that
The conditions that W is contractible and satisfies (3.1) imply that W c := Int(H \ W ) is also contractible and satifies (3.1), where Int(−) stands for the interior.
Let K be an object of FS cyl,k W c (π E ). Each object L of FS cyl,n W (π E ) determine a object L K in FS cyl,n+k (π E ) given by adding to L the Lagrangians in K.
, which on objects is given by sending L to L K.
Proof. By (3.1), W contains a neighborhood of [R, ∞) or (−∞, R]. Therefore, we can choose the Hamiltonian term in the Floer data such that each element in X (L 0 K, L 1 K) is a tuple given by adjoining an element in X (L 0 , L 1 ) with another element in X (K, K). It establishes a bijective correspondence between the generators in CF (L 0 K, L 1 K) and CF (L 0 , L 1 ) ⊗ CF (K, K). We can arrange the Floer data such that CF 0 (K, K) is rank one and hence generated by a chain level representative e of the cohomological unit.
The A ∞ functor K has first order term sending x ∈ CF (L 0 , L 1 ) to the generator in CF (L 0 K, L 1 K) corresponding to x ⊗ e, and has no higher order terms an an A ∞ functor. To check that K is indeed an A ∞ functor, it suffices to show that there is a bijective correspondence between the moduli governing the A ∞ structures in FS cyl,n W (π E ) and the moduli governing the A ∞ structures in FS cyl,n+k (π E ) with all inputs being of the form x ⊗ e.
By projecting to H • via π E and appealing to the open mapping theorem, we know that every solution v : Σ → E contributing to the A ∞ structure of FS cyl,n W (π E ) has image inside W . These solutions persist when we consider the Floer theory in FS cyl,n+k (π E ). Moreover, the domain Σ and the branched covering map π Σ determine a complex structure on the disc, and hence an element S ∈ R d+1 . Adjoining these solutions u with the rigid map from S to E asymptotic to e at all punctures, we get precisely all the solutions contributing to the moduli governing the A ∞ structure of FS cyl,n+k (π E ) with all inputs being of the form x ⊗ e. This finishes the proof.
Using Corollary 3.4, we can inductively construct a plethora of exact triangles. For example, we will prove in Section 5.1 that our setup applies to the standard Lefschetz fibration π E on the A m−1 -Milnor fiber E. There are well-known exact triangles relating matching spheres and thimbles in FS cyl,1 (π E ) = FS(π E ). We can obtain exact triangles in FS cyl,n (π E ) by adjoining matching spheres/thimbles to the exact triangles in FS cyl,1 (π E ) (see Figure 3.1). 3.3. Serre functor. An important observation due to Kontsevich and Seidel is that the global monodromy τ induces the Serre functor on the Fukaya-Seidel cayegory up to degree shift [37,36,39]. In our context, the global monodromy of π E induces an auto-equivalence of FS cyl,n (E) for each n. By formally the same argument: Proposition 3.5. The global monodromy of π E induces the Serre functor on FS cyl,n (E) up to a degree shift by −2n. Sketch of proof. As a graded symplectomorphism, we require that τ acts as the identity on the trivialization of the bicanonical bundle in a compact region containing the critical points. It means that for each compact exact graded Lagrangian L in E, τ (L) = L as graded objects. On the chain level, there is a canonical isomorphism (see Figure 3.2) given by sending an intersectin of L + 0 ∩ L 1 to the corresponding intersection point of τ (L 0 ) ∩ L + 1 but viewing the latter as a generator in the dual group CF (L 1 , τ (L 0 )). Here, the superscript + means positive isotopy near infinity. In particular, when L 0 is compact, a generator x of CF i (L 0 , L 1 ) is mapped to the linear dual of the corresponding generator ). The canonical isomorphism (3.2) lifts to the canonical isomorphism for Lagrangin tuples Moreover, under the canonical isomorphism (3.3), there is a bijective correspondence between the moduli computing higher A ∞ operations In other words, on the object level, the global monodromy τ [−2n] sends L to τ (L)[−2n], whose Yoneda image is in turn isomorphic to CF (L, −) ∨ . This is the first piece of geometric information that enters into Seidel's argument.

Comparing Fukaya-Seidel categories
In this section, we discuss the relation between FS cyl,n (π E ) and the Fukaya-Seidel category of the Lefschetz fibration on Hilb n (E) \ D r induced by π E (we recall that this is a 'weak' Lefschetz fibration, in the sense that it may have critical points at infinity, but it still has a Fukaya-Seidel category of Lagrangians proper over the base, as constructed in [35]). This allows us to translate our subsequent study of FS cyl,n (π E ) back to the usual Fukaya-Seidel category.

4.1.
A directed subcategory. Let c 1 , . . . , c m be the set of critical values of π E . By applying a compactly supported diffeomorphism of H • , we can assume that c k := k + √ −1 ∈ H • for k = 1, . . . , m. For simplicity, we assume that there is exactly one critical point lying above a critical value. By possibly applying a further compactly supported diffeomorphism of H • , we can assume that the symplectic parallel transport is well-defined everywhere in is not a critical value of π E for all t = 0, 1, and such that the vanishing cycles from the critical points lying above c a to c b match up under symplectic parallel transport along γ to give a Lagrangian matching sphere L γ in E. A thimble path γ : [0, 1] → H of π E : E → H • is a smooth path from c a , for some a ∈ {1, . . . , m}, to a point on the real line such that γ(t) ∈ H • \ {c b |b = 1, . . . , m} for t = 0, 1, and the symplectic parallel transport of the vanishing cycle from c a gives a Lagrangian disc (thimble) L γ in E.
For an n-tuple Γ = {γ 1 , . . . , γ n } of pairwise disjoint embedded curves in H • such that each curve is either a matching path or a thimble path of π E : E → H • , we can define the corresponding n-tuple of Lagrangians In this case, we call Γ an admissible tuple.
For r ∈ R and k ∈ {1, . . . , m}, let l r,k be the straight line joining r and c k . Every l r,k is a thimble path.
x x x x Let I be the set of cardinality n subsets of {1, . . . , m}. We define a partial ordering on I, called the Bruhat order, as follows: For I 0 , I 1 ∈ I, I 0 ≤ I 1 if and only if there is a bijection f : Let f : I → I aff (see Section 2.2.6) be a function such that Given f and I = {i 1 < · · · < i n } ∈ I, we define Figure 4.1).
Let T I := L Γ I ∈ L cyl,n . For a generic perturbation of ω E inside a compact subset, which changes the symplectic connection but keeps the symplectic Lefschetz fibration structure, we have L γ I,k L γ I ,k for all (I, k) = (I , k ). (4.6) Moreover, by applying a compactly supported diffeomorphism of H • again, we can assume that there exists 0 < η < 1 such that π E is symplectically locally trivial in π −1 E ({z ∈ H • |im(z) < 1 − η}) and for any γ I,k = γ I ,k ∈ Γ, we have γ I,k ∩ γ I ,k ⊂ {z ∈ H • |im(z) < 1 − η or im(z) = 1}. (Since we have applied a compactly supported diffeomorphism of H • , the symplectic form on the base becomes the pullback of ω H • in place of ω H • , but this does not affect the current discussion.) We are interested in the subcategory of FS cyl,n (π E ) that is (split-)generated by {T I |I ∈ I}.
Proof. For the first statement, it suffices to note that if I 0 ≮ I 1 then we can find an isotopy of thimbles Figure 4.2). It implies that CF (T I 0 1 , T I 1 ) = HF (T I 0 1 , T I 1 ) = 0, but by Lemma 3.2, T I 0 1 is quasi-isomorphic to T I 0 as objects in FS cyl,n (π E ), so the first statement follows.
For the second one, when I 0 = I 1 = I, we can find an isotopy of thimbles T I t ∈ L cyl,n such that is a singleton. By Lemma 3.2 again, it implies that HF (T I , T I ) has rank one, so is generated by the cohomological unit.
Lemma 4.2. Let I 0 < I 1 < · · · < I d . For the pairs (T I 0 , T I d ), and (T I j−1 , T I j ) for j = 1, . . . , d, we can choose Floer cochain data (A, H, J) such that A = 0 and H ≡ 0. Moreover, for x 0 ∈ X (T I 0 , T I d ) and x j ∈ X (T I j−1 , T I j ) for j = 1, . . . , d, we can also choose the Floer data (A S , K, J) such that A S = 0 and K ≡ 0. Therefore, regularity of R d+1,h (x 0 ; x d , . . . , x 1 ) can be achieved by generic J.
When there is no Hamiltonian perturbation, the regularity of u ∈ R d+1,h (x 0 ; x d , . . . , x 1 ) is equivalent to the regularity of the corresponding map v : Σ → E from Lemma 2.15, so it suffices to address the regularity of v restricted to each connected component. By abuse of notation, we denote the resriction to a connected component Σ 0 of Σ by v.
There are two cases, namely whether or not π E •v is a constant map. If π E •v is not a constant, then generic J can be used to achieve regularity. If π E • v is a constant, then by the boundary conditions, π Σ | Σ 0 must have degree 1, so Σ 0 must be a disc with d + 1 boundary punctures. We denote the Lagrangian boundary label on ∂Σ 0 by L 0 , . . . , L d so that L j ∈ T I j for j = 0, . . . , d.
as a holomorphic map with boundary on π E (L j ), then λ L 0 > λ L 1 implies that the slope of π E (L 0 ) is larger than that of π E (L 1 ), and one can check that the moduli of such maps has virtual dimension 0 and π E • v is regular. The cokernel of the Fredholm operator D v of v sits inside a short exact sequence where D π E •v is the Fredholm operator for π E • v and D fiber is the Fredholm operator of v viewed as a map to the fiber. By regularity of π E • v, we have coker(D π E •v ) = 0. On the other hand, coker(D fiber ) can also be made 0 by a generic choice of J. This implies the regularity of v.
Finally, we need to consider the case that π E • v is a constant map to a point c k for some k. In this case, d is not necessarily 1 but, by (4.3), we still have We can assume that L i ∩ L j = p for all i, j, where p is the critical point lying above c k . Let B p be a Darboux ball centered at p. We can assume that J is integrable near p so that B p can be identified with a ball in C 2 . Moreover, by (4.8), we can assume that T p L i ∩T p B p are pairwise transversally intersecting Lagrangian planes in T p B p with strictly decreasing Kähler angles. The regularity of the constant map to p can hence be studied locally as the constant map to the origin in C 2 with Lagrangian boundary conditions being a sequence of Lagrangian planes with strictly decreasing Kähler angles. This problem has virtual dimension the same as the dimension of R d+1,0 and if we fix the conformal structure of the domain, then it becomes virtual dimension 0 and the constant solution is regular. In our case, the conformal structure of the domain S is fixed by the other connected components Σ 1 of Σ and the map This proves the regularity of all maps involved and hence finishes the proof.
By Lemma 4.2, we can define the A ∞ operations without introducing a Hamiltonian term K, whenever I 0 < I 1 < · · · < I d . In a standard way, we can extend the A ∞ operations by adding idempotents. Corollary 4.3. Let R = ⊕ I∈I Ke I and e 2 I = e I . We have a strictly unital A ∞ algebra R ⊕ (⊕ I 0 <I 1 CF (T I 0 , T I 1 )) (4.10) with unit I∈I e I .
Proof. See [37,Section 7] for how the A ∞ structure is defined after adjoining the strict units e I of the idempotents.
Proof. This follows from Lemmas 4.1 and 4.2, Corollary 4.3 and homological perturbation.

4.2.
Fukaya-Seidel embeds into cylindrical Fukaya-Seidel. In this section, we show that is a Lefschetz fibration, and identify the subcategory of the associated Fukaya-Seidel category D π FS(π Y E ) generated by thimbles with the category of perfect modules over the A ∞ -algebra (4.10).
Proof. Without loss of generality, it suffices to show that f [n] is regular at a 0-dimensional length n subscheme z supported at 0. We can also assume that f (x, y) = x.
Let z t be a family of 0-dimensional length n subschemes such that z 0 = z and z t is a length n subscheme supported at (t, 0). Then Proof. If the support of z ∈ Hilb n (E) contains a point p that is not a critical point of π E , then we can apply Lemma 4.5 locally near p to show that z is not a critical point of π E , consider whether the support is a disjoint union of n points or not. We consider the former case first. For I ∈ I, we use z I ∈ Conf n (E) ⊂ Hilb n (E) to denote the subscheme with support being the union of the critical points lying above {c i |i ∈ I}. E and Sym(T I ) is a Lefschetz thimble. Proof. Since z I consists of pairwise distinct points, near z I , π [n] E is locally given by ( is also a straight line so it is a Lefschetz thimble.
For the other case, we have a local lemma.
For n ≥ 2 and z ∈ Hilb n (C 2 ) supported at the origin of C 2 , the length of the projection of z by π is strictly less than n.
Proof. Let C = Spec(C[w]) and f : C[w] → C[u, v] be the algebra homomorphism sending w to u 2 +v 2 . Let I be a length n ideal of C[u, v] supported at the origin, which corresponds to the point z in the statement. We want to show that the ideal f −1 (I) in C[w] has length strictly less than n.
Since dim C (C[u, v]/I) = n, we know that u a v b ∈ I if a, b, ≥ 0 and a + b ≥ n. It means that when n is even (resp. odd), we have ( ∈ I). Therefore, z n 2 ∈ f −1 (I) when n is even (resp. z n+1 2 ∈ f −1 (I) when n is odd) which in turn implies that f −1 (I) has length no greater than n+1 2 .
Corollary 4.9. The critical points of (4.11) are precisely {z I } I∈I and they are Lefschetz.
Proof. By Lemma 4.8, if the support of z has multiplicity 2 at some critical point of π E , then z ∈ D r which is not in Y E = Hilb n (E) \ D r . The result then follows from Corollary 4.6 and Lemma 4.7.
Remark 4.10. Even though (4.11) has only Lefschetz critical points, it is in general not symplectically locally trivial near the horizontal boundary, because there are critical points of π In light of Remark 4.10, we should clarify what we mean by FS(π Y ). Objects of the Fukaya-Seidel category FS(π Y ) are restricted to be Lefschetz thimbles Sym(T I ) for I ∈ I. As in Corollary (4.4) (see [37,Section 7]), the endomorphism A ∞ algebra of the direct sum of the objects is defined to be where the Floer cochains are taken in Hilb n (E) \ D r and no Hamiltonian perturbation is put on the Floer equations defining the A ∞ structure. As in [37], this definition is quasi-isomorphic to the directed subcategory of the vanishing cycles in the distinguished fiber.
Proof. There is an obvious bijective correspondence of objects and generators between (4.10) and (4.12). Note that all pseudo-holomorphic maps involved in defining the µ d operations in (4.10) are contained in Hilb n (E) \ D r .
For the definition of the µ d operations in (4.12), we can use moduli of pseudo-holomorphic maps with no Hamiltonian perturbation term as in (4.10)  E outside a compact subset of Hilb n (E) \ D r and generic inside the compact subset, and we count the the corresponding moduli of maps to define µ d operations in (4.12).
Given the difference of domain moduli for (4.12) and in (4.10), we cannot directly compare the µ d operations. The standard method to circumvent this is to use the 'total Fukaya category' trick from [35,Section 10a].
Briefly, one can show that µ d operations in (4.10) yield a category quasi-isomorphic to another one in which the operationsμ d are defined by the domain moduli ∪ d,h R d+1,h but with the domain dependent J chosen independent of the interior marked points, equal to J E outside a compact subset of Hilb n (E) \ D r and generic inside a compact subset. With these data,μ d is defined by counting the corresponding moduli of maps u with the additional condition that u(z) ∈ D HC if and only if z ∈ mk(S). In this modification, the interior marked points are merely decorative, and one can canonically identify the moduli spaces with the ones defining the µ d operations in (4.12).
It is natural to ask when the thimbles {T I } I∈I split-generate D π FS cyl,n (π E ). We next show that this holds when π E is the standard Lefschetz fibration on the type-A Milnor fibre.

Type A geometry
In this section, we apply the results in Sections 2-4 to the case in which E = A m−1 is a type A Milnor fiber. When 2n ≤ m, the corresponding Y E = Hilb n (E) \ D r is the generic fiber of the adjoint quotient map restricted to a nilpotent (or Slodowy) slice associated to a nilpotent with two Jordan blocks, as studied in [41,29] in the context of symplectic Khovanov cohomology. The main result of the section is that the embedding in Proposition 4.11 is essentially surjective in this case.

5.1.
A m−1 Milnor fibers. We recall the symplectic geometry of type A Milnor fibers, with an emphasis on verifying the assumptions made in the setup in Section 2. The following model is a slight modification of the one in [17,Section 7]. Let m > 1 be an integer. Let . Let M be the subvariety given by the equations Let π M : M → C be the projection to the x coordinate. For x = 1, . . . , m, the fiber at x is given by which is a smooth rational curve. For x = i ∈ {1, . . . , m}, is a union of two irreducible smooth rational curves. One can check that π M is a Lefschetz fibration. Consider the following sections and define D : Proof. For a dense subset, the identification is given by We leave the rest to the reader.
Lemma 5.2 (cf. Lemma 7.2 of [17]). Let ω X be a product symplectic form on X that tames the complex structure. Then ω M := ω X | M is a symplectic form and P x is ω M -orthogonal to D for all x.
Proof. Both assertions are clear. For the second one, observe that T D and T P x lie in the first and second factor of T C ⊕ (T (CP 1 ) m+1 ) = T X, respectively.
Let ω X be the standard symplectic form on X and define ω M : which is still a holomorphic Lefschetz fibration, and a symplectic Lefschetz fibration with respect to ω M R := ω M | M R . We equip the base unit disc B 1 with the hyperbolic area form ω hyp .
Proof. For > 0 small, we have a holomorphic embedding be an increasing function such that there exists δ > 0 so that g(r) = r for r near 1 − , g is strictly increasing when r ∈ (1 − , 1 − δ), and g(r) = 1 for r ≥ 1 − δ. It induces a smooth map φ g : It is straight forward to check that the assumptions made in Section 2.2.4 are satisfied. Most notably, is symplectically locally trivial and every holomorphic map CP 1 → E has positive algebraic intersection number with D E .
Remark 5.4. There is an additional feature in this setting that is not assumed in Section 2.
and g be its Lie algebra. The adjoint quotient map χ : g → h/W = Sym m (C) take an element A ∈ g to the coefficients of its characteristic polynomial, i.e. the symmetric functions of the eigenvalues (with multiplicities) of A. The set of regular values is given by Conf m (C) ⊂ Sym m (C) = C m . Let S n,m ⊂ g be the affine subspace consisting of matrices The affine subspace S n,m is a nilpotent (Slodowy) slice of the nilpotent element of Jordan type (n, m − n). The restriction χ| Sn,m : S n,m → Sym m (C) admits a simultaneous resolution by Grothendieck, and Example 5.5. When n = 1, where P τ (t) is the degree m monic polynomial with roots given by elements in τ . Note that, if τ = {1, . . . , m}, then Y 1,τ is biholomorphic to We denote the projection to the z co-ordinate by π 1,τ : Y 1,τ → C.
Lemma 5.6 ( [28]). When 2n ≤ m, there is a holomorphic open embedding j : Y n,τ → Hilb n (Y 1,τ ). The complement of the image is the relative Hilbert scheme of the projection π 1,τ , i.e. the subschemes whose projection to C z does not have length n.
Proof. Although Manolescu only considers the case m even, the same proof goes through in general. More precisely, for A ∈ S n,m , it is easy to check that Let τ = {1, . . . , m} and identify Y 1,τ and π 1,τ with M and π M from Section 5.1, respectively. By the dictionary 5.15 and Lemma 5.6, we know that Y n,m := Hilb n (E) \ D r is an open subset of Y n,τ when 2n ≤ m. Moreover, Y n,m exhausts Y n,τ as R goes to infinity. 7 Lemma 5.8. When 2n ≤ m, the map π 1,τ | j(Yn,τ ) is given by sending a matrix A ∈ Y n,τ to the top left entry a 1 .
Proof. By [28, Remark 2.8], the composition is given by the roots of A(t) (in the notation of the proof of Lemma 5.6). Therefore the map π 1,τ • j : Y n,τ → C is the sum of the roots of A(t), which is a 1 .

Sliding invariance.
If L 0 and L 1 are not connected by a path L t ∈ L cyl,n , then the quasiisomorphism types of L 0 and L 1 in FS cyl,n (π E ) are in general different. However, we present three special cases where the quasi-isomorphism type is unchanged by a move not arising from an isotopy through admissible tuples.
The first case is the analogue of [41, Lemma 49].
Lemma 5.9. Let Γ := {γ 1 , . . . , γ n } be an admissible tuple. Suppose γ i and γ j are mathing paths and i = j. Let γ be the matching path obtained by sliding γ i across γ j . If Γ is the admissible tuple obtained by replacing γ i by γ , then L Γ is quasi-isomorphic to L Γ .
Proof. First consider the case that n = 2, i = 1, j = 2 and m ≥ 2n. In this case, it is proved in [41,Lemma 49] that Sym(L Γ ) and Sym(L Γ ) are Hamiltonian isotopic in Y n,m (they treat the case m = 2n but the proof works for all m ≥ 2n). In particular, we can find a ∈ HF (Sym(L Γ ), Sym(L Γ )) and b ∈ HF (Sym(L Γ ), Sym(L Γ )) such that µ 2 (b, a) = 1 HF (Sym(L Γ ),Sym(L Γ )) and µ 2 (a, b) = 1 HF (Sym(L Γ ),Sym(L Γ )) (5.21) As in the proof of Proposition 4.11, when we turn off the Hamiltonian perturbation, there is a bijective correspondence between the J-holomorphic curves in Y n,m and the corresponding curves in A m−1 . In particular, we can find a ∈ HF (L Γ , L Γ ) and b ∈ HF (L Γ , L Γ ) such that which implies that L Γ and L Γ are quasi-isomorphic objects in FS cyl,n (A m−1 ). Moreover, by the open mapping theorem, the projection to H • of the curves contributing to µ 2 in (5.22) is contained in the disc bounded by γ 1 and γ (see Figure 5.1). Now, we consider the general n with 2n ≤ m. Without loss of generality, we continue to assume i = 1 and j = 2. If Γ has no thimble path, then we do not need to impose Hamiltonian perturbations in the Floer equation when we compute the differential and product. In this case, the Floer solutions contributing to µ 2 in (5.22) persist because they lie above the disc bound by γ 1 and γ , which is not altered when adding the other Lagrangian components. Together with the constant triangles on the other Lagrangian components representing µ 2 (e, e) = e, we conclude that there exists a ∈ HF (L Γ , L Γ ) and b ∈ HF (L Γ , L Γ ) such that (5.22) holds, and hence L Γ is quasi-isomorphic to L Γ (see Figure 5.1).
If Γ contains some thimble paths, then Hamiltonian perturbation terms in the Floer equation are necessary. However, the Hamiltonian terms can be taken to be zero over the disc bound by γ 1 and γ and hence the Floer solutions contributing to µ 2 in (5.22) persist. The only other relevant triangles are the constants on the other Lagrangian components, so the result follows as before.
Finally, we explain the remaining case where 2n > m. From the nature of the proof for the case when 2n ≤ m, we can see that if we consider an embedding from A m−1 to A 2n−1 by adding critical points far away, then the added critical points are not relevant to the Floer theory and the existence of the holomorphic curves in A 2n−1 contributing to the relevant Floer product implies the existence of the corresponding curves in A m−1 . Therefore, the result follows. . . , γ n } be an admissible tuple. Suppose γ i is a thimble path and γ j is a mathing path. Let γ be the thimble path obtained by sliding γ i across γ j . If Γ is the admissible tuple obtained by replacing γ i with γ , then L Γ is quasi-isomorphic to L Γ .
Proof. We first consider the case that n = 2. Let γ 1 be the thimble path and γ 2 be the matching path. We can assume that the thimble path γ only intersects γ 1 at the starting point, which we denote by c. Along with γ , we consider two more auxiliary thimble paths γ 1,w and γ w , which are obtained by positively wrapping γ 1 and γ along the real line, respectively. Without loss of generality (by possibly switching the role of γ 1 and γ ), we can assume that γ 1,w intersects γ at two points c and c , and any other pair of thimble paths amongst γ 1 , γ , γ 1,w , γ w intersect only at c (see Figure 5.2). Let Γ w := {γ 1,w , γ 2 } and Γ w := {γ w , γ 2 }. It is clear that L Γ is quasi-isomorphic to L Γw and L Γ is quasi-isomorphic to L Γ w . To show that L Γ is quasi-isomorphic to L Γ , it suffices to find a ∈ HF (L Γ , L Γ ), b ∈ HF (L Γw , L Γ ), and a ∈ HF (L Γ w , L Γw ) such that µ 2 (a, b) = 1 HF (L Γw ,L Γ ) and µ 2 (b, a ) = 1 HF (L Γ w ,L Γ ) (5. 23) and such that a is identified with a under the continuation map (with respect to positive wrapping along the real line). In the given positions of the Lagrangians, we do not need to use Hamiltonian terms on the Floer multiplication maps To compute these maps and hence verify (5.23), we rely on (5.22) and a restriction argument as follows.
We add a critial valueĉ and extend the thimble paths γ 1 , γ , γ 1,w , γ w to run intoĉ. The thimble paths become matching pathsγ 1 ,γ ,γ 1,w ,γ w and we defineΓ :  The degree 0 generator for each of the Floer cochain groups above is given by an intersection point lying above either c or c , tensored with the degree 0 generator of CF (L γ 2 , L γ 2 ). For example, the cochain complex CF (Lγ 1,w , Lγ ) is generated by one degree 0 and one degree 1 generator lying over c (arising from Morsifying the clean S 1 -intersection locus), and transverse degree 2 intersection points lying over each of c andĉ. Since each of the CF 0 groups involved has rank one,â,b andâ are the Since the previous computation is local, the general case where n > 2 can be treated as in the proof of Lemma 5.9.
The last case involves sliding a thimble path across another thimble path. The proof is again an adaption of Lemma 5.9, extending the thimble paths in order to apply the same kind of restriction argument as in the proof of Lemma 5.10, so we omit it.
Lemma 5.11. Let Γ := {γ 1 , . . . , γ n } be an admissible tuple. Suppose γ i and γ j are thimble paths and i = j. Let γ be the thimble path obtained by sliding γ i across γ j . If Γ is the admissible tuple obtained by replacing γ i with γ , then L Γ is quasi-isomorphic to L Γ .

Generation.
We now show that when E is the A m−1 -Milnor fiber, the embedding in Proposition 4.11 is essentially surjective. In other words, we want to show that the split-closure A of the thimbles T I is the entire D π FS cyl,n (π E ). We first recall some general facts for A ∞ /triangulated categories.
Lemma 5.12. The full subcategory generated by an exceptional collection is admissible (i.e. admits right and left adjoints). In particular, A is an admissible subcategory of D π FS cyl,n (π E ).
Proof. Recall that a category admitting a full exceptional collection is already split-closed [35,Remark 5.14]. The result then follows from e.g. [19,Lemma 1.58].
Lemma 5.13. Let S be the Serre functor of D π FS cyl,n (π E ). If S(X), S −1 (X) ∈ A for all X ∈ A, then D π FS cyl,n (π E ) = A ⊕ A ⊥ .
Proof. If Y ∈ D π FS cyl,n (π E ) satisfies Hom(Y, X) = 0 for all X ∈ A, then Hom(X, S(Y )) ∨ = 0 for all X ∈ A. It means that Hom(S −1 (X), Y ) = 0 for all X ∈ A. By assumption, this is equivalent to Hom(X, Y ) = 0 for all X ∈ A. Similarly, Hom(X, Y ) = 0 for all X ∈ A implies that Hom(Y, X) = 0 for all X ∈ A. As a result, the left-orthogonal of A coincides with its right-orthogonal, so D π FS cyl,n (π E ) splits as a direct sum A ⊕ A ⊥ .
Lemma 5.14. Suppose T = {T 1 , . . . , T n } ∈ L cyl,n is a Lagrangian tuple such that each T i is a thimble of π E . Then T is generated by {T I } I∈I .
Proof. First note that the braid group acts transitively (up to isotopy) on all the T = {T 1 , . . . , T n } such that T i is a thimble for each i. Therefore, it suffices to show that for each simple braid σ and the associated symplectomorphism φ σ , the images φ σ (T I ) and φ −1 σ (T I ) are generated by {T I } I∈I . Let σ be the positive half twist swapping c j and c j+1 . There are 4 cases of T I to consider, namely, whether c j and/or c j+1 is contained in γ I,k for some k = 1, . . . , n or not. For each of these 4 cases, one can apply the exact triangles from Corollary 3.4 and the sliding invariance property in Lemma 5.11 to show that for all I ∈ I, φ σ (T I ) and φ −1 σ (T I ) are generated by {T I } I∈I (see Figure 5.4 for some examples).
x x x x x x x x - Figure 5.4. The top row is an exact triangle so the third term is generated by the first and second terms, which are in turn generated by {T I } I∈I . The second row represents two quasi-isomorphic objects.
Proof. By Proposition 3.5, the Serre functor of D π FS cyl,n (π E ) is given by the global monodromy τ up to grading shift. Therefore, it suffices to prove that for each thimble T I , the images τ (T I ) and τ −1 (T I ) are split-generated by the collection of thimbles {T I } I∈I . This is the content of Lemma 5.14.
Proof. It suffices to show that for each L ∈ L cyl,n , there is an object T in A such that Hom(T , L) = 0. By definition, we have L = {L 1 , . . . , L n } and π E (L i ) ⊂ U i for some contractible U i . We can assume that U i ∩ ∂H is connected and non-empty so that H • \ U i is also contractible.
For each i, there is a thimble T i of π E such that π E (T i ) ⊂ U i and HF (T i , L i ) = 0. This follows from the fact that thimbles generate FS cyl,1 , the usual Fukaya-Seidel category of the Milnor fibre, see [35] (note that when n = 1 there are no critical points at infinity). Let T = {T 1 , . . . , T n }. It is clear that there is a cochain isomorphism which implies that HF (T , L) = 0. By Lemma 5.14, we have T ∈ A, concluding the proof.
Proof. When n = m, there is only one object in A up to quasi-isomorphism and it is an exceptional object. Therefore, the result follows from Proposition 5.16.

The extended symplectic arc algebra
In this section, we will introduce a particular collection of admissible tuples, and hence the corresponding collection of objects in L cyl,n ; these objects are motivated by the diagrammatics in [44,13]. The A ∞ endomorphism algebra of this collection of objects will be the 'extended symplectic arc algebra', and will contain the symplectic arc algebra from [1] as a subalgebra. The corresponding Lagrangian products in Hilb n (A m−1 )\D r would not be cones over Legendrian submanifolds at infinity, which is why it is important to be able to study these Lagrangians in the cylindrical model. 6.1. Weights and projective Lagrangians. To introduce the collection of admissible tuples, we start with some terminology. Without loss of generality, we assume that the critical values are c k := k + √ −1 for k = 1, . . . , m. A weight of type (n, m) is a function λ : {1, . . . , m} → {∧, ∨} such that |λ −1 (∨)| = n. Let Λ n,m be the set of all weights of type (n, m). For λ ∈ Λ n,m , let c λ,1 < · · · < c λ,n be the integers such that λ(c λ,j ) = ∨. For each c λ,j , if there exists c ∈ λ −1 (∧) with c > c λ,j and such that |{c ∈ {1, . . . , m}|λ(c) = ∨, c λ,j < c < c }| = |{c ∈ {1, . . . , m}|λ(c) = ∧, c λ,j < c < c }| (6.1) then we call c λ,j a good point of λ; the minimum of all c satisfying (6.1) is denoted by c ∧ λ,j . If c λ,j is not a good point of λ, then we call it a bad point (see Figure 6.1 for an example).
By iteratively applying Lemmas 5.9, 5.10 and 5.11, we obtain the following (see Figures 6.3, 6.4, 6.5): When only the quasi-isomorphism type is important, we denote either of L λ or L λ by L λ . which is well-defined up to quasi-isomorphism. We want to choose a basis for the cohomology of K symp n,m as follows. Let λ ∪ λ be the union of all the paths in λ and λ . By definition, λ ∪ λ is a union of embedded circles and arcs; some circles might be nested inside one another. It will be helpful to consider alternative admissible tuples which avoid such nesting (but for which the quasi-isomorphism type of the associated Lagrangian tuple is unchanged).
There is an admissible tupleλ such that • if γ ∈ λ is not contained in a circle of λ ∪ λ (for example, when γ is a thimble path), then γ ∈λ • if γ ∈ λ is contained in a circle C of λ ∪ λ , then there is a matching pathγ ∈λ with the same end points of γ such thatγ is enclosed in C • λ ∪λ is a union of embedded circles and arcs such that none are nested. If γ is contained in a circle C, then each critical value enclosed in C is an end point of a matching path of λ (and also a matching path of λ), directly from the definitions (6.2), (6.3) (resp. (6.5), (6.6)). Thus, a suitableλ can be obtained by iteratively applying Lemma 5.9 to λ , ensuring that L λ is quasi-isomorphic to Lλ (see Figure 6.6). Note that the definition ofλ depends on the pair (λ, λ ), and not just on λ .
x x x x x x x x x x x x Figure 6.6. Here λ is red,λ is grey, and λ is black.
With the grading conventions of Figure 6.2, the pure degree elements in CF (L λ , Lλ ) are concentrated either in odd or even degree, so the Floer differential vanishes (compare to [1,Proposition 5.12]). Therefore, the cohomology of K symp n,m is given by K symp n,m = ⊕ λ,λ ∈Λn,m HF (L λ , Lλ ) = ⊕ λ,λ ∈Λn,m CF (L λ , Lλ ) (6.10) as a graded vector space. We call a basis B symp for K symp n,m 'geometric' if each basis element in B symp is concentrated at a single intersection point in L λ ∩ Lλ under the isomorphism (6.10). Once aλ has been chosen for each pair (λ, λ ), two different geometric bases can differ only by signs.
6.2. Extended arc algebra. We briefly recall the diagrammatic extended arc algebra K alg n,m . Details can be found in [13,44], to which we defer for many details. For each weight λ ∈ Λ n,m , there is an associated cup diagram λ alg and a cap diagram λ alg as follows. The cup diagram λ alg can be obtained by adding a thimble path to λ from a + √ −1 to a, for each a ∈ {1, . . . , m}, such that a + √ −1 is not contained in any of the paths in λ (see Figure 6.7). For the cap diagram λ alg , we need to replace the thimble paths from c + √ −1 to 6m − c in λ by vertical rays from c + √ −1 to c + √ −1∞, and in addition, for each a ∈ {1, . . . , m} such that a + √ −1 is not contained in any of the paths in λ, add a thimble path from a + √ −1 to a + √ −1∞ (see Figure 6.7). In this paper, the only cup and cap diagrams we will encounter are given by λ alg or λ alg for some λ ∈ Λ n,m . A cap diagram is defined to be the reflection β r of a cup diagram β along the line {im(z) = 1}. If α is a cap diagram and λ is a weight, then we say that λα an oriented cap diagram if α r λ is an oriented cup diagram.
The union of a cup diagram β and a cap diagram α is denoted by β ∪ α and is called a circle diagram. This is a union of embedded circles and arcs in the upper half-plane. An orientation of a circle diagram β ∪ α is a weight λ such that λα and βλ is an oriented cap diagram and oriented cup diagram, respectively. Given such a λ, we denote the resulting oriented circle diagram by βλα.
To see that Φ λ 0 ,λ 1 is well-defined, we observe that all the thimble paths contained in λ alg but not in λ are on the left of the thimble paths (if any) in λ, and the same is true for λ alg and λ. Therefore, λ x satisfies (6.12). On the other hand, since each Lagrangian component contains exactly one of the x i , it means that λ x also satisfies (6.11).
The algebra structure on K alg n,m is defined by applying an appropriate diagrammatic TQFT. We briefly recall one possible definition of this algebra, and some crucial properties that we will use later, in Section 6.4 below, and refer the readers to [13,Sections 3 & 4] for a detailed exposition.
6.3. Compact subalgebra. We call a weight λ ∈ Λ n,m a compact weight if λ (and hence λ) consists only of matching paths. Let Λ c n,m ⊂ Λ n,m be the subset of compact weights. We define H symp n,m := ⊕ λ,λ ∈Λ c n,m CF (L λ , L λ ) (6.15) which is well-defined up to quasi-isomorphism. As in (6.10), its cohomology is given by On the diagrammatic side, we can define the corresponding subalgebra H alg n,m of K alg n,m , which is generated by oriented circle diagrams such that the underlying cap and cup diagrams are given by λ alg and λ alg for some λ, λ ∈ Λ c n,m . It is clear that the graded vector space isomorphism in Lemma 6.3 induces a graded vector space isomorphism between H symp n,m and H alg n,m . Previous study has focussed on the case m = 2n. In this case, H alg n,2n is generated by oriented circle diagrams whose underlying diagram only contains circles (and the corresponding Lagrangian submanifolds of Hilb n (A 2n−1 ) are compact, being products of spheres rather than products of spheres and thimbles). The algebra H alg n,2n is also known as Khovanov's arc algebra.
Theorem 6.4 ([1, 2]). The A ∞ algebra H symp n,2n is formal. Moreover, there is an isomorphism between H alg n,2n and H symp n,2n , sending the oriented circle diagram basis of H alg n,2n to a geometric basis of H symp n,2n .
Sketch of proof. Formality of H symp n,2n is the main result of [1], whilst a basis-preserving algebra isomorphism between H alg n,2n and H symp n,2n is one of the main results of [2]. We now recall the basis for H symp n,2n chosen in [2], and explain why it is a geometric basis in our sense. The essential point [2, Corollary 5.5] is the compatibility of the basis with various Künneth-type functors and decompositions (of the cohomology of products of spheres and thimbles with the cohomologies of the constituent factors). The natural action of the braid group Br 2n on A 2n−1 and on Y n,2n factors through an action of the symmetric group Sym 2n on cohomology. Furthermore, there are inclusions In particular, the image of {e i |1 ≤ i ≤ 2n − 1} in H 2 (Y n,2n ) forms a basis. The restriction map H 2 (Y n,2n ) → H 2 (Sym(L λ )) is surjective for each λ ∈ Λ c n,2n , and by identifying the range with the cohomology of suitable multi-diagonals in (P 1 ) 2n , it is proved in [2] that the chosen basis has the property that it induces a well-defined basis for H 2 (Sym(L λ )), i.e. there is a basis of the latter such that each e i maps to a basis element or to zero. Now, we translate the basis of H 2 (Y n,2n ) to a basis of H 2 (A 2n−1 ). Denote by s 1 , . . . , s 2n−1 the homology classes of the 2n − 1 standard matching spheres in A 2n−1 , i.e. the ones that lie above {im(z) = 1}; here they are labelled from left to right, and the spheres are oriented as the complex curves in the resolution of the A 2n−1 -surface singularity. We denote the image of e i under (6.17) by v i ∈ H 2 (A 2n−1 ) for 1 ≤ i ≤ 2n − 1. They satisfy where * stands for linear dual. The set {v j } 2n−1 j=1 is the corresponding basis for H 2 (A 2n−1 ). It induces a well-defined basis for HF 2 (L λ , L λ ) via restriction where the first map is defined by regarding L λ as the disjoint union of the matching spheres it contains, and hence as a submanifold of A 2n−1 . This basis for H 2 (L λ ) naturally corresponds to the basis of H 2 (Sym(L λ )). Monomial products of the resulting elements give a basis for HF * (L λ , L λ ).
When HF (L λ , L λ ) = {0}, it has rank one in minimal degree. Let a min be a minimal degree generator of HF (L λ , L λ ) over the integers Z. Then HF (L λ , L λ ) is generated by a min as a module over each of HF (L λ , L λ ) and HF (L λ , L λ ). A basis in HF (L λ , L λ ) can therefore be defined by taking products of a min with the bases for either of HF (L λ , L λ ) or HF (L λ , L λ ); these act via the action of H * (Y n,2n ), which acts centrally, so there is no ambiguity (aside from a choice of sign of a min ). This finishes recalling the basis for H symp n,2n , which we call a 'convenient' basis. We claim this convenient basis is geometric. Let S λ,1 , . . . , S λ,n be the matching spheres that L λ contains. The first observation is that v j (S λ,k ) ∈ {0, ±1} for each j, and furthermore, for each j there is exactly one k for which v j (S λ,k ) = 0. This means that the convenient basis of H 2 (Sym(L λ )) induced from H 2 (Y n,2n ) coincides with the product basis on Sym(L λ ) as a product of matching spheres. The geometric basis in CF (L λ , Lλ) also coincides with the product basis, becauseλ is a Hamiltonian pushoff of λ, and λ ∪λ consists of n embedded pairwise non-nested circles. That suffices to show that the convenient basis on HF (L λ , L λ ) is geometric.
On the other hand, the minimal degree generator of HF (L λ , L λ ) is forced to be geometric, because the minimal degree subspace has rank 1. Since λ ∪λ is a union of embedded and non-nested circles, the product map HF (L λ , Lλ ) ⊗ HF (L λ , L λ ) → HF (L λ , Lλ ) (6.20) decomposes into pieces, one for each circle C in λ ∪λ . More precisely, we have isomorphisms HF (L λ , Lλ ) = ⊗ C H * (S 2 ) (6.21) where the tensor product is over all circles C in λ ∪λ , and m C is the number of paths of λ that lie in C. The product map (6.20) decomposes into the tensor product over all circles C of maps for which a typical local model is given by Figure 6.8. The product of the degree 0 geometric generator in H * (S 2 ) and a degree 2 geometric generator in (H * (S 2 )) ⊗m C is, up to sign, the degree 2 geometric generator in H * (S 2 ), because H 2 (S 2 ) has rank 1, the product is (by definition) a basis element of the convenient basis, and [2, Section 5] proves that the convenient basis is a basis for the cohomology groups over Z. We conclude that the convenient basis is geometric.
Corollary 6.5. After possibly changing the signs of certain basis elements, the graded vector space isomorphism Φ| H symp n,2n is an algebra isomorphism between H symp n,2n and H alg n,2n .
From now on, we assume Φ| H symp n,2n has been chosen to be an algebra isomorphism. 6.4. From the compact arc algebra to the extended arc algebra. In this section, we summarise the multiplication rule for K alg n,m and H alg n,m from [13, Sections 3 & 4]. Our approach is dictated by two considerations. First, the easiest (though not the original) description of the algebra structure on K alg n,m is to realise K alg n,m as a quotient algebra of H alg m,2m . Second, in the next section, we precisely compute K symp n,m by realising it as a quotient of H symp m,2m by an ideal which, under the isomorphism H alg m,2m = H symp m,2m (Theorem 6.4), we identify with the corresponding ideal on the algebraic side. The main purpose of this section is therefore to recall this quotient description of K alg n,m . For completeness, we briefly recall the multiplication rule for H alg m,2m from [13, Section 3, multiplication]. Let dλc and bµa be two oriented circle diagrams with a, b, c, d ∈ Λ c n,m . If b = c, then the product is defined to be zero. If b = c, then we put the diagram bµa above the diagram dλc and apply the TQFT surgery procedure -based on the Frobenius algebra underlying H * (S 2 ; Z), and described e.g. in [13, Section 3, the surgery procedure] iteratively to convert it into a disjoint union of diagrams each of which has no cups/caps in the 'middle section'. After this, the middle sections are unions of line segments; we shrink each such line segment to a point to obtain a disjoint union of some new oriented circle diagrams. The product (dλc)(bµa) is then defined to be the sum of the corresponding basis vectors of H alg m,2m . One distinguished feature of this TQFT surgery procedure is that the output circle (resp. disjoint union of two circles) is (resp. are) oriented according to the following rules, where 1 ⇔ anti-clockwise orientation, and x ⇔ clockwise orientation: Each TQFT operation yields a disjoint union of zero, one or two new oriented diagrams replacing the old diagram (the iterative application may yield larger linear combinations of diagrams). This completes our résumé of H alg m,2m as an algebra. We now recall how to realise K alg n,m as a quotient algebra of H alg m,2m ; for an independent definition of K alg n,m , see [13]. For λ ∈ Λ n,m , we define its closure cl(λ) ∈ Λ c m,2m by Intuitively, cl(λ) can be regarded as putting m − n many ∨'s and n many ∧'s to the left and right of λ, respectively, to make cl(λ) a compact weight. For any oriented circle diagram bλa ∈ K alg n,m , we define cl(bλa) := cl(b) cl(λ)cl(a) (6.27) which is an oriented circle diagram of H alg m,2m . Lemma 6.6. [13, Lemma 4.2] The map bλa → cl(bλa) is a degree preserving bijection between the set of oriented circle diagrams in K alg n,m with underlying weight λ ∈ Λ n,m and the set of oriented circle diagrams in H alg m,2m with underlying weight cl(λ).

More algebra isomorphisms
We will next identify K symp n,m and K alg n,m as algebras for every n, m. The proof of Theorem 6.4 in [2] relied in an essential way on the fact that the Floer product for a triple of Lagrangians meeting pairwise cleanly can be understood, via 'plumbing models', as a possibly sign-twisted convolution product, and that for every triple of Lagrangians associated to compact weights, one could find non-vanishing Floer products which factored through products associated to triples with plumbing models. By contrast, for the extended algebra, there are triples of weights and corresponding Lagrangians for which the product of minimal degree generators cannot be written as a product of minimal degree generators between interpolating Lagrangians with plumbing models. Instead of mimicking the previous strategy of proof, we will instead reduce the isomorphism of extended arc algebras (in stages) to Theorem 6.4 by restriction-type arguments similar to those appearing in the proof of Lemma 5.10. The crucial point, as explained in Section 6.4, is that all the extended arc algebras can be understood as sub-quotients of the algebras H alg n,2n .
7.1. The compact cases. We first explain how to compute the algebra structure on H symp n,m for all n, m. We assume 2n ≤ m because H symp n,m = 0 otherwise. Let c : Λ c n,m → Λ c m−n,2(m−n) be the injection given by (6.37). Let such that π E • u is holomorphic. We claim that if the output of u maps to 1 + √ −1 under π E , so do all the inputs of u. This is because if π E • u restricted to the boundary labelled by L 0 is not a constant, then the image of u must have non-empty intersection with the unbounded region of H \ (l 0 ∪ l 1 ∪ l 2 ) by the holomorphicity of π E • u. This in turn implies that π E • u is not relatively compact by the open mapping theorem, a contradiction. As a result, one can show that π E • u restricted to the boundary labelled by L 0 is a constant. Inductively applying this argument, one can show that the restriction of π E • u to the whole boundary is constant, and hence u itself is a constant map. The choices of matchings depend on the ordered pair we start with, cf. the discussion around Figure  6.6; to emphasise this dependence, we denote the matchings by c(λ 0 ) λ 0 ,λ 1 and c(λ 1 ) λ 0 ,λ 1 respectively.
Proof. It suffices to show that if the x 0 -coefficient of µ 1 (x 1 ) is non-zero for some x 0 ∈ G, then x 1 ∈ G.
By definition, x 0 ∈ G implies that {1, . . . , m − 2n} + √ −1 ⊂ π(x 0 ). Let u : Σ → A 2k−1 be a J-holomorphic map contributing to the x 0 -coefficient of µ 1 (x 1 ) such that near an output puncture ξ 0 , u is asymptotic to the output lying above 1 + √ −1. We project u to H • by π E and apply the open mapping theorem. The condition (7.2) when a = 1 forces at least one of the boundary components of Σ adjacent to ξ 0 to be mapped constantly to 1 + √ −1 by π E • u. Let the two boundary components of Σ adjacent to ξ 0 be ∂ i Σ and ∂ j Σ, respectively, and suppose π E • u| ∂ j Σ = 1 + √ −1. As a result, the other puncture ξ that is adjacent to ∂ j Σ is also mapped to 1 + √ −1 by π E • u. Let the other boundary component of Σ that is adjacent to ξ be ∂ j Σ.
By the Lagrangian boundary conditions, u(∂ i Σ) and u(∂ j Σ) are both contained in the Lagrangian components of L c(λ 0 ) λ 0 ,λ 1 (or L c(λ 1 ) λ 0 ,λ 1 ). However, each of L c(λ 0 ) λ 0 ,λ 1 and L c(λ 1 ) λ 0 ,λ 1 has only one component whose projection to H • contains 1 + √ −1. Therefore ∂ i Σ = ∂ j Σ, and the two punctures adjacent to ∂ i Σ are both mapped to 1+ √ −1 under π E •u. By the open mapping theorem, we conclude that the restriction of π E • u to the connected component of Σ that contains ξ 0 is a constant map and that connected component of Σ is a bigon.
By the last paragraph of the proof of Lemma 7.2, there is a canonical isomorphism between the chain complexes which on generators is given by (7.5), and on differentials is given by incorporating or removing the m − 2n constant bigon components from the proof of Lemma 7.2.
Lemma 7.4. The subspace I λ 0 ,λ 1 is independent of the choices of c(λ 0 ) λ 0 ,λ 1 and c(λ 1 ) λ 0 ,λ 1 , provided Proof. Let c(λ 0 ) a and c(λ 0 ) b be two different choices of c(λ 0 ) λ 0 ,λ 1 such that (7.2) is satisfied for the By interpolating slopes, there exists another choice c(λ 0 ) c of c(λ 0 ) λ 0 ,λ 1 such that ( λ 0 ,λ 1 ). We need to show that the isomorphism sends I a to I c . By Corollary 7.3 and for dimension reasons, it suffices to show that the image of I a under µ 2 (−, θ) is contained in I c . The same will then be true when we replace c(λ 0 ) a by c(λ 0 ) b , so the result will follow.
By examining the slopes of the matching paths at 1 + √ −1 again, either ∂ i Σ or ∂ j Σ is mapped to the constant value 1 + √ −1 by π • u. Say ∂ j Σ is mapped this way; then the next boundary components of Σ and ∂ i Σ are mapped to components of the same Lagrangian tuple (i.e. L c(λ 0 ) c , L c(λ 0 ) a or L c(λ 1 ) λ 0 ,λ 1 ) under u. As a result, these two boundary components co-incide, π • u restricted to the component of Σ containing ξ 0 is a constant map, and this component is a triangle.
In this case, the exact same argument as in the proof of Lemma 7.4 shows that if the x 0 -coefficient of µ 2 (x 2 , x 1 ) is non-zero for some x 0 ∈ G 0,2 , then x 1 ∈ G 0,1 and x 1 ∈ G 1,2 , where G i,j is the respective G for CF (L c(λ i ) , L c(λ j ) ). This precisely says that I is an ideal in H c n,m . Moreover, we also know that any u contributing to the x 0 -coefficient of µ 2 (x 2 , x 1 ) has m − 2n components of constant triangles. Therefore, under the identification of generators (7.5) in Corollary 7.3, the moduli spaces of holomorphic triangles defining the products µ 2 on the two sides are also canonically identified. The result follows.
Next, we want to understand the ideal I in terms of the geometric basis of H c n,m . Consider the cochain model (cf. 6.16, (7.1)).
Lemma 7.7. There is an algebra isomorphism from ⊕HF (L ) sending I to I arc .
Without loss of generality, we can assume that c(λ 0 ) λ 0 ,λ 1 = c(λ 0 ). Note that we can choose c(λ 1 ) such that the pair (c(λ 1 ) λ 0 ,λ 1 , c(λ 1 )) also satisfies (7.2). By the same reasoning as in Lemmas 7.4 and 7.6, the image of I under (7.10) is contained in I arc . Since µ 2 (θ λ 0 ,λ 1 , −) is an isomorphism on cohomology and the dimensions spanned by I and I arc in cohomology are the same, the image of I is precisely I arc .
From Lemmas 7.6 and 7.7, we have an algebra isomorphism ⊕CF (L c(λ 0 ) , L c(λ 1 ) )/I arc H symp n,m . We want to show that this isomorphism respects the geometric basis.
We can assume that λ 0 ,λ 1 and λ 1 are obtained, respectively, by removing the leftmost m − 2n matching paths of c(λ 0 ) , c(λ 1 ) and c(λ 1 ), and translating by −(m−2n). Let θ −1 be a quasi-inverse of θ. We can choose θ to be image of θ −1 under the canonical inclusion CF (L λ 1 , Lλ In this case, the composition (7.11) coincides with the canonical cochain isomorphism essentially the same map as in (7.5), (7.6)). As a consequence, there is an algebra isomorphism from H symp n,m to H c n,m /I respecting the geometric basis.
For the combinatorial arc algebra, we have exactly the same quotient description for H alg n,m (see (6.39)). To conclude, we have 7.2. All cases. We show that Φ is an algebra isomorphism from K symp n,m to K alg n,m for all n, m in this section. The strategy is similar to the previous section.
Proof. Let x 1 ∈ B. We want to show that if the x 0 -coefficient of µ 1 (x 1 ) is non-zero then x 0 ∈ B.
By definition, x 1 ∈ G implies that m + a + √ −1 ⊂ π(x 1 ) for some a = 1, . . . , n. Let u : Σ → A 2k−1 be a J-holomorphic map contributing to the x 0 -coefficient of µ 1 (x 1 ) such that near an input puncture ξ, u is asymptotic to the element of x 1 lying above m + a + √ −1. We project u to H • by π and apply the open mapping theorem. The condition (7.14) at m + a + √ −1 forces at least one of the boundary components of Σ adjacent to ξ to be mapped constantly to m + a + √ −1 by π • u. It immediately implies that m + a + √ −1 ∈ π(x 0 ).
Proof. For j = 0, 1, by forgetting the points {1, . . . , n} + m + √ −1 and 'extending' the matching paths of e(λ j ) λ 0 ,λ 1 that contain m + a + √ −1 for some a = 1, . . . , n to thimble paths, we get an upper matching λ j of λ j . There is then an obvious bijective correspondence Moreover, there is a canonical isomorphism between the chain complexes which on generators is given by (7.16); the differentials on the left and right agree by the proof of Lemma 7.9.
Let J a and J c be the respective J λ 0 ,λ 1 for HF (L e(λ 0 ) a , L e(λ 1 ) λ 0 ,λ 1 ) and HF (L e(λ 0 ) c , L e(λ 1 ) λ 0 ,λ 1 ). We need to show that the isomorphism sends J a to J c . By Corollary 7.10 and a dimension count, it suffices to show that the image of J a under µ 2 (−, θ) is contained in J c .
The proof that µ 2 (J a , θ) ⊂ J c is similar to the proof of Lemma 7.9. Let B a and B c be the respective B λ 0 ,λ 1 for CF (L e(λ 0 ) a , L e(λ 1 ) λ 0 ,λ 1 ) and CF (L e(λ 0 ) c , L e(λ 1 ) λ 0 ,λ 1 ). By slight abuse of notation, we denote a chain level lift of θ to CF (L e(λ 0 ) c , L e(λ 0 ) a ) by θ. It suffices to show that if x 1 ∈ B a and the x 0coefficient of Let u : Σ → A 2k−1 be a J-holomorphic map contributing to the x 0 -coefficient of µ 2 (x 1 , θ) such that near an input puncture ξ, u is asymptotic to the element of x 1 lying above m + a + √ −1. We project u to H • by π and apply the open mapping theorem. Since (e(λ 0 ) a , e(λ 1 ) λ 0 ,λ 1 ) satisfies (7.14), it forces at least one of the boundary components of Σ adjacent to ξ to be mapped constantly to m + a + √ −1 by π • u. Let the two boundary components of Σ adjacent to ξ be ∂ i Σ and ∂ j Σ, with Lagrangian labels L e(λ 0 ) a and L e(λ 1 ) λ 0 ,λ 1 , respectively. If π • u| ∂ j Σ = m + a + √ −1, then we are done because we must have m + a + √ −1 ∈ π(x 0 ). If π • u| ∂ i Σ = m + a + √ −1, then we denote the next boundary component of Σ adjacent to ∂ i Σ by ∂ i Σ, which is equipped with the Lagrangian label L e(λ 0 ) c . Using the fact that (e(λ 0 ) c , e(λ 1 ) λ 0 ,λ 1 ) satisfies (7.14), we conclude that either ∂ j Σ or ∂ i Σ is mapped to m + a + √ −1 under π • u. In either case, we have m + a + √ −1 ∈ π(x 0 ).
By very similar arguments, the analogues of Lemma 7.6 and 7.7 hold. In particular, we obtain algebra isomorphisms )/J arc (7.19) where J arc is generated by (7.15). There is also exactly the same quotient description for K alg n,m (see (6.47)). As a result, the analog of Proposition 7.8 is also true.
Proposition 7.12. The map Φ of Lemma 6.3 gives an algebra isomorphism from K symp n,m to K alg n,m .

An nc-vector field
The algebra K symp n,m in principle carries a non-trivial A ∞ -structure. Seidel gave a necessary and sufficient criterion for formality of an A ∞ -algebra: it should admit a degree one Hochschild cohomology class which acts by the Euler field, see Theorem 8.7 (a proof is given in [1]). Following the language of non-commutative geometry, we call a degree one Hochschild cocycle a non-commutative vector field or nc-vector field: the motivating example is a global vector field V ∈ H 0 (T Z ) ⊂ HH 1 (D b (Coh(Z)) on an algebraic variety Z. In this section, we apply the method from [1] to construct an nc-vector field by counting holomorphic discs with prescribed conormal-type conditions at infinity. 8.1. Moduli spaces of maps revisited. We first explain how to modify the moduli spaces of maps used in [1] to construct an nc-vector field in our setting. Let R d+1,h (0,1) be the moduli space of unit discs S with d + 1 boundary punctures {ξ i } d i=0 , h unordered interior marked points mk(S) and 2 ordered interior marked points s 0 , s 1 such that s 1 lies on the hyperbolic geodesic arc between s 0 and ξ 0 (but s 1 = s 0 and s 1 = ξ 0 ). We allow s 0 , s 1 ∈ mk(S) so R d+1,h (0,1) is also contractible and we have dim(R d+1,h (0,1) ) = dim(R d+1,h ) + 3. We call the h unordered interior marked points type-1 interior marked points, and the 2 ordered interior marked points type-2 interior marked points. We denote mk(S) ∪ {s 0 , s 1 } by mk(S) + .
We can compactify R d+1,h (0,1) to R d+1,h (0,1) as in [1, Section 3.6] (which treats the case h = 0). Informally, the compactification includes stable broken configurations as considered in [1,Section 3.6], with an extra condition that there are a total of h type-1 interior marked points (counted with multiplicity) lying in the interiors of the irreducible components of the configuration. In particular, the codimension one boundary facets of R d+1,h (0,1) are given by (1) ({ξ i+1 , . . . , ξ i+j move together) A nodal domain in which a collection of input boundary punctures bubble off: When j = 1, we need h 2 > 0 so that the domain is stable.
(2) (s 1 → s 0 ) A domain with a sphere bubble carrying the two type-2 interior marked points and some type-1 interior marked points attached to a disc carrying the remaining type-1 interior marked points and the d + 1 boundary punctures. Letting M h 0,3 denote the moduli space of spheres with 3 type-2 (hence ordered) marked points and h unordered type-1 interior marked points, and R d+1,h 1 denote the moduli space of unit discs with d + 1 boundary punctures, h type-1 interior marked points and 1 type-2 interior marked point, this component is The attaching point is understood to be the type-2 interior marked point on the disc and the last type-2 marked point of the sphere.
Let D 0 ⊂ Hilb n (E) is the divisor of ideals whose support meets D E . We now assume that D 0 is moveable, and the base locus of its linear system contains no rational curve, and (8.5) there is a holomorphic volume form with simple poles on D 0 (so c 1 (Hilb n (E)) = P D(D 0 )) (8.6) Let D 0 be a divisor in Hilb n (E) linearly equivalent to, but sharing no common irreducible component with, D 0 . Let B 0 = D 0 ∩ D 0 be the base locus of the corresponding pencil; we assume that it contains no rational curve. Note that these conditions will hold in the case of type A Milnor fibres (see [1,Section 6]), and more generally (8.5) holds when D E ⊂ E is moveable and (8.6) holds when there is a holomorphic volume form on E with simple poles on D E (see [1,Lemma 6.3]). Given x 0 ∈ X (H 0 , L 0 , L d ) and x j ∈ X (H j , L j−1 , L j ) for j = 1, . . . , d, we define to be the moduli of all maps u : S → Hilb n (E) such that for z ∈ mk(S), the multiplicity of intersection with D HC equals mult(z) u(z) ∈ Sym(L j ) for z ∈ ∂ j S lim s→±∞ u( j (s, ·)) = x j (·) uniformly (8.8) where J, K should be understood as their extension to Conf n (E) (see Remark 2.13). Note that Lemma 2.18 and 2.19 remain true for u ∈ R d+1,h We want to discuss the regularity and compactification of R d+1,h (0,1) (x 0 ; x d , . . . , x 1 ) following Section 2.2.7, 2.2.9, 2.2.10 and [1].
First, we pick a consistent choice of (A S , J, K) for all elements in R d+1,h (0,1) and all possible cylindrical Lagrangian labels and in/outputs. In particular, this implies that if S ∈ R d+1,h (0,1) has a sphere component, then over that component K ≡ 0 and the complex structure is J E . We can then introduce the following moduli spaces of maps that are relevant for the compactification of R d+1,h (0,1) (x 0 ; x d , . . . , x 1 ): The moduli (8.9) consists of maps u : S → Hilb n (E) such that S ∈ R d+1,h 1 , and for z ∈ mk(S), the multiplicity of intersection with D HC equals mult(z) u(z) ∈ Sym(L j ) for z ∈ ∂ j S lim s→±∞ u( j (s, ·)) = x j (·) uniformly. (8.11) It has an evaluation map using the type-2 interior marked point to Hilb n (E). The virtual dimension of (8.9) is where 2d 0 comes from c 1 (Hilb n (E)) = P D(D 0 ) and d comes from dim(R d+1 1 ), For S ∈ M h 0,3 , we can rigidify the domain and assume the type-2 marked points are 0, 1, ∞. The moduli (8.10) consists of maps u : S → Hilb n (E) \ D r such that S ∈ M h 0,3 , E -holomorphic for z ∈ mk(S), the multiplicity of intersection with D HC equals mult(z).
. . , x 1 ) is 1 dimensional, it can be compactified by adding the following boundary strata, which are themselves regular.
In the rest of this section, we establish some regularity and compactness statements and prove Proposition 8.1.
8.1.1. Moduli of Chern one spheres. We first discuss M h (Hilb n (E)\D r |1). Let C be a Chern 1 rational curve in E and q 2 , . . . , q n ∈ E be n − 1 pairwise distinct points such that at most one of q 2 , . . . , q n lies in C. The product determines a Chern 1 rational curve in Sym n (E) which meets ∆ E at at most one point. Therefore, it uniquely lifts to an irreducible Chern 1 rational curveC in Hilb n (E). We call C a Chern 1 rational curve of product type. In fact, these are essentially all the irreducible Chern 1 rational curves in Hilb n (E).
Lemma 8.2 (Lemma 6.4 of [1]). Let C be an irreducible rational curve in Hilb n (E) with Chern number 1 and not contained in D HC . Then C is of product type.
Proof. By assumption, π HC (C) is an irreducible Chern 1 rational curve in Sym n (E). By the tautological correspondence, we get a (possibly disconnected) closed complex curve C in E of Chern number 1. It means that C lies inside a finite union of fibers of π E . Having Chern number 1 means that [C ]·[D E ] = 1 so C is a union of a Chern 1 rational curve C and n−1 points in E. By the assumption that C is not contained in D HC (or equivalently π HC (C) is not contained in ∆ E ), we know that the points are pairwise disjoint and at most one of them lies inside C . Proof. When h > 0, by Lemma 8.2, M h (Hilb n (E) \ D r |1) is actually empty so there is nothing to prove.
When h = 0, the regularity of M h (Hilb n (E) \ D r |1) follows from the explicit description of Lemma 8.2 and the fact that every direct summand of the normal bundle of a product-type Chern one rational curve in Hilb n (E) \ D r has Chern number ≥ −1. Therefore, automatic regularity for these somewhere injective curves applies.
To define a pseudocycle, the codimension 2 subset to be added is the union of the image of stable Chern 1 rational curves in Hilb n (E) that meets D r non-trivially, which is denoted by B r in [1]. It is of codimension 2 because every such stable Chern 1 rational curve maps to a point in Sym n (H • ) that lies inside ∆ H • ∩ D where D is the divisor in Sym n (H • ) consisting of subschemes of H • whose support meets critical values of π E .
We denote the pseudocycle by GW 1 .

8.1.2.
Moduli of discs. The following regularity statement follows as in Lemma 2.14. For an element u in the moduli space (8.7) or (8.9), we can define E(u) by (2.78). Even though E is not exact, we can still define ω geom K and ω top K on Conf n (E) × S. They differ by R K which, by the same reasoning as in Lemma 2.26, is uniformly bounded independent of u. Moreover, the integration of ω top K over the graph of u is bounded a priori by the Lagrangian boundary conditions and the intersection numbers with D 0 , D r , because Hilb n (E) \ (D 0 ∪ D r ) is exact. To conclude, we have Lemma 8.5 (Energy). For a fixed choice of {x j } d j=1 and a moduli of maps (8.7) or (8.9), there exists T > 0 such that for all elements u in the moduli, we have E(u) < T .
We also have the corresponding statement for positivity of intersections. Lemma 8.6 (Positivity of intersection). Let u be an element in (8.7) or (8.9). If u(z) ∈ D 0 (resp. u(z) ∈ D r ), then the intersection u(z) between Im(u) and D 0 (resp. D r ) contribute positively to the algebraic intersection number.
Proof. For both cases (D 0 and D r ), it follows from the fact that the Hamiltonian vector field in the perturbation term is tangent to (every stratum of) the divisor. For D 0 , this is exactly Lemma 2.32. For D r , the argument is similar to that of Lemma 2.19 and 2.34.
Proof of Proposition 8.1. Let u k : S k → Hilb n (E) be a sequence of maps in (8.7). First consider the case that there is a subsequence of S k converging to S ∈ R d+1,h (0,1) . Without loss of generality, we can assume S k = S for all k.
The analog of Lemma 2.30 holds. More precisely, if there is energy concentration at a point in S but outside ν(mk(S) + ), then it will produce a sphere bubble or a disc bubble in E which intersects D E . By Lemma 8.6, this will imply that for large k, the algebraic intersection number between u k and D 0 is greater than 1. Contradiction.
Energy concentration cannot happen at a point outside mk(S) + . If energy concentration happens at mk(S) + , then it is either at {s 0 , s 1 } or not. If not, the resulting sphere bubble (tree) has Chern number 0 so it is contained in Hilb n (E) (recall that c 1 (Hilb n (E)) = P D(D 0 )). Therefore, it intersects D r positively by Lemma 2.5 and gives a contradiction. If energy concentration happens at s 0 or s 1 , say s 0 , then the bubble tree has either Chern number zero or Chern number one. We get a contradiction in the former case as above. For the latter case, the Chern 1 bubble intersects D 0 , so the algebraic intersection number between u and D 0 is at least 2 in total. This again gives a contradiction.
Energy concentration at strip like ends give parts of (8.15) and (8.16). This finishes the bubbling analysis when S k has a subsequence converging to S ∈ R d+1,h (0,1) . Now, if S k has a subsequence converging to the boundary strata (8.1), (8.3), we get the remaining terms in (8.15) and (8.16).
If S k has a subsequence converging to the boundary strata (8.4), then for k large, u has algebraic intersection at least 2 with D 0 because each disc component in the limit contributes at least one.
Next, we consider the case that S k has a subsequence converging to the boundary strata (8.2). Without loss of generality, we can assume S k = S lies in the boundary strata (8.2) for all k. The analog of Lemma 2.30 still holds. That is, there is no energy concentration at a point in S \ν(mk(S) + ), and hence no energy concentration at a point in S\mk(S) + . If energy concentration happens at mk(S), we get a Chern 0 bubble and the usual contradiction. If energy concentration happens at s 0 or s 1 , say s 0 , then at least one of the spheres (including the original one in the domain) has vanishing Chern number, so meets D r . Therefore, the tree of spheres define a stable Chern 1 rational curve in Hilb n (E) which lies in a real codimension 4 subset of Hilb n (E) (see Corollary 8.3). This can be avoided by genericity of the choice of data. Energy concentration of solutions in (8.2) at strip-like ends is a higher codimension phenomenon.
Finally, when there is no energy concentration, the boundary strata (8.2) correspond precisely to (8.17) and (8.18).

Pure
Lagrangians. An nc-vector field of an A ∞ category A is a cocycle b ∈ CC 1 (A, A). Given an nc-vector field b, a b-equivariant object is a pair (L, c L ) such that L ∈ Ob(A), c L ∈ hom 0 A (L, L) and b 0 | L = µ 1 (c L ). In this case, (L, c L ) is called a b-equivariant lift of L. Given two b-equivariant objects (L, c L ) and (L , c L ), the map is a chain map from hom A (L, L ) to itself. An nc-vector field is called pure if every object L ∈ Ob(A) admits a b-equivariant lift (L, c L ) and for every pair (L, c L ), (L , c L ) of b-equivariant lifts, the endomorphism on H(hom A (L, L )) induced by (8.19) is given by a → |a|a (8.20) for all pure degree elements a ∈ H(hom A (L, L )), where | · | denotes degree. In other words, purity asserts that (8.19) agrees with the Euler vector field. Theorem 8.7 (Seidel). If an A ∞ category A over a field of characteristic 0 admits a pure nc-vector field, then A is formal.
To compensate for these strata, we consider the closed-open maps which are the direct analogs of the ones in [1, Section 3.5]. The only difference is that we added the type-1 interior marked points and hence use R d+1,h defines an nc-vector field for FS cyl,n (π E ).
We now specialise to the case where E = A m−1 is the type A Milnor fiber. It is shown in [1] that both GW 1 and [B 0 ] are null-homologous so Proposition 8.8 applies. The Lagrangian spheres and thimbles in E have trivial first cohomology and so (by exactness) trivial first Floer cohomology, so admit equivariant structures. Lemma 8.9. For each λ ∈ Λ n,m , L λ is pure.
Proof. The argument of [1, Lemma 6.9] generalises to our situation: we briefly recall the key points, adapted to the current setup. By Lemma 2.22, the rigid count of R d+1,h (0,1) (x 0 ) is 0 when h > 0. When h = 0, the domain is a disjoint union of n discs which are mapped to different Lagrangian components of L λ . Since the intersection with D 0 is 1, exactly one of these n discs hits D E transversely once, and the others are disjoint from D E and are hence constant maps by exactness. The key point in the argument of [1, Lemma 6.9] is that, over each point t ∈ γ of a fibred Lagrangian L γ ⊂ E, there are exactly two holomorphic discs with boundary on L γ (the hemispheres of a P 1 -fiber of E). When γ is a matching path, this algebraic count gives a → 2a under the map (8.19) for the degree 2 generator a ∈ HF (L γ ). Moreover, by [1, Lemma 2.12] and the paragraph thereafter, (8.19) necessarily vanishes on degree 0 generators of HF (L γ ) whenever HF 0 (L γ ) has rank 1 (which holds when γ is a matching path or a thimble path). The result follows.

Formality
By Proposition 7.12, we know that on the cohomological level, the extended symplectic arc algebra K symp n,m argees with the extended arc algebra K alg n,m . In this section, we prove that the extended symplectic arc algebra is formal when working over a field of characteristic 0, by proving that the nc-vector field constructed in 8.2.1 is pure. To ease notation, in this section, given a weight λ, we use λ and λ to denote λ alg and λ alg , respectively.
Proof. We give a proof when i = 0; the case i = 1 can be dealt with similarly.
Recall that, schematically, the product in the combinatorial arc algebra is computed by stacking an oriented λ 0 ∪ λ 0 on top of an oriented λ 1 ∪ λ 0 , and resolving / removing the middle levels by a sequence of elementary cobordisms, which induce maps by the action of an underlying TQFT.
The diagram λ 1 ∪ λ 0 consists of circles and lines. Let C 1 , . . . , C k be the circles, and define for j = 1, . . . , k Γ o,j := {γ ∈ λ 0 |γ is contained in the circle C j }. The minimal degree generator x min corresponds to orienting all the circles in λ 1 ∪λ 0 counterclockwise (note that the orientations of the arcs in λ 1 ∪ λ 0 are uniquely determined by the rule (6.12), namely, the remaining ∨ must occupy the rightmost available positions). On the other hand, let C s 1 , . . . , C st be the circles in λ 1 ∪ λ 0 that are oriented clockwise by x. For each j ∈ {s 1 , . . . , s t }, we pick a single γ j ∈ Γ o,j . Now we give λ 0 ∪ λ 0 the orientation y that is uniquely determined by the property that a circle C in λ 0 ∪ λ 0 is oriented clockwise if and only if C contains γ j for some j ∈ {s 1 , . . . , s t }.
One feature of the Khovanov TQFT is that a counterclockwise circle times a counterclockwise (resp. clockwise) circle is a counterclockwise (resp. clockwise) circle. The analog is true for the Khovanovtype TQFT from [13,44]. One can check, by applying the rules underlying the Khovanov-type TQFT from [13,44], that our prescription ensures that we have µ 2 (x min , y) = x (see Figure 9.1 for an example). In particular, the product map Figure 9.1. From left to right: y (top) and x min (bottom) before taking product; successively applying two TQFT operations to the first picture; the product x. Clockwise orientations on circles are indicated by . We choose γ j to be the outermost upper arcs in the 2 circle components of x; this forces the indicated s on y.
Proof. Note that λλa, bλλ and bλa are oriented circle diagrams so the LHS and RHS of (9.3) are well-defined. Now we need to justify (9.3). If γ ∈ λ is a half-circle and the component in λλa containing γ is a circle C, then by the definition of λ, we know that C is oriented counterclockwise. Similarly, if γ * ∈ λ is a half-circle and the component in bλλ containing γ * is a circle C , then C is oriented counterclockwise. In this case, when we do the resolution between γ and γ * involved in the product, the outcome is also oriented counterclockwise. In other words, the resolution cobordism move does not change the orientation (see Figure 9. On the other hand, if the component of λλa containing γ is a line, then we need to first complete it to a circle before computing the product. Moreover, the circle to which it is completed is oriented counterclockwise (see Figure 9.3). The same holds for bλλ.
Therefore, each type of resolution, and hence the product obtained from a composition of such, does not change the orientations. The result follows. The set of weights Λ n,m has a natural 'Bruhat' partial ordering (see Section 4.1). Intuitively, it is given by exchanging pairs ∨ and ∧; one increases in the Bruhat order by moving ∨'s to the right, see [13,Section 2]. Given λ ∈ Λ n,m , there is a unique maximal element w λ in the set {w ∈ Λ n,m | λwλ is an oriented circle diagram}. (9.5) Explicitly, the maximal element is given by putting a ∨ on the right end-points of all the half-circles and lines in λ (or λ) and a ∧ otherwise. Lemma 9.3. Let λ 0 , λ 1 ∈ Λ n,m . At least one of the following statements holds.
Since we have assumed that (2) fails, so we get a contradiction and hence, λ 1 ∪ λ 0 is a diagram with only line components (and there is only one orientation given by η = λ 1 ).
Let s 1 < · · · < s k be the integers which are the left end points of half-circles of λ 0 . By definition, λ 0 (s i ) = ∨ for all i. If λ 1 (s i ) = ∧ for all i, then we are done, because then λ 1 has ∨ on all the right end points of half-circles and lines of λ 0 , and hence λ 1 = w λ 0 .
If this is not the case, then let s j be the largest among s 1 , . . . s k such that λ 1 (s j ) = ∨. Let s j + t be the right end point of the half-circle of λ 0 with left end point s j . By definition of λ 0 , exactly half of {s j + 1, . . . , s j + t − 1} are labelled by ∧ and half by ∨ with respect to λ 0 . Moreover, all of them lie in a half-circle of λ 0 .
v Proof. By Lemma 8.9, L λ is pure for each λ ∈ Λ n,m . Any partial order admits a total order refinement. We choose a total ordering (≤ T ) on Λ n,m that is compatible with the Bruhat partial ordering ≤. We make choices of b-equivariant structure on the L λ with decreasing order of λ.
Let η ∈ Λ n,m and suppose that a b-equivariant structure on L λ is chosen for all λ > T η, such that b is pure. We want to choose a b-equivariant structure on L η such that b is pure on {L λ } λ≥ T η .
We choose a b-equivariant structure on L η such that the minimal degree element of HF (L η , L wη ) has weight equal to the degree. By Lemma 9.1, HF (L η , L wη ) is cyclic as a module over HF (L wη , L wη ) so the weight equals the degree for all pure degree elements in HF (L η , L wη ).
By Lemma 9.2, we have This means that λλ η in HF (L η , L λ ) has weight equal to its degree. By Lemma 9.1 again, we know the minimal degree elements in HF (L η , L λ ) have weight equal to degree, hence so do all elements in HF (L η , L λ ). The same is true for HF (L λ , L η ) because Lemma 9.2 also implies that µ 2 (ηλ λ , λ λ λ) = ηλ λ (9.8) By induction over the total order, the result follows.
Proof of Theorem 1.1. Recall the notation for the tuple of Lagrangian thimbles T I from Section 4. It is easy to see that T (λ) −1 (∨) can be generated by {L λ |λ ∈ Λ n,m } using the exact triangles constructed in Section 3.2. Conversely, L λ can be generated by {T (λ) −1 (∨) |λ ∈ Λ n,m }, either again by directly applying the exact triangles, or by appeal to Proposition 5.16. Therefore, by Proposition 4.4, we have quasi-equivalences D π FS(π n,m ) perf-(End(⊕T (λ) −1 (∨) )) perf-(K symp n,m ) = perf-(K alg n,m ) (9.9) Remark 9.5. Theorem 1.1 and Proposition 3.5 together give a new proof of a result of [7], namely that the Serre functor of the constructible derived category of the Grassmannian is given by the action of the centre of the braid group.
There is an algebra isomorphism K alg n,m (K alg n,m ) op , given on basis elements by sending an oriented circle diagram βλα to its reflection α r λβ r (see [13,Equation (3.9)]). Together with Theorem 1.1, this implies that Corollary 9.6. For all λ, λ ∈ Λ n,m , we have HF (L λ , L λ ) = HF (L λ , L λ ) (9.10) as graded vector spaces. This is already non-trivial, because L λ , L λ are not compact Lagrangians, so it is not a priori clear why they should satisfy this Poincaré duality type of equality (because of the non-trivial wrapping in constructing the category, the underlying cochain groups are typically not equal).
The algebra isomorphism K alg n,m (K alg n,m ) op can be understood geometrically: we give a sketch of the argument. By a hyperbolic isometry between the upper half plane and the unit disc, we can assume the target of π E is the unit disc B. We also assume that the critical values lie on the real line, and that infinity in the upper half plane is mapped to √ −1 ∈ ∂B. Let ι B : B → B be the reflection along the real line. We can assume that there is an anti-symplectic involution ι E : E → E covering ι B .
Let u : S → Y E be a solution that contributes to the A ∞ structure of FS cyl,n (π E ). Let ι S : S → S be reflection in the real diameter of the disc, and S the image of S (equipped with the push-forward data by ι S ). Let ι Y E : Y E → Y E be the anti-symplectic involution induced by ι E . In this case, ι Y E • u • ι S : S → Y E is tautologically a solution to the push-forward equation, which is itself a perturbed pseudo-holomorphic equation. Note, however, that the ordering of the Lagrangian boundary conditions is reversed on ∂S.
This illustrates that, for appropriate choices of Floer data, the involution ι E induces an A ∞equivalence where FS cyl,n (π E ) † is quasi-equivalent to FS cyl,n (π E ) but the wrapping at infinity for CF (?, ??) in FS cyl,n (π E ) † is defined to be wrapping the latter entry clockwise, with stop at − √ −1, whilst the wrapping in FS cyl,n (π E ) is wrapping the former entry counterclockwise with stop at √ −1 (see Figure  9.5). For more details of (quasi-)equivalences of the kind of (9.11), see [42,Appendix]. To identify FS cyl,n (π E ) † with FS cyl,n (π E ), we need to pick a quasi-equivalence Ψ between them. We choose the one that corresponds to moving the 'stop' − √ −1 for FS cyl,n (π E ) † to √ −1 clockwise along the boundary of B and keeping a compact region containing the critical points unchanged (see Figure 9.5). This choice of quasi-equivalence has the property that for all λ ∈ Λ n,m .
Applying Ψ • ι E to the hom space between the Lagrangians associated to weights in Λ n,m , we get Corollary 9.6: where the first equality comes from (9.11) and the second one comes from (9.12). Similarly, applying Ψ • ι E to the multiplication maps, we can deduce K symp n,m (K symp n,m ) op , so (9.11) can be viewed as a geometric interpretation of the isomorphism K alg n,m (K alg n,m ) op .
Proof of Corollary 1.5. The statement follows from Theorem 1.1 and the fact that K alg n,m = (K alg m−n,m ) op (see [13,Equation (3.10)]). The origin of this equality is the Schubert-cells compatible identification of the Grassmannians Gr(n, m) = Gr(m − n, m) but one can give a concrete isomorphism as follows (since this is not required for the result, we will give the map, but leave the verification that it is indeed an isomorphism to the reader).
Let P D : Λ n,m → Λ m−n,m be the bijection In words, P D is obtained by rotating λ by π (see Figure 9.6). Each basis element of K alg n,m is an oriented circle diagram, for which we can perform a π-rotation to obtain another oriented circle diagram, and hence a basis element of (K alg m−n,m ) op . This gives a bijection between basis elements of K alg n,m and (K alg m−n,m ) op . Since clockwise cup becomes clockwise cap, and vice versa, this bijection preserves grading. One can check from the defining TQFT that it induces an algebra isomorphism.

Dictionary between Lagrangians and modules
In this section, we summarise the dictionary between certain objects in FS cyl,n (A m−1 ) (or their Yoneda images as modules over K symp n,m ) and modules over K alg n,m . Modules in this section are right modules, which differs from the convention used in [13]. However, as remarked previously, there is a canonical algebra isomorphism K alg n,m (K alg n,m ) op , which identifies the right K alg n,m -modules used below with the left K alg n,m -modules used in [13]. For an A ∞ or dg category/algebra C, we use perf-C and C-perf to denote the dg category of perfect right, and respectively left, C modules. We use [C, D] denote the dg category of C-D bimodules. Since K alg n,m is homologically smooth, every proper (cohomologically finite) module or bimodule is perfect.
10.1. Indecomposable projectives. For each λ ∈ Λ n,m , we define P (λ) to be the submodule of K alg n,m generated by the oriented circle diagrams of the form P (λ) := ⊕Kλ alg µα (10.1) The collection of all P (λ) (together with their grading shifts) is the set of all indecomposable projective modules of K alg n,m . Under the isomorphism K symp n,m K alg n,m , P (λ) is the same as the Yoneda embedding of L λ as a right K symp n,m module. Thus, the indecomposable projectives are the Lagrangians associated to weights.
By [13,Theorem 5.3], the set of all indecomposable injective modules of K alg n,m is given by P (λ) * := Hom(P (λ), K) (10.2) where the right module structure is given by f a(m) := f (ma * ) for a ∈ K alg n,m , m ∈ P (λ) and f ∈ P (λ) * . In other words, P (λ) * is obtained by pulling back the left module Hom(P (λ), K) via the algebra isomorphism K alg n,m = (K alg n,m ) op . On the symplectic side, up to a grading shift, Hom(P (λ), K) is given by the Yoneda embedding of τ −1 (L λ ) as a left K symp n,m module, where τ is the global monodromy (see Proposition 3.5), so P (λ) * corresponds to the pull-back of τ −1 (L λ ) via the algebra isomorphism K symp n,m = (K symp n,m ) op . In view of the geometric explanation of the equivalence K symp n,m = (K symp n,m ) op given in (9.11), P (λ) * is given by the right Yoneda embedding of Ψ • ι E • τ −1 (L λ ).
10.2. Standard modules. For each λ ∈ Λ n,m , there is a standard module V (λ) defined in [13,Equation (5.11)]. As a graded vector space, V (λ) is generated by oriented cap diagrams of the form λα. The collection of all V (λ) gives a full exceptional collection for the dg category of right K alg n,m modules and Ext i (V (λ 0 ), V (λ 1 )) = 0 for some i only if λ 0 < λ 1 , where < is with respect to the Bruhat partial ordering.
An inductive construction of a projective resolution of V (λ) is described in [12,Theorem 5.3]. By identifying P (λ) with the Yoneda image of L λ and imitating that inductive construction, we have Proposition 10.1. The standard module V (λ), as an A ∞ module over K alg n,m = K symp n,m , is quasiisomorphic to the Yoneda image of T λ −1 (∨) up to grading shift.
If λ is not the maximal weight, then there is k ∈ {1, . . . , m} such that λ(k) = ∨ and λ(k + 1) = ∧. Let λ be the weight given by 'swapping positions k and k + 1'. That is Let λ ∈ Λ n−1,m−2 be the weight given by forgetting the positions k, k + 1.That is We define ∪ k,k+1 λ to be the admissible tuple in {im(z) ≤ 1} consisting of 1 matching path connecting k + √ −1 and k + 1 + √ −1 and n − 1 thimble paths from i + √ −1 to i for each i ∈ λ −1 (∨) \ {k}. The notation suggests that it is obtained from applying a 'cup functor' to λ in position k, k + 1 in the sense of [2].
The construction of the projective resolution comes from iteratively applying the exact sequence of modules in [12,Equation 5.8]. Therefore, it is sufficient to prove that the exact triangle in perf-K alg n,m induced from the exact sequence of modules in [12,Equation 5.8] corresponds to the exact triangle of Yoneda modules (10.5) proved in Section 3.2. This correspondence can be proved by induction on m, induction on the partial ordering of weights (from high to low; note that λ > λ) and the fact that HF (T (λ ) −1 (∨) [−1], L ∪ k,k+1 λ ) has rank one, so up to quasi-isomorphism there is only one non-trivial mapping cone. We leave the details to readers.
Thus, the standard modules are the Lefschetz thimbles.
Remark 10.2. In constrast to the collection of indecomposable projective modules, the collection of standard modules does not have formal endomorphism algebra when n > 1. A minimal model for the algebra when n = 2 is given in [23], which conjectured that this minimal model is not formal; we give a proof of that conjecture in the Appendix.
By [13,Theorem 5.3], the set of all costandard modules of K alg n,m are given by V (λ) * := Hom(V (λ), K) (10.6) where the right module structure is given by f a(m) := f (ma * ) for a ∈ K alg n,m , m ∈ V (λ) * and f ∈ V (λ) * . In other words, up to grading shift, it is given by the Yoneda embedding of τ −1 (T λ −1 (∨) ) as a right (K symp n,m ) op module after pull-back via the algebra isomorphism K symp n,m = (K symp n,m ) op . In view of (9.11) again, V (λ) * is given by the right Yoneda embedding of Ψ In fact, the Ψ • ι E • τ −1 (T λ −1 (∨) ) form the right Koszul duals of the thimbles (i.e. HF (T λ −1 0 (∨) , Ψ • ι E • τ −1 (T λ −1 1 (∨) )) = K if and only if λ 0 = λ 1 , and equal to 0 otherwise). It is well-known that costandard modules are right Koszul dual to standard modules (see for example [9]). The collection of all L(λ) is the set of all irreducible modules of K alg n,m . It is not immediately clear in our context which Lagrangians correspond to the irreducible modules. In view of the work of [16], and the 'topological model' for the compact core of the space Y n,m arising from its description as a quiver variety (see e.g. [2]), it is reasonable to believe that one can enlarge the category FS cyl,n (E) to allow Lagrangian discs that intersect exactly one of the {L λ } λ once and are disjoint from the others. These would be the analogues of 'cocore discs' over the components of the skeleton of a plumbing of cotangent bundles, and the Yoneda images of these Lagrangian discs are natural candidates for the irreducible modules.

Symplectic annular Khovanov homology
This section gives an application of our main results to link homology theories in the sense of Khovanov. Suppose m = 2n. A braid β ∈ Br n in the n-string braid group defines bimodules P (j) β over each of the extended arc algebras K alg j,n with 0 ≤ j ≤ n. One can combine a theorem of Roberts [34] with deep recent work of Beliakova, Putyra and Wehrli [8] to infer that there is a spectral sequence ⊕ j HH * (P It is by now classical that Khovanov homology can be understood as a certain morphism group Ext perf-H alg n,2n (P, (β × id)(P )) in the derived category of modules over the compact arc algebra H alg n,2n , see [21], where β × id ∈ Br 2n belongs to the 2n-string braid group and P is a particular projective. From the viewpoint of the (extended) arc algebras, the existence of such a spectral sequence is rather mysterious. The purpose of this section is to give a transparent account of why it should exist, starting from a new semi-orthogonal decomposition of perf -K alg n,2n which we shall derive from the geometric viewpoint afforded by Theorem 1.1.
We assume m = 2n in Sections 11 and 12 unless stated otherwise.
The corresponding tuple L hs is called the 'horseshoe' Lagrangian tuple and the Lagrangian Sym(L hs ) ⊂ Y n,m is the horseshoe Lagrangian from [41,1]. Let Br n be the braid group on n strands. The paper [41] associates to each element β ∈ Br n a symplectomorphism φ The main result of [41] is that Kh symp (κ(β)) is a link invariant, i.e. it is independent of the representation of κ as a braid closure.
One can give an equivalent definition of symplectic Khovanov cohomology using L hs instead of its product Sym(L hs ). Let E := A m−1 denote the Milnor fiber. To each simple braid element σ i we associate the symplectomorphism φ σ i : E → E given by the Dehn twist along the i th matching sphere (lying above the line joining i + √ −1 and i + 1 + √ −1). This defines a representation Br m → π 0 Symp ct (E). Let Br n → Br m be the embedding of the left n strands. We have the restricted representation Br n → π 0 Symp ct (E) so each braid β ∈ Br n determines a symplectomorphism φ β of E that acts as the identity on the 'right half'.
Definition 11.1. The symplectic annular Khovanov homology of the braid β ∈ Br n is the direct sum of Hochschild homology groups where, by convention, FS cyl,j (π E 1/2 ) = K and P (j) β = K is the diagonal bimodule when j = 0 (cf. Remark 1.2).
Symplectic annular Khovanov homology is clearly a braid invariant. It is not an invariant of the link closure of the braid. Its definition is motivated by the fact that combinatorial annular Khovanov homology, originally defined by a diagrammatic calculus for links in a solid torus [3], is itself isomorphic to a direct sum of Hochschild homologies of braid bimodules over the extended arc algebras, a recent theorem of [8] establishing a conjecture due to [6]. Indeed, Theorem 1.1 and the same formality arguments for bimodules as in [2] would prove that Proposition 11.2. AKh symp (β) is isomorphic to annular Khovanov homology when K has characteristic 0.
We will prove Theorem 1.9 by constructing a spectral sequence AKh symp (β) ⇒ Kh symp (κ(β)). Remark 11.3. One could also define a braid invariant as the direct sum of fixed point Floer homologies of φ β on Y j,n over j ∈ {0, . . . , n} (where 'partial wrapping' would remove fixed points at infinity). It seems likely that this geometric definition would recover that of Definition 11.1, but establishing such an isomorphism is beyond the scope of this paper. See [38] for closely related results.
Remark 11.4. There is an sl 2 action on AKh symp (β), by combining [18] and Proposition 11.2, which implies a rank inequality rank(HH * (FS cyl,j (π E 1/2 ), P for all j < n/2. Coupled with Remark 11.3, this predicts a non-trivial existence result for periodic points of the symplectomorphisms of Y j,n associated to braids. It would be interesting to construct the sl 2 action symplectic-geometrically.
11.2. Embedding of algebras. Let E := A m−1 and π E be the standard Lefschetz fibration. Let W 1 := {re(z) < n + 1 2 } and W 2 := {re(z) > n + 1 2 }. Note that FS cyl,j W 1 (π E ) = FS cyl,j (π E 1/2 ). As explained in Section 3.2, there is a faithful A ∞ functor K : FS cyl,j W 1 (π E ) → FS cyl,n (π E ) for each object K of FS cyl,n−j W 2 (π E ). By essentially the same argument, we can prove that for L i ∈ FS cyl,j W 1 (π E ) and K i ∈ FS cyl,j W 2 (π E ). Varying either the L i or K i , Floer multiplication amongst the corresponding groups also respects the tensor product structure. Let 0 ≤ j ≤ n be an integer. For each λ ∈ Λ j,n , one can define an object L λ in FS cyl,j W 1 (π E ) by forgetting the right n critical values (see Figure 11.1). Our convention is FS cyl,j W 1 (π E ) = K when j = 0, i.e. that the category contains a unique object whose endomorphism algebra is the ground field concentrated in degree zero; the object L λ is by definition the unique non-zero object in the category in this case. For the other extreme, when j = n, the object L λ is the tuple of Lagrangian thimbles (cf. Corollary 5.17). Similarly, for each µ ∈ Λ n−j,m−n , one can define an object K µ in FS cyl,n−j W 2 (π E ) by forgetting the left n critical values. We have an algebra isomorphism HF (L λ 0 K µ 0 , L λ 1 K µ 1 ) = K symp j,n × K symp n−j,m−n (11.5) where the sum is over all λ 0 , λ 1 ∈ Λ j,n and µ 0 , µ 1 ∈ Λ n−j,m−n . Figure 11.1. Lagrangian tuples L λ (blue, left) and K µ (black, right) Let A be the A ∞ full subcategory of FS cyl,n (π E ) consisting of objects L λ K µ over all λ ∈ Λ j,n , µ ∈ Λ n−j,m−n and j = 0, . . . , n. For each integer 0 ≤ j ≤ n, we have the full subcategory A j of A given by the objects L λ K µ over all λ ∈ Λ j,n and µ ∈ Λ n−j,m−n .
Lemma 11.5. For each integer 0 ≤ j ≤ n, the endomorphism algebra of the objects in A j is formal, and hence quasi-isomorphic to the right hand side of (11.5).
Sketch of proof. This is a version of the Künneth theorem, i.e. that A ∞ structure on a product Lagrangian is quasi-isomorphic to the tensor product of a dg model for each factor.
Alternatively, in our case, we observe that each L λ K µ is pure with respect to the nc vector field. Therefore, one can repeat the strategy in Section 9. The key point is that once we know that each individual object is pure, it is possible to deduce a collection of objects is pure by investigating only the cohomological algebra of the collection of objects (this was exactly the strategy in Section 9); the crucial point is to find enough non-vanishing products for purity to be propagated from the subalgebras associated to the idempotents to the entire algebra. Since the cohomological algebra of the collection of L λ K µ splits as a tensor product, the strategy from Section 9 can be implemented, to conclude that this corresponding collection of Lagrangian tuples is formal.
It is not clear whether the endomorphism algebra of the direct sum of all the objects in A is formal or not. However, the dg category of perfect modules perf-A is formal in the following weaker sense.
Lemma 11.6. perf-A is quasi-equivalent to perf-K alg n,m .
Proof. By iteratively applying exact triangles, we see that the objects in A j can generate thimbles T = {T 1 , . . . , T n } in FS cyl,n (π E ) such that j of the T i 's are contained in π −1 E (W 1 ), and the remaining n − j are contained in π −1 E (W 2 ). Therefore, every thimble of FS cyl,n (π E ) can be generated by objects in A, so the result follows.
Proof. It is clear that CF (X 0 , X 1 ) = 0 if j 0 < j 1 , because of the definition of Floer cochains via positive isotopies along R = ∂H.
Theorem 11.8. perf-K alg n,m admits a semi-orthogonal decomposition perf-A n , . . . , perf-A 0 , where for each j, perf-A j is quasi-equivalent to perf-(K alg j,n × K alg n−j,m−n ). Proof. This immediately follows from Lemmas 11.5, 11.6 and 11.7. Theorem 11.8 categorifies the identity m n = n j=0 n j m−n n−j for ranks of Grothendieck groups. Remark 11.9. Even though we are primarily interested in the case m = 2n in this section, Theorem 11.8 holds for all n < m. This algebraic result seems to be new and may be of independent interest.
On the other hand, A ! shares many features with A. By the same arguments as in Lemma 11.5 and 11.6, we know that perf-A ! is quasi-equivalent to perf-K alg n,m and for j = 0, . . . , n, the category A ! j is formal with endomorphism algebra The analogues of Lemma 11.11 and Theorem 11.8 are Lemma 11.11. Let X 0 ∈ A ! j 0 and X 1 ∈ A ! j 1 . If HF (X 0 , X 1 ) = 0, then j 0 ≤ j 1 . As a result, A ! 0 , . . . , A ! n is a semi-orthogonal decomposition of A ! .
Theorem 11.12. perf-K alg n,m admits a semi-orthogonal decomposition perf-A ! 0 , . . . , perf-A ! n , where for each j, perf-A ! j is quasi-equivalent to perf-(K alg j,n × K alg n−j,m−n ). One consequence of Theorems 11.8 and 11.12 is that, when m = 2n, perf-(K j,n × K n−j,m−n ) is the same as [(K n−j,n ) op , K j,n ]. By the isomorphism (K n−j,n ) op = K j,n (see Corollary 1.5), this in turn is isomorphic to [K j,n , K j,n ]. Similarly, (K j,n × K n−j,n )-perf is the same as [K j,n , (K n−j,n ) op ], which in turn is isomorphic to [K j,n , K j,n ]. 11.3. A Beilinson-type spectral sequence. We next explain why a semi-orthogonal decomposition of an A ∞ category C induces a spectral sequence with target a given morphism group in C. From now on, for an object X, we use X r and X l to denote its right and left Yoneda modules, respectively.
Let C be a split-closed trianglated A ∞ category with a semi-orthogonal decomposition C = C n , . . . , C 0 (11.8) A Koszul dual semi-orthogonal decomposition is a semi-orthogonal decomposition C = C ! 0 , . . . , C ! n (11.9) such that for X ∈ C ! i and Y ∈ C k H(hom C (X, Y )) = 0 only if i = k (11.10) Without loss of generality, we can assume each summand in the semi-orthogonal decomposition is split-closed trianglated. Note that Koszul dual semi-orthogonal decompositions always exist [24].
Let ι j : C j → C and ι ! j : C ! j → C be the embeddings. Let π j : C → C j and π ! j : C → C ! j be the projection functors (i.e. the unique functors such that π j • ι j = id C j and π j • ι i = 0 for i = j). These functors induce pull back functors on modules.
Proof. Let X ∈ C ! j . By (11.10), we have H(hom C (X, X )) = 0 for all X ∈ C i such that i = j. It implies that, up to quasi-isomorphism, we have X l = (π j ) * • (ι j ) * (X l ) (11.11) so the result follows.
Since Yoneda embedding and (π j ) * are both cohomologically full and faithful, so is the functor (ι j ) * ((−) l ) : C ! j → C j -perf (11.12) By (11.10) again, we can see that is the 0 functor when i = j.
Proof. Combining the fact that (11.13) is the 0 functor and that both Yoneda embedding and (ι j ) * are essentially surjective, we know that (11.12) is essentially surjective, so the result follows.
Let ψ j : C j -perf → C ! j be a quasi-inverse.
Lemma 11.15. Let X ∈ C be quasi-isomorphic to an iterated mapping cone of the form where X j ∈ C ! j for j = 0, . . . , n. Then ψ j • (ι j ) * (X l ) is quasi-isomorphic to X j .
Proof. By applying the functor ψ j • (ι j ) * ((−) l ) to the iterated mapping cone (11.14), and using the fact that (11.13) is a 0 functor, we see that only the object X j contributes and hence By definition, ψ j • (ι j ) * ((−) l ) is quasi-isomorphic to the identity functor on C j so the right hand side of (11.15) is in turn quasi-isomorphic to X j . Proposition 11.16. Let X, Y ∈ C. There is a spectral sequence to H(hom C (X, Y )) from where ψ j • (ι j ) * (X l ), as an object in C ! j , is regarded as a left C ! j module via the Yoneda embedding.
Proof. By (11.8), X is quasi-isomorphic to an iterated mapping cone of the form (11.14). An expression of X as such a mapping cone induces a filtration on Y r (X), with direct sum of graded pieces being ⊕ n j=0 Y r (X j ). (11.17) Thus we have a spectral sequence from the cohomology of (11.17) to H(hom C (X, Y )).
On the other hand, we have quasi-isomorphisms (11.20) where the second quasi-isomorphism comes from Lemma 11.15, and the other quasi-isomorphisms come from the definitions of the objects involved.
Remark 11.17. In the extreme case where each C j (and C ! j ) is an exceptional object, the semiorthogonal decomposition is a full exceptional collection and Proposition 11.16 reduces to the wellknown Beilison type spectral sequence (cf. [35, Section (5l)]).

From annular to ordinary Khovanov homology
In this section, we explain how to apply Proposition 11.16 to the semi-orthogonal decompositions in Theorem 11.8, 11.12 to obtain a spectral sequence from annular to ordinary (symplectic) Khovanov homology.
Recall that for an A ∞ or dg category/algebra C, the Hochschild homology of a C-C bimodule P is defined to be HH * (C, P ) := H(∆ C ⊗ C-C P ) (12.1) where ∆ C is the diagonal bimodule and ⊗ C-C is the derived tensor product over [C, C]. Equivalently, ∆ C defines a right C ⊗ C op module and P defines a left C ⊗ C op module, and we have The fact that Hochschild homology groups appear in the first page of a spectral sequence to symplectic Khovanov cohomology arises from a relation between the horseshoe Lagrangian and the diagonal bimodule, that we explain next. More precisely, in section 12.1, we compute (ι j ) * (L l hs ) and (ι ! j ) * (φ β (L hs ) r ) for the embeddings ι j : A j → FS cyl,n (π E ) and ι ! j : A ! j → FS cyl,n (π E ). Then, we describe the quasi-inverse ψ j and complete the proof of Theorem 1.9 in Section 12.2.
12.1. Horseshoe Lagrangian tuple and diagonal bimodule. Recall that we have the pull-back functor The following is the key technical result.
Proposition 12.1. Up to grading shift, (ι j ) * (L l hs ) is quasi-isomorphic to the diagonal K j,n -bimodule. We are going to divide the proof into two steps. First, we will show that (ι j ) * (L l hs ) coincides with the diagonal bimodule (up to grading shift) on the cohomological level. Then, we will explain how to adapt the strategy in Section 9 to prove that this bimodule is formal. For all the Lagrangian tuples in the proof, we continue to use our grading convention (6.8) and the cost is that we will see eventually that (ι j ) * (L l hs ) corresponds to the diagonal bimodule shifted by n−j instead of the diagonal bimodule. This arises from the fact that there are n − j matching paths of (ι j ) * (L l hs ) being oriented clockwise in the sense of the algebraic extended arc algebra (see Remark 12.5).
Lemma 12.2. For λ ∈ Λ j,n and µ ∈ Λ n−j,n , we have vector space isomorphisms HF k+n−j−s (L hs , ι j (L λ , K µ )) = HF k (L µ + , L λ µ − ) (12.9) for all k. (Here n − j − s is the number of thimble paths in K µ .) The importance of Lemma 12.2 is that the right hand side of (12.9) is part of the symplectic extended arc algebra. We will see later how to use (12.9), and the identification of the symplectic and algebraic extended arc algebras (Proposition 7.12), to compute the bimodule multiplication map of (ι j ) * (L l hs ). Proof of Lemma 12.2. We use an upper matching to represent L hs and a lower matching to represent K µ (see the first picture of Figure 12.1). Now, we apply a symplectomorphism to 'move the right half to the top' and make all the matching paths of L hs be straight line segments (see the second picture of Figure 12.1). For each matching path of K µ , we apply a further symplectomorphism to 'bend it to the right' and make the two end points become n + 1 − c µ,i l + 1 3 + √ −1 and n + 1 − c µ,i l + 2 3 + √ −1 (see the third picture of Figure 12.1). Then, we remove all critical values contained in the thimble paths of K µ and all the paths that contain these critical values (see the fourth picture of Figure 12.1). Finally, for those matching paths of L hs that do not intersect with a matching path of K µ , we bend them to the right and turn them into thimble paths (see the fifth picture of Figure 12.1). Turning L hs (red) and K µ (black) to L µ + (red) and L µ − (black), respectively After this procedure, the set of critical values (the crosses and black dots in Figure 12.1) is exactly S, and we identify it with {1, . . . , n + 2s} by f . The Lagrangian tuples L hs and K µ become L µ + and L µ − respectively.
If we now add in a lower matching of L λ that does not intersect with L µ − , then together with L µ − we obtain a representative of L λ µ − (see the second picture of Figure 12.2). Moreover, for degree reasons (either all generators have odd degree, or all have even degree), both the cochain model CF (L hs , ι j (L λ , K µ )) and CF (L µ + , L λ µ − ) have vanishing differentials. Furthermore, there is a identification of the generators by sending a generator x of CF (L µ + , L λ µ − ) to the generator y of CF (L hs , ι j (L λ , K µ )) such that π E (y) is the union of π E (x) and the end points of the thimble paths of K µ (Figure 12.2). This identification increases the grading by the number of points added, which is n − j − s. It completes the proof.
To prove Lemma 12.2, it suffices to consider a lower matching of L λ to give a representative of L λ µ − . However, for this representative, there may be nested circles in the union of matching and thimble paths of L µ + and L λ µ − , which will be inconvenient later on. By possibly sliding the lower matching of L λ cross some matchings of L µ − (and L λ itself), we can choose a representative L λ µ − of L λ µ − with no nested circles (see the second picture of Figure 12.3). This corresponds to choosing another representative L λ µ of ι j (L λ , K µ ), by sliding across some matching paths of K µ (see the first picture of Figure 12.3). For these representatives, we still have the (grading shifting) cochain isomorphism with 0 differential which is again given by sending a generator x of CF (L µ + , L λ µ − ) to the generator y of CF (L hs , L λ µ ) such that π E (y) is the union of π E (x) and the end points of the thimble paths of K µ . The geometric generators on the right hand side of (12.10) are geometric basis elements of K symp j+s,n+2s , which are identified with the corresponding diagrammatic basis elements of K alg j+s,n+2s under the map Φ of Proposition 7.12. Next, we want to compare the multiplication map The second and third terms of (12.11) and (12.12) are identified via (12.10), and the identification of the first terms is defined similarly (i.e. adding to a generator x of CF (L λ 0 µ − , L λ 1 µ − ) the intersection points of CF (K µ , K µ ) that lie above those critical values that are contained in the thimble paths of K µ , see Figure 12.4, note that this identification is degree-preserving, because the points being added now have degree 0).
By the open mapping theorem, all solutions u contributing to the map (12.11) must include n−j −s constant triangles which, under π E , map to the n − j − s critical values that are contained in the thimble paths of K µ . Removing these constant triangles (and removing the thimble paths of K µ ), we can identify the moduli spaces defining the maps (12.11) and (12.12), so the maps (12.11) and (12.12) agree under the identification above. Let CF (L † λ 0 µ − , L † λ 1 µ − ) be a cochain model for the pair (L λ 0 µ − , L λ 1 µ − ) which contains the matching paths of µ − used above, and such that the union of matching and thimble paths has no nested circles (see the first picture of Figure 12.5). In particular, as the matching paths in µ − are not nested, the Floer cochain complex canonically splits where, for j = 0, 1, L † λ j is obtained by removing the matchings corresponding to µ − (see the second picture of Figure 12.5). Moreover, as there is no nesting, the geometric generators are geometric basis elements of K symp j,n ⊗ K symp s,2s . By Proposition 7.12, these geometric generators can in turn be identified with diagrammatic basis elements in K alg j,n ⊗ K alg s,2s . Let be a continuation map (the induced map on cohomology is independent of choices). By Proposition 7.12, we can completely describe µ 2 (κ(−), −) : CF (L † λ 0 µ − , L † λ 1 µ − ) × CF (L µ + , L λ 0 µ − ) → CF (L µ + , L λ 1 µ − ) (12.15) using the algebraic extended arc algebra. In particular, by applying (12.13) to the first term of (12.15), we completely understand the map µ 2 (κ(− ⊗ e µ − ), −) : CF (L † λ 0 , L † λ 1 ) × CF (L µ + , L λ 0 µ − ) → CF (L µ + , L λ 1 µ − ) (12.16) where e µ − ∈ CF (L µ − , L µ − ) is the identity element. On the other hand, we now consider CF (L λ 0 , L λ 1 ) × CF (L P D(µ) , L λ 0 ) → CF (L P D(µ) , L λ 1 ) (12.17) where, for j = 0, 1, L λ j is defined by removing the matching paths (and their end points) of L λ j µ − that correspond to µ − , and L P D(µ) is an upper matching such that there are no nested circles in the union of the matching paths of L P D(µ) and L λ j (see Figure 12.6). Analogously, we have a continuation map κ : CF (L † λ 0 , L † λ 1 ) → CF (L λ 0 , L λ 1 ) (12.18) which we also know completely because the geometric generators are all geometric basis elements and we can apply Proposition 7.12.
The outcome of this discussion is the following: Proof. After applying Proposition 7.12 to the geometric basis elements for all the Floer cochain groups involved, the maps (12.16) and (12.19) can be computed by the algebraic extended arc algebra in K alg j+s,n+2s and in K alg j,n , respectively. The upshot is that, for (12.16), elements of the form − ⊗ e µ − correspond to orienting the s circles in λ 1 µ − alg ∪ λ 0 µ − alg associated to µ − counterclockwise. Applying TQFT type multiplication to these circles does nothing [13,Equation (3.4)], and the rest of the diagrammatic calculus operations (for 12.16 and (12.19)) are obviously identified.
Remark 12.5. Another way to interpret the grading shift is as follows. We have seen from Lemma 6.3 (or Proposition 7.12) that the map from the symplectic extended arc algebra to the algebraic extended arc algebra is given by putting a ∨ to the points π E (x), where x is a geometric basis element. Each generator x of CF (L hs , L λ µ ) (see the first picture of Figure 12.3) must have j points lying above {1, . . . , n} and n − j points lying above {n + 1, . . . , 2n}. That corresponds to making j counterclockwise caps and n − j clockwise caps for the oriented cap diagram associated to L hs . This is where the extra n − j in the grading comes from.
Note that, after applying P D (which is grading preserving), (12.27) would become µ 2 (−, −) : HF (L P D(µ 0 ) , L λ ) × HF (L P D(µ 1 ) , L P D(µ 0 ) ) → HF (L P D(µ 1 ) , L λ ) (12.29) By Corollary 12.4 and 12.6, and the fact that every basis element of HF (ι j (L λ 0 , K µ 0 ), ι j (L λ 1 , K µ 1 )) can be written as the product of x ⊗ e K ∈ HF (ι j (L λ 0 , K µ 0 ), ι j (L λ 1 , K µ 0 )) and (12.30) e L ⊗ y ∈ HF (ι j (L λ 1 , K µ 0 ), ι j (L λ 1 , K µ 1 )) (12.31) for some x, y, we now know that the multiplication map HF (ι j (L λ 0 , K µ 0 ), ι j (L λ 1 , K µ 1 )) × HF (L hs , ι j (L λ 0 , K µ 0 )) → HF (L hs , ι j (L λ 1 , K µ 1 )) (12.32) can be identified with the left and right multiplication maps µ 2 (−, µ 2 (−, −)) : HF (L λ 0 , L λ 1 ) × HF (L P D(µ 0 ) , L λ 0 ) × HF (L P D(µ 1 ) , L P D(µ 0 ) ) → HF (L P D(µ 1 ) , L λ 1 ) (12.33) In other words, we have now shown that (ι j ) * (L l hs ) coincides with the diagonal bimodule (shifted by n − j) on the cohomological level. Having this, we can now finish the proof of Proposition 12.1 Completion of the proof of Proposition 12.1. In the previous discussion, we have shown that (ι j ) * (L l hs ) coincides with the diagonal K symp j,n -K symp j,n bimodule on the cohomological level. It remains to show that (ι j ) * (L l hs ) is formal as a bimodule over K symp j,n , or formal as a left module over A j . As in the proof of formality in Section 9, it suffices to show that we can choose a consistent choice of b-equivariant structures for the collection of Lagrangians in A j together with the single additional Lagrangian L hs . As explained in Lemma 11.5, the existence of a consistent choice of b-equivariant structures for the collection of Lagrangians in A j follows from Section 9 so we now only need to argue the consistency for the pairs (L hs , ι j (L λ , K µ )).
Then, we can run the proof of Proposition 9.4. We choose a b-equivariant structure on L hs such that the rank 1 vector space HF (L hs , ι j (L λmax , K µmax )) is pure, where λ max and µ max are the unique maximal weight in Λ j,n and Λ n−j,n , respectively. Fixing K µmax , we want to inductively argue that HF (L hs , ι j (L λ , K µmax )) is pure for all λ. Let λ be such that the purity of HF (L hs , ι j (L λ , K µmax )) has been verified for all λ > T λ, where > T is the total ordering of weights in Proposition 9.4; then we need to verify purity for HF (L hs , ι j (L λ , K µmax )). By Lemma 9.2, and the identification between (12.32) and (12.33), there exists a weight λ > T λ such that we have a non-trivial product (this is the analog of (9.7)) HF (ι j (L λ , K µmax ), ι j (L λ , K µmax )) × HF (L hs , ι j (L λ , K µmax )) → HF (L hs , ι j (L λ , K µmax )) (12.34) Together with cyclicity, it implies that HF (L hs , ι j (L λ , K µmax )) is pure. Thus HF (L hs , ι j (L λ , K µmax )) is pure for all λ.
Similarly, we can fix L λ and apply the inductive argument to HF (L hs , ι j (L λ , K µ )) with varying µ. All together, we conclude that HF (L hs , ι j (L λ , K µ )) is pure for all λ, µ and hence (ι j ) * (L l hs ) is formal as a left module over A j .
Proof. We can apply the same method as in the proof of Proposition 12.1 to (φ β ) * (ι j ) * (φ β (L hs )) l , where (φ β ) * : perf-A j → perf-A j is the pull-back functor under φ β . This is because, geometrically, CF (φ β (L hs ), φ β (L) K) can be canonically identified with CF (L hs , L K), as can all holomorphic polygons contributing to the A ∞ structure amongst tuples of such Floer groups. It may be worth emphasising that at this point in the argument we use the fact that, working with pullback data for Hamiltonians, almost complex structures, etc, the symplectomorphism φ β gives a canonical identification of moduli spaces of polygons; the corresponding isomorphism, from the algebraic viewpoint, is much less transparent.
We have the analogous results for the embedding ι ! j : A ! j → FS cyl,n (π E ) and the induced pull back (ι ! j ) * : perf-FS cyl,n (π E ) → perf-A ! j [K j,n , K j,n ] (12.35) Proposition 12.8. Up to grading shift, (ι ! j ) * (L r hs ) is quasi-isomorphic to the diagonal bimodule. Sketch of proof. A completely parallel argument applies (see Figure 12.7 and 12.8 for illustration). If one keeps track of the grading and follows our grading convention, then (ι ! j ) * (L r hs ) is quasiisomorphic to the diagonal bimodule shifted by n − j (cf. Remark 12.5).
Remark 12.11. By taking morphisms with an appropriate object in A i for i = j, one can see that (L λ K µ ) l is not quasi-isomorphic to (L ! λ K ! µ ) l . By Lemma 12.10, we know that the quasi-inverse ψ j is given, on the object level, by A further inspection of the proof of Lemma 12.10 shows that the quasi-isomorphism (12.37) is functorial when we vary λ ∈ Λ j,n and µ ∈ Λ n−j,m−n . Together with the fact that L λ K µ is quasi-isomorphic to L λ K µ in FS cyl,n (π E ) (so the former can be regarded as an object in A j ), we have a commutative diagram where the horizontal isomorphisms are the ones in (11.5), (11.7), which in particular send L λ K µ ∈ A j to L ! λ K ! µ ∈ A ! j . This implies that ψ j sits inside the commutative diagram Proof of Theorem 1.9. By Proposition 11.16, we have a spectral sequence to Kh(κ(β)) from ⊕ n j=0 H((ι ! j ) * (φ β (L hs ) r ) ⊗ A ! j ψ j • (ι j ) * (L l hs )) (12.41) By Proposition 12.1, Lemma 12.9 and the commutative diagram above, we have, up to grading shift, β ⊗ K j,n −K j,n ∆ K j,n ) (12.42) = HH * (K j,n , P (j) β ) (12.43) which is what we want.
Appendix A. Non-formality of standard modules/thimbles Here we explain that the endomorphism algebra of the Lefschetz thimbles in FS(π n,m ) may be non-formal when n > 1. The non-formality follows rather directly from the minimal model computed in the thesis of Klamt [23]. Since this is not required for our main results, our treatment is somewhat brief.
When (n, m) = (1, m) the algebra is known to be formal; by duality, the same holds when n = m−1.
The cohomological algebra is given by the following quiver with relations, by Proposition 10.1 and [23, Section 5.2.8, 5.2.9]. P (4|3) The subscript and superscript of the name of the arrow indicate the target and the source of the arrow, respectively. The definition of Id For the A ∞ structure, Klamt computed a minimal model with µ 3 = 0 but µ d = 0 for d > 3. The list of non-zero µ 3 is given in [23, Table 5.25] that for any a ∈ Ext(P (4|1), P (4|3)) and b ∈ Ext(P (2|1), P (4|2)), the sum F Remark A.2. Lemma A.1 shows that if one tries to apply the strategy in Section 9 to prove that the collection of thimbles is formal, then one would fail because the cohomological algebra of the thimbles imposes restrictions which preclude the existence of a consistent choice of equivariant structures.