Continuous and coherent actions on wrapped Fukaya categories
CContinuous and coherent actions on wrapped Fukaya categories
Yong-Geun Oh † and Hiro Lee Tanaka (cid:63) † Center for Geometry and Physics (IBS), Pohang, Korea & Department of Mathematics,POSTECH, Pohang Korea. (cid:63)
Department of Mathematics, Texas State UniversityOctober 6, 2020
Abstract
We establish the continuous functoriality of wrapped Fukaya categories with respect to Li-ouville automorphisms, yielding a way to probe the homotopy type of the automorphism groupof a Liouville sector. These methods prove Liouville and monotone cases of a conjecture ofTeleman from the 2014 ICM. In the case of a cotangent bundle, we show that the Abouzaidequivalence between the wrapped category and the ∞ -category of local systems intertwines ouraction with the action of diffeomorphisms of the zero section. In particular, our methods yielda typically non-trivial map from the rational homotopy groups of Liouville automorphisms tothe rational string topology algebra of the zero section. Contents b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 For both . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 A ∞ -categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Homotopy types from categories of simplices . . . . . . . . . . . . . . . . . . . . . . . 223.3 Continuation maps and Floer theory in Liouville bundles . . . . . . . . . . . . . . . 231 a r X i v : . [ m a t h . S G ] O c t The wrapped Fukaya categories 24 C ∗ P (families of local system categories) . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 From Fukaya categories to local systems (the bundled Abouzaid functor) . . . . . . . 335.3 The Abouzaid functor descends to the wrapped category . . . . . . . . . . . . . . . . 365.3.1 Toward a proof of Theorem 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 365.3.2 Proof of Theorem 5.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Tw W (cid:39) C ∗ P HF quad unchanged . . . . . . . . . . . . . . . . . . . . . . . 546.3 Comparing the non-wrapped Abouzaid map to the quadratically wrapped Abouzaidmap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.4 Proof of Theorem 6.0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 C ∗ P is compatible with the diffeomorphism action . . . . . . . . . . . . . . . . . . . 587.2 Proof of Theorem 1.0.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 Introduction
Liouville manifolds, and Liouville sectors more generally, are fruitful objects of study in symplecticgeometry. They give rise to interesting geometric questions whose solutions are amenable to flexible,and often topological , techniques. However, there are fundamental questions about Liouville sectorswhose answers are wanting in the literature.One question concerns the study of automorphisms of Liouville sectors. For compact symplecticmanifolds the automorphism groups Symp and Ham enjoy enticing formal properties—Ham issimple, for example, and its group isomorphism class is a complete invariant of a compact symplecticmanifold. Our knowledge of these groups is quite sparse in most examples. The same is trueconcerning automorphisms of Liouville sectors, though it should be mentioned that the communityhas identified non-trivial collections of automorphisms in many examples (often by manipulatingDehn twists about Lagrangian spheres), and progress has been made in studying spaces of compactlysupported symplectic automorphisms of various Liouville manifolds. See for example [Kea17, Kea14,DRE15, Eva11].A second question concerns what continuous dependence is enjoyed by wrapped Fukaya cate-gories of Liouville sectors. While it is known that stop removal and open inclusions of Liouvillesectors induce functors on their wrapped Fukaya categories [GPS17, GPS18b], it has remainedunclear to what extent wrapped categories display a dependence on families of embeddings—thatis, on isotopies.We prove here that such dependencies exist with as much coherence as one could hope for. Thisof course broadens the use of Floer-theoretic invariants in studying automorphism groups.
Theorem 1.0.1.
Let M be a Liouville sector, and let Aut o ( M ) denote the topological group ofLiouville automorphisms (Definition 2.0.3). Then Aut o ( M ) acts coherently on the wrapped Fukayacategory of M . That is, there exists an A ∞ homomorphismAut o ( M ) → Aut( W ( M ))from the space of Liouville automorphisms of M to the space of automorphisms of the wrappedFukaya category of M .The word “continuous” in the title of this paper refers to the fact that the map in Theorem 1.0.1can be realized as a map of topological spaces—i.e., is continuous. So for example, it induces amap on homotopy groups (Corollary 1.0.7). The word “coherent” refers to the fact that the map inTheorem 1.0.1 is a map of A ∞ -algebras in spaces—the coherence is exhibited in the compatibilityof the higher A ∞ -homotopies.Our methods apply equally well to the compact monotone setting (see Section 1.1.2). Thus,the present work establishes (in the Liouville and monotone settings) a widespread expectation inthe community: that well-behaved actions on M induce continuous actions on the Fukaya categoryof M . For example, this is stated as a conjecture in Teleman’s 2014 ICM talk [Tel14, Conjecture2.9]. One key departure from Teleman’s proof proposal is that we do not employ Lagrangiancorrespondences. Teleman also sought to utilize a theorem equating group actions on dg-categorieswith certain maps of E -algebras; in our construction, the map of E -algebras is exhibited after exhibiting the group action. (See Corollary 1.0.5 and Section 1.1.2.) Moreover, our proof avoidsa great deal of Floer-theoretic set-up by utilizing the categorical technique of localizations, twice.(See Section 1.2.)Before we go on, we should be explicit about what decorations we put on M —this affectswhat kind of category W ( M ) is. For example, Theorem 1.0.1 as stated is only true for W ( M )3eing two-periodically graded and linear over Z / Z . This is because a Liouville automorphism φ ∈ Aut o ( M ) does not “know” how to respect any choice of grading gr , nor of background class b ∈ H ( M ; Z / Z )—and such data would be required to form a Z -graded and Z -linear Fukayacategory. One should thus also consider the natural automorphism group consisting of Liouvilleautomorphisms equipped with data respecting these decorations (Section 2.3). Notation 1.0.2.
We will writeAut o ( M ) , Aut gr,b ( M ) , Aut( M )to respectively denote the group of Liouville automorphisms, of Liouville automorphisms with datarespecting a chosen gr and b , and of Liouville automorphisms with data respecting some unspecifieddecorative choices.Any statement about the non-superscripted Aut is true for all possible decorations, while astatement about Aut gr,b is only true for the wrapped category of a Liouville sector M equippedwith a grading and background class, and a statement about Aut o ( M ) is true for the undecoratedcase.We will write W ( M ) for the wrapped category, suppressing dependence on decorations.Then our methods also yield: Theorem 1.0.3.
For any Liouville sector M equipped with decorations, there exists an A ∞ ho-momorphism Aut( M ) → Aut( W ( M ))from the space of Liouville automorphisms of M (equipped with data respecting chosen decorations)to the space of automorphisms of the wrapped Fukaya category of M .In Theorem 1.0.3, the wrapped category on the right depends on the decorations, but wesuppress this dependence from the notation (following Notation 1.0.2). Remark 1.0.4.
Often, this decorated automorphism group is not too homotopically differentfrom Aut o ( M )—for example, when the decorations are a grading gr and a background class b ∈ H ( M ; Z / Z ), the map Aut gr,b ( M ) → Aut o ( M ) induces an isomorphism on π ≥ (Proposition 2.3.8).Now take based loop spaces and apply Dunn additivity to obtain: Corollary 1.0.5.
There exists a map of E -algebrasΩ Aut( M ) → Ω Aut( W ( M )) . Remark 1.0.6.
By generalities concerning A ∞ -categories, one can identify the endomorphismsof the identity functor of any A ∞ -category C with the (topological space associated to the) non-positive truncation of the Hochschild cochains of C . The based loop space at the identity functor id C is precisely the space of those invertible endomorphisms—i.e., natural equivalences of the identityfunctor. Put another way, the based loop space Ω Aut( W ( M )) is the space of units of the Hochschildcochain algebra. The E -algebra structure of Ω Aut( W ( M )) coincides with the E -algebra structureinherited from the Hochschild cochains of W ( M ). Given any cochain complex C , the non-positive truncation τ ≤ C is the cochain complex whose i th group is givenby C i if i <
0, by 0 if i >
0, and by ker d if i = 0. The space associated to a non-positive cochain complex isconstructed via the Dold-Kan correspondence.
4y taking homotopy groups of domain and target, we have the following:
Corollary 1.0.7.
The above maps induce group homomorphisms π k +1 Aut( M ) → (cid:40) HH ( W ( M )) × k = 0 HH − k ( W ( M )) k ≥ . That is, the homotopy groups of Aut( M ) map to the Hochschild cohomology groups of the wrappedFukaya category. Remark 1.0.8.
Here, HH ( W ( M )) × indicates the units of the multiplication of degree 0 Hochschildcohomology elements, and the homomorphism is with respect to this multiplication in degree 0.For k ≥
1, the group structure on the target is additive. That the homotopy groups of Aut( W ( M ))may be identified with certain Hochschild cohomology groups is an A ∞ -categorical version of aresult of T¨oen [To¨e07]; see Theorem 3.3.1 of [OT20a]. Remark 1.0.9.
To put the above corollary in context, recall that the Hochschild cohomology ofthe Fukaya category of M is isomorphic to the quantum cohomology ring of M in various contexts:when M is monotone [She16], and when M is toric [AFO + ], for example. So one can view theabove results as an analogue of the Seidel homomorphism π Ham( M ) → QH ( M ) × [Sei97], butgeneralized to higher homotopy groups of the appropriate analogue of Ham in the Liouville setting.Indeed, one of our main geometric constructions is inspired by work of Savelyev who generalizedthe Seidel homomorphism to higher homotopy groups of Ham in the monotone setting [Sav13]. Wealso conjecture a connection with the work of Lekili and Evans [EL19]; see Conjecture 1.1.4 below.Moreover, if a Lie group G acts smoothly on M by Liouville automorphisms respecting relevantdecorations, we have an induced homomorphism G → Aut( M ), hence an E -algebra map Ω G → Ω Aut( W ( M )). By noting that this map factors through the Hurewicz map for Ω G by adjunction(Notation 7.1.9), we conclude that the map C ∗ Ω G → Hoch ∗ ( W ( M )) to the Hochschild cochaincomplex is a map of E -algebras in chain complexes. Thus Corollary 1.0.10.
Let G act smoothly on M by Liouville automorphisms respecting relevantstructures. Then the action induces a map of Gerstenhaber algebras H ∗ (Ω G ) → HH ∗ ( W ( M )) . We suspect that this E -map is precisely the sought-after E -map in Teleman’s ICM address; seefor example Theorem 2.5 and the surrounding discussion in [Tel14], and see also Conjecture 1.1.4below.The latter half of this work applies the above results to the special case when M = T ∗ Q is acotangent bundle. We fix an orientation on Q and we choose b ∈ H ( M ; Z / Z ) to be pulled backfrom the second Stiefel-Whitney class of Q . We also fix the canonical grading on T ∗ Q .Then a result of Abouzaid [Abo12] states that the resulting wrapped Fukaya category W ( T ∗ Q )is equivalent to the A ∞ -category Loc ( Q ) of local systems on the zero section Q . By abstractnon-sense, Loc ( Q ) is equivalent to Mod ( C ∗ (Ω Q )), modules over the A ∞ -algebra of chains on thebased loop space. We thus have a quasi-isomorphism of chain complexes from the Hochschild chaincomplex to chains on the free loop space: C Hoch ∗ ( W ( T ∗ Q )) (cid:39) C Hoch ∗ ( Mod ( C ∗ Ω Q )) (cid:39) C ∗ ( L Q ) . Q results in anisomorphism of graded commutative algebras HH ∗ ( C ∗ Ω Q ) (cid:39) H ∗ +dim Q ( C ∗ L Q )from the Hochschild cohomology groups of the A ∞ -algebra C ∗ Ω Q to the shifted homology of thefree loop space. We thus have the following: Corollary 1.0.11.
Let Q be compact and orientable, and choose an orientation of Q . The abovemaps induce group homomorphisms π k +1 Aut gr,b ( T ∗ Q ) → (cid:40) H dim Q ( L Q ) × k = 0 H k +dim Q ( L Q ) k ≥ gr,b ( T ∗ Q ) map to the homology of the free loop space of thezero section.(As before, when k = 0, the group structure on the target is multiplicative with respect tothe string multiplication on C ∗ L Q , and when k ≥
1, the group structure is the additive one onhomology.)Note that because Aut gr,b → Aut o induces an isomorphism on homotopy groups in degrees ≥ o tothe homology of the free loop space of Q .To state our final results, we note that there is a natural action of the diffeomorphism groupDiff( Q ) on Loc ( Q ). On the other hand, any diffeomorphism induces a Liouville automorphism of T ∗ Q . This induced map can be lifted to naturally respect the choices of b and gr , so we have amap Diff( Q ) → Aut gr,b ( T ∗ Q ). Theorem 1.0.12.
The diagram Diff( Q ) (cid:47) (cid:47) (cid:15) (cid:15) Aut gr,b ( T ∗ Q ) Thm 1 . . (cid:15) (cid:15) Aut(
Loc ( Q )) ∼ (cid:47) (cid:47) Aut( W ( T ∗ Q ))commutes up to homotopy. Here, the bottom arrow is induced by the Abouzaid equivalence [Abo12]and the left vertical arrow is induced by the action of Diff( Q ) on Loc ( Q ).Informally, Theorem 1.0.12 states that the Abouzaid equivalence W ( T ∗ Q ) (cid:39) Loc ( Q ) can bemade equivariant with respect to the natural Diff( Q ) action on both categories, and that thisaction factors through our construction from Theorem 1.0.3. Corollary 1.0.13.
Let [ α ] ∈ π k +1 Diff( Q ) and suppose the image of [ α ] is non-zero under the map π k +1 Diff( Q ) → HH − k ( C ∗ Ω Q ) from (1.1). Then the image of [ α ] is non-zero in π k +1 Aut gr,b ( T ∗ Q ) . In particular, such [ α ] detect non-trivial elements in the homotopy groups of Aut gr,b ( T ∗ Q ). See for example Malm [Mal11]; this is in fact an isomorphism of BV algebras with the string topology operationson the right-hand side, and the natural BV structure on Hochschild cohomology on the left-hand side. Q )denote the space of those continuous maps f : Q → Q that happen to be homotopy equivalences.By taking based loops, and noting that the action of Diff on Loc factors through hAut, we havethe following:
Corollary 1.0.14.
The following diagram commutes: π k +1 Diff( Q ) (cid:47) (cid:47) (cid:15) (cid:15) π k +1 Aut gr,b ( T ∗ Q ) (cid:15) (cid:15) π k +1 hAut( Q ) (cid:47) (cid:47) H dim Q + k ( L Q ) . Example 1.0.15.
Let Q = S and take R = Z . We must contemplate the composite Z ∼ = π Ω Diff( S ) (cid:39) π Ω hAut( S ) → ( HH ( C ∗ Ω S )) × ∼ = Z × Z / Z . To understand the last map, we note the equivalence C ∗ Ω S (cid:39) Z [ x ± ] as an A ∞ algebra. The 0thHochschild cohomology is (the center of) this ring, and its units are the monomials of the form ± x i for i ∈ Z ; the homomorphism Z ∼ = π hAut( S ) → ( HH ) × is given by i (cid:55)→ x i .In particular, Theorem 1.0.12 shows that π Diff( S ) → π Aut gr,b ( T ∗ S ) is an injection. Whileless is known about the space of diffeomorphisms of tori T n , we regardless produce non-trivialelements in π Aut gr,b ( T ∗ T n ) straightforwardly using the same method.As pointed out to us by Sylvan, because both the inclusion Q → T ∗ Q and the projection T ∗ Q → Q are homotopy equivalences, we have a factorizationDiff( Q ) → Aut o ( T ∗ Q ) → hAut( Q ) . (1.2)(It does, however, take a little bit of effort to render the second map in (1.2) a map of of A ∞ spaces.) It is thus natural to try to detect homotopy groups of Aut o ( T ∗ Q ) by identifying elementsthat survive the map Diff( Q ) → hAut( Q ). For example, a result of Felix-Thomas [FT04] statesthat when Q is simply-connected, the map π k +1 hAut( Q ) ⊗ Q → H dim Q + k ( L Q ; Q )is an injection. So we can immediately deduce non-trivial elements in the rational homotopy groupsof Aut o ( T ∗ Q ) if we can detect non-trivial elements of π • (Diff( Q )) ⊗ Q that survive in the stringalgebra. Indeed, this gives a shorter proof of Corollary 1.0.13. When Q is compact and oriented, the map Ω hAut( Q ) → C ∗ ( L Q )[ − n ] (with homological shiftingconventions) has another description. At the level of homotopy/homology groups, it is equal to thecomposition π ∗ Ω hAut( Q ) → H ∗ Ω hAut( Q ) [ Q ] −−→ H ∗ Ω hAut( Q ) ⊗ H n ( Q ) ev −→ H ∗ + n ( L Q ) . Q ) × Q → L Q . The verification of thispresentation is independent of any Floer theory. (See also [FT04].)In other words, outside of π , the information our actions obtains about Aut factors throughthe Hurewicz map of hAut, and in fact of Aut as well. This is not satisfying, and at the very least,one would hope for a construction that factors through the stable homotopy type (which sees moreabout the topology of a space than its homology). This is another good reason to seek situationswhen we can define wrapped Fukaya categories over the sphere spectrum. Let us change settings for a bit and consider a compact monotone X . The methods of the presentwork carry over straightforwardly to this setting—in fact, the cumbersome need to verify someof our C estimates vanishes. We would like to emphasize that the essential geometry of such aconstruction was established in the work of Savelyev [Sav13] and is not due to us. The methodsthere result in a functor from Simp ( B Ham( X )) to C at A ∞ (see Section 1.2), and by realizing thehomotopy type of B Ham( X ) as a localization of Simp ( B Ham( X )), we have the following analogueof Theorem 1.0.1: Theorem 1.1.1.
One has a map of A ∞ algebrasHam( X ) → Aut(
Fukaya ( X )) . Then any Lie group G with a Hamiltonian action on X enjoys a continuous homomorphism G → Ham( X ). Composing with the theorem above, we have: Corollary 1.1.2.
A Hamiltonian action of a Lie group G on a monotone X induces a map of A ∞ -algebras G → Aut(
Fukaya ( X )).The above proves [Tel14, Conjecture 2.9] in the monotone setting. The next result establishes(in the monotone setting) a conjectural E -algebra map sought in Theorem 2.5 of loc. cit., thoughour methods do not utilize Lagrangian correspondences. Corollary 1.1.3.
Any Hamiltonian action of G on a monotone X induces a map C ∗ (Ω G ) → Hoch ∗ ( Fukaya ( X )) . (1.3)of E -algebras. Proof.
By taking based loop spaces, we have a map of E -algebrasΩ G → Ω Aut(
Fukaya ( X )) . By passing to chain complexes, we obtain an E -algebra map. Let us explain this a little further,as it is formal based on ∞ -categorical arguments: The passage from spaces to spectra (otherwiseknown as Σ ∞ ) is lax symmetric monoidal, and so is the passage from spectra to chain complexes(otherwise known as tensoring/smashing with H Z ).Of course, it is unsatisfactory not to compare the methods of the present work to the methodsproposed in other works. To this end, recall that Lekili and Evans [EL19] construct a maphom W ( T ∗ G ) ( T ∗ e G, T ∗ e G ) → hom Fukaya ( X − × X ) (∆ X , ∆ X ) . (1.4)8he construction is as follows: One notes that the graph of the Hamiltonian action Γ ⊂ G × X → X quantizes to a Lagrangian C ⊂ T ∗ G × X − × X , essentially by lifting ( g, x, g ( x )) to the value of themoment map at g ( x ). A famous paradigm expects that Lagrangian correspondences give rise tofunctors and bimodules; Lekili and Evans realize this paradigm in this example by constructing afunctor from (the full subcategory consisting of the cotangent fiber of) the wrapped Fukaya categoryof T ∗ G to the Fukaya category of X − × X . It is trivial to verify that C geometrically composeswith the cotangent fiber T ∗ e G at the identity to yield the diagonal ∆ X ⊂ X − × X ; the map (1.4)is the induced map of A ∞ -algebras encoding this functor.Importantly, taking cohomology on both sides of (1.4), we obtain a map H ∗ (Ω G ) → HH ∗ ( Fukaya ( X ))from the homology of the based loop space of G to the Hochschild cohomology of the Fukayacategory of X . We have seen a map like this before! We conjecture the following:
Conjecture 1.1.4. As A ∞ algebra maps, the map (1.3) is homotopic to the map (1.4).That is, a map constructed from the geometry of correspondences and quilts in [EL19] is equiv-alent to a map constructed using the geometry of bundles and the algebra of localizations here.We warn the reader that the codomains of the two maps must be identified; part of the conjectureis that the equivalence is intertwined by a natural chain-level, A ∞ -algebra equivalence betweenHochschild cochains and endomorphisms of the diagonal.In particular, a proof of Conjecture 1.1.4 would prove (by Corollary 1.1.3) that the correspondences-style construction (1.4) can be lifted to an E -algebra map. We enumerate natural avenues of pursuit through a sequence of remarks:
Remark 1.1.5 (Other notions of automorphisms) . While we have taken Aut( M ) and Aut o ( M )to be natural notions of automorphism group of a Liouville sector M , it is not clear that these arethe only such natural choice (even up to homotopy equivalence). In another direction, one couldcontemplate the automorphism group of the skeleton of a Liouville manifold; the correct notionof automorphism presumably depends both on the stratified homotopy type of the skeleton, andon some tubular or infinitesimal differential-geometric data attached to the skeleton (to be able torecover the equivalence type of its wrapped Fukaya category, for example). Remark 1.1.6 (Filtered enhancements) . When M = T ∗ Q , all our computations “factor” throughthe homotopy type of M (cid:39) Q ; this is unsurprising given that the wrapped Fukaya category of T ∗ Q depends only on the homotopy type of Q and our techniques incurably pass through wrapped Floerconstructions. However, we suspect that by filtering automorphism groups in a way compatiblewith the action filtration on Floer complexes, one can glean far richer symplectic data. Remark 1.1.7 (Is symplectic geometry helping differential topology, or vice versa?) . Even forthe case of M = T ∗ Q with Q compact and oriented, we note that the factorization π ∗ Diff( Q ) → π ∗ hAut( Q ) → H ∗ +dim Q ( L Q ) from Corollary 1.0.14 is difficult to study using tools of homotopytheory. It is not yet clear in which direction information will naturally flow in the future: Whether Here, we are identifying the endomorphisms of the diagonal with Hochschild cohomology of the Fukaya categoryfor monotone symplectic manifolds—this equivalence usually passes through quantum cohomology of X . See alsoRemark 1.0.9. Remark 1.1.8.
At present, the relation between Diff and hAut is best understood rationally, andthe best-understood examples are the even-dimensional “genus g ” manifolds ( S n × S n ) g obtainedby taking the connect sum of S n × S n g times. Even better understood are the manifolds obtainedby removing a small open disk from ( S n × S n ) g . The obvious low-hanging fruit is to detect rationalhomotopy groups of Aut of the cotangent bundles of these manifolds (with boundary). Remark 1.1.9.
Let us state two further reasons we take an interest in the continuous functorialityof Fukaya categories. First, and independently of any symplectic considerations, many useful al-gebraic invariants now have concrete geometric interpretations. The study of symplectic geometryhas benefitted from the congruence of the geometry of symplectic phenomena with the geometryof algebraic phenomena. For example, the circle action on framed E -algebras, known as the BVoperator in characteristic 0, is visible in the Fukaya categories having Calabi-Yau like propertiessimply by geometric rotation of Reeb orbits. Another example is the ability to encode the para-cyclic structure of the s-dot construction of Fukaya categories using configurations of points on theboundary of a disk [Tan19]. Our Corollary 1.0.5 follows this storyline.Second, the moduli space of embeddings M → M (cid:48) also encodes a part of the moduli spaceof Lagrangian correspondences, or more generally of bimodules between Fukaya categories. Viamirror symmetry, understanding this moduli space on the A model is expected to yield insightsabout the moduli space of bimodules between derived categories of sheaves on the B model. We make heavy use of ∞ -categorical machinery. We refer the reader to [OT20a] for backgroundand Section 3.1 for the results we utilize. Choice 1.2.1.
We fix a base ring R . We let C at A ∞ denote the ∞ -category of A ∞ -categories over R . (See [OT20a]). Informally, its objects are R -linear A ∞ -categories, and two objects A and B enjoy a space of morphisms between them—combinatorially, this space can be described as having vertices given by functors, edges given byhomotopies of functors, and higher simplices given by homotopies between homotopies. In C at A ∞ ,all A ∞ equivalences are also invertible up to homotopy.By choosing appropriate structures on M and appropriate brane structures on our Lagrangians,we assume W ( M ) is R -linear. (For example, with the usual relative Pin structures on our branes,one can take R to be Z . When gradings are chosen, we can take C at A ∞ to consist of usual A ∞ -categories, while when we have no gradings, we must demand that our A ∞ -categories and functorsare 2-periodic.)We let Aut( W ( M )) denote the space of R -linear automorphisms of W ( M ).Given a group G , one can construct a category B G as follows: B G has a unique object, andthe endomorphisms of this object are defined to be G . If G has a topology, one can demand thatthe category “remembers” the topology on this space of endomorphisms—or, at least, remember10he homotopy type of G . We call the resulting construction the classifying ∞ -category of G . Onecan perform this construction even more generally when G is not a group, but is an A ∞ -algebra inspaces.We are interested in the case G = Aut( M ), the space of Liouville automorphisms of M re-specting the choices made on M in Choice 1.2.1. (See Section 2.3 for examples.) We let B Aut( M )denote the classifying ∞ -category of Aut( M ).The following is a precise formulation of Theorem 1.0.1 and Theorem 1.0.3. Theorem 1.2.2.
There exists a functor B Aut( M ) → C at A ∞ (1.5)sending a distinguished base point of the domain to the wrapped Fukaya category W ( M ).A functor induces a map of morphisms spaces, and in particular, of endomorphism spaces.Because every morphism in the domain is invertible, the functor sends morphisms in the domainto equivalences in the codomain. Thus, we conclude: Corollary 1.2.3 (Precise form of Theorem 1.0.1 and Theorem 1.0.3.) . One has a mapAut( M ) → Aut( W ( M ))of group-like A ∞ -spaces. Remark 1.2.4. “Group-like A ∞ -spaces” may seem a mouthful. The intuition is that the mapAut( M ) → Aut( W ( M )) is morally a continuous group homomorphism.To construct the functor (1.5) we use an algebraic version of the following tautology: Thebarycentric subdivision of a space is homotopy equivalent to the original space. This saves us anenormous amount of analytic legwork.To explain this, let us denote by B Aut( M ) the classifying space. Let
Simp ( B Aut( M )) denotethe following category (in the classical sense): An object is a continuous map j : | ∆ n | → B Aut( M )from an n -simplex to the classifying space. A morphism from j to j (cid:48) is the data of an injectiveposet map [ n ] → [ n (cid:48) ] such that the induced composition | ∆ n | → | ∆ n (cid:48) | j (cid:48) −→ B Aut( M ) is equal to j .To see why we call this a model for a barycentric subdivision, we encourage the reader to fix some j and enumerate all the objects admitting a map to j . We note that the only invertible morphismsare the identity morphisms.Then the algebraic version of the above tautology is as follows: If we localize Simp ( B Aut( M ))—i.e., if we invert all morphisms of Simp ( B Aut( M ))—one obtains B Aut( M ). (See Section 3.2.) Thus,by the universal property of localizations, to construct a functor B Aut( M ) → C at A ∞ as in (1.5),we need only construct a functor W : Simp ( B Aut( M )) → C at A ∞ (1.6)for which every morphism of the domain is sent to an equivalence of A ∞ -categories. Warning 1.2.5.
Our geometric constructions require the composition | ∆ n | → B Aut( M ) → B Aut o ( M ) to be smooth , not just continuous; we ignore this for now. See Section 3.2 for more. Note that font distinguishes the ∞ -category B Aut( M ) from the space B Aut( M ). W j . There is a tautological M -bundle over B Aut( M ). Givena smooth map j : | ∆ n | → B Aut( M ), we obtain a smooth M -bundle over | ∆ n | by pulling backalong j . We define an A ∞ -category denoted by O j whose objects are branes living in the fibers above the vertices of the n -simplex | ∆ n | , and mor-phisms are defined by choosing parallel transports over the edges of the n -simplex and intersectingbranes. The A ∞ -operations are defined by counting certain holomorphic maps into the M -bundlecompatible with connections. We refer readers to [OT20b], Section 3.3, and Section 4.1 for details.Then, given a smooth map j : | ∆ n | → B Aut( M ), W j is defined by localizing O j along non-negative continuation maps. This localization idea adaptsAbouzaid-Seidel’s unpublished work (as utilized in [GPS17]) to the bundle setting. There are somesubtleties to point out: • We are forced to utilize continuation maps defined by counting holomorphic disks with one boundary puncture (not strips). • That one can define a “family” wrapped Fukaya category over | ∆ n | for n (cid:54) = 0 relies onSavelyev’s observation [Sav13] that one can operadically map the tautological family of holo-morphic disks to | ∆ n | . This also specifies which holomorphic maps we are counting to definethe A ∞ operations. • As usual, one needs to choose auxiliary data (connections and almost-complex structures) tocount holomorphic curves. Our definition of O j is inductive on the dimension of the domain | ∆ n | of j , and the inductive step requires us to extend data defined on ∂ | ∆ n | to the interiorof | ∆ n | . We use that j maps to B Aut( M ), and not B Aut o ( M ), so that j encodes alreadythe homotopical information needed to extend auxiliary data to the interior. • To obtain the functor (1.6), one must then verify that inclusions of simplices define A ∞ -functors O j → O j (cid:48) , and that the induced maps W j → W j (cid:48) are equivalences. This requires away to compute morphisms in the wrapped categories, and we do so using methods analogousto [GPS17]: A “sequential colimit” construction of wrapped Floer cochains recovers themorphisms in W j (Lemma 4.4.2).This concludes the summary of the construction of the functor W in (1.6), and hence of thefunctor in Theorem 1.2.2. Remark 1.2.6.
Note that the construction involves two distinct localizations—one to pass from afamily of non-wrapped “directed” Fukaya categories O to a family of wrapped Fukaya categories W ,and the other to pass from a category Simp ( B Aut( M )) to the classifying ∞ -category B (Aut( M )).Both localizations are used to avoid difficult or tedious analytical constructions.Now we explain the proof of Theorem 1.0.12 which, in the example of M = T ∗ Q , verifies thatthe above construction yields non-trivial actions.Fix a smooth manifold Q . B Diff( Q ) has a tautological Q -bundle over it, and we can pull the Q -bundle back along any simplex j : | ∆ n | → B Diff( Q ). By assigning to each j the A ∞ -category12f local systems on the pulled back bundle, we obtain a functor Tw C ∗ P : Simp ( B Diff( Q )) → C at A ∞ . On the other hand, there is a natural inclusion Diff( Q ) → Aut gr,b ( T ∗ Q ), and this induces a map oftheir classifying spaces (and hence of their categories of simplices). Call this induced map D , andconsider the composite Simp ( B Diff( Q )) D −→ Simp ( B Aut gr,b ( T ∗ Q )) O −→ C at A ∞ . We thus have two functors O ◦ D and Tw C ∗ P from Simp ( B Diff( Q )) to C at A ∞ . In Section 5.2, weconstruct a natural transformation from the latter to the former. This natural transformation is anon-wrapped, family-friendly version of the construction Abouzaid utilized in [Abo12].The next step is to show that the natural transformation O ◦ D → Tw C ∗ P factors through thelocalization W ◦ D . This requires us to prove the following geometric theorem, which we state ingreater generality (beyond the case of the zero section of the cotangent bundle): Theorem 1.2.7 (Theorem 5.3.1) . Let M be a Liouville sector, and c : L → L a continuationelement associated to a non-negative isotopy. We also fix a compact test brane X ⊂ M . Then themap of twisted complexes c ∗ : ( L ∩ X, D ) → ( L ∩ X, D )—induced by the Abouzaid functor from O ( M ) to Tw C ∗ P ( X )—is an equivalence.By the universal property of localization, we conclude that the natural transformation O ◦ D → Tw C ∗ P induces a natural transformation W ◦ D → Tw C ∗ P . (1.7)The remaining key step is to prove that this natural transformation is in fact a natural equivalence.This is a family version of the Abouzaid equivalence, and we prove it in Theorem 6.0.1. Weutilize Abouzaid’s original result in our proof, and in particular, we must relate the quadraticdefinition of the wrapped complex to the localization definition. This is accomplished througha combination of Proposition 6.1.9 and a standard result in computing morphism complexes oflocalizations (Recollection 3.1.1(2)).As a consequence we have a diagram of ∞ -categories B Diff( Q ) (cid:47) (cid:47) Tw C ∗ P (cid:38) (cid:38) B Aut gr,b ( T ∗ Q ) W (cid:119) (cid:119) C at A ∞ . The homotopy commutativity of the diagram is exhibited by the natural equivalence (1.7). Notethat the left-hand map sends a distinguished object of B Diff( Q ) to Tw C ∗ P .The final step in proving Theorem 1.0.12 is taken in Proposition 7.1.13. There, we show thatthis left-hand functor is naturally equivalent to the functor exhibiting the natural Diff( Q ) action on Loc ( Q ) (cid:39) Tw C ∗ P . Then, because the right-hand functor sends a vertex to W ( Q ), Theorem 1.0.12follows by taking based loops of each ∞ -category in the above commutative triangle. (Equivalently,by recording the effect that the functors have on endomorphism spaces.) The notation Tw C ∗ P is an artifact of a particular presentation of the A ∞ -category of local systems we utilize inSection 5.1. This presentation turns out to play well with the construction from [Abo12], so we will lug around thenotation for this pay-off. emark 1.2.8. A consequence of Theorem 6.0.1 is that the localization-style definition of thewrapped Fukaya category of a cotangent bundle is equivalent to Abouzaid’s quadratic definitionin [Abo12], but this equivalence is not proven by writing an explicit, analytically defined functorfrom one to the other. In the present work, it is rather obtained by comparing both A ∞ -categoriesto the A ∞ -category of local systems, and even relies on the (independent) generation results ofAbouzaid and of Ganatra-Pardon-Shende [GPS18b]. Our proof that W (cid:39) Tw C ∗ P differs from thatof [GPS18a] in that we compute the equivalence of endomorphism algebras in a way that passesthrough and relies on Abouzaid’s construction, while [GPS18a] does not.Of course, in principle one need not pass through the local system category; for more generalLiouville sectors, it is possible to write down a comparison functor from the cofinally wrappedcategory to the quadratically wrapped category directly, but we do not do this here. Here we make explicit our Fukaya-categorical conventions.
Remark 1.3.1.
We follow the conventions of almost all the literature concerning wrapped cat-egories, with one notable exception. The work [GPS17] reads boundary branes clockwise withrespect to the boundary of a disk, while we use the more standard counterclockwise reading (Con-vention 1.3.4(3)).This does result in some (very minor) differences in the algebra, which we enumerate here forthe reader’s convenience:1. In our work, a key filtered colimit will be of the form CF ∗ ( X, Y ) → CF ∗ ( X, Y ) → . . . ,induced by a sequence of morphisms Y → Y → . . . (see Lemma 4.4.2). In [GPS17], however,the analogous colimit is of the form CF ∗ ( X , Y ) → CF ∗ ( X , Y ) → . . . , induced by maps . . . → X → X .2. When we define our directed A ∞ -category O , we will define hom O j (( i, L, w ) , ( i (cid:48) , L (cid:48) , w (cid:48) )) to bezero if w > w (cid:48) ; this is opposite the convention utilized in [GPS17]. Convention 1.3.2 (Strip coordinates and positive/negative punctures) . We will often denote anelement of the infinite strip R × [0 ,
1] by ( τ, t ). Every boundary-punctured holomorphic disk S willbe equipped with strip-like ends—i.e., holomorphic embeddings (cid:15) : [0 , ∞ ) × [0 , → S, (cid:15) : ( −∞ , × [0 , → S, (called positive and negative, respectively) that converge as τ → ±∞ to the boundary puncturesof S . We will call these boundary punctures positive and negative accordingly.In all our applications, there will exactly one negative boundary puncture of S , while all otherboundary punctures will be positive. Convention 1.3.4 (Morphisms and µ k ) . The operations µ , µ , . . . of a Fukaya category will bedefined as usual by counting (pseudo)holomorphic maps u : S → M satisfying certain boundary conditions. 14 lob:https://vectr.com/a0b0ee2c-4584-504a-a8a7-a8a...1 of 1 10/18/19, 10:53 PM −∞ + ∞ + ∞ L L L k − L k . . . Figure 1.3.3.
The labeling of a holomorphic disk contributing to a µ k operation as in (1.8).1. (Generating chords.) The generators of the Floer cochain complex CF ∗ ( L , L ) are in bijec-tion with certain chords [0 , → M from L to L . In particular, in strip-like end coordinates,the limiting chords lim τ →±∞ u ◦ (cid:15) ( τ, − )will be read as a morphism from the brane labeled at time t = 0 ∈ [0 ,
1] to the brane labeledat time t = 1 ∈ [0 , constant in our applications, but it will be healthy to conjurethis convention as a rule of thumb.2. (Inputs and outputs.) The negative puncture defines an output, while the positive puncturesare inputs. More precisely, a strip-like end of a negative puncture will be decorated byboundary conditions giving rise to an output of the µ k operations, and the positive strip-likeends will be decorated by boundary conditions encoding the inputs of the µ k operations.3. (Ordering the boundary arcs.) The branes decorating the boundary arcs of S will be readin an order given by the boundary orientation of ∂S induced from the standard holomorphicorientation of S —in particular, when we draw a picture of S , we read the brane labelscounterclockwise.4. (Orienting the boundary arcs.) Likewise, when we equip a boundary arc of S with a movingboundary condition, the “positivity” of the moving boundary isotopy of a brane will be withrespect to the boundary orientation of the arc. (See for example Definition 2.2.4.)5. (Composition order.) Counting holomorphic disks with boundary conditions L , . . . , L k (readcounter-clockwise from the negative puncture) gives rise to the operation µ k : CF ( L k − , L k ) ⊗ . . . ⊗ CF ( L , L ) → CF ( L , L k ) . (1.8)For a summary of these conventions, see Figure 1.3.3. Example 1.3.5.
Fix a brane L ⊂ M and fix an isotopy from L to L (cid:48) . If the isotopy is non-negative(Definition 2.2.4), one obtains an element of CF ( L, L (cid:48) ) by counting holomorphic disks with a singleboundary puncture, and with moving boundary condition dictated by the isotopy; this is detailedin [OT20b]. See also Figure 1.3.6. 15 lob:https://vectr.com/a0b0ee2c-4584-504a-a8a7-a8a...1 of 1 10/18/19, 10:53 PM −∞ LL (cid:48) L Figure 1.3.6.
A holomorphic disk with one boundary puncture and with a moving boundarycondition given by a non-negative isotopy L . The count of such disks gives rise to an element of CF ( L, L (cid:48) ). Example 1.3.7.
Fix a brane L ⊂ M and fix an isotopy L from L to L (cid:48) . Fix also a brane K ⊂ M .If the isotopy is non-negative (Definition 2.2.4), one obtains a chain map CF ( K, L ) → CF ( K, L (cid:48) )by counting holomorphic strips with moving boundary condition at t = 1. See Figure 1.3.8; we alsorefer to [OT20b] for more details. blob:https://vectr.com/a0b0ee2c-4584-504a-a8a7-a8a...1 of 1 10/18/19, 10:53 PM −∞ + ∞ K L LL (cid:48) Figure 1.3.8.
A holomorphic strip with moving boundary condition L at t = 1 and fixed boundarycondition K at t = 0, counts of which define a continuation map CF ( K, L ) → CF ( K, L (cid:48) ). Notethat the moving boundary condition places L near τ = ∞ , and places L (cid:48) near τ = −∞ . (Inparticular, the isotopy evolves in the − ∂/∂τ direction.) We would like to thank Gabriel Drummond-Cole, Rune Haugseng, Sheel Ganatra, Sander Kupers,John Pardon, Yasha Savelyev, and Zack Sylvan for helpful conversations.The first author is supported by the IBS project IBS-R003-D1. The second author was sup-ported by IBS-CGP in Pohang, Korea and the Isaac Newton Institute in Cambridge, England,during the preparation of this work. This material is also based upon work supported by the Na-tional Science Foundation under Grant No. DMS-1440140 while the second author was in residenceat the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019semester. 16
Geometric background
We assume the reader is familiar with the definition of Liouville sector. (Background is givenin [GPS17] and in [OT20b].) Informally, a Liouville sector is an exact symplectic manifold M with boundary, with two important features: (i) Outside a compact subset of M , the Liouvilleflow induces an isomorphism of M with the symplectization of a compact, contact manifold withboundary, and (ii) The boundary ∂M satisfies a “barrier condition” guaranteeing that holomorphiccurves with boundary bounded away from ∂M will have interiors bounded away from ∂M . Notation 2.0.1.
1. We let Z denote the Liouville vector field on M .2. We let θ denote the Liouville form on M .3. We let ∂ ∞ M denote the contact boundary of M . This is a contact manifold with boundary,and is well-defined up to co-oriented contact diffeomorphism.4. We will often choose a proper exact embedding ι : ∂ ∞ M × R ≥ → M from the symplectizationof the contact boundary. We will let r denote the R ≥ coordinate, and set r = e s , i.e., s = log r . Definition 2.0.2 (Liouville automorphisms) . Let M i , i = 0 ,
1, be Liouville sectors. A
Liouvilleisomorphism from M to M is a diffeomorphism φ : M → M satisfying φ ∗ θ = θ + df for some compactly supported smooth function f : M → R . If M = M , we call φ a Liouville automorphism . Definition 2.0.3.
Let M be a Liouville sector. We letAut o ( M )denote the topological group of Liouville automorphisms of M .We endow Aut o ( M ) with the smallest topology satisfying the following properties:1. The topology contains the weak Whitney topology inherited from C ∞ ( M, M ), and2. The map Aut o ( M ) → Aut( ∂ ∞ M ) to the space of smooth contact diffeomorphisms of ∂ ∞ M is continuous.We also endow Aut o ( M ) with a diffeology, so that we may speak of smooth maps into Aut o ( M ).We impose the smallest diffeology satisfying the following:1. It contains the subspace diffeology inherited from the diffeology of C ∞ ( M, M ),2. The map Aut o ( M ) → Aut( ∂ ∞ M ) to the space of smooth contact diffeomorphisms of ∂ ∞ M is smooth.We note that because ∂ ∞ M is compact, we can unambiguously endow the contact automorphismspace Aut( ∂ ∞ M ) with the usual group diffeology inherited from the diffeomorphism group of ∂ ∞ M . Remark 2.0.4.
Having endowed Aut o ( M ) with a diffeological group structure, the Milnor classi-fying space B Aut o ( M ) is naturally endowed with a diffeological space structure, by a constructionof Christensen-Wu [CW17]. Remark 2.0.5.
The details of the above topology and diffeology will not play an explicit rolein this paper; all that will matter is that the results of Section 3.2 are true thanks to the abovetopology and diffeology. 17 .1 Liouville bundles
Definition 2.1.1 (Liouville bundle) . Fix a Liouville sector M . A Liouville bundle with fiber M is the choice of a smooth M -bundle p : E → B , together with a smooth reduction of the structuregroup from Diff( M ) to Aut o ( M ).Our main examples will be the universal Liouville bundle E Aut o ( M ) → B Aut o ( M ) (wherethe smoothness of reduction of structure group is understood in the diffeological sense) and thoseLiouville bundles pulled back along smooth maps j : | ∆ ne | → B Aut o ( M ). Let M be a Liouville sector. Definition 2.2.1.
A subset A ⊂ M is called conical near infinity if for some (and hence all) θ ∈ [ θ ] Liou , and for some compact subset K , the complement A \ K is closed under the positiveLiouville flow.There are standard decorations one should put on Liouville sectors and their Lagrangians toobtain a Z -graded, Z -linear Fukaya category—for example, gradings and Pin structures. We assumethese structures to be chosen throughout. To that end: Definition 2.2.2.
Let M be a Liouville sector. A brane is a conical-near-infinity Lagrangian L ⊂ M equipped with the relevant decorations. Example 2.2.3.
So for example, if L is a compact Lagrangian, then L (when equipped with theappropriate decorations) is a compact brane. Note also that our branes have no boundary—bydefinition of Lagrangian (submanifold), L is locally diffeomorphic to Euclidean space, and henceboundaryless. Definition 2.2.4 (Non-negative isotopy) . Now fix an exact Lagrangian isotopy j : L × [0 , t → M through conical-near-infinity Lagrangians. (In particular, this induces an isotopy of Legendrians in-side ∂ ∞ M .) We say this is a non-negative wrapping , or a non-negative isotopy (of the Lagrangians)if for some (and hence any) choice of Liouville form θ on M , we have the following outside a compactsubset of L : θ ( Dj ( ∂ t )) ≥ . Put another way, the flow of L in ∂ ∞ M is non-negative with respect to the contact form inducedby θ . Definition 2.2.5 (Cofinal sequence of non-negative wrappings) . Now suppose one has chosen asequence of conical-near-infinity branes L (0) , L (1) , . . . together with a non-negative wrapping from L ( i ) to L ( i +1) for every i . We say this is a cofinalsequence of non-negative wrappings if the following holds: For any non-negative wrapping of L (0) to another conical-near-infinity brane L (cid:48) , there exists1. w ∈ Z and2. a non-negative wrapping from L (cid:48) to L ( w ) In [GPS17], this notion is called a positive wrapping (see Definition 3.20 of loc. cit.). L (0) → L (cid:48) → L ( w ) is homotopic to the composite isotopy L (0) → L (1) → . . . → L ( w ) through non-negative isotopies. Remark 2.2.6.
The non-negativity of the wrapping allows us to define so-called continuationelements (see Section 3.3); these yield in particular cohomology classes c ∈ HF ∗ ( L ( i − , L ( i ) ) . For any brane K transversal to the L ( i ) , we will hence be able to define a sequence of cohomologygroups . . . → HF ∗ ( K, L ( i − ) c ∗ −→ HF ∗ ( K, L ( i ) ) → . . . by using the (cohomology-level) µ operation. The directed limit (i.e., colimit) of this sequencewill be isomorphic to the cohomology of the morphism complexes in our family wrapped categories(Lemma 4.4.2). We now describe automorphisms groups of Liouville manifolds that “respect” particular decorationsthat are extrinsic to the data of [ θ ], focusing on the examples of gradings and background class b ∈ H ( M ; Z / Z ). The methods here carry over to other decorations we anticipate will be of usein Floer theory, especially when one must trivialize more than det ( T M ). A grading on M is the data of a trivialization det ( T M ) ∼ = C × M as a complex line bundle. Onemay equivalently encode this data in the following homotopy-coherent diagram: EU (1) (cid:39) ∗ (cid:15) (cid:15) M det ( T M ) (cid:47) (cid:47) (cid:53) (cid:53) BU (1) (cid:39) K ( Z , . Explicitly, the line bundle det ( T M ) is classified by a map to BU (1) (cid:39) K ( Z , φ : M → M ; it is also the data of a higher homotopy coherent diagram: ∗ (cid:15) (cid:15) M φ (cid:32) (cid:32) (cid:42) (cid:42) (cid:52) (cid:52) M (cid:47) (cid:47) (cid:62) (cid:62) BU (1)The space of such data is encoded as a homotopy fiber product:19 efinition 2.3.1 (Aut gr ) . We let Aut gr ( M ) denote the space of Liouville automorphisms of M respecting gradings. It is defined to be the homotopy pullback:Aut gr ( M ) (cid:47) (cid:47) (cid:15) (cid:15) hom T op / ( ∗→ BU (1)) ( M, M ) (cid:15) (cid:15) Aut o ( M ) (cid:47) (cid:47) hom T op /BU (1) ( M, M ) (2.1)
Remark 2.3.2.
Let us explain the maps in (2.1).
T op /BU (1) is the slice ∞ -category of topological spaces equipped with a map to BU (1). hom T op /BU (1) is the morphism space in this ∞ -category. Likewise, hom T op / ( ∗→ BU (1)) is the morphism space in theslice ∞ -category of spaces equipped with a map to BU (1) along with a null-homotopy of said map.aLet us also explain the bottom horizontal arrow. This is most efficiently encoded by first observinga homotopy equivalence Aut o ( M ) ∼ ←− Aut comp ( M )from the space of those Liouville automorphisms of M equipped with a homotopy between choicesof ω -compatible almost complex structures. The forgetful map is a homotopy equivalence becausethe space of almost-complex structures compatible with ω is contractible. On the other hand,Aut comp ( M ) has a natural map to hom T op /BU (1) ( M, M ) where the choice of almost-complex struc-ture J on T M defines a map from M to BU (1) (classifying det ( T M )), and the homotopy between J and φ ∗ J determines the homotopy between M → BU (1) and the composite M φ −→ M → BU (1).In summary, the bottom horizontal arrow is determined by considering the compositeAut o ( M ) ∼ ←− Aut comp → hom T op /BU (1) ( M, M )and choosing a homotopy inverse (together with a homotopy exhibiting the homotopy inverseness—this is a choice in a contractible space of choices) to the left-hand arrow.
Remark 2.3.3.
All the spaces in (2.1) are endomorphism spaces of particular ∞ -categories. Forexample, Aut o ( M ) is the endomorphism space of M , considered as an object of the topologicallyenriched category of Liouville sectors with morphisms being Liouville isomorphisms. Aut gr ( M ) isthe endomorphism space of a corresponding fiber product category; thus it is an A ∞ -space (andin fact, group-like). Because the forgetful map Aut gr ( M ) → Aut o ( M ) is now seen to arise from afunctor, it gives rise to a map of A ∞ -algebras (in the ∞ -category of spaces).We are interested in studying Aut o ( M ), so it will be useful to know how far away the homotopytype of Aut gr ( M ) is from that of Aut o ( M ).Because the forgetful map is a homomorphism (see Remark 2.3.3), it suffices to compute thefiber over the identity map φ = id M : M → M . So it suffices to compute the homotopy fiber of themap hom T op / ( ∗→ BU (1)) ( M, M ) → hom T op /BU (1) ( M, M )over the identity morphism. This homotopy fiber is straightforwardly seen to be homotopyequivalent to hom( M, Ω BU (1)) (cid:39) hom( M, U (1)) . That is, we have a fiber sequencehom(
M, U (1)) → Aut gr ( M ) → Aut o ( M ) . ≥ Proposition 2.3.4.
The forgetful map Aut gr ( M ) → Aut o ( M ) induces an isomorphism on homo-topy groups π k for k ≥
3. The induced map on π is an injection. b Likewise, fix an element b ∈ H ( M ; Z / Z ). This is classified by a map ˜ b : M → K ( Z / Z , b once and for all. Definition 2.3.5 (Aut b ( M )) . We let Aut b ( M ) denote the space of Liouville automorphisms of M equipped with data respecting b . It is defined via the following homotopy pullback square:Aut b ( M ) (cid:47) (cid:47) (cid:15) (cid:15) hom T op /K ( Z / Z , (( M, ˜ b ) , ( M, ˜ b )) (cid:15) (cid:15) Aut o ( M ) (cid:47) (cid:47) hom T op ( M, M )Informally, an element of Aut b ( M ) is the data of a Liouville automorphism φ : M → M , togetherwith a homotopy from ˜ b ◦ φ to ˜ b . The same observation as in Remark 2.3.3 shows Aut b ( M ) is agroup-like A ∞ -algebra in spaces, and Aut b ( M ) → Aut o ( M ) lifts to an A ∞ map.Moreover, the fiber of this map can be computed as the fiber of the maphom T op /K ( Z / Z , (( M, ˜ b ) , ( M, ˜ b )) → hom T op ( M, M )which is given by hom
T op ( M, Ω K ( Z / Z , (cid:39) hom T op ( M, R P ∞ ) . As before, we conclude:
Proposition 2.3.6.
The forgetful map Aut b ( M ) → Aut o ( M ) induces an isomorphism on homotopygroups π k for k ≥
3. The induced map on π is an injection. (Aut gr,b ( M )) . We define Aut gr,b ( M ) as the homotopy pullbackAut gr,b ( M ) (cid:47) (cid:47) (cid:15) (cid:15) Aut gr ( M ) (cid:15) (cid:15) Aut b ( M ) (cid:47) (cid:47) Aut o ( M ) . Proposition 2.3.8.
The forgetful map Aut gr,b ( M ) → Aut o ( M ) induces an isomorphism on homo-topy groups π k for k ≥
3. The induced map on π is an injection. Proof.
Combine Proposition 2.3.4 and 2.3.6. 21
Imported results
Our paper depends on three results from other papers, so we recall them here for the reader’sbenefit. A ∞ -categories Let A be an A ∞ -category, and let H A denote its 0th cohomology category. We fix a collectionof morphisms C in H A . The localization of A along C , denoted A [ C − ], is a new A ∞ -categoryobtained by freely adjoining inverses to elements of C . So a functor out of A [ C − ] is the samething as a functor out of A sending every element of C to an equivalence (see Recollection 3.1.1(3)below).Localizations arise in this paper because they have been shown to be a useful and computabletechnique for defining wrapped Fukaya categories—this follows idea of Abouzaid-Seidel, as utilizedin [GPS17]. Recollection 3.1.1.
We recall some facts about A ∞ -categories and their localizations. We referthe reader to [OT20a] for more precise details.1. The ∞ -category C at A ∞ (see Choice 1.2.1) can be obtained from the usual category A ∞ Cat of A ∞ -categories by formally inverting the functors that are equivalences. Moreover, functorspaces in C at A ∞ may be computed using known model category techniques. (See Section 3of [OT20a].)2. When A enjoys certain algebraic properties—properties that O j will satisfy—one can computehom complexes in A [ C − ] as a filtered colimit of a diagram built from morphism complexesof hom A . (This is Lemma 4.6.1 of [OT20a].)3. For any A ∞ -category B , the natural map of functor spaceshom C at A ∞ ( A [ C − ] , B ) → hom C at A ∞ ( A , B )is an inclusion of connected components—the essential image may be identified with thosefunctors out of A that send elements of C to equivalences in B . (See Section 4 of [OT20a].)4. There is a formal way to enlarge an A ∞ -category A so that the enlargement contains allmapping cones. This is called the category of twisted complexes of A , and is denoted Tw A .5. We will also use that any A ∞ -category defines an ∞ -category by taking the A ∞ -nerve ; this is aconstruction due to the second author [Tan16] and Faonte [Fao17] independently, generalizinga dg-construction due to Lurie [Lur12]. Notation 3.2.1 (Simplices) . Fix an integer d ≥
0. We let | ∆ d | denote the standard topological d -dimensional simplex, given by the subset of those ( t , . . . , t d ) ∈ R d +1 satisfying t i ≥ (cid:80) t i = 1.More generally, given any linear order A , we let | ∆ A | denote the subset of R A given by those ( t a ) a ∈ A satisfying t a ≥ (cid:80) a ∈ A t a = 1. We will sometimes refer to | ∆ A | as the geometric realizationof A . The extended d -simplex is the space | ∆ de | ⊂ R d +1 of those ( t , . . . , t d ) ∈ R d +1 satisfying (cid:80) t i = 1. It is abstractly homeomorphic to R d ¿ 22hen dealing with infinite-dimensional entities such as B Aut or Aut, one must specify whatwe mean by a smooth map j . We utilize the framework of diffeological spaces, and in particular, wewill study smooth maps from | ∆ ne | ∼ = R n to (cid:92) B Aut( M ), where the “hat” notation denotes that wehave put a diffeological space structure on B Aut o ( M ). We refer the reader to [OT20c] for details. Notation 3.2.2.
We define a category
Simp ( (cid:92) B Aut o ( M )). Objects are smooth simplices j : | ∆ ne | → (cid:92) B Aut o ( M ) (for n ≥
0) and morphisms are simplicial maps | ∆ ne | → | ∆ n (cid:48) e | that are compatible with j and j (cid:48) . Recollection 3.2.3.
Here are the facts we will need. All of these are detailed in [OT20c].1. The localization of
Simp along all morphisms is an ∞ -category homotopy equivalent to theclassifying space B Aut o ( M ). We remind the reader that there is a natural way to con-vert a topological space into an ∞ -category, called the singular complex functor; the re-sulting ∞ -category is denoted Sing ( B Aut o ( M )). The more precise result is that the map Simp ( (cid:92) B Aut o ( M )) → Sing ( B Aut o ( M )) is a localization.The same result holds for the group Diff( Q ) as well, so that Sing ( B Diff( Q )) (cid:39) B Diff( Q ) is alocalization of Simp ( (cid:92) B Diff( Q )).2. (cid:92) B Aut( M ) has a tautological smooth M -bundle over it, and one may pull back this bundlealong smooth maps j . If the domain of j is a smooth manifold in the usual sense, the pullbackis a smooth bundle in the usual sense. (See for example [CW17].)3. Finally, for any simplicial set S , on has the barycentric subdivision subdiv ( S ). One has anatural map subdiv ( S ) → S , and this map induces an equivalence of localizations—that is, ofthe ∞ -groupoids obtained by inverting all morphisms in subdiv ( S ) and in S . In particular,if S is already a Kan complex, the map subdiv ( S ) → S exhibits S as the Kan completion of subdiv ( S ). Our main constructions rely on counting certain holomorphic curves in Liouville bundles . Let ussummarize the foundations we laid for such counts.
Recollection 3.3.1.
All the results below can be found in [OT20b]:1. The usual Gromov compactness results hold when analyzing holomorphic curves in Liouvillebundles. In fact, one can perform straightforward generalizations of the usual C estimates inthe Liouville setting (so that holomorphic curves remain in an a compact subset determinedby the boundary conditions) and of the usual energy estimates (so that the usual finiteenergy assumptions may be applied to encode how nodal curves may develop). This wasutilized in [OT20b] to construct a non-wrapped Fukaya category associated to a Liouvillebundle E → B ; the main theorem is that this non-wrapped Fukaya category is indeed an A ∞ -category. We apply this in the present work by associating an A ∞ -category O j to eachsimplex j : | ∆ n | → B (Aut( M )); we will recall more in Section 4.1.2. As mentioned after Warning 1.2.5, we utilize two types of continuation maps in the presentwork: those defined by counting holomorphic strips, and those defined by counting holomor-phic disks with one boundary puncture. The count of continuation strips is homotopic tothe count of continuation once-punctured disks when the latter is post-composed by the µ operation. 23 The wrapped Fukaya categories
We fix a Liouville bundle E → B . The first agenda of this section is to define the wrappedFukaya category W j associated to a smooth simplex j : | ∆ ne | → B (Definition 4.3.3). We do thisby localizing the non-wrapped Fukaya categories O j along non-negative continuation maps. (Theintuition is that in any wrapped Fukaya category, a non-negative finite wrapping should induce anequivalence.) We denote the resulting A ∞ -category by W j . This is a bundle version of an ideaoriginally due to Abouzaid and Seidel; see also [GPS17], where we learned of the idea, and whosenotation we largely follow.We then prove that the assignment j (cid:55)→ W j is locally constant (i.e., forms a local system ofwrapped Fukaya categories), meaning any inclusion of simplices j ⊂ j (cid:48) induces an equivalence of A ∞ -categories (Proposition 4.5.3). This allows us to prove Theorem 1.2.2. In this subsection, we briefly recall the definition of O j given in [OT20b]. While we will apply ourconstructions to the universal example of the universal bundle over B = (cid:92) B Aut o ( M ), we state ourconstructions with the generality of an arbitrary base B . Choice 4.1.1 ( L b and cofinal wrapping sequences.) . For every point b ∈ B , we choose a countablecollection L b of eventually conical branes in the fiber E b such that the following holds: For everyeventually conical brane L (cid:48) ⊂ E b , there exists an element L ∈ L b such that L admits a non-negativewrapping to L (cid:48) , or L (cid:48) admits a non-negative wrapping to L .Then, for every L ∈ L b , we choose a cofinal wrapping sequence L = L (0) → L (1) → . . . . Finally, because this totality of choices is a countable collection, we may assume that if L ( w ) and L (cid:48) ( w (cid:48) ) are in the same fiber, then they are either transverse, or L = L (cid:48) and w = w (cid:48) . Notation 4.1.2 (The wrapping index w ) . In the cofinal wrapping sequence of Choice 4.1.1, wewill often denote the superscript index by ( w ). The w stands for “wrapping index.”Fix a smooth map j : | ∆ ne | → B . (Note | ∆ ne | is an extended simplex as in Definition 3.2.1.)Then we consider the non-wrapped Fukaya category O j associated to j whose definition giveninductively on n —we first define O j for all j having domain of dimension ≤ n , then for those j with domain having dimension n + 1. (See [OT20b, Section 6] for the details.) Notation 4.1.3 ( b i and L b i ) . For every 0 ≤ i ≤ n , let b i be the image of the i th vertex of | ∆ ne | under j . Recall we have chosen a countable collection of branes and a cofinal wrapping sequence ofthese (Choice 4.1.1). In particular, L b i denote the countable collection of branes associated to b i . Definition 4.1.4 (Objects) . An object of O j is a triplet ( i, L, w ) where i ∈ { , . . . , n } , L ∈ L b i , w ∈ Z ≥ . Notation 4.1.5 ( L ( w ) ) . One can informally think of the triplet ( i, L, w ) as the brane L ( w ) inside E b i . For this reason, we will soon denote an object simply by L ( w ) , omitting i . (See for exampleDefinition 4.1.13.) Definition 2.2.4. Definition 2.2.5 otation 4.1.6 (Parallel transport Π) . Fix a pair of objects ( L , i , w ) and ( L , i , w ). Theintegers i and i define a simplicial map β : | ∆ | → | ∆ n | ⊂ | ∆ ne | sending the initial vertex of | ∆ | to i and the final vertex to i .We let h = j ◦ β . One also has an underlying ordered pair of branes (cid:126)L = ( L , L ). By choosingFloer data Θ (cid:126)L (see [OT20b, Section 5.4]), we have a parallel transport taking the initial fiber of h ∗ E (i.e., the fiber above the initial vertex of | ∆ | ) to the final fiber of h ∗ E .We let Π i ,i denote this parallel transport. Definition 4.1.7 (Morphisms) . For given two objects ( i , L , w ) and ( i , L , w ) of O j , we definethe graded abelian group hom O j (( i , L , w ) , ( i , L , w ))to be (cid:76) x ∈ Π i ,i ( L ( w ) ∩ L ( w o x [ −| x | ] . w < w R ( i , L , w ) = ( i , L , w )0 otherwise . Here, Π i ,i is the parallel transport map (Notation 4.1.6). We also note that o x is the orientation R -module of rank one associated to the intersection point x , and | x | is the Maslov index associatedto the brane data. Remark 4.1.8.
We have rendered O j to be directed in the w index; this means that the morphismcomplex from ( i, L, w ) to ( i (cid:48) , L (cid:48) , w (cid:48) ) will be zero unless w < w (cid:48) , or ( i, L, w ) = ( i (cid:48) , L (cid:48) , w (cid:48) ) (in whichcase the morphism complex is just the ground ring R in degree 0). Remark 4.1.9.
The set x ∈ Π i ,i ( L ) ∩ L is also in bijection with the set of flat sections of h ∗ E → | ∆ | (with respect to Θ ( L ,L ) ) beginning at L ( w )0 and ending at L ( w )1 . (See Notation 4.1.6.)Higher operations µ d for d ≥ Definition 4.1.10 ( µ d for the non-wrapped categories) . As usual, fix a smooth map j : | ∆ ne | → B .For d ≥
1, fix a collection (cid:126)L = { ( i , L , w ) , . . . , ( i d , L d , w d ) } . We may assume w < . . . < w d by Definition 4.1.7 (otherwise µ d is forced to be 0) .Note that the integers i , . . . , i d induce a simplicial map β : | ∆ d | → | ∆ n | ⊂ | ∆ ne | by sendingthe a th vertex of | ∆ d | to the i a th vertex of | ∆ n | . (This assignment, of course, need not be order-preserving.) For a given collection of intersection points x a ∈ Π i a − ,i a (cid:16) L ( w a − ) a − (cid:17) ∩ L ( w a ) a ( a = 1 , . . . , d )and x ∈ Π i ,i d (cid:16) L ( w )0 (cid:17) ∩ L ( w d ) d , we define M ( x d , . . . , x ; x )to be the moduli space of holomorphic sections u E (cid:15) (cid:15) S r ⊂ (cid:47) (cid:47) u (cid:50) (cid:50) S ◦ d +1 ν β (cid:47) (cid:47) | ∆ d | β (cid:47) (cid:47) | ∆ n | ⊂ | ∆ ne | j (cid:47) (cid:47) B satisfying the obvious boundary conditions. 25 emark 4.1.11. The map ν β : S ◦ d +1 → | ∆ n | is defined in [OT20b, Subsection 5.3]. These aredegree one maps from the ( d + 1)st universal family of disks to the d -simplex and their existence isdue originally to Savelyev [Sav13].As usual, the brane structures on the L ( w ) allow us to orient these moduli spaces, and predicttheir dimension based on the degrees of the x a . We define µ d ( x d , . . . , x ) = (cid:88) x M ( x d , . . . , x ; x ) x where the number M is counted with sign. In case our branes are not Z -graded, we as usualwe declare the x coefficient of µ d to be zero when there is no zero-dimensional component of M ( x d , . . . , x a ; x ). Remark 4.1.12.
Given an ordered ( d +1)-tuple of objects in O j with underlying branes (cid:126)L , considerthe induced map β : | ∆ d | → | ∆ n | ⊂ | ∆ ne | . The A ∞ -operations are defined by moduli spacesdepending only on h = j ◦ β . Definition 4.1.13.
Fix j : | ∆ ne | → B . We let O j denote the A ∞ -category where • an object is as in Definition 4.1.4, • hom O j is as in Definition 4.1.7, and • The operations µ d are as in Definition 4.1.10. Remark 4.1.14.
When a µ d operation involves an element of an endomorphism hom-complexhom O j ( L, L ) = R , the operation is fully determined by demanding that the generator of the basering R (Choice 1.2.1) be a strict unit.We then have the following result from [OT20b]: Theorem 4.1.15 ([OT20b]) . O j is an A ∞ -category. Construction 4.1.16 (Functors O j → O j (cid:48) ) . Now suppose we have a map α : [ n ] → [ n (cid:48) ], a smoothmap j (cid:48) : | ∆ n (cid:48) e | → B , and consider the induced diagram | ∆ ne | α (cid:47) (cid:47) j (cid:33) (cid:33) | ∆ n (cid:48) e | j (cid:48) (cid:124) (cid:124) B. (Equivalently, consider a morphism in the category Simp ( B ).) Then there are induced assignmentsas follows:1. An object ( i, L, w ) of O j is sent to the object ( α ( i ) , L, w ) inside O j (cid:48) . Here, we are identifyingthe fiber of ( j (cid:48) ) ∗ E above α ( i ) with the fiber of j ∗ E above i in the obvious way.2. If α is an injection, then compatibility of the Floer data over the above diagram (see (Θ2) of[OT20b, Section 5.4] gives an isomorphism of graded abelian groupshom O j (( i , L , w ) , ( i , L , w )) → hom O j (cid:48) (( α ( i ) , L , w ) , ( α ( i ) , L , w )) . Otherwise, factoring α as a surjection followed by an injection, we have the same isomorphismby identifying the fiber above a vertex i (cid:48) ∈ | ∆ n (cid:48) | with a fiber above any point in the preimage α − ( i (cid:48) ). 26e call this assignment α ∗ . Proposition 4.1.17.
The assignment α ∗ from Construction 4.1.16 is an A ∞ -functor O j (cid:48) → O j .In fact, the assignments j (cid:55)→ O j and α (cid:55)→ α ∗ define a functor Simp ( B Aut( M )) → A ∞ Cat to the (strict) category of R -linear A ∞ -categories and R -linear functors between them, with theusual (strictly associative) composition of functors. Proof.
Remark 4.1.12 shows that the µ d operations are respected on the nose, as one is countingsections over two bundles over | ∆ d | that admit an isomorphism respecting all boundary conditionsand choice of J and A . This shows α ∗ is a functor of A ∞ -categories, simply by defining the functorto have higher homotopies equaling zero.Now suppose we have a commutative diagram of smooth maps | ∆ ne | α (cid:47) (cid:47) j (cid:35) (cid:35) | ∆ n (cid:48) e | α (cid:48) (cid:47) (cid:47) j (cid:48) (cid:15) (cid:15) | ∆ n (cid:48)(cid:48) e | j (cid:48)(cid:48) (cid:123) (cid:123) B .
We must show that ( α (cid:48) ◦ α ) ∗ = α (cid:48)∗ ◦ α ∗ . This is straightforward from Construction 4.1.16. Remark 4.1.18.
It follows immediately from the isomorphism observed in Construction 4.1.16itself that when α : [ n ] → [ n (cid:48) ] is a surjection, α ∗ is an equivalence of A ∞ -categories. (i.e., α ∗ isessentially surjective and induces a quasi-isomorphism of hom-complexes). We now consider a Liouville bundle E → | ∆ | . Then for any brane X ⊂ E in the fiber above 0and any brane Y ⊂ E in the fiber above 1, there are two natural Floer complexes to associate:1. The Floer complex whose µ term is defined by counting holomorphic sections of the Liouvillebundle E . This is hom O ( X, Y ).2. The Floer complex obtained by first parallel transporting X to E along the edge | ∆ | , thencomputing the usual Floer complex CF ∗ (Π , X, Y ) in E . (If Π , X is an object of O , thischain complex is equivalent to hom O (Π , X, Y ).)There are obvious isomorphismshom O ( X, Y ) ∼ = CF ∗ (Π , X, Y ) (4.1)and (for any
X, X (cid:48) ∈ E ) hom O ( X, X (cid:48) ) ∼ = CF ∗ (Π , X, Π , X (cid:48) ) . (4.2)By considering a Liouville bundle E → | ∆ | over a two simplex, we have: Lemma 4.2.1.
The maps (4.1) and (4.2) are quasi-isomorphisms. Moreover, for any pair
X, X (cid:48) ∈{ X i } , the diagram H ∗ hom O ( X (cid:48) , X ) ⊗ H ∗ hom O ( X, Y ) µ (cid:47) (cid:47) (4.2) ⊗ (4.1) (cid:15) (cid:15) H ∗ hom O ( X (cid:48) , Y ) (4.1) (cid:15) (cid:15) HF ∗ (Π , X (cid:48) , Π , X ) ⊗ HF ∗ (Π , X, Y ) µ (cid:47) (cid:47) HF ∗ (Π , X (cid:48) , Y )commutes. 27e leave the proof to the reader, as it is standard given the techniques of [OT20b].We previously knew how to speak of continuation elements of two branes in the same fiber (suchas Π , X and Y ; see Section 3.3). Given (4.1) and the compatibility of Lemma 4.2.1, it makes senseto speak of continuation elements between objects of O j in possibly different fibers: Definition 4.2.2.
Given a Liouville fibration E → B , fix a smooth map j : | ∆ n | → B andconsider the non-wrapped Fukaya category O j (Definition 4.1.13). Fix two objects ( L , i , w ) and( L , i , w ).A continuation element from ( L , i , w ) to ( L , i , w ) is any element of H ∗ hom O j (( L , i , w ) , ( L , i , w ))arising from a continuation element of HF ∗ (Π , L ( w )0 , L ( w )1 ) under the isomorphism (4.1). Fix a smooth map j : | ∆ ne | → B and consider the A ∞ -category O j from Definition 4.1.13. Thereare two families of localizing morphisms one can consider: Definition 4.3.1 ( C and C Π ) . We let C Π denote the collection of morphisms in O j arising ascontinuation maps ( i, L, w ) → ( i (cid:48) , L (cid:48) , w (cid:48) ) for w (cid:48) > w (see Definition 4.2.2).On the other hand, we let C ⊂ C Π denote the collection of continuation maps with i = i (cid:48) and L = L (cid:48) .It turns out we can localize with respect to either C or C Π from Definition 4.3.1, and end upwith the same localization. Lemma 4.3.2.
The natural map O j [ C − ] → O j [ C − ] between localizations is an equivalence of A ∞ -categories. Proof.
Let A be any A ∞ -category, and let W be any class of morphisms. Consider the localizationfunctor A → A [ W − ], and let W ⊂ A denote the subcategory of morphisms that are mapped toequivalences under the localization functor. It is immediate that A [ W − ] (cid:39) A [ W − ].So it suffices to show that C ⊃ C Π . Suppose[ c ] ∈ H hom O (( i , L , w ) , ( i , L , w ))is a non-negative continuation element. Then choose a large enough wrapping index w (cid:48) of ( i , L , w )so that one can find a non-negative continuation element c (cid:48) from ( i , L , w ) to ( i , L , w (cid:48) ). Thenchoose a positive enough wrapping index w (cid:48) so that one can find a non-negative continuation ele-ment c (cid:48) (cid:48) from ( i , L , w (cid:48) ) to ( i , L , w (cid:48) ). (Note that our assumption that the ( L, w ) form a cofinalsequence allows for this.) Then one has a commutative diagram( i , L , w (cid:48) )( i , L , w (cid:48) ) [ c (cid:48) (cid:48) ] (cid:55) (cid:55) ( i , L , w ) [ c (cid:48) ] (cid:103) (cid:103) [ c (cid:48) ] (cid:79) (cid:79) ( i , L , w ) [ c ] (cid:55) (cid:55) [ c (cid:48) ] (cid:79) (cid:79)
28n the cohomology category of O , hence of O [ C − ] and of O [ C − ]. In O [ C − ], the cohomologycategory admits a unique inverse to [ c (cid:48) ] and [ c (cid:48) ]; then it follows that all the diagonal maps (andin particular, [ c ]) are equivalences in O [ C − ]. In particular, any c ∈ C Π is contained in C . Definition 4.3.3 ( W j ) . Fix j : | ∆ ne | → B a smooth map. We let W j denote the localization O j [ C − ], and we call it the partially wrapped Fukaya category associated to j . Example 4.3.4. If n = 0 and j : | ∆ | → B simply chooses a point b ∈ B , then O j [ C − ] isequivalent to the partially wrapped Fukaya category associated to the fiber E b (as in [GPS17]). Now we compute hom-complexes hom W j ( X, Y ) for two branes X and Y in the same fiber. Remark 4.4.1.
We note that this also computes hom-complexes when the branes are in twodifferent fibers of E → B : Simply find Y (cid:48) in the same fiber as X and consider a parallel-transportcontinuation Y → Y (cid:48) . In W j , this map induces an equivalence Y (cid:39) Y (cid:48) by Lemma 4.3.2, so we havehom W j ( X, Y ) (cid:39) hom W j ( X, Y (cid:48) ). Lemma 4.4.2.
The hom-complex hom W j ( X, Y ) has cohomology given by the wrapped Floer co-homology (computed in the fiber containing X and Y ) as defined in [AS10, Section 3.7]. Proof.
This is essentially the same proof as in Lemma 3.37 of [GPS17] (which in turn is due toAbouzaid-Seidel’s unpublished work). The main difference is that—because of our counterclockwiseorientation of the boundary of disks—our continuation maps behave covariantly, as opposed tocontravariantly.Let Y (0) → . . . denote a cofinal sequence (Definition 2.2.5). Without loss of generality, we mayassume Y = Y (0) . We first claim that if c : L → L (cid:48) is a map in C (in particular, L and L (cid:48) are inthe same fiber), then the maphocolim i hom O j ( L (cid:48) , Y ( i ) ) → hocolim i hom O j ( L, Y ( i ) ) (4.3)is a quasi-isomorphism. To verify this claim, note that µ is respected at the level of cohomology(Recollection 3.3.1(2)). So we have induced maps of cohomology groups H ∗ (hocolim i hom O j ( L (cid:48) , Y ( i ) )) ∼ = colim i H ∗ hom O j ( L (cid:48) , Y ( i ) ) → colim i H ∗ hom O j ( L, Y ( i ) ) (4.4) ∼ = H ∗ (hocolim i hom O j ( L, Y ( i ) )) . There are isomorphisms in the lines above—this is because a filtered colimit of cohomology isnaturally isomorphic to the cohomology of a filtered homotopy colimit. Now we note that the colimitof the cohomology groups is a definition for wrapped Floer cohomology (in the fiber containing L and Y ) employed in Abouzaid-Seidel’s paper [AS10, Section 3.7]. Moreover, in this definition ofthe wrapped Fukaya category, L and L (cid:48) are exhibited as equivalent objects by a continuation map(this is proven in [BKO19]). So the arrow (4.4) is an isomorphism of groups. This shows that (4.3)is a quasi-isomorphism.Because (4.3) is a quasi-isomorphism for any choice of Y ( i ) and any choice of c : L → L (cid:48) in C ,a general fact about localizations (Recollection 3.1.1(2)) shows that the natural maphocolim i hom O j ( X, Y ( i ) ) → hocolim i hom O j [ C − ] ( X, Y ( i ) )29s an equivalence for any X . On the other hand, the latter homotopy colimit is indexed by asequence of quasi-isomorphisms, because the maps Y ( i ) → Y ( i +1) are already in C , being non-negative continuation maps. Thus we havecolim i H ∗ hom O j ( X, Y ( i ) ) ∼ = colim i H ∗ hom O j [ C − ] ( X, Y ( i ) ) ∼ = H ∗ hom O j [ C − ] ( X, Y (0) )while the left-hand side is the colimit definition of wrapped Floer cohomology. This completes theproof.
We first note that α ∗ (Construction 4.1.16) respects C (Definition 4.3.1). That is, consider adiagram | ∆ ne | α (cid:47) (cid:47) j (cid:33) (cid:33) | ∆ n (cid:48) e | j (cid:48) (cid:124) (cid:124) B. where α is induced by an order-preserving injection [ n ] → [ n (cid:48) ]. By definition, the continuationmaps C of O j are sent to (some of the) continuation maps C (cid:48) of O j (cid:48) , so we have an induced functoron the localizations α ∗ : W j → W j (cid:48) . (4.5)(Note that we have abused notation by using α ∗ again.)Moreover, because O : Simp ( B ) → A ∞ Cat is a functor respecting each C , the naturality oflocalizations implies the following: Proposition 4.5.1.
The assignment j (cid:55)→ W j and α (cid:55)→ α ∗ induces a functor of ∞ -categories W : N ( Simp ( B )) → C at A ∞ . Remark 4.5.2.
Note that the target of W is the ∞ -category C at A ∞ , rather than the (strict) cate-gory A ∞ Cat . Indeed, the former is where we can articulate the universal property of localizations;see Section 3.1.
Proposition 4.5.3 (Local triviality) . The map α ∗ : W j → W j (cid:48) in (4.5) is an equivalence of A ∞ -categories. Proof.
We note that it suffices to prove the proposition when j : | ∆ | → B is a 0-simplex in B .It follows that α ∗ is essentially surjective from the proof of Lemma 4.3.2. Now we must provethat the map is fully faithful (i.e., induces a quasi-isomorphism on morphism complexes). Thisfollows from Lemma 4.4.2, as α induces the obvious identity map on wrapped Floer cohomologygroups. Now we are ready to complete the proof of Theorem 1.2.2.
Proof of Theorem 1.2.2.
By Proposition 4.5.1, we have a functor
Simp ( B Aut( M )) → A ∞ Cat. A ∞ -categories;hence by the universal property of localization of ∞ -categories, we have an induced diagram ∆ → C at A ∞ as follows: N ( Simp ( B Aut( M ))) W (cid:47) (cid:47) (cid:15) (cid:15) C at A ∞ Sing ( B Aut( M )) (cid:54) (cid:54) (Informally, the above is a homotopy-commutative diagram of functors between ∞ -categories.)Here, the left vertical map is the localization map along every edge of N ( Simp ( B Aut( M )). (Rec-ollection 3.2.3(1).) The dashed arrow is the induced map on localizations, and the map we seek,because of the obvious equivalence B Aut( M ) (cid:39) Sing ( B Aut( M )) . This completes the proof. 31
Local systems and a bundle version of the Abouzaid map
Fix a Liouville domain M and a compact brane Q ⊂ M . In [Abo12], Abouzaid constructed afunctor from the quadratically wrapped category of M to the category of local systems on Q . Inthis section we construct a bundle version of this functor. Remark 5.0.1.
For concreteness, we construct this functor for the case of the Liouville bundleover B Diff( Q ) with fiber M = T ∗ Q ; but one can do this for any Liouville bundle whose structuregroup has been reduced in such a way as to preserve the set Q ⊂ M .Let us outline this section’s contents. Fix a smooth fiber bundle E (cid:48) → B (cid:48) with fiber given by asmooth, compact manifold Q (possibly with boundary). To this data we assign a collection of localsystem categories—specifically, to each j : | ∆ ne | → B (cid:48) , we also associate a dg-category P j whosetwisted complex category is quasi-equivalent to a category of local systems on j ∗ E (cid:48) . In particular,the assignment j (cid:55)→ Tw P j is a local system of local system categories. Our main interest in thisconstruction is the universal example when B (cid:48) = B Diff( Q ).Setting M = T ∗ Q , we again utilize the result that the localization of a subdivision recoversthe original homotopy type (Recollection 3.2.3(1)). This implies the existence of a functor of ∞ -categories Tw P : B Diff( Q ) → C at A ∞ . Then, the main result of the present section is to show that one can make the diagram B Diff( Q ) (cid:47) (cid:47) Tw P (cid:37) (cid:37) B Aut gr,b ( T ∗ Q ) W (cid:119) (cid:119) C at A ∞ commute up to a natural transformation from W to Tw P (Corollary 5.3.3). That is, we have aDiff( Q )-equivariant functor from W ( T ∗ Q ) to Tw P ( Q ).To accomplish this goal, we first construct a natural transformation from O to Tw P (Propo-sition 5.2.6). Then, the main aim is to prove that the non-negative continuation maps in O aresent to equivalences in the category Tw C ∗ P of local systems (Theorem 5.3.1)—this is the mostgeometrically involved component of our arguments. By the universal property of the localization W , we conclude that the bundle version of the Abouzaid functor descends to the wrapped categories(Corollary 5.3.3).We will prove in the next section that this natural transformation is a natural equivalence—i.e.,that this is a Diff( Q )-equivariant equivalence (Theorem 6.0.1). C ∗ P (families of local system categories) Remark 5.1.1.
As in Section 3.2, we can endow the topological space B Diff( Q ) with a diffeologicalspace structure. Because Q is compact, B Diff( Q ) satisfies smooth approximation. So, for example,if Sing ( B Diff( Q )) denotes the usual singular complex of continuous simplices | ∆ n | → B Diff( Q ),and if Sing C ∞ ( B Diff( Q )) denote the simplicial set whose n -simplices are smooth maps j : | ∆ ne | → B Diff( Q ) from extended simplices, then the inclusion of Sing C ∞ ( B Diff( Q )) to Sing ( B Diff( Q )) isa homotopy equivalence of simplicial sets (Recollection 3.2.3(1)). Notation 5.1.2 ( E Q ) . Note that B Diff( Q ) carries a principal Diff( Q ) bundle—the universal one—and hence an associated fiber bundle with fibers Q . We denote this fiber bundle by E Q → B Diff( Q ).32 otation 5.1.3 ( Q a ) . Let j : | ∆ ne | → B Diff( Q ) be a smooth map. For any a ∈ [ n ], we let Q a denote the fiber of j ∗ E Q above the a th vertex of | ∆ n | ⊂ | ∆ ne | .Now we give some notation to the Moore path space category modeling the ∞ -groupoid Sing ( j ∗ E Q ). Construction 5.1.4 ( P j ) . Let j : | ∆ ne | → B Diff( Q ) be a smooth map. We let P j denote thetopologically enriched category defined as follows:We declare the object set to be the disjoint union of fibers P j := (cid:97) a ∈ [ n ] Q a . Given q a ∈ Q a , q b ∈ Q b , we declare hom P j ( q a , q b ) to be the topological space of continuous maps γ : [0 , ∞ ] → j ∗ E Q such that γ is compactly supported (i.e., constant beyond some finite time t ∈ [0 , ∞ ]) and such that γ (0) = q a and γ ( ∞ ) = q b . Composition is defined in the obvious way: If t γ is the smallest time for which γ is constant, γ (cid:48) ◦ γ is defined by setting( γ (cid:48) ◦ γ )( t ) = (cid:40) γ ( t ) t ≤ t γ (cid:48) ( t − t ) t ≥ t . Notation 5.1.5.
Construction 5.1.4 defines a functor P : Simp ( B Diff( Q )) → Cat
T op , j (cid:55)→ P j which we denote (as indicated) by P . (It has the obvious effect on morphisms.) Here, Cat
T op isthe category of categories enriched in topological spaces. For the notation
Simp ( B Diff( Q )), seeNotation 3.2.2. Remark 5.1.6.
Given any map | ∆ ne | → | ∆ n (cid:48) e | → B Diff( Q ), the induced map P j → P j (cid:48) is anequivalence of topologically enriched categories; this follows by noting that the inclusion j ∗ E Q → ( j (cid:48) ) ∗ E Q is a homotopy equivalence. Notation 5.1.7 ( Tw C ∗ P ) . Recall that the functor C ∗ sending a topological space P to its sin-gular chain complex C ∗ P is lax monoidal. As a result, applying C ∗ to the morphism spaces to atopologically enriched category D , we obtain a dg-category C ∗ D .We let C ∗ P j denote the dg-category associated to P j . We denote the composite functor C ∗ P : Simp ( B Diff( Q )) P −→ Cat
T op C ∗ −→ dgCat. We also denote by Tw C ∗ P the composite of C ∗ P with the Tw functor (Recollection 3.1.1(4)). Given any diffeomorphism φ : Q → Q , one has an induced exact symplectomorphism D φ : T ∗ Q → T ∗ Q by pushing forward the effect of φ on cotangent vectors. Because this clearly respects thediffeological smooth structures, and there exist natural lifts of D respecting the choices of gr and b = w ( Q ), we have an induced functor D : Simp ( B Diff( Q )) → Simp ( B Aut gr,b ( T ∗ Q )) . By an equivalence of topologically enriched categories, we mean an essentially surjective functors whose maps onmorphism spaces are weak homotopy equivalences. O ◦ D to Tw C ∗ P (Proposition 5.2.6). We do so by utilizing a non-wrapped, family-friendly version of a construction we learned fromAbouzaid’s paper [Abo12].
Remark 5.2.1.
In [Abo12], the author constructs a functor whose domain is a quadraticallywrapped Fukaya category—i.e., one whose morphisms are defined by using quadratic Hamiltonians.Here, we instead use our non-wrapped Fukaya categories O j as the domain. Aside from this detail,the main ideas remain unchanged—in particular, the analytic input for the functor utilized in ourpresent work is somewhat simpler.And, as we show here, the construction carries through successfully when M is a Liouville sector(not necessarily a Liouville manifold). Let us remark that our bundle-version of the constructioncarries over to a setting in which the structure group of Aut( M ) is reduced in such a way thatevery fiber admits a distinguished brane Q ; such is the case we are in, as Diff( Q ) preserves the zerosection of T ∗ Q . Construction 5.2.2 (The Abouzaid functor on objects and morphisms) . Fix a smooth map j : | ∆ ne | → B Diff( Q ) and consider the associated fibration P j = j ∗ B Diff( Q ) → | ∆ ne | . For an integer a ∈ { , . . . , n } , let M a be the fiber above the a th vertex of | ∆ en | , and let L a ⊂ M a be a brane.We assume L a intersects the zero section Q a ⊂ M a transversally. To L a we associate the followingobject of Tw C ∗ P ( Q a ): (cid:77) x a ∈ L a ∩ Q a x a [ −| x a | ] , D . (5.1)That is, as a local system, the object is generated by free R -modules in degrees | x a | , subject to adifferential D . The differential is given as follows: Given two intersection points x a , x (cid:48) a ∈ L a ∩ Q a ,we let H ( x a , x (cid:48) a )be the compactified moduli space of (possibly broken) holomorphic sections u : R × [0 , → R × [0 , × M a with boundary on L a and Q a , converging to x a and x (cid:48) a . Given a map u , consider the restriction of u to the boundary line R × { } ⊂ R × [0 ,
1] mapping to Q a . The resulting map R → M a admits anarc-length parametrization (we use the Riemannian metric induced by the choice of almost-complexstructure on M a ), and in particular, any u determines an arc-length parametrized path in Q a . Thisinduces a map from H ( x a , x (cid:48) a ) to the space of paths in Q a from x a to x (cid:48) a , and in particular a mapof chain complexes F : C ∗ H ( x a , x (cid:48) a ) → hom P j ( x a , x (cid:48) a ) . Then, one chooses fundamental chain classes for H , and pushes them forward. These elements ofthe hom-complexes of C ∗ P j determine the differential D . We refer the reader to [Abo12, (2.27)]for details. Construction 5.2.3 (The rest of the Abouzaid functor) . We now explain how the above assign-ment on objects extends to an A ∞ -functor O j → Tw C ∗ P j .Fix an integer d ≥
1. For each i ∈ { , , . . . , d } , choose also an integer a i ∈ { , . . . , n } and abrane L a i above the a i th vertex. The choices of a i determine a unique simplicial map from the The functor O : Simp ( B Aut gr,b ( T ∗ Q )) → A ∞ Cat is from Proposition 4.1.17. -simplex to the n -simplex (by sending the i th vertex to the a i th vertex). Let us take a cone onthis map—specifically, the map | ∆ d +1 | → | ∆ n | , (cid:40) i (cid:55)→ a i i ∈ { , . . . , d } ,i (cid:55)→ a d i = d + 1 . Given generators y i,i +1 ∈ hom O j ( L a i , L a i +1 ), we let H ( y , , . . . , y ( d − ,d ; Q a d )denote the moduli space of holomorphic sections uj ∗ E | S r (cid:47) (cid:47) j ∗ E (cid:15) (cid:15) S r ⊂ S ◦ d +1+1 ν (cid:47) (cid:47) u (cid:79) (cid:79) | ∆ d +1 | → | ∆ d | (cid:47) (cid:47) | ∆ n | satisfying the following boundary conditions: On the strip like ends from the i th arc to the ( i + 1)starc, u converges to the parallel transport arc given by y i,i +1 , while the ( d + 1)st boundary arc—from the ( d + 1)st puncture to the 0th puncture—is constrained by the parallel transport boundarycondition from Q a d to Q a . We note that ν is the same map as in Definition 4.1.10.Given such a u , one can measure the arclength of the restriction of u to the ( d + 1)st boundaryarc. (For example, by taking its vertical velocity; i.e., by using the fiberwise Riemannian metrics.)By the assumption of general position, an intersection x ∈ L a ∩ Q a and x d ∈ L a d ∩ Q a d arenot related by parallel transport—we do not have triple intersections. So this guarantees that thevertically measured arclength is non-zero. (We point this out, as if the vertical velocity were zerothe arc length parametrization would not be defined, and hence we would not have a map to hom P .)As before we have a map from H to the space of paths in j ∗ E from Q a to Q a d , and thisextends continuously to the compactification H . Pushing forward fundamental chains, we obtainthe desired A ∞ functor maps. Remark 5.2.4.
We note one subtlety in the construction; it is somewhat unnatural to utilize thecone map | ∆ d +1 | → | ∆ n | , as then verifying the A ∞ functor relations forces us to use (for example)the fact that for k ≤ d , the moduli space H ( y , , . . . , y ( k − ,k ; Q a d )is cobordant to the moduli space H ( y , , . . . , y ( k − ,k ; Q a k )so that the usual compactification of the moduli spaces yield the desired A ∞ functor relations. Notation 5.2.5 (The Abouzaid functor F .) . We will denote by F the A ∞ functor defined in Constructions 5.2.2 and 5.2.3. We may write F j to denote the dependenceon the simplex j , but will largely leave this dependence implicit.This construction clearly respects inclusions | ∆ ne | → | ∆ n (cid:48) e | . As such, we have: Proposition 5.2.6.
The non-wrapped Abouzaid construction F induces a natural transformation Simp ( B Diff( Q )) O ◦ D (cid:45) (cid:45) Tw C ∗ P (cid:49) (cid:49) (cid:11) (cid:19) A ∞ Cat. .3 The Abouzaid functor descends to the wrapped category Proposition 5.2.6 constructs a map from the non-wrapped, directed family of Fukaya categories tofamilies of local system categories. To descend our construction to the wrapped setting, the maingeometric result we must verify is the following.
Theorem 5.3.1.
Let M be a Liouville sector, and c : L → L a continuation element associatedto a non-negative isotopy. We also fix a compact test brane X ⊂ M . Then the map on twistedcomplexes c ∗ : ( X ∩ L , D ) → ( X ∩ L , D )—induced by the Abouzaid functor from O ( M ) to Tw C ∗ P ( X )—is an equivalence.The theorem immediately implies: Corollary 5.3.2.
Let M = T ∗ Q and X = Q . Then the map on twisted complexes c ∗ : ( Q ∩ L , D ) → ( Q ∩ L , D ) is an equivalence.By the universal property of localization and Lemma 4.3.2, we also conclude: Corollary 5.3.3.
The natural transformation from Proposition 5.2.6 induces a natural transfor-mation from W ◦ D to Tw C ∗ P : Simp ( B Diff( Q )) W ◦ D (cid:44) (cid:44) Tw C ∗ P (cid:50) (cid:50) (cid:11) (cid:19) C at A ∞ By the universal property of Tw , this in turn induces a natural transformation Tw W ◦ D → Tw C ∗ P . Remark 5.3.4.
The functors W ◦ D and C ∗ P both send morphisms of Simp ( B Diff( Q )) to equiva-lences in C at A ∞ . Thus they both induce a functor Sing ( B Diff( Q )) (cid:39) B Diff( Q ) → C at A ∞ by Recollection 3.2.3(1). In particular, these exhibit Diff( Q ) actions on both W ( T ∗ Q ) and C ∗ P ( Q ).Corollary 5.3.3 says that the map Tw W ( T ∗ Q ) → Tw C ∗ P ( Q ) is Diff( Q )-equivariant. Because Theorem 5.3.1 is a statement contained in single fiber of a Liouville bundle (and in par-ticular, is a statement about a single Liouville sector), we now work in a Liouville sector M . Wewill also fix a compact test brane X ⊂ M ; this X plays the role of the zero section Q above. Forsimplicity of exposition (and without loss of generality), we will assume that X is connected.We fix a non-negative Hamiltonian isotopy L ( t ) := φ tF ( L ) , L (0) = L , L (1) = L so that L and L are transverse to X . By definition, the Hamiltonian vector field X F is, outsidesome compact region, equal to βZ for some constant β ≥ Notation 5.3.5 ( n χ L ( y ) and c χ L ) . Let L ( t ) , t ∈ [0 ,
1] be a Hamiltonian isotopy generated by aHamiltonian F such that F ( r, y ) = θr on the cylindrical regions of L and L for a positiveconstant θ > c χ L ∈ CF ( L , L ) the continuation element induced by the isotopy L and a choice of elongation function χ . c χ L is obtained by counting holomorphic disks with oneboundary puncture, and with moving boundary condition dictated by χ and L . See Figure 1.3.6and Section 3.3. We let n χ L ( y ) denote the number of such disks with output y .36he idea of the proof of Theorem 5.3.1 is to replace the given isotopy L = { L ( t ) } t ∈ [0 , byanother isotopy L (cid:48) = { L (cid:48) ( t ) } t ∈ [0 , satisfying the following properties:(i) The isotopy preserves the twisted complexes ( X ∩ L , D ) and ( X ∩ L , D )—that is, ( X ∩ L ( t ) , D (cid:48) ) = ( X ∩ L (cid:48) ( t ) , D (cid:48) ) for t = 0 , F ( c χ L (cid:48) ) is homotopic to F ( c χ L ) (these maps have the same domain and codomain in light of(i)), and(iii) L (cid:48) (1) can be isotoped back to L (0) via a compactly supported Hamiltonian isotopy.Property (iii) allows us to prove that the map F ( c χ L (cid:48) ) admits a homotopy inverse, essentiallyconstructed by counting continuation strips induced by the (compactly supported!) reverse isotopyof L (cid:48) . Theorem 5.3.1 will then follow from (ii) .By tracing through the definition of the Abouzaid functor in Construction 5.2.2, we see thatthe map F : CF ∗ ( L , L ) → Tw C ∗ P ( X ) is given by F ([ y ]) = (cid:77) x,x (cid:48) ( − | y | +( | x | +1)( | y | + | x (cid:48) | ) ev ∗ ([ H ( x, y, x (cid:48) )])for y ∈ X ( L , L ) and x ∈ X ∩ L , x (cid:48) ∈ X ∩ L . (See [Abo12, (2.27)] for more details on this formula;here, [ H ( x, y, x (cid:48) )] is a choice of fundamental class for H ( x, y, x (cid:48) ).) The map c ∗ : ( X ∩ L , D ) → ( X ∩ L , D ) is defined by applying F to the continuation element c χ L , so to prove Theorem 5.3.1,we must prove that F ( c χ L ) is an equivalence in Tw C ∗ P ( X ).By the definition of c χ L (Notation 5.3.5), we have F ( c χ L ) = F (cid:88) y ∈ L ∩ L ; | y | =0 n χ L ( y ) (cid:104) y (cid:105) = (cid:88) y ∈ L ∩ L ; | y | =0 (cid:77) x,x (cid:48) ( − | y | +( | x | +1)( | y | + | x (cid:48) | ) n χ L ( y ) ev ∗ ([ H ( x, y, x (cid:48) )])= (cid:77) x,x (cid:48) ( − | y | +( | x | +1)( | y | + | x (cid:48) | ) ev ∗ (cid:88) y ∈ L ∩ L ; | y | =0 n χ L ( y )[ H ( x, y, x (cid:48) )] . (5.2) L y L L xx (cid:48) X Figure 5.3.6.
The F term of the Abouzaid functor applied to a continuation element.We first introduce the following notation. 37 otation 5.3.7. Let M ( D \ { z } ; L χ , y ) be the moduli space counting once-punctured disks without put y and boundary dictated by L and χ , as in Notation 5.3.5. Then let M ( D \ { z } ; L χ , y ) H ( x, y, x (cid:48) ) := M ( D \ { z } ; L χ , y ) ev z × ev z H ( x, y, x (cid:48) ) . (5.3)Then by definition, we have the following (see Figure 5.3.6.): Proposition 5.3.8. ev ∗ (cid:88) y ∈ L ∩ L ; | y | =0 n χ L ( y )[ H ( x, y, x (cid:48) )] = ev ∗ (cid:88) y ∈ L ∩ L ; | y | =0 [ M ( D \ { z } ; L χ , y ) H ( x, y, x (cid:48) )] . (5.4)Our next goal is to prove that the righthand side of (5.4) is equal to a chain obtained by pushingforward the fundamental class of a different moduli space. Let us define the moduli space. Notation 5.3.9.
We denote: • The lower semi-disc by D − = { z ∈ D | Im z ≤ } • by ∂ + D − ∼ = [ − , ∂D − with Im z = 0, and • we write ∂ − D − = ∂D − \ Int ∂ + D − = {| z | = 1 & Im z ≤ } where Int ∂ + D − ∼ = ( − ,
1) under the above identification.
Notation 5.3.10 ( M ( D − ; L χ , x, x (cid:48) )) . We consider the equation u : D − → M∂ J u = 0 , (cid:82) D − | du | < ∞ ,u ( z ) ∈ X for z ∈ ∂ − D − u ( z ) ∈ L χ ( z ) for z ∈ ∂ + D − (5.5)where χ : ∂ + D − → [0 ,
1] is a monotone function such that χ ( z ) = 1 and χ ( z ) = 0 near − , ∈ ∂ + D − respectively. We let M ( D − ; L χ , x, x (cid:48) )denote the moduli space of solutions to (5.5) for which u limits to x ∈ X ∩ L and x (cid:48) ∈ X ∩ L inthe obvious way. Lemma 5.3.11.
For all x, x (cid:48) , we have thatev ∗ (cid:88) y ∈ L ∩ L ; | y | =0 [ M ( D \ { z } ; L χ , y ) H ( x, y, x (cid:48) )] = ev ∗ (cid:0) [ M ( D − ; L χ , x, x (cid:48) )] (cid:3) . roof. We will smoothen corners of the elements in the fiber product M ( D \{ z } ; L χ , y ) H ( x, y, x (cid:48) )and construct a compact one dimensional moduli space such that the fiber product becomes a partof its boundary. Explanation of this construction is in order.We start with explicit description of the domain curves of the two moduli spaces appearing inthe fiber product (5.3). Notation 5.3.12 (Θ − and Z ) . We denote by Θ − the domain (equipped with the strip-like co-ordinates) of the relevant moduli spaces, and denote by Θ − Z the nodal curve obtained by theobvious grafting. (See Figure 5.3.6 for the image of the grafted domain.) We mention that we haveconformal equivalences Θ − ∼ = D \ { z } and Z ∼ = D \ { z , z , z } . We take the following explicitmodel for Θ − : Consider the domain { z ∈ C | | z | ≤ , Im z ≥ } ∪ { z ∈ C | | Re z | ≤ , Im z ≤ } and take its smoothing around Im z = 0 that keeps the reflection symmetry about the y -axis of thedomain. Then we take Z = { z ∈ C | ≤ Im z ≤ } \ { (0 , } . (5.6)Again we equip Z with a strip-like coordinate at z = (0 ,
1) that keeps the reflection symmetry.
Notation 5.3.13 ( r ) . We denote by Θ − Z the nodal curve obtained from Θ − and Z by the obvious grafting. Note that we have conformal equivalences Θ − ∼ = D \ { z } and Z ∼ = D \ { z , z , z } . Using a given strip-like coordinate around { (0 , } in Z \ { (0 , } , we constructone-parameter family of the glued domains which we denote byΘ − r Z for all sufficiently large r >
0. We perform this grafting while maintaining the above-mentionedreflection symmetry.By gluing the defining equations of the two moduli spaces on the glued domain Θ − r Z forall sufficiently large r > D − (adjustingthe positively moving boundary condition accordingly), we arrive at (5.5). Thus, by a standardgluing-deformation and compactness argument, the result follows. Remark 5.3.14.
Note that the moduli space of solutions to (5.5) is equivalent to the moduli spacedefining a continuation map using holomorphic strips (not disks) with moving boundary conditions.This is because under the conformal equivalence D − \ { , − } ∼ = D \ { , − } ∼ = R × [0 , (cid:40) ∂u∂τ + J ( ρ ( τ ) ,t ) ∂u∂t = 0 u ( τ, ∈ X, u ( τ, ∈ L ρ (1 − τ ) . (5.7)after a suitable choice of nondecreasing elongation function ρ : R → [0 ,
1] satisfying ρ ( τ ) = (cid:40) τ ≥ a τ ≤ a . 39 emark 5.3.15. By assumption on the isotopy, the isotopy is non-negative outside a compact set K . By compactness of X , we may enlarge K so that X ⊂ Int K without loss of generality. Thisallows us to apply the strong maximum principle to deriveimage u ⊂ K. (5.9)(See [BKO19, section 13] for similar application of strong maximum principle.)Next we need to establish a uniform energy bound. Let h X : X → R and h s : L s → R beLiouville primitives of X and L s respectively. (So for example, dh X = θ | X . For a given exactLagrangian isotopy generated by a Hamiltonian H , a smooth family of Liouville primitives of L s isgiven by h s = h + (cid:90) s ( (cid:104) θ, X H (cid:105) − H ) ◦ φ tH dt. (5.10)(See [Oh15a, Proposition 3.4.8], for example.) Then we have the following energy identity. Lemma 5.3.16.
Let X ⊂ Int K , let L be a nonnegative (near infinity) isotopy, and let H be aHamiltonian, linear outside K , generating the isotopy L . Then for any finite energy solution u of (5.7) with u ( ±∞ ) ≡ x ± , we have E J ( u ) = − ( h X ( x + ) − h X ( x − )) + ( h ( x + ) − h ( x − )) − (cid:90) ∞−∞ ρ (cid:48) ( τ ) H ( u (1 − τ, dτ. (5.11)where x − ∈ X ∩ L and x + ∈ X ∩ L . In particular, E J ( u ) ≤ C ( X, L ; K ) (5.12)for some constant C ( X, L ; K ) depending only on X, L and K but independent of u . Proof.
While what follows mostly parallels the proof of [AOOdS18, Lemma 7.3], there are twodifferences: • The current case treats the general case of Lagrangian branes in general Liouville sectors, while[AOOdS18, Lemma 7.3] treats the case of the zero section X = Q and the fiber L = 0 = T ∗ q Q for which we can take h X = 0 = h . Because of this, the statement of the current lemma ismore general including [AOOdS18, Lemma 7.3] as a special case. • There are differences in the details of the proof due to differences in conventions.Because of these, we include a complete proof here for the reader’s convenience.Consider the following family of the action functional A X,L s ( γ ) = − (cid:90) [0 , γ ∗ θ + h X ( γ (1)) − h s ( γ (0)) , (5.13)where γ ∈ P ( X, L s ). Obviously, we have A X,L ( x + ) − A X,L ( x − ) = (cid:90) ∞−∞ ddτ (cid:16) A X,L ρ (1 − τ ) ( u ( τ )) (cid:17) dτ. We derive ddτ ( A X,L ρ (1 − τ ) ( u ( τ ))) = ddτ (cid:32)(cid:90) [0 , − ( u ( τ )) ∗ θ − h X ( u ( τ, (cid:33) − ddτ (cid:0) h ρ (1 − τ ) ( u ( τ, (cid:1) = (cid:90) [0 , (cid:12)(cid:12)(cid:12) ∂u∂τ (cid:12)(cid:12)(cid:12) J t dt − (cid:28) θ, ∂u∂τ ( τ, (cid:29) − ddτ (cid:0) h ρ (1 − τ ) ( u ( τ, (cid:1) (5.14)40sing the first variation of the action functional with free boundary condition (see [Oh15b, Equation(12.1.1)]), together with the fact that u is a solution of (5.5). Therefore by combining the abovecalculations and integrating over −∞ < τ < ∞ , we have obtained A X,L ( x + ) − A X,L ( x − ) = E J ( u ) − (cid:90) ∞−∞ (cid:28) θ, ∂u∂τ ( τ, (cid:29) dτ − ( h ( x − ) − h ( x + )) . Noticing that the boundary condition u ( τ, ∈ L ρ (1 − τ ) implies that u ( τ,
1) = φ ρ (1 − τ ) H ( v ( τ )) forsome curve v ( τ ) ∈ L , we can write h ρ (1 − τ ) ( u ( τ, (cid:101) h ρ (1 − τ ) ( v ( τ ))for (cid:101) h ρ (1 − τ ) = h ρ (1 − τ ) ◦ φ ρ (1 − τ ) H : L → R . Then we compute ddτ ( h ρ (1 − τ ) ( u ( τ, d (cid:101) h ρ (1 − τ ) (cid:18) dvdτ (cid:19) − ρ (cid:48) (1 − τ ) (cid:32) d (cid:101) h s ds (cid:33) (cid:12)(cid:12)(cid:12)(cid:12) s = ρ (1 − τ ) ( v ( τ )) (5.15)= dh ρ (1 − τ ) ◦ dφ ρ (1 − τ ) H (cid:18) dvdτ (cid:19) − ρ (cid:48) (1 − τ ) (cid:16) (cid:104) θ, X H (cid:105) ( φ ρ (1 − τ ) H ( v ( τ ))) − H ( φ ρ (1 − τ ) H ( v ( τ ))) (cid:17) . It follows from the definitions that ∂u∂τ ( τ,
1) = dφ ρ (1 − τ ) H (cid:18) dvdτ (cid:19) − ρ (cid:48) (1 − τ ) X H ( u ( τ, . Plugging this in the previous equation and using the definition of Liouville primitive we obtain ddτ ( h ρ (1 − τ ) ( u ( τ, (cid:28) θ, ∂u∂τ ( τ, (cid:29) + ρ (cid:48) (1 − τ ) H ( u ( τ, . Substituting this into (5.14), we obtain ddτ ( A H ; ρ (1 − τ ) ( u ( τ ))) = (cid:90) [0 , (cid:12)(cid:12)(cid:12) ∂u∂t (cid:12)(cid:12)(cid:12) J t dt − ρ (cid:48) (1 − τ ) H ( u ( τ, (cid:90) [0 , (cid:12)(cid:12)(cid:12) ∂u∂t (cid:12)(cid:12)(cid:12) J t dt = ddτ ( A H ; ρ (1 − τ ) ( u ( τ ))) + ρ (cid:48) (1 − τ ) H ( u ( τ, . By integrating this over τ ∈ R , we obtain E J ( u ) = A X,L ( x − ) − A X,L ( x + ) + (cid:90) ∞−∞ ρ (cid:48) (1 − τ ) H ( u ( τ, dτ = A X,L ( x − ) − A X,L ( x + ) − (cid:90) ∞−∞ ρ (cid:48) ( τ ) H ( u (1 − τ, dτ. Then we evaluate A X,L ( x − ) − A X,L ( x + ) = − ( h ( x + ) − h X ( x + )) + ( h ( x − ) − h X ( x − ))= ( h X ( x + ) − h X ( x − )) − ( h ( x + ) − h ( x − )) . Combining the two, we have proved (5.11). 41inally we prove the uniform energy bound (5.12). Recall the support bound image u ⊂ K (5.9).Therefore | h ( x − ) − h ( x − ) | ≤ max K |(cid:104) θ, X H (cid:105)| + (cid:107) H (cid:107) K where (cid:107) H (cid:107) K := sup x ∈ K | H ( x ) | . Next we get the bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞∞ ρ (cid:48) ( τ ) H ( u (1 − τ, dτ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) ∞−∞ ρ (cid:48) ( τ ) | H ( u (1 − τ, | dτ ≤ (cid:107) H (cid:107) K . Combining these with the energy identity (5.11), we derive E J ( u ) ≤ max K |(cid:104) θ, X H (cid:105)| + 2 (cid:107) H (cid:107) K + (max h X − min h X ) =: C ( X, L , H ; K ) . (5.16)Finally we take C ( X, L ; K ) := inf H { C ( X, L , H ; K ) | L ( t ) = φ tH ( L ) ∀ t ∈ [0 , } (5.17)where the inf is taken over all Hamiltonians H that are linear outside K . This finishes the proof. Remark 5.3.17.
We examine the nature of the upper bound C ( X, L , H ; K ) given in (5.16). Clearlythe term (max h X − min h X ) does not depend on L but only on X . Under the hypotheses of Lemma5.3.16, we have the support property image u ⊂ K (see (5.9)). This implies that we have only toexamine the constant max K |(cid:104) θ, X H (cid:105)| + 2 (cid:107) H (cid:107) K (5.18)when we take the infimum of C ( X, L , H ; K ) as we vary over L that is linear outside K . We alsoremark that, for fixed K , the map L (cid:55)→ C ( X, L ; K ) is continuous with respect to the fine C topology on the space of isotopies L . This section will be occupied by the proof of Theorem 5.3.1. We choose
R > K ⊂ r − (( −∞ , R/ L to L (cid:48) = { L (cid:48) ( t ) } via L para = { L s } s ∈ [0 , with L s = { L s ( t ) } t ∈ [0 , and L = L , L = L (cid:48) so that the following hold: • the hypotheses of Lemma 5.3.16 holds for all s ∈ [0 , • we have L (cid:48) ( t ) ∩ r − (( −∞ , R ]) = L ( t ) ∩ r − (( −∞ , R ]) , • L (cid:48) ( t ) ∩ (cid:0) M \ r − (( −∞ , R ]) (cid:1) = L ∩ (cid:0) M \ r − (( −∞ , R ]) (cid:1) , and we suitably interpolate on the region r − ([ R, R ]). Therefore it follows from Remark 5.3.17that the constant C ( X, L , H ; K ) for L s is uniformly bounded over s ∈ [0 ,
1] by considering C ( X, L para ; K ) := sup s ∈ [0 , C ( X, L s ; K ) . In particular L (cid:48) ( t ) ∩ X = L ( t ) ∩ X for all t ∈ [0 ,
1] and L (cid:48) (1) ∩ (cid:0) M \ r − (( −∞ , R ]) (cid:1) = L ∩ (cid:0) M \ r − (( −∞ , R ]) (cid:1) . L (cid:48) ( t ) ≡ L ( t ) on the region r − ([ R/ , R ]) and so the strong maximum principle can beapplied which prevents any trajectory associated to L (cid:48) continued from M ( X, L ) from penetratinginto r − ([ R/ , R ]). Furthermore we have X ∩ L ( t ) = X ∩ L (cid:48) ( t )and we may assume this intersection to be contained in the compact region K .Define twisted complexes T = ( X ∩ L , D ) , D = { ev ∗ ([ H ( X, L ; x , x )]) } x ,x ∈ X ∩ L ,T (cid:48) = ( X ∩ L , D ) , D (cid:48) = { ev ∗ ([ H ( X, L ; x (cid:48) , x (cid:48) )]) } x (cid:48) ,x (cid:48) ∈ X ∩ L . and two morphisms between them by S = ev ∗ (cid:88) x ∈ X ∩ L ,x (cid:48) ∈ X ∩ L [ M ( X, L ρ ; x, x (cid:48) )] , S (cid:48) = ev ∗ (cid:88) x ∈ X ∩ L ,x (cid:48) ∈ X ∩ L [ M ( X, L (cid:48) ρ ; x, x (cid:48) )] Here we use the moduli space of solutions of (5.7) with elongated (moving) Lagrangian boundaryand asymptotic boundary conditions.
Proposition 5.3.18. [ S (cid:48) ] = [ S ]in µ C ∗ P ) -cohomology.Assuming this proposition for the moment, we proceed with: Proof of Theorem 5.3.1.
We deform L (cid:48) (1) to L via a compactly supported isotopy L (cid:48) . Since acompactly supported isotopy can be composed with its inverse so that their composition (throughan isotopy of compactly supported isotopies) is isotopic to the constant isotopy (cid:98) L , we obtain (cid:77) x,x (cid:48) ev ∗ ([ M ( X, L (cid:48) L (cid:48) ; x, x (cid:48) )]) ∼ (cid:77) x,x (cid:48) ∈ X ∩ L ev ∗ ([ M ( X, (cid:98) L ; x, x (cid:48) ]) = id ( X ∩ L ,D ) . Thus we have id ( X ∩ L ,D ) ∼ (cid:77) x,x (cid:48) ev ∗ ([ M ( X, L (cid:48) L (cid:48) ; x, x (cid:48) )]) ∼ (cid:77) x,z (cid:77) y ev ∗ ([ M ( X, L (cid:48) ; x, y )]) · ev ∗ ([ M ( X, L (cid:48) ; y, z )]))= (cid:77) x,x (cid:48) ev ∗ ([ M ( X, L (cid:48) ; x, x (cid:48) )]) · (cid:77) z,z (cid:48) ev ∗ ([ M ( X, L (cid:48) ; z, z (cid:48) )]) . So the morphism (cid:16)(cid:76) x,x (cid:48) ev ∗ ([ M ( X, L (cid:48) ; x, x (cid:48) )]) (cid:17) admits a right homotopy inverse. Tracing throughthe same work for L (cid:48) L (cid:48) shows that the morphism admits a left homotopy inverse as well; that is,43he morphism is an equivalence in Tw C ∗ P ( X ). On the other hand, we have (cid:77) x,x (cid:48) ev ∗ ([ M ( X, L (cid:48) ; x, x (cid:48) )]) = (cid:77) x,x (cid:48) ev ∗ ([ M ( X, L (cid:48) ; x, x (cid:48) )]) = ev ∗ (cid:0) [ M ( D − ; L χ , x, x (cid:48) )] (cid:3) = ev ∗ (cid:88) y ∈ L ∩ L ; | y | =0 [ M ( D \ { z } ; L χ , y ) H ( x, y, x (cid:48) )] = ev ∗ (cid:88) y ∈ L ∩ L ; | y | =0 n χ L ( y )[ H ( x, y, x (cid:48) )] = F ( c χ L ) . The first equality follows from the uniform energy bound (depending only on the behavior of L and L (cid:48) inside K ), guaranteeing that the count of continuation maps yields equiavlent maps. The nextequalities are given by Remark 5.3.14, by Lemma 5.3.11, by (5.4), then by (5.2).Combining the above, we have proved F ( c χ L ) is an equivalence. This finishes the proof ofTheorem 5.3.1. Proof of Proposition 5.3.18.
We consider one-parameter family L s (i.e. a homotopy of isotopies)with 0 ≤ s ≤ K but deforms outside r − (( −∞ , R/ M para(0) ( X, { L s } ; x − , x + ) = (cid:97) s ∈ [0 , (cid:48) | x − | = | x + | { s } × M ( X, L s ; x − , x + )for the pairs ( x − , x + ) with | x − | = | x + | with L = L and L = L (cid:48) . Here we introduce the modulispaces M para( k ) ( X, { L s } ; x − , x + )in general where the integer k appearing in the subindex ( k ) of the moduli space stands for thedegree of the relevant operators which is the same as | x − | − | x + | . This is one smaller than thedimension of the relevant parameterized moduli space.We know that: • the relevant Hamiltonians defining L s are only non-linear in a compact region contained in r − (( −∞ , R ]), • there exists a uniform energy bound, and • there is no bubbling,so the boundary of M para(0) ( X, { L s } ; x − , x + ) consists of the types M para( − ( X, { L s } ; x − , y ) H ( X, L ; y, x + ) , H ( X, L ; x − , z ) M para( − ( X, { L s } ; z, x + ) , M para ( X, { L s } ; x − , x + ) | s =0 , M para( − ( X, { L s } ; x − , x + ) | s =1 . ∗ ([ M ( X, L (cid:48) ; x − , x + )]) − ev ∗ ([ M ( X, L ; x − , x + )])= (cid:77) y ∈∈ X ∩ L ; | y | = | x + | +1 ev ∗ (cid:16) M para(0) ( X, { L s } ; x − , y ) H ( X, L ; y, x + ) (cid:17) + (cid:77) z ∈∈ X ∩ L ; | z | = | x − |− ev ∗ (cid:16) H ( X, L ; x − , z ) M para(0) ( X, { L s } ; z, x + ) (cid:17) . (5.19)Now we introduce a collection of moduli spaces M para( − ( X, { L x } ) = { M para ( X, { L s } ; x, x (cid:48) ) } s ∈ [0 , ,x,x (cid:48) ; | x (cid:48) | = | x |− and define a 2-cochain H := (cid:110) ev ∗ (cid:16) [ M para( − ( X, { L s } ; x, x (cid:48) )] (cid:17)(cid:111) s ∈ [0 , ,x,x (cid:48) ; | x (cid:48) | = | x |− . Then (5.19) gives rise to ev ∗ ([ M (0) ( X, L (cid:48) )]) − ev ∗ ([ M (0) ( X, L )])= ev ∗ (cid:16) [ M para( − ( X, { L s } )] (cid:17) · ev ∗ ([ H ( X, L )]) − ev ∗ ([ H ( X, L )]) · ev ∗ (cid:16) [ M para( − ( X, { L s } )] (cid:17) (5.20)which can be rewritten into [ S (cid:48) ] − [ S ] = µ P ) ([ H ]) :By the definition of the µ on the twisted complex, we have µ P ) ( H ) = µ P ([ H ]) + µ P ([ H ] , D ) + µ P ( D , [ H ]) . In the current case, we have µ P ([ H ]) = ∂ ([ H ]) = 0 since H is a finite sum of zero-dimensional chains.This finishes the proof. 45 The equivalence Tw W (cid:39) C ∗ P The main result of this section is
Theorem 6.0.1.
The natural transformation Tw W ◦ D → Tw C ∗ P from Corollary 5.3.3 is a naturalequivalence. That is, for every smooth j : | ∆ n | → B Diff( Q ), the map Tw W D ◦ j → Tw C ∗ P j is anequivalence of A ∞ -categories.The proof requires some preliminary results that verify:1. An isomorphism between two definitions of wrapped Floer cohomology—one using a colimitindexed over a non-negative sequence of wrappings (as in [AS10]) and the other using aHamiltonian quadratic near infinity (as in [Abo12]). This is Proposition 6.1.9.2. That quadratically wrapped Floer cohomology does not change under continuation maps oflinear-near-infinity non-negative Hamiltonians. This is Lemma 6.2.2.3. A compatibility between the non-wrapped Abouzaid map (Proposition 5.2.6) and the quadrat-ically wrapped Abouzaid map from [Abo12] when mediated by the isomorphism from thejust-mentioned Proposition 6.1.9. This compatibility is expressed in Corollary 6.3.2.These ingredients will be mixed in Section 6.4 to give a proof of Theorem 6.0.1.As it turns out, the above ingredients can be proven in large generality, so that is what we willdo; moreover, we only need to verify these ingredients in a single fiber of a Liouville bundle, so inwhat follows, we will fix some Liouville sector M . Remark 6.0.2.
Let us comment on the proof of Theorem 6.0.1, which shows that our familyof wrapped Fukaya categories is equivalent to a family of local system categories. A non-trivialaspect of proving this equivalence is that Abouzaid’s construction in [Abo12] utilized quadratic
Hamiltonians to define wrappings; this is in contrast to the definition of W in the present work(following [GPS17]), which is a result of localizing with respect to non-negative, linear Hamiltoniancontinuation maps. In particular, morphisms of W are not so tractable using pure geometry.Put another way, we must confront the fact that there are multiple definitions of the wrappedFukaya category in the literature.While possible, it is non-trivial to write down the analysis (and in particular, compactnessarguments) to see that one has an A ∞ algebra map between the different versions of wrapped endo-morphisms, and we do not do this. Instead, we formally conclude that such an algebra map existsby the universal property of localizations—i.e., by making use of category theory. The analyticallegwork, via this strategy, is reduced to checking that the underlying map of endomorphism com-plexes is a quasi-isomorphism, which one can do by straightforward arguments invoking the actionfiltration and relating a cofinal sequence of linear wrappings to a single quadratic wrapping.We refer the reader also to Section 2.2 of [Syl19] for a separate approach.When we do apply the general results for our purposes, we will state this application as acorollary, and we will apply our general results in the following setting: Choice 6.0.3 (Choice for proving Theorem 6.0.1.) . For any simplex j : | ∆ ne | → B Diff( Q ) and forany 0 ≤ a ≤ n , we set M = T ∗ Q a to be the fiber above the a the vertex of | ∆ ne | .We also choose a point q a ∈ Q a in the zero section above the a th vertex of | ∆ ne | . Choose also acofinal sequence for the cotangent fiber L = L (0) = T ∗ q a Q a (Definition 2.2.5).We can arrange so that a cotangent fiber T ∗ q a Q a and all its cofinal wrappings are transverse to Q a and have only a single intersection point with Q a , so that the natural transformation inducedby the Abouzaid map sends T ∗ q a Q a to the object q a ∈ P j . We will assume so.46elow, N A ∞ referes to the A ∞ -nerve of an A ∞ category. (See Recollection 3.1.1(5).) Lemma 6.0.4.
Fix a Liouville sector M and a cofinal sequence of wrappings for a brane L = L (0) (Definition 2.2.5). Then the continuation maps induce a functor of ∞ -categories Z ≥ → N A ∞ ( O ( M )) as follows: L (0) → L (1) → . . . . Proof of Lemma 6.0.4.
The continuation maps determine, for every i ∈ Z ≥ , a morphism in O ( M )from the brane L ( i ) to the brane L ( i +1) ; in particular, for each i we have an edge in N A ∞ ( O j ). Bythe weak Kan property of ∞ -categories, and because Z ≥ is a poset, this sequence of edges lifts toa unique (up to contractible choice) functor from the ∞ -category Z ≥ to N A ∞ ( O ( M )). We have already shown that the colimit of a cofinal sequence of Floer cohomologies computes thehom-complex of the wrapped category W (Lemma 4.4.2). In this section, we show that this colimitalso computes the quadratically wrapped Floer cohomology (Proposition 6.1.9).We first set some notation. Notation 6.1.1 ( CF ∗ ( L, L (cid:48) ; H ) , HF ∗ ( L, L (cid:48) ; H )) . Let M be a Liouville sector and let L, L (cid:48) ⊂ M be branes. Fix a smooth function H : M → R . We denote the set of Hamiltonian chords of H from L to L (cid:48) Chord(
L, L (cid:48) ; H ) = { γ ∈ P ( L, L (cid:48) ) | γ (0) ∈ L, γ (1) ∈ L (cid:48) } and the associated the Floer cochain complex by CF ∗ ( L, L (cid:48) ; H )whose differential is defined by considering the perturbed equation ∂ J,H ( u ) = 0, i.e., (cid:40) ∂u∂τ + J (cid:0) ∂u∂t − X H ( t, u ) (cid:1) = 0 u ( τ, ∈ L, u ( τ, ∈ L (cid:48) . (6.1)The cohomology of this complex (i.e., the Floer cohomology) will be denoted HF ∗ ( L, L (cid:48) ; H ) . Remark 6.1.2.
We recall that there exists a natural chain isomorphism CF ( L, L (cid:48) ; H ) → CF ( L, φ H ( L (cid:48) ); 0)induced by the correspondence γ ∈ P ( L, L (cid:48) ) (cid:55)→ (cid:101) γ ∈ P ( L, φ H ( L (cid:48) )) , (cid:101) γ ( t ) := φ tH ( γ ( t )) , and J t (cid:55)→ (cid:101) J t defined by (cid:101) J t = ( φ tH ) ∗ J t which transforms the equation ∂ J,H ( u ) = 0 to ∂ (cid:101) J, ( v ) = 0 , v ( τ, t ) = φ tH ( u ( τ, t ))) . .1.1 HamiltoniansDefinition 6.1.3 (Quadratic near infinity) . Let M be a Liouville domain. A smooth function H : M → R is called quadratic near infinity if H = 12 r outside a compact subset of M . (Here, we are using the coordinate r from Notation 2.0.1.) Remark 6.1.4.
Our wish is to compare CF ( L, L ; H )—with H quadratic near infinity—to a colimitof Floer complexes constructed from a cofinal sequence L (0) → L (1) → . . . .Because each L ( w ) in this sequence is conical near infinity, we may assume that each is the resultof a Hamiltonian isotopy by a linear-near-infinity Hamiltonian F ( w ) : M → R . In what follows,we will choose a particular model for such a sequence of Hamiltonians—namely, we will set F ( w ) to be obtained by increasing the slopes of a standard linear-near-infinity Hamiltonian (see (6.2)).The reader may make the necessary adjustments to the following proofs for a more general cofinalsequence of linear-near-infinity Hamiltonians. (For example, by altering (6.8) to interpolate H witha given F ( w ) , rather than with the sequence of F s we choose in (6.2).) Notation 6.1.5 ( H and F ) . In this section, we will use the symbol H to denote an autonomousHamiltonian that is quadratic near infinity (Definition 6.1.3).We will use the symbol F to denote a Hamiltonian that is autonomous and is linear near infinity,i.e., F = ar + b outside a compact subset for some constant a, b with a >
0. Note that φ vF ( L ) = φ vF ( L ) (6.2)is still linear near infinity, and in particular outside { r ≤ R K } ⊃ K for any R K large enough. Notation 6.1.6 ( HF ∗ quad ) . Given a quadratic-near-infinity Hamiltonian H , we define the notation HF ∗ quad ( L, L (cid:48) ) = HF ∗ ( L, φ H ( L (cid:48) ); 0) . Note that the dependence on H is suppressed on the left-hand side. (See also Notation 6.1.1 forthe right-hand side.) Lemma 6.1.7.
Fix a Liouville embedding ι : [0 , ∞ ) × ∂ ∞ M → M and set r = e s as in Nota-tion 2.0.1. Let H : M → R be quadratic near infinity (Definition 6.1.3). Then there exists someconstant C = C ( ι, H ) > H − θ ( X H ) ≥ − C. (6.3) Proof.
Outside a large compact subset of M , we have θ ( X H ) = θ ( rX r ) = rdr ( − J X r ) = rdr (cid:18) ∂∂r (cid:19) = r so H − θ ( X H ) = r − r > r >
2. The lemma follows because the (closure of) thecomplement of the image of ι is compact. Definition 2.2.5. otation 6.1.8 (The chain maps φ ) . Consider the interpolating homotopy(1 − s ) vF + H, s ∈ [0 , C and energyestimates (Recollection 3.3.1(1)), we have induced chain maps φ vv (cid:48) : CF ( L, L (cid:48) ; vF ) → CF ( L, L (cid:48) ; v (cid:48) F )for all v < v (cid:48) and φ v : CF ( L, L (cid:48) ; vF ) → CF ( L, L (cid:48) ; H )for any v ∈ Z + . (See Notation 6.1.1.)Passing to cohomology, the maps φ from Notation 6.1.8 induce a commutative diagram HF ( L, L (cid:48) ; F ) (cid:47) (cid:47) (cid:15) (cid:15) HF ( L, L (cid:48) ; 2 F ) (cid:47) (cid:47) (cid:15) (cid:15) · · · (cid:47) (cid:47) HF ( L, L (cid:48) ; vF ) (cid:47) (cid:47) (cid:15) (cid:15) · · · HF ( L, L (cid:48) ; H ) (cid:47) (cid:47) HF ( L, L (cid:48) ; H ) (cid:47) (cid:47) · · · (cid:47) (cid:47) HF ( L, L (cid:48) ; H ) · · · (6.4)and hence a homomorphism ϕ ∞ : colim v →∞ HF ∗ ( L, L (cid:48) ; vF ) → HF ∗ ( L, L (cid:48) ; H ) ∼ = HF ∗ quad ( L, L (cid:48) ) . (The last isomorphism is Remark 6.1.2.)The main result we seek to prove now is: Proposition 6.1.9. ϕ ∞ is an isomorphism.Before proving Proposition 6.1.9, let us state the following consequence, which one obtains bysetting L = L (cid:48) ⊂ T ∗ Q to be a cotangent fiber to a point a ∈ Q . Corollary 6.1.10.
In the setting of Choice 6.0.3, the induced mapcolim w H ∗ hom O j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) → HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q a )is an isomorphism. Proof.
Observe that hom O j ( L, ( L (cid:48) ) ( w ) ) is naturally isomorphic to CF ∗ ( L, L (cid:48) ; wF ) when one choosesthe cofinal sequence to be given by wF . (See Remark 6.1.2 and Remark 6.1.4.) One then observesthat given two cofinal sequences of wrappings ( L (cid:48) ) ( w ) and ( L (cid:48) ) ( v ) , there is a natural category whoseobject set is given by the union { L ( w ) } w ∪ { L ( v ) } v , and a morphism is given by an isotopy class ofa non-negative Hamiltonian isotopy between two objects. Each of the original cofinal sequences iscofinal in this category, so the colimits of the Floer cohomology groups agree. Before proving Proposition 6.1.9, we set some notation.
Notation 6.1.11 (Action and energy) . The holomorphic strip equation (6.1) may be interpretedas a gradient flow equation of the action functional A ( γ ) = A L,L (cid:48) ( γ ) = − (cid:90) γ ∗ θ + (cid:90) H ( γ ( t )) dt + h L (cid:48) ( γ (1)) − h L ( γ (0)) (6.5)where h L : L → R is a function satisfying θ | L = dh L . We have the basic energy identity E ( J,H ) ( u ) = A ( u ( ∞ )) − A ( u ( −∞ )) . otation 6.1.12. Each chain of CF ( L (cid:48) , L ; H ) is a linear combination α = (cid:88) a z (cid:104) z (cid:105) ∈ CF ( L, L (cid:48) ; H ) , z ∈ Chord(
L, L (cid:48) ; H ) (6.6)where each z is a Hamiltonian chord from L to L . We denotesupp α = { z ∈ Chord(
L, L (cid:48) ; H ) | a z (cid:54) = 0 in (6.6) } and define the action (cid:96) ( α ) of the cycle by (cid:96) ( α ) = max { A ( z ) | z ∈ supp α } . Notation 6.1.13 (Action filtration) . In terms of this action, we have an increasing filtration CF ≤ c ( L, L (cid:48) ; H ) := { α ⊂ CF ( L, L (cid:48) ; H ) | (cid:96) ( α ) ≤ c } with CF ≤ c ( L, L (cid:48) ; H ) ⊂ CF ≤ c (cid:48) ( L, L (cid:48) ; H ) for c < c (cid:48) . Each CF ≤ c ( L, L (cid:48) ; H ) is a subcomplex of CF ( L, L (cid:48) ; H ). By definition, we have CF ( L, L (cid:48) ; H ) = colim c CF ≤ c ( L, L (cid:48) ; H ) . (6.7) We start with surjectivity. Let a ∈ HF ( L, L (cid:48) ; H ) and choose a cycle α representing it, i.e., µ ( α ) =0. Denote c = (cid:96) ( α ) (Notation 6.1.12). Now we consider the following linear adjustment of H : H ( v ) ( x ) = (cid:40) H ( x ) if r ( x ) ≤ v ( v + 1) r ( x ) − ( v + 1) if r ( x ) ≥ v + 1with suitable smooth interpolation in between. We fix a sequence of integers { v k } k ∈ N diverging to ∞ and consider the sequence H ( v k ) of linear near infinity Hamiltonians. We note that { H ( v k ) } k ∈ N is a monotone sequence of Hamiltonians converging to H uniformly on M cpt ∪ { r ≤ R } for anylarge R > ϕ ∞ . Lemma 6.1.14.
There exists b ∈ HF ( L, L (cid:48) ; H ( v k ) ) such that a = ϕ ∞ ( b ) . Proof.
Let α = (cid:80) ki =1 a i (cid:104) z i (cid:105) be a cycle representing a with z i ∈ Chord(
L, L (cid:48) ; H ).Clearly z i ∈ Chord(
L, L (cid:48) ; H ( v k ) ) as long as v k (cid:96) ( α ) so large thatsupp α ⊂ H − (( −∞ , v k ])where we abuse the notation supp α by also denoting it as ∪ z ∈ supp α image z ⊂ M . Therefore α canbe regarded as a chain in CF ( L, L (cid:48) ; H ( v k ) ). Denote the resulting chain of H ( v k ) by β k . We nowprove β k is a cycle of H ( v k ) , if we choose k even larger if necessary.To avoid confusion, we denote by µ H and µ H ( vk ) be the µ -map for H and H ( v k ) respectively.Then standing hypothesis is µ H ( α ) = 0 and we want to prove µ H ( vk ) ( β k ) = 0 This is because we put the output of the differential at τ = −∞ , not at τ = ∞ .
50y choosing a larger k if necessary. By definition, we have µ H ( α ) = k (cid:88) i =1 a i µ H ( (cid:104) z i (cid:105) )where µ H ( (cid:104) z i (cid:105) ) = (cid:88) y ∈ Chord(
L,L (cid:48) ; H ) n ( J,H ) ( z i , y ) (cid:104) y (cid:105) with n ( J,H ) ( z i , y ) = M ( z i , y ; J, H ) where M ( z i , y ; J, H ) is the moduli space of solutions u of (6.1)satisfying u ( −∞ ) = z i , u ( ∞ ) = y . We rearrange the sum into µ H ( α ) = (cid:88) y ∈ Chord(
L,L (cid:48) ; H ) (cid:32) k (cid:88) i =1 a i n ( J,H ) ( z i , y ) (cid:33) (cid:104) y (cid:105) . Therefore µ H ( α ) = 0 is equivalent to k (cid:88) i =1 a i n ( J,H ) ( z i , y ) = 0for all y ∈ Chord(
L, L (cid:48) ; H ).The same formula with H replaced by H ( v k ) holds and so µ H ( vk ) ( β k ) = (cid:88) y ∈ Chord(
L,L (cid:48) ; H ( vk ) ) (cid:32) k (cid:88) i =1 a i n ( J,H ( vk ) ) ( z i , y ) (cid:33) (cid:104) y (cid:105) . Therefore it remains to prove k (cid:88) i =1 a i n ( J,H ( vk ) ) ( z i , y ) = 0 (6.8)for all y ∈ Chord(
L, L (cid:48) ; H ( v k ) ) by choosing k sufficiently large. This will follow if we establish M ( z i , y ; J, H ( v k ) ) = (cid:40) M ( z i , y ; J, H ) if M ( z i , y ; J, H ) (cid:54) = ∅∅ if M ( z i , y ; J, H ) = ∅ (6.9)for all i and y . Sublemma 6.1.15.
There are finitely many y ∈ CF ( L, L (cid:48) ; H ) such that M ( z i , y ; J, H ) (cid:54) = ∅ forsome i = 1 , . . . , k . Proof.
By the energy identity, we have A ( y ) ≤ (cid:96) ( α ) . On the other hand, for any Hamiltonian chord y of H , we derive A ( y ) = − (cid:90) y ∗ θ + (cid:90) H ( y ( t )) dt = (cid:90) H ( y ( t )) − θ ( X H ( y ( t ))) dt > (cid:90) ( − C ) dt = − C
51y Lemma 6.1.7. Therefore under the given hypothesis, we have − C < A ( y ) ≤ (cid:96) ( α ) . By the nondegeneracy assumption on H , this finishes the proof.Set R = max y { r ( y ) | y is as in the above sublemma } . Then it follows from the main C estimate in [OT20b] that there exists a sufficiently large k suchthat max r ◦ u ≤ R + C (cid:48) for u ∈ M ( z i , y ; J, H ) where C (cid:48) depends only on inf H > −∞ .The same discussion still applies to H ( v k ) since we still have H ( v k ) − θ (cid:16) X H ( vk ) (cid:17) ≥ − C and max r ◦ u ≤ R + C (cid:48) for u ∈ M ( z i , y ; J, H ( v k ) for the same constant C, C (cid:48) above respectively. Combining the two, wehave proved (6.9) and so µ H ( vk ) ( β k ) = 0.Next we would like to prove [ φ v k ( β k )] = [ α ] = a. For this we have only to know that (1 − s ) H ( v k ) + sH ≡ H on r − ( −∞ , R + C (cid:48) ]) and the same C -estimate as [OT20b] applies for the continuation equation for H = { H s = (1 − s ) H ( v k ) + sH } . This implies that any solution u of continuation equation satisfies(6.1) provided we choose v k sufficiently large so that H s ≡ H for all s ∈ [0 ,
1] on M \ ι ([ R + C (cid:48) , ∞ ).This in fact implies φ v k ( β k ) = α in chain level and hence proves [ φ v k ( β k )] = [ α ].This finishes the proof of surjectivity.For the proof of injectivity, let β k be a sequence of H ( v k ) -cycle such that [ β k +1 ] = [ φ k ( k +1) ( β k )]and [ φ v k ( β k )] = 0 in HF ( L, L (cid:48) ; H ) for all k ≥ k with k sufficiently large. Then φ v k ( β k ) = µ H ( α (cid:48) k )for some H -chain α (cid:48) k or each k ≥ k . Denote λ = (cid:96) ( α (cid:48) k ). Under this hypothesis, by the similarargument given in the surjectivity proof, we can find a sufficiently large (cid:96) = (cid:96) ( k , λ ) > k suchthat φ v k v (cid:96) ( β k ) = µ H ( v(cid:96) ) ( α (cid:48) (cid:96) ) . (6.10)Now we consider a conformally symplectic dilation f : M → M defined by the Liouville flow fortime log( ρ ) with ρ = v (cid:96) v k which becomes f ( x ) = ( ρr, y )52or x = ( r, y ) ∈ M end; ι . The isotopy t (cid:55)→ f ◦ φ tH ( v(cid:96) ) ◦ f − is still a Hamiltonian isotopy generated bythe Hamiltonian v k v (cid:96) H ( v (cid:96) ) ◦ f =: G k (cid:96) We note that v k v (cid:96) H ( v (cid:96) ) ◦ f ( r, y ) = v k r for any x = ( r, y ) such that H ( v (cid:96) ) ( r, y ) = v (cid:96) r . Therefore wecan find a chain isomorphism η ∗ : CF ( L, L (cid:48) ; G k (cid:96) ) → CF ( L, L (cid:48) ; H ( v k ) )associated to the isotopy η : s (cid:55)→ φ − sG k (cid:96) ◦ φ sH ( vk which is compactly supported.We then define a map ψ k (cid:96) : CF ( L, L (cid:48) ; H ( v (cid:96) ) ) → CF ( L, L (cid:48) ; H ( v k ) )as the composition ψ k (cid:96) = ( η ) ∗ ◦ f ∗ which is a quasi-isomorphism. Furthermore we also have Sublemma 6.1.16.
The map ψ k (cid:96) ◦ φ v k v (cid:96) = ( η ) ∗ ◦ f ∗ ◦ φ v k v (cid:96) is chain homotopic to id on CF ( L, L (cid:48) ; H ( v k ) ). Proof.
We have only to notice that the isotopy g s := η (1 − s ) ◦ f − s ◦ ( φ − sH ( vk φ sH ( v(cid:96) ) ) : M → M and g = η ◦ f ◦ φ H ( vk and g = id. In particular, we obtain ψ k (cid:96) ◦ φ v k v (cid:96) − id : CF ( L, L (cid:48) ; H ( v k ) ) → CF ( L, L (cid:48) ; H ( v k ) )is chain homotopic to 0. This finishes the proof.Now we apply the map ψ k (cid:96) to (6.10) and get ψ k (cid:96) ◦ φ v k v (cid:96) ( β k ) = ψ k (cid:96) ◦ µ H ( v(cid:96) ) ( α (cid:48) (cid:96) ) . The left-hand side can be written as ψ k (cid:96) ◦ φ v k v (cid:96) ( β k ) = β k + µ H ( vk ( γ )for some chain γ of H ( v k ) and the right-hand side coincides with µ H ( vk ◦ ψ k (cid:96) ( α (cid:48) (cid:96) )by the chain property of ψ k (cid:96) . Combining the two, we have derived β k = µ H ( vk ◦ ψ k (cid:96) ( α (cid:48) (cid:96) ) − µ H ( vk ( γ ) = µ H ( vk (cid:0) ψ k (cid:96) ( α (cid:48) (cid:96) ) − γ (cid:1) . This proves [ β k ] = 0. By the compatibility of the sequence [ β k ], this proves lim k [ β k ] = 0 and hencethe injectivity of the map ϕ ∞ . This finishes the proof of the proposition.53 .2 Positive wrappings leave HF quad unchanged Notation 6.2.1 ( H F ) . Given two (possibly time-dependent) Hamiltonians H and F , we definea time-dependent Hamiltonian H F : R × M → R by H F ( t, x ) = H ( t, x ) + wF ( φ tH ( x )) . Lemma 6.2.2.
Suppose that ( L (cid:48) ) ( w ) is obtained from L (cid:48) by a non-negative Hamiltonian isotopy,and let H be a quadratic-near-infinity Hamiltonian. Then the continuation map HF ( L, L (cid:48) : H ) → HF ( L, ( L (cid:48) ) ( w ) ; H )is an isomorphism. Proof.
Let F be a linear-near-infinity Hamiltonian inducing a cofinal sequence of nonnegative iso-topies ( L (cid:48) ) ( v ) . Choose a linear-near-infinity Hamiltonian G whose Hamiltonian flow realizes theisotopy from L (cid:48) to ( L (cid:48) ) ( w ) . We have a commutative diagram of Floer cohomology groups HF ( L, L (cid:48) ; vF ) (cid:47) (cid:47) (cid:15) (cid:15) . . . → HF ( L, L (cid:48) ; v (cid:48) F ) → . . . (cid:47) (cid:47) (cid:15) (cid:15) HF ( L, L (cid:48) ; H ) (cid:15) (cid:15) HF ( L, L (cid:48) ; vF G ) (cid:47) (cid:47) . . . → HF ( L, L (cid:48) ; v (cid:48) F G ) → . . . (cid:47) (cid:47) HF ( L, L (cid:48) ; H G ) (6.11)where arrows are given by continuation maps, and v < v (cid:48) . Moreover, the sequences { vF } v and { vF G } v are both cofinal in the spliced diagram of Hamiltonians { vF } v ∪ { vF G } v . Thus the colimits colim v →∞ HF ( L, L (cid:48) ; vF ) , colim v →∞ HF ( L, L (cid:48) ; vF G )are equivalent. On the other hand, the top row and the bottom row of (6.11) are colimit diagramsby Proposition 6.1.9. Thus the rightmost vertical arrow of (6.11) is an isomorphism.On the other hand, we have the obvious isomorphism HF ( L, ( L (cid:48) ) ( w ) ; H ) ∼ = HF ( L, L (cid:48) ; H G ) . This completes the proof.Now suppose that L = L (cid:48) is a cotangent fiber of T ∗ Q at a point a ∈ Q , and choose a cofinalsequence for L : Corollary 6.2.3.
In the setting of Choice 6.0.3, we havecolim w HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) ∼ = HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q a ) . For later reference, we record explicitly the following fact, which follows from the commutativityof (6.11):
Lemma 6.2.4.
In the setting of Choice 6.0.3, the diagram of cohomology groups H ∗ hom O j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) (cid:15) (cid:15) (cid:47) (cid:47) HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) (cid:15) (cid:15) H ∗ hom O j ( T ∗ q a Q a , T ∗ q a Q ( w +1) a ) (cid:47) (cid:47) HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q ( w +1) a )is commutative. Here, all arrows are induced by continuation maps.54 .3 Comparing the non-wrapped Abouzaid map to the quadratically wrappedAbouzaid map For this section, we let | ∆ n | = | ∆ | , so that the Liouville bundle E → | ∆ | is simply a choice ofLiouville sector M . In the following lemma, we thus drop the j variable: Lemma 6.3.1.
Let M be a Liouville sector and Q ⊂ M a compact exact brane. Fix objects X, L ( w ) ∈ Ob O ( M ). The Abouzaid functor defines an object of Tw C ∗ P ( Q ) by (5.1) from anyobject L of O ( M ), and we abbreviate this object as L ∩ Q. Then the diagram H ∗ hom Tw C ∗ P ( Q ) ( Q ∩ X, Q ∩ L ( w ) ) H ∗ hom O ( X, L ( w ) ) Cor . . (cid:47) (cid:47) P rop . . (cid:52) (cid:52) HF ∗ quad ( X, L ( w ) ) [ Abo12 ] (cid:79) (cid:79) commutes. Here, the horizontal map is a continuation map, while the two other maps are thenon-wrapped (Proposition 5.2.6) and wrapped (Section 4 of [Abo12]) versions of the Abouzaidmap.Moreover, at the level of cohomology, these diagrams are compatible with the filtered diagramfrom Lemma 6.0.4. Proof of Lemma 6.3.1.
Because we are working at the level of cohomology, we may model theisotopy from L to L ( w ) by some linear-near-infinity Hamiltonian F , and choose the quadraticHamiltonian H in such a way that H = F on the region where L and L ( w ) intersect.Choosing a non-negative interpolating isotopy from F to H identifies H ∗ hom O j ( L, ( L (cid:48) ) ( w ) ) = HF ∗ ( L, L (cid:48) ; F ) → HF ∗ quad ( L, ( L (cid:48) ) ( w ) )as a subcomplex of chords with action bounded by some A .Both the non-wrapped and wrapped maps count holomorphic triangles one of whose verticeslimit to Hamiltonian chords, and we find that the non-wrapped map counts precisely such trianglesrestricted to the subcomplex of chords with action less than or equal to A . This shows that thediagram commutes.That the diagram is natural in w follows straightforwardly by choosing the quadratic Hamil-tonian to be equal to the Hamiltonian defining L ( w ) and L ( w (cid:48) ) . (Note that while these choicescertainly affect the chain-level maps, we may choose these Hamiltonians to have no effect at thelevel of cohomology.)Now we set M = T ∗ Q = T ∗ Q a and set L = L (cid:48) to be a cotangent fiber at some point q a ⊂ Q a . Corollary 6.3.2.
In the setting of Choice 6.0.3, for every w , the diagram H ∗ hom C ∗ P j ( q a , q ( w ) a ) H ∗ hom O j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) Cor 6.1.10 (cid:47) (cid:47)
Prop 5.2.6 (cid:51) (cid:51) HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) [ Abo12 ] (cid:79) (cid:79) q ( w ) a is the intersection point of T ∗ q a Q ( w ) a with the zero section. (We have chosen our cofinal wrapping sothat there is only one intersection point.)Moreover, at the level of cohomology, these diagrams are compatible with the filtered diagramfrom Lemma 6.0.4. Proof of Theorem 6.0.1.
Because the cotangent fiber T ∗ q a Q a and the object q a ∈ Q a generate the A ∞ -categories in which they reside, all that remains is to show that the map on endomorphismcomplexes of these objects—induced by Corollary 5.3.3—is a quasi-isomorphism.We first claim the diagram below commutes: H ∗ hom W j ( T ∗ q a Q a , T ∗ q a Q a ) Cor 5 . . (cid:47) (cid:47) H ∗ hom C ∗ P j ( q a , q a )colim w H ∗ hom O j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) Cor 6 . . ∼ = (cid:47) (cid:47) Lem 4 . . ∼ = (cid:79) (cid:79) Prop 5 . . (cid:51) (cid:51) HF ∗ quad ( T ∗ q a Q a , T ∗ q a Q a ) ∼ = [ Abo12 ] (cid:79) (cid:79) (6.12)Let us explain the diagram.We first note that, for a fixed w , there is a homotopy commutative diagram of chain complexeshom W j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) (cid:47) (cid:47) hom C ∗ P j ( q a , q ( w ) a )hom O j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) (cid:53) (cid:53) (cid:79) (cid:79) where the diagonal map is the Abouzaid construction for O (Proposition 5.2.6), and the upper,dashed, horizontal map is induced by the universal property of localization (Corollary 5.3.3). Here, q ( w ) a is the intersection point of T ∗ q a Q ( w ) a with the zero section. (We have chosen our cofinal wrappingso that there is only one intersection point.)Moreover, there is a homotopy coherent functor Z ≥ → O j by Lemma 6.0.4, so the lower leftcorner of the triangle coheres into a homotopy-coherent sequential diagram indexed by w . Sincewe have functors O j → W j and O j → C ∗ P j , we have an induced homotopy-coherent diagram ofthe colimits: colim w hom W j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) (cid:47) (cid:47) colim w hom C ∗ P j ( q a , q ( w ) a )colim w hom O j ( T ∗ q a Q a , T ∗ q a Q ( w ) a ) (cid:52) (cid:52) (cid:79) (cid:79) We note that the two sequential colimits in the top horizontal line is a colimit of isomorphismsupon passage to cohomology—this is because continuation maps are sent to equivalences in W j (by definition of localization) and in C ∗ P j (by Theorem 5.3.1). Thus, for both items in the tophorizontal line, the cohomology of the colimit is isomorphic to the cohomology of the w = 0 term.This explains the upper-left triangle in (6.12). For Tw C ∗ P j this is obvious, while for Tw W j , this follows from Abouzaid’s Theorem and Proposition 4.5.3. w -indexed colimit (at the level ofcohomology) to the triangle in Corollary 6.3.2. We observe that the filtered colimits on the rightvertical edge consists of maps that are all equivalences, so in this way we may identifycolim w H ∗ hom C ∗ P j ( q a , q ( w ) a ) ∼ = H ∗ hom C ∗ P j ( q a , q a ) , and the lower-right corner of the triangle arises by using Corollary 6.2.3. This completes theexplanation of the commutative diagram (6.12).Referring again to (6.12), note that the left-hand vertical arrow was verified to be an isomor-phism at the level of cohomology in Lemma 4.4.2, the bottom horizontal arrow was verified to be anisomorphism in Corollary 6.1.10, and the right-hand vertical arrow is an isomorphism by [Abo11].Because these ∼ =-labeled arrows are isomorphisms, it follows that the top horizontal arrow is alsoan isomorphism, which is what we sought to prove.57 The diffeomorphism action on Loc
Notation 7.0.1 (Diff) . Fix Q an oriented, compact manifold. We let Diff( Q ) denote the topologicalgroup of orientation-preserving diffeomorphisms of Q .The goal of this section is to prove that the natural action of Diff( Q ) on Loc ( Q ) is compatiblewith the action of Aut gr,b ( T ∗ Q ) on W ( T ∗ Q ) from Theorem 1.0.1.Given the results of our previous section, the only thing left to do is to verify that the diffeomor-phism action on Tw C ∗ P is the standard action of the diffeomorphism group on the A ∞ -category oflocal systems. This is proven in Proposition 7.1.13. C ∗ P is compatible with the diffeomorphism action Our eventual goal is to prove that the (orientation-preserving) diffeomorphism group action on the ∞ -category of local systems is compatible with its action on the wrapped Fukaya category of acotangent bundle. So first let us show that our construction C ∗ P encodes the usual action on the ∞ -category of local systems.For this, recall that the ∞ -category of local systems on a space B with values in an ∞ -category D is equivalent to the ∞ -category Fun ( Sing ( B ) , D )of functors from Sing ( B ) to D . The evident action of hAut( B ) on B —and hence on Sing ( B )—exhibits the action of hAut( B ) on the ∞ -category of local systems.On the other hand, our construction of C ∗ P passes through a combinatorial trick that replaces B Diff( Q ) by a category of simplices in B Diff( Q ), which one can informally think of as the categoryencoding the barycentric subdivision of Sing ( B Diff( Q )). We must show that this combinatorialtrick allows us to recover the natural action of Diff( Q ) on Q ; this is the content of Corollary 7.1.4below. Construction 7.1.1.
Let p : E → B be a Kan fibration of simplicial sets. We let subdiv ( B ) denotethe subdivision simplicial set associated to B (Recollection 3.2.3(3)).We have an induced functor p − : subdiv ( B ) → K an , ( j : ∆ k → B ) (cid:55)→ p − ( j )to the ∞ -category of Kan complexes. Indeed, realizing subdiv ( B ) to be the nerve of a category, theabove is induced by an actual functor to the category of simplicial sets, sending an object j to thesimplicial set j ∗ E . Remark 7.1.2.
Moreover, because p is a Kan fibration, every edge in subdiv ( B ) is sent to anequivalence in K an ; thus the above functor factors through the localization of subdiv ( B ). More-over, we know the localization to be equivalent as an ∞ -category to the Kan complex B (Recollec-tion 3.2.3(3)). We draw this factorization as follows: subdiv ( B ) p − (cid:47) (cid:47) (cid:36) (cid:36) K an B F (cid:61) (cid:61) That is, p − induces some functor F : B → K an of ∞ -categories.58n the other hand, the Kan fibration p : E → B classifies a functor of ∞ -categories from B to K an by the straightening/unstraightening correspondence. (See 3.2 of [Lur09].) Our main goal isto prove: Lemma 7.1.3.
The functor classified by p : E → B admits a natural equivalence to the functor F (induced by p − in Construction 7.1.1).Given the lemma, we have Corollary 7.1.4.
Let EQ be the tautological Q bundle over B Diff( Q ). We then have a Kanfibration p : Sing ( EQ ) → Sing ( B Diff( Q )), and the induced functor N ( Simp ( B Diff( Q ))) → K an , ( j : | ∆ a | → B ) (cid:55)→ Sing ( j ∗ EQ ) . Consider the induced functor F from Remark 7.1.2: N ( Simp ( B Diff( Q ))) (cid:47) (cid:47) (cid:41) (cid:41) K anSing ( B Diff( Q )) . F (cid:55) (cid:55) Then F is naturally equivalent to the functor sending a distinguished vertex of Sing ( B Diff( Q )) to Sing ( Q ), and exhibiting the action of Sing (Diff( Q )) on Sing ( Q ). Proof of Corollary 7.1.4.
The fibration EQ → B Diff( Q ) classifies the functor Sing ( B Diff( Q )) → K an exhibiting the Diff( Q ) action on Q . Now apply Lemma 7.1.3.We need to recall a few tools before proving the lemma. Recollection 7.1.5 (Relative nerve) . Fix a functor f : C → sSet from a category C to the categoryof simplicial sets. Then one can construct a coCartesian fibration N f ( C ) → N ( C ) called the relativenerve of f . (See Section 3.2.5 of [Lur09].) N f ( C ) is a simplicial set defined as follows: For any finite, non-empty linear order I , an elementof N f ( C )( I ) is given by the data of: • A simplex φ : ∆ I → C (where ∆ I ∼ = ∆ n for n = | I | − • For every subset I (cid:48) ⊂ I , setting i (cid:48) = max I (cid:48) , a simplex τ I (cid:48) : ∆ I (cid:48) → f ( φ ( i (cid:48) )).These data must satisfying the following condition: • For any I (cid:48) ⊂ I (cid:48)(cid:48) , the diagram of simplicial sets∆ I (cid:48) τ I (cid:48) (cid:47) (cid:47) (cid:15) (cid:15) f ( φ ( i (cid:48) )) (cid:15) (cid:15) ∆ I (cid:48)(cid:48) τ I (cid:48)(cid:48) (cid:47) (cid:47) f ( φ ( i (cid:48)(cid:48) ))must commute. 59 xample 7.1.6. Let us parse what the relative nerve N p − ( subdiv ( B )) is, where p − is the functorfrom Construction 7.1.1. An n -simplex in N p − ( subdiv ( B )) is the data of • A collection of inclusions of simplices∆ a (cid:44) → ∆ a (cid:44) → . . . (cid:44) → ∆ a n together with a map j : ∆ a n → B , and • A map τ : ∆ n → j ∗ E , such that • For every i ∈ { , . . . , n } , the i th vertex of τ [ n ] must be a vertex in j ∗ E | ∆ ai .Because p : E → B is assumed to be a Kan fibration, it follows that the forgetful map N f ( subdiv ( B )) → subdiv ( B ) is a coCartesian fibration (in fact, a left fibration)–see Proposition 3.2.5.21 of [Lur09]. Remark 7.1.7.
Moreover, it is proven in [Lur09] that the fibration N f ( C ), when straightened,classifies a functor naturally equivalent to f . See again Proposition 3.2.5.21 of [Lur09]. Notation 7.1.8.
On the other hand, note that we have a natural mapmax : subdiv ( B ) → B for any simplicial set B . On vertices, it sends j : ∆ a → B to the vertex j (max[ a ]) ∈ B . Thisinduces the obvious map on higher simplices.Thus we have, for any Kan fibration E → B , the pulled back Kan fibration max ∗ E → subdiv ( B )as follows: max ∗ E (cid:47) (cid:47) (cid:15) (cid:15) E p (cid:15) (cid:15) subdiv B max (cid:47) (cid:47) B Proof of Lemma 7.1.3 .
We have the map of coCartesian fibrations ∗ max E → N p − ( subdiv ( B ))which, on the fiber above j : ∆ a → B , includes the simplicial set of all maps whose τ lands inthe fiber above j (max[ a ]). This is obviously a weak homotopy equivalence along the fibers because E → B is a Kan fibration. Thus we have a diagram N p − ( subdiv ( B )) (cid:15) (cid:15) max ∗ E (cid:47) (cid:47) (cid:111) (cid:111) (cid:15) (cid:15) E p (cid:15) (cid:15) subdiv ( B ) subdiv ( B ) max (cid:47) (cid:47) B where each horizontal arrow is a weak homotopy equivalence (i.e., an equivalence in the modelstructure for Kan complexes). Thus every square in this diagram—upon passage to Kan com-plexes, i.e., their localizations—exhibits an equivalence of Kan fibrations. Because the straighten-ing/unstraightening construction sends equivalences of fibrations to natural equivalences of functors,the result follows.Now we make use of the Quillen adjunction employing Lurie’s dg nerve construction. Seealso [BD19]. 60 otation 7.1.9 (Nerve and its adjoint) . Fix a base ring R . Consider the adjunction R [ − ] : sSet ⇐⇒ dgCat : N dg . Here, N dg is Lurie’s dg nerve. (See Construction 1.3.1.6 of [Lur12].) It is a functor sending anydg category to an ∞ -category, and any dg-functor to a map of simplicial sets. We denote its leftadjoint by R [ − ]. Remark 7.1.10.
The adjunction of Notation 7.1.9 can be promoted to a Quillen adjunction. TheQuillen adjunction is with respect to the Joyal model structure for simplicial sets, and the Tabuadamodel structure for dg-categories.Note that these two model structures have simplicial localizations equivalent to C at ∞ and C at A ∞ , respectively. For the fact that the model structure on dg-categories recovers the ∞ -categoryof A ∞ -categories, see [OT20a]. Remark 7.1.11.
When C is an ∞ -groupoid—for example, Sing ( B ) for some topological space B —then R [ C ] is equivalent to the dg-category C ∗ P . Notation 7.1.12 ( R Q ) . Now let R Q denote the following composition: N ( Simp ( B Diff( Q )) p − −−→ K an (cid:44) → sSet R [ − ] −−−→ C at A ∞ . Proposition 7.1.13.
There exists a natural equivalence N ( Simp ( B Diff( Q )) C ∗ P (cid:45) (cid:45) R Q (cid:49) (cid:49) (cid:11) (cid:19) C at A ∞ . Here, R Q is the functor from Notation 7.1.12 and C ∗ P is the functor from Notation 5.1.7. Proof.
For every j : | ∆ n | → B Diff( Q ), let D ( j ) ⊂ p − ( j ) denote the full subcategory spannedby those 0-simplices of j ∗ E that are contained in a fiber above one of the vertices of | ∆ n | . Thenthe natural transformation induced by the inclusion D ( j ) → p − ( j ) is essentially surjective andobviously fully faithful. On the other hand, D ( j ) is equivalent to the path category P j . Thus thecomposite natural equivalences p − ( j ) ← D → P exhibits the natural equivalence we seek by choosing an inverse to either equivalence. Remark 7.1.14.
The above proposition accomplishes the goal of seeing that C ∗ P exhibits theaction of Diff( Q ) on the ∞ -category of local systems. To see this, consider the composite N ( Simp ( B Diff( Q )) max −−→ Sing C ∞ ( (cid:92) B Diff( Q )) ∼ −→ Sing ( B Diff( Q )) ∼ −→ B Sing (Diff( Q )) ι −→ K an R [ − ] −−−→ C at A ∞ . (7.1)Here, 61 max is the natural map from a subdivision to the underlying simplicial set; the next arrow isthe natural map from smooth, extended simplices to continuous simplices. • The next arrow identifies the singular complex of the classifying space B Diff( Q ) with the ∞ -category with one object, whose endomorphism space is given by Sing (Diff( Q )). The arrow ι is the natural inclusion of this subcategory into K an —the unique object of B ( Sing (Diff( Q )))is sent to the Kan complex Sing ( Q ), and we have the obvious map on morphisms. Finally, R [ − ] is the left adjoint to the dg-nerve from Notation 7.1.9.By the universal property of localization, R Q factors as in the below diagram: N ( Simp ( B Diff( Q )) (cid:41) (cid:41) R Q (cid:47) (cid:47) C at A ∞ Sing ( B Diff( Q )) (cid:55) (cid:55) We know from Lemma 7.1.3 that the dashed arrow in the diagram is equivalent to our compos-ite (7.1). Thus, by the natural equivalence of Proposition 7.1.13, we conclude that the functor
Sing ( B Diff( Q )) → C at A ∞ induced by C ∗ P is also equivalent to the composite map of (7.1). Thiswas our goal. Proof of Theorem 1.0.12.
Among functors from N Simp ( B Diff( Q )) to C at A ∞ , we have the followingnatural equivalences: Tw R Q Prop 7.1.13 −−−−−−−→ Tw C ∗ P Thm 6.0.1 −−−−−−−→ Tw W ◦ D . Because all three of the above functors— Tw R Q , Tw C ∗ P , and Tw W ◦ D —map morphisms in N Simp ( B Diff( Q )) to equivalences in C at A ∞ , each induces a functor from the Kan completionof N Simp ( B Diff( Q )). This Kan completion is an ∞ -groupoid equivalent to Sing ( B Diff( Q )), andhence to B Diff( Q ), by Recollection 3.2.3(1).By Remark 7.1.14, the functor induced by Tw R Q classifies the Diff( Q ) action on C ∗ P ( Q ). Onthe other hand, Tw W ◦ D by construction classifies the action of Diff( Q ) on W ( M ) induced by theaction of Aut on W ( M ). This completes the proof. References [Abo11] Mohammed Abouzaid,
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