Iterations of symplectomorphisms and p-adic analytic actions on Fukaya category
IITERATIONS OF SYMPLECTOMORPHISMS AND p -ADICANALYTIC ACTIONS ON FUKAYA CATEGORY YUSUF BARIS¸ KARTAL
Abstract.
Inspired by the work of Bell on dynamical Mordell-Lang conjec-ture, and by the family Floer homology, we construct p -adic analytic familiesof bimodules on the Fukaya category of a monotone or negatively monotonesymplectic manifold, interpolating the bimodules corresponding to iterates ofa symplectomorphism φ isotopic to identity. We consider this family as a p -adic analytic action on the Fukaya category. Using this, we deduce that theranks of Floer homology groups HF ( φ k ( L ) , L (cid:48) ; Λ) are constant in k ∈ Z , withfinitely many possible exceptions. We also prove an analogous result withoutthe monotonicity assumption for generic φ isotopic to identity by showing howto construct a p -adic analytic action in this case. Contents
1. Introduction 1Acknowledgments 92. Background on Fukaya categories 103. Families of bimodules and symplectomorphisms 164. Comparison with the action of φ tα and proof of Theorem 1.1 275. Generic α and the proof of Theorem 1.5 36Appendix A. Semi-continuity statements for chain complexes 44References 471. Introduction
Motivation and main results.
In [Bel06], Bell proves the following theorem:let Y be an affine variety over a field of characteristic 0 and f : Y → Y be anautomorphism. Consider a subvariety X ⊂ Y and a point x ∈ Y . Then, the set { n ∈ N : f n ( x ) ∈ X } is a union of finitely many arithmetic progressions and a setof finitely many numbers. In [Sei16], Seidel conjectures a symplectic version of thisstatement. Namely, given symplectic manifold ( M, ω M ), a symplectomorphism φ and two Lagrangians L, L (cid:48) ⊂ M (for which Floer homology is well-defined), the set(1.1) { n ∈ N : f n ( L ) and L (cid:48) are Floer theoretically isomorphic } form a union of finitely many arithmetic progressions and a set of finitely manynumbers. Key words and phrases. categorical dynamics, Fukaya category, dynamical Mordell-Lang, p-adic analytic action. a r X i v : . [ m a t h . S G ] S e p YUSUF BARIS¸ KARTAL
The purpose of this paper is to prove a version of this statement when φ ∈ Symp ( M, ω M ), i.e. φ is isotopic to identity through symplectomorphisms. Ourmain theorem holds when ( M, ω M ) is monotone or negatively monotone. Namely: Theorem 1.1.
Assume ( M, ω M ) is a symplectic manifold and L, L (cid:48) ⊂ M aremonotone Lagrangians such that Assumption 1.2 is satisfied. Then, the rank of HF ( φ k ( L ) , L (cid:48) ; Λ) is constant except for finitely many k . Here, HF ( L, L (cid:48) ; Λ) denotes the Lagrangian Floer homology group defined withcoefficients in the Novikov field Λ = Q (( T R )). We will sometimes omit Λ from thenotation and denote this group by HF ( L, L (cid:48) ). Assumption 1.2.
The monotone Fukaya category F ( M ; Λ) is smooth, and it isgenerated by a set of Lagrangians L , . . . , L m with minimal Maslov number at least satisfying either • image of π ( L i ) → π ( M ) is torsion • L i is Bohr-Sommerfeld monotoneAlso, L, L (cid:48) have minimal Maslov number at least and they are also Bohr-Sommerfeldmonotone in the latter case. We assume L and L (cid:48) are Bohr-Sommerfeld as the proof is more geometric in thiscase; however, as we will explain at the end of Section 4, this assumption can bedropped. We should also note that in the examples we have L i satisfy the latterassumption and not the former. Example 1.3.
One can let M = Σ g to be a surface of genus greater than or equalto 2. Finite generation of F (Σ g , Λ) is shown in [Sei11] and [Efi12], and that thiscategory is homologically smooth follows from the fact that matrix factorization cat-egories are homologically smooth (see [Dyc11]). Alternatively see [AS20, Lemma2.18]. That one can let generators to be Bohr-Sommerfeld monotone follows fromthe fact that every non-separating curve has such a representative in its isotopyclass (see [Sei11], note the author uses the term balanced for Bohr-Sommerfeldmonotone). Let (cid:96) ⊂ Σ g be a non-separating simple closed curve with primitivehomology class in Σ g (in particular it is not null-homologous). One can let (cid:96) to beone of the meridians in a decomposition Σ g = ( T ) g . Let φ be a symplectomor-phism with small flux that disjoints (cid:96) from itself and let (cid:96) = φ ( (cid:96) ), (cid:96) = φ ( (cid:96) )(also assume (cid:96) ∩ (cid:96) = ∅ ). Consider L = (cid:96) and L (cid:48) = (cid:96) ∪ (cid:96) equipped with Spinstructures. Applying Theorem 1.1, we see that the rank of HF ( φ k ( L ) , L (cid:48) ; Λ) is con-stant except finitely many k . In this example, the finite exceptional set is k = 1 , (cid:96) ∪ (cid:96) cannot be Bohr-Sommerfeld monotone. Example 1.4.
Let M = Σ × Σ . In addition to (cid:96) , (cid:96) , (cid:96) consider a non-separatingcurve (cid:96) ⊂ Σ with primitive homology class, that is fixed by φ , and that intersecteach of (cid:96) , (cid:96) and (cid:96) exactly at one point. Let φ (cid:48) = φ × φ . Consider the Lagrangiantori (cid:96) × (cid:96) and (cid:96) × (cid:96) . These tori intersect at one point, defining a morphism (cid:96) × (cid:96) → (cid:96) × (cid:96) (of non-zero degree possibly). Let L be a Lagrangian representingthe cone of this morphism, which can be obtained by Lagrangian surgery. We let L (cid:48) = (cid:96) × (cid:96) . Clearly, φ (cid:48)− k ( L (cid:48) ) has non-vanishing Floer homology with (cid:96) × (cid:96) only TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY3 when k = 1 and with (cid:96) × (cid:96) only when k = 2. Therefore, HF ( φ (cid:48) k ( L ) , L (cid:48) ; Λ) is non-zero only at k = 1 ,
2. One can produce examples with more sophisticated finitesets of exceptional k in this way. Presumably, the images of L and (cid:96) × (cid:96) ∪ (cid:96) × (cid:96) in Sym (Σ ) also illustrate a case when the rank jumps twice.To define Bohr-Sommerfeld monotonicity, we need to fix some data on M . See[Sei11] for the case of higher genus surfaces and [WW10] for more general manifolds.Throughout the paper, we use the word monotone to refer to both monotone andnegatively monotone. In other words, ( M, ω M ) is monotone if [ c ( M )] = λ [ ω M ], forsome λ (cid:54) = 0. Under Assumption 1.2, the counts of marked discs defining the Fukayacategory with objects L i are finite (see [Oh93], [She16], [WW10]); therefore, theFukaya category can be defined over the field of Novikov polynomials. Let F ( M, Λ)denote the Fukaya category spanned by L i that is defined over the Novikov fieldΛ = Q (( T R )). One can define the Fukaya category over a field extension of Q that is generated by { T E ( u ) } , where u ranges over marked pseudo-holomorphicdiscs with boundary on L i and asymptotic to intersection points. The set of allpossible energies is in the span of a finitely generated subgroup G ⊂ R ; therefore,the extension of Q by all such T E ( u ) is in a finitely generated extension of Q byelements of the form T g , g ∈ G . In other words, the Fukaya category can be definedover a finitely generated field extension of Q generated by such elements. Moreover,Assumption 1.2 implies disc counts defining the Yoneda modules h L , h L (cid:48) are alsofinite and by adding energies of these into G ⊂ R , we can ensure these modulesare also defined over the Fukaya category with coefficients in the finitely generatedextension above. Fix brane structures, as well as Floer data to define the Fukayacategory and these modules (see [Sei08] for definitions).After the proof of Theorem 1.1, we seek ways to drop the assumption of monotonic-ity. We do not expect Theorem 1.1 to hold in general; for instance, for M = T witha fixed area form, there exists one periodic symplectic flows on M with non-zeroflux. More specifically, if (cid:96) ⊂ T is a meridian, one can let φ to be the rotation of T in the orthogonal direction by π/
3. Then, HF ( φ k ( (cid:96) ) , (cid:96) ; Λ) is not constant in k even after finitely many k , although it is still periodic. However, one can still provethe following: Theorem 1.5.
Assume ( M, ω M ) is a symplectic manifold with two Lagrangianbranes L, L (cid:48) ⊂ Λ satisfying Assumption 1.6. Given generic φ ∈ Symp ( M, ω M ) , therank of HF ( φ k ( L ) , L (cid:48) ; Λ) is constant in k ∈ Z with finitely many possible exceptions. By generic, we mean that the flux of an isotopy from 1 to φ is generic. This will beexplained further in this section and in Section 5. Assumption 1.6. F ( M, Λ) is an (uncurved, Z or Z / Z -graded) A ∞ -category over Λ , it is smooth and proper, generated by L , . . . , L m . Also, L, L (cid:48) are Lagrangianswith brane structure that bound no Maslov discs (so they define objects of Fukayacategory). We let F ( M, Λ) denote the category spanned by the generators L , . . . , L m , asbefore. Theorem 1.5 is valid in great generality. As examples of such M , one canconsider T and T × T . YUSUF BARIS¸ KARTAL
Remark 1.7.
Bell’s theorem mentioned at the beginning is a case of the dynamicalMordell-Lang conjecture. In [BSS17], the authors prove a version of this conjecturefor coherent sheaves. Theorem 1.1 and Theorem 1.5 are closer to this statement inspirit. Note however, our techniques extend to prove a statement that is closer toSeidel’s conjecture: namely, under Assumption 1.2, k ∈ N such that φ k ( L ) and L (cid:48) are stably Floer theoretically isomorphic form a set that is either finite or cofinite(hence either a singleton or everything). By stably isomorphic, we mean L is adirect summand of L (cid:48)⊕ q , for some q (cid:29)
0, and vice versa. An immediate corollaryof this is if φ k ( L ) is Hamiltonian isotopic to L (cid:48) for two different k , then it is stablyisomorphic for all k . Let α be a closed 1-form such that φ = φ α . By replacing φ by φ /nα , where n ∈ N , we see that φ k/nα ( L ) is stably isomorphic to L (cid:48) for all k ; therefore, φ fα ( L ) and L (cid:48) are stably isomorphic for f ∈ Q . One can presumablyshow that for small t ∈ R , HF ( L, φ tα ( L )) = 0; hence, L and φ tα ( L ) are not stablyisomorphic unless α | L = 0. This implies φ tα ( L ) is Hamiltonian isotopic to L (cid:48) for all t .1.2. Summary of the proofs.
Fix a path from 1 M to φ through symplectomor-phisms and assume the flux of the path is α where α is a closed 1-form. Everyclosed 1-form generates a symplectic isotopy φ tα as the flow of vector field X α sat-isfying ω ( · , X α ) = α and φ α is Hamiltonian isotopic to φ by Banyaga’s theorem.Therefore, the action of φ and φ α on Floer homology are the same and we mayassume φ = φ α . The isotopy φ tα generate a family of bimodules by:(1.2) ( L i , L j ) (cid:55)→ HF ( φ tα ( L i ) , L j )We give a quasi-isomorphic description of this family inspired by family Floer ho-mology (see [Abo14]) and quilted Floer homology (as in [Ma’15], [Gan12]). Thesimplest version is the following: consider the ring Q ( T R )[ z R ]. To every pair ofLagrangians, associate(1.3) ( L i , L j ) (cid:55)→ M Λ α ( L i , L j ) = CF ( L i , L j ) ⊗ Q ( T R )[ z R ]with differential given by(1.4) x (cid:55)→ (cid:88) ± T E ( u ) z α ([ ∂ h u ]) .y where the sum ranges over pseudo-holomorphic strips from x to y and [ ∂ h u ] denotesthe class of “one side of the boundary of u ”. When, L i satisfy the condition that Im ( π ( L i ) → π ( M )) are torsion, these z -terms are actually 1. However, one candefine the higher structure maps of the bimodule by the rule(1.5) ( x , . . . , x k | x | x (cid:48) , . . . , x (cid:48) l ) (cid:55)→ (cid:88) ± T E ( u ) z α ([ ∂ h u ]) .y where the sum ranges over pseudo-holomorphic marked discs with input x , . . . , y, . . . x (cid:48) l and output y . Without the z -term, this is just the definition of A ∞ -structure maps,and the count is finite by monotonicity. [ ∂ h u ] ∈ H ( M, Z ) denotes the “part ofboundary of disc u ” from the bimodule input x to output y (see also Figure 3.1.For a precise definition of the class [ ∂ h u ], we will fix a base point on M and homo-topy classes of paths from this base point to generators of Floer homology groups).Now, the z -term does not have to be trivial, and this gives a deformation of the di-agonal bimodule of F ( M ). That the maps (1.5) satisfy the A ∞ -bimodule equationis immediate. Moreover, it is possible to show this gives the bimodule correspond-ing to φ fα , when we plug z = T f for small f ∈ R . There are no convergence issues TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY5 as in [Abo14] as the count is finite, but for larger f , the bimodule M Λ α | z = T f has noa priori relation to φ fα . On the other hand, one can show that for any Lagrangianbrane ˜ L , the Yoneda modules satisfy(1.6) h φ fα (˜ L ) (cid:39) h ˜ L ⊗ F ( M, Λ) M Λ α | z = T f when | f | is small (we use | f | to denote ordinary absolute value). Similarly, if | f | and | f (cid:48) | are small, then(1.7) M Λ α | z = T f + f (cid:48) (cid:39) M Λ α | z = T f ⊗ F ( M, Λ) M Λ α | z = T f (cid:48) See Lemma 4.6 and Lemma 3.5 respectively. For arbitrary, f >
0, there exists asequence of numbers 0 = s < s < · · · < s r = f such that(1.8) h φ fα (˜ L ) (cid:39) h ˜ L ⊗ F ( M, Λ) M Λ( s − s ) α ⊗ F ( M, Λ) M Λ( s − s ) α · · · ⊗ F ( M, Λ) M Λ( s r − s r − ) α Similarly, for f <
0, there exists 0 = s > s > · · · > s r = f such that (1.8) holds.Above we mentioned that Fukaya category can be defined over Q ( T g : g ∈ G ), where G is a finitely generated additive subgroup of R containing all possible energies ofpseudo-holomorphic discs. Without loss of generality add α ( H ( M, Z )) into G , thenthe coefficients defining (1.4) would be in Q ( T g : g ∈ G ), and one can “evaluate”(1.4) at z = T f , for f ∈ Z . In other words, one can define a similar family M Kα over F ( M, K )- the Fukaya category with coefficients over K , where K ⊂ Q (( T R ))is a field containing Q ( T g : g ∈ G ). We will add some roots of T α ( C ) to K .Let p > Q ( T g : g ∈ G ). By choosing a finite number of elements from 1 + p Z p that satisfy noalgebraic relation over Q , one can find an embedding of Q ( T g : g ∈ G ) into Q p such that T g (cid:55)→ p Z p . Moreover, as we will see in bigger generality, for any f ∈ Z ( p ) = { ab : a, b ∈ Z , p (cid:45) b } , and v ∈ p Z p , one can define a canonical f th root via(1.9) v f = ∞ (cid:88) i =0 (cid:18) fi (cid:19) ( v − i ∈ p Z p Therefore, we can write a field homomorphism(1.10) µ : K := Q ( T g/b : g ∈ G, p (cid:45) b ∈ Z ) → Q p that sends T g/b into 1 + p Z p . Denote the image of T g/b by T g/bµ .It is easy to see that the exponentiation (1.9) extends to an element v t ∈ Q p (cid:104) t (cid:105) simply by replacing f by t . Q p (cid:104) t (cid:105) is the Tate algebra and it can be thought as theanalytic functions on p -adic unit disc Z p . Therefore, one can define a p -adic familyof bimodules exactly in the same way, except we replace formula (1.5) by(1.11) ( x , . . . , x k | x | x (cid:48) , . . . , x (cid:48) l ) (cid:55)→ (cid:88) ± T E ( u ) µ T tα ([ ∂ h u ]) µ .y We denote this family by M Q p α and think of it as an “analytic map from Z p to auto-equivalence group of the Fukaya category”. Our first claim is that this map behaveslike a group homomorphism on a ( p -adically) small neighborhood p n Z p ⊂ Z p of 0.In other words, it is group-like. More precisely, there exists a morphism(1.12) π ∗ M Q p α ⊗ π ∗ M Q p α → ∆ ∗ M Q p α YUSUF BARIS¸ KARTAL of families over Q p (cid:104) t , t (cid:105) that induce a quasi-isomorphism on a small neighborhoodof t = t = 0. To explain the notation, we identify Q p (cid:104) t , t (cid:105) with (completed)tensor product of Q p (cid:104) t (cid:105) with itself. Then π k correspond to projection maps and∆ to group multiplication for the group Sp ( Q p (cid:104) t (cid:105) ) (note that we make no formalreference to affinoid domain and work entirely with Tate algebra, which is also aHopf algebra in this case, ∆ is the coproduct map). The map (1.12) induces a wellknown quasi-isomorphism at t = t = 0 and the semi-continuity property of quasi-isomorphisms imply the same in a small p -adic neighborhood. This proves that M Q p α is group-like after base change under Q p (cid:104) t (cid:105) ⊂ Q p (cid:104) t/p n (cid:105) . The latter can beseen as the algebra of analytic functions on p n Z p , and one can heuristically think of M Q p α | Q p (cid:104) t/p n (cid:105) as a Sp ( Q p (cid:104) t/p n (cid:105) ) action on F ( M, Q p ). One can presumably show thegroup-like property over Q p (cid:104) t (cid:105) using deformation class computations as in [Sei14],[Kar18]; however, we do not follow this way. Note that the F ( M, K )-bimodule M Kα | z = T f base changes to M Q p α | t = f , for any f ∈ Z ( p ) ; therefore,(1.13) M Kα | z = T f + f (cid:48) (cid:39) M Kα | z = T f ⊗ F ( M,K ) M Kα | z = T f (cid:48) M Λ α | z = T f + f (cid:48) (cid:39) M Λ α | z = T f ⊗ F ( M, Λ) M Λ α | z = T f (cid:48) for any f, f (cid:48) ∈ p n Z ( p ) = { ab : a, b ∈ Z , p n | a, p (cid:45) b } (i.e. when f and f (cid:48) are p -adicallysmall).When f ∈ p n Z ( p ) , one can choose 0 = s < s < · · · < s r = f in (1.8) such that all s i ∈ p n Z ( p ) . Therefore, by letting ˜ L = L (cid:48) and using (1.13), we see that(1.14) h φ fα ( L (cid:48) ) (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ α | z = T f By replacing f by − f and applying ⊗ F ( M, Λ) h L on the right, we obtain(1.15) CF ( L, φ − fα ( L (cid:48) )) (cid:39) h φ − fα ( L (cid:48) ) ⊗ F ( M, Λ) h L (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ α | z = T − f ⊗ F ( M, Λ) h L By assumption, h L (cid:48) and h L are definable over F ( M, K ); hence,(1.16) h L (cid:48) ⊗ F ( M,K ) M Kα | z = T − f ⊗ F ( M,K ) h L is well-defined and has cohomology of dimension same as HF ( L, φ − fα ( L (cid:48) )) ∼ = HF ( φ fα ( L ) , L (cid:48) ).It also extends to(1.17) h L (cid:48) ⊗ F ( M, Q p ) M Q p α | t = − f ⊗ F ( M, Q p ) h L under base change along µ : K → Q p .One can show that(1.18) h L (cid:48) ⊗ F ( M, Q p ) M Q p α | Q p (cid:104) t/p n (cid:105) ⊗ F ( M, Q p ) h L is quasi-isomorphic to a finite complex of finitely generated Q p (cid:104) t/p n (cid:105) -modules. Thisimplies that its cohomology has constant rank at all f ∈ p n Z ( p ) , with finitely manyexceptions. In other words, HF ( φ fα ( L ) , L (cid:48) ) has constant rank among all f ∈ p n Z ( p ) ,with finitely many possible exceptions.Choose f i ∈ Z ( p ) for i = 1 , . . . , p n − | f i | is small and f i ≡ i ( mod p n ).We want to replace L (cid:48) by φ − f i α ( L (cid:48) ) in (1.18); however, we need definability of h φ − fiα ( L (cid:48) ) over F ( M, K ), thus over F ( M, Q p ). For this purpose, we use h algφ − fiα ,L (cid:48) , TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY7 an A ∞ -module over F ( M, K ) that becomes quasi-isomorphic to h φ − fiα ( L (cid:48) ) after ex-tending the coefficients from K to Λ and that is obtained by an algebraic de-formation of h L (cid:48) . One can replace h L (cid:48) by h algφ − fiα ,L (cid:48) in (1.18), and the same rea-soning proves that the newly obtained complex has cohomology of constant rankamong all f ∈ p n Z ( p ) with finitely many possible exceptions. In other words, HF ( L, φ − f − f i α ( L (cid:48) )) ∼ = HF ( φ f + f i α ( L ) , L (cid:48) ) has constant dimension for f ∈ p n Z ( p ) ,with finitely many possible exceptions. Since { f i } i ∪ { } cover all classes modulo p n , we can conclude that the rank of HF ( φ fα ( L ) , L (cid:48) ) is p n -periodic except finitelymany, in other words, if f, f (cid:48) ∈ Z ( p ) , then HF ( φ fα ( L ) , L (cid:48) ) and HF ( φ f + p n f (cid:48) α ( L ) , L (cid:48) )has the same dimension, except possibly a finite number of f .We did not have any restriction on p >
2, and one can replace it by another prime p (cid:48) to conclude that the dimension of HF ( φ fα ( L ) , L (cid:48) ) is also p (cid:48) n (cid:48) -periodic for f (cid:48) ∈ Z ( p (cid:48) ) ,with finitely many possible exceptions. As p n and p (cid:48) n (cid:48) are coprime, this proves thatthe rank of HF ( φ fα ( L ) , L (cid:48) ) is constant among f ∈ Z , with finitely many possibleexceptions, i.e. Theorem 1.1 follows. Observe that the proof implies the constancyof dimension of HF ( φ fα ( L ) , L (cid:48) ) for the dense subset Z ( p ) ∩ Z ( p (cid:48) ) = Z ( pp (cid:48) ) = { ab : a, b ∈ Z , p (cid:45) b, p (cid:48) (cid:45) b } of R with finitely many exceptions.As mentioned above, one can actually drop the Bohr-Sommerfeld assumption on L and L (cid:48) . For the proof to go through, we need definability of h L (cid:48) , h L over F ( M, K ) or F ( M, ˜ K ), where ˜ K is a finitely generated extension of K . This may not be alwayspossible, but as one can represent L and L (cid:48) as elements of tw π ( F ( M, Λ)) and as thedata to define a twisted complex is finite, there exists a finitely generated extension˜ K of K inside Λ, and F ( M, ˜ K )-modules h (cid:48) L (cid:48) , h (cid:48) L that become quasi-isomorphic to h L (cid:48) and h L (cid:48) after base change along ˜ K → Λ. Since ˜ K is finitely generated, one canextend µ : K → Q p to a map ˜ µ : ˜ K → Q (cid:48) p , where Q (cid:48) p is a finite (not only finitelygenerated) extension of Q p , which automatically carries a discrete valuation. Thisallows us to define the complex h (cid:48) L (cid:48) ⊗ F ( M, Q (cid:48) p ) M Q (cid:48) p α ⊗ F ( M, Q (cid:48) p ) h (cid:48) L over Q (cid:48) p (cid:104) t/p n (cid:105) , whoserank at t = kp n is still HF ( φ − kp n α ( L ) , L (cid:48) ; Λ). As before, this implies that the rankof HF ( φ kp n α ( L ) , L (cid:48) ; Λ) is constant in k with finitely many exceptions. By replacing L (cid:48) by φ iα ( L (cid:48) ), where i = 0 , . . . , p n −
1, we conclude p n -periodicity of the rank of HF ( φ kα ( L ) , L (cid:48) ; Λ), and by switching primes, we conclude Theorem 1.1 without theBohr-Sommerfeld assumption on L and L (cid:48) .After the proof of Theorem 1.1, we turn to the proof of Theorem 1.5. The proof isanalogous to Theorem 1.1, and the main difference is in finding a smaller field ofdefinition K g ⊂ Λ analogous to K with an embedding into Q p such that analogousformulae define a group-like p -adic family of bimodules. For the former, we need K g to contain all series (cid:80) ± T E ( u ) defining the coefficients of the A ∞ -structure of F ( M, Λ). For any such field, we have a category F ( M, K g ) that base change to F ( M, Λ) under the inclusion K g → Λ. We also need to find a map µ g : K g → Q p such that the series(1.19) (cid:88) ± µ g ( T E ( u ) ) µ g ( T α ([ ∂ h u ]) ) t used to define (1.11) are well-defined and converge in Q p (cid:104) t (cid:105) . For the former, weneed µ g ( T α ([ ∂ h u ]) ) t to be well-defined, whereas the latter would follow from an YUSUF BARIS¸ KARTAL assumption ensuring val p ( µ g ( T E ( u ) )) > µ g ( T α ([ ∂ h u ]) ) t mean µ g ( T α ([ ∂ h u ]) ) ≡ mod p ).For the p -adic convergence, one can prove that there exists rationally independentpositive real numbers E , . . . , E n ∈ ω M ( H ( M, (cid:83) L i ∪ L ∪ L (cid:48) ; Q )) such that themonoid spanned by them contain all possible energies of pseudo-holomorphic discswith boundary on L i , L, L (cid:48) . By defining the map µ g : K g → Q p such that it sends T E i into p Z p , we guarantee p -adic convergence as well, as the length of expressionof E ( u ) in terms of { E i } also goes to infinity by Gromov compactness.In summary, we need a map that sends T E i into p Z p and T α ([ C ]) into 1 + p Z p .Observe, however, that these conditions can be inconsistent, as there may be non-trivial rational linear relations between E i and the periods α ([ C ]). The only waywe avoid this situation is to assume α is generic , i.e.(1.20) α ( H ( M, Z )) ∩ ω M ( H ( M, (cid:91) L i ∪ L ∪ L (cid:48) ; Q )) = { } (we actually call α generic if it satisfies a weaker condition, but (1.20) also hold foralmost all α ). Once we have the genericity property, it is possible to define such amap from the field generated by T E i and T α ([ C ]) into Q p .As mentioned, we need K g to contain some Novikov series such as those definingthe A ∞ -structures. Previously, this was automatic as these series were finite sumsas a result of monotonicity. Similarly, we need to add series defining the module h algφ fα ,L (cid:48) or the bimodule M Λ α | z = T f (that are defined over F ( M, Λ) when | f | is small).But, there are only countably many elements of Λ that we need to add, and weadd them to K g by hand. We are able to construct a map µ g from the countablygenerated field K g to Q p . After this step, the proof of Theorem 1.1 applies almostverbatim. As before, we obtain p n periodicity of the rank of HF ( φ k ( L ) , L (cid:48) ; Λ), forsome n (cid:29)
0. Since we have freedom to switch to another prime, we conclude thatthe rank is constant with finitely many possible exceptions.1.3.
Applications, comments and possible generalizations:
Since we defineFukaya category using Bohr-Sommerfeld monotone Lagrangians the coefficients ofthe A ∞ -structure are actually finite. In this case one can set T = e − , or anyother real number. In an earlier version of this paper, we have also proven aweaker version of Theorem 1.1 when one sets Novikov parameter to be a small realnumber, namely the rank of HF ( φ k ( L ) , L (cid:48) ; R ) is periodic with finitely many possibleexceptions. Note that this theorem holds under a strong assumption of convergence:one has to assume convergence of all A ∞ -coefficients when φ fα ( L ) and φ fα ( L (cid:48) ) areallowed as boundary conditions, which does not follow from the assumption that L and L (cid:48) are Bohr-Sommerfeld monotone. This assumption is needed even forthe statement to make sense, otherwise HF ( φ k ( L ) , L (cid:48) ; R ) may not be well-defined.One has analogues of Corollary 3.24 without the convergence assumption though.In other words, one can construct an action by bimodules and prove the analogoustheorem stated in this language. TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY9 One possible application of Theorem 1.1 is to categorical entropy. This notion isdefined in [DHKK14] and it can be computed via the formula(1.21) lim sup k k log (cid:0) (cid:88) i,j dimHF ( φ k ( L i ) , L j ; Λ) (cid:1) The conclusion of Theorem 1.1 is that the categorical entropy vanishes when φ ∈ Symp ( M, ω M ). This is not very surprising: in the case of surfaces the logarithmicgrowth of fixed point Floer homology can be computed as the smallest topologicalentropy in the mapping class and this confirms a categorical analogue for the iden-tity component. We believe our techniques can be useful in showing deformationinvariance of the categorical entropy for other symplectic mapping classes as well.The most crucial property used in the proof of Theorem 1.1 is the finiteness ofthe sums defining the A ∞ -structure. On the other hand, there is a large class ofnon-degenerate symplectic manifolds with a finite set of generators satisfying thisproperty: Weinstein manifolds. In this case, the Fukaya category is replaced bythe wrapped Fukaya category and lacks the properness. Most of the arguments inthis paper can be shown to work though, as the hom-sets of the wrapped Fukayacategory admit an action filtration with finite dimensional quotients. The p -adicsemi-continuity argument fails on the other hand and we cannot prove analogousgroup-like property. It is currently work in progress to extend the result to includewrapped Fukaya categories, but this requires different techniques such as the de-formation classes of Seidel [Sei14]. The argument in this paper works for partiallywrapped Fukaya categories, as long as they are smooth and proper. In either ofthese situation, we confine ourselves to compact L and L (cid:48) .1.4. Outline of the paper.
In Section 2, we recall the basics of Fukaya categoriesas well as related homological algebra. We also introduce the field K ⊂ Λ, overwhich both the Fukaya category and the family of invertible bimodules are defined.In Section 3, we recall the notion of families and their homological algebra. Wealso construct the Novikov and p -adic families M Λ α , resp. M Q p α , and we establishthe group-like property of M Q p α over Q p (cid:104) t/p n (cid:105) . Section 4 is devoted to comparisonof the algebraically constructed bimodule above to the Floer homology groups (seeProposition 4.1). We then use this to conclude proof of Theorem 1.1. We alsoexplain how to drop Bohr-Sommerfeld assumption on L and L (cid:48) . In Section 5,we prove Theorem 1.5. More attention is paid to the new constructions than theresults with prior analogues in this paper. In Appendix A, we establish some semi-continuity results for complexes and family Floer homology. We postpone theseinto an appendix so as not to interrupt the flow of the paper. Acknowledgments
We would like to thank John Pardon for suggesting the semi-continuity argumentin Proposition 3.20, Umut Varolg¨une¸s and Mohammed Abouzaid for suggesting toconsider negatively monotone and monotone symplectic manifolds, and Ivan Smithfor helpful conversations and email correspondence. We also thank to Sheel Ganatrafor pointing out to a reference and some suggestions. Background on Fukaya categories
Reminders and remarks on Fukaya categories and related homolog-ical algebra.
In this section, we will recall the basics of Fukaya categories and re-lated homological algebra, and we will explain some of our conventions. Throughoutthe paper Λ denotes the Novikov field with rational coefficients and real exponents,i.e. Λ = Q (( T R )).Let L , . . . , L m ⊂ M be monotone Lagrangians with minimal Maslov number 3 thatare oriented and equipped with Spin -structures. Assume L i are pairwise transverse.To define the Fukaya category whose objects are given by L , . . . , L m , one countsmarked holomorphic discs. More precisely, define(2.1) hom ( L i , L j ) = CF ( L i , L j ; Λ) = Λ (cid:104) L i ∩ L j (cid:105) if i (cid:54) = j A generic choice of almost complex structure lets one to endow hom ( L i , L j ) with adifferential, defined by the formula µ ( x ) = (cid:80) ± T E ( u ) .y , where the sum runs overpseudo-holomorphic strips with boundary on L i ∪ L j and asymptotic to x and y .Here, E ( u ) denotes the energy of the strip, which is equal to its symplectic area.More generally, given ( L i , . . . L i q ) that are pairwise distinct and x j ∈ L i j − ∩ L i j ,define(2.2) µ q ( x q , . . . , x ) = (cid:88) ± T E ( u ) .y where y runs over intersection points y ∈ L i ∩ L i q and u runs over rigid markedpseudo-holomorphic discs with boundary on (cid:83) j L i j , and asymptotic to { x j } and y near the markings. Spin structures on the Lagrangians allow one to orient themoduli of such discs, determining the signs in (2.1) and (2.2). Also, see Remark2.3. By standard gluing and compactness arguments, this defines a Z / Z -graded A ∞ -structure over Λ. The condition that the Maslov numbers of L i are at least 3imply that A ∞ structure has no curvature.To include hom ( L i , L i ) (as well as situation when two of the Lagrangians L i q and L i r coincide), one can follow different options: the one that we take here is theapproach via count of pearly trees. Our main references are [Sei11, Section 7] and[She11, Section 4]. Fix Morse-Smale pairs ( f i , g i ) on each L i , and define(2.3) hom ( L i , L i ) = CM ( f i ; Λ) = Λ (cid:104) crit ( f i ) (cid:105) if i = j The differential on hom ( L i , L i ) is defined via the count of Morse trajectories. Todefine more general structure maps, one has to consider “holomorphic pearly trees”,i.e. Morse flow lines connected by pseudo-holomorphic “pearls”. See [She11, Section4] for more details. We assume the perturbation data of the pearly trees havevanishing Hamiltonian terms. The energy of a holomorphic pearly tree is definedto be the sum of energies of all holomorphic pearls. See also [CL06] and [BC09].We will abuse the notation and denote the hom-sets by CF ( L i , L j ; Λ) even when L i = L j .The reason we prefer this model of Fukaya categories over the one in [Sei08] isthat it gives us better control over the topological energy of the discs. Namely,the topological energies of the discs all belong to the finitely generated group ω M ( H ( M, (cid:83) L i ; Z )). Another reason we use this model in the convenience in TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY11 applying Fukaya’s trick (see for instance Lemma 4.3). One could use the model in[FOOO09] as well.Even though formally we are counting pseudo-holomorphic pearly trees, we willrefer them as “pseudo-holomorphic discs” throughout the paper, by abuse of ter-minology. Similarly, to avoid confusion our figures will present discs, rather thanpearly trees. Note 2.1.
We must warn that in the upcoming figures such as Figure 3.1 or Figure3.4, we use wavy lines going through the disc. This has nothing to do with the Morsetrajectories of the pearly trees, rather they represent the homotopy class of a pathin M going from one input to the output. The meaning of this path is also clearfor pearly trees. Remark 2.2.
We must add that it is still possible to use the model presented in[Sei08]. Namely, we still assume the Lagrangians L i are pairwise transverse. Hence,when we choose Floer data for a pair ( L i , L j ) we assume its Hamiltonian termvanishes, unless i = j . In this case, one weights the holomorphic discs with T E top ( u ) ,where E top ( u ) is the topological energy (see [AS10] for a definition). E top ( u ) isdifferent from the symplectic area, but the difference only depends on the inputsand the output of the disc. Indeed, one can rescale the generators to get rid ofthe extra terms in the topological area (i.e. one obtains an A ∞ -category where the A ∞ -coefficients are given by the same count of discs, but weighted with T ω M ([ u ]) as before). In this case, one again has the property that the set of possible energieslies in a finitely generated subgroup of R . It is also possible to apply Fukaya’s trickas in Lemma 4.3. See Remark 4.4Let ˜ L ⊂ M be another oriented, monotone Lagrangian brane of minimal Maslovnumber 3 and equipped with a Spin -structure. Assume ˜ L (cid:116) L i for all i . Then thereexists a right, resp. left, A ∞ -module h ˜ L , resp. h ˜ L such that(2.4) h ˜ L ( L i ) = CF ( L i , ˜ L ; Λ), resp. h ˜ L = CF ( ˜ L, L i ; Λ)where the structure maps are defined analogously. A short way to define them isas follows: extend F ( M, Λ) by adding ˜ L , and these are the corresponding Yonedamodules. We denote the restriction of right and left Yoneda modules correspondingto ˜ L to F ( M, Λ) by h ˜ L and h ˜ L respectively. If ˜ L is not transverse to all L i , onecan apply a small Hamiltonian perturbation. Different Hamiltonian perturbationsgive rise to quasi-isomorphic modules over F ( M, Λ).
Note 2.3.
Throughout the paper, we will omit the signs and write ± , as theyare standard, similar to above where we wrote (cid:80) ± T E ( u ) for the coefficients ofthe A ∞ -structure maps. Most of the sums we have are merely deformations ofstandard formulas and the signs do not change. For example, in the sums (3.11),(3.24), (5.13) and (5.16), one obtains the diagonal bimodule by putting z = 1 (or t = 0), and the signs are the same as those of the diagonal bimodule (and those ofthe A ∞ -structure coefficients). Similarly, the sums (4.4), and (4.7) share the samesigns as the formulas defining the right Yoneda module.Throughout the paper, we will work with smaller fields of definition for the Fukayacategory. In other words, if K ⊂ Λ is a subfield containing all the coefficients (cid:80) ± T E ( u ) defining the A ∞ -structure, then one could as well follow the definitionabove to obtain a K -linear A ∞ -category, which we denote by F ( M, K ). By basechange along the inclusion map K → Λ, one obtains the original category F ( M, Λ).If K is a smaller field of definition and µ : K → Q is a field extension, one obtainsa category via base change, and we denote this category by F ( M, Q ), omitting µ from the notation (for us, Q will be the field of p -adics for some prime p or a finiteextension of it).Similarly, if the coefficients defining the modules h ˜ L and h ˜ L belong to K , onecan define right, resp. left modules over F ( M, K ). Via base change along µ : K → Q , one obtains modules over F ( M, Q ). We keep the notation h ˜ L and h ˜ L forthese modules though. It is not necessarily true that these modules are invariantunder Hamiltonian perturbations of ˜ L . The coefficients defining the continuationmorphisms may not belong to smaller subfield K .For later use, fix a base point on M and given any generator of the Fukaya categoryor of the modules h L (cid:48) and h L , fix a relative homotopy class of paths on M fromthe base point to the generator.Since F ( M, Λ) is not built to contain all Lagrangians, the following clarification isneeded:
Definition 2.4.
We say { L i } split generate ˜ L , if ˜ L , as an object of the Fukayacategory with objects { L i }∪{ ˜ L } is quasi-isomorphic to an element of tw π ( F ( M, Λ)).We say { L i } split generate the Fukaya category , if this holds for any ˜ L asabove.Throughout the paper, F ( M, Λ) will always consist of objects { L i } split generatingthe Fukaya category. Remark 2.5. If { L i } split generate ˜ L , then the modules h ˜ L and h ˜ L are perfect, i.e.they can be represented as a summand of a complex of Yoneda modules of F ( M, Λ).Equivalently, any closed module homomorphism from h ˜ L , resp. h ˜ L , to a direct sumof right, resp. left, A ∞ -modules factor through a finite sum (in cohomology, i.e. upto an exact module homomorphism).One way to ensure split generation of the Fukaya category is the non-degeneracy of M , i.e. { L i } split generate the Fukaya category if the open-closed map from theHochschild homology of the category spanned by { L i } hits the unit in the quantumcohomology. See [Abo10] for the version for wrapped Fukaya categories. Anotherimplication of non-degeneracy is (homological) smoothness: Definition 2.6. An A ∞ -category is called (homologically) smooth , if its diag-onal bimodule is perfect, equivalently the diagonal bimodule can be represented asa direct summand of a twisted complex of Yoneda bimodules. An A ∞ -category iscalled proper , if the hom-complexes have finite dimensional cohomology. Similarly,an A ∞ -module is called proper if the complexes associated to every object havefinite dimensional cohomology. TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY13 In other words, if M is non-degenerate, then F ( M, Λ) is homologically smooth see[Gan12, Theorem 1.2]. It is also proper by definition. Similarly, the modules h ˜ L and h ˜ L are proper. Remark 2.7.
Smoothness of a category implies that the category is split generatedby finitely many objects. Together with properness, it also implies that propermodules over the category are perfect, i.e. they can be represented as a directsummand of a complex of Yoneda modules (see Lemma 3.15).We will make frequent use of the following easy lemma:
Lemma 2.8.
Given a smooth and proper A ∞ -category B over a field K of char-acteristic , and given proper (left/right/bi-) modules N , N , if hom B mod ( N , N ) has finite dimensional cohomology, then this dimension does not change under thebase change under a field extension K ⊂ Q . Moreover, the cohomologies are relatedby ordinary base change under K ⊂ Q .Proof. As we will see later in Lemma 2.8, properness of N and N imply they areactually perfect, i.e. they can be represented as Yoneda modules corresponding totwisted complexes with idempotents. Call these X , X ∈ tw π ( B ). As a result ofYoneda Lemma,(2.5) dim K H ∗ ( hom B mod ( N , N )) = dim K H ∗ ( hom tw π ( B ) ( X , X ))The latter clearly remains the same under base change. (cid:3) Remark 2.9.
Proof of Lemma 2.8 actually implies that dim K H ∗ ( hom B mod ( N , N ))is finite. However, presumably in the form it is stated, the lemma holds withoutthe smoothness assumption on B . One attempt to prove it can be made as follows:replace B by a quasi-equivalent dg algebra, and N i by dg modules. Assume N is cofibrant (free), so that the complex hom B mod ( N , N ) of A ∞ -pre morphisms isquasi-isomorphic to the complex of dg-module maps (see [Kel06]). Choosing a basisfor N over B , one can identify the vector space underlying the latter complex withthe maps from the basis to N , which turns into the complex of maps from thebasis to ( N ) Q after base change. Therefore, the cohomology of this complex isrelated to the cohomology of hom B mod ( N , N ) by base change and this completesthe proof without the smoothness of B . Implicitly, this proof still assumes finitegeneration of B .As mentioned, we will often work with smaller fields of definition; however, smooth-ness does not depend on the coefficient field as long as the category is also proper.In other words: Lemma 2.10.
Let B be a proper A ∞ -category over a field K ⊂ Λ . If the basechange B Λ := B ⊗ K Λ is smooth, then so is B .Proof. The smoothness is equivalent to perfectness of the diagonal bimodule. Itsuffices if one shows that every closed morphism of bimodules(2.6) f : B → (cid:77) η Y η where {Y η } is a collection of twisted complexes of Yoneda bimodules over B , factorsin cohomology through a finite direct sum. We know this holds after base changeto Λ, due to smoothness of B Λ . In other words, a projection f (cid:48) of (2.6) to a cofinitesub-sum (cid:76) η (cid:48) Y η (cid:48) becomes exact after base change to Λ. Assume f (cid:48) ⊗ K d ( h ).Choose a basis for Λ over K that include 1, and write every component of h in thisbasis. If we throw away the parts with basis elements other than 1, we obtain amorphism ˜ h ⊗ K
1, whose differential is still equal to f (cid:48) ⊗ K
1. Therefore, there is amorphism ˜ h : B → (cid:76) η (cid:48) Y η (cid:48) , whose differential is f (cid:48) . (cid:3) Therefore, the category F ( M, K ) is smooth, whenever it is defined (i.e. the A ∞ -structure maps lie in K ). More generally: Corollary 2.11. If B is smooth, proper, and L , . . . , L m is a set of objects of B that split generate B Λ , then they split generate B .Proof. Consider the part of the bar resolution of diagonal bimodule of B only in-volving objects L i . This resolution can be filtered by finite twisted complexes, i.e.there exists an infinite sequence Y ⊂ Y ⊂ . . . obtained by (stupid truncations) ofthe bar resolution. Let f k : Y k → B denote the restriction of the resolution map to Y k . Then, L i split generate B if and only if f k is split for some k , i.e.(2.7) RHom ( B , Y k ) f k ◦ −−→ RHom ( B , B )is surjective. This holds after extending the coefficients to Λ; therefore, it holdsover K as well, by Lemma 2.8. (cid:3) Going back to Fukaya categories, we have the following simple observation, thatwill be used regularly:
Lemma 2.12.
Assume ˜ L and ˜ L (cid:48) are split generated by { L i } . Then, H ∗ ( h ˜ L (cid:48) ⊗ F ( M, Λ) h ˜ L ) ∼ = HF ( ˜ L, ˜ L (cid:48) ) .Proof. By Hamiltonian perturbations, one can ensure ˜ L and ˜ L (cid:48) are transverse toeach other and to all L i . Then, it is possible to extend F ( M, Λ) by adding these.Denote this extension by ˜ F ( M, Λ). Standard homological algebra shows that(2.8) HF ( ˜ L, ˜ L (cid:48) ; Λ) ∼ = H ∗ ( hom ˜ F ( M, Λ) ( ˜ L, ˜ L (cid:48) )) ∼ = H ∗ ( h ˜ L (cid:48) ⊗ ˜ F ( M, Λ) h ˜ L )But by the split generation statement tw π ( ˜ F ( M, Λ)) = tw π ( F ( M, Λ)), i.e. F ( M, Λ)and F ( M, Λ) are Morita equivalent and(2.9) H ∗ ( h ˜ L (cid:48) ⊗ ˜ F ( M, Λ) h ˜ L ) ∼ = H ∗ ( h ˜ L (cid:48) ⊗ F ( M, Λ) h ˜ L ) (cid:3) Remark 2.13.
Under the assumptions of the lemma, if M is a F ( M, Λ)-bimodule,then it is the restriction of a bimodule (cid:102) M over the larger category ˜ F ( M, Λ), and (cid:102) M ( ˜ L, ˜ L (cid:48) ) (cid:39) h ˜ L (cid:48) ⊗ ˜ F ( M, Λ) M ⊗ ˜ F ( M, Λ) h ˜ L . We will prefer to work with h ˜ L (cid:48) ⊗ ˜ F ( M, Λ) M ⊗ ˜ F ( M, Λ) h ˜ L , since in general the extension of M is abstract, and should not beconfused with the concrete constructions we are going make. TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY15 We now explain the notion of Bohr-Sommerfeld monotonicity which appeared inAssumption 1.2. We borrow the definition of this notion from [WW10, Remark4.1.4]. To define this notion, we need to assume ω M is rational, i.e. the monotonicityconstant is rational. Let [ ω M ] = c ( M ) for simplicity. Then there exists a (negative)pre-quantum bundle, i.e. a line bundle L with a unitary connection ∇ whosecurvature is equal to − πiω M , and a bundle isomorphism L ∼ = K − , where thelatter denotes the anti-canonical bundle. The restriction of ( L , ∇ ) to a Lagrangian˜ L is flat, and K − | ˜ L carries a natural non-vanishing “Maslov section”. We call aLagrangian ˜ L Bohr-Sommerfeld monotone if • ( L , ∇ ) | ˜ L has trivial monodromy • under the induced identification L| ˜ L ∼ = K − | ˜ L , the Maslov section is homo-topic to a non-vanishing flat sectionFor us the crucial implication of this condition is the following lemma that followsfrom [WW10, Lemma 4.1.5] and Gromov compactness: Lemma 2.14.
If each of L i , L and L (cid:48) are Bohr-Sommerfeld monotone, then thereare only finitely many pseudo-holomorphic marked discs (in the -dimensional mod-uli) with boundary on these Lagrangians and with fixed asymptotic conditions at themarkings. The conclusion of the lemma also follows from the first option in Assumption 1.2,as we remarked in Section 1.2.2.
Energy spectrum and definability of Fukaya category over smallersubfields.
As remarked, the monotonicity of the generators L i imply that thecoefficients of the A ∞ -structure are finite, i.e. the Fukaya category can be definedover the field of Novikov polynomials Q ( T R ). The assumptions on L, L (cid:48) imply thatthe Yoneda modules h L (cid:48) and h L are also defined over Q ( T R ). The purpose of thissection is to find smaller fields of definition for Fukaya category.Since the boundary of marked discs used to define F ( M, Λ) are all on various L i , theenergy of such discs would take values in the image of ω M : H ( M, (cid:83) i L i ; Z ) → R .In other words, as we construct the Fukaya category using pearl complex for theimmersed Lagrangian (cid:83) i L i , we see that the energies of all discs involved lie in thefinitely generated group ω ( H ( M, (cid:83) i L i ; Z )). Hence, there exists a finitely generatedadditive subgroup G pre ⊂ R that contains all possible energies. In the statementof Theorem 1.1, we used two other monotone Lagrangians denoted by L and L (cid:48) .Without loss of generality, assume the discs with boundary conditions on L, L (cid:48) inaddition to L i also have topological energy inside G pre .In this case, the Fukaya category of M with objects L i is defined over Q ( T g : g ∈ G pre ) ⊂ Λ. Because of the last assumption, the left/right modules correspondingto
L, L (cid:48) are also defined over Q ( T g : g ∈ G pre ) (in the first option in Assumption1.2, we still do not need infinite series, as the result of [Oh93] still holds when thereis a single boundary component of the disc mapping to either L or L (cid:48) , i.e. the sumsdefining module structure are finite). Remark 2.15. G pre is not invariant under Hamiltonian perturbations. Hence, theinvariance of the Fukaya category holds only after base change to a larger field.As mentioned, the formula (1.4) will be used to define family Floer homology. Inparticular, we will evaluate at z = T f for a small rational number f . Therefore, wedefine: Definition 2.16.
Let G ⊂ R be the additive subgroup spanned by G pre and α ( C )where C is an integral 1-cycle in M , and α is the closed 1-form fixed in Section 1satisfying φ = φ α . Given prime p , let G ( p ) be the set { gm : g ∈ G, m ∈ Z , p (cid:45) m } .Since G is finitely generated and torsion free, one can find a basis of G over Z . Thisbasis induce a basis of G ( p ) over Z ( p ) = { nm : n, m ∈ Z , p (cid:45) m } , and G ( p ) is a free Z ( p ) -module.Since G ⊂ G ( p ) are ordered groups, Q (( T G )) ⊂ Q (( T G ( p ) )) are defined in thestandard way, i.e. they are Novikov series that involve only T g -terms such that g ∈ G , resp. g ∈ G ( p ) . Fix the following notation: Notation.
Let K = Q ( T G ( p ) ) be the field of rational functions in T g , g ∈ G ( p ) .This field is not finitely generated over Q but it can be obtained by adding rootsto finitely generated field Q ( T G ). A corollary of the remarks on the energy of discsdefining Fukaya category and Yoneda modules h L , h L (cid:48) imply that the coefficientsof the structure maps are in K . Therefore, the Fukaya category, as well as thestructure maps are defined over K . In other words, we have a proper A ∞ -category F ( M, K ) defined over K that bases changes (strictly) to F ( M, Λ), as well as A ∞ -modules over F ( M, K ) still denoted by h L , h L (cid:48) . Remark 2.17.
By Remark 2.15, F ( M, K ) is not invariant under Hamiltonianperturbations either: the continuation maps are defined only after base change toa slightly larger field that depend on the continuation data.3.
Families of bimodules and symplectomorphisms
Family of bimodules over the Novikov field.
Recall that α is a fixedclosed 1-form on M such that φ α = φ , where φ tα denote the flow of X α , which isthe vector field satisfying ω ( · , X α ) = α . One can see φ tα as a family of symplec-tomorphism and up to some technicalities it defines a class of bimodules by therule(3.1) ( L i , L j ) (cid:55)→ HF ( L i , φ t ( L j ))Our first goal in this section is to give another description of this family inspiredby family Floer homology and quilted Floer homology (see [Ma’15], [Gan12]). Thenotion of family we use is essentially due to Seidel (see [Sei14]). He allows affinecurves as the parameter space of the family. For our purposes, this is insufficient.A natural “space” one can could work with has ring of functions(3.2) Λ { z R } [ a,b ] := { (cid:88) a r z r : , where r ∈ R , a r ∈ Λ } TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY17 where the series satisfy the convergence condition val T ( a r ) + rν → ∞ , for all ν ∈ [ a, b ]. The isomorphism type of this ring is independent of a and b as longas a < b , and we consider it as a non-Archimedean analogue of the interval [ a, b ].For instance, given f ∈ [ a, b ], there exists an evaluation map Λ { z R } [ a,b ] → Λ givenby z (cid:55)→ T f (and z r (cid:55)→ T fr , which does not follow automatically). More will beexplained in Appendix A, but we note that we often omit a and b from the notation,and use Λ { z R } to denote Λ { z R } [ a,b ] for some a < < b (therefore, “ z = 1” can bethought as a point of the heuristic parameter space).On the other hand, by monotonicity, we will only need finite series, except for somesemi-continuity statements. Therefore, until Section 5, where the monotonicityassumption is dropped, we instead consider the ring(3.3) Λ[ z R ] = { finite sums (cid:88) a r z r : , where r ∈ R , a r ∈ Λ } , and analogously defined K [ z R ], as heuristic ring of functions of our parameter space.Note that we will not attempt to associate a geometric spectrum to these rings,and we often refer to them as the parameter space, by abuse of terminology.Let B be a smooth and proper A ∞ -category over K ⊂ Λ or Λ.
Definition 3.1. A Novikov family M of bimodules over B is an assignmentof a free ( Z / Z )-graded K [ z R ]-module, resp. Λ[ z R ]-module, M ( ˜ L, ˜ L (cid:48) ) to every pairof objects together with K [ z R ]-linear, resp. Λ[ z R ]-linear, structure maps B ( L (cid:48) , L (cid:48) ) ⊗ . . . B ( L (cid:48) m , L (cid:48) m − ) ⊗ M ( L n , L (cid:48) m ) ⊗ . . . B ( L , L )(3.4) → M ( L , L (cid:48) )[1 − m − n ](3.5)satisfying the standard A ∞ -bimodule equations. A (pre-)morphism of two fam-ilies M and M (cid:48) is a collection of K [ z R ]-linear, resp. Λ[ z R ]-linear, maps f m | | n : B ( L (cid:48) , L (cid:48) ) ⊗ . . . B ( L (cid:48) m , L (cid:48) m − ) ⊗ M ( L n , L (cid:48) m ) ⊗ . . . B ( L , L )(3.6) → M (cid:48) ( L , L (cid:48) )[ − m − n ](3.7)The category of Novikov families form a K [ z R ]-linear, resp. Λ[ z R ]-linear, pre-triangulated dg category, where the differential and composition are given by stan-dard formulas for bimodules. A morphism of families means a closed pre-morphism.The cone of a morphism is defined as the cone of underlying bimodules, equippedwith the obvious family structure ( K [ z R ]-linear, resp. Λ[ z R ]-linear, structure) itself.We use the term Novikov family for both K and Λ. Which type of family we areworking with will be clear from the notation.For an additive subgroup G ⊂ R such that T g ∈ K for all g ∈ G , one can alsoconsider the ring K [ z G ], sums of monomials z g , g ∈ G . Definition 3.2.
We define the family M Kα of F ( M, K )-bimodules via(3.8) ( L i , L j ) (cid:55)→ M Kα ( L i , L j ) = CF ( L i , L j ; K ) ⊗ K K [ z R ]where CF ( L i , L j ; K ) = K (cid:104) L i ∩ L j (cid:105) . To define the differential, consider the pseudo-holomorphic strips with boundary on L i and L j defining the Floer differential.Recall that we chose a base point on M and relative homotopy classes of paths Figure 3.1.
The counts defining M Λ α and M Kα from this point to generators of CF ( L i , L j ; Λ). Concatenate the chosen path fromthe base point to the input chord of the strip, the L i side of the boundary and thereverse of the path from the base point to output. Denote this class by [ ∂ h u ], where u is the Floer strip. Then, define the differential for (3.8) via the formula(3.9) µ ( x ) = (cid:88) ± T E ( u ) z α ([ ∂ h u ]) .y where x and y are generators of CF ( L i , L j ; Λ) and u ranges over the Floer stripswith given boundary conditions, with input x and output y . We obtain the moregeneral structure maps of the family of bimodules by deforming the structure mapsfor the diagonal bimodule. Namely, the structure maps for diagonal bimodule send( x k , . . . , x | x | x (cid:48) , . . . x (cid:48) l ) to signed sum(3.10) (cid:88) ± T E ( u ) .y where the sum ranges over the discs with input x (cid:48) l . . . x, . . . , x k and with output y .We define the structure maps for the family M Kα via the formula(3.11) (cid:88) ± T E ( u ) z α ([ ∂ h u ]) .y where [ ∂ h u ] denotes the class obtained by concatenating the chosen path from thebase point to x , u ◦ γ where γ is a path in the marked disc from the markedpoint corresponding to x to marked point corresponding to y , and the reverse ofthe chosen path from the base point to y (See Figure 3.1, [ ∂ h u ] is obtained byconcatenating the wavy line in the figure with paths to base point). Definition 3.3.
Let M Λ α denote the Λ[ z R ]-linear Novikov family of F ( M, Λ)-bimodules that is obtained by replacing K by Λ in Definition 3.2. Equivalently, thisfamily can be obtained by extension of the coefficients of M Kα along the inclusionmap K → Λ. Note 3.4.
Let ˜ L and ˜ L (cid:48) be two Lagrangians that satisfy the conditions on L and L (cid:48) in Assumption 1.2. As mentioned in Remark 2.13, one can abstractly extend theFukaya category to include ˜ L and ˜ L (cid:48) and the bimodule M Kα to (cid:103) M Kα . However, thisextension by abstract means, whereas the notation (cid:103) M Kα ( ˜ L, ˜ L (cid:48) ) suggests the sameconcrete definition as (3.8). Therefore, we will use the complex(3.12) h ˜ L (cid:48) ⊗ F ( M,K ) M Kα ⊗ F ( M,K ) h ˜ L TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY19 instead, which is well-defined whenever the modules h ˜ L (cid:48) and h ˜ L are defined over F ( M, K ).Define M Λ fα := M Λ α | z = T f and M Kfα := M Kα | z = T f , i.e. the base change of therespective family under the map Λ[ z R ] → Λ, resp. K [ z R ] → K that sends z r to T fr . The latter makes sense only for f ∈ G ( p ) ⊂ R . These are bimodules over F ( M, Λ), resp. F ( M, K ).Later we will show M K − fα , together with the left-module corresponding to L andright module corresponding to L (cid:48) can be used to recover the groups HF ( φ fα L, L (cid:48) )for some f , via the formula(3.13) HF ( φ fα ( L ) , L (cid:48) ) ∼ = H ∗ ( h L (cid:48) ⊗ F ( M,K ) M Λ − fα ⊗ F ( M,K ) h L ) ⊗ K Λwhere h L (cid:48) denotes the right Yoneda module corresponding to L (cid:48) and h L denotesthe left Yoneda module corresponding to L .We also note: Lemma 3.5.
For f, f (cid:48) ∈ R such that | f | , | f (cid:48) | are small, M Λ( f + f (cid:48) ) α (cid:39) M Λ fα ⊗ F ( M, Λ) M Λ f (cid:48) α . The same statement holds for Λ replaced by K if f, f (cid:48) ∈ Z ( p ) . We will prove a p -adic version of this statement in Proposition 3.20. A similarsemi-continuity argument works in this case too, only one has to consider small f, f (cid:48) with respect to Archimedean absolute value. More precisely, one can write amap(3.14) M Λ fα ⊗ F ( M, Λ) M Λ f (cid:48) α → M Λ( f + f (cid:48) ) α varying continuously in f and f (cid:48) that is similar to (3.25) and that restrict to aquasi-isomorphism at f = f (cid:48) = 0. The main addition to ideas involved in the proofof Proposition 3.20 are about more general semi-continuity statements of chaincomplexes over Λ[ z R ] (or over the ring Λ { z R } ). We give a sketchy proof of Lemma3.5 in Appendix A. Remark 3.6.
One can give another proof of this statement for f, f (cid:48) ∈ Z ( p ) withsmall p -adic absolute value. See Corollary 3.24.3.2. p-adic arcs in Floer homology. Let p > Q ( T G ) (cid:44) → Q p such that elements of the form T g map to elements of 1 + p Z p . More precisely, fix an integral basis g , . . . , g k of thegroup G . Let µ , . . . , µ k ∈ Z p be algebraically independent over Q . Define a map g i (cid:55)→ pµ i from Q ( T G ) → Q p . This is well defined since T g i are algebraicallyindependent. We will extend it to a map Q ( T G ( p ) ) → Q p using Definition 3.7.Recall that Q p (cid:104) t (cid:105) = { (cid:80) n a n t n : n N , a n ∈ Q p , val p ( a n ) → ∞} is the Tate algebraover Q p with one variable, and it can be thought as the set of analytic functions onthe p -adic unit disc Z p : Definition 3.7. [[BSS17]] Let v = 1 + ν ∈ p Z p . Define v t ∈ Q p (cid:104) t (cid:105) to be thefunction(3.15) (1 + ( v − t = (1 + ν ) t := ∞ (cid:88) i =0 (cid:18) ti (cid:19) ν i The convergence of (3.15) on Z p is clear, see also [BSS17, Proposition 2.1]. Let uslist some properties, mainly following [BSS17]:(1) v t is the n th power of v when t = n ∈ N (2) v t + t (cid:48) = v t v t (cid:48) ∈ Q p (cid:104) t, t (cid:48) (cid:105) (3) ( v v ) t = v t v t Proof. (1) follows from the binomial theorem. To see (3), check it first on N ⊂ Z p using (1). A functional equation that holds on a dense (or just infinite) subset of Z p holds over Q p (cid:104) t (cid:105) by Strassman’s theorem (see [Kat07, Theorem 3.38]).Similarly, to see (2), check it first on N × N ⊂ Z p × Z p using (1), and concludeby density that the equation holds on Z p × Z p . By an iterated application ofStrassman’s theorem, we obtain (2). (cid:3) We can define the map from µ : Q ( T G ( p ) ) → Q p as the map satisfying(3.16) T g i /n (cid:55)→ (1 + pµ i ) /n where p (cid:45) n and 1 /n th power is taken by specialization of (1 + pµ i ) t to t = 1 /n .Therefore, the elements of type T a map to elements of 1 + p Z p . Since µ i arealgebraically independent, this gives a well-defined map Q ( T G ( p ) ) → Q p . To seethis, first notice Q [ T G ( p ) ] → Q p is well-defined as the former is the group algebra.If an element of Q [ T G ( p ) ] maps to 0 ∈ Q p , this gives a non-trivial algebraic relationover Q of elements (1 + pµ i ) /N for some large N . This is impossible as it wouldimply an algebraic relation between 1 + pµ i . We will denote µ ( T a ) also by T aµ ,where a ∈ G ( p ) .This defines a map K → Q p . Let F ( M, Q p ) denote the category obtained byextending the coefficients of F ( M, K ) to Q p .The following is the p -adic analogue of Definition 3.1: Definition 3.8.
For a given smooth and proper A ∞ -category B over Q p , a p -adic family M of bimodules over B is an assignment of a free ( Z / Z )-graded Q p (cid:104) t (cid:105) -module M ( L, L (cid:48) ) to every pair of objects together with Q p (cid:104) t (cid:105) -linear structuremaps B ( L (cid:48) , L (cid:48) ) ⊗ . . . B ( L (cid:48) m , L (cid:48) m − ) ⊗ M ( L n , L (cid:48) m ) ⊗ . . . B ( L , L )(3.17) → M ( L , L (cid:48) )[1 − m − n ](3.18)satisfying the standard bimodule equations. A (pre)-morphism of two families M and M (cid:48) is a collection of Q p (cid:104) t (cid:105) -linear maps f m | | n : B ( L (cid:48) , L (cid:48) ) ⊗ . . . B ( L (cid:48) m , L (cid:48) m − ) ⊗ M ( L n , L (cid:48) m ) ⊗ . . . B ( L , L )(3.19) → M (cid:48) ( L , L (cid:48) )[ − m − n ](3.20)As before, the category of p -adic families form a Q p (cid:104) t (cid:105) -linear pre-triangulated dgcategory, where the differential and composition are given by standard formulas forbimodules, and a morphism of families means a closed pre-morphism. The coneof a morphism is defined as the cone of underlying map of bimodules, equippedwith the natural Q p (cid:104) t (cid:105) -linear structure. TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY21 Definition 3.8 easily generalizes to other non-Archimedean fields extending Q p aswell as to Tate algebra with several variables Q p (cid:104) t , . . . t n (cid:105) . Let M be a family and q be a Q p -point of Sp ( Q p (cid:104) t (cid:105) ), i.e. a continuous ring homomorphism Q p (cid:104) t (cid:105) → Q p .Note that these ring homomorphisms are in correspondence with elements of Z p .One can define the restriction M | t = q as M ⊗ Q p (cid:104) t (cid:105) Q p . This is an A ∞ -bimodule over B . Example 3.9.
For any bimodule M over B , one can define a p-adic family by M ( L, L (cid:48) ) = M ( L, L (cid:48) ) ⊗ Q p Q p (cid:104) t (cid:105) with the structure maps obtained by base change.This type of family has the same restrictions at every point. In particular, one canlet M to be a Yoneda bimodule h L (cid:2) h L (cid:48) (i.e. the exterior tensor product of leftand right Yoneda modules, see [Gan12, (2.83),(2.84)] for its structure maps). Wecall such a family a constant family of Yoneda bimodules .One can define convolution of two families. First, recall the convolution of bimod-ules over B : Definition 3.10.
Let M and M be two bimodules over B . Then, M ⊗ B M isthe bimodule defined by(3.21) ( L, L (cid:48) ) (cid:55)−→ (cid:77) M ( L k , L (cid:48) ) ⊗ B ( L k − , L k ) ⊗ · · · ⊗ B ( L , L ) ⊗ M ( L, L )[ k ]where the direct sum is over all ordered sets ( L , . . . , L k ) for all k ∈ Z ≥ . Thedifferential is given by( m ⊗ b ⊗ · · · ⊗ b f ⊗ m ) (cid:55)→ (cid:88) ± µ M ( m ⊗ b ⊗ . . . ) ⊗ · · · ⊗ m +(3.22) (cid:88) ± m ⊗ · · · ⊗ µ M ( · · · ⊗ m ) + (cid:88) ± m ⊗ · · · ⊗ µ B ( . . . ) ⊗ · · · ⊗ m and other structure maps are defined similarly. Definition 3.11.
Given families M and M over Sp ( Q p (cid:104) t (cid:105) ), one can endow M ⊗ B M with the structure of a family over Q p (cid:104) t , t (cid:105) . One obtains a family over Sp ( Q p (cid:104) t (cid:105) ) via base change along the (co)diagonal map Q p (cid:104) t , t (cid:105) → Q p (cid:104) t (cid:105) , t , t (cid:55)→ t .We denote this family by M ⊗ rel B M .One can also construct M ⊗ rel B M by performing the construction in Definition3.10 Q p (cid:104) t (cid:105) -linearly. One can see M ⊗ rel B M as the fiberwise convolution of twofamilies, i.e. as the tensor product relative to base; hence, the notation. Example 3.12.
Let M and M be two constant families associated to bimodules M and M over B . Then, M ⊗ rel B M is the constant family associated to M ⊗ B M . In particular, if M = h L (cid:2) h L (cid:48) and M = h L (cid:2) h L (cid:48) , then M ⊗ rel B M is theconstant family associated to B ( L , L (cid:48) ) ⊗ ( h L (cid:2) h L (cid:48) ).By Morita theory, a family of bimodules can be thought as a family of endomor-phisms of the category. Therefore, if the parameter space of the family is a group,one can study “actions of this group on the category”. Observe Q p (cid:104) t (cid:105) is the ringof functions of a group, and is itself a Hopf algebra over Q p with comultiplicationgiven by ∆ : t (cid:55)→ t ⊗ ⊗ t (the counit is given by e : t (cid:55)→ t (cid:55)→ − t ). Figure 3.2.
The counts defining M Q p α Let π i : Q p (cid:104) t (cid:105) → Q p (cid:104) t , t (cid:105) denote the map t (cid:55)→ t i for i = 1 ,
2. Given p -adic family M , one can extend the coefficients along π , π and ∆ to define three 2-parameter p -adic family of bimodules denoted by π ∗ M , π ∗ M and ∆ ∗ M (we identify Q p (cid:104) t , t (cid:105) with a suitable completion of Q p (cid:104) t (cid:105) ⊗ Q p (cid:104) t (cid:105) such that t ⊗ t , ⊗ t = t ). Define: Definition 3.13. A p -adic family M of bimodules over B is called group-like if∆ ∗ M (cid:39) π ∗ M ⊗ rel B π ∗ M and if the restriction to counit M | t = e is quasi-isomorphicto diagonal bimodule.Clearly, if M is group-like, and f , f ∈ Z p , then M | t = f + f (cid:39) M | t = f ⊗ B M | t = f .We want to construct an explicit group-like p -adic family M Q p α of bimodules over F ( M, Q p ). To this end, associate(3.23) ( L i , L j ) (cid:55)→ M Q p α ( L i , L j ) = CF ( L i , L j ; Q p ) ⊗ Q p (cid:104) t (cid:105) The structure maps are defined via the formula(3.24) ( x , . . . , x k | x | x (cid:48) , . . . , x (cid:48) l ) (cid:55)→ (cid:88) ± T E ( u ) µ T tα ([ ∂ h u ]) µ .y where the sum ranges over the marked discs with input x (cid:48) l . . . , x, x k , . . . , x andoutput y . The class [ ∂ h u ] ∈ H ( M ; Z ) is defined as before, and α ([ ∂ h u ]) ∈ G by definition of G ; therefore, T α ([ ∂ h u ]) µ = µ ( T α ([ ∂ h u ]) ) ∈ p Z p is defined. Let T tα ([ ∂ h u ]) µ ∈ Q p (cid:104) t (cid:105) be its “ t th power” as in Definition 3.7. The sum is finite andit is easy to check the bimodule equation is satisfied (see Figure 3.3 for instance).It is immediate that the restriction to t = 0 is isomorphic to diagonal bimodule of F ( M, Q p ). See also Figure 3.2.To prove that this family is group-like, our next task to write a closed morphismof families(3.25) π ∗ M Q p α ⊗ rel F ( M, Q p ) π ∗ M Q p α → ∆ ∗ M Q p α such that the restriction of (3.25) to t = t = 0 to the quasi-isomorphism F ( M, Q p ) ⊗ F ( M, Q p ) F ( M, Q p ) → F ( M, Q p ) from the convolution of diagonal bimodule with itself todiagonal bimodule. TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY23 Figure 3.3.
Some degenerations of discs as in Figure 3.2Given an A ∞ -category B , it is a general result that B ⊗ B B (cid:39) B . Moreover, thebimodule quasi-isomorphism from left hand side to the right is given by(3.26) g k | | l : ( x , . . . , x k | x ⊗ b ⊗ · · · ⊗ b f ⊗ x (cid:48) | x (cid:48) , . . . x (cid:48) l ) (cid:55)→± µ B ( x , . . . , x k , x, b , . . . b f , x (cid:48) , x (cid:48) , . . . x (cid:48) l )Here, x ⊗ b ⊗ · · · ⊗ b f ⊗ x (cid:48) is the element of B ⊗ B B . When B is the Fukaya category,this map is geometrically given by the count of marked discs as usual. To deformit, we define the following cohomology classes: given a pseudo-holomorphic disc u with output y and with input given by generators x , . . . , x k , x , b . . . , b f , x (cid:48) , x (cid:48) , . . . x (cid:48) l (in counter-clockwise direction after output), define [ ∂ α ] ∈ H ( M ; Z ) tobe the path obtained by concatenating the fixed path from the base point of M to generator x , the image under u of a path from the input marked point for x tooutput marked point, and the reverse of the path from the base point to y . Wethink of this class as the portion of boundary of u from x to y . Similarly define[ ∂ u ] ∈ H ( M ; Z ) by replacing x with x (cid:48) . The class [ ∂ u ] can be thought as theportion of boundary from x (cid:48) to y . In other words, the paths [ ∂ u ] and [ ∂ u ] areobtained by concatenating the u -image of the respective wavy line in Figure 3.4with chosen paths from the base point to the generator. To define the map (3.25),fix input x , . . . , x k , x, b . . . , b f , x (cid:48) , x (cid:48) , . . . x (cid:48) l as above. The coefficient of y underthe map (3.25) is given by(3.27) (cid:88) ± T E ( u ) µ T t α ([ ∂ u ]) µ T t α ([ ∂ u ]) µ .y where u range over the pseudo-holomorphic discs with given input and output. SeeFigure 3.4. The input x , resp. x (cid:48) is associated to marked point labeled as π ∗ M Q p α ,resp. π ∗ M Q p α , and the output y is associated to ∆ ∗ M Q p α .Recall that T α ([ ∂ u ]) µ := µ ( T α ([ ∂ u ]) ) ∈ p Z p . As before T t α ([ ∂ u ]) µ ∈ Q p (cid:104) t (cid:105) ⊂ Q p (cid:104) t , t (cid:105) is its t th -power as in Definition 3.7. T t α ([ ∂ u ]) µ is defined similarly. It iseasy to check this defines a map of bimodules (3.25). See for instance Figure 3.5.Moreover, it restricts to the standard quasi-isomorphism of diagonal bimodules at t = t = 0 defined by (3.26).Our next task is to prove (3.25) is a quasi-isomorphism. This relies on a semi-continuity argument for which we need a properness result for the domain of (3.25), Figure 3.4.
The counts defining (3.25)
Figure 3.5.
Some degenerations of moduli of discs as in Figure 3.4i.e. we need to show that this family is a cohomologically finitely generated Q p (cid:104) t (cid:105) -module at every pair of objects ( L, L (cid:48) ). We need some technical preparation forthis:
Definition 3.14.
A family M is called perfect if it is quasi-isomorphic to a directsummand of a twisted complex of constant families of Yoneda bimodules in thecategory of families. It is called locally perfect if there is an admissible cover of Sp ( Q p (cid:104) t (cid:105) ) (or any other parameter space we are using) such that each restriction of M is perfect. M is called proper if the cohomology of M ( L, L (cid:48) ) is finitely generatedover Q p (cid:104) t (cid:105) for all L, L (cid:48) .Clearly, perfect implies locally perfect and if B is a proper category, locally perfectimplies proper. We will see that proper implies perfect for a smooth, proper A ∞ -category B . For simplicity, we start with the following: Lemma 3.15.
Let B be a smooth and proper A ∞ -category over a field of charac-teristic . A proper right/left-module or a bimodule over B is perfect.Proof. Let M be a proper right module over B . Then, M ⊗ B B (cid:39) M as rightmodules. One can represent the diagonal bimodule B in terms of Yoneda bimodules h L (cid:2) h L (cid:48) , where h L denote the left Yoneda module, h L (cid:48) denote the right Yonedamodule, and (cid:2) is for exterior tensor product (see [Gan12, (2.83),(2.84)]). Observe(3.28) M ⊗ B ( h L (cid:2) h L (cid:48) ) (cid:39) M ( L ) ⊗ h L (cid:48) TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY25 Therefore, M can be written as direct a summand of a twisted complex (iteratedcone) of modules of type M ( L ) ⊗ B h L (cid:48) , but the latter is quasi-isomorphic to finitelymany copies of h L (cid:48) as M is proper. This concludes the proof.The proof is the same for left modules, and for bimodules one uses(3.29) M (cid:39) B ⊗ B M ⊗ B B together with the finite resolution of the diagonal on both sides. (cid:3) Assume B is smooth and proper A ∞ -category over Q p (or over a subfield of Q p ).Lemma 3.15 immediately generalizes to: Lemma 3.16.
A proper p -adic family of bimodules over B is perfect.Proof. Let M be a proper family. The quasi-isomorphism(3.30) M (cid:39) B ⊗ B M ⊗ B B still holds true, and by using the representation of the diagonal bimodule as adirect summand of twisted complex of Yoneda bimodules, we see that M is quasi-isomorphic to a direct summand of a twisted complex of families of the form(3.31) ( h L (cid:2) h L (cid:48) ) ⊗ B M ⊗ B ( h L (cid:2) h L (cid:48) ) (cid:39) M ( L , L (cid:48) ) ⊗ ( h L (cid:2) h L (cid:48) )Therefore, it suffices to show the last type of family is perfect. Note that here weconsider M ( L , L (cid:48) ) as a chain complex over Q p (cid:104) t (cid:105) , and ( h L (cid:2) h L (cid:48) ) is the Yonedabimodule as before. The family structure on their tensor product is obvious.By assumption, M ( L , L (cid:48) ) has finitely generated cohomology over Q p (cid:104) t (cid:105) (or whicheverparameter space we are using). By [Ked04, Proposition 6.5], every finitely generatedmodule over the affinoid domain has a finite free resolution, and this immediatelyimplies the existence of finitely generated free complex C of modules over the affi-noid domain quasi-isomorphic to M ( L , L (cid:48) ). It is easy to see that(3.32) C ⊗ ( h L (cid:2) h L (cid:48) ) (cid:39) M ( L , L (cid:48) ) ⊗ ( h L (cid:2) h L (cid:48) )is perfect. This completes the proof. (cid:3) Remark 3.17.
The notions of p -adic family of left/right modules can be definedsimilarly. Then, Lemma 3.16 still holds for such families. Corollary 3.18.
Let M and M be two proper p -adic families (over an affinoiddomain as before). Then the convolution M ⊗ rel B M is proper.Proof. By Lemma 3.16, both families are perfect; therefore, they can be representedas summands of complexes of constant families of Yoneda bimodules. It follows fromExample 3.12 that the convolution of two constant families of Yoneda bimodulesis perfect; hence, proper. Therefore, M ⊗ rel B M can be represented as the directsummand of a twisted complex (iterated cone) of proper modules and it is properitself. (cid:3) Corollary 3.19.
Let N denote the cone of the morphism (3.25). Then, H ∗ ( N ( L i , L j )) is a finitely generated module over Q p (cid:104) t , t (cid:105) for all L i , L j , i.e. N is proper. Proof.
By construction, π ∗ M Q p α and π ∗ M Q p α are both proper modules; therefore,by Corollary 3.18, π ∗ M Q p α ⊗ rel F ( M, Q p ) π ∗ M Q p α is also proper. Since ∆ ∗ M Q p α is propertoo, the cone of a morphism(3.33) π ∗ M Q p α ⊗ rel F ( M, Q p ) π ∗ M Q p α → ∆ ∗ M Q p α is proper. This completes the proof. (cid:3) Proposition 3.20. H ∗ ( N ) vanishes on the smaller affinoid domain Q p (cid:104) t /p n , t /p n (cid:105) for a sufficiently large n . Therefore, M Q p α | Q p (cid:104) t/p n (cid:105) is group-like.Proof. By Lemma 3.19, the cohomology of N is finitely generated over Q p (cid:104) t , t (cid:105) ,and it vanishes at t = t = 0 (as (3.25) is a quasi-isomorphism at t = t = 0). Asin the proof of Lemma 3.16, [Ked04, Proposition 6.5] implies that each N ( L i , L j )is quasi-isomorphic to a free finite complex of Q p (cid:104) t , t (cid:105) -modules. Then, the resultfollows from the following standard result, applied to a free finite complex quasi-isomorphic to (cid:76) i,j N ( L i , L j ). (cid:3) Lemma 3.21.
Let ( C, d ) be a free, finite complex of Q p (cid:104) t , t (cid:105) modules whoserestriction to t = t = 0 is acyclic. Then, for sufficiently large n , the restrictionof ( C, d ) to Q p (cid:104) t /p n , t /p n (cid:105) is acyclic.Proof. By choosing a trivialization of the graded module C , one can see d as asquare matrix of elements of Q p (cid:104) t , t (cid:105) . The rank of H ∗ ( C, d ) at a point ( p , p ) isthe same as(3.34) rank ( C ) − rank ( d | t = p ,t = p )The rank of d is maximal and is equal to rank ( C ) / t = t = 0, and there isa square submatrix of d of size rank ( C ) /
2, whose determinant is non-vanishing at t = t = 0. Let f ( t , t ) ∈ Q p (cid:104) t , t (cid:105) denote the determinant of this submatrix,normalized by a constant so that f (0 ,
0) = 1.As f (0 ,
0) = 1, f has an inverse defined on a formal neighborhood of (0 , f ( t , t ) = 1 + (cid:80) a k,l t k t l , where a k,l ∈ Q p . Then val p a k,l → ∞ as f convergesover Z p × Z p . Therefore, all but finitely many of a k,l are O ( p ), i.e. a k,l ∈ p Z p . Bychoosing n large enough, we can ensure other terms except the constant term arealso O ( p ); hence, f takes non-zero values over p n Z p × p n Z p and its inverse convergeover this set. In other words, f is invertible in the ring Q p (cid:104) t /p n , t /p n (cid:105) .This implies that for any ring homomorphism Q p (cid:104) t /p n , t /p n (cid:105) → Q to a field, therank of H ∗ ( C ⊗ Q, d ⊗ Q ) is 0, i.e. H ∗ ( C ⊗ Q, d ⊗ Q ) vanishes. Then, given curve S ⊂ Sp ( Q p (cid:104) t /p n , t /p n (cid:105) ) and a ring homomorphism O an ( S ) → Q to a field wehave a spectral sequence with E -page(3.35) T or p ( H − q ( C | O an ( S ) , d | O an ( S ) ) , Q ) ⇒ H − p − q ( C ⊗ Q, d ⊗
1) = 0where the
T or is also over O an ( S ), and it is supported at p = 0 , S is a curve.Therefore, this spectral sequence degenerates at E -page. Hence, H − q ( C | O an ( S ) , d | O an ( S ) ) ⊗ TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY27 Q = 0, for all such Q , which implies H − q ( C | O an ( S ) , d | O an ( S ) ) = 0.This holds forany curve S . Likewise, consider the spectral sequence(3.36) T or p ( H − q ( C, d ) , O an ( S )) ⇒ H − p − q ( C | O an ( S ) , d | O an ( S ) ) = 0This time, T or is over Q p (cid:104) t /p n , t /p n (cid:105) , and T or p ( · , O an ( S )) is supported at p = 0 , S is of codimension 1. As before, this lets us conclude H − q ( C, d ) ⊗ O an ( S ) forall such S , and this implies H ∗ ( C | Q p (cid:104) t /p n ,t /p n (cid:105) , d | Q p (cid:104) t /p n ,t /p n (cid:105) ) = 0, finishing theproof. (cid:3) Remark 3.22.
It is easy to see that Q p (cid:104) t/p n (cid:105) is the set of power series in t thatconverge over p n Z p . Therefore, the Berkovich spectrum Sp ( Q p (cid:104) t/p n (cid:105) ) is actually asmaller subdomain inside Sp ( Q p (cid:104) t (cid:105) ); however, contrary to Archimedean geometry,it is also a subgroup. In particular, the iterates of this “neighborhood of (0 , Q p (cid:104) t (cid:105) . Remark 3.23.
Presumably, using the machinery of deformation classes of [Sei14],one can show that the family M Q p α is already group-like without restricting to Sp ( Q p (cid:104) t/p n (cid:105) ).Even though Proposition 3.20 is about the family Q p , it still allows us to con-clude a weak group-like statement for the family M Λ α . Recall the notation M Kfα := M Kα | z = T f , for f ∈ R . Corollary 3.24.
Let f, f (cid:48) ∈ p n Z ( p ) ⊂ Q . Then, M Kfα ⊗ F ( M,K ) M Kf (cid:48) α (cid:39) M K ( f + f (cid:48) ) α .The same statement holds if K is replaced by Λ .Proof. One can define the map(3.37) M Kfα ⊗ F ( M,K ) M Kf (cid:48) α → M K ( f + f (cid:48) ) α similar to (3.25). More precisely, one would need to replace (3.27) by(3.38) (cid:88) ± T E ( u ) T fα · [ ∂ u ] T f (cid:48) α · [ ∂ u ] .y and it is easy to check this defines a bimodule map. Moreover, after the basechange along µ : K → Q p , the map (3.37) becomes same as (3.27) evaluated at t = f, t = f (cid:48) considered as elements of p n Z p ⊂ Q p . Same holds for the cone of(3.37). By Proposition 3.20, this cone vanishes after base-change, implying it is 0before the base change as well. Therefore, (3.37) is a quasi-isomorphism before thebase change as well. One can then apply base change along the inclusion K → Λto conclude M Λ fα ⊗ F ( M, Λ) M Λ f (cid:48) α (cid:39) M Λ( f + f (cid:48) ) α . (cid:3) Comparison with the action of φ tα and proof of Theorem 1.1 We need the following proposition to conclude the proof of Theorem 1.1:
Proposition 4.1.
For any f ∈ p n Z ( p ) , (4.1) HF ( φ fα ( L ) , L (cid:48) ) ∼ = HF ( L, φ − fα ( L (cid:48) )) ∼ = H ∗ ( M Λ α | z = T − f ( L, L (cid:48) )) and this group has the same dimension over Λ as (4.2) dim K (cid:0) H ∗ (cid:0) h L (cid:48) ⊗ F ( M,K ) M Kα | z = T − f ⊗ F ( M,K ) h L (cid:1)(cid:1) = dim Q p (cid:0) H ∗ (cid:0) h L (cid:48) ⊗ F ( M, Q p ) M Q p α | t = − f ⊗ F ( M, Q p ) h L (cid:1)(cid:1) As before, we denote M Λ α | z = T f by M Λ fα as well. We want to show the bimodules M Λ fα act on the perfect modules coming from Lagrangians in the expected way.Recall that for a given ˜ L ⊂ M satisfying some assumptions (such as ˜ L (cid:116) L i forall i ), we have a right F ( M, Λ)-module h ˜ L that satisfies h ˜ L ( L i ) = CF ( L i , ˜ L ) = CF ( L i , ˜ L ; Λ). The differential and structure maps are defined by counting markeddiscs with one boundary component on ˜ L , and with other boundary components onvarious L i . We will deform this module algebraically to give a simpler descriptionof h φ fα (˜ L ) for small f : Definition 4.2.
Fix homotopy classes of paths from the base point of M to inter-section points ˜ L ∩ L i . Let h algφ fα , ˜ L denote the right F ( M, Λ)-module defined by(4.3) L i (cid:55)→ CF ( L i , ˜ L )and whose differential/structure maps are given by(4.4) (cid:88) ± T E ( u ) T fα ([ ∂ u ]) .y where u range over holomorphic curves with one boundary component on ˜ L (theone on the clockwise direction from the output) and other components on L i . Here, y is the output marked point and [ ∂ u ] ∈ H ( M ; Z ) denote the homology class ofthe path obtained by concatenating the fixed paths from the base point with ( u -image of) the wavy path in Figure 4.1 from the module input to module output y (this is very related to class previously denoted by [ ∂ u ], and we use the samenotation as no confusion can arise).Clearly, h algφ α , ˜ L = h ˜ L ; therefore, one can see h algφ fα , ˜ L as a deformation of h ˜ L . The proofof the following Lemma implies (4.4) converge T -adically, i.e. h algφ fα , ˜ L is well-defined,for small | f | : Lemma 4.3.
For f ∈ R such that | f | is small, we have h algφ fα , ˜ L (cid:39) h φ fα (˜ L ) . In otherwords, h algφ fα , ˜ L gives what is expected geometrically.Proof. This is an application of Fukaya’s trick, see [Abo14], for instance. Namely,if f is small enough, the intersection points φ fα ( ˜ L ) ∩ L i can be identified with˜ L ∩ L i , and this holds throughout the isotopy. Consider a smooth isotopy ψ f of M fixing all L i setwise and mapping ˜ L to φ fα ( ˜ L ). Choose the almost complexstructures for ( L i , ˜ L ) and ( L i , φ fα ( ˜ L )) to be related by ψ f . Similarly, choose theperturbation data (family of almost complex structures) for discs with boundaryon ( L i , . . . , L i k , ˜ L ) and ( L i , . . . , L i k , φ fα ( ˜ L )) to be related by ψ f . For small | f | ,the tameness and regularity will be preserved. Then, one can identify the moduliof pseudo-holomorphic discs labeled by ( L i , . . . , L i k , ˜ L ) with pseudo-holomorphic TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY29 Figure 4.1.
The counts defining h algφ fα , ˜ L , which gives a simplerquasi-isomorphic description of h φ fα (˜ L ) discs labeled by ( L i , . . . , L i k , φ fα ( ˜ L )), where the identification is via the compositionby ψ f . One has the energy identity(4.5) E ( ψ f ◦ u ) = E ( u ) + f α ([ ∂ u ]) − g ( f, x ) + g ( f, y )where x is the input, y is the output and g ( f, y ) is a real number that only dependson f and y (it may depend on the homotopy class of the isotopy, but the isotopy isgiven in our situation). See [Abo14, Lemma 3.2] for a version of this identity. In oursituation, this is still a similar application of Stokes theorem, namely if one movesthe disc by ψ f , then the energy difference can be measured as the area traced bythe part of boundary labeled by ˜ L (which is homotopic to wavy path in Figure 4.1).This energy difference can be measured by f α ([ ∂ u ]), except one has to correct itby the areas traced by fixed paths from the base point of M to intersection points x, y . The correction can be written in the form g ( f, x ) − g ( f, y ), where g ( f, x ) is anumber that depend on x and continuously on f (we neither need nor attempt tocompute it).After rescaling the generators via x (cid:55)→ T g ( f,x ) x , one can identify the structure mapsof h φ fα (˜ L ) with respect to above almost complex structure, and the structure mapsof h algφ fα , ˜ L defined in (4.4). The quasi-isomorphism class of h φ fα (˜ L ) is independent ofthe almost complex structure; therefore, h algφ fα , ˜ L is well-defined and quasi-isomorphicto h φ fα (˜ L ) . (cid:3) Remark 4.4.
As mentioned in Remark 2.2, it is possible to apply Fukaya’s trickeven if one uses the model of Fukaya category described in [Sei08], namely in thepresence of non-vanishing Hamiltonian terms. In this case, as ˜ L is assumed to betransverse to all L i , we can choose the Floer data for the pair ( L i , ˜ L ) with vanishingHamiltonian term (we do not need Floer data for ( ˜ L, ˜ L ) in order to define h ˜ L ).When we choose the smooth isotopy ψ f as above, we have to make sure it fixes aneighborhood of the intersection points and Hamiltonian chords between various L i . Instead of almost complex structures, we deform Floer data and perturbationdata by ψ f , and we also have to make sure that for small time regularity is preservedand no new Hamiltonian chords are introduced between various L i . Then it is easyto prove an analogue of the energy identity (4.5) and the rest of the argument worksin the same way. Figure 4.2.
The counts defining (4.8) h algφ fα , ˜ L comes from a Novikov family of right modules over F ( M, Λ):
Definition 4.5.
Let h algφ α , ˜ L be the family of right F ( M, Λ)-modules defined by(4.6) L i (cid:55)→ CF ( L i , ˜ L ) ⊗ Λ { z R } and whose differential/structure maps are given by(4.7) (cid:88) ± T E ( u ) z α ([ ∂ u ]) .y where the sum is analogous to (4.4).Recall that Λ { z R } denotes the ring Λ { z R } [ a,b ] for some a < < b , which weintroduced in Section 3 briefly, and explain more in Appendix A. The series (4.7)belong to Λ { z R } [ a,b ] for small | a | and | b | (this is equivalent to convergence of (4.4)for f ∈ [ a, b ]). One can replace this ring by Λ[ z R ] when ˜ L is Bohr-Sommerfeldmonotone. Clearly, h algφ fα , ˜ L = h algφ α , ˜ L | z = T f .Using Lemma 4.3, we can prove: Lemma 4.6.
For f ∈ R with small | f | , h ˜ L ⊗ F ( M, Λ) M Λ fα (cid:39) h φ fα (˜ L ) and h ˜ L (cid:39) h φ fα (˜ L ) ⊗ F ( M, Λ) M Λ − fα (recall M Λ fα := M Λ α | z = T f ).Proof. By Lemma 4.3, it suffices to prove the corresponding statement for h algφ fα , ˜ L .We define a map(4.8) h ˜ L ⊗ F ( M, Λ) M Λ fα → h algφ fα , ˜ L in a way very similar to (3.26). Namely, send(4.9) ( x ⊗ x ⊗ · · · ⊗ x k ⊗ x (cid:48) ; x (cid:48) , . . . x (cid:48) l )to the sum(4.10) (cid:88) ± T E ( u ) T fα ([ ∂ u ]) .y where [ ∂ u ] denote the u -image of the part of boundary of the disc from M Λ fα -input x to output y concatenated with paths from base point of M . See Figure 4.2 (thewavy line is homotopic to mentioned part of the boundary). One can check that TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY31 (4.8) is an A ∞ -module homomorphism in a way similar to (3.25) by consideringthe degenerations of discs in Figure 4.2 analogous to Figure 3.5. When f = 0, (4.8)is a quasi-isomorphism as its cone is the standard bar resolution of h ˜ L . Also, (4.8)can be thought as the specialization of a map of families(4.11) h ˜ L ⊗ F ( M, Λ) M Λ α → h algφ α , ˜ L at z = T f , which can be defined by replacing (4.10) by (cid:80) ± T E ( u ) z α ([ ∂ u ]) .y .We want to apply a similar semi-continuity argument as in the proof that (3.25) isa quasi-isomorphism to show (4.8) is a quasi-isomorphism for small | f | . For this,we want to apply Lemma A.5. Note the quasi-isomorphism of families(4.12) h ˜ L ⊗ F ( M, Λ) M Λ α (cid:39) h ˜ L ⊗ F ( M, Λ) M Λ α ⊗ F ( M, Λ) F ( M, Λ)where the last F ( M, Λ) denotes the diagonal bimodule of the category denoted inthe same way. Since the category F ( M, Λ) is smooth, its diagonal bimodule isquasi-isomorphic to a direct summand of an iterated cone of Yoneda bimodules;hence, (4.12) is quasi-isomorphic to a direct summand of an iterated cone of rightmodules of the form(4.13) h ˜ L ⊗ F ( M, Λ) M Λ α ⊗ F ( M, Λ) ( h L i ⊗ h L j ) (cid:39) M Λ α ( L i , ˜ L ) ⊗ h L j where h L i ⊗ h L j is a Yoneda bimodule, and M Λ α ( L i , ˜ L ) is a free finite complex overΛ { z R } (we can extend the category F ( M, Λ) by ˜ L after making it transverse to all L i and M Λ α also naturally extends in the same concrete way). Therefore, one canrepresent the cone of (4.11) evaluated at any object L k of F ( M, Λ) as a complexsatisfying the assumptions of Lemma A.5. Since we know this cone vanishes at z = 1, we conclude the proof that (4.11) is a quasi-isomorphism when evaluated at z = T f for small | f | , i.e. (4.8) is a quasi-isomorphism by Lemma A.5.Now, define(4.14) h algφ fα , ˜ L ⊗ F ( M, Λ) M Λ − fα → h ˜ L similarly by replacing (4.10) by (cid:80) ± T E ( u ) T fα ([ ∂ l,m u ]) .y , where we define [ ∂ l,m u ] ∈ H ( M ; Z ) to be the class obtained by concatenating the path from h algφ fα , ˜ L input( h ˜ L input in Figure 4.2) to M Λ − fα input ( M Λ α input in Figure 4.2) with the fixedpaths to the base point. It is not hard to check that (4.14) is also an A ∞ -modulehomomorphism, and it is a quasi-isomorphisms when f = 0. Similarly, (4.14)extends to a map of families(4.15) h algφ α , ˜ L ⊗ F ( M, Λ) M Λ − α → h ˜ L where M Λ − α is defined by the obvious modification of M Λ α . The same proof appliesto show (4.14) is a quasi-isomorphism for small | f | . The only difference is that onehas to use the resolution of the diagonal bimodule F ( M, Λ) on both appearancesin the following(4.16) h algφ α , ˜ L ⊗ F ( M, Λ) F ( M, Λ) ⊗ F ( M, Λ) M Λ − α ⊗ F ( M, Λ) F ( M, Λ) (cid:3) An immediate corollary of Lemma 4.6 is the following:
Corollary 4.7.
The Yoneda module h φ fα ( L (cid:48) ) is definable over F ( M, K ) for f ∈ Z ( p ) such that | f | is sufficiently small. In other words, there exists a perfect module over F ( M, K ) such that one obtains a module quasi-isomorphic to h φ fα ( L (cid:48) ) after basechange to Λ (we do not make any uniqueness statement though).Proof. Consider the perfect module h L (cid:48) ⊗ F ( M,K ) M Kα | z = T f over F ( M, K ), whichis well-defined as f ∈ Z ( p ) and T fα ( C ) ∈ K . Once one extends the coefficients toΛ, one obtains h L (cid:48) ⊗ F ( M, Λ) M Λ fα , which is quasi-isomorphic to h φ fα ( L (cid:48) ) by Lemma4.6. (cid:3) Remark 4.8.
One could use Lemma 4.3 directly to prove the corollary. Since thesum defining h L (cid:48) is assumed to be finite, (4.4) also remains finite. Therefore, if f ∈ Z ( p ) , h algφ fα ,L (cid:48) is definable over F ( M, K ) in the sense above, and by Lemma 4.3,it becomes quasi-isomorphic to h φ fα ( L (cid:48) ) after extending the coefficients K ⊂ Λ.We can drop the assumption that | f | is small from Lemma 4.6 after modifying itas follows: Lemma 4.9.
For any f > and ˜ L , one can find a sequence of numbers s s > · · · > s r = f such that(4.17) is satisfied. Moreover, if f ∈ Z ( p ) , resp. f ∈ p n Z ( p ) then one can assume all s i ∈ Z ( p ) , resp. s i ∈ p n Z ( p ) .Proof. Without loss of generality, assume f > φ sα ( ˜ L ) , s ∈ [0 , f ]. By Lemma 4.6, any s ∈ [0 , f ] has an open neighborhood ( s − (cid:15), s + (cid:15) ) suchthat(4.18) h φ sα (˜ L ) (cid:39) h φ s (cid:48) α (˜ L ) ⊗ F ( M, Λ) M Λ( s − s (cid:48) ) α h φ s (cid:48)(cid:48) α (˜ L ) (cid:39) h φ sα (˜ L ) ⊗ F ( M, Λ) M Λ( s (cid:48)(cid:48) − s ) α for s (cid:48) , s (cid:48)(cid:48) ∈ ( s − (cid:15), s + (cid:15) ).Assume (cid:15) is small enough so that Lemma 3.5 is also satisfied for | f | , | f (cid:48) | < (cid:15) , i.e. M Λ fα ⊗ F ( M, Λ) M Λ f (cid:48) α (cid:39) M Λ( f + f (cid:48) ) α for | f | , | f (cid:48) | < (cid:15) . Then h φ s (cid:48)(cid:48) α (˜ L ) (cid:39) h φ sα (˜ L ) ⊗ F ( M, Λ) M Λ( s (cid:48)(cid:48) − s ) α (cid:39) (4.19) h φ s (cid:48) α (˜ L ) ⊗ F ( M, Λ) M Λ( s − s (cid:48) ) α ⊗ F ( M, Λ) M Λ( s (cid:48)(cid:48) − s ) α (cid:39) (4.20) h φ s (cid:48) α (˜ L ) ⊗ F ( M, Λ) M Λ( s (cid:48)(cid:48) − s (cid:48) ) α (4.21)for any s (cid:48) , s (cid:48)(cid:48) ∈ ( s − (cid:15), s + (cid:15) ). The last equivalence comes from Lemma 3.5.Call such s (cid:48) , s (cid:48)(cid:48) algebraically related. One can cover [0 , f ] by finitely many suchsmall intervals, and this gives a sequence 0 = s < s < · · · < s r = f such that s i and s i +1 are algebraically related, concluding the proof. We could choose any( r + 1)-tuple 0 = s (cid:48) < · · · < s (cid:48) r = f sufficiently close to 0 = s < s < · · · < s r = f ,in particular we can assume they are in Z ( p ) , resp. p n Z ( p ) . (cid:3) TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY33 Corollary 4.10. If f ∈ p n Z ( p ) , then (4.22) h φ fα ( L (cid:48) ) (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ fα Proof.
Assume f > f ∈ p n Z ( p ) , we can choose0 = s < · · · < s r = f in Lemma 4.9 from p n Z ( p ) . Therefore, s i +1 − s i ∈ p n Z ( p ) and M Λ fα ⊗ F ( M, Λ) M Λ f (cid:48) α (cid:39) M Λ( f + f (cid:48) ) α holds when f, f (cid:48) ∈ p n Z ( p ) by Corollary 3.24.This fact, together with (4.17) implies the corollary. (cid:3) Observe that if one confines themselves to the case f ∈ p n Z ( p ) , one can proveLemma 4.9 and Corollary 4.10 by using Corollary 3.24, rather than Lemma 3.5. Proof of Proposition 4.1.
Let f ∈ p n Z ( p ) . Then by Corollary 4.10(4.23) h φ − fα ( L (cid:48) ) (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ − fα Applying ( · ) ⊗ F ( M, Λ) h L to both sides of (4.23), we obtain(4.24) h φ − fα ( L (cid:48) ) ⊗ F ( M, Λ) h L (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ − fα ⊗ F ( M, Λ) h L Moreover,(4.25) CF ( L, φ − fα ( L (cid:48) )) (cid:39) h φ − fα ( L (cid:48) ) ⊗ F ( M, Λ) h L by Lemma 2.12. Combining these, we get(4.26) CF ( φ fα ( L ) , L (cid:48) ) (cid:39) CF ( L, φ − fα ( L (cid:48) )) (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ − fα ⊗ F ( M, Λ) h L which proves (4.1) as(4.27) h L (cid:48) ⊗ F ( M, Λ) M Λ − fα ⊗ F ( M, Λ) h L (cid:39) M Λ − fα ( L, L (cid:48) )The statement about the dimension is straightforward, namely the proper module(4.28) h L (cid:48) ⊗ F ( M,K ) M Kα | z = T − f ⊗ F ( M,K ) h L is well defined. By extending coefficients under K (cid:44) → Λ one obtains(4.29) h L (cid:48) ⊗ F ( M, Λ) M Λ α | z = T − f ⊗ F ( M, Λ) h L on the one hand, and by extending coefficients under K (cid:44) → Q p , one obtains(4.30) h L (cid:48) ⊗ F ( M, Q p ) M Q p α | t = − f ⊗ F ( M, Q p ) h L Base change under field extensions do not change the dimensions of cohomologygroups, and this finishes the proof. (cid:3)
Remark 4.11.
One can generalize Proposition 4.1 to all f ∈ Z ( p ) by using anargument that will also be used in the proof of Theorem 1.1.We can now prove the main theorem: Theorem 1.1.
Under given assumptions, the rank of HF ( φ k ( L ) , L (cid:48) ) , k ∈ N isconstant except for finitely many k . Proof.
Recall that we assume φ = φ α without loss of generality. By Proposition4.1, the rank of HF ( φ k ( L ) , L (cid:48) ) is equal to the rank of(4.31) H ∗ (cid:0) h L (cid:48) ⊗ F ( M, Q p ) M Q p α | t = − k ⊗ F ( M, Q p ) h L (cid:1) as long as k ≡ mod p n ). On the other hand, the following is a finitely generatedgraded module over Q p (cid:104) t (cid:105) :(4.32) H ∗ (cid:0) h L (cid:48) ⊗ F ( M, Q p ) M Q p α ⊗ F ( M, Q p ) h L (cid:1) whose restriction to t = − k gives (4.31). More precisely, let C be a free, finitecomplex over Q p (cid:104) t (cid:105) that is quasi-isomorphic to h L (cid:48) ⊗ F ( M, Q p ) M Q p α ⊗ F ( M, Q p ) h L (which exists by [Ked04, Proposition 6.5] as before). Then, (4.31) is isomorphicto H ∗ ( C − k −−→ C ), i.e. the cohomology of the cone of multiplication by − k , andit is an extension of H ∗ ( C ) / ( − k ) H ∗ ( C ) and ker ( − k ) ⊂ H ∗ ( C )[1] (one can useresolution Q p (cid:104) t (cid:105) − k −−→ Q p (cid:104) t (cid:105) of the structure sheaf of − k , or Tor spectral sequence,or equivalently the universal coefficient theorem). Since the Tate algebra Q p (cid:104) t (cid:105) isa PID (see for instance [Bos14, Section 2, Cor 10]), H ∗ ( C ) admits a description asthe sum of a free module and a torsion module supported at finitely many points.Hence, the rank of both H ∗ ( C ) / ( − k ) H ∗ ( C ) and ker ( − k ) are constant except forfinitely many k ∈ Z p (thus, the rank of (4.31) also satisfies this). Combiningthis with the previous statement, we conclude that the rank of HF ( φ k ( L ) , L (cid:48) ) isconstant among k ∈ p n Z ( p ) with finitely many exceptions.Now choose f , f , . . . , f p n − ∈ Z ( p ) such that f i ≡ i ( mod p n ) and | f i | is small sothat Lemma 4.3 and Lemma 4.6 hold for ˜ L = L (cid:48) and f = − f i . In other words, h algφ − fiα ,L (cid:48) (cid:39) h φ − fiα ( L (cid:48) ) (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ − f i α . In particular, h φ − fiα ( L (cid:48) ) is definable over F ( M, K ), and therefore over F ( M, Q p ) (see Corollary 4.7). Given f ≡ f i ( mod p n ),one can prove(4.33) h φ − fα ( L (cid:48) ) (cid:39) h L (cid:48) ⊗ F ( M, Λ) M Λ − f i α ⊗ F ( M, Λ) M Λ( − f + f i ) α as a corollary of Lemma 4.9, similar to Corollary 4.10. Then one can simply followthe proof of Proposition 4.1 to prove that the rank of HF ( φ fα ( L ) , L (cid:48) ) is the sameas the rank of(4.34) H ∗ (cid:0) h L (cid:48) ⊗ F ( M, Q p ) M Q p − f i α ⊗ F ( M, Q p ) M Q p ( − f + f i ) α ⊗ F ( M, Q p ) h L (cid:1) i.e. the rank of of Q p (cid:104) t (cid:105) -module (or Q p (cid:104) t/p n (cid:105) -module, after restriction)(4.35) H ∗ (cid:0) h L (cid:48) ⊗ F ( M, Q p ) M Q p − f i α ⊗ F ( M, Q p ) M Q p α ⊗ F ( M, Q p ) h L (cid:1) at point t = − f + f i . This allows one to conclude the rank of HF ( φ fα ( L ) , L (cid:48) ) isconstant among all but finitely many f satisfying f ≡ f i ( mod p n ). Therefore, therank of HF ( φ fα ( L ) , L (cid:48) ) , f ∈ Z is periodic of period (dividing) p n , except for finitelymany f ∈ Z .Notice, we can replace p by another prime p (cid:48) >
2, to conclude this sequence isperiodic of period ( p (cid:48) ) n (cid:48) for some n (cid:48) , except finitely many terms. Since, p and p (cid:48) are coprime, this implies that rank of HF ( φ fα ( L ) , L (cid:48) ) , f ∈ Z is constant outsidefinitely many f . (cid:3) TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY35 Remark 4.12.
The proof actually implies that the rank of HF ( φ fα ( L ) , L (cid:48) ) is con-stant except for finitely many f ∈ Z ( pp (cid:48) ) = { ab : a, b ∈ Z , p (cid:45) b, p (cid:48) (cid:45) b } . Applying thisversion to rα, r ∈ R in place of α implies that rank of HF ( φ rfα ( L ) , L (cid:48) ) is constantexcept for finitely many f ∈ Z ( pp (cid:48) ) . However, from this, we cannot immediatelyconclude that the rank of HF ( φ fα ( L ) , L (cid:48) ) , f ∈ R is constant except for finitely many f ∈ R , as the union of all these finite sets corresponding to classes in R / Z ( pp (cid:48) ) maystill be infinite. Remark 4.13.
Under the assumption that L and L (cid:48) are Bohr-Sommerfeld mono-tone, one can assume they are two of the fixed generators { L i } without loss of gener-ality. Therefore, it is possible to calculate the rank of h L (cid:48) ⊗ F ( M, Q p ) M Q p α ⊗ F ( M, Q p ) h L as the dimension of the cohomology of M Q p α ( L, L (cid:48) ) or M Kα ( L, L (cid:48) ). In particular, if L ∩ L (cid:48) = ∅ , then this complex is 0, and Proposition 4.1 implies HF ( φ k ( L ) , L (cid:48) ; Λ) = 0for all k ∈ p n Z ( p ) . As mentioned in Remark 3.23, it is likely that the group-likeproperty holds over the larger base Sp ( Q p (cid:104) t (cid:105) ), that would let one to conclude thesame for all k ∈ Z ( p ) . Therefore, one would have HF ( φ α ( L ) , L (cid:48) ; Λ) = 0 for all α .This may sound to contradict the theorem; for instance, by letting L (cid:48) = φ α ( L ).However, L and φ α ( L ) cannot be Bohr-Sommerfeld at the same time, i.e. thisremark does not apply in this case. As we will explain, one can get rid of Bohr-Sommerfeld assumption on L and L (cid:48) by passing to a finitely generated extension K ⊂ ˜ K over which h L (cid:48) and h L are definable. However, this does not mean thatthe modules h L (cid:48) and h L (defined in the usual way as in Section 2, not as Yonedamodules of twisted complexes over F ( M, Λ)) have coefficients in K or ˜ K . Dropping the assumption that L and L (cid:48) are Bohr-Sommerfeld monotone. We briefly explain how to drop the assumption that L and L (cid:48) are Bohr-Sommerfeld,while holding the assumption that they are monotone and have minimal Maslovnumber 3. The idea is simple: one can represent the modules h L (cid:48) and h L as iteratedcones/twisted complexes of Yoneda modules of { L i } . Since the data to definea twisted complex is finite, these modules are definable over a finitely generatedextension of K , if not K itself.More precisely, represent L (cid:48) as an element of tw π ( F ( M, Λ)), i.e. L (cid:48) (cid:39) ( (cid:76) k L i k , σ, π ),where σ is the differential of the twisted complex and π is the idempotent. Writecomponents of σ and π as linear combinations of the generators of CF ( L i , L j ). It iseasy to see that one must add only finitely many elements from Λ to the subfield K to include the coefficients of these linear expressions. In other words, there existsa finitely generated extension K ⊂ ˜ K such that σ and π , and hence this twistedcomplex is defined over F ( M, ˜ K )— the category obtained from F ( M, K ) via basechange along K → ˜ K . Let h (cid:48) L (cid:48) denote the image of this twisted complex underYoneda embedding. By construction, h (cid:48) L (cid:48) turns into a module quasi-isomorphicto h L (cid:48) , when we base change under ˜ K ⊂ Λ. Denote this module by h (cid:48) L (cid:48) as well.Similarly, h L is also definable over a finitely generated extension of K , i.e. by fur-ther extending ˜ K by finitely many elements, one can ensure that there exists a left F ( M, ˜ K )-module h (cid:48) L that becomes quasi-isomorphic to h L after base change along˜ K ⊂ Λ. Since ˜ K is finitely generated and countable, one can find a finite (not only finitelygenerated) extension Q (cid:48) p of Q p and a map ˜ µ : ˜ K → Q (cid:48) p extending µ : K → Q p . Forthis one can first extend µ to a map to Q p from a maximal purely transcendentalextension of K inside ˜ K (by choosing a set of elements of Q p that are algebraicallyindependent over µ ( K )). Then, ˜ K is finite over this extension of K , and thereexists a finite extension Q (cid:48) p of Q p and a map ˜ µ : ˜ K → Q (cid:48) p , whose restriction to K is equal to µ . Q (cid:48) p carries a unique discrete valuation extending that on Q p (see [Shi10, Theorem9.5]), and one can use the base change under Q p (cid:104) t (cid:105) → Q (cid:48) p (cid:104) t (cid:105) to obtain a family M Q (cid:48) p α over the latter that is group-like over Q (cid:48) p (cid:104) t/p n (cid:105) . The proof of Proposition 4.1still applies and we have(4.36) dim Λ (cid:0) HF ( φ k ( L ) , L (cid:48) ) (cid:1) = dim Q (cid:48) p (cid:0) H ∗ (cid:0) h (cid:48) L (cid:48) ⊗ F ( M, Q (cid:48) p ) M Q (cid:48) p α | t = − f ⊗ F ( M, Q (cid:48) p ) h (cid:48) L (cid:1)(cid:1) for all k ∈ p n Z ( p ) . This dimension is constant for all but finitely many k as before.Replacing L (cid:48) by φ − i ( L (cid:48) ), where i = 0 , , . . . p n −
1, we see that dim Λ (cid:0) HF ( φ k ( L ) , L (cid:48) ) (cid:1) is constant for all but finitely many k ∈ i + p n Z ( p ) . Therefore, dim Λ (cid:0) HF ( φ k ( L ) , L (cid:48) ) (cid:1) is p n periodic for all but finitely many k ∈ Z , and by replacing p by another prime,we see that this dimension is actually constant among k ∈ Z except finitely many.This concludes the proof of Theorem 1.1 without the Bohr-Sommerfeld assumptionon L and L (cid:48) . Remark 4.14.
Observe that we did not need to use f i ∈ i + p n Z ( p ) such that | f i | is small as we did not need an algebraic replacement for h φ − i ( L (cid:48) ) .5. Generic α and the proof of Theorem 1.5 In this section, we will explain a way to construct group-like p -adic families ofbimodules (i.e. “analytic Z p -actions”) without the assumption of monotonicity of M at the cost of assuming α is generic. Using this, we will deduce Theorem 1.5.The main reason we had to assume monotonicity was that, we had no way ofensuring convergence for an infinite series of the form (3.24) as well as (3.27) in Q p . One could try to choose the map µ : K → Q p so that µ ( T E ( u ) ) ∈ p Z p and µ ( T α ([ ∂ h u ]) ) ∈ p Z p , but such a map may not exist if there are algebraic relationsbetween the elements of the form T E ( u ) and T α ([ ∂ h u ]) . On the other hand, forgeneric [ α ] ∈ H ( M, R ), there are no such algebraic relations. Therefore, one candefine the embedding of the field of definition into Q p such that the images of T E ( u ) converge in p -adic topology, just like T E ( u ) converging in T -adic topology byGromov compactness.Our construction works in the general setting as long as one assumes the Fukayacategory is well defined with coefficients in Λ, and it is smooth and proper. Weassume there exists a finite set of Lagrangians L , . . . , L m that are all tautologicallyunobstructed and that generate the Fukaya category (we actually make the strongerassumption that L i bound no disc with Maslov index 2). We denote the span of L , . . . , L m by F ( M, Λ). Two natural examples of such M are an elliptic curve TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY37 and product of two elliptic curves. Let L, L (cid:48) be two branes satisfying Assumption1.6, which are represented by elements of tw π ( F ( M, Λ)) by generation assumption.Assume L i , L, L (cid:48) are pairwise transverse.Fix an almost complex structure on M . Our first goal is to embed the additivesemi-group spanned by E ( u ) > p Z p . For this,we prove that there exists a discrete free submonoid of R ≥ that contains energiesof all pseudo-holomorphic curves in M with boundary on L i , L, L (cid:48) : Lemma 5.1.
There exists rationally independent elements (5.1) E , . . . , E n ∈ R > ∩ ω M ( H ( M, L ∪ L (cid:48) ∪ (cid:91) i L i ; Q )) such that the monoid spanned by them contain the energy of every non-constantpseudo-holomorphic curve.Proof. For simplicity, we work with curves with boundary on a single Lagrangian L . Let J denote the compatible almost complex structure. Let ω , . . . , ω n (cid:48) denoteclosed 2-forms that are obtained by small perturbations of ω M and that satisfy:(1) ω i are still symplectic and tamed by J (2) ω i | L = 0 for all i and { [ ω i ] } form a basis of H ( M, L ; Q ) (in particular theyare rational)(3) [ ω M ] ∈ H ( M, L ; R ) is in the convex hull of the rays generated by [ ω i ](1) can be ensured since M is compact; therefore, small perturbations of ω M remainsymplectic and are still tamed by J . For other conditions, first choose closed 2-forms θ = ω M , θ , . . . , θ n (cid:48) representing a basis of H ( M, L ; R ) and identify thespace (cid:76) i R .θ i with H ( M, L ; R ). Consider a small open cone around ω M insidethis finite dimensional space, and choose ω i among 1-forms sufficiently close to ω M and representing rational classes (rational classes of 1-forms are dense). Then, it iseasy to make this choice so that (2) and (3) are satisfied.Since each ω i is symplectic and tamed by J , the energy of a non-constant pseudo-holomorphic curve representing class β ∈ H ( M, L ; Z ) with respect to the metric g i ( v, w ) = ( ω i ( v, Jw ) + ω i ( w, Jv )) is still strictly positive and can be calculated as ω i ( β ). Furthermore, there exists a natural number N such that ω i ( H ( M, L ; Z )) ⊂ N Z . Choose a common N for all ω i .Consider the set(5.2) { β ∈ H ( M, L ; Q ) : ω i ( β ) > ω i ( β ) ∈ N Z for all i } ∪ { } If we use ( ω i ) i to identify H ( M, L ; Q ) with Q n (cid:48) , then (5.2) corresponds to ( N Z + ) n (cid:48) ∪{ } . Therefore, (5.2) is a finitely generated submonoid of H ( M, L ; Q ). As re-marked, it contains the classes of all J -holomorphic curves. Also, as ω M can berepresented as a positive linear combination of ω i , the image of (5.2) under ω M isa submonoid of R ≥ , and it is finitely generated as (5.2) is. We conclude the proofby applying Lemma 5.2 below to the image of (5.2) under ω M . (cid:3) Lemma 5.2.
Every finitely generated additive submonoid of R ≥ can be extendedin its rational span to a submonoid generated by positive, rationally independentelements.Proof. We proceed by induction on the number of generators. Assume the state-ment holds when there are less than n generators. Consider an additive submonoidof R ≥ generated by x , . . . , x n >
0. By induction hypothesis for x , . . . , x n − , wecan find a monoid in their rational span generated by rationally independent ele-ments y , . . . y k > x , . . . , x n − . If x n is rationally independent of { y i } , then y , . . . , y k , x n is a rationally independent set of generators of a monoidcontaining x , . . . , x n .Otherwise, one can write x n as x n = a y i + · · · + a q y i q − b y j − · · · − b r y j r ,where a i ∈ Q + , b j ∈ Q ≥ , such that the sets { y i . . . , y i q } and { y j , . . . , y j r } forma partition of { y , . . . , y k } (i.e. they are disjoint, and their union contains all y i ).Let b y j + · · · + b r y j r be denoted by η . Since x n >
0, there exists λ , . . . , λ q ∈ Q ≥ such that λ + · · · + λ q = 1, and a h y i h − λ h η > η > a h y i h = ι h η , where ι h >
0. Thus, (cid:80) ι h >
1, choose λ h to be a rational partitionof 1 such that ι h − λ h > y j g and a h y i h − λ h η . For instance, let τ ∈ Z + be the productof numerators of all a h , and let τ ∈ Z + be the product of denominators of all λ h and b g . Then, it is easy to see that the monoid spanned by positive rationallyindependent elements τ ( a h y i h − λ h η ) , h = 1 , . . . , q , τ τ y j g , g = 1 , . . . , r contain y , . . . , y k , x n , and thus it contains x , . . . , x n . This finishes the proof. (cid:3) Remark 5.3.
Lemma 5.1 is still true when we allow the almost complex structureto vary, as in [Sei08], as long as ω M is rational. If we want it to hold for moregeneral ω M , we have to let almost complex structures vary in a bounded way. Forinstance, one can choose a small neighborhood of [ ω M ] in H ( M, R ) whose elementsare represented by symplectic forms deforming ω M , and consider only the almostcomplex structures tamed by these forms. Remark 5.4.
Gromov compactness imply that the set of energies of non-constantpseudo-holomorphic curves is a discrete subset of R + ; however, we are unable touse this statement to give a simpler proof of Lemma 5.1. The main difficulty is thefinite generation statement, it is not true that an additive submonoid of R ≥ thatlies in a finitely generated subgroup of R is always contained in a finitely generatedsubmonoid of R ≥ . For instance, consider the monoid generated by 1 and √
2. It isa discrete monoid, and we can expand it to another discrete monoid by adding anelement of the set Z + √ Z that is outside the original monoid (such as 3 √ − Z + √ Z from outside the monoid,to expand it further. We continue in this way, and the union of this nested sequenceof monoids is a monoid that is not finitely generated. If we assume the elementadded at the n th step is larger than n , then this monoid is discrete.Let E + ⊂ R ≥ denote the monoid spanned by E , . . . , E n . Since these generatorsare rationally independent, there is no algebraic relation between T E i . Definition 5.5. α is called generic if α ( H ( M, Z )) intersects the group generatedby E + only at 0. TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY39 This condition is weaker than(5.3) α ( H ( M, Z )) ∩ ω M ( H ( M, L ∪ L (cid:48) ∪ (cid:91) i L i ; Q )) = { } which is entirely topological. Clearly, there are abundance of generic 1-forms.Our next goal is to find a field K g ⊂ Λ, that contain all the coefficients of A ∞ -structure maps, and that we can embed into Q p . Using this embedding, we obtaina category over Q p , which we still denote by F ( M, Q p ). We also want to define asimilar p -adic family of F ( M, Q p )-bimodules over Q p (cid:104) t (cid:105) such that the restriction to t = f comes from a natural F ( M, K g )-bimodule.Fix a prime p >
2. For a generic α , there are no non-trivial algebraic relationsbetween various T α ( C ) and T E i . Consider the Novikov series(5.4) (cid:88) k a k T E ( k )+ fα ( k ) where a k ∈ Z , E ( k ) ∈ E + , α ( k ) ∈ α ( H ( M ; Z )), f ∈ Z ( p ) satisfying(1) E ( k ) + f α ( k ) → ∞ (2) E ( k ) → ∞ The first condition is for convergence in Λ. Denote the set of such series by R big ⊂ Λ.It is easy to see that R big is a subring of Λ. To write a ring homomorphism R big → Z p , choose a basis { α j : j = 1 , . . . l } ⊂ α ( H ( M ; Z )) ⊂ R . Choose a setof algebraically independent elements of Z p corresponding to each E i and each α j .Call these elements µ i and µ (cid:48) j ∈ Z p . Define µ big : R big → Z p as follows: we send(5.5) T α j → pµ (cid:48) j , T E i → pµ i , T fα j → (1 + pµ (cid:48) j ) f The image of (5.4) converges in Z p as E ( k ) → ∞ ; therefore, one can extend it to R big . More precisely, the total number of E i in E ( k ) ∈ E + goes to ∞ ; therefore, thep-adic valuation of their image goes to ∞ . It is possible to show that the coefficientsdefining F ( M, Λ) are in R big ; therefore, one can define the Fukaya category over R big and base change along µ big , giving us a category over Z p and eventually over Q p . Then, it is easy to construct p -adic families of bimodules, and prove similarproperties such as group-like property. On the other hand, when making dimensioncomparisons in the previous sections, we implicitly took advantage of the flatnessof the map K → Q p . We do not know this for R big → Q p , it is not always true thatthe map µ big is injective, and hence extends to a map from the field of fractions of R big to Q p . Presumably, generic choice of µ i , µ (cid:48) j can ensure injectivity; however, wehave not been able to prove this. Instead, we will construct a countably generatedsubring of R big that satisfies this property.Let P be a countable set of series(5.6) F = (cid:88) k a k X ν ( k )1 . . . X ν n ( k ) n Y fη ( k )1 . . . Y fη l ( k ) l =: (cid:88) a k X ν ( k ) Y fη ( k ) where a k , f ∈ Z ( p ) , X i , Y j are new variables corresponding to E i , α j , and ν i ( k ) ∈ Z ≥ , η j ( k ) ∈ Z . The exponential factor f ∈ Z ( p ) does not vary in a single series,and we include it as part of the data defining (5.6). Assume X i , Y j , Y − j ∈ P . Define E · ν ( k ) := (cid:80) i E i ν i ( k ) ∈ E + and α · η ( k ) := (cid:80) j α j η j ( k ) ∈ α ( H ( M, Z )). Wealso assume:(1) E · ν ( k ) + f α · η ( k ) → ∞ (2) E · ν ( k ) → ∞ as k → ∞ for all F ∈ P . The former is a condition for the convergence of(5.7) F ( T E , . . . T E n , T α , . . . , T α l ) = (cid:88) a k T E · ν ( k )+ fα · η ( k ) in Λ. One has F ( T E , . . . T E n , T α , . . . , T α l ) ∈ R big as a result of (1) and (2).Let R P denote the subring of R big generated over Z by T E i , T ± α j and other F ( T E , . . . T E n , T α , . . . , T α l ), for F ∈ P . Note 5.6.
It is also possible to evaluate F ∈ P at X i = pµ i , Y j = 1 + pµ (cid:48) j . We canevaluate Y fη j ( k ) j at Y j = 1 + pµ (cid:48) j via p -adic interpolation formula (3.15), and extendto other series F ∈ P in the obvious way. One way to make this more formal isas follows: given F ∈ P as in (5.6), with an exponential factor f ∈ Z ( p ) , one maydefine a series ˜ F in X i , Y ± j with coefficients in Q p (and with integral exponents) byreplacing Y fη ( k )1 . . . Y fη l ( k ) l with(5.8) (cid:88) a ,...,a l ∈ Z ≥ (cid:18) fa (cid:19) . . . (cid:18) fa l (cid:19) ( Y η ( k )1 − a . . . ( Y η l ( k ) l − a l Then, ˜ F ( pµ , . . . , pµ n , pµ (cid:48) , . . . , pµ (cid:48) l ) is “the evaluation of F at X i = pµ i , Y j =1+ pµ (cid:48) j ”. For instance, for F = X Y f Y f , this series would be (cid:80) a,b ∈ Z ≥ (cid:0) fa (cid:1)(cid:0) fb (cid:1) X ( Y − a ( Y − b . It is easy to see that ˜ F converge at X i = pµ i , Y j = 1 + pµ (cid:48) j . Thisfollows from the condition E · ν ( k ) → ∞ , which implies ν ( k ) + · · · + ν n ( k ) → ∞ as k goes to infinity.Let P denote Z [ P ] \{ } , which is still countable (here Z [ P ] denote the ring generatedby the series in P ). For a generic choice of µ i , µ (cid:48) j ∈ Z p , we have F ( pµ , . . . , pµ n , pµ (cid:48) , . . . , pµ (cid:48) l ) (cid:54) = 0, for any F ∈ P . One way to see this as follows: the expression(5.9) F ( pµ , . . . , pµ n , pµ (cid:48) , . . . , pµ (cid:48) l )defines a non-zero analytic function of µ i , µ (cid:48) j . Therefore, its zero set is an analytichypersurface inside Z n + lp . Such an hypersurface has measure 0 with respect tothe Haar measure on Z n + lp (this is obvious in dimension 1 as the zero set of ananalytic function is finite by Strassman’s theorem, and for higher dimensions it canbe proven by induction using Fubini’s theorem). Hence, the union of countablymany of them also has measure 0.Hence, the map Z [ P ] → Q p that sends F to F ( pµ , . . . , pµ n , pµ (cid:48) , . . . , pµ (cid:48) l ) isinjective. Observe that R P ∼ = Z [ P ] (as α is generic), and this map corresponds torestriction of µ big to R P . Hence, µ big | R P is injective and factors through the fieldof fractions of R P .We will let P to be a set of series such that the set { F ( T E , . . . T E n , T α , . . . , T α l ) : F ∈ P} includes the series defining the Fukaya category F ( M, Λ), the modules
TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY41 h L , h L (cid:48) , the analogues of the bimodules M Λ α | z = T f , and the map (3.14). This willallow us to define the field of definition K g as the field of fractions of R P and µ g : K g → Q p as the extension of the map above. Then, one can prove analoguesof Proposition 4.1 and Theorem 1.1, following the same reasoning.Given pseudo-holomorphic curve u , one can write E ( u ) uniquely as a positive in-tegral linear combination of E i . Let ν ( u ) = ( ν ( u ) , . . . , ν ( u ) n ) denote the co-efficients (i.e. E ( u ) = E · ν ( u ) in the notation above). Similarly, given [ C ] ∈ H ( M, Z ), one can write α ([ C ]) as an integral linear combination of α j , let η ([ C ]) =( η ([ C ]) , . . . , η l ([ C ])) be its coefficients (i.e. α ([ C ]) = α · η ([ C ]) in the notationabove). We let P to be the following set of series:(1) X i , i = 1 , . . . , n and Y j , j = 1 , . . . , l (2) the signed sums (cid:80) ± X ν ( u ) where u ranges over pseudo-holomorphic curveswith fixed inputs and one fixed output, and with boundary on L i , L , and L (cid:48) (3) the signed sums (cid:80) ± X ν ( u ) where u ranges over pseudo-holomorphic curveswith fixed inputs, one fixed output and fixed [ ∂ h u ] ∈ H ( M ; Z ), and withboundary on L i (4) the signed sums (cid:80) ± X ν ( u ) Y fη ([ ∂ u ]) where u ranges over the same set ofpseudo-holomorphic curves as (4.4) with fixed inputs and one fixed output,where ˜ L = L (cid:48) , and f ∈ Z ( p ) is sufficiently small so that the series convergeat X i = T E i , Y j = T α j (5) the signed sums (cid:80) ± X ν ( u ) Y fη ([ ∂ h u ]) where u ranges over the same set ofpseudo-holomorphic curves as (1.5), (3.11) and (3.24) with fixed inputsand one fixed output, and f ∈ Z ( p ) is sufficiently small so that the seriesconverge at X i = T E i , Y j = T α j (6) the signed sums (cid:80) ± X ν ( u ) Y f η ([ ∂ u ]) Y f η ([ ∂ u ]) where u ranges over thesame set of pseudo-holomorphic curves as (3.27) with fixed inputs and onefixed output, and f , f ∈ Z ( p ) are sufficiently small so that the seriesconverge at X i = T E i , Y j = T α j The condition (2) ensures the coefficients (cid:80) ± T E ( u ) of the A ∞ -structure, and ofthe modules h L , h L (cid:48) are among { F ( T E , . . . T E n , T α , . . . , T α l ) : F ∈ P} . Thanksto condition (4), the module analogous to h algφ fα ,L (cid:48) is definable over R P and its fieldof fractions. Similarly, condition (5) guarantees that the analogous bimodule to M Λ α | z = T f is definable over R P and its field of fractions, and condition (6) is for thedefinability of the bimodule homomorphisms(5.10) M α | Λ z = T f ⊗ F ( M, Λ) M α | Λ z = T f → M α | Λ z = T f f Finally, the purpose of the condition (3) is that the coefficients of the family M Λ α lie in R P and in K g so that one can define the family M K g α (analogous to M Kα defined before). This condition is not strictly necessary, but it is more convenientto include it. Note 5.7.
The signs in conditions (2)-(6) come from the orientation of the cor-responding moduli space of discs, and they are identical to signs in the analogoussums defined before. For instance, the signs of the summands of (2) (and its subsum (3)) are the same as the signs in the sums defining the A ∞ -structure. Similarly,the signs in (5) are the same as the signs in (1.5), (3.11) and (3.24). As mentionedin Note 2.3 these signs are standard and omitted throughout the paper. Remark 5.8.
The convergence assumptions in (4)-(6) are needed, since we areno longer in the monotone setting, and these sums are not finite. However, usingFukaya’s trick, one can show that the convergence holds for sufficiently small non-zero | f | , resp. | f | , | f | .Recall that we define K g ⊂ Λ to be the field of fractions of R P , and µ g : K g → Q p to be the map induced by µ big | R P .After this set-up, it is possible to follow the steps of the proof of Theorem 1.1. Wedescribe these steps in less detail. First, we define Novikov and p -adic families.Since, we hit the convergence issue, we let the base of Novikov family to be the ringΛ { z R } defined in Section 3 and further elaborated in Appendix A. In other words,(5.11) Λ { z R } := (cid:26) (cid:88) a r z r : , where r ∈ R , a r ∈ Λ (cid:27) where the series satisfy the convergence condition val T ( a r ) + rν → ∞ , for all ν ∈ [ a, b ]. Here, a < < b are fixed numbers with small absolute value. Recallthat the isomorphism type of the ring does not depend on a and b , but the co-efficients of series we will encounter will belong to this ring for small | a | and | b | only (c.f. convergence in family Floer homology). We like to think of Λ { z R } as anon-Archimedean analogue of a small closed interval containing 0. Let K g { z R } bethe set of elements of Λ { z R } where the coefficients of all z r -terms are in K g . Definition 5.9.
Let M K g α denote the family of F ( M, K g )-bimodules defined via(5.12) ( L i , L j ) (cid:55)→ M K g α ( L i , L j ) = CF ( L i , L j ; K g ) ⊗ K g K g { z R } and with structure maps(5.13) ( x k , . . . , x | x | x (cid:48) , . . . x (cid:48) l ) (cid:55)→ (cid:88) ± T E ( u ) z α ([ ∂ h u ]) .y where the sum ranges over pseudo-holomorphic discs as in (3.11) (see also Note2.3). Define the family of F ( M, Λ)-bimodules M Λ α by replacing K g by Λ.For the series (cid:80) ± T E ( u ) z α ([ ∂ h u ]) to be in Λ { z R } , one needs(5.14) E ( u ) + f α ([ ∂ h u ]) → ∞ for small | f | . This holds and can be proved using Fukaya’s trick. When we re-strict the families M K g α and M Λ α to z = T f , we implicitly assume | f | is small andconvergence holds. Also, note for small f ∈ Z ( p ) M K g α | z = T f is defined over K g byconstruction, thanks to condition (5).We also define: Definition 5.10.
Let M Q p α be the p -adic family of F ( M, Q p ) modules defined via(5.15) ( L i , L j ) (cid:55)→ M Q p α ( L i , L j ) = CF ( L i , L j ) ⊗ K g Q p (cid:104) t (cid:105) TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY43 and with structure maps(5.16) ( x , . . . , x k | x | x (cid:48) , . . . , x (cid:48) l ) (cid:55)→ (cid:88) ± µ g ( T E ( u ) ) µ g ( T α ([ ∂ h u ]) ) t .y The convergence of the series (5.16) is guaranteed by the fact that val p ( µ g ( T E ( u ) )) →∞ . The following lemma follows from the definitions: Lemma 5.11.
Given f ∈ Z ( p ) such that | f | is small, the bimodule M K g α | z = T f becomes M Q p α | t = f under base change along µ g : K g → Q p . The analogue of Lemma 3.5 also holds:
Lemma 5.12.
For f , f ∈ R such that | f | , | f | are small, M Λ α | z = T f f (cid:39) M Λ α | z = T f ⊗ F ( M, Λ) M α | Λ z = T f . The same statement holds for M K g α if f , f ∈ Z ( p ) .Proof. The proof for M Λ α is identical. To prove the statement about M K g α , oneneeds to ensure the coefficients (cid:80) ± T E ( u ) T f α ([ ∂ u ]) T f α ([ ∂ u ]) defining the map(5.17) M Λ α | z = T f ⊗ F ( M, Λ) M α | Λ z = T f → M Λ α | z = T f f are in K g . This follows from construction of K g , due to condition (6). (cid:3) Similarly, we have the analogue of Proposition 3.20:
Lemma 5.13.
For n (cid:29) , the restriction M Q p α | Q p (cid:104) t/p n (cid:105) is group-like.Proof. The proof of Proposition 3.20 applies verbatim. To write the map (3.25),one has to ensure convergence of the series analogous to (3.27). This also followsfrom the fact that val p ( µ g ( T E ( u ) )) → ∞ . (cid:3) As observed, the modules h L (cid:48) and h L are defined over K g . Similarly, given f ∈ Z ( p ) ,the deformation h algφ fα ,L (cid:48) is defined over K g , whenever | f | is small and the convergenceholds for the defining series. Therefore, there exists a complex(5.18) h L (cid:48) ⊗ F ( M, Q p ) M Q p α | Q p (cid:104) t/p n (cid:105) ⊗ F ( M, Q p ) h L of Q p (cid:104) t/p n (cid:105) -modules, which has finitely generated cohomology and whose cohomo-logical rank at t = − f ∈ p n Z ( p ) is the same as HF ( φ fα ( L ) , L (cid:48) ; Λ). The proof of thisis identical to Proposition 4.1. By the same reasoning the rank of HF ( φ fα ( L ) , L (cid:48) ; Λ)is constant in f ∈ p n Z ( p ) with finitely many possible exceptions. Replacing h L (cid:48) by h algφ fiα ,L (cid:48) , where f i ∈ i + p n Z ( p ) for i = 0 , . . . , p n − | f i | is small, we concludethe same for HF ( φ fα ( L ) , L (cid:48) ; Λ), f ∈ i + p n Z ( p ) . In other words, HF ( φ kα ( L ) , L (cid:48) ; Λ)has p n -periodic rank in k ∈ Z with finitely many exceptions, and since we have norestriction on p , we can replace it by another prime to conclude p (cid:48) n (cid:48) -periodicity,and thus constancy in k . In other words, we have: Theorem 1.5.
Suppose Assumption 1.6 holds. Given generic φ ∈ Symp ( M, ω M ) ,the rank of HF ( φ k ( L ) , L (cid:48) ; Λ) is constant in k ∈ Z with finitely many possible ex-ceptions. Remark 5.14.
It may be possible to prove a version of Theorem 1.5 by the follow-ing method: it is expected that HF ( φ fα ( L ) , L (cid:48) ; Λ) , f ∈ R forms a coherent analyticsheaf over R . Therefore, one can attempt to check that the rank of such a sheafjumps only at a discrete set, and for a generic r ∈ R , the sequence { r.k : k ∈ Z } in-tersects this set only at finitely many points. Hence, the rank of HF ( φ r.kα ( L ) , L (cid:48) ; Λ)is constant in k ∈ Z , with finitely many possible exceptions, for generic r . On theother hand, using this method, we do not see how to describe the generic set moreconcretely. Appendix A. Semi-continuity statements for chain complexes
In this Appendix, we collect the semi-continuity statements required for Lemma3.5 and Lemma 4.6. We start with the following definition:
Definition A.1.
Given b ≤ c , let Λ { z R } [ b,c ] denote the ring consisting of series(A.1) (cid:88) a r z r where r ∈ R , a r ∈ Λ, and(A.2) val T ( a r ) + rν → ∞ for any ν ∈ [ b, c ]. Only countably many a r can be non-zero, but we do not imposeany other condition on the set of r satisfying a r (cid:54) = 0 (e.g. they can accumulate).We think of this ring as the ring of functions on the universal cover of { x ∈ Λ : b ≤ val T ( x ) ≤ c } . Another heuristic is that one can think of this ring as a non-Archimedean analogue of the interval [ b, c ]. Observe that if b < c , Λ { z R } [ b,c ] isindependent of b and c , due to isomorphism(A.3) Λ { z R } [ b,c ] → Λ { z R } [ b (cid:48) ,c (cid:48) ] z r (cid:55)→ T αr z βr where β, α are such that βν + α is the increasing linear bijection [ b (cid:48) , c (cid:48) ] → [ b, c ].We will often omit b, c from the notation and denote this ring by Λ { z R } . We willassume 0 ∈ ( b, c ) unless stated otherwise.One can replace Λ[ z R ] in Definition 3.1 by Λ { z R } . Throughout this appendix, wewill refer to this modified notion. In particular, we use the notation M Λ α to meana Novikov family in the latter sense. For instance, M Λ α ( L, L (cid:48) ) = CF ( L, L (cid:48) ; Λ) ⊗ Λ Λ { z R } .The ring Λ { z R } is not Noetherian. However, for our purposes, we only need arestricted set of exponents for z . In other words, our examples will be defined overthe subring Λ { z ± r , . . . z ± r k } for a fixed set r i of real numbers that are linearlyindependent over Q . This is the subring of series in Λ { z R } whose exponents belongto integral span of { r i } and it is isomorphic to Λ { z ± , . . . z ± k } , the ring of powerseries in variables z , . . . , z k with a similar convergence condition. Being a quotientof the Tate algebra, the latter ring is Noetherian. Example A.2.
One can let r , . . . , r k to be a basis for an additive subgroup of R that contains α ( H ( M, Z )). TERATIONS OF SYMPLECTOMORPHISMS AND p -ADIC ANALYTIC ACTIONS ON FUKAYA CATEGORY45 Definition A.3.
Let b ≤ a ≤ c . Then, there is a ring homomorphism(A.4) ev T a : Λ { z R } [ b,c ] → Λ (cid:80) a r z r (cid:55)−→ (cid:80) a r T ar We call this map evaluation map at z = T a . Given f ∈ Λ { z R } [ b,c ] , ev T a ( f ) isalso denoted by f ( T a ), and the base change of a Λ { z R } [ b,c ] -module M along ev T a is denoted by M | z = T a . One can also think of ev T a as a Λ-point of Λ { z R } [ b,c ] .Even though we use real exponents for both Λ and Λ { z R } , there is no a priorirelation between the topology of these rings and the topology in R . For instance, t (cid:55)→ T t is not a continuous assignment. Nevertheless, we can still prove somesemi-continuity results as t ∈ R varies continuously. First we prove: Lemma A.4.
Let f ( z ) ∈ Λ { z R } be such that f (1) (cid:54) = 0 . Then there exists δ > such that f ( T t ) (cid:54) = 0 for | t | < δ .Proof. Let f ( z ) = (cid:80) a r z r . Without loss of generality, assume f (1) = 1. Bythe convergence condition, there exists an (cid:15) > val T ( a r ) − (cid:15) | r | → ∞ .Therefore, val T ( a r ) − (cid:15) | r | ≥ r . We split f as f + f where f isa finite sub-sum (cid:80) r ∈ F a r z r of (cid:80) a r z r , and f is the sum of remaining terms suchthat the coefficients of f have val T ( a r ) − (cid:15) | r | ≥
1. In particular, f (1) = O ( T ),i.e. val T ( f (1)) ≥
1, and f (1) = 1 + O ( T ). Also, by assumption, f ( T t ) = O ( T )for t ∈ [ − (cid:15), (cid:15) ].Let a r = (cid:80) α a r ( α ) T α be the expansion of a r . Then, f ( z ) = (cid:80) r ∈ F,α a r ( α ) T α z r and (cid:80) r ∈ F a r (0) = 1 as f (1) = 1 + O ( T ). Consider f ( T t ) = (cid:80) r ∈ F,α a r ( α ) T α T tr .As F is finite, there is a positive gap between the valuations of a r (0) T tr terms of thesum and the terms a r ( α ) T α + tr with α (cid:54) = 0, as long as | t | is small. Therefore, thesetwo types of terms of the sum cannot cancel out each other. Furthermore, for small | t | , a r (0) T tr have valuation less than 1, so these terms cannot cancel out with termsof f ( T t ). As (cid:80) r ∈ F a r (0) T tr remains non-zero and have different valuation thanthe remaining terms of f ( T t ), for small variation of t , f ( T t ) remains non-zero. (cid:3) We can use this to prove:
Lemma A.5.
Let C be a finite complex of free modules of finite rank over Λ { z R } . If C | z =1 have vanishing cohomology, then so does C | z = T t for small | t | . More generally,the rank at point T t is upper semi-continuous.Proof. Choosing bases for C i , one can see C as families of matrices d such that d = 0. Consider C i − d i − −−−→ C i d i −→ C i +1 . The rank of cohomology at a specificpoint z = T t is given by rank ( C i ) − rank ( d i − | z = T t ) − rank ( d i | z = T t ). Choose asquare sub-matrix of d i − , resp. d i , such the restriction to z = 1 is non-singularand of size equal to rank ( d i − | z =1 ), resp. rank ( d i | z =1 ). Let f ( z ) ∈ Λ { z R } be theproduct of determinants of these matrices. As f (1) (cid:54) = 0, Lemma A.4 implies that f ( T t ) (cid:54) = 0 for small | t | . Therefore, rank ( d i − | z = T t ) + rank ( d i | z = T t ) is at least rank ( d i − | z =1 ) + rank ( d i | z =1 ). This finishes the proof of Lemma A.5. (cid:3) It is easy to define multivariable versions of Λ { z R } , and prove analogues of LemmaA.4. For instance, let b < c and b < c be real numbers. Define Λ { z R , z R } to bethe series of the form(A.5) (cid:88) a r z r z r where r = ( r , r ) ∈ R , a r ∈ Λ, and satisfying(A.6) val T ( a r ) + r ν + r ν → ∞ for any ν ∈ [ b , c ] , ν ∈ [ b , c ]. As before, this ring does not depend on b There is a ring homomorphism Λ { z R } ⊗ Λ { z R } → Λ { z R , z R } suchthat f ⊗ g (cid:55)→ f g . One can presumably topologize the rings Λ { z R } , Λ { z R , z R } andprove the map induces an isomorphism from the completed tensor product. We donot need this for our current purposes.We have given the proof of Lemma 4.6 already. We would like to give: Sketch of proof of Lemma 3.5. One can write the morphism (3.14) of bimodules,which comes from a morphism of families(A.7) π ∗ M Λ α ⊗ rel F ( M, Λ) π ∗ M Λ α → ∆ ∗ M Λ α This is a map of families over Λ { z R , z R } . One defines (A.7) by a formula similar to(3.27), namely(A.8) (cid:88) ± T E ( u ) z α · [ ∂ u ]1 z α · [ ∂ u ]2 .y is the coefficient of y , and the sum is over the same set of discs as in (3.27). At z = z = 1, this defines the canonical quasi-isomorphism (3.26). We want to applyLemma A.5, and for this we need to find a quasi-isomorphic family to the cone of(A.7) that gives a free finite module over Λ { z R , z R } when evaluated at any pair ofobjects of F ( M, Λ). This is automatic for the target of the morphism (A.7) and weonly need this for its domain. Similar to the proof of Lemma 4.6, consider(A.9) π ∗ M Λ α ⊗ rel F ( M, Λ) π ∗ M Λ α (cid:39) π ∗ M Λ α ⊗ rel F ( M, Λ) F ( M, Λ) ⊗ rel F ( M, Λ) π ∗ M Λ α Since the diagonal bimodule F ( M, Λ) is perfect, it is a direct summand of aniterated cone of Yoneda bimodules; therefore, (A.9) is a direct summand of aniterated cone of bimodules of the form(A.10) π ∗ M Λ α ⊗ rel F ( M, Λ) ( h L i ⊗ h L j ) ⊗ rel F ( M, Λ) π ∗ M Λ α (cid:39) π ∗ M Λ α ( L i , · ) ⊗ Λ { z R ,z R } π ∗ M Λ α ( · , L j )The last term is naturally isomorphic to M Λ α ( L i , · ) ⊗ Λ M Λ α ( · , L j ), and it could bewritten as π ∗ M Λ α ( L i , · ) ⊗ rel Λ π ∗ M Λ α ( · , L j ) consistently with the previous notation.When evaluated at any pair of objects, the latter family of bimodules satisfy theassumption of Lemma A.5 (its multivariable version to be precise), and we canalso apply this lemma to the cone of (A.7), with the domain replaced by a quasi-isomorphic direct summand of an iterated cone of bimodules as in (A.10). 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