MMAURER-CARTAN DEFORMATION OF LAGRANGIANS
HANSOL HONGA bstract . The Maurer-Cartan algebra of a Lagrangian L is the algebra that encodes thedeformation of the Floer complex CF ( L , L ; Λ ) as an A -algebra. We identify the Maurer-Cartan algebra with the 0-th cohomology of the Koszul dual dga of CF ( L , L ; Λ ) . Makinguse of the identification, we prove that there exists a natural isomorphism between theMaurer-Cartan algebra of L and a certain analytic completion of the wrapped Floer coho-mology of another Lagrangian G when G is dual to L in the sense to be defined. In viewof mirror symmetry, this can be understood as specifying a local chart associated with L in the mirror rigid analytic space. We examine the idea by explicit calculation of theisomorphism for several interesting examples. C ontents
1. Introduction 12. Maurer-Cartan space of a Lagrangian 53. Preliminaries on homological algebra in filtered setting 114. Maurer-Cartan space of a Lagrangian and Koszul dual dga 195. Dual pairs in wrapped Fukaya categories 276. Examples: local models for SYZ fibrations 35Appendix A. Double Koszul dual of an algebra 46References 501. I ntroduction
Deformation of a Lagrangian submanifold has been a central problem in symplec-tic geometry. The geometric deformation of a Lagrangian L is locally parametrized by H ( L ) , and McLean’s classical result [McL98] showed that the deformation of a special La-grangian is unobstructed in the sense that deformation parametrized by H ( L ) automat-ically satisfies a special Lagrangian condition. Such a deformation problem draws moreattention after Strominger-Yau-Zaslow [SYZ96] asserts that the mirror complex manifoldof a symplectic manifold should be obtained as a complexified deformation space of aspecial Lagrangian torus.On the other hand, in view of homological mirror symmetry, a Lagrangian (togetherwith some additional structure) may be identified as an object of the Fukaya category.In this sprit, one can instead study deformation of the endomorphism algebra of a La-grangian regarded as an object of the A -category, which is, in fact, the core of La-grangian Floer theory of the Lagrangian. Such a deformation may admit further nontriv-ial obstructions from holomorphic disks bounded by the Lagrangian. More specifically,for given a compact Lagrangian L , one can study deformation of the A -algebra CF ( L , L ) following [FOOO09]. The deformation is parametrized by b P CF ( L , L ) – H ( L ) (using a r X i v : . [ m a t h . S G ] S e p AURER-CARTAN DEFORMATION OF LAGRANGIANS 2 the canonical model) that solves the following nonlinear equation m ( ) + m ( b ) + m ( b , b ) + m ( b , b , b ) + ¨ ¨ ¨ = Maurer-Cartan equation, and b satisfying the equation kills the obstruction m for L being an well-deinfed object in the Fukaya category. The set of solutions, denotedby MC ( L ) , is called the Maurer-Cartan space of L .Through a series of papers [CHL17], [CHL19] and [CHL18], we propose a mirrorconstruction by gluing via quasi-isomorphisms local mirrors obtained as Maurer-Cartanspaces of Lagrangians (or strictly speaking, weak Maurer-Cartan spaces to produce aLandau-Ginzburg mirror). It often happens that a single Lagrangian already has bigenough deformation to produces a full mirror, especially when we deform the Floertheory of an immersed Lagrangian L by generators from self-intersections. While theMaurer-Cartan deformation of immersed Lagrangians can be nicely formulated in therealm of immersed Floer theory [AJ10], their geometric deformation seems a very deli-cate problem that has not been fully understood.Another benefit of using MC ( L ) is that the construction is essentially algebraic oncethe necessary part of the A -structure on the Floer complex is known. This enables us toextend the deformation into noncommutative directions without much further efforts, yetfind new interesting geometric phenomena (see [CHL]). In this regard, it is more naturalto extract the Maurer-Cartan algebra A L of L from (1.1) (see Definition 2.3) rather than thespace, which supposedly serves as the ring of functions on MC ( L ) .The first half of the article aims to reformulate the mirror construction via MC ( L ) interms of well-known homological algebra tools, bar/cobar (denoted by B / Ω ) construc-tion. The upshot is to realize the Maurer-Cartan algebra A L as the Koszul dual of the A -algebra CF ( L , L ) . Theorem 1.1 (Proposition 4.1 and Proposition 4.3) . Let L be a compact graded Lagrangianin a symplectic manifold M. Then the Maurer-Cartan algebra A L can be identified as the -thcohomology of the dga A L : = hom CF ( L , L ) ( Λ , Λ ) . This dga can be computed equivalently as either Ω ( CF ( L , L ) _ ) or ( B CF ( L , L )) _ , where ( ´ ) _ is the topological dual of a normed vector space. For an A -algebra A over a base ring k , the dga hom A ( k , k ) refers to the Koszul dual,following [LPWZ08]. There are many different versions of Koszul duals in different gen-eralities, and the one presented here is most adapted to filtered A -setting. In particular,we use the topological dual as mentioned in the statement, and this is closely related withconvergence in T -adic topology (for T P Λ ).Observe that (1.1) is an infinity sum in general. Therefore any construction rooted fromthe Maurer-Cartan deformation (of a compact Lagrangian) is necessarily accompaniedwith the convergence issue. In Floer theory, it is usually handled by introducing the field Λ with a non-Archimedean valuation, recording the areas of contributing holomorphicdisks as exponents of T . Moreover, b in (1.1) should have a positive T -adic valuationto ensure the convergence. Accordingly, the Koszul dual will be carried out over Λ -coefficient in our setting, keeping track of the T -adic topology induced by the valuationon Λ and modules over it. This may seem as merely changing the coefficient ring, butwe will see that the areas of holomorphic disks provide crucial information about thelocation and the size of MC ( L ) in the mirror space. In particular, it turns MC ( L ) into ananalytic neighborhood in the mirror space for immersed L , not just a formal one that onewould obtain over C .Under some additional assumptions, an A -algebra and its Koszul dual dga shareimportant homological algebraic properties such as equivalences between certain module AURER-CARTAN DEFORMATION OF LAGRANGIANS 3 categories over these algebras. We expect that purely algebraic results known for Koszulduality may lead to some nontrivial geometric observation on Lagrangian Floer theoryand mirror symmetry. We remark that there is an interpretation of the mirror symmetryvia Koszul duality between sheaves constructed from Lagrangian torus fibers. See [Tu15]for more details.In the second half of the article, we focus on a pair of Lagrangians ( G , L ) in a Liouvillemanifold M such that their Floer complex (over Λ ) has a nontrivial component of rank 1 indegree 0, only. These Lagrangians are viewed as objects in the wrapped Fukaya categorydefined over Λ (see for e.g, [RS17] or [LP12]). By introducing a semi-simple coefficientring k Λ : = ‘ ki = Λ x π i y , our setting includes the case when G = Y ri = G i and L = Y ri = L i consist of the same number of irreducible components such that | G i X L j | = δ ij . For givensuch a pair ( G , L ) , L determines an augmentation ϵ L : CF ( G , G ) Ñ k Λ by identifying CF ( G , L ) – k Λ in m : CF ( G , G ) b CF ( G , L ) Ñ CF ( G , L ) . (Here, CF ( G , G ) is the wrappedFloer cohomology of G in Λ -coefficient,)The key observation is that the augmentation ϵ L can be regarded as specifying a pointin the “spec” of CF ( G , G ) . To make more sense of it, suppose that CF ( G , G ) is con-centrated at degree 0 so that it is simply an algebra (over k Λ ) for degree reason. If G generates the Fukaya category of M , then ˇ M : = “Spec” CF ( G , G ) (or more precisely awould-be space whose function ring is CF ( G , G ) ) should give a mirror of M . Pascaleff[Pas19] considered a similar situation in which M is a Log Calabi-Yau surface, and G aLagrangian section.On the other hand, in sprit of SYZ mirror symmetry, the mirror space ˇ M is supposedto be a deformation space of a Lagrangian, or the moduli of Lagrangians (in some loosesense). Hence each point of ˇ M should correspond to some Lagrangian. In this point ofview, it is natural to think that L sits in ˇ M as the point corresponding to the maximalideal ker ϵ L in of the algebra CF ( G , G ) .Having identified L with a point in the mirror ˇ M (constructed from the noncompactgenerator G ), the next question to ask is how to describe its neighborhood in ˇ M . Recallthat we already have an intrinsic deformation space of L , which is the Maurer-Cartanspace MC ( L ) . The following theorem answers how MC ( L ) sits in ˇ M as a neighborhoodof the point corresponding to L or dually, it describes the relation between CF ( G , G ) andthe Maurer-Cartan algebra A L . Theorem 1.2 (Theorem 5.2) . If two Lagrangians L = ‘ ri = L i and G = ‘ ri = G i in a Liouvillemanifold M satisfy | G i X L j | = δ ij , then there is a natural A -algebra homomorphism κ : CF ( G , G ) Ñ A L , where A L is the Koszul dual of CF ( L , L ) .If G generates L in the wrapped Fukaya category of M, and CF ( G , G ) = HF ( G , G ) , then κ descended to the -th cohomology induces the isomorphism between a completion of CF ( G , G ) (ina certain T-adic sense) and the Maurer-Cartan algebra A L . We believe that the condition CF ă ( G , G ) = κ admits an explicit formula ÿ m k ( Z , P , ˜ b , ¨ ¨ ¨ , ˜ b ) = P ¨ κ ( Z ) ,for Z P CF ( G , G ) , where P generates CF ( G , L ) and ˜ b = ř x i X i is the formal linear com-bination of all generators X i of CF ( L , L ) except the unit. Making use of its compatibility AURER-CARTAN DEFORMATION OF LAGRANGIANS 4 with algebraic structures, κ can be explicitly computed in many cases, at least on the levelof cohomology.We provide detailed computations for several important classes of examples in Section6. Observe that two typical examples of such a pair ( G , L ) arises in ‚ M , a cotangent bundle with a cotangent fiber G and the zero section L , ‚ M with a SYZ fibration where G is a Lagrangian section and L a torus fiber.Etg ¨u and Lekili [EL17b] have given a nice explanation on the former case related withKoszul duality, and analyzed in detail the case of plumbing of spheres in this context. Ourfocus is more on the latter one, and in particular, the case when the compact Lagrangian L has nontrivial H ( L ) so that its Maurer-Cartan deformation is nonempty. This is anessential reason why we need to deal with the convergence issue more carefully. Namely,the elements in H ( L ) can be inserted arbitrarily many times to A -operations withoutchanging the degree of the output. Keeping track of the degree changes in algebraicconstructions, H ( L ) is responsible for the 0-th cohomology of A L , the Maurer-Cartanalgebra A L .With help of the theorem, we shall make an atlas of the mirror rigid analytic varietiesobtained from the generating Lagrangian G of several local SYZ models, locating theMaurer-Cartan deformation space of important classes of Lagrangians (as well as torusfibers) in the mirror space. This procedure takes the opposite way to gluing constructionin [HL18] or [CHL18] in the sense that we describe local charts in terms of global coordinates on the mirror given G .The examples include the cotangent bundle of T n (or a trivial SYZ fibration), a conicbundle with a nodal torus fiber, and a deformed conifold. Purpose of the last exam-ple is to examine the noncommutative (quiver) situation. Interestingly, we will see thatimmersed Lagrangians (such as nodal torus fiber) tend to have bigger Maurer-Cartanneighborhoods than embedded ones (such as torus fibers). Intuitively, this means: themore singular the Lagrangian is, the more Lagrangians it can produce by deformation. Itis, of course, plausible since one can smooth out singularity to make the Lagrangian lesssingular.On the contrary, torus fibers have very small Maurer-Cartan neighborhood. Note thatdifferent torus fibers do not intersect. This implies that their Maurer-Cartan neighborhooddo not overlap, since the Maurer-Cartan deformation is based on Lagrangian intersectionFloer theory. For this reason, one should take into account every fibers to obtain a fulldeformation space of Lagrangian torus fibers. This is along the same line of thoughts asin the family Floer program [Abo14] or the family of valuations in [FOOO16] .Here are some future directions. One potential application of Theorem 1.2 is the com-putation of the Maurer-Cartan algebra of L from the wrapped Floer cohomology of itsdual G . Usually, the Maurer-Cartan algebra involves nonlinear relations from higher m k -operations that makes it hard to compute by direct count of holomorphic disks. On theother hand, in many examples including conic bundles, the wrapped Floer cohomology isconcentrated at degree 0, and is relatively easy to compute by explicit perturbations. Alsothere is a nice generation result of the wrapped Fukaya category provided by [CRGG17]. There are some cases such as toric manifolds where the symplectic information is concentrated only atcritical fibers, in which case it may be enough to look at their weak Maurer-Cartan deformation to constructa mirror [CHL19].
AURER-CARTAN DEFORMATION OF LAGRANGIANS 5
It would be also interesting to extend the result into weak Maurer-Cartan deformationand its associated Landau-Ginzburg mirror. In 6.2, we shall perform some relevant con-struction for a certain weakly unobstructed immersed circle in the pair-of-pants, whichmay give us a hint for a more general formulation.
Notations
We will work over several different coefficient rings depending on the geometric situa-tion. The most frequently used is the universal Novikov field over C defined as Λ : = ÿ i = a i T A i | a i P C , lim i Ñ8 λ i + + . (1.2)We also use the following subrings of Λ often: Λ : = ÿ i = a i T A i | A i ě + + , Λ + : = ÿ i = a i T A i | A i ą + + .Finally, the following is the set of elements in Λ which has multiplicative inverses: Λ U : = t a + λ : a P C ˆ and λ P Λ + u ,which can be though of as unitary elements in Λ . In addition, we will also use theirsemi-simple generalizations, whose definition is to be given later. Aknowledgement
The author express his gratitude to Junwu Tu for explanation on the constructions inhomological algebra relevant to the paper. He also thanks Cheuk Yu Mak and Cheol-Hyun Cho for valuable discussions. The work of the first named author is supportedby the Yonsei University Research Fund of 2019 (2019-22-0008) and the National Re-search Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2020R1C1C1A01008261).2. M aurer -C artan space of a L agrangian We briefly recall Lagrangian Floer theory following [FOOO09], especially the Maurer-Cartan deformation of a Lagrangian, which is the main subject in the paper. As we are torelate this with an endomorphism of a noncompact Lagrangians in later applications, ashort review on wrapped Floer theory will be also given. Along the way, we present theprecise T -adic setting for both of Floer theories in 2.2 that we work with throughout thepaper, which will be particularly useful in later applications.2.1. Fukaya A -algebra and wrapped Fukaya category. Let us start with a compact un-obstructed Lagrangian L in a symplectic manifold M . We allow L to be immersed withclean intersections. The unobstructedness of L means that the evaluation image of themoduli space of holomorphic discs bounding L with one marked point vanishes, whichresults in the absence of m -term in the A -structure below. For instance, if there isno nonconstant disc bounding L , then L is automatically unobstructed. We will use thecanonical model for the Floer complex CF ( L , L ; Λ ) that is modeled on H ˚ ( L ; Λ ) . When L is immersed, we additionally include two copies of cocycles on the self-intersection lociinto CF ( L , L ; Λ ) . Each of them formally spans rank-1 Λ -vector space in CF ( L , L ; Λ ) whoseelements will be called immersed generators . AURER-CARTAN DEFORMATION OF LAGRANGIANS 6
Theorem 2.1 ([FOOO09]) . There exist a sequence of multilinear mapsm k : CF ( L , L ; Λ ) b k Ñ CF ( L , L ; Λ ) for k ě which satisfies ÿ i , k + k = k + ( ´ ) ˚ m k ( X , ¨ ¨ ¨ , X i , m k ( X i + , ¨ ¨ ¨ , X i + k ) , X i + k + , ¨ ¨ ¨ , X k ) , (2.1) where ˚ = ( ´ ) | X | + ¨¨¨ + | X i | , and | X i | denotes the shifted degree | X i | = | X i | ´ . A vector space V with a family of multilinear operations t m k : V b k Ñ V u k ě satisfying2.1 is called an A -algebra, and hence the theorem simply tells us that CF ( L , L ; Λ ) admitsan A -structure. If L is obstructed, (2.1) should additionally involve a certain nonzerocohomology class usually called m ( ) . Each m k can be further decomposed as m k = ÿ β P π ( M , L ) m k , β T ω ( β ) (2.2)which can be an infinite sum, but Gromov compactness ensures that this operation con-verges over Λ .Now we recall the wrapped Floer cohomology first defined in [AS10], but we will fol-low the version given in [Abo10] that uses a quadratic Hamiltonian. Let M be a Liouvillemanifold with an exact symplectic form ω = d θ . Let us take a Liouville vector field Z such that ι Z ω = θ . We require M to be symplectomorphic to ( ) ˆ Σ outside a com-pact region for some contact hypersurface Σ . More precisely, ( ) ˆ Σ is equipped with ω = d ( r α ) where α : = θ | Σ and r is the standard coordinate on ( ) . A Lagrangian L issaid to be conical at infinity if L intersects Σ transversally and Θ | L ” L conical at infinity, we define its (self-)wrapped Floercomplex as follows. We take a quadratic Hamiltonian H : M Ñ R in the sense that H = r on the region symplectomorphic to ( ) ˆ Σ . Then the wrapped Floer complex CW ( L , L ) = CW ( L , L ; H ) is formally generated over C by the points in ϕ H ( L ) X L , orequivalently the time-1 Hamiltonian chords from L to itself. The operations m k : CW ( L , L ) b k Ñ CW ( L , L ) (2.3)are defined by counting disks which satisfy the holomorphic equation, but slightly per-turbed by Hamiltonian terms. Indeed, a raw output would naturally lie in CW ( L , L ; ( k ´ ) H ) , and one needs a certain rescaling trick to map it back to the original space CW ( L , L ; H ) .Unlike (2.2), the output of (2.3) is a finite sum of intersection points. We refer readers to[Abo10] for more details. t m k u defines an A -structure, i.e., the operations satisfy therelation (2.1).More generally, for a ( k + ) -tuple of a Lagrangian L , ¨ ¨ ¨ , L k , one can analogouslydefine m k : CW ( L , L ) b ¨ ¨ ¨ b CW ( L k ´ , L k ) Ñ CW ( L , L k ) . (2.4)Here, CW ( L i , L j ) = CW ( L i , L j ; H i , j ) also involves a perturbation H i , j , and CW ( L i , L j ) canbe equivalently defined to be either the C -vector space freely generated by Hamiltonianchords from L i to L j , or that generated by the intersection points between ϕ H i , j ( L i ) and L j .Finally, the wrapped Fukaya category W Fuk ( M ) is defined as the C -linear categorywhose objects are exact Lagrangians (possibly noncompact) and morphism space between L and L is given by hom W Fuk ( M ) ( L , L ) = CW ( L , L ) . W Fuk ( M ) is naturally an A -category with operations m k (2.4). AURER-CARTAN DEFORMATION OF LAGRANGIANS 7
Monotone wrapped Fukaya category over Λ . One can extend the wrapped Fukaya categoryso that it includes possibly nonexact monotone Lagrangians. The A -operations in thiscase should be supplemented by multiplying the energy term analogously to (2.2). Sucha construction was thoroughly carried out in [RS17]. The resulting category over Λ willbe denoted by W Fuk Λ ( M ) .[LP12, Lemma 1.2] gives a fully faithful embedding W Fuk ( M ) b C Λ ã Ñ W Fuk Λ ( M ) (2.5)onto the subcategory of W Fuk Λ ( M ) consisting of exact Lagrangians. The functor (2.5)is defined as follows. For exact Lagrangians L and L , suppose an intersection point P P ϕ H ( L ) X L gives rise to the generator r P P hom W Fuk ( M ) ( L , L ) = CW ( L , L ; H ) . Thenthe above functor sends r P regarded as a morphism in W Fuk ( M ) to T f L ( P ) ´ f L ( P ) ´ H ( P ) P where f L and f L are such that d θ L = d f L and d θ L = d f L ,and the same symbol P now represents the standard generator of the morphism space in W Fuk Λ ( M ) supported at the intersection point P .2.2. Valuations on Floer complexes.
We will mainly work on W Fuk Λ ( M ) throughoutthe paper. For notational simplicity, let us write W C and W Λ for respectively W Fuk ( M ) and W Fuk Λ ( M ) from now on. Once a perturbation H is fixed, one can assign a quantitycalled the ( T -adic) valuation ν H ( P ) : = H ( P ) + f L ( P ) ´ f L ( P ) (2.6)to a geometric generator P P ϕ H ( L ) X L of a morphism space hom W Λ ( L , L ) in this cat-egory, which extends to a non-Archimedean valuation ν H : hom W Λ ( L , L ) Ñ R that de-pends on the perturbation datum. We postpone the more detailed discussion on non-Archimedean normed spaces to (3.1).By abuse of notation, we denote the normalized generator T f L ( P ) ´ f L ( P ) ´ H ( P ) P by r P re-garded as a morphisms in W Λ , and it satisfies ν H ( r P ) =
0. From the above discussion,it is precisely the image of the standard generator (written by the same notation r P ) in W Fuk ( M ) supported at P . We refer to it as an exact generator for this reason.If L P W Λ is a compact object, then one may alternatively use Morse-Bott model H ˚ ( L ; Λ ) (or its variants) for hom W Λ ( L , L ) , which coincides with CF ( L , L ; Λ ) describedearlier. In this case, the valuation ν (notice the absence of the subscript since there is noperturbation term) is much simpler. Namely, we set valuations of elements in H ˚ ( L ; C ) to be zero, and extend to H ˚ ( L ; Λ ) in an obvious manner. For non-immersed generatorsin H ˚ ( L ; Λ ) , this is somewhat consistent with (2.6) since H = L = L for an endomorphism space.On the other hand, immersed generators have slightly different features in this regard,though their valuations are still defined to be zero. Let X be an immersed generator ofthe (Morse-Bott) Floer complex CF ( L , L ; Λ ) of an exact Lagrangian immersion ι : L í L ,supported at the self-intersection ι ( X + ) = ι ( X ´ ) . Suppose that the generator X representsthe branch jump from X ´ to X + in Floer theory of L . While ν ( X ) is still set to be zero, onecan assign another real number to X analogously to (2.6) as follows. Let f L be a chosenprimitive of ι ˚ θ for the exact brane L . Measuring the difference of the values of f L at thebranch jump X gives f L ( X + ) ´ f L ( X ´ ) P R . This quantity will play an important role,indicating the position of L in the global moduli of Lagrangian. As before, if we use the AURER-CARTAN DEFORMATION OF LAGRANGIANS 8 scaling r X = T f L ( X + ) ´ f L ( X ´ ) X for immersed generators, then the A -operation does notinvolve the area term, which is essentially going back to the C -linear category W C . Insummary, an exact immersed generator r X can have nonzero valuation ν ( r X ) ‰ W Λ counts holomorphic polygons which has positive en-ergies, the output of an operation can still involve negative power of T in the coefficientsdue to the unavoidable Hamiltonian terms. We illustrate this in the following simpleexample of M = T ˚ S . Example 2.2.
Let G be a cotangent fiber in M : = T ˚ S . If we identify T ˚ S = R ˆ S anduse ( s , t ) as local coordinates (with s P R and t P [
0, 1 ] /0 „ ), the symplectic form is given by ω = dsdt with Θ = sdt. (In view of X = C ˆ , ( t , s ) and z P C ˆ are related by log z = s + it.)A Hamiltonian perturbation H for hom W C ( G , G ) = CW ( G , G ; H ) can be taken to be H ( s , t ) = s , and the intersection between ϕ H ( G ) and G occurs at ( s , t ) = ( i , 0 ) for each i P Z , which wedenote by Z i . With respect to our notation fixed above, hom W C ( G , G ) is generated by r Z i (i P Z )supported at Z i with the multiplicationm ( r Z i , r Z j ) = r Z i + j , and no higher operations. Hence it is isomorphic to C [ z , z ´ ] where r Z j is identified with z j .For the above choice of perturbation H, the morphism r Z i in W C corresponds toT ´ i Z i (2.7) where Z i is now viewed as the standard generator of hom W Λ ( G , G ) associated with the intersec-tion point ( s , t ) = ( i , 0 ) (denoted by the same symbol). Thus (2.7) leads tom ( T ´ i Z i , T ´ j Z j ) = T ´ ( i + j ) Z i + j , and we have m ( Z i , Z j ) = T ´ ij Z i + j . Hence, it is possible to have negative powers of T appearingin the output of A -operations even among geometric generators. Nevertheless, this does createany convergence issue since the output is always a finite sum. Maurer-Cartan equation.
Let L be a graded unobstructed immersed Lagrangian,which is compact. We assume that L consists of a single irreducible component forsimplicity. Consider its Lagrangian Floer complex CF ( L , L ; Λ ) which is naturally an A -algebra. By passing to the minimal model, we assume that the degree-0 component CF ( L , L ; Λ ) is generated by the unit class. To be more concrete, we will take the under-lying vector space of CF ( L , L ; Λ ) to be H ˚ ( L ; Λ ) (adjoined with immersed generators if L is immersed). Throughout, we assume that CF ( L , L ; Λ ) is supported on nonnegativedegrees.In [CHL17], the localized mirror associated to L was constructed as the space ofMaurer-Cartan deformation of L defined in the following way . Let X , ¨ ¨ ¨ , X l form abasis of the degree-1 component CF ( L , L ; Λ ) , whose valuations are zero. For (free) for-mal variables x , ¨ ¨ ¨ , x l , we consider a linear combination b = ř x i X i , and compute m ( b ) + m ( b , b ) + ¨ ¨ ¨ (2.8)where we use the convention m k ( x i X i , ¨ ¨ ¨ , x i k X i k ) = x i k ¨ ¨ ¨ x i m k ( X i , ¨ ¨ ¨ , X i k ) . (2.9)Notice that elements in Λ with positive valuations can only be plugged into formal vari-ables x , ¨ ¨ ¨ , x l in order to to make sense of possibly an infinite sum (2.8). [CHL17] mainly deals with the weak Maurer-Cartan deformation
AURER-CARTAN DEFORMATION OF LAGRANGIANS 9
Since CF ( L , L ; Λ ) is Z -graded, the outcome must be a linear combination of degree-2generators of CF ( L , L ; Λ ) with coefficient being noncommutative formal power series in x , ¨ ¨ ¨ , x l , i.e., m ( b ) + m ( b , b ) + ¨ ¨ ¨ = f ( x , ¨ ¨ ¨ , x l ) X l + + ¨ ¨ ¨ + f N ( x , ¨ ¨ ¨ , x l ) X N where X l + , ¨ ¨ ¨ , X N generate CF ( L , L ; Λ ) . Therefore we see that if x i ’s are taken fromthe algebra Λ tt x , ¨ ¨ ¨ , x l uuxx f , ¨ ¨ ¨ , f N yy , (2.10)then the associated b solves the Maurer-Cartan equation m ( b ) + m ( b , b ) + ¨ ¨ ¨ =
0. (2.11)Here, Λ tt x , ¨ ¨ ¨ , x l uu is the space of (noncommutative) formal power series over Λ withbounded coefficients. Namely, a series ř I P ( Z ą ) l T λ I x I arranged in the length-increasingorder belongs to Λ tt x , ¨ ¨ ¨ , x l uu if and only if the set t λ I : I P ( Z ą ) l u ( Ă R ) is boundedbelow. (Here, x I = x i ¨ ¨ ¨ x i l for I = ( i , ¨ ¨ ¨ , i l ) .) Thus inf t λ I : I P ( Z ą ) l u is a well-definedreal number for any element of Λ tt x , ¨ ¨ ¨ , x l uu , and will be called the ( T -adic) valuation or simply the energy of ř I P ( Z ą ) l T λ I x I . As we will see more in detail later, Λ tt x , ¨ ¨ ¨ , x l uu can be obtained as the continuous dual of the tensor algebra of CF ( L , L ) (over Λ ) withrespect to a natural T -adic topology. Correspondingly, xx f , ¨ ¨ ¨ , f N yy is the closure of theideal generated by f , ¨ ¨ ¨ , f N . That is, it consists of elements of the form ÿ k = g k f i k h k (2.12)where, for any fixed integer m ą
0, there exists k such that the length of (all monomialsin) g k f i k h k is greater than m for k ě k . In addition, the sum of valuations of g k , f i k and h k is required to be bounded above by a constant independent of k . Definition 2.3.
The Maurer-Cartan algebra of L is defined to beA L : = Λ tt x , ¨ ¨ ¨ , x l uuxx f , ¨ ¨ ¨ , f N yy .Intuitively, the Maurer-Cartan space from L is the (possibly noncommutative) spacewhose function ring is the Maurer-Cartan algebra of L . For a Maurer-Cartan solution b (that is, b satisfying (2.11)) with coefficients in Λ + , ( L , b ) defines an unobstructed objectwhose Floer cohomology is well defined. Therefore, the Maurer-Cartan space can bethought of as a local moduli formed by objects in the Fukaya category near L (see, forinstance, [HL18]). Example 2.4.
Let us consider ( C ˚ ) n = T ˚ T n , the cotangent bundle of the n-dimensional torus L : = T n . Since A -operations admits no contribution from nonconstant discs, we only need totake into account the cup product on H ˚ ( L ) . Since H ˚ ( L ) is an exterior algebra generated byd θ , ¨ ¨ ¨ , d θ n , we see that the Maurer-Cartan equation for b = ř x i θ i givesm ( e b ) = ÿ i , j ( x i x j ´ x j x i ) d θ i ^ d θ j , AURER-CARTAN DEFORMATION OF LAGRANGIANS 10 and hence the Maurer-Cartan algebra of L is the symmetric algebra on n-letters, x , ¨ ¨ ¨ x n . Takinginto account the convergence issue over Λ , the Maurer-Cartan algebra is given by Λ tt x , ¨ ¨ ¨ , x n uuxx x i x j ´ x j x i | ď i ‰ j ď n yy (2.13) which can be thought of as a certain completion of the polynomial ring over Λ in n-variables, andone can think of x , ¨ ¨ ¨ , x n as variables for ( Λ + ) n . There is a way to include energy-0 Maurer-Cartan deformation, which is to use Λ U -connection ρ on L . Fix a generator X , ¨ ¨ ¨ , X l of H ( L ; Z ) , and suppose ρ has holonomy ρ i P Λ U along PD ( X i ) . The Maurer-Cartan deformation by ř x i X i with x i P Λ + canbe enhanced using ρ by introducing a new variable z i = ρ i e x i . Here, the appearance ofthe expression e x i is natural in the sense that x i appears in A -operations deformed by b = ř x i d θ i as an exponential due to the divisor axiom. However, the change of variables x i into e x i has a dramatic effect on the valuation, since e x i always has valuation zero forany x i P Λ + . Example 2.5.
Applying the above discussion to Example 2.4 with b = ( ρ , ř i x i d θ i ) , we obtainan enlarged deformation space whose Maurer-Cartan algebra is given by Λ t z i : i P Z ux z i z j = z i + j : i , j P Z y (2.14) where Λ t z i : i P Z u consists of an infinite sum ř k = a k z i k with lim val ( a k ) = . One does notneed to take the closure of the ideal x z i z j = z i + j : i , j P Z y since it is automatically closed due tosome general fact about Tate algebras, see [BGR84] .Notice that (2.13) describes the ring of convergent (analytic) functions on ( Λ + ) n , whereas (2.14) is that of ( Λ U ) n . The exponential coordinate change z i = ρ i e x i = ρ i + ρ i x i + ¨ ¨ ¨ clearlyexplains the necessity of the condition lim val ( a k ) = , since otherwise ř k = a k z i k would notgive a well-defined element in (2.13) after coordinate change back to x i . The last example concerns the Maurer-Cartan deformation by immersed generators ofa Lagrangian, which can be intuitively thought of as smoothing out the correspondingself-intersections.
Example 2.6.
Consider the deformed conifold t ( u , v , u , v ) P C : u v ´ u v = ϵ u with ϵ ‰ . It admits a double conic fibration by writing the defining equation as u v = z ´ a andu v = z ´ b (with b ´ a = ϵ ). Let M : = t ( u , v , u , v ) P C : u v ´ u v = ϵ uzt z = u ,which is the anti-canonical divisor complement of a deformed conifold. The projection to z-planedefines a double-conic fibration, and two matching cycles L and L over the paths drawn inFigure 1 are Lagrangian 3-spheres.Consider the union L : = L ‘ L regarded as an immersed Lagrangian. Floer theory on L hasbeen examined in [CPU16] using a certain simplicial model. L has the Maurer-Cartan algebra Λ tt x , y , z , w uuxx xyz ´ zyx , yzw , wzy , zwx ´ xwz , wxy ´ yxw , xz , zx , yw , wy yy where x , y , z , w are taken from the path algebra Γ Q forQ : ‚ x (cid:28) (cid:28) z (cid:37) (cid:37) ‚ y (cid:101) (cid:101) w (cid:92) (cid:92) . AURER-CARTAN DEFORMATION OF LAGRANGIANS 11 F igure
1. A double conic fibration on the deformed conifold with twosingular fibers
Here, the relations xz = zx = yw = wy = come from composability of arrows in Q. In Example 2.6, we can alternatively use the semi-simple coefficient ring k Λ : = Λ π ‘ Λ π with π i ¨ π i = π i and π ¨ π = π ¨ π = k Λ tt x , y , z , w uu as a replacement for Λ tt x , y , z , w uu with x , y , z , w P Γ Q , where π x = x π = x , x π = π x = z π = z π =
0, and similar for y , z , w . In what follows, we will work over such a coefficient ring to efficiently include thecase when L is given as a union of several Lagrangians.3. P reliminaries on homological algebra in filtered setting We review basic algebraic constructions that are relevant to our geometric applicationsbelow. The statements in this section must be well-known in homological algebra to someextent, but we do not know if there is any literature whose setting precisely fits into ourgeometric setup. A good reference is [EL17a, Section 2] whose geometric setting has muchsimilarity to ours, but T -adic convergence was not considered as they mostly work over C . A related construction in the filtered (and curved) setting seems to be carried out in[DL18]. In our geometric situation, the coefficient field is always Λ , and the constructioncomes with a convergence issue in most of steps. We will try to make the expositions asdetailed and explicit as possible for this reason.To begin with, we fix once and for all our base ring k to be a semisimple ring k Λ = ‘ ri = Λ π i such that π i ¨ π i = π i and π i ¨ π j = i ‰ j . Here, Λ is the universal Novikovfield over C defined in (1.2). There exists a valuation val : Λ Ñ R such that val ( T λ ) = λ ,which has an obvious extension to k Λ . We set k C = ‘ ri = C π i so that k Λ = k C b C Λ . Forboth k Λ and k C , the sum ř ď i ď r π i of idempotents defines the unit, which we denote by .3.1. Non-Archimedean topology on k Λ -modules. In what follows, the underlying space V of any algebraic structures we are to consider will be of the following form. V is a free k Λ -bimodule endowed with a valuation ν : V Ñ R Y t8u such that ‚ ν ( cX ) = val ( c ) + ν ( X ) for c P k and X P V ; ‚ ν ( X + Y ) ě min t ν ( X ) , ν ( Y ) u for X , Y P V ; ‚ ν ( X ) = if and only if X = X ÞÑ e ´ ν ( X ) defines a non-Archimedean norm, and hence a topology on V . Infact, it is a slight generalization to a semi-simple base ring of a non-Archimedean normedvector space, which frequently appears in Floer theory. AURER-CARTAN DEFORMATION OF LAGRANGIANS 12
Let us consider a completion ˆ V of V with respect to e ´ ν . By definition, ˆ V contains allformal infinite sums ÿ i = X i for X i P V (3.1)with ν ( X i ) Ñ 8 , in which case the sum is said to converge with respect to ν . Since V is a free k Λ -module, one can find a set of elements t Z i P V : i P I , ν ( Z i ) = u in V forsome index set I such that any element of ˆ V can be written as ř i = c i Z i with c i P k Λ and val ( c i ) Ñ 8 . The valuation function naturally extends to the completion ˆ V . When V hasa finite rank over k Λ , (3.1) can be rewritten as a finite linear combination, say ř Ni = c i Z i with c i P k Λ , and hence allowing infinite sums does not give any difference in this case,i.e., V = ˆ V for a finitely generated V .The valuation ν on V gives rise to a filtration of V defined by F λ V : = t X P V : ν ( X ) ě λ u that satisfies ‚ F λ V Ă F λ V for λ ą λ , ‚ c F λ V Ă F λ + val ( c ) V for c P Λ .This induces a filtration on the k -fold tensor product T k V : = V b k of V (over k Λ ) by F λ T k V = F λ ( V b ¨ ¨ ¨ b V ) : = ď λ + ¨¨¨ + λ k = λ F λ V b ¨ ¨ ¨ b F λ k V (3.2)We denote by ˆ T k V or V ˆ b k the completion of T k V with respect to this filtration. Thecompletion allows an infinite sum ÿ i , ¨¨¨ , i k ě X i b ¨ ¨ ¨ b X i k whenever ν ( X i ) + ¨ ¨ ¨ + ν ( X i k ) Ñ 8 as i + ¨ ¨ ¨ + i n Ñ 8 . Note that ˆ T k V is strictly largerthan T k ˆ V in general. If V is finitely generated over k Λ , we have T k V = ˆ T k V = T k ˆ V .For later use, we denote by ˆ TV the completion of TV = ‘ k V b k with respect to thefiltration F λ TV : = ď k ď λ + ¨¨¨ + λ k = λ F λ V b ¨ ¨ ¨ b F λ k V naturally extending (3.2). It consists of formal sums ÿ k X k for X k P F λ k ˆ T k V with λ k Ñ 8 . Such a completion is typically used in Lagrangian Floer theory for compactLagrangians. See, for e.g., [FOOO09, 3.2]. One can also consider the length-completion of TV induced by the filtration TV Ą ‘ k ě T k V Ą ‘ k ě T k V Ą ‘ k ě T k V Ą ¨ ¨ ¨ .Hence the length-completion consists of (noncommutative) formal power series in X i ’sover k Λ . AURER-CARTAN DEFORMATION OF LAGRANGIANS 13 A -algebras and coalgebras in Novikov setting. An A -algebra ( V , t m k u k ě ) over k Λ is a free Z -graded k Λ -bimodule V equipped with a sequence of multilinear operations m k : V b k Ñ V of degree 2 ´ k which satisfies the relations (2.1). In addition, we assume that V isequipped with a valuation function ν , and admits the structure described in 3.1. Ac-cordingly, we assume that for each d P Z , there exists λ d P R satisfying m k ( F λ V i , ¨ ¨ ¨ , F λ k V i k ) Ă F λ + ¨¨¨ + λ k + λ d V d + ´ k for all k and i , ¨ ¨ ¨ i k with i + ¨ ¨ ¨ + i k = d . Notice that this is slightly stronger thanthe degreewise boundedness of individual m k , as λ d above should not depend on d . Itis rather related to the boundedness of the differential of the bar construction of V tobe introduced shortly. The condition is vacuous for finitely generated V , so the Floercomplex of a compact immersed Lagrangians provides a typical example of such. Also,the (wrapped) Floer complex of exact Lagrangians also satisfy the condition since A -operations preserve the valuation in this case. (Recall from 2.2 that the Floer complexadmits a natural valuation in either case.) Remark 3.1.
We use the following sign change that turns an A -algebra with m k ě ” into adga d ( X ) = m ( X ) , X ¨ Y = ( ´ ) | X | m ( X , Y ) . (See for e.g., [CHL18, Appendix A] .) Adding extra signs is necessary due to difference betweenA - and dg-sign conventions. To avoid a potential confusion, an A -algebra ( V , m , m , m ě ” ) will be referred to as an A -dga. For two A -algebras ( V , ν ) and ( V , ν ) , an A -homomorphism ϕ is defined as asequence of multilinear maps ϕ k : ˆ T k V Ñ V of degree 1 ´ k satisfying ÿ k ÿ i , ¨¨¨ , i k ě m V l ( ϕ i ( X ( ) ) , ¨ ¨ ¨ , ϕ i l ( X ( l ) )) = ÿ ( ´ ) | X ( ) | ϕ k ( X ( ) , m V k ( X ( ) ) , X ( ) ) .where we write X P B k V as X = X ( ) b ¨ ¨ ¨ b X ( l ) on the left hand side with X ( i ) P T j V forsome j , and the decomposition on the right hand side is similarly defined.An A -right module ( E , ν E ) over V is a k Λ -bimodule that is equipped with a sequenceof multilinear maps t n | k u , n k | : E b T k V Ñ E , satisfying ÿ ( ´ ) | Y | + | X ( ) | n | k ( Y , X ( ) , m k ( X ( ) ) , X ( ) ) + ÿ n | k ( n | k ( Y , X ( ) ) , X ( ) )) = A -operations, we assume for each d P Z that n | k ( F λ E i , F λ V i , ¨ ¨ ¨ , F λ k V i k ) Ă F λ + λ + ¨¨¨ + λ k + λ d E d + ´ k for all k and i + i + ¨ ¨ ¨ + i k = d . Left modules are similarly defined.Let E and E be two A -right modules over V . A degree- d pre- A -morphism f between two A -modules over V is given by a sequence of multilinear maps t ψ k ,1 u , ψ k ,1 : T k V b E Ñ E of degree 1 ´ k + d which satisfies for each d P Z , ψ k ,1 ( F λ V i , ¨ ¨ ¨ , F λ k V i k , F λ k + ( E ) i k + ) Ă F λ + ¨¨¨ + λ k + λ k + + λ d ( E ) d + ´ k + d (3.3)for any k and i + ¨ ¨ ¨ + i k + = d . We denote by hom dV ( E , E ) the set of all degree- d pre- A -module homomorphisms from E to E , and set hom V ( E , E ) : = ‘ d hom dV ( E , E ) . AURER-CARTAN DEFORMATION OF LAGRANGIANS 14
There is a differential M (in A -convention) on hom V ( E , E ) defined by M ( ψ ) | k ( Y , X , ¨ ¨ ¨ , X k ) : = ř ( ´ ) | ψ | + | Y | + | X ( ) | ψ | k ( Y , X ( ) , m k ( X ( ) ) , X ( ) )+ ř ( ´ ) | ψ | ψ | k ( n E | k ( Y , X ( ) ) , X ( ) )+ ř n E | k ( ψ | k ( Y , X ( ) ) , X ( ) ) . (3.4)It is elementary to check that M = M -closed pre-morphisms are usually called A -module homomorphisms. When E = E = E , we can compose (in A -convention) twomorphisms t ψ | k u and t ϕ | k u from E to itself by M ( ϕ , ψ ) | k ( Y , X , ¨ ¨ ¨ , X k ) = ÿ ( ´ ) | ϕ | ϕ | k ( ψ | k ( Y , X ( ) ) , X ( ) ) , (3.5)and ( hom V ( E , E ) , M , M ) defines an A -dga in this case.We next discuss an A -coalgebra C , which is the dual notion to an A -algebra. Definition 3.2.
An A -coalgebra C over k Λ is a free Z -graded k Λ -module equipped with a familyof maps ∆ k : C Ñ C b k of degree k ´ such that, for each k ě , ÿ r + s + t = ku = r + + t ( ´ ) ˚ ( b r b ∆ s b b t ) ∆ u =
0. (3.6)
For two A -coalgebras C and C , a morphism between them is a sequence of linear maps t ϕ k : C Ñ C k u satisfying ÿ ( ´ ) ˚ ( id b k b ∆ k b id b k ) ˝ ϕ k + k + k = ÿ ( ϕ i b ϕ i b ¨ ¨ ¨ b ϕ i m ) ˝ ∆ m .The signs in (3.6) follow the Koszul convention (with respect to the shifted degrees).For instance, in (3.6), ( ´ ) ˚ ( b r b ∆ s b b t )( c ( ) , c ( ) , c ( ) ) = ( ´ ) | c ( ) | c ( ) b ∆ s ( c ( ) ) b c ( ) where c ( ) b c ( ) b c ( ) = ( c b ¨ ¨ ¨ b c r ) b ( c r + b ¨ ¨ ¨ b c r + s ) b ( c r + s + b ¨ ¨ ¨ b c r + s + t ) .We again assume that C is equipped with a valuation function ν : C Ñ R , and thestructure maps t ∆ k u k ě satisfies ∆ k ( F λ C d ) Ă F λ + λ d ( C b k ) d + k ´ (3.7)for any k . In particular, if there exists c P C with ∆ i ( c ) ‰ i ’s, then wecan find λ P R such that ν ( ∆ k ( c )) ą λ for all k ě A -coalgebra and their morphisms,which will not be used in the paper. They can be defined by simply dualizing the corre-sponding notions for A -algebras.A dg-coalgebra ( C , d , ∆ : C Ñ C b C ) is a special case of an A -coalgebra with ∆ ě ” d and ∆ differ from ∆ and ∆ by d ( c ) = ∆ ( c ) , ∆ ( c ) = ř ( ´ ) | c | c b c when ∆ ( c ) = ř c b c . The sign change makes ( d , ∆ ) satisfy the usual graded co-Leibnitzrule. (This is dual to sign changes given in Remark 3.1.) We will call ( C , ∆ , ∆ , ∆ ě ” ) (without sign change) an A -dg-coalgebra to avoid confusion. AURER-CARTAN DEFORMATION OF LAGRANGIANS 15
Bar construction.
We next recall bar/cobar constructions in our setting. A dg-coalgebra is closely related with an A -algebra structure via non-reduced bar construc-tion. (The bar construction will refer to the reduced one in this paper.) Namely, A -structure on V over k Λ is equivalent to a dg-coalgebra structure on r BV : = T ( sV ) = ‘ k T k ( sV ) , which is a non-reduced bar construction of V (see for e.g. [Kel01]). Here, sV is a suspension of V which shifts degree by ´
1, and hence for x P V , | sx | = | x | ´
1. Ifthere is no danger of confusion, we will use the same notation x for sx in TV . The shifteddegree | x | is nothing but the degree of sx . In summary, we have | x | = | sx | = | x | ´ t m k u define an A -structure on a vector space V . Then we define ∆ and d on TV to be ∆ : r BV Ñ r BV b r BV x b ¨ ¨ ¨ b x k ÞÑ ÿ i ( x b ¨ ¨ ¨ b x i ) b ( x i + b ¨ ¨ ¨ b x k ) (3.8) d : r BV Ñ r BV x b ¨ ¨ ¨ b x k ÞÑ ÿ i , l ( ´ ) | x | + ¨¨¨ + | x i | x b ¨ ¨ ¨ b m l ( x i + , ¨ ¨ ¨ , x i + l ) b ¨ ¨ ¨ b x k .(3.9)One can easily check that this defines a dg-coalgebra (with respect to the shifted degree),and that the condition d = A -relations among t m k u k ě .In this paper, we will mainly consider the reduced bar construction, or simply the barconstruction, for an augmented A -algebra, which is given as follows. Let V be anaugmented A -algebra, i.e., an A -algebra together with an A -homomorphism ε : V Ñ k Λ . One can decompose V into V = ¯ V ‘ k Λ where ¯ V : = ker ε is a (non-unital) A -algebra.We further assume that ε is strict, in the sense that it does not involve higher compo-nents V b k Ñ k Λ for k ě
2. This implies that ¯ V is preserved by any A -operations. Forinstance, CF ( L , L ; Λ ) for a compact unobstructed Lagrangian L admits a (strict) augmen-tation ε after passing to the minimal model if necessary, where ε is the projection to theunit component. Definition 3.3.
The bar construction of an augmented A -algebra ( V , ε ) over k Λ is a dg-coalgebra on BV = T ( s ¯ V ) where ¯ V = ker ε and the operations d and ∆ are given by the formulas (3.9) and (3.8) (with V replaced by ¯ V).
Note that BV includes k Λ as a component, and is coaugmented via the inclusion of thiscomponent. In general, the coaugmentation of an A -coalgebra C is an A -coalgebrahomomorphism η : k Λ Ñ C .3.4. Cobar construction.
To define cobar construction of an A -algebra over k Λ , we needanother type of a completion of the tensor algebra. For a k Λ -bimodule C with a valuationfunction ν , let us consider TC = ‘ k T k C . Let TC be the subspace of the length-completionof TC consisting of ÿ i = c i x ⃗ α i ( c i P k Λ and ⃗ α i = ( α i ,1 , ¨ ¨ ¨ , α i , k i ) P I k i for some k i ě ) such that it is supported on finitely many degrees, i.e., |t deg x ⃗ α i : i =
1, 2, ¨ ¨ ¨ u| ă 8 , andthat, for all i =
1, 2. ¨ ¨ ¨ val ( c i ) + k i ÿ l = ν ( x α i , l ) ą λ (3.10)for some λ P R . AURER-CARTAN DEFORMATION OF LAGRANGIANS 16
Given an A -coalgebra ( C , ∆ k ) over k Λ with the coaugmentation η : k Λ Ñ C , there isa natural way to obtain a dg-algebra, called the cobar construction. We assume η is strictin the sense that η k ” k ě Definition 3.4.
Let C be a coaumented A -coalgebra over k Λ , and ¯ C : = C / η ( k ) . Its cobarconstruction is the dg-algebra Ω C : = T ( s ´ ¯ C ) , where the differential δ : = d Ω C is given by δ ( x b ¨ ¨ ¨ b x k ) = ÿ i , j ( ´ ) | x | + ¨¨¨ + | x i ´ | x b ¨ ¨ ¨ b x i ´ b ∆ j ( x i ) b x i + b ¨ ¨ ¨ b x k (3.11) The multiplication ¨ on Ω C is defined to be the usual concatenation of tensors.
One can check that δ satisfies the graded Leibnitz rule (with respect to the degreeshifted by s ´ ) and δ =
0, and hence Ω C defines a dga. Also, Ω C is augmented via theprojection to the component k Λ in T ( s ´ ¯ C ) . Remark 3.5.
In general, (3.11) can be an infinite sum, and hence requires a certain convergencecondition. From the condition (3.7) imposed on t ∆ k u , if ∆ k ( x ) = ř i , | ⃗ α i | = k c i x ⃗ α i , then t val ( c i ) + ř k i l = ν ( x α i , l ) : i =
1, 2, ¨ ¨ ¨ u is bounded below. Thus δ ( x ) gives a well-defined element in T ( s ´ ¯ C ) by definition of T ( ´ ) . In many literatures, some finiteness condition on the coalgebra structure isassumed when defining the cobar construction so as to avoid the completion procedure. Dual of A -algebras and A -coalgebras. In our T -adic setting, the most naturalway to take the dual of a vector space over Λ is to consider bounded linear maps. Namely,linear functionals are required to satisfy certain boundedness conditions with respect thevaluation functions. We first discuss the dual of a k Λ -module.Let us consider a non-graded k Λ -bimodule V over k Λ with a valuation ν : V Ñ R Y t8u .Let V _ denote the set of all right k Λ -module maps V Ñ k Λ which are bounded. Recallthat a k Λ -linear map f : V Ñ Λ is said to be bounded ifinf t val ( f ( X )) P k Λ : X P V with ν ( X ) = u ‰ ´8 . (3.12)In this case, one can define ν _ : V _ Ñ R Y t8u to be this infimum. Such f gives acontinuous linear map between the two non-Archimedean normed spaces ( V , e ´ ν ) and ( k Λ , e ´ val ) , and V _ is usually referred to as the topological (or continuous) dual of V . Onecan check that e ´ ν _ defines a non-Archimedean norm on V _ . Since (3.12) is equivalent toinf t val ( f ( X )) ´ ν ( X ) P k Λ : X P V u , we have val ( f ( X )) ě ν _ ( f ) + ν ( X ) for any X P V . The k Λ -bimodule structure on V _ is defined by ( a ¨ f ¨ b )( X ) = a ¨ f ( b ¨ X ) for a , b P k Λ . One can analogously define the left dual _ V of V , the set of bounded left k Λ -module maps from V to k Λ , and _ V becomes a k Λ -bimodule via ( a ¨ f ¨ b )( X ) = f ( X ¨ a ) ¨ b .Consider a k Λ -module V given in the form of V = V C b C Λ for some k C -module V C or V = V Λ b Λ Λ for some k C -module V C or k Λ ( : = k C b C Λ ) -module V Λ . This happens,for example, when V is free. In this case, the topological dual can also be described in thefollowing algebraic terms. Lemma 3.6.
For a (non-graded) k Λ -module V, we haveV _ = hom k Λ ( V Λ , k Λ ) b Λ Λ = hom k C ( V C , k Λ ) b Λ Λ . where in the last term, k Λ is regarded as a k C -module in an obvious way. AURER-CARTAN DEFORMATION OF LAGRANGIANS 17
Proof.
It suffice to prove the first equality since the second is simply the base change.Since an element in hom k Λ ( V Λ , k Λ ) extends by linearity to a map V Ñ k Λ whose normis nonnegative, we have V _ Ă hom k Λ ( V Λ , k Λ ) b Λ Λ . For the other direction, observethat any bounded linear map V Ñ k Λ can be made to have a nonnegative norm aftermultiplying suitable power of T . □ For a graded k Λ -module V = ‘ d P Z V d , we define V _ = ‘ d ( V d ) _ , where we set thegrading on ( V d ) _ to be ´ d so that ( V _ ) d = ( V ´ d ) _ . This resembles taking the gradeddual of V , but the usual algebraic dual of each graded piece now being replaced by thetopological dual. Let t X i : i P I u freely generate the degree- d component V d over k Λ forsome index set I (which is possibly an infinite set). We assume further that for each i , X i P π β i ¨ V d ¨ π α i for some α i and β i (if they exist, they must be unique). For this choice ofgenerators, one can define the coordinate function x i P π α i ¨ ( V d ) _ ¨ π β i Ă ( V d ) _ of degree ´ d by x i ( X j ) : = δ ij π α i .and extending it k Λ -linearly. Obviously, x i is bounded, and ν _ ( x i ) = ´ ν ( X i ) . In general,an element of ( V _ ) d can be written as a possibly infinite linear combination ÿ j = b i x i b i P k Λ (3.13)satisfying val ( b i ) + ν _ ( x i ) ą λ for all i and some fixed constant λ P R . If V d is finitelygenerated, then (3.13) reduces to a finite linear combination over k Λ . A general elementof V _ is a finite sum of series like (3.13) with different degrees. Note that we can actuallyevaluate an element of V _ at a point in ˆ V , not only in V .3.5.1. Dual of a tensor algebra.
We make a few remarks about the relation between thetensor algebra and the (topological) dual. For a free graded k Λ -bimodule ( V , ν ) , thedegree- d piece of TV is given by ( TV ) d = à k à d + ¨¨¨ + d k = d V d b ¨ ¨ ¨ b V d k .Recall that TV = ‘ d ( TV ) d is equipped with a filtration induced by ν , and it makes senseto take its topological dual ( TV ) _ . Namely, it consists of linear maps f : TV Ñ k Λ whichsupport on finitely many degrees and satisfy f ( F λ TV ) Ă F λ + λ f k Λ We always have a k Λ -bimodule homomorphism ( V d ) _ b ¨ ¨ ¨ b ( V d k ) _ Ñ ( V d k b ¨ ¨ ¨ b V d ) _ (3.14)by letting ( y b ¨ ¨ ¨ b y k ) ( Z k b ¨ ¨ ¨ b Z ) : = y ( ¨ ¨ ¨ y k ´ ( y k ( Z k ) Z k ´ ) ¨ ¨ ¨ Z ) , which clearlydefines a bounded map. Taking the direct sum over ( d , ¨ ¨ ¨ , d k ) with k fixed, the inducedmap on the length- k component T k ( V _ ) = À d , ¨¨¨ , d k V _ d b ¨ ¨ ¨ b V _ d k of T ( V _ ) looks as T k ( V _ ) Ñ à d , ¨¨¨ , d k ( V d k b ¨ ¨ ¨ b V d ) _ Ă à d à ř ki = d i = d V d k b ¨ ¨ ¨ b V d _ = ( T k V ) _ .This is an isomorphism when V is finitely generated. Similarly, the composition ( T ( V _ )) ´ d = à n à d i = d ( V d ) _ b ¨ ¨ ¨ b ( V d n ) _ Ñ à n à ř d i = d ( V d n b ¨ ¨ ¨ b V d ) _ Ñ (( TV ) d ) _ AURER-CARTAN DEFORMATION OF LAGRANGIANS 18 induces a map T ( V _ ) Ñ ( TV ) _ . For the left-dual _ V , we instead use ( y b ¨ ¨ ¨ b y k ) ( Z k b¨ ¨ ¨ b Z ) = y k ( Z k y k ´ ( Z k ´ ¨ ¨ ¨ y ( Z ))) , and proceed analogously. From now on we willmainly consider the right dual to simplify the exposition. Lemma 3.7.
If V is finitely generated graded module over k Λ , then ( TV ) _ – T ( V _ ) , whereT ( V _ ) is the subspace of the length-completion spanned by homogeneous formal series each ofwhose terms are bounded by a fixed constant (see (3.10) ).Proof. Let us fix a basis t X i : i P I u of V with X i P π β i V π α i , and consider associatedcoordinate functions t x i : i P I u . V b ¨ ¨ ¨ b V admits a basis t X i b ¨ ¨ ¨ b X i k : k ě α i l = β i l + u . Therefore an element of ( TV ) _ is completely determined by its value at theseelement. Namely, it can be expressed as ÿ k = ÿ v P I k c v ⃗ x v (3.15)where ⃗ x v = x v b ¨ ¨ ¨ b x v k . For this to be an element of ( TV ) _ , it has to be a boundedmap, and c v is nonzero for finitely many degrees only. Namely, the set of values of (3.15)at normalized basis elements ! T ´ ř k val ( X vk ) X v b ¨ ¨ ¨ b X v k : ( v , ¨ ¨ ¨ , v k ) P I k , k =
1, 2, ¨ ¨ ¨ ) with valuation zero should be bounded below. Thus there exists λ P R such that val ( c v ) ´ k ÿ l = ν ( X v l ) = val ( c v ) + k ÿ l = ν ( x v l ) ą λ for all v . Together with the fact that (3.15) supports only finitely many degrees, this isprecisely a description of an element of T ( V _ ) . □ Dual of a finitely generated A -algebra over k Λ . Now suppose we are given an A -algebra ( V , t m k u k ě ) . As discussed, the natural map V _ b ¨ ¨ ¨ b V _ Ñ ( V b ¨ ¨ ¨ b V ) _ (3.16)is an isomorphism only for a finitely generated A -algebra ( V , t m k u k ě ) . Thus we restrictourselves to duals of such A -algebras. For f P V _ , we define ∆ k ( f )( X b ¨ ¨ ¨ b X k )) = ( ´ ) | X | + ¨¨¨ + | X k | f ( m k ( X , ¨ ¨ ¨ , X k ))) , (3.17)which is a priori an element of ( V b ¨ ¨ ¨ b V ) _ . Since A -operations t m k u on V satisfies ν ( m i ( X , ¨ ¨ ¨ , X k )) ą ř ki = ν ( X i ) + λ d for some λ d P R for any k and | X | + ¨ ¨ ¨ | X k | = d , val ( ∆ k ( f )( X b ¨ ¨ ¨ b X k )) = val ( f ( m k ( X , ¨ ¨ ¨ , X k ))) ě ν _ ( f ) + ν ( m k ( X , ¨ ¨ ¨ , X k )) ą ν _ ( f ) + k ÿ i ν ( X i ) + λ .which shows that ∆ k ( f ) is a bounded linear map on ( V b k ) d , and hence ∆ k ( f ) P ( V b k ) _ – V _ b ¨ ¨ ¨ b V _ using the fact that V is finitely generated. Since t ∆ k u k ě are dual operationsto t m k u k ě , it satisfies the relations (3.6) for A -coalgebra structure. Furthermore, thecondition (3.7) holds for t ∆ k u k ě as λ d above does not depend on k in m k . Therefore weobtain the following. AURER-CARTAN DEFORMATION OF LAGRANGIANS 19
Lemma 3.8.
For a finitely generated A -algebra over k Λ , its dual V _ admits a structure of anA -coalgebra over k Λ . In particular, for a graded unobstructed compact (possibly immersed) Lagrangian L , CF ( L , L ; Λ ) _ is an A -coalgebra. We will provide a more explicit description of thecoalgebra structure on CF ( L , L ; Λ ) _ in 4.1.3.5.3. Dual of an A -coalgebra over k Λ . Let us next consider an A -coalgebra C . This time,we do not assume C to be finitely generated. As before, consider the dual C _ of C , theset of bounded k Λ -right-linear maps on C .As in (3.14), k elements f , ¨ ¨ ¨ , f k in C _ induce a map f b ¨ ¨ ¨ b f k on C b ¨ ¨ ¨ b C givenby ( f b ¨ ¨ ¨ b f k )( x k b ¨ ¨ ¨ b x ) : = f ( ¨ ¨ ¨ f k ´ ( f k ( x k ) x k ´ ) ¨ ¨ ¨ x ) .In terms of T -adic topology, it has the following estimate: val (( f b ¨ ¨ ¨ b f k )( x k b ¨ ¨ ¨ b x )) = ÿ val ( f i ( x i )) ě ÿ i ( ν _ ( f i ) + ν ( x i )) .Also the family t ∆ k u k ě satisfies ∆ k ( F λ C d ) Ă F λ + λ d ( T k C ) d + k ´ for some λ P R not depending on k . Therefore, if we define m k ( f , ¨ ¨ ¨ , f k )( x ) : = ( ´ ) | f | + ¨¨¨ + | f k | ( f b ¨ ¨ ¨ b f k )( ∆ k ( x )) , (3.18)then m k ( f , ¨ ¨ ¨ , f k ) is a well-defined element in C _ = ‘ ( C d ) _ . Finally, t m k u satisfies A -relations by dualizing the relations among t ∆ k u . Hence we obtain Lemma 3.9.
The dual of an A -coalgebra over k Λ is naturally an A -algebra over k Λ . Remark 3.10.
The duality between A -algebra and coalgebra (3.17) and (3.18) involves nontriv-ial signs, and so does the relation between operations on the A -dga ( V , m , m , m geq ” ) andthose on the A -dg-coalgebra ( V _ , ∆ , ∆ , ∆ geq ” ) . On the other hand, when viewed as a dgaand a dg-coalgebra respectively, ( V , d , ¨ ) and ( V _ , d , ∆ ) (the two d’s are different) are related inthe following way: ( d f )( x ) = ( ´ ) | f | f ( dx ) , ( ∆ f )( x b y ) = f ( x ¨ y ) for f P V _ and x P V. This is compatible with the sign difference between A and dg-conventionsin our setup.
4. M aurer -C artan space of a L agrangian and K oszul dual dga In this section, we show that the localized mirror obtained by Maurer-Cartan formalismof a Lagrangian L (see Section 2) can be identified with the Koszul dual of the Floercomplex of L . Consider a graded unobstructed compact Lagrangian L in a symplecticmanifold M . It is possible that L is a finite union of several irreducible Lagrangians, L = ‘ i P Γ L i , in which case we take the base ring k to be k Λ = ‘ i P Γ Λ π i . Thus the Floercomplex CF ( L , L ; Λ ) is naturally a bimodule over k Λ .Throughout, we will assume CF ( L , L ; Λ ) is minimal, and is supported in nonnega-tive degrees only. In particular, we have CF ( L , L ) – k Λ spanned by the unit class,or more precisely the sum of units for irreducible components of L . The projectionto the unit component CF ( L , L ; Λ ) Ñ CF ( L , L ; Λ ) defines a canonical augmentation ϵ : CF ( L , L ; Λ ) Ñ k Λ . AURER-CARTAN DEFORMATION OF LAGRANGIANS 20
The Maurer-Cartan algebras of Lagrangian revisited.
For simplicity, let us write V L = CF ( L , L ; Λ ) from now on, which is an A -algebra over k Λ as in the setting of theprevious section. Recall from 2.1 that V L is equipped with a non-Archimedean valua-tion which vanishes on H ˚ ( L ; C ) and the standard geometric generators associated withself-intersections. We further assume that V L is supported on nonnegative degrees only.To make the exposition more explicit, we fix generators X , ¨ ¨ ¨ , X N of CF ą ( L , L ) with ν ( X i ) = V L = span x X , ¨ ¨ ¨ , X l y V ě L = span x X l + , ¨ ¨ ¨ , X N y .Recall from Lemma 3.8 that the linear dual CF ( L , L ) _ is endowed with a structure ofan A -coalgebra. For simplicity, we write C L for CF ( L , L ) _ in what follows. C L hascounit and coaugmentation by dualizing those of V L . We write η : k Λ Ñ C L for thecoaugmentation, whose image is the (scalar multiples of) coordinate function for the unitclass in V L . We have a splitting C L = ¯ C L ‘ k Λ , where ¯ C L = C L / η ( k Λ ) To fix generators of ¯ C L , we take the coordinate function x i for X i . Recall that thedegree of x i is taken to be | x i | : = ´| X i | . The same notation x i has already been used fora coefficient in b in the Maurer-Cartan algebra in 2.3, but we will see that the two can benaturally identified. For instance, if L is a Lagrangian torus, a degree-1 generator x i canbe thought of as a coordinate for the 1-cocycles d θ i whereas other x j ’s are dual to wedgeproducts among d θ i ’s.The coalgebra structure on C L has the following concrete description. Suppose that wehave nontrivial coefficient c v P k Λ of X i in the A -operation on v : = X a b ¨ ¨ ¨ b X a k : m k ( X a , ¨ ¨ ¨ , X a k ) = ¨ ¨ ¨ + c v X i + ¨ ¨ ¨ . (4.1)For each such a tuple v , we have one summand x a k b ¨ ¨ ¨ b x a in ∆ k ( x i ) : ∆ k ( x i ) = ¨ ¨ ¨ + c v ( x a k b ¨ ¨ ¨ b x a ) + ¨ ¨ ¨ .Note that val ( c v ) ě v , and t ∆ k u k ě satisfies the condition (3.7).The cobar construction of ( C L , η ) produces a dg algebra Ω C L . Its underlying k Λ -module T ( s ´ ¯ C L ) is generated by formal (noncommutative) series in x i ’s with a fixeddegree such that valuations of coefficients are bounded below. Note that in T ( s ´ ¯ C L ) ,the degree of x i should be inverse-shifted, that is, | x i | : = | x i | +
1. We will often omitthe tensor product symbol, and simply write x a k b ¨ ¨ ¨ b x a : = x a k ¨ ¨ ¨ x a to denote thecorresponding element in ( s ´ C L ) b k . Then δ is given by δ ( x ) = ÿ k ÿ v c v x a k ¨ ¨ ¨ x a (4.2)where the inner sum is taken over all v = X a b ¨ ¨ ¨ b X a k such that m k ( X a , ¨ ¨ ¨ , X a k ) = ¨ ¨ ¨ + c v X + ¨ ¨ ¨ with c v ‰ Ω C L can be thought of as the formal function ringof the k Λ -bimodule CF ( L , L ) consisting of, a priori, noncommutative power series. Noticethat constant functions are also included as T ( s ´ ¯ C L ) = k ‘ (cid:0) ‘ k ě ( s ´ ¯ C L ) b k (cid:1) . We showthat the dg-structure on Ω C L encodes the obstruction of a Maurer-Cartan deformation ofthe A -algebra CF ( L , L ) . Proposition 4.1.
Let L be a graded unobstructed Lagrangian. Consider the dg-algebra Ω C L forC L = CF ( L , L ) _ . Then the 0-th cohomology H ( Ω C L , δ ) can be identified with the Maurer-Cartan algebra A L of L . AURER-CARTAN DEFORMATION OF LAGRANGIANS 21
Proof.
We prove this for L with a single irreducible component only, and it can be eas-ily generalized to other cases. By definition, the degree-0 component of Ω C L is givenby Ts ´ (cid:0) CF ( L , L ) (cid:1) _ , which consists of infinite series in x , ¨ ¨ ¨ , x l with bounded coeffi-cients, i.e. ÿ i = k ÿ v Pt ¨¨¨ , l u k c v x v with inf t c v : i =
1, 2, ¨ ¨ ¨ , v P t ¨ ¨ ¨ , l u k u ą ´8 .This is precisely Λ tt x , ¨ ¨ ¨ , x l uu in (2.10).Let X l + , ¨ ¨ ¨ , X N be the degree-2 generators of CF ( L , L ) . Then the degree-(-1) com-ponent of Ω C L is the set of infinite series consisting of words in x , ¨ ¨ ¨ , x l , x l + , ¨ ¨ ¨ , x N which have exactly one x j for l + ď j ď N . On the other hand, it is obvious from theconstruction that δ ( x j ) = f j ( x , ¨ ¨ ¨ , x l ) P ( Ω C L ) for x j with l + ď j ď N if and only ifthe following holds: m ( ) + m ( b ) + m ( b , b ) + ¨ ¨ ¨ = f ( x , ¨ ¨ ¨ , x l ) X l + + ¨ ¨ ¨ + f N ( x , ¨ ¨ ¨ , x l ) X N where b = ř li = x i X i , and we linearly expand the left hand side by (2.9).Now, a general degree-1 element f P ( Ω C L ) can be written as f = ÿ k = g k ( x , ¨ ¨ ¨ , x l ) x i k h k ( x , ¨ ¨ ¨ , x l ) for some polynomials g , h and l + ď i k ď N , where the lengths of terms become greaterthan any fixed number after certain stage, and T -adic valuations of terms are boundedbelow. Since δ ( g k ) = δ ( h k ) = δ ( f ) hasprecisely the same description as the series (2.12) after applying the Leibnitz rule. Hence,the image of ( Ω C L ) under δ coincides with the closure of the two-sided ideal generatedby δ ( x l + ) , ¨ ¨ ¨ , δ ( x N ) . Therefore the 0-th δ -cohomology computes Λ tt x , ¨ ¨ ¨ , x l uuxx δ ( x l + ) , ¨ ¨ ¨ , δ ( x N ) yy = k tt x , ¨ ¨ ¨ , x l uuxx f , ¨ ¨ ¨ , f N yy ,which is exactly the Maurer-Cartan algebra of L . □ In general, Ω ( V _ L ) may carry nontrivial information in higher degree component, andhence taking the 0-th cohomology A L may lose some geometric information of the La-grangian L . For instance, if L is simply connected, then A L is trivial.The curved version of the dga Ω ( V _ L ) has been already considered in [CHL], whichincludes the superpotential as a curvature term. It was viewed as the extended (weakMaurer-Cartan) deformation of L in the sense that it uses higher degree elements of V L additionally. Definition 4.2.
We will call Ω C L the Maurer-Cartan dga of L , which we denote by A L . The Maurer-Cartan dga has another description using the bar construction. In general,one has the following purely algebraic statement.
Proposition 4.3.
For a finitely generated A -algebra V, Ω ( V _ ) is isomorphic to ( BV ) _ .Proof. Lemma 3.7 shows that both Ω ( V _ ) and ( BV ) _ have the same underlying vectorspace, which is the subspace of the length-completion of T ( V _ ) formed by boundedseries. The remainder of the proof is comparing the algebraic operations and their signs,which is elementary. □ AURER-CARTAN DEFORMATION OF LAGRANGIANS 22
Applying this to V L = CF ( L , L ) , we see that the Maurer-Cartan dga of L can bealso expressed as ( BV L ) _ , and we will mostly use this alternative description in whatfollows. The main advantage of doing so is that ( BV ) _ makes sense even for an infinitedimensional V . Mimicking the proof of Proposition 4.1, one can derive a simple formulafor the differential d on this dga, whose proof is left as an exercise. Lemma 4.4.
For an A -algebra V, let X , ¨ ¨ ¨ , X N freely span V over its coefficient ring, and for-mally write ˜ b : = ř Ni = x i X i . Then d on ( BV ) _ is implicitly given by m ( e ˜ b ) = ř Ni = ( ´ ) | x i | d ( x i ) X i . Koszul dual algebras and generalized Maurer-Cartan algebras.
For a unital algebra A over a field k with an augmentation ϵ : A Ñ k , its Koszul dual algebra is defined by A ¡ = Ext ˚ ( k , k ) equipped with a Yoneda product. Here, the A -module structure on k isinduced by ϵ which splits the unit k Ñ A . The main question around A ¡ is whether ornot taking Koszul dual twice comes back to A itself. Such algebras has been extensivelystudied, and generalized into various directions since it first appeared in [Pri70].Our interest lies in the A -(or dg-) version of this construction, especially on the chainlevel rather than the cohomology Ext-algebras. Koszul duality in this context has alreadybeen investigated in many literatures such as [LPWZ08] or more categorical approach[Pos11], etc. In this section, we recall the Koszul dual of A -algebras adapted to ourfiltered setting, and explain its relationship with our Maurer-Cartan formalism.We begin with ( V , ε ) , a unital augmented A -algebra over k Λ with a valuation ν ,and set ¯ V : = ker ε . As before, k Λ can be regarded as an A -module over V via theaugmentation ϵ . Consider the sethom V ( k Λ , k Λ ) = tt f k ,1 u k ě : f k ,1 : k Λ b V b k Ñ k Λ u (4.3)of pre- A homomorphisms from k Λ to itself. Recall from 3.4 and 3.5 that hom V ( k Λ , k Λ ) is an A -dga with a differential M and the multiplication M . Definition 4.5.
For an augmented A -algebra ( V , ϵ ) over k Λ , hom V ( k Λ , k Λ ) (or its A quasi-isomorphism class) is called the Koszul dual of V, where the algebraic structure may follow eitherA - or dg-sign conventions. We will denote it by E ( V ) . Ignoring the valuation for a moment, the 0-cohomology of hom A ( k , k ) for an algebra A over k is precisely A ¡ = Ext ˚ A ( k , k ) , which justifies its name. As we shall see below, theKoszul dual for an A -algebra V = CF ( L , L ) indeed coincides with the Maurer-Cartandga of L .Recall from 3.2 that in our T -adic setting, hom V ( k Λ , k Λ ) consists of multilinear mapsthat decompose into finitely many homogeneous components, each of which is bounded.Putting aside the boundedness issue (which does not create additional difficulty here),the following is well-known. See, for e.g., [EL17a, Proposition 14] for the same statementin unfiltered setting. Proposition 4.6. E ( V ) = hom V ( k Λ , k Λ ) is quasi-isomomrphic to ( BV ) _ . Thus it is quasi-isomorphic to Ω ( V _ ) when V is finitely generated.Proof. Since V b k b k Λ – V b k and the action of unit is pre-determined by the unital prop-erty, a homogenous element t f k ,1 : k Λ b V b k Ñ k Λ u k ě of degree 1 ´ k + d in E ( V ) isequivalent to a bounded linear map BV = T ( s ¯ V ) Ñ k Λ of degree d . (Formally, onecan write f = ‘ k f k ,1 .) Taking this model, we obtain a bijective correspondence betweenhom V ( k Λ , k Λ ) and ( BV ) _ .We next compare the algebraic operations. The differential M on BV _ is given by M ( f )( X b ¨ ¨ ¨ b X k ) = d ( f )( X b ¨ ¨ ¨ b X k ) = ( ´ ) | f | f ( d ( X b ¨ ¨ ¨ b X k )) AURER-CARTAN DEFORMATION OF LAGRANGIANS 23 = ( ´ ) | f | f (cid:16) ÿ ( ´ ) | X ( ) | X ( ) b m ( X ( ) ) b X ( ) (cid:17) which agrees with (3.4) under our correspondence above, since the last two terms onthe right hand side of (3.4) vanish for inputs from the augmentation ideal. Note that thehidden factor k Λ absorbed in V b k has a nontrivial shifted degree in view of hom V ( k Λ , k Λ ) ,so | f | here plays a role of | ψ | in (3.4). Finally, taking into account the sign change (3.10)between A - and dg-conventions, the product M on BV _ is given by M ( f , g )( X b ¨ ¨ ¨ b X k ) = ÿ ( ´ ) | f | ( f b g ) (cid:16) X ( ) b ( X ( ) (cid:17) = ÿ ( ´ ) | f | f ( g ( X ( ) ) X ( ) ) ,and this matches with (3.5) under the correspondence. □ Applying this to V L = CF ( L , L ) for a graded unobstructed Lagrangian L , we concludethat the Maurer-Cartan dga A L is nothing but (a dga model of) the Koszul dual of the A -algebra CF ( L , L ) .Going back to classical Koszul duality theory for algebras, the important class of objectsis, roughly speaking, a graded algebra A over k such that hom A ( k , k ) (graded with respectto our convention) has a trivial cohomology at every nonzero degree. In this case, A iscalled a Koszul algebra, and its double Koszul dual gets back to A .On the other hand, recall that the Maurer-Cartan algebra A L of a Lagrangian L neglectsnonzero-degree part of the cohomology of hom V L ( k Λ , k Λ ) . We speculate that the localizedmirror functor which can be understood as a functor V L mod Ñ A L mod (4.4)establishes an equivalence if and only if the Floer complex V L satisfies an analogouscondition to the Koszulity of an algebra, for e.g., the cohomology of hom V L ( k Λ , k Λ ) beingsupported at degree 0, only. Here, the left hand side of (4.4) can be thought of as thesubcategory of the compact Fukaya category generated by L . For a general L , one mayneed to consider the extended mirror functor V L mod Ñ A L dg-mod M Ñ hom V L ( k Λ , M ) which reduces to (4.4) by regarding hom V L ( k Λ , M ) as a module over the 0-th cohomologyof hom V L ( k Λ , k Λ ) .We remark that in most of existing applications of the localized mirror functor, L istaken to be weakly unobstructed, and the resulting localized mirror is given as a Landau-Ginzburg model. Thus the above formulation does not directly apply. It would be moreinteresting to seek for a reasonable condition in the curved setting which guarantees themirror functor to be an equivalence.4.3. Dual pairs of objects in A -categories. Consider two objects L = ‘ ri = L i and G = ‘ ri = G i in a unital (filtered) A -category C over the field Λ . Here, each L i and G i areobjects of C , and one may need to replace C by its additive enlargement if necessary, inorder to make sense of the direct sum. Thus, morphism spaces between L and G as wellas their endomorphisms are naturally modules over k Λ = ‘ i Λ x π i y .Suppose G and L satisfy the following conditions. ‚ hom C ( G , L ) – k Λ as hom C ( G , G ) -modules. ‚ hom ą C ( G , G ) = ă C ( L , L ) = ‚ hom C ( L , L ) is finitely generated over k Λ with hom C ( L , L ) – k Λ . AURER-CARTAN DEFORMATION OF LAGRANGIANS 24
Such ( G , L ) can be thought of as a Koszul dual pair in the sense we explain now.By degree reason, the left hom C ( G , G ) -module structure on hom C ( G , L ) given as for k ě n k | : = m k + : hom i C ( G , G ) b ¨ ¨ ¨ b hom i k C ( G , G ) b hom C ( G , L ) Ñ hom C ( G , L ) must be trivial unless k = i = ¨ ¨ ¨ i k = C is unital, we have n | ( G , P ) = P for P P hom C ( G , L ) where G is the unit on hom C ( G , G ) . The similar is true for the righthom C ( L , L ) -module hom C ( G , L ) , and in this case, the only nontrivial is the action of ascalar multiple of the unit in hom C ( L , L ) . Lemma 4.7.
Fix a generator P of hom C ( G , L ) so that hom C ( G , L ) = k Λ x P y . Define ε = ε L :hom C ( G , G ) Ñ k Λ by m ( Z , P ) = ε ( Z ) P . Then ε is a strict augmentation on hom C ( G , G ) .Proof. We show that t ε k u k ě : hom C ( G , G ) Ñ k Λ defined by ε : = ε , ε k ě ” A -homomorphism. From the hom C ( G , G ) -module structure on hom C ( G , L ) , it is easy tosee that ε ( Z ) = Z =
0. Also, we have ε ( m ( Z )) = ε ( m ( Z )) P = m ( m ( Z ) , P ) ,and the right hand side equals m ( m ( Z , P )) ´ m ( Z , m ( P )) , each of which vanishes since m ( P ) =
0. It now suffices to check the following identity: ε ( m ( Z , Z )) = ε ( Z ) ¨ ε ( Z ) .To see this, observe that m ( m ( Z , Z ) , P ) = m ( Z , m ( Z , P )) ´ m ( m ( Z ) , Z , P ) + m ( Z , m ( Z ) , P )= m ( Z , m ( Z , P ))= ε ( Z ) m ( Z , P ) = ( ε ( Z ) ε ( Z )) P .Here, m ( m ( Z ) , Z , P ) = m ( Z , m ( Z ) , P ) = m ( Z ) = m ( Z ) = □ One can similarly define an augmentation on hom C ( L , L ) , but it is easy to see fromdegree reason that the resulting augmentation is the projection to the unit componenthom C ( L , L ) – k Λ .Although hom C ( G , L ) is trivial regarded as either a left hom C ( G , G ) - or a right hom C ( L , L ) -module, its ( hom C ( G , G ) , hom C ( L , L )) -bimodule structure is still quite rich. Explicitly, itis given by the A -operations t m k u k ě m l + k + : hom C ( G , G ) b l b hom C ( G , L ) b hom C ( L , L ) b k Ñ hom C ( G , L ) (4.5)on C defined by ( y , ¨ ¨ ¨ y l , P , X , ¨ ¨ ¨ , X k ) ÞÑ m k + l + ( y , ¨ ¨ ¨ , y l , P , X , ¨ ¨ ¨ , X k ) .For notational simplicity, denote A -algebras hom C ( L , L ) and hom C ( G , G ) by V and W ,respectively, As before, we write ¯ V : = hom ą C ( L , L ) (the augmentation kernel) and ¯ W : = ker ε . Since hom C ( G , L ) – k Λ , the above map can be thought of as a paring between T ( sW ) and T ( sV ) . Here, the degree-shift makes the paring Z -graded (i.e., it has degreezero). We can alternatively view this pairing in the following perspective. AURER-CARTAN DEFORMATION OF LAGRANGIANS 25
Koszul map.
One can first take direct sum over k after removing the unit component of V to get ˜ κ l : W b l b BV Ñ k Λ , (4.6)where BV = ˆ T ( s ¯ V ) is the bar-construction of V . Recall that BV has the component k Λ ,and hence ˜ κ l should contain the component W b l b k Λ = W b l Ñ k Λ .Keeping track of the corresponding part in (4.5), it coincides with the augmentation ε . Bydualizing (4.6), we obtain κ l : W b l Ñ ( BV ) _ (4.7)(4.7) is well-defined due to the boundedness condition on m k -operations which impliesthat (4.6) gives a bounded linear map for each fixed element of W b l . This will be moreclear once we have explicit formula for κ below.Recall that ( BV ) _ = hom V ( k Λ , k Λ ) is the koszul dual dga of V . The map κ = t κ l u l ě will be referred to as the Koszul map . The Koszul map admits the following explicitdescription. Let t X , ¨ ¨ ¨ , X N u freely generate V , and suppose X i P π β i ¨ V ¨ π α i . Denotetheir dual variables by x , ¨ ¨ ¨ , x N which can be naturally identified as elements of BV _ ,and take a formal linear combination ˜ b = ř Ni = x i X i (which should not be confused with b in (2.8) consisting of degree-1 elements only). Then ÿ k m l + k + ( Z , ¨ ¨ ¨ , Z l , P , ˜ b , ¨ ¨ ¨ , ˜ b ) = P κ l ( Z , ¨ ¨ ¨ , Z l ) , (4.8)where we use the convention of pushing the formal variables to the back, i.e., m k ( Z , ¨ ¨ ¨ , Z l , P , x i X i , ¨ ¨ ¨ , x i m X i m ) = m k ( Z , ¨ ¨ ¨ , Z l , P , X i , ¨ ¨ ¨ X i m ) x i m ¨ ¨ ¨ x i for k = l + m +
1. This makes κ into a k Λ -module homomorphism, though it is slightlydifferent from our earlier convention (2.9) in which we put x i ’s before Floer generators.To make formulas look more consistent, we could locate formal variables back to Floergenerators, for instance ř X i x i , from the very beginning, which would not cause anyconflict since the Maurer-Cartan algebra is not affected by this change.More concretely, the left hand side of (4.8) computes ÿ k ÿ i , ¨¨¨ , i k m ( ⃗ Z , P , x i X i , ¨ ¨ ¨ , x i k X i k ) = ÿ k ÿ i , ¨¨¨ , i k m ( ⃗ Z , P , X i , ¨ ¨ ¨ , X i k ) x i k ¨ ¨ ¨ x i , = P ÿ k ÿ i , ¨¨¨ , i k c i , ¨¨¨ , i k x i ¨ ¨ ¨ x i k (4.9)where ⃗ Z = Z b ¨ ¨ ¨ b Z l and we put m ( ⃗ Z , P , X i , ¨ ¨ ¨ , X i k ) = P c i , ¨¨¨ , i k for c i , ¨¨¨ , i k P k Λ .Therefore κ l ( Z , ¨ ¨ ¨ , Z l ) = ř k ř i , ¨¨¨ , i k c i , ¨¨¨ , i k x i ¨ ¨ ¨ x i k .We next show that the Koszul map κ : = t κ l u l ě is an A -algebra homomorphism. Letus begin by checking the degree of κ . Lemma 4.8.
The degree of κ l (4.7) is given by ´ l.Proof. If x i ¨ ¨ ¨ x i m nontrivially appears in κ l ( Z , ¨ ¨ ¨ , Z l ) , then the corresponding m k -operationsmust be nonzero. That is, we have to look at the case when the operation m l + m + ( Z , ¨ ¨ ¨ , Z l , P , X i , ¨ ¨ ¨ , X i m ) can have a nonzero output. Since the only possible output is P itself, we see that | Z | + ¨ ¨ ¨ + | Z l | + | X i | + ¨ ¨ ¨ + | X i m | + ´ ( l + m + ) = AURER-CARTAN DEFORMATION OF LAGRANGIANS 26 (since | P | = | x i | + ¨ ¨ ¨ + | x i m | = ( ´ | X i ) + ¨ ¨ ¨ + ( ´ | X i m | )= n + | Z | + ¨ ¨ ¨ + | Z l | + ´ ( l + m + )= | Z | + ¨ ¨ ¨ + | Z l | + ( ´ l ) as desired. □ Proposition 4.9. κ : W Ñ ( BV ) _ is an A -algebra homomorphismProof. We write M and M for the operations of the A -dga ( BV ) _ , i.e., M ( x ) = d ( x ) , M ( x , y ) = ( ´ ) | x | x b y .We have to verify the identity M ( κ l ( Z , ¨ ¨ ¨ , Z l )) + ÿ l + l = l M ( κ l ( Z , ¨ ¨ ¨ , Z l ) , κ l ( Z l + , ¨ ¨ ¨ , Z l ))= ÿ k + k = l + ( ´ ) | Z | + ¨¨¨ + | Z j | κ k ( Z , ¨ ¨ ¨ , Z j , m k ( Z j + , ¨ ¨ ¨ , Z j + k ) , Z j + k + ¨ ¨ ¨ , Z l ) .Let us consider the A -relations for the tuple ( Z , ¨ ¨ ¨ , Z l , P , e ˜ b ) where e ˜ b = + ˜ b + ˜ b b ˜ b + ˜ b b ˜ b b ˜ b + ¨ ¨ ¨ ,and ˜ b b i is inserted to A -operations as m k ( ´ , ¨ ¨ ¨ , ´ , i hkkkikkkj ˜ b , ¨ ¨ ¨ , ˜ b ) . The relation is equivalentto ÿ ( ´ ) | ⃗ Z | m ( ⃗ Z , P , e ˜ b , m ( e ˜ b ) , e ˜ b ) + ÿ ( ´ ) | ⃗ Z ( ) | ´ m ( ⃗ Z ( ) , m ( ⃗ Z ( ) , P , e ˜ b ) , e ˜ b )= ÿ ( ´ ) | ⃗ Z ( ) | m ( ⃗ Z ( ) , m ( ⃗ Z ( ) ) , ⃗ Z ( ) , P , e ˜ b ) ,where we omit the subscript k in m k for simplicity. Here, | ⃗ Z | is the sum of the shifteddegrees of factors in ⃗ Z . The right hand side equals ÿ ( ´ ) | ⃗ Z ( ) | m ( ⃗ Z ( ) , m ( ⃗ Z ( ) ) , ⃗ Z ( ) , P , e ˜ b ) = P ÿ ( ´ ) | ⃗ Z ( ) | κ ( ⃗ Z ( ) , m ( ⃗ Z ( ) ) , ⃗ Z ( ) ) ,and the second term on the left hand side equals ÿ ( ´ ) | ⃗ Z ( ) | ´ m ( ⃗ Z ( ) , m ( ⃗ Z ( ) , P , e ˜ b ) , e ˜ b ) = ÿ ( ´ ) | ⃗ Z ( ) | ´ m ( ⃗ Z ( ) , P κ ( ⃗ Z ( ) ) , e ˜ b )= ÿ ( ´ ) | ⃗ Z ( ) | ´ m ( ⃗ Z ( ) , P , e ˜ b ) κ ( ⃗ Z ( ) )= ÿ ( ´ ) | κ ( ⃗ Z ( ) ) | P κ ( ⃗ Z ( ) ) ¨ κ ( ⃗ Z ( ) )= P ÿ M ( κ ( ⃗ Z ( ) ) , κ ( ⃗ Z ( ) )) since κ l has degree 1 ´ l . Lastly, applying m ( e ˜ b ) = ř Ni = ( ´ ) | x i | d ( x i ) X i (see Lemma 4.4),the first term on the left hand side becomes ( ´ ) | ⃗ Z | m ( ⃗ Z , P , e ˜ b , m ( e ˜ b ) , e ˜ b )= ÿ k ÿ i , ¨¨¨ , i k ÿ ď a ď k ( ´ ) | ⃗ Z | + | x ia | m ( ⃗ Z , P , x i X i , ¨ ¨ ¨ , x i a ´ X i a ´ , d ( x i a ) X i a , x i a + X i a + , ¨ ¨ ¨ , x i k X i k )= ÿ k ÿ i , ¨¨¨ , i k ÿ ď l ď k ( ´ ) | ⃗ Z | + | x ia | + ˚ m ( ⃗ Z , P , X i , ¨ ¨ ¨ , X i a , ¨ ¨ ¨ , X i k ) x i k ¨ ¨ ¨ x i a + d ( x i a ) x i a ´ ¨ ¨ ¨ x i , AURER-CARTAN DEFORMATION OF LAGRANGIANS 27 where ˚ = | x i | + ¨ ¨ ¨ + | x i a ´ | comes from the Koszul convention since | d ( x i a ) X i a | = ´ | ⃗ Z | + | x i a | + ˚ = ( ´| X i | ´ ¨ ¨ ¨ ´ | X i k | ´ ) + | x i a | + ˚ = ( | x i | + ¨ ¨ ¨ | x i k | ´ ) + | x i a | + ( | x i | + ¨ ¨ ¨ + | x i a ´ | ) ” | x i a + | + ¨ ¨ ¨ + | x i k | mod 2Comparing with (4.9), we see that m ( ⃗ Z , P , e ˜ b , m ( e ˜ b ) , e ˜ b ) = P d ( κ ( ⃗ Z )) = P M ( κ ( ⃗ Z )) ,which completes the proof. □ The A -algebra homomorphism κ : hom C ( G , G ) Ñ A L = ( B hom C ( L , L )) _ induces amap κ : H hom C ( G , G ) Ñ A L = H ( B hom C ( L , L )) _ ) on the 0-th cohomology, whichwill be of our main interest in the rest of the paper.5. D ual pairs in wrapped F ukaya categories We apply the homological algebra tool developed so far to the Fukaya A -algebra, andstudy the Koszul duality between A -algebras from two Lagrangians in some specialgeometric relation. Let ( M , ω = d Θ ) be a Liouville manifold, and consider an exactcompact Lagrangian L = ‘ ri = L i which, together with suitable Floer data, gives an objectin W C : = W Fuk ( M ) . Since L consists of r irreducible components, we are naturally towork over the semi-simple ring k C or k Λ . Analogously to the algebraic setting in 4.3,suppose that there exists another Lagrangian G = ‘ ri = G i in W C so that the pair ( G , L ) admits the following properties. Assumption 5.1.
We assume G and L satisfy ‚ hom W C ( G , L ) – k C as hom W C ( G , G ) -modules; ‚ hom ă W C ( L , L ) = and hom W C ( L , L ) – k C ; ‚ hom i W C ( G , G ) = for i ‰ , and G generates L in W C . The first condition implies that (after rearranging L i ’s in L = ‘ ri = L i suitably) L i in-tersects G i exactly at one point, say P i , with degree-0, and L i X G j = j ‰ i . Thenhom W C ( G , L ) is spanned by P = ř P i . We also remark that the second condition only con-straints the degree of immersed generators of L . Note, in particular, that G and L form aKoszul dual pair in W in the sense of 4.3. Our goal is to compare the wrapped Floer co-homology of G and the Maurer-Cartan algebra of L using the Koszul map κ algebraicallydefined in 4.3Our setting slightly generalizes the geometric one in [EL17b, Introduction] in the sensethat an individual component L i can be non-simply-connected. For this reason, it iscrucial to introduce a certain completion of hom W Λ ( G , G ) = CW ( G , G ; Λ ) (related withthe T -adic topology) to compare with the Maurer-Cartan algebra of L . We will alsoexamine in 5.3 the case of non-exact tori L sitting as fibers of a Lagrangian torus fibation,which will be useful in applications to local SYZ examples later. Requiring hom ą W C ( G , G ) = A L = H ( A L ) and H hom W C ( G , G ) AURER-CARTAN DEFORMATION OF LAGRANGIANS 28 W C ( G , G ) and the Maurer-Cartan algebra A L in C -coefficient. Let us first com-pare Floer theory of G and L over C -coefficient. We will lift our argument to the filteredsetting (over Λ -coefficient) in 5.2.Suppose that two objects G and L in W C = W Fuk ( M ) satisfy Assumption 5.1. Follow-ing [EL17b, Introduction], we begin with the Yoneda map for G ,hom W C ( L , L ) Ñ hom hom W C ( G , G ) ( hom W C ( G , L ) , hom W C ( G , L )) = hom hom W C ( G , G ) ( k C , k C ) ,(5.1)which is a quasi-isomorphism since G generates L . We remark that k C above is taken tobe a left module over hom W C ( G , G ) . [EL17a, Proposition 14] (or Proposition 4.6 adaptedto C -coefficient setting) shows thathom hom W C ( G , G ) ( k C , k C ) – ( B hom W C ( G , G )) where ( ´ ) is the graded dual (over k C ), that is, the direct sum of the sets of left k C -module maps from individual graded pieces to k C . The bar construction B in this case isthe one for k C -module, taken with respect to the (strict) augmentation ε C : hom W C ( G , G ) Ñ k C given by m ( r Z , r P ) = ε C ( r Z ) r P , which is related with ε given in Lemma 4.7 via basechange. Recall that r Z and r P denote the exact generators of morphism spaces in W C associated with the actual geometric intersection point Z and P . The augmentation ε C depends significantly on L .For simplicity, let us write V C and W C for hom W C ( L , L ) and hom W C ( G , G ) respectively.Taking the bar construction on the A quasi-isomorphism V C Ñ ( BW C ) , we have adg-coalgebra quasi-isomorphim BV C » ÝÑ B (cid:0) ( BW C ) (cid:1) , or dually, ( B ( BW C )) ÝÑ ( BV C ) (5.2)since Ext group vanishes over a divisible group. We write ( A L ) C for H ( BV C ) , the 0-thcohomology of the right hand side of (5.2). It can be thought of as the “formal” Maurer-Cartan algebra of L in which the Maurer-Cartan relations are given as formal powerseries, not involving any Novikov parameters.On the other hand, notice that by our assumption on the grading on G , W C is simply analgebra over k C . Using deformation theory argument, it is shown in [Seg08, Theorem 2.14]that the natural map W C Ñ ( B ( BW C )) induces an isomorphism ˆ W ϵ C C Ñ H ( B ( BW C )) where ˆ W ϵ C C is the completion of W C with respect to ker ε C . We conclude thatˆ W ϵ C C – ( A L ) C (5.3)as algebras, with the isomorphism explicitly given by κ C : ˆ W ϵ C C Ñ ( A L ) C ÿ i ě m W C i + ( r Z , r P , i hkkkkikkkkj b , b , ¨ ¨ ¨ , b ) = r P ¨ κ C ( r Z ) (5.4)where b = ř ˜ x i r X i is a formal linear combination of degree-1 exact generators r X i of V C = hom W C ( L , L ) coupled with their dual coordinate functions ˜ x i . Note that ř x i X i = ř ˜ x i r X i since the valuations of x i and X i compensate each other. However the sum in (5.4) shouldbe expanded in terms of ˜ x i ’s, as otherwise the expansion would involve some nontrivialNovikov number. It is easy to see that κ C ( Z ) above is well-defined as a formal (noncom-mutative) power series in ˜ x i ’s. AURER-CARTAN DEFORMATION OF LAGRANGIANS 29 the Koszul map κ in Floer theory (over Λ ). We would like to upgrade (5.3) into our T -adic setting, which will turn out to give us some benefit in studying the local structureof the mirror space, later. For instance, this will allow us to analyze possibly nonexacttorus fibers in a SYZ fibration and their corresponding points in the mirror space (see5.3).Over Λ -coefficient, the following will substitute for the completion on the left handside of (5.3). Let ( A C , ϵ C ) be an augmented algebra over k C , and consider the inducedaugmented algebra ( A = A C b Λ , ϵ = ϵ C b Λ ) over k Λ . For such ( A , ϵ ) , we define ˆ A ϵ byˆ A ϵ : = ˆ A ϵ Λ b Λ Λ (5.5)where ˆ A ϵ Λ is the usual completion of A Λ = A C b C Λ with respect to the filtration A Λ Ą ¯ A Λ Ą ¯ A Λ ¨ ¯ A Λ Ą ¯ A Λ ¨ ¯ A Λ ¨ ¯ A Λ Ą ¨ ¨ ¨ for ¯ A Λ : = ker ( ε C b Λ ) . Its element f consists of the data f = ( f , f , f , ¨ ¨ ¨ ) such that f i P W Λ / ( ¯ W Λ ) i + and f i = f i + in W Λ / ( ¯ W Λ ) i + . In spirit of Lemma 3.6, ˆ A ϵ can bethought of as the subset of the completion of A with respect to the filtration A Ą ¯ A Ą ¯ A ¨ ¯ A Ą ¯ A ¨ ¯ A ¨ ¯ A Ą ¨ ¨ ¨ which consists of bounded series in T -adic valuation, where ¯ A : = ker ϵ . We shall see inactual applications that this type of completion can produce a fairly big open neighbor-hood of the point corresponding to the maximal ideal ¯ A .We are now ready to state the main theorem in this section. For M a Liouville manifold,recall that we write W C : = W Fuk ( M ) and W Λ : = W Fuk Λ ( M ) . Theorem 5.2.
If two exact Lagrangian branes L = ‘ ri L i and G = ‘ ri G i in W C satisfy As-sumption 5.1, then the Maurer-Cartan algebra A L is isomorphic to the completion ˆ W ϵ L of W : = hom W Λ ( G , G ) given as in (5.5) , where ϵ L is the augmentation on W determined by L.Moreover, the isomorphism is explicitly given by a natural extension of the Koszul map κ :hom W Λ ( G , G ) Ñ A L , ÿ m k ( Z , P , b , ¨ ¨ ¨ , b ) = P ¨ κ ( Z ) (5.6) where b = ř x i X i is a formal linear combination of degree-1 generators.Proof. Our starting point is a quasi-isomorphism BV C » Ñ B ( BW C ) of dg-coalgebras over k C , where BW C is taken with respect to the natural augmentation ϵ C : W Ñ k C induced by L . Since Λ is torsion-free, taking ( ´ ) b C Λ preserves a quasi-isomorphism by universalcoefficient theorem. Thus we have à k (cid:16) ( sV C ) b C k b Λ (cid:17) » ÝÑ à k (cid:18)(cid:16) s ( BW C ) (cid:17) b C k b Λ (cid:19) .We next apply hom k Λ ( ´ , k Λ ) to each graded pieces of the left hand side consideredas a right k Λ -module (this preserves an quasi-isomorphism too, since k Λ is divisible) toobtain à ˜ d = ´ d hom Λ (cid:16) ‘ k (cid:16) T k s ( BW C ) (cid:17) d b Λ , Λ (cid:17) » ÝÑ à ˜ d = ´ d hom Λ ( ‘ k ( T k sV C ) d b Λ , Λ ) ,arranged accordingly with the degree ˜ d = ´ d ď
0. Here, ( T k ´ ) d is the degree- d com-ponent of the k -fold tensor product (over C in the above situation). We will use d as a AURER-CARTAN DEFORMATION OF LAGRANGIANS 30 running index instead of the actual degree ˜ d in what follows. By base change, à d hom C (cid:16) ‘ k (cid:16) T k s ( BW C ) (cid:17) d , Λ (cid:17) » ÝÑ à d hom C ( ‘ k ( T k sV C ) d , Λ ) .Here, Λ in the second slot is viewed as a C -module in an obvious way. After takingtensor product ( ´ b Λ Λ ) , the right hand side recovers the graded topological dual of BV by Lemma 3.6, which is the Maurer-Cartan dga A L of L .Let us pay attention to the left hand side. More concretely, it can be written as à d hom C (cid:32) à i + ¨¨¨ + i k = d + k (cid:16) ¯ W b C i C (cid:17) b ¨ ¨ ¨ b (cid:16) ¯ W b C i k C (cid:17) , Λ (cid:33) (5.7)where the symbol “ b ” is introduced to indicate the tensor product used for T k in T k s ( BW C ) so that one can distinguish it from “ b ” from BW C . In Appendix A, it is shown that the0-th cohomology of the complex (5.7) is isomorphic to ˆ W ϵ Λ for ϵ = ϵ C b Λ , which com-pletes the proof. It is not difficult to check that the resulting isomorphism ˆ W ϵ – BV _ is given by extending the formula (5.6), which seems to be essentially a unique algebrahomomorphism one can construct in this situation. □ Observe that the map (5.6) makes sense already on the chain level. Taking the formallinear combination ˜ b including generators of all degrees, we have an A -homomorphism κ : CW ( G , G ; Λ ) Ñ A L . (5.8)We conjecture that (5.8) induces a quasi-isomorphism between the Maurer-Cartan dga A L and a certain completion of the A -algebra CW ( G , G ; Λ ) under the weaker assumption CW ą ( G , G ) = igure
2. Compatibility between the differentials on CW ( G , G ) and A L The relationship between algebraic structures on CW ( G , G ) and the Maurer-Cartanalgebra A L can be heuristically seen by a usual cobordism argument in Floer theory. Forinstance, Figure 2 describes the cobordism that implies compatibility of the chain-levelKoszul map (5.8) with the differential on CW ( G , G ) and A L . Note that the differential on A L is essentially the Maurer-Cartan equation m ( e b ) = κ with products can be intuitively seen as in Figure 3. AURER-CARTAN DEFORMATION OF LAGRANGIANS 31 F igure
3. Compatibility between the products on CW ( G , G ) and A L (local) SYZ fibration: a generating section and fibers. We next consider the follow-ing geometric setting for a local SYZ fibration. Let ( M , ω = d Θ ) be a Liouville manifoldand, suppose it admits a Lagrangian torus fibration π : M Ñ B . We will deal withthe monotone fibers π ´ ( b ) with minimal Maslov number 0 so that they become gradedobjects in the monotone Fukaya category W Λ = W Fuk Λ ( M ) .Suppose that there exists an exact Lagrangian torus fiber L of π , and a Lagrangiansection G of π that generates L . By Theorem 5.2, the Maurer-Cartan algebra of L can beobtained by completing (in the sense of (5.5)) the endomorphism algebra hom W Λ ( G , G ) at the augmentation ideal for ε = ε L : hom W Λ ( G , G ) Ñ k Λ . We are interested in thebehavior of nearby torus fibers L and their associated Koszul maps κ : hom W Λ ( G , G ) Ñ A L . They are in general nonexact.Let Θ be an 1-form such that d Θ = ω and Θ | L = d f for some function f : L Ñ R .In our local setting, one can use action-angle coordinates u , ¨ ¨ ¨ u n and ρ , ¨ ¨ ¨ ρ n to findsuch Θ . We may assume ( u , ¨ ¨ ¨ , u n ) = ( ¨ ¨ ¨ , 0 ) for points in L . Set a i : = ş β i Θ | L for β i a loop along the direction of ρ i . Note that ( u , ¨ ¨ ¨ , u n ) = ( a , ¨ ¨ ¨ , a n ) gives the actioncoordinates for L , due to Stokes’ theorem since d Θ = ω .We take contractible open sets U , U ( Ă B ) with U Ă U , both of which contain ( ¨ ¨ ¨ , 0 ) and ( a , ¨ ¨ ¨ , a n ) where we use u i as coordinates on the base B of π . Thenwe choose compactly supported functions h i ( u i ) for 1 ď i ď n such that h i | U ” a i and h i | B z U ”
0. They give rise to an 1-form r Θ : = Θ ´ ř h i ( u i ) d ρ i on M . We assume that | h i | is small enough for d r Θ = ω ´ ř h i ( u i ) du i d ρ i = ř ( ´ h i ( u i )) du i d ρ i to be still symplec-tic. This is possible for all the examples considered in the paper, but may impose someadditional condition on L and L , since for this to hold true, the gap between U and U should not be too narrow.Notice that d r Θ and ω differ from each other by a compactly supported exact 2-form.Hence Moser’s trick applies to the situation to produce a (compactly supported) diffeo-morphism ψ such that ψ ˚ ( d r Θ ) = ω . As ω = d ˜ Θ on U that contains L , ψ preserves L . In particular, if we set Θ : = ψ ˚ r Θ , then d Θ = ω and Θ | L = r Θ | L is exact since itsintegration over any loops in L vanishes by construction. By construction, the close form Θ | G ´ Θ | G vanishes away from a contractible subset of G , and hence is exact. Thus G isan exact Lagrangian with respect to Θ , and the pair ( G , L ) of Θ -exact Lagrangians fits AURER-CARTAN DEFORMATION OF LAGRANGIANS 32 into the setting of Theorem 5.2. Therefore the Maurer-Cartan algebra of L can be obtainedas a suitable completion of hom W Λ ( G , G ) .We can make the above discussion more concrete by relating Floer theory of ( G , L ) withthat of ( G , L ) as follows. The closed 1-form α : = Θ ´ Θ induces a symplectomorphism ϕ by integrating the symplectic vector field X α given by ι X α ω = α . Since α = ´ ř a i d ρ i on π ´ ( U ) , the symplectomorphism ϕ simply translates L back to L via ( u , ¨ ¨ ¨ , u n ) ÞÑ ( u ´ a , ¨ ¨ ¨ , u n ´ a n ) . On the other hand, π ´ ( U ) X G lies in some contractible open setsince G intersects each torus fiber at one point, and the 1-form α admits a well-definedprimitive, say r F , over this set. We extend r F outside π ´ ( V ) using suitable bump functionsto get F : M Ñ R . Thus, restricting ourselves to G , we have ϕ ( G ) = ϕ F ( G ) since dF and α agrees with each other on a region near G .From the above discussion, we conclude that Floer theory of the pair ( G , L ) can be iden-tified with that of ( ϕ ( G ) , ϕ ( L )) = ( ϕ F ( G ) , L ) , and hence that of ( G , L ) as F is a compactlysupported Hamiltonian. Suppose that we make the wrapping for both G and ϕ F ( G ) hap-pen away from the support of F so that hom W Λ ( G , G ) and hom W Λ ( ϕ F ( G ) , ϕ F ( G )) haveprecisely the same set of geometric generators. However, these set-theoretically identicalgenerators have different valuations on the two Λ -vector space since ϕ F doe not preserve Θ in general. In practice,hom W Λ ( G , G ) – Ñ hom W Λ ( ϕ F ( G ) , ϕ F ( G )) simply scales these geometric intersection points by suitable power of T . In this case, theKoszul map hom W Λ ( G , G ) Ñ A L can be easily obtained from that for ( G , L ) after suchscaling generators of hom W Λ ( G , G ) .5.4. Example: T ˚ S . Let us first look into the simplest case of M = T ˚ S as a warm-up example. We use coordinate ( t , s ) P T ˚ S = S ˆ R where t P [
0, 2 π ] /0 „ π as inExample 2.2, so we have ω = dsdt with Θ = sdt . The zero section L : = S is exact withrespect to Θ , and clearly, L and the cotangent fiber G satisfy Assumption 5.1. Recall thatgenerators of hom W C ( G , G )(= CW ( G , G ; H = s )) can be taken to be exact ones, r Z i for i P Z with the multiplication m ( r Z i , r Z j ) = r Z i + j being the only nontrivial A -operation among them. Under the base changehom W C ( G , G ) b C Λ = hom W Λ ( G , G ) , r Z i maps to T ´ i Z i where Z i denotes the geometric generator supported at the intersectionpoint Z i . Notice that hom W Λ ( G , G ) – Λ [ Z , Z ´ ] is precisely the function ring of Λ ˆ ,where Z : = r Z serves as the standard coordinate, and r Z = Z is the unit (i.e., thefunction on Λ ˆ whose value is constantly 1).We write P for the unique intersection point between G and L , and at the same time, P denotes the corresponding geometric generator of hom W Λ ( G , L ) . In this example, theassociated exact generator r P of hom W Λ ( G , L ) agrees with P since the primitives of Θ | L and Θ | G are both zero.From direct counting, we have m ( Z , P ) = T P for t P u : = ϕ H ( G ) X L , but we mayassume that the difference between P and P is negligible. Hence m ( Z , P ) = T P , or m ( r Z , r P ) = r P (no T appearing since it is written in terms of exact generators), and theaugmentation ϵ : hom W Λ ( G , G ) Ñ Λ associated with L sends r Z to 1. Consequently, ε has AURER-CARTAN DEFORMATION OF LAGRANGIANS 33 the kernel ker ε = t r Z i ´ r Z : i P Z u = t T ´ i Z i ´ Z : i P Z u ,and in particular, the corresponding point in Λ ˆ is Z =
1. On the other hand, the Maurer-Cartan algebra A L is a ring Λ t x u consisting of bounded power series, where x is lineardual to dt P H ( S ) . Setting b = xdt , the Koszul map κ computes P κ ( Z ) = ÿ k ě m k + ( Z , P , b , ¨ ¨ ¨ , b ) = ÿ k = k ! x k m ( Z , P ) = P e x T ,and hence κ ( Z ) = κ ( r Z ) = κ ( T ´ Z ) = e x . The contributing triangles to κ ( Z ) and κ ( Z ´ ) are shown in Figure 4.F igure
4. The zero section L and a cotangent fiber G in T ˚ S Remark 5.3.
We can deform CF ( L , L ) by Λ U -flat connections ∇ z whose holonomy is parametrizedby z P Λ U . The associated κ can be computed asP κ ( Z ) = m ∇ z ( Z , P ) = P zT . Thus κ ( Z ) = z, which agrees with the earlier computation if we set z = e x . By Theorem 5.2, the Maurer-Cartan space of L can be obtained by completing Λ [ Z , Z ´ ] at the maximal ideal generated by Z = c + c ( Z ´ ) + c ( Z ´ ) + ¨ ¨ ¨ (5.9)for c i P Λ , as long as ´8 ă inf t val ( c i ) : i =
0, 1, 2, ¨ ¨ ¨ u . In particular, x can be identifiedwith ( Z ´ ) ´ ( Z ´ ) + ( Z ´ ) ´ ¨ ¨ ¨ = log (( Z ´ ) + ) ,where “log” on the right hand side means a priori a formal expansion. In order for sucha series (5.9) to be well-defined as a genuine function on Λ ˆ , Z must be lie in 1 + Λ + .Notice that Z = P Λ ˆ is precisely the point corresponding to L . We conclude that theMaurer-Cartan space of L sits in Λ ˆ as B (
1, 1 ) , the open ball centered at 1 with radius 1(which is merely a different way of describing the same set 1 + Λ + ).One can further deform L by equipping it with a fixed flat connection ∇ α with holo-nomy α P Λ U . Repeating the same computation, we see that the object ( L , ∇ α ) sits at apoint Z = α in Λ ˆ , and the corresponding Maurer-Cartan space is the open ball B ( α , 1 ) centered at this point with radius 1.We next consider a general torus fiber L r : = t ( t , s ) P S ˆ R : s = r u . Since L r is nolonger exact, we cannot reduce the computation to C -coefficient case. Let P r and P r bethe unique intersection points in G X L r and ϕ H ( G ) X L r , respectively. They also denote AURER-CARTAN DEFORMATION OF LAGRANGIANS 34 the corresponding geometric generators of the Floer complexes. Observe from the picturethat the difference between P r and P r cannot be neglected this time.Let us first compute the augmentation ε L r of hom W Λ ( G , G ) associated with L . Elemen-tary counting shows that m ( Z , P r ) = T r P r .This together with the unital property of Z implies that P r and T r P r should be identifiedfor hom W Λ ( G , G ) . Then we have m ( Z , P r ) = T ( ´ r ) P r = T ´ r P r ,and hence ε L r ( Z ) = T ´ r , or ε L r ( Z ) = T ´ r In terms of the exact generator Z = Ă Z .Therefore the maximal ideal ker ε r indicates the point Z = T ´ r in Λ ˆ .The computation of κ is also similar, except that one has to measure the areas of con-tributing polygons explicitly. For instance, ÿ k ě m k + ( Z , P r , b , ¨ ¨ ¨ , b ) = ÿ k = k ! x k m ( Z , P r ) = T ( ´ r ) e x P r = T ´ r e x P r where b = xdt on L r , which leads to κ ( Z ) = T ´ r e x . Thus κ ( Z ) = T ´ r e x , and thiscompletely determines κ since it is a ring homomorphism.Setting Z r : = T r Z , one can go back to the same situation as in L = L , except that Z r has a nontrivial valuation. To recover x , one can take the formal expansion ( Z r ´ ) ´ ( Z r ´ ) + ( Z r ´ ) ´ ¨ ¨ ¨ = log (( Z r ´ ) + ) .Notice that this is consistent with 5.3 which says that the Koszul map for L r can beobtained from that of L after scaling Z to Z r = T r Z . On the level of algebra, the Maurer-Cartan algebra of L r is isomorphic to the the completion of Λ [ Z r , Z ´ r ] at the ideal x Z r ´ y as in (5.5).Let us describe the corresponding region in Λ + . In the original coordinate Z , one has T r ( Z ´ T ´ r ) ´ T r ( Z ´ T ´ r ) + T r ( Z ´ T ´ r ) ´ ¨ ¨ ¨ = log (cid:0) T r ( Z ´ T ´ r ) + (cid:1) ,which is valid as a function Z for Z P T ´ r + T ´ r Λ + . Therefore we conclude that theMaurer-Cartan deformation space of L r sits in the global mirror Λ + as the open ball B ( T ´ r , e r ) .We remark that the balls obtained by varying r are mutually disjoint. This is consistentwith the fact that torus fibers do not intersect each other, and hence Floer theory betweentwo different fiber is trivial.5.5. Λ U -flact connections on L . As one can see in the previous example, in Maurer-Cartan deformation, Λ U -connections serve identically as degree one cocycles in L pairedwith C ˚ -flat connections after exponentiating the variable, as long as the divisor-typeaxiom holds. Therefore, in practice, one may take ( Λ U ) b parametrizing Λ U -flat connec-tions on L for related computations, instead of the coordinate functions on H ( L ) . Here, b denotes the first Betti number of L )When ⃗ z varies over ( Λ U ) b , the associated Koszul map is defined by P κ l ( Z , ¨ ¨ ¨ , Z l ) = m ∇ ⃗ z l + ( Z , ¨ ¨ ¨ , Z l , P ) , (5.10) AURER-CARTAN DEFORMATION OF LAGRANGIANS 35 and it is essentially determined by the augmentation ϵ L (once we know the boundaryclasses of contributing disks). By our assumption on the degrees, (5.10) is nontrivial onlywhen l =
1, so it reduces to an algebra homomorphism H hom W Λ ( G , G ) Ñ Λ t z ˘ , ¨ ¨ ¨ , z ˘ b u (5.11)where the right hand side consists of infinite Laurent series ř i = a i z v i with lim i Ñ8 val ( a i ) = . (See Example 2.5 and the paragraph above it for the related discussions.)In the applications in Section 6, we will proceed mainly with connections rather thandealing with divisors on L . Readers are warned that one should stick to the original vari-able x i (dual to X i P H ( L ) ) to apply Theorem 5.2, or it is possible that H hom W Λ ( G , G ) can be completed in a different way for (5.11) to become an isomorphism.6. E xamples : local models for SYZ fibrations
We examine the main idea of Theorem 5.2 using simple, but illustrative geometricexamples, which are typical local models for SYZ fibrations. When there is no singularfiber, then the fibration is locally the same as ( C ˆ ) n = T ˚ T n , which is nothing but theproduct of Example 5.4 that we have already analyzed in detail. Like before, the Maurer-Cartan space of each torus fiber in ( C ˆ ) n forms a small chart in ( Λ ˆ ) n . The latter space isSpec of the endomorphism algebra of the Lagrangian G – R n in W Fuk Λ (( C ˆ ) n ) . Thesecharts are indeed disjoint small balls in ( Λ ˆ ) n . More precisely, the Maurer-Cartan spaceof the Lagrangian fiber L ⃗ r : = t ( z , ¨ ¨ ¨ , z n ) : log | z i | = r i u equipped with the holonomy ∇ ⃗ c = ( c , ¨ ¨ ¨ , c n ) P ( Λ U ) n sits in the global space ( Λ ˆ ) n asa polydisk B ( T ´ r c , e r ) ˆ ¨ ¨ ¨ ˆ B ( T ´ r n c n , e r n ) about the point ( T ´ r c , ¨ ¨ ¨ , T ´ r n c n ) .Before we proceed to our main examples, let us make a short remark on the cotangentbundle of a simply connected manifold and related works. Cotangent bundles.
The wrapped Floer cohomology of a cotangent fiber in T ˚ L has beenshown to be quasi-isomorphic to C ´˚ ( Ω L ) by Abouzaid [Abo12] where Ω L is thebased loop space. On the other hand, recall that the Maurer-Cartan dga A L of L canbe expressed as Ω H ´˚ ( L ; Λ ) , where we identify the continuous dual of H ˚ ( L ; Λ ) with H ´˚ ( L ; Λ ) (which is possible since everything is finite dimensional).Suppose L is simply connected. Then the A -coalgebra structure on H ´˚ ( L ; Λ ) isfinite by degree reason, i.e, ∆ i ( c ) = i ’s. Thus one can avoidcompletion procedure in 3.4 for the cobar construction of H ´˚ ( L ; Λ ) , and the resultingdga is quasi-isomorphic to C ´˚ ( Ω L ) by Adams [Ada56]. Combined with the result ofAbouzaid, the Maurer-Cartan dga of L can be identified with a suitable completion of thewrapped Floer cohomology of a cotangent fiber. (This point has been already discussedimplicitly in [EL17a].) Based on the above observation, we speculate that Theorem 5.6generalizes to the nonzero degree components under some milder assumptions on thegrading of G and L .We now begin to explore more nontrivial examples of local SYZ fibrations that comeswith typical type of singularities. We will be interested in the Maurer-Cartan spaces ofsingular fibers as well . AURER-CARTAN DEFORMATION OF LAGRANGIANS 36
Local model for I -singular fiber. Let us consider the following exact symplecticmanifold M = C zt z z = ϵ u ,which is a toy model for the 2-dimensional SYZ fibration that has a unique nodal singularfiber. The fibration is given by ( z , z ) ÞÑ ( | z | ´ | z | , | z z ´ ϵ | ) (6.1)and its Floer theory and mirror symmetry have been studied and fully understood throughmany literatures for e.g., [Aur07]. The singular fiber denoted by L occurs when | z z ´ ϵ | = ϵ and | z | ´ | z | =
0, and L is an immersed sphere with a transversal double pointat z = z =
0. It is easy to visualize the Lagrangians in our interest using the conic fibra-tion w : M Ñ C ˚ , w ( z , z ) = z z ´ ϵ . The projection of torus fibers draws a concentriccircle about the origin, as does that of L in particular (see Figure 6).The Maurer-Cartan algebra A L is the ring of bounded power series ring on two vari-ables u and v , A L = Λ tt u , v uu / uv = vu ,associated with the immersed generator U and V supported at the nodal point. See fore.g., [Sei] or [HKL18] for more details. In fact, A L can be computed from the wrappedFloer theory of the dual Lagrangian and the Koszul map κ .The geometric generators U and V have nontrivial valuations depending on the size ofthe base circle in w -plane that L projects to. Since U and V are complementary to eachother supported at the same self-intersection, we have ν ( U ) + ν ( V ) =
0. As before, wedenote by r U and r V the associated exact generators, i.e., r U = T ´ ν ( U ) U and r V = T ´ ν ( V ) V .One technical subtlety here is that we actually need to identify L with a certain twistedcomplex ˜ L δ Ñ ˜ L built out of the thimble ˜ L emanating from ( z , z ) = (
0, 0 ) in orderto rigorously include L as an object of W C while the immersed Floer theory of L itselfperfectly makes sense thanks to [AJ10] (or [CRGG17, Section 4, 6] is closer to our setting).The Floer computation with L and ˜ L δ Ñ ˜ L should not be much different from each other,and we will proceed with L itself, here.Let us next consider G , a Lagrangian section of (6.1) that projects to the positive real axisin w -plane. Over each point in the positive real axis lies C ˚ – T ˚ S given by z z = const .and G is isomorphic to a cotangent fiber of T ˚ S along this direction. G generates thewrapped Fukaya category of M , and hence Theorem 5.2 applies to the situation in thisexample.The wrapped Floer cohomology of G can be computed by counting holomorphic sec-tions of the fibration w following the idea of [Pas14, Proposition 4.5]. We remark that[CU13] computed the wrapped Floer cohomology of analogous Lagrangians in moregeneral situation of A n . Our computation below uses a Hamiltonian similar to “ H ”appearing in [CPU16, Section 6]. If we denote a perturbation G by a quadratic Hamil-tonian by ϕ H ( G ) , the intersection points in G X ϕ H ( G ) can be parametrized by C a , b where ( a , b ) P Z indicates the location of the intersection point in w -plane and the conic fiber,respectively (see Figure 5). Each of them produces a degree-0 exact generator denoted as r C a , b , and in particular, hom W Λ ( G , G ) is simply an algebra without higher operations. Theproduct m is given by m ( r C a , b , r C a , b ) = k ÿ (cid:18) ki (cid:19) r C a + a , b + b + i (6.2) AURER-CARTAN DEFORMATION OF LAGRANGIANS 37 F igure
5. Generators of hom W Λ ( G , G ) where k is given by k = " min t| a | , | a |u if a and a have different signs0 otherwise.We see that hom W Λ ( G , G ) is a commutative algebta. It is elementary to deduce from (6.2)that two elements r C ˘ generate hom W Λ ( G , G ) as an algebra, or it will be more clear afterwe compare this with A L via the Koszul map κ .F igure
6. Contributing polygons to κ ( r C ˘ ) Let us now consider κ : hom W Λ ( G , G ) Ñ A L . Shaded regions in Figure 6 show holo-morphic sections of the conic fibration that contribute to κ ( r C ˘ ) . (In the figure, G is notfully wrapped, but only minimal number of times to show the generators C ˘ as inter-section points.) This implies κ ( r C ´ ) = r u and κ ( r C ) = r v . The outputs do not involve AURER-CARTAN DEFORMATION OF LAGRANGIANS 38 any Novikov numbers since we work with exact generators and variables. The fact that A L is commutative can be deduced from this without actually solving the Maurer-Cartanequation for L .Since κ is a ring homomorphism, we can compute κ ( r C a , b ) for other a , b P Z using (6.2).For example, one has κ ( r C ) = ˜ u ˜ v ´ m ( r C ´ , r C ) = r C + r C . The general formula is given by κ ( r C a , b ) = " ( r u r v ´ ) b r u | a | a ă ( r u r v ´ ) b r v | a | a ě U = ˜ C ´ and V = ˜ C , and it gives rise toa simpler and more familiar presentation hom W Λ ( G , G ) = Λ [ U , V , UV ´ ] (the sameexpression also appears in [Pas19]). Namely, it corresponds to the function ring of ˇ M : = Λ zt UV = u . Most importantly, U and V can be thought of as global coordinates onˇ M . The crucial reason behind it is the finiteness of the A -operations on wrapped Floertheory.To the contrary, recall that the Maurer-Cartan algebra of L is given as A L = Λ tt u , v uu uv = vu which allows an infinite sum of monomials in u , v as long as the valuation is boundedbelow. Thus A L is the function ring of ( Λ + ) . (The maximal subset of Λ on which everyelements in A L can be evaluated is ( Λ + ) .) In other words, the Maurer-Cartan space MC L is isomorphic to ( Λ + ) with coordinates u and v . This is along the same line as thecoefficients for the bounding cochains in [FOOO09].Nevertheless MC L is not formal unlike the impression one may get from the term“completion” in the statement of Theorem 5.2. Let us examine how the local chart MC ( L ) (dual to A L ) sits in the global mirror ˇ M = Λ zt UV = u . First of all, the chart is aroundthe origin ( U , V ) = (
0, 0 ) since their augmentation values are zero. Recall that the exactvariables ˜ u = T ν ( U ) u and ˜ v = T ν ( V ) v precisely match with the exact generators U and V in hom W Λ ( G , G ) under κ . The constraint that val ( u ) ą val ( v ) ą ( u , v ) P ( Λ + ) )translated into the condition val ( U ) ą ν ( U ) , val ( V ) ą ν ( V ) . We see that MC ( L ) embedsas the polydisk U L : = t ( U , V ) : || U || ă e ´ ν ( U ) , || V || ă e ´ ν ( V ) u into ˇ M = Λ zt UV = u .(Here, we are using the norm e ´ ν induced from the valuation on hom W Λ ( G , G ) .)One can vary L by choosing non-concentric circle about the origin, while still enclosingthe origin and passing through ´ ϵ . Such a deformation is certainly not a Hamiltonianisotopy, and hence produce non-isomorphic objects in the Fukaya category. We investigatethe change of MC ( L ) in this case. Let us assume ν ( U ) = ν ( V ) = L ,which can be achieved by adjusting ϵ suitably. U L is then given as the polydisk with eachfactor having radius 1.Take an immersed sphere L obtained by taking a circle smaller than | w | = ϵ , and write U and V for the corresponding immersed generator for L . Stokes formula implies that ν ( U ) ă ν ( V ) ą
0. Therefore MC ( L ) is embedded in Λ zt UV = u as a polydisk U L with radius of U -factor bigger than e ´ =
1, and that of V -factor less than 1. On theother hand, an immersed sphere L with sitting over a circle in w -plane with a biggerenclosed area satisfy ν ( U ) ą ν ( V ) ă
0. In Figure 7 (a), we depict three differentcharts U L , U L and U L by plotting the norm ( x , y ) = ( e ´ ν ( U ) , e ´ ν ( V ) ) . One corner of eachchart moves along the curve ab = AURER-CARTAN DEFORMATION OF LAGRANGIANS 39 F igure
7. Regions in the mirror space occupied by (a) immersed La-grangian spheres and (b) Lagrangian toriF igure
8. Four different types of tori in M distinguished by wall-crossingand valuations: (a) L stdChek , (b) L stdCli f , (c) L bigChek and (d) L smallCli f We next turn our attention to various Lagrangian tori in M . We will only considerthe tori sitting over circles in w -plane with height | z | ´ | z | =
0. These can be dividedinto four crucially different families of Lagrangian tori, which can be distinguished fromeach other by their locations relative to wall (or that of L ) and “sizes” of their projectionsrelative to that of L . The original fibers of the fibration (6.1) already has Chekanov andClifford tori, denoted by L stdChek and L stdCli f in different sides of the wall. In addition, onecan consider a Chekanov torus L bigChek which lies above a circle in w -plane bigger than | w | = ϵ , or a Clifford torus L smallCli f sitting over a circle smaller than | w | = ϵ . Notice that the AURER-CARTAN DEFORMATION OF LAGRANGIANS 40 latter two can be Hamiltonian isotopic to none of fibers of (6.1). Figure 8 shows typicalmembers of these four different families.Let us equip the tori with Λ U -connections parametrized by their corresponding holonomies ( z , z ) . Here, z is the holonomy along a circle in the conic fiber, and z is that of a circlein | w | -plane. (See [HKL18] for more details.) One can compute κ for these tori by directlycounting holomorphic sections with appropriate boundary conditions. Alternatively, wemay use the fact that κ is invariant under the quasi-isomorphism in the Fukaya category,since the construction of κ is purely categorical. In either way, one has, for L Chek , z = r C = UV ´ z = r C ´ = T a U ñ U = T ´ a z P T ´ a Λ U , V = T a ( z + ) P " T a Λ + z P ´ + Λ + T a Λ U otherwisefor some a such that a ă L stdChek and a ą L bigChek . Therefore, e ´ ( U ) = e a and0 ă e ´ ( V ) ď e ´ a , which draws vertical lines below the graph of xy = y -axis as the size of the w -projection of the Chekanovtorus becomes bigger. Likewise, plotting Clifford tori in ( x , y ) -planes give horizontal linesleft to xy = | z | ´ | z | ‰ xy = A pair-of-pants.
Let M be a pair-of-pants. M admits a torus fibration with the “Y”-shaped central fiber, whose Floer theory cannot be defined. Instead of considering theSYZ mirror, we look into the wrapped Floer theory of three noncompact Lagrangian G , G and G shown in Figure 9. Any two of them already generate the wrapped Fukayacategory. The endomorphism algebra of ‘ G i is built upon the path algebra of a quiverwith more than one vertex, and hence is noncommutative. It also has nonzero degreecomponents, so it is hard to extract the conventional mirror out of it.We take the following alternative approach. let us first explain the wrapped cohomol-ogy for each of G i ’s. Wrapping G near two punctures it asymptotes, we obtain a similarpicture as what we have seen for T ˚ S , except that the third puncture appears in the mid-dle of the cylinder. If we denote the intersection points between G and its perturbation ϕ H ( G ) by U , U , U , ¨ ¨ ¨ and V , V , V , ¨ ¨ ¨ (with U = V being the unit) as indicated inFigure 9, then their corresponding exact generators have degree 0, and satisfy m ( ˜ U i , ˜ U j ) = ˜ U i + j , m ( ˜ V i , ˜ V j ) = ˜ V i + j , m ( ˜ U i , ˜ V j ) = i , j ě +
13] for more details). Setting U : = ˜ U and V : = ˜ V , we see that thewrapped Floer cohomology of G can be identified with W F ( G , G ; k ) = k [ U , V ] x UV y (6.4)where k could be C or Λ depending on which wrapped Fukaya category we are dealingwith. Thus the corresponding space is the union of two coordinate axes in k , or t ( U , V ) P k | UV = u .We then consider the circle L in Figure 9 and its deformation by xd θ . Let us assume forsimplicity that L is exact so as not to have any Novikov numbers in the formula below. Bydirect counting, the kernel of the resulting augmentation on (6.4) is the ideal generatedby U ´ V (or the point corresponding to L is ( U , V ) = (
1, 0 ) ). In particular, V doesnot survive in the completion of 6.4 with respect to this ideal, since V = ( ´ ) n ( U ´ ) n V for any n . AURER-CARTAN DEFORMATION OF LAGRANGIANS 41 F igure
9. Noncompact Lagrangians G and G and their endomorphismalgebras interacting with the Maurer-Cartan algebra of the circle L Through a similar calculation to 5.4, one can also show that the Maurer-Cartan spaceof L lies in (6.4) via κ G , L ( U ) = e x and κ G , L ( V ) = V lies in the kernel of κ , it becomes zero in the completion. Thus it does notviolate Theorem 5.2. (Note that G generates L .) Namely, A L = Λ t x u is isomorphic tothe completion of (6.4) at the ideal x U ´ V y (additionally requiring usual boundednesscondition in Λ -coefficient case). MC ( L ) – Λ + sits in the U -axis ( Ă t ( U , V ) P k | UV = u ) as the unit open disk around ( U , V ) = (
1, 0 ) .On the other hand, exactly the same procedure applying to the pair ( G , L ) identifiesthe Maurer-Cartan space of L as a subset of the coordinate axis W = κ G , L ( U ) = e x and κ G , L ( W ) = k [ U , W ] / x U W y of the wrapped Floer cohomology of G as before. (Here, U is the exact generator occurring near the same puncture as U for G .)Repeating the same computation for other circles in the same leg of M , we conclude thatin view of Lagrangian moduli, the two coordinate axes t ( U , 0 ) | U P k ˚ u (from G ) and t ( U , 0 ) | U P k ˚ u (from G ) should be identified by letting ( U , 0 ) = ( U , 0 ) , as indicatedin Figure 9.Observe that the origin U = V = L , since theexponential e x is always nonzero. We next seek for the object in the Fukaya categorythat should correspond to the origin. Let us consider the immersed circle L with threeself-intersection points X , Y and Z described in Figure 10. This immersed Lagrangian hasbeen used to study homological mirror symmetry of surfaces in many literatures, sincefirst introduced by Seidel [Sei11].Let X , Y , Z and X , Y , Z denote degree 0 and 1 generators, respectively. Observe thatfor ω = d Θ , ż L Θ = ż B ∆ F Θ ´ ż B ∆ B Θ = ż ∆ F ω ´ ż ∆ B ω (6.5)where ∆ F and ∆ B are two triangles bounded by L with corners X , Y and Z and theirboundaries are positively oriented. By imposing the reflection symmetry on L so that(6.5) vanishes, L can be made exact, or ι ˚ Θ = d f for some function f on the domain AURER-CARTAN DEFORMATION OF LAGRANGIANS 42 F igure
10. The Maurer-Cartan algebra of the Seidel Lagrangian and theendomorphism algebra of G S of the immersion ι : S í L . Thus each immersed generator naturally comes withits valuation ν , and one can assign exact generators to X , Y , Z , X , Y , Z by multiplying T ´ ν ( ´ ) . The Maurer-Cartan equation for L in terms of these exact generators is given as m ( e b ) = ( ˜ y ˜ z ´ ˜ z ˜ y ) r X + ( ˜ z ˜ x ´ ˜ x ˜ z ) r Y + ( ˜ x ˜ y ´ ˜ y ˜ x ) r Z + ˜ x ˜ y ˜ z L (6.6)where b = ˜ x r X + ˜ y r Y + ˜ z r Z (= xX + yY + zZ ) . (For instance, r X = T ´ ν ( X ) X and ˜ x = T ν ( X ) x .)Notice that there is no Novikov numbers due to the usage of exact generators. Remark 6.1.
Strictly speaking, L does not completely fit into our setting in Section 5 sinceit is only Z -graded, and a lower (de Rham) degree term such as ˜ x ˜ y ˜ z L possibly appears in theMaurer-Cartan equation (6.6) . To keep the Z -graded setting, one could introduce an extra variablee with deg e = encoding Maslov indices, which turns the last term in (6.6) to ˜ x ˜ y ˜ ze L .On the other hand, the coefficient W : = ˜ x ˜ y ˜ z = T ν ( X )+ ν ( Y )+ ν ( Z ) xyz is called the potentialfunction defined on the weak Maurer-Cartan space of L , which give a local Landau-Ginzburgmirror. To deal with Lagrangians more systematically, one may need a generalization of κ intoweak Maurer-Cartan deformations, which we leave for future research. Therefore the relations are generated by commutators among variables together with xyz , and hence the (strict) Maurer-Cartan algebra in Λ -coefficient can be represented as A L = Λ tt x , y , z uuxx [ x , y ] , [ y , z ] , [ z , x ] , xyz yy .The associated space MC ( L ) is given by t xyz = u Ă ( Λ + ) . One can also use C -coefficient, in which case the corresponding Maurer-Cartan deformation of L gives aformal neighborhood the origin in the union of three coordinate planes in C .Another feature of L slightly off the setting in 5 is that G i and L intersect at two points,say P of degree 0 and Q of degree 1. Despite the existence of an additional intersectionpoint Q , the map κ G i , L defined by the precisely same formula (5.6) still produces analgebra homomorphism modulo the ideal x r PQ ( x , y , z ) , r QP ( x , y , z ) y where r PQ and r QP are given by m b ( Q ) = r QP ( x , y , z ) P nad m b ( P ) = r PQ ( x , y , z ) Q . To see this, observe that AURER-CARTAN DEFORMATION OF LAGRANGIANS 43 the A -relation m ( multiple of Q hkkkkkkkkikkkkkkkkj m ( Z , Z , P , e b ) , e b ) + m ( m ( Z , Z ) , P , e b ) ´ m ( Z , m ( Z , P , e b ) , e b ) + m ( Z , Z , m b ( P ) , e b ) = Z , Z P hom W Λ ( G i , G i ) reduces to κ ( m ( Z , Z )) = κ ( Z ) κ ( Z ) mod x r PQ , r QP y since the first and the last terms are multiple of r QP and r PQ , respectively. It is easy to seethat r PQ ( x , y , z ) = z and r PQ ( x , y , z ) = xy up to scaling in this case. Remark 6.2.
The pair ( r PQ , r QP ) gives the matrix factorization of W mirror dual to G in viewof the Landau-Ginzburg mirror of M mentioned in Remark 6.1. See [CHL18] for more details. As a result, one obtains an algebra homomorphism κ G , L : Λ [ U , V ] x UV y Ñ A L xx r PQ , r QP yy – Λ tt x , y uuxx xy yy . (6.7)The homomorphism is determined by images of U and V , and the shaded holomorphicdisks drawn in Figure 10 shows κ G , L ( U ) = ˜ x = T ν ( X ) x , κ G , L ( V ) = ˜ y = T ν ( Y ) y .In particular, κ G , L becomes an isomorphism after taking completion of the left handside. Setting x = y =
0, (6.7) reduces to the augmentation U , V ÞÑ
0, from which one candeduce that L (without boundary deformation) sits at the origin ( U , V ) = (
0, 0 ) .By the same argument, we see also that L corresponds to the origin ( U , W ) = (
0, 0 ) in view of G . Accordingly, one has to identify the two “origins”. Combined with theprevious identification ( U , 0 ) = ( U , 0 ) for U , U ‰
0, we conclude that U -axis in t UV = u from G and U -axis in t U W = u from G should be glued together.The upshot is the union of three coordinate axes t ( U , 0, 0 ) u Y t ( V , 0 ) u Y t (
0, 0, W ) u (6.8)in Λ , that is obtained by keeping track of locations of point-like objects (in the Fukaya cat-egory) in the “Spec” of W F ( G , G ) and W F ( G , G ) . (We could also include W F ( G , G ) ,but it is redundant here.)Putting together (6.7) and its analogues for G , G , we see that the subspace t ( x , y , z ) P Λ + | xy = yz = zx = u of the Maurer-Cartan space MC ( L ) embeds into (6.8) via ( U , V , W ) = ( T ν ( X ) x , 0, 0 ) or ( T ν ( Y ) y , 0 ) or (
0, 0, T ν ( Z ) z ) ,which describes a certain neighborhood V L of the origin in (6.8). Note that V L is thecritical loci of the potential W : U L Ñ Λ ( x , y , z ) ÞÑ T ν ( X )+ ν ( Y )+ ν ( Z ) xyz ,and hence, it can be thought of as the set of nonzero objects ( L , b ) for b P MC ( L ) . AURER-CARTAN DEFORMATION OF LAGRANGIANS 44
Smoothing of conifold.
The last example is the (divisor complement of) deformedconifold discussed in Example 2.6 M : = t ( u , v , u , v , z ) P C ˆ C ˚ | u v = z ´ a , u v = z ´ b uzt z = u .The projection to z -plane defines a double conic fibration on M , whose fibers degenerateover z = a and z = b . Near a or b , M is locally isomorphic to the product of the examplein 6.1 with T ˚ S , and as before, it admits a torus fibration with fibers lying over concentriccircles about the origin in z -plane.Take two Lagrangians L and L to be matching spheres lying over paths drawn inFigure 11. As in the picture, G i for i =
0, 1 is a Lagrangian section of the torus fibrationwhich intersects L i exactly once. More details on Floer theory of these Lagrangians canbe found in [CPU16]. We finally set G = G ‘ G and L = L ‘ L .F igure
11. two Lagrangian spheres L , L and their dualsThe computation of κ for G and L can be done in a similar manner to what we didin 6.1, and we do not present much details. Following [CPU16], the intersection pointsbetween G and ϕ H ( G ) can be parametrized by t P ia , b , c : a , b , c P Z u for i =
0, 1 spanning theendomorphism of G i , t Q a , b , c : a P + Z , b , c P Z u spanning hom ( G , G ) , and t R a , b , c : a P + Z , b , c P Z u spanning hom ( G , G ) . Analogous to the way of indexing in 6.1, a and ( b , c ) in P ia , b , c indicate the wrapping along the base and the double-conic fiber directions,respectively. Q a , b , c and R a , b , c are similarly defined, except that they lie between the twodifferent Lagrangians. One can easily guess from Figure 12 how they are arranged. Wedenote their associated exact generators by r P ia , b , c (for i =
0, 1), r Q a , b , c and r R a , b , c . r P i is theunit in the corresponding component of hom ( G , G ) .In [CPU16], m between these generators has been explicitly computed. For instance, m ( r P a , b , c , r P a , b , c ) = k ÿ i = k ÿ j = (cid:18) k i (cid:19)(cid:18) k j (cid:19) r P a + b , a + b + i , a + b + j .where k = min t| a | , | a |u if a and a have different signs, and k = m ( r Q a , b , c , r R a , b , c ) = k ÿ i = k ÿ j = (cid:18) k i (cid:19)(cid:18) k j (cid:19) r P a + b , a + b + i , a + b + j .where k = min t| a | ´ | a | ´ u + k = t| a | ´ | a | ´ u if a ă ă a ,and k = min t| a | ´ | a | ´ u and k = t| a | ´ | a | ´ u + a ă ă a ( k = k = a and a have the same sign). Formulas for other products have similarpatterns, and we omit. AURER-CARTAN DEFORMATION OF LAGRANGIANS 45
Recall from Example 2.6 that the Maurer-Cartan algebra A L is a quiver algebra gen-erated by four arrows x , y , z , w , with relations xyz = zyx and its permutations. Thesearrows are dual to immersed generators X , Y , Z , W taken as in Figure 12. ( Y and W arecomplementary to X and Z , respectively.)F igure
12. The Koszul map κ for generators of hom W ( G , G ) We can compute κ : hom ( G , G ) Ñ A L using the same strategy as in 6.1. Namely, wefirst find the images of generators of hom ( G , G ) with low indices by directly countingholomorphic disks (for e.g., the contributing disks to κ ( Q ˘ ,0,0 ) presented in Figure 12),and then apply the known formula of m above. The general formula for κ is given asfollows:For 0 ă a P Z , κ ( r P a , b , c ) = ( ˜ w ˜ x ) a ( ˜ y ˜ x ´ ) b ( ˜ w ˜ z ´ ) c , κ ( r P ´ a , b , c ) = ( ˜ y ˜ z ) a ( ˜ yx ´ ) b ( ˜ w ˜ z ´ ) c , κ ( r P a , b , c ) = ( ˜ x ˜ w ) a ( ˜ x ˜ y ´ ) b ( ˜ z ˜ w ´ ) c , κ ( r P ´ a , b , c ) = ( ˜ z ˜ y ) a ( ˜ x ˜ y ´ ) b ( ˜ z ˜ w ´ ) c .For 0 ă a P + Z , κ ( r Q a , b , c ) = ˜ x ( ˜ w ˜ x ) a ´ ( ˜ y ˜ x ´ ) b ( ˜ w ˜ z ´ ) c , κ ( r Q ´ a , b , c ) = ˜ z ( ˜ y ˜ z ) ´ a + ( ˜ y ˜ x ´ ) b ( ˜ w ˜ z ´ ) c , κ ( r R a , b , c ) = ˜ w ( ˜ x ˜ w ) a ´ ( ˜ x ˜ y ´ ) b ( ˜ z ˜ w ´ ) c , κ ( r R ´ a , b , c ) = ˜ y ( ˜ z ˜ y ) ´ a + ( ˜ x ˜ y ´ ) b ( ˜ z ˜ w ´ ) c .The proof is the direct count of some simple disks from the picture followed by a tedious,but elementary algebraic computation, and we leave it as an exercise for readers.Negative powers of a polynomial in the formula should be interpreted as series, fore.g., ( ˜ x ˜ y ´ ) ´ = ´ ´ ˜ x ˜ y ´ ˜ x ˜ y ˜ x ˜ y ´ ¨ ¨ ¨ . Note that such infinite sums are legitimate in A L .Also, the loops based at the same vertex commute in A L due to the relations, and hencethe order of factors can be switched with some minor effect. For instance,˜ w ( ˜ x ˜ w ) a ´ ( ˜ x ˜ y ´ ) b ( ˜ z ˜ w ´ ) c = ( ˜ w ˜ x ) a ´ ˜ w ( ˜ z ˜ w ´ ) c ( ˜ x ˜ y ´ ) b holds in A L , and the orders of factors in the above formulas are arbitrarily chosen forconvenience. AURER-CARTAN DEFORMATION OF LAGRANGIANS 46 A ppendix A. D ouble K oszul dual of an algebra Let W be a k Λ -algebra given as W = W C b Λ for a k C -module W C . Suppose an aug-mentation ϵ : W C Ñ k C is given, and write ¯ W C : = ker ϵ . We assume that ¯ W C / ¯ W C isfinitely generated (or finite dimensional as a C -vector space). Intuitively, ¯ W C / ¯ W C is the(Zariski) tangent space of “Spec” W C at the point corresponding to the maximal ideal ¯ W C .We then consider the intermediate k Λ -module W Λ : = W C b Λ , and set ¯ W Λ to be thekernel of the linearly extended augmentation ε b Λ . It induces a filtration W Λ = W C b Λ Ą ¯ W Λ Ą ¯ W Λ ¨ ¯ W Λ Ą ¯ W Λ ¨ ¯ W Λ ¨ ¯ W Λ Ą ¨ ¨ ¨ ,and correspondingly, the completion ˆ W Λ : = lim ÐÝ i W Λ ( ¯ W Λ ) i = k Λ ‘ lim ÐÝ i ¯ W Λ ( ¯ W Λ ) i of W Λ .Thus an element f of the completion carries the data f = ( f , f , f , ¨ ¨ ¨ ) such that f i P W Λ / ( ¯ W Λ ) i + and f i = f i + in W Λ / ( ¯ W Λ ) i + (in particular f P k Λ ). Our goal is toshow that the 0-th cohomology of the complex à d hom C (cid:32) à i + ¨¨¨ + i k = d + k (cid:16) ¯ W b C i C (cid:17) b ¨ ¨ ¨ b (cid:16) ¯ W b C i k C (cid:17) , Λ (cid:33) (A.1)(that equals (5.7)) computes the completion ˆ W Λ of W Λ which will complete the proof ofTheorem 5.2.We first introduce the following auxiliary complex, whose completion computes ˆ W Λ at the 0-th cohomology: A = k Λ ‘ ¯ W Λ ‘ ( ¯ W Λ b ¯ W Λ ) ‘ ¨ ¨ ¨ A ´ = s ¯ W b Λ ‘ (cid:16) ( ¯ W Λ b s ¯ W b Λ ) À ( s ¯ W b Λ b ¯ W Λ ) (cid:17) ‘ ¨ ¨ ¨ ... A ´ i = s i ¯ W b ( i + ) Λ ‘ (cid:16) À i + i = i + s i ´ ¯ W b i Λ b s i ´ ¯ W b i Λ (cid:17) ‘ ¨ ¨ ¨ where all the tensor products are taken over Λ . Its length-completion (by the number of“ b ”s) can be regarded as the cobar construction of BW Λ adapted to Λ -coefficient,ˆ A ‚ = Ω BW Λ = ¯ T Λ s ´ (cid:16) T Λ s ¯ W Λ (cid:17) = ź k (cid:16) s ´ T Λ s ¯ W Λ (cid:17) b k = ź k (cid:32) s ´ (cid:32) à l s ¯ W b l Λ (cid:33)(cid:33) b k .It is consistent with the cobar construction in 3.4, in the sense that the valuation on Λ isbounded. In particular, A ‚ is equipped with a natural differential d (as well as product ´ b ´ ) given as follows. If we write A ‚ as the double complex k Λ ‘ ¯ W C s ¯ W b C (cid:79) (cid:79) (cid:47) (cid:47) ¯ W C b ¯ W C s ¯ W b C (cid:79) (cid:79) (cid:47) (cid:47) s (cid:16) ( ¯ W b C b ¯ W C ) ‘ ( ¯ W C b ¯ W b C ) (cid:17) (cid:79) (cid:79) (cid:47) (cid:47) ¯ W C b ¯ W C b ¯ W C (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) (cid:79) AURER-CARTAN DEFORMATION OF LAGRANGIANS 47 then d is a combination of the vertical map a b b Ñ a ¨ b and horizontal map a b b Ñ a b b (for a , b P ¯ W Λ ) with appropriate Koszul signs. Lemma A.1. H ( ˆ A ‚ ) is isomorphic to ˆ W Λ .Proof. Let A ‚ = F A ‚ Ą F A ‚ Ą ¨ ¨ ¨ denote the length-filtration. Observe first that H ( A ‚ ) – W Λ since up to im d , any higher tensors a b ¨ ¨ ¨ b a i is equivalent to a ¨ ¨ ¨ a i .It can be also deduced by the bar-cobar adjunction, but “cobar” in this context is the onebefore taking length-completion like here. A similar argument proves that H ( F A ‚ ) = ¯ W Λ b ¯ W Λ / x a ¨ b b c ´ a b b ¨ c y , and the image of H ( F A ‚ ) Ñ H ( A ‚ ) – W Λ is pre-cisely ¯ W Λ in view of the identification H ( A ‚ ) – W Λ , and that of H ( F j A ‚ ) Ñ H ( A ‚ ) is ¯ W j + Λ . Therefore the inverse limit of the system t H ( A ‚ ) / H ( F j A ‚ ) u j = ¨¨¨ computesthe completion ˆ W Λ .On the other hand, ˆ A is the inverse limit of t ( B j ) ‚ : = A ‚ / F j A ‚ , u j = ¨¨¨ of cochaincomplexes, where the differential d j on ( B j ) ‚ is naturally induced by d on A ‚ . For each j ,we have an exact sequence,0 Ñ im d j Ñ ker d j Ñ H ˚ ( A ‚ / F j A ‚ ) Ñ Ñ lim ÐÝ j im d j Ñ lim ÐÝ j ker d j Ñ lim ÐÝ j H ˚ ( A ‚ / F j A ‚ ) Ñ d j satisfies the Mittag-Leffler condition (transition maps for t ( B j ) ‚ u and hencefor t im d j u are all surjective). Therefore it only remains to prove that H ( A ‚ / F j A ‚ ) isisomorphic to H ( A ‚ ) / H ( F j A ‚ ) , which can be verified using the exact sequence0 Ñ F j A ‚ Ñ A ‚ Ñ A ‚ / F j A ‚ since A ‚ is supported at nonpositive degrees only. □ The following lemma combined with Lemma (A.1) proves that the 0-th cohomology of(A.1) computes ˆ W Λ . Lemma A.2.
Let Φ be the dga map Φ : ˆ A ‚ Ñ à d hom C (cid:32) à i + ¨¨¨ + i k = d + k (cid:16) ¯ W b C i C (cid:17) b ¨ ¨ ¨ b (cid:16) ¯ W b C i k C (cid:17) , Λ (cid:33) (A.2) defined by extending the evaluationW Λ = W C b Λ Ñ hom C ( ¯ W C , Λ ) a b c ÞÑ ( ev a b c : f Ñ f ( a ) c ) using b on both sides, and the natural map ¯ W C b ¨ ¨ ¨ b ¯ W C Ñ ( ¯ W C b ¨ ¨ ¨ b ¯ W C ) definedanalogously to (3.16) . Then Φ induces an isomorphism on the -th cohomology. It is elementary to check that Φ is a well-defined chain map. AURER-CARTAN DEFORMATION OF LAGRANGIANS 48
Proof.
The right hand side of (A.2) is obtained by applying (degree-wise) hom C ( ´ , Λ ) tothe double complex k C ‘ ¯ W C d (cid:15) (cid:15) s ´ ( ¯ W b C ) d (cid:15) (cid:15) ¯ W C b ¯ W C d (cid:15) (cid:15) d (cid:111) (cid:111) s ´ ( ¯ W b C ) (cid:15) (cid:15) s ´ ( ¯ W b C ) b ¯ W C À ¯ W C b s ´ ( ¯ W b C ) d (cid:111) (cid:111) (cid:15) (cid:15) ¯ W C b ¯ W C b ¯ W C d (cid:111) (cid:111) (cid:15) (cid:15) where d is dual to the multiplication on ¯ W C , and d is induced by the natural map ¯ W C b ¯ W C Ñ ( ¯ W C b ¯ W C ) , but b on the right hand side is replaced by b . Accordingly,the differential on the right hand side of (A.2) can be decomposed into the vertical andhorizontal ones denoted by δ and δ , respectively. Its degree-0 component is the set of k C -linear maps from k C ‘ ¯ W C ‘ ( ¯ W C b ¯ W C ) ‘ ( ¯ W C b ¯ W C b ¯ W C ) ‘ ¨ ¨ ¨ Ñ k Λ .By restricting such a map ϕ to first finitely many components ϕ i : = ϕ | : k C ‘ ¯ W C ‘ ¨ ¨ ¨ ‘ i hkkkkkkkkkkkikkkkkkkkkkkj (cid:16) ¯ W C b ¨ ¨ ¨ b ¯ W C (cid:17) Ñ k Λ ,we obtain a sequence ( ϕ , ϕ , ϕ , ¨ ¨ ¨ ) of k C -linear maps. (Conversely, ϕ can be understoodas the limit of this sequence since for each i , ϕ i + restricts to ϕ i on the common domain.)We will use this identification frequently, below. For simplicity, we assume ϕ = k Λ Ă ˆ A ‚ is precisely matched with the component where ϕ lives.Let Φ denote the map induced from Φ on the 0-th cohomology. We first prove that Φ issurjective. Let ϕ = ( ϕ , ϕ , ¨ ¨ ¨ ) P hom C ( B ( BW C ) , k Λ ) (5.7) of degree zero be given. Con-sider the subspace ¯ W C = t f : ¯ W C Ñ k C : f | ¯ W C ” u Ă ¯ W C . It is naturally the algebraicdual of ¯ W C / ¯ W C , and hence, is finite dimensional. Take a , ¨ ¨ ¨ a l such that [ a ] , ¨ ¨ ¨ , [ a l ] generate ¯ W C / ¯ W C as basis. Then there exists v = k C x a , ¨ ¨ ¨ , a l y b Λ whose associatedevaluation map equals the restriction ϕ | : ¯ W C Ñ k Λ .Consider an element h = dg in the image of d : ¯ W C Ñ ( ¯ W C b ¯ W C ) , and set ψ ( h ) = ϕ ( g ) ´ g ( v ) P k Λ This gives a well-defined map ψ : im d Ñ k Λ , since we have g ´ g P ¯ W C for h = dg = dg which implies ϕ ( g ´ g ) = g ( v ) ´ g ( v ) . Let us take a linear extension of ψ to ( ¯ W C b ¯ W C ) Ñ k Λ , and denote it by ψ . It gives a map ˜ ψ : = δ ψ : ¯ W C b ¯ W C Ñ k Λ bypre-composing ¯ W C b ¯ W C Ñ ( ¯ W C b ¯ W C ) .We next consider ϕ ´ ˜ ψ : ¯ W C b ¯ W C Ñ k Λ , AURER-CARTAN DEFORMATION OF LAGRANGIANS 49 and its restriction to the subspace (cid:0) ¯ W C b ¯ W C (cid:1) defined as (cid:16) ¯ W C b ¯ W C (cid:17) : = ! f P ¯ W C b ¯ W C : ˜ f | ( ¯ W C b ¯ W ) ‘ ( ¯ W C b ¯ W C ) ” ) where ˜ f denotes the image of f under ¯ W C b ¯ W C Ñ ( ¯ W C b ¯ W C ) . As before, ˜ f can beequivalently viewed as a map from ( ¯ W C / ¯ W C ) b to k Λ , and we obtain a map (cid:16) ¯ W C b ¯ W C (cid:17) Ñ (cid:0) ¯ W C / ¯ W C b ¯ W C / ¯ W C (cid:1) (A.3)which is obviously surjective. It is elementary to check that the middle map ¯ W C b ¯ W C Ñ ( ¯ W C b ¯ W C ) is injective (for e.g., one can argue by choosing a C -basis of ¯ W C ), and hence(A.3) is an isomorphism. Thus we can find v : = ř i v i b v i P k Λ x a , ¨ ¨ ¨ , a l y b suchthat ϕ ´ ˜ ψ | ( ¯ W C b ¯ W C ) agrees with the evaluation at v . The finite dimensionality of theright hand side of (A.3) is crucial, here.For an element h = ř i ( dg i b g i ) ‘ ř i ( g i b dg i ) in (cid:16) ¯ W b C b ¯ W C (cid:17) ‘ (cid:16) ¯ W C b ¯ W b C (cid:17) ,we define ψ ( h ) : = ( ϕ ´ ˜ ψ ) (cid:32) ÿ i g i b g i (cid:33) ´ ÿ g i ( v ) ¨ g i ( v ) (A.4)To see this is well-defined, assume h can be written as h = ř i ( d ˜ g i b ˜ g i ) ‘ ř i ( ˜ g i b d ˜ g i ) for other ˜ g ij ’s, and observe (cid:32) ÿ i g i b g i ´ ÿ i ˜ g i b ˜ g i (cid:33) ( a ¨ b ) b c = (cid:32) ÿ i dg i b g i ´ ÿ i d ˜ g i b ˜ g i (cid:33) ( a b b ) b c = (cid:32) ÿ i g i b g i ´ ÿ i ˜ g i b ˜ g i (cid:33) a b ( b ¨ c ) = (cid:32) ÿ i g i b dg i ´ ÿ i ˜ g i b d ˜ g i (cid:33) a b ( b b c ) = ř i g i b g i ´ ř i ˜ g i b ˜ g i P (cid:0) ¯ W C b ¯ W C (cid:1) , on which the right hand side of(A.4) vanishes.Finally, we extend ψ to the whole (cid:16) ¯ W b C b ¯ W C (cid:17) ‘ (cid:16) ¯ W C b ¯ W b C (cid:17) , and apply δ tothe extended ψ to get ˜ ψ . The repeat the above procedure to ϕ ´ ˜ ψ . Inductively, onecan construct a sequence ψ , ψ , ψ , ¨ ¨ ¨ such that δ ψ i + = ϕ i + ´ ev v i + δ ψ i . δ ψ = ϕ ´ ev v δ ψ = ϕ ´ ev v ´ δ ψ δ ψ = ϕ ´ ev v ´ δ ψ ...or equivalently, δψ = ϕ ´ ev v for ψ = ( ψ , ψ , ¨ ¨ ¨ ) and v = v b ř v i b v i b ¨ ¨ ¨ .To show Φ is injective, suppose that Φ ( v ) for v P ˆ A is the image of a degree- ( ´ ) element under the differential. This implies that Φ ( v ) vanishes for any d -closed elementin À i (cid:0) ¯ W C (cid:1) b i . Let us write v : = ( v , v = ř i v i b v i , ¨ ¨ ¨ ) such that v i P ¯ W b i Λ . Anelement f P ¯ W C is d -closed if f | ¯ W C ”
0, and hence we have Φ ( v )( f ) = f ( v ) = f . Strictly speaking, f ( v ) is the evaluation of the extension f : ¯ W C b Λ Ñ k Λ at v .Using finite dimensionality of ¯ W C / ¯ W C , we conclude that [ v ] = (cid:0) ¯ W C / ¯ W C (cid:1) b C Λ – AURER-CARTAN DEFORMATION OF LAGRANGIANS 50 ¯ W Λ / ¯ W Λ . Thus we can express v as v = ř i x i x i for some x i , x i P ¯ W Λ . Notice that α : = ř x i b x i maps to v under the differential modulo ¯ W b ě C .By the same reason, we have f ( v ) = f = f ‘ f P ¯ W C ‘ ( ¯ W C b ¯ W C ) suchthat d f ( w b w ) = f ( w ¨ w ) and f i | ¯ W C ” f i | ¯ W C ”
0, where we write f = ř j f j b f j .Note that d f is essentially the same as ˜ f , the image of f under ¯ W C b ¯ W C Ñ ( ¯ W C b ¯ W C ) except that b on the right hand side is replaced by b . Therefore f ( v ) = = f ( v ) + ˜ f ( v ) = ÿ i f ( x i x i ) + f ( v ) = ˜ f (cid:16) ÿ x i b x i + v (cid:17) Namely, g ( ř x i b x i + v ) = g P (cid:0) W C / W C (cid:1) b (cid:0) W C / W C (cid:1) . Note that (cid:0) W C / W C (cid:1) b (cid:0) W C / W C (cid:1) – (cid:0) W C / W C b W C / W C (cid:1) – (cid:16) ¯ W C b ¯ W C / ( ¯ W b C b ¯ W C ) à ( ¯ W C b ¯ W b C ) (cid:17) .where we again used the finite dimensionality of ¯ W C / ¯ W C . We conclude that ÿ x i b x i + v = ÿ ( y i ¨ y i ) b z i + ÿ y i b ( z i ¨ z i ) for some y i , z i , y i , y i , z i , z i P ¯ W Λ . So far, we have found an element α : = ř x i b x i + ř ( y i b y i ) b z i + ř y i b ( z i b z i ) whose differential gives v modulo ¯ W b ě C .Repeating this procedure, one can construct α such that d ( α ) = v , which completes theproof. □ R eferences [AAE +
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