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Augmentations Are Sheaves AUGMENTATIONS ARE SHEAVES LENHARD NG, DAN RUTHERFORD, VIVEK SHENDE, STEVEN SIVEK, AND ERIC ZASLOW ABSTRACT. We show that the set of augmentations of the Chekanov–Eliashberg algebra of a Leg- endrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC14], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ14], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry. CONTENTS 1. Introduction............................................................................. 2 2. Background............................................................................. 7 2.1. Contact geometry.................................................................... 7 2.2. The LCH differential graded algebra................................................... 7 2.3. A1 categories....................................................................... 11 2.4. Constructible sheaves................................................................. 14 3. Augmentation category algebra............................................................ 18 3.1. Differential graded algebras and augmentations......................................... 18 3.2. Link grading......................................................................... 20 3.3. A1-categories from sequences of DGAs............................................... 23 3.4. A construction of unital categories..................................................... 27 4. The augmentation category of a Legendrian link............................................. 32 4.1. Definition of the augmentation category................................................ 32 4.2. DGAs for the perturbations and unitality of Aug+ ....................................... 36 4.3. Invariance........................................................................... 42 4.4. Examples........................................................................... 47 5. Properties of the augmentation category.................................................... 55 5.1. Duality and long exact sequences...................................................... 55 5.2. Dictionary and comparison to previously known results.................................. 59 5.3. Equivalence of augmentations......................................................... 60 6. Localization of the augmentation category.................................................. 66 7. Augmentations are sheaves................................................................ 73 7.1. The Morse complex category.......................................................... 76 7.2. Local calculations in the augmentation category......................................... 79 7.3. Local calculations in the sheaf category................................................ 92 7.4. Augmentations are sheaves............................................................ 97 8. Some exact sequences................................................................... 98 References.................................................................................. 101 2 LENHARD NG, DAN RUTHERFORD, VIVEK SHENDE, STEVEN SIVEK, AND ERIC ZASLOW 1. INTRODUCTION A powerful modern approach to studying a Legendrian submanifold Λ in a contact manifold V is to encode Floer-theoretic data into a differential graded algebra A(V; Λ), the Chekanov–Eliashberg DGA. The generators of this algebra are indexed by Reeb chords; its differential counts holomorphic disks in the symplectization R×V with boundary lying along the Lagrangian R×Λ and meeting the Reeb chords at infinity [Eli98, EGH00]. Isotopies of Legendrians induce homotopy equivalences of algebras, and the homology of this algebra is called Legendrian contact homology. A fundamental insight of Chekanov [Che02] is that, in practice, these homotopy equivalence classes of infinite dimensional algebras can often be distinguished by the techniques of algebraic geometry. For instance, the functor of points commutative rings ! sets r 7! fDGA morphisms A(V; Λ) ! rg=DGA homotopy is preserved by homotopy equivalences of algebras A(V; Λ) [FHT01, Lem. 26.3], and thus furnishes an invariant. Collecting together the linearizations (“cotangent spaces”) ker /(ker )2 of the aug- mentations (“points”) : A(V; Λ) ! r gives a stronger invariant: comparison of these linearizations as differential graded vector spaces is one way that Legendrian knots have been distinguished in practice since the work of Chekanov. As the structure coefficients of the DGA A(V; Λ) come from the contact geometry of (V; Λ), it is natural to ask for direct contact-geometric interpretations of the algebro-geometric constructions above, and in particular to seek the contact-geometric meaning of the – a priori, purely algebraic – augmentations. In some cases, this meaning is known. As in topological field theory, exact Lagrangian cobordisms between Legendrians give rise (contravariantly) to morphisms of the cor- responding DGAs [EGH00, Ekh12, EHK]. In particular, exact Lagrangian fillings are cobordisms from the empty set, and so give augmentations. However, not all augmentations arise in this manner. Indeed, consider pushing an exact filling surface L of a Legendrian knot Λ in the Reeb direction: on the one hand, this is a deformation of L inside T ∗L, and so intersects L – an exact Lagrangian – in a number of points which, counted with signs, is −χ(L). On the other hand, this intersection can be computed as the linking number at infinity, or in other words, the Thurston–Bennequin number of Λ: tb(Λ) = −χ(L). Now there is a Legendrian figure eight knot with tb = −3 (see e.g. [CN13] for this and other examples); its DGA has augmentations, and yet any filling surface would necessarily have genus −2. This obstruction has a categorification, originally due to Seidel and made precise in this context by Ekholm [Ekh12]. Given an exact filling (W; L) of (V; Λ) (where we will primarily focus on 3 4 the case V = R and W = R ), consider the Floer homology HFt(L; L), where the differential accounts only for disks bounded by L and a controlled Hamiltonian perturbation of L for time < t, i.e. loosely those disks with action bounded by t. There is an inclusion HF−"(L; L) ! HF1(L; L). The former has generators given by self-intersections of L with a small perturbation of itself, and the latter has generators given by these together with Reeb chords of Λ. The quotient of these chain complexes leads to what is called “linearized contact cohomology” in the literature; for reasons to be made clear shortly, we write it as Hom−(, )[1]. That is, we have: 1 (1.1) HF−"(L; L) ! HF1(L; L) ! Hom−(, )[1] −! : AUGMENTATIONS ARE SHEAVES 3 4 ∼ Finally, since the wrapped Fukaya category of R is trivial, we get an isomorphism Hom−(, ) = ∼ ∗ r HF−"(L; L). On the other hand, HF−"(L; L) = Hc (L; ). In particular, taking Euler characteristics recovers: ∗ r −tb(Λ) = χ(Hom−(, )) = χ(Hc (L; )) = χ(L): One could try to construct the missing augmentations from more general objects in the derived Fukaya category. To the extent that this is possible, the above sequence implies that the categor- ical structures present in the symplectic setting should be visible on the space of augmentations. An important step in this direction was taken by Bourgeois and Chantraine [BC14], who define a non-unital A1 category which we denote Aug−. Its objects are augmentations of the Chekanov– 0 Eliashberg DGA, and its hom spaces Hom−(, ) have the property that the self Homs are the linearized contact cohomologies. The existence of this category was strong evidence that augmen- tations could indeed be built from geometry. On the other hand, when V = T 1M is the cosphere bundle over a manifold, Λ ⊂ V is a Legen- drian, and r is a commutative ring, a new source of Legendrian invariants is provided by the category Sh(M; Λ; r) of constructible sheaves of r-modules on M whose singular support meets T 1M in Λ [STZ14]. The introduction of this category is motivated by the microlocalization equivalence of the category of sheaves on a manifold with the infinitesimally wrapped Fukaya category of the cotangent bundle [NZ09, Nad09]: ∼ ∗ µ : Sh(M; r) −! F uk"(T M; r): In particular, to a Lagrangian brane L ⊂ T ∗M ending on Λ, there corresponds a sheaf µ−1(L) with the property that −1 −1 ∗ ∗ r HomSh(M)(µ (L); µ (L)) = HomF uk(T M)(L; L) = HF+"(L; L) = H (L; ); and we write Sh(M; Λ; r) := µ−1(L). Like ordinary cohomology, the category Sh(M; Λ; r) is unital; like compactly supported coho- mology, the Bourgeois–Chantraine augmentation category Aug−(Λ; r) is not. In an augmentation category matching the sheaf category, the Hom spaces would fit naturally into an exact sequence 1 (1.2) HF+"(L; L) ! HF (L; L) ! Hom+(, )[1] −! : Together these observations suggest the following modification to the Bourgeois–Chantraine con- struction. As noted in [BC14], Aug− can be defined from the n-copy of the Legendrian, ordered with respect to the displacement in the Reeb direction. To change the sign of the perturbations, in the front diagram of the Legendrian we re-order the n-copy from top to bottom, instead of from bottom
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