Abstract.
Inspired by the geometry of wrapped Fukaya categories, we introduce the notion of wrapped microlocal sheaves. We show that traditional microlocal sheaves are equivalent to functionals on wrapped microlocal sheaves, in analogy with the expected relation of infinitesimal to wrapped Fukaya categories. As an application, we calculate wrapped microlocal sheaves on higher-dimensional pairs of pants, confirming expectations from mirror symmetry.

1.1 Wrapped microlocal sheaves

1.2 Mirror symmetry for pairs of pants

1.3 Microlocal sheaves on surfaces

1.4 Acknowledgements

2 Landau-Ginzburg BݐµBitalic_B-model

2.1 Preliminaries

2.2 Matrix factorizations

2.3 Coordinate hyperplanes

2.4 Descent page_content

3 Microlocal Aݐ´Aitalic_A-model

3.1 Setup

3.2 Sheaves

3.3 Singular support

3.4 Microlocal sheaves

3.5 Microstalks

3.6 Wrapped microlocal sheaves

3.7 Wrapped constructible sheaves

3.7.1 Local systems

3.7.2 Stratifications

3.8 Duality

4 Surfaces

4.1 Combinatorial Fukaya categories

4.2 Microlocal sheaves via gluing

4.3 Example: punctured spheres

5 Pairs of pants

5.1 Liouville manifolds

5.2 Tailored pairs of pants

5.3 Contactification and symplectification

5.4 Mirror symmetry

The aim of this paper is twofold: to introduce a notion of wrapped microlocal sheaves parallel to wrapped Fukaya categories as introduced by Abouzaid-Seidel [3] and studied by Auroux [4], and to establish a homological mirror symmetry equivalence for higher-dimensional pairs of pants extending the results of Seidel [31], Sheridan [32], and the collaboration [2]. More speculatively, we are also motivated by the expectation that wrapped microlocal sheaves will offer a good model for Aݐ´Aitalic_A-branes in a Betti version of Geometric Langlands.

Wrapped microlocal sheaves arise naturally from both geometric and categorical considerations, and consequently enjoy many appealing features. Most notably, traditional microlocal sheaves are equivalent to functionals on wrapped microlocal sheaves (see Theorem 1.6 below) in analogy with the expected relation of infinitesimal to wrapped Fukaya categories, or the relation of perfect complexes with compact support to coherent sheaves [5]. Returning to a more classical setting, one might also keep in mind the identification of compactly-supported cohomology with functionals on Borel-Moore homology.

To provide a collection of simple examples of the theory, we first calculate wrapped microlocal sheaves on exact symplectic surfaces equipped with skeleta. In particular, in the case of punctured spheres, we obtain versions (see Theorem 1.15 below) of the mirror symmetry results of [2]. We then turn to our main application and calculate wrapped microlocal sheaves supported along natural skeleta within higher-dimensional pairs of pants. We establish their equivalence with the expected Landau-Ginzburg BݐµBitalic_B-models in their guise as matrix factorizations (see Theorem 1.13 below). This can be viewed as a generalization, in the language of microlocal sheaves, of results of [2] to higher dimensions, and the results of [31, 32] from compact to arbitrary branes. It also provides the “opposite direction” of homological mirror symmetry to that undertaken in [24, 25].

1.1. Wrapped microlocal sheaves

Roughly speaking, to an exact symplectic manifold Mݑ€Mitalic_M equipped with a Lagrangian skeleton Lݐ¿Litalic_L, there are associated two versions of the Fukaya category: the infinitesimal Fukaya-Seidel category [30] with Lagrangian branes in Mݑ€Mitalic_M running along Lݐ¿Litalic_L, and the wrapped Fukaya category [3] with Lagrangian branes in Mݑ€Mitalic_M transverse to Lݐ¿Litalic_L. (When the skeleton is noncompact, the wrapped variant is often called partially wrapped [4].)

There is a broad expectation that the infinitesimal Fukaya-Seidel category may be modeled by microlocal sheaves on Mݑ€Mitalic_M supported along Lݐ¿Litalic_L. While there is not yet a general account of such an equivalence, there is a convincing and rapidly growing body of evidence, originally motivated by the intersection theory of Lagrangian cycles, and including the far from exhaustive list of references [6, 9, 20, 27, 33]. (We include below a brief review of microlocal sheaves, but for further details, all roads lead to the pioneering work of Kashiwara-Schapira [14].)

In this paper, we propose a notion of wrapped microlocal sheaves which we can verify in many situations similarly models the wrapped Fukaya category. We primarily focus on the traditional microlocal setting of conic open subspaces Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z of the cotangent bundle of a real analytic manifold ZݑZitalic_Z. This suffices for the study of wrapped microlocal sheaves on exact symplectic manifolds Mݑ€Mitalic_M that arise from such conic open subspaces Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z by Hamiltonian reduction. In particular, it is the approach we take to calculate wrapped microlocal sheaves on higher-dimensional pairs of pants. We also calculate wrapped microlocal sheaves on exact symplectic surfaces equipped with skeleta where a more general approach via the gluing of local constructions is not too involved and fits within the scope of this paper.

Throughout, we fix an algebraically closed field kݑ˜kitalic_k of characteristic zero (though more general settings are possible). We work with kݑ˜kitalic_k-linear differential graded (dg) categories, kݑ˜kitalic_k-linear differential ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-graded (ℤ/2ℤ2\mathbbZ/2blackboard_Z / 2-dg) categories, and derived functors, though our language may not explicitly reflect this. For example, by a kݑ˜kitalic_k-module, we will mean a dg kݑ˜kitalic_k-module, and by a perfect kݑ˜kitalic_k-module, we will mean a dg kݑ˜kitalic_k-module with finite-dimensional cohomology.

Given the cotangent bundle T*Zsuperscriptݑ‡ݑT^{*}Zitalic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z of of a real analytic manifold ZݑZitalic_Z, let us begin by listing some prominent features wrapped microlocal sheaves enjoy. (We recommend the analogy with coherent sheaves discussed in Remark 1.7 as an organizing framework.) Fix a closed conic Lagrangian subvariety Λ⊂T*ZΛsuperscriptݑ‡ݑ\Lambda\subset T^{*}Zroman_Λ ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z which we will refer to as a support Lagrangian. To a conic open subspace Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z, there is a dg category μݑ†ℎΛw(Ω)ݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛΩ\mu\mathitSh^w_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) of wrapped microlocal sheaves on ΩΩ\Omegaroman_Ω supported along ΛΛ\Lambdaroman_Λ. Given an inclusion of conic open subspaces Ω′⊂ΩsuperscriptΩ′Ω\Omega^\prime\subset\Omegaroman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ roman_Ω, there is a natural corestriction functor μݑ†ℎΛw(Ω′)→μݑ†ℎΛw(Ω)→ݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛsuperscriptΩ′ݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛΩ\mu\mathitSh^w_\Lambda(\Omega^\prime)\to\mu\mathitSh^w_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) → italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ). These assignments assemble into a cosheaf μݑ†ℎΛwݜ‡subscriptsuperscriptݑ†ℎݑ¤Λ\mu\mathitSh^w_\Lambdaitalic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT of dg categories supported along ΛΛ\Lambdaroman_Λ. There exists a stratification of ΛΛ\Lambdaroman_Λ such that the restriction of μݑ†ℎΛwݜ‡subscriptsuperscriptݑ†ℎݑ¤Λ\mu\mathitSh^w_\Lambdaitalic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT to each stratum is locally constant. The stalk of μݑ†ℎΛwݜ‡subscriptsuperscriptݑ†ℎݑ¤Λ\mu\mathitSh^w_\Lambdaitalic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT at a smooth point of ΛΛ\Lambdaroman_Λ is (not necessarily canonically) equivalent to perfect kݑ˜kitalic_k-modules. Given a closed embedding of support Lagrangians Λ⊂Λ′ΛsuperscriptΛ′\Lambda\subset\Lambda^\primeroman_Λ ⊂ roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, there is a natural localization μݑ†ℎΛ′→μݑ†ℎΛ→ݜ‡subscriptݑ†ℎsuperscriptΛ′ݜ‡subscriptݑ†ℎΛ\mu\mathitSh_\Lambda^\prime\to\mu\mathitSh_\Lambdaitalic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT → italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT.

Example 1.1.

For the zero-section Λ=ZΛݑ\Lambda=Zroman_Λ = italic_Z and the entire cotangent bundle Ω=T*ZΩsuperscriptݑ‡ݑ\Omega=T^{*}Zroman_Ω = italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z, one can interpret μݑ†ℎZw(T*Z)ݜ‡subscriptsuperscriptݑ†ℎݑ¤ݑsuperscriptݑ‡ݑ\mu\mathitSh^w_Z(T^{*}Z)italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_Z end_POSTSUBSCRIPT ( italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z ) as a dg category of certain locally constant sheaves. More precisely, it is equivalent to the dg category of perfect modules over chains on the Poincaré ∞\infty∞-groupoid of ZݑZitalic_Z. In particular, if ZݑZitalic_Z is connected, it is equivalent to the dg category of perfect modules over chains on the based loop space of ZݑZitalic_Z. (One could compare with the parallel result for the wrapped Fukaya category proved by Abouzaid [1].) In contrast, the traditional dg category of local systems is equivalent to the dg category of those modules over chains on the Poincaré ∞\infty∞-groupoid of ZݑZitalic_Z whose underlying kݑ˜kitalic_k-module is perfect.

Example 1.2.

In the papers [24, 25], we calculated the Landau-Ginzburg Aݐ´Aitalic_A-model M=ℂn,W=z1⋯znformulae-sequenceݑ€superscriptℂݑ›ݑŠsubscriptݑ§1⋯subscriptݑ§ݑ›M=\mathbbC^n,W=z_1\cdots z_nitalic_M = blackboard_C start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT , italic_W = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⋯ italic_z start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT taking its branes in the form of microlocal sheaves. In the most basic formulation, we took as support Lagrangian the “singular thimble” Lc=Cone(Tn-1)⊂Msubscriptݐ¿ݑݐ¶ݑœݑ›ݑ’superscriptݑ‡ݑ›1ݑ€L_c=\mathitCone(T^n-1)\subset Mitalic_L start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = italic_C italic_o italic_n italic_e ( italic_T start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT ) ⊂ italic_M given by the cone over a compact torus in a regular fiber. We established a mirror equivalence between traditional microlocal sheaves on Mݑ€Mitalic_M supported along Lcsubscriptݐ¿ݑL_citalic_L start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and perfect complexes with proper support on the (n-2)ݑ›2(n-2)( italic_n - 2 )-dimensional pair of pants Pn-2subscriptݑƒݑ›2P_n-2italic_P start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT. If one repeats the arguments of [24, 25] with the wrapped microlocal sheaves of this paper, one will arrive at a mirror equivalence with all coherent complexes on Pn-2subscriptݑƒݑ›2P_n-2italic_P start_POSTSUBSCRIPT italic_n - 2 end_POSTSUBSCRIPT.

The essential nature of wrapped microlocal sheaves is perhaps best captured by their relation with traditional microlocal sheaves. To pursue this, to a conic open subspace Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z, let us denote by μݑ†ℎΛ♢(Ω)ݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh^\diamondsuit_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) the cocomplete dg category of large microlocal sheaves on ΩΩ\Omegaroman_Ω whose microstalks are arbitrary kݑ˜kitalic_k-modules supported along ΛΛ\Lambdaroman_Λ. Thus we require objects of μݑ†ℎΛ♢(Ω)ݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh^\diamondsuit_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) to be geometrically tame in the sense that their singular support lies within ΛΛ\Lambdaroman_Λ, but we do not impose any algebraic restriction on their size.

We will focus on two natural ways to cut out a small dg subcategory of μݑ†ℎΛ♢(Ω)ݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh^\diamondsuit_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) by imposing finiteness conditions.

First, there is the full dg subcategory

xymatrixxymatrix\xymatrix
of objects whose microstalks are perfect kݑ˜kitalic_k-modules supported along ΛΛ\Lambdaroman_Λ. We refer to these as traditional microlocal sheaves, though more technically they might also be termed microlocally constructible. They are the version of microlocal sheaves closely related to the infinitesimal Fukaya-Seidel category, and can be represented locally by constructible sheaves.

Traditional microlocal sheaves enjoy the following contravariant versions of the features of wrapped microlocal sheaves listed above. Given an inclusion of conic open subspaces Ω′⊂ΩsuperscriptΩ′Ω\Omega^\prime\subset\Omegaroman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ roman_Ω, there is a natural restriction functor μݑ†ℎΛ(Ω)→μݑ†ℎΛ(Ω′)→ݜ‡subscriptݑ†ℎΛΩݜ‡subscriptݑ†ℎΛsuperscriptΩ′\mu\mathitSh_\Lambda(\Omega)\to\mu\mathitSh_\Lambda(\Omega^\prime)italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) → italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ). These assignments assemble into a sheaf μݑ†ℎΛݜ‡subscriptݑ†ℎΛ\mu\mathitSh_\Lambdaitalic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT of dg categories supported along ΛΛ\Lambdaroman_Λ. There exists a stratification of ΛΛ\Lambdaroman_Λ such that the restriction of μݑ†ℎΛݜ‡subscriptݑ†ℎΛ\mu\mathitSh_\Lambdaitalic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT to each stratum is locally constant. The stalk of μݑ†ℎΛݜ‡subscriptݑ†ℎΛ\mu\mathitSh_\Lambdaitalic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT at a smooth point of ΛΛ\Lambdaroman_Λ is (not necessarily canonically) equivalent to perfect kݑ˜kitalic_k-modules. Given a closed embedding of support Lagrangians Λ⊂Λ′ΛsuperscriptΛ′\Lambda\subset\Lambda^\primeroman_Λ ⊂ roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, there is a natural fully faithful embedding μݑ†ℎΛ→μݑ†ℎΛ′→ݜ‡subscriptݑ†ℎΛݜ‡subscriptݑ†ℎsuperscriptΛ′\mu\mathitSh_\Lambda\to\mu\mathitSh_\Lambda^\primeitalic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT → italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT.

Second, we can cut out a small dg subcategory of μݑ†ℎΛ♢(Ω)ݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh^\diamondsuit_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) by imposing the standard categorical finiteness of compact objects. Recall that an object c∈ݒžݑݒžc\in\mathcalCitalic_c ∈ caligraphic_C of a stable dg category is compact if and only if the functor it corepresents Hom(c,-):ݒž→Modk:Homݑ→ݒžsubscriptModݑ˜\operatorpage_seo_titleHom(c,-):\mathcalC\to\operatorpage_seo_titleMod_kroman_Hom ( italic_c , - ) : caligraphic_C → roman_Mod start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT preserves coproducts. If ݒžݒž\mathcalCcaligraphic_C is cocomplete, in particular it contains coproducts, then we may recover it as the ind-category ݒž≃Indݒžcsimilar-to-or-equalsݒžIndsubscriptݒžݑ\mathcalC\simeq\operatorpage_seo_titleInd\mathcalC_ccaligraphic_C ≃ roman_Ind caligraphic_C start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT of its full dg subcategory of compact objects.

Definition 1.3.

The dg category of wrapped microlocal sheaves is the full dg subcategory

xymatrixxymatrix\xymatrix
of compact objects within the cocomplete dg category of all microlocal sheaves.

The above definition is useful for the clean characterization it provides, but there is an equivalent and more geometric way to approach wrapped microlocal sheaves that better illuminates their relation to the wrapped Fukaya category.

To explain this, let us briefly recall the geometry of microstalks. For simplicity, let us focus on microstalks at a generic point (z,ξ)∈Λݑ§ݜ‰Λ(z,\xi)\in\Lambda( italic_z , italic_ξ ) ∈ roman_Λ, where z∈Zݑ§ݑz\in Zitalic_z ∈ italic_Z and ξ∈Tz*Zݜ‰subscriptsuperscriptݑ‡ݑ§ݑ\xi\in T^{*}_zZitalic_ξ ∈ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_Z, so that in particular Λ⊂T*ZΛsuperscriptݑ‡ݑ\Lambda\subset T^{*}Zroman_Λ ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z is a smooth Lagrangian submanifold near to (z,ξ)ݑ§ݜ‰(z,\xi)( italic_z , italic_ξ ).

Choose a small open ball B⊂ZݐµݑB\subset Zitalic_B ⊂ italic_Z around z∈Zݑ§ݑz\in Zitalic_z ∈ italic_Z, and a smooth function f:B→ℝ:ݑ“→ݐµℝf:B\to\mathbbRitalic_f : italic_B → blackboard_R such that f(z)=0ݑ“ݑ§0f(z)=0italic_f ( italic_z ) = 0, and dfz=ξݑ‘subscriptݑ“ݑ§ݜ‰df_z=\xiitalic_d italic_f start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = italic_ξ. The graph L=Γdf⊂T*Zݐ¿subscriptΓݑ‘ݑ“superscriptݑ‡ݑL=\Gamma_df\subset T^{*}Zitalic_L = roman_Γ start_POSTSUBSCRIPT italic_d italic_f end_POSTSUBSCRIPT ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z is a small Lagrangian ball centered at (z,ξ)ݑ§ݜ‰(z,\xi)( italic_z , italic_ξ ). Let us assume it intersects Λ⊂T*ZΛsuperscriptݑ‡ݑ\Lambda\subset T^{*}Zroman_Λ ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z transversely at the single point (z,ξ)ݑ§ݜ‰(z,\xi)( italic_z , italic_ξ ).

Then for Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z a conic open subspace containing (z,ξ)ݑ§ݜ‰(z,\xi)( italic_z , italic_ξ ), the microstalk along L⊂Ωݐ¿ΩL\subset\Omegaitalic_L ⊂ roman_Ω of a microlocal sheaf ℱ∈μݑ†ℎΛ♢(Ω)ℱݜ‡superscriptsubscriptݑ†ℎΛ♢Ω\mathcalF\in\mu\mathitSh_\Lambda^\diamondsuit(\Omega)caligraphic_F ∈ italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT ( roman_Ω ) is the vanishing cycles

xymatrixxymatrix\xymatrix
where the sheaf ℱ~~ℱ\tilde\mathcalFover~ start_ARG caligraphic_F end_ARG represents the restriction of the microlocal sheaf ℱℱ\mathcalFcaligraphic_F to a small conic open neighborhood of (z,ξ)ݑ§ݜ‰(z,\xi)( italic_z , italic_ξ ).

By abstract formalism, there is an object ℱL∈μݑ†ℎΛ♢(Ω)subscriptℱݐ¿ݜ‡superscriptsubscriptݑ†ℎΛ♢Ω\mathcalF_L\in\mu\mathitSh_\Lambda^\diamondsuit(\Omega)caligraphic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ∈ italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT ( roman_Ω ), which we call a microlocal skyscraper, corepresenting the microstalk in the sense of a natural equivalence

We have the following geometric alternative to the above categorical definition.

Lemma 1.4 (Lemma 3.15 below).

The microlocal skyscrapers ℱL∈μݑ†ℎΛ♢(Ω)subscriptℱݐ¿ݜ‡superscriptsubscriptݑ†ℎnormal-Λnormal-♢normal-Ω\mathcalF_L\in\mu\mathitSh_\Lambda^\diamondsuit(\Omega)caligraphic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ∈ italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT ( roman_Ω ) form a collection of compact generators, and thus wrapped microlocal sheaves

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form the full dg subcategory split-generated by the microlocal skyscrapers.

Remark 1.5.

From the above geometric characterization, we can see more clearly the relation with the wrapped Fukaya category. The microlocal skyscraper ℱL∈μݑ†ℎΛw(Ω)subscriptℱݐ¿ݜ‡superscriptsubscriptݑ†ℎΛݑ¤Ω\mathcalF_L\in\mu\mathitSh_\Lambda^w(\Omega)caligraphic_F start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ∈ italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT ( roman_Ω ) corresponds to the object of the wrapped Fukaya category given by the Lagrangian ball L⊂Ωݐ¿ΩL\subset\Omegaitalic_L ⊂ roman_Ω transverse to ΛΛ\Lambdaroman_Λ equipped with a suitable brane structure.

The morphisms between microlocal skyscrapers involve the potentially complicated global geometry of the exact symplectic manifold ΩΩ\Omegaroman_Ω and support Lagrangian ΛΛ\Lambdaroman_Λ. This is in parallel with the potentially complicated global dynamics of the wrapping Hamiltonian H:Ω→ℝ:ݐ»→ΩℝH:\Omega\to\mathbbRitalic_H : roman_Ω → blackboard_R found in the construction of the wrapped Fukaya category.

Now let us focus on how the above versions of microlocal sheaves are related. Since the full dg subcategory μݑ†ℎΛw(Ω)⊂μݑ†ℎΛ♢(Ω)ݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛΩݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh^w_\Lambda(\Omega)\subset\mu\mathitSh^\diamondsuit_% \Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) ⊂ italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) of wrapped microlocal sheaves comprises the compact objects, we can recover all microlocal sheaves from it by forming the ind-category

It turns out we can also recover the small dg category μݑ†ℎΛ(Ω)⊂μݑ†ℎΛ♢(Ω)ݜ‡subscriptݑ†ℎΛΩݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh_\Lambda(\Omega)\subset\mu\mathitSh^\diamondsuit_\Lambda% (\Omega)italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) ⊂ italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) of traditional microlocal sheaves from wrapped microlocal sheaves (though not vice versa, see Remark 1.8). The natural hom-pairing between μݑ†ℎΛ(Ω)ݜ‡subscriptݑ†ℎΛΩ\mu\mathitSh_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) and the opposite dg category μݑ†ℎΛw(Ω)opݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛsuperscriptΩݑœݑ\mu\mathitSh^w_\Lambda(\Omega)^opitalic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT lands in perfect kݑ˜kitalic_k-modules. We prove this is in fact a perfect pairing in the following sense.

Theorem 1.6 (Theorem 3.21 below).

Remark 1.7.

With mirror symmetry in mind, we advocate the following informal analogy.

To the conic open subspace Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z and support Lagrangian Λ⊂T*ZΛsuperscriptݑ‡ݑ\Lambda\subset T^{*}Zroman_Λ ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z, let us imagine assigning a variety XΛ(Ω)subscriptݑ‹ΛΩX_\Lambda(\Omega)italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ). We can think of wrapped microlocal sheaves μݑ†ℎΛw(Ω)ݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛΩ\mu\mathitSh^w_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) as coherent sheaves Coh(XΛ(Ω))Cohsubscriptݑ‹ΛΩ\operatorpage_seo_titleCoh(X_\Lambda(\Omega))roman_Coh ( italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) ), and thus all microlocal sheaves μݑ†ℎΛ♢(Ω)ݜ‡subscriptsuperscriptݑ†ℎ♢ΛΩ\mu\mathitSh^\diamondsuit_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT ♢ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) as ind-coherent sheaves IndCoh(XΛ(Ω))IndCohsubscriptݑ‹ΛΩ\operatorpage_seo_titleInd\operatorpage_seo_titleCoh(X_\Lambda(\Omega))roman_Ind roman_Coh ( italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) ). In line with the results of [5], Theorem 1.6 says we can compatibly think of traditional microlocal sheaves μݑ†ℎΛ(Ω)ݜ‡subscriptݑ†ℎΛΩ\mu\mathitSh_\Lambda(\Omega)italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) as perfect complexes with proper support PerfݑݑŸݑœݑ(XΛ(Ω))subscriptPerfݑݑŸݑœݑsubscriptݑ‹ΛΩ\operatorpage_seo_titlePerf_\mathitprop(X_\Lambda(\Omega))roman_Perf start_POSTSUBSCRIPT italic_prop end_POSTSUBSCRIPT ( italic_X start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) ).

Let us further extend the analogy to the natural functoriality in the conic open subspace Ω⊂T*ZΩsuperscriptݑ‡ݑ\Omega\subset T^{*}Zroman_Ω ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z and support Lagrangian Λ⊂T*ZΛsuperscriptݑ‡ݑ\Lambda\subset T^{*}Zroman_Λ ⊂ italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_Z. To an open inclusion Ω′⊂ΩsuperscriptΩ′Ω\Omega^\prime\subset\Omegaroman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⊂ roman_Ω, and closed embedding Λ⊂Λ′ΛsuperscriptΛ′\Lambda\subset\Lambda^\primeroman_Λ ⊂ roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT, we can think of assigning a correspondence

xymatrixxymatrix\xymatrix
where pݑpitalic_p is proper and Gorenstein, and qݑžqitalic_q is smooth. Then we can think of the natural functors μݑ†ℎΛ′w(Ω′)→μݑ†ℎΛw(Ω)→ݜ‡subscriptsuperscriptݑ†ℎݑ¤superscriptΛ′superscriptΩ′ݜ‡subscriptsuperscriptݑ†ℎݑ¤ΛΩ\mu\mathitSh^w_\Lambda^\prime(\Omega^\prime)\to\mu\mathitSh^w_% \Lambda(\Omega)italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) → italic_μ italic_Sh start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) and μݑ†ℎΛ(Ω)→μݑ†ℎΛ′(Ω′)→ݜ‡subscriptݑ†ℎΛΩݜ‡subscriptݑ†ℎsuperscriptΛ′superscriptΩ′\mu\mathitSh_\Lambda(\Omega)\to\mu\mathitSh_\Lambda^\prime(\Omega^% \prime)italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) → italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( roman_Ω start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) as corresponding respectively to functors

xymatrixxymatrix\xymatrix
Note that these are the restrictions of adjoint functors on all ind-coherent sheaves.

Remark 1.8.

While objects of μݑ†ℎΛw(Ω)ݜ‡superscriptsubscriptݑ†ℎΛݑ¤Ω\mu\mathitSh_\Lambda^w(\Omega)italic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_w end_POSTSUPERSCRIPT ( roman_Ω ) similarly give functionals on μݑ†ℎΛ(Ω)opݜ‡subscriptݑ†ℎΛsuperscriptΩݑœݑ\mu\mathitSh_\Lambda(\Omega)^opitalic_μ italic_Sh start_POSTSUBSCRIPT roman_Λ end_POSTSUBSCRIPT ( roman_Ω ) start_POSTSUPERSCRIPT italic_o italic_p end_POSTSUPERSCRIPT, it is not true that they produce all possible functionals. For example, take Z=T1ݑsuperscriptݑ‡1Z=T^1italic_Z = italic_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT to be the circle, Λ=T1Λsuperscriptݑ‡1\Lambda=T^1roman_Λ = italic_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT the zero-section, and Ω=T*T1Ωsuperscriptݑ‡superscriptݑ‡1\Omega=T^{*}T^1roman_Ω = italic_T start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_T start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT the entire cotangent bundle. Then we have equivalences

xymatrixxymatrix\xymatrix
and by [5], the hom-pairing gives an equivalence

xymatrixxymatrix\xymatrix
compatibly with Theorem 1.6. But clearly there are more functionals on PerfݑݑŸݑœݑ(ݔ¾m)subscriptPerfݑݑŸݑœݑsubscriptݔ¾ݑš\operatorpage_seo_titlePerf_\mathitprop(\mathbbG_m)roman_Perf start_POSTSUBSCRIPT italic_prop end_POSTSUBSCRIPT ( blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) than those coming from Coh(ݔ¾m)Cohsubscriptݔ¾ݑš\operatorpage_seo_titleCoh(\mathbbG_m)roman_Coh ( blackboard_G start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) alone. For example, one could take the hom-pairing with a direct sum of skyscraper sheaves at infinitely many points.

The proof we give of Theorem 1.6 is an application of the theory of arboreal singularities developed in [22, 23]. Let us sketch the argument here. First, by abstract formalism, it suffices to prove the theorem locally in ΛΛ\Lambdaroman_Λ, so we may focus on the germ of ΛΛ\Lambdaroman_Λ at a point. Then applying [23], we may non-characteristically deform the germ of ΛΛ\Lambdaroman_Λ to a nearby conic Lagrangian subvariety ΛݑŽݑŸݑsubscriptΛݑŽݑŸݑ\Lambda_\mathitarbroman_Λ start_POSTSUBSCRIPT italic_arb end_POSTSUBSCRIPT with arboreal singularities. Thus microlocal sheaves along the germ are equivalent to microlocal sheaves along ΛݑŽݑŸݑsubscriptΛݑŽݑŸݑ\Lambda_\mathitarbroman_Λ start_POSTSUBSCRIPT italic_arb end_POSTSUBSCRIPT. Once again, by abstract formalism, it suffices to prove the theorem locally in ΛݑŽݑŸݑsubscriptΛݑŽݑŸݑ\Lambda_\mathitarbroman_Λ start_POSTSUBSCRIPT italic_arb end_POSTSUBSCRIPT, so we may focus further on the germ of ΛݑŽݑŸݑsubscriptΛݑŽݑŸݑ\Lambda_\mathitarbroman_Λ start_POSTSUBSCRIPT italic_arb end_POSTSUBSCRIPT at a point. Now applying [22], we find that large microlocal sheaves on the germ of ΛݑŽݑŸݑsubscriptΛݑŽݑŸݑ\Lambda_\mathitarbroman_Λ start_POSTSUBSCRIPT italic_arb end_POSTSUBSCRIPT are equivalent to modules over a directed tree ݒ¯ݒ¯\mathcalTcaligraphic_T. Here the perfect and coherent modules coincide, and form a smooth and proper dg category, and thus the assertion of the theorem holds.

1.2. Mirror symmetry for pairs of pants

Now let us turn to a concrete application in the setting of homological mirror symmetry. We will introduce natural skeleta within pairs of pants. Calculate wrapped microlocal sheaves supported along them. We can not immediately invoke the preceding theory since pairs of pants are not conic open subspaces of cotangent bundles. Thus we will first pass to the symplectification of the contactification of neighborhoods of their skeleta. Identify these with conic open subspaces of cotangent bundles. Since wrapped microlocal sheaves are invariant under this modification, it provides a natural avenue to the preceding theory.

Let ℙℂn+1=Projℂ[z0,…,

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