Conic Microsheaves in Symplectic Geometry

• Conic microsheaves are a categorical and microlocal quantization framework for conic Lagrangians that unifies symplectic geometry with microlocal sheaf theory.
• They establish explicit functorial correspondences linking Floer theory and Fukaya categories, enabling the construction of microsheaf quantizations.
• Applications include advances in geometric representation theory and the geometric Langlands program, highlighted by orthogonality and precise endomorphism algebra properties.

A conic microsheaf is a categorical and microlocal quantization of (eventually) conic Lagrangians within exact symplectic or Weinstein manifolds, where the “conicity” encodes invariance under natural scaling actions on cotangent fibers. This framework synthesizes microlocal sheaf theory, symplectic geometry, and Floer theory. Conic microsheaves supply bridges between sheaf theory and Fukaya categories, and have proven central in the categorical paper of geometric representation theory, including the geometric Langlands program and symplectic topology.

1. Definition and Structure of Conic Microsheaves

Given a smooth manifold $M$, the derived category $Sh(M)$ of sheaves with coefficients in a stable $\infty$-category admits a notion of microsupport $ss(F) \subset T^*M$, a closed, conic, coisotropic subset classifying singular support directions of $F$. For each open conic $\underline\Omega\subset T^*M$, the category

$$\text{pre-}\mu\!Sh(\underline\Omega) := Sh(M) / Sh_{T^*M\setminus\underline\Omega}(M)$$

sheafifies to a sheaf of stable categories $\mu Sh_M$ on $T^*M$ with sections $\mu Sh(\underline\Omega)$. For a conic subset $\Lambda\subset T^*M$, one sets the global sections $\mu Sh_\Lambda=\Gamma(\mu Sh_M|_\Lambda)$.

In a Weinstein manifold $W$ (i.e., an exact symplectic manifold with Liouville vector field gradient-like for a proper Morse function), these definitions extend to sheaves of categories $\mu Sh_W\to W$, with $\mu Sh_K=\Gamma(\mu Sh_W|_K)$ for any conic $K\subset W$. The core $\frc_W\subset W$ (points not escaping under the Liouville flow) plays a central role.

An immersed exact Lagrangian $\bar L\looparrowright W$ is called eventually conic if, outside a compact set, its Legendrian lift projects to a compact Legendrian in the contact boundary, equivalently satisfying $$\bar L\cap(\partial W\times \mathbb{R})\subset \Lambda\times t_0$$ for compact $\Lambda\subset \partial W$ (Li et al., 25 Nov 2025).

2. Floer–Microsheaf Correspondence and Quantization

Ganatra–Pardon–Shende established an equivalence between the wrapped Fukaya category of a Weinstein manifold $W$ and the category of microsheaves on its Liouville skeleton $L_W$:

$$\text{Ind Fuk}(W)\simeq\mathfrak{Sh}(W)=\Gamma(L_W,\mu sh)$$

For $W=\mathrm{Higgs}_G^s(C)_d$ (moduli space of stable Higgs bundles), this underpins the construction of microsheaf quantizations: a canonical functor

$$\mathfrak{F}: Fuk(\mathrm{Higgs}_G^s(C)_d)\to \mathfrak{Sh}(\mathrm{Higgs}_G^s(C)_d)$$

assigns to each unobstructed Lagrangian $L\subset W$ a microsheaf $$\mathcal{F}_L\in\Gamma(\mathcal{N}_G^s(C)_d,\mu sh)$$ whose microsupport lies in the global nilpotent cone $\Lambda_{nilp}$ (Shende, 2021). The functoriality is reflected in the isomorphism of morphism complexes:

$$CF^*(L,L')\cong \operatorname{Hom}_{\mathfrak{Sh}}(\mathcal{F}_L, \mathcal{F}_{L'})$$

For $L$ a smooth Hitchin fiber $F$, $\mathcal{F}_F$ is a conic microsheaf supported on $\Lambda_{nilp}$.

In general symplectic manifolds $(W,\lambda)$, conic microsheaf quantization arises from an explicit functor assigning local systems on eventually-conic exact Lagrangians $L$ to sections of $\mu Sh_{W}$ supported on $\frc_W^L$, an extended core incorporating the Lagrangian cone of $L$ (Li et al., 25 Nov 2025).

3. Composition, Orthogonality, and Functoriality

Kashiwara–Schapira's microlocal theory of sheaves ensures that for $F_{12}\in Sh(M_1\times M_2)$ and $F_{23}\in Sh(M_2\times M_3)$, the convolution $F_{23}\circ F_{12}$ has microsupport

$$ss(F_{23}\circ F_{12})\subset ss(F_{23})\circ ss(F_{12})$$

This induces a convolution product on conic microsheaf categories:

$$\circ: \mu Sh_{\Lambda_{12}}(\Lambda_{12})\otimes \mu Sh_{\Lambda_{23}}(\Lambda_{23})\longrightarrow\mu Sh_{\Lambda_{23}\circ\Lambda_{12}}(\Lambda_{23}\circ\Lambda_{12})$$

In Weinstein settings, the composition of Lagrangian correspondences $L_{13}=L_{23}\circ L_{12}$ admits compatible functorial quantizations at the microsheaf level. Under positive gappedness or disjoint-at-infinity conditions, the relevant functorial squares commute. For embedded correspondences, the quantization recovers the geometric operation at the Lagrangian level (Li et al., 25 Nov 2025).

Distinct smooth Hitchin fibers, as compact holomorphic Lagrangian tori, are disjoint; thus, their associated conic microsheaves are orthogonal:

$$\operatorname{Hom}_{\mathrm{Sh}}(\mathcal{F}_F, \mathcal{F}_{F'})=0 \qquad (F\neq F')$$

This categorical orthogonality mirrors the vanishing of Floer complexes $CF^*(F,F')$ (Shende, 2021).

4. Algebra of Endomorphisms and Microlocal Support

The endomorphism algebra of a conic microsheaf $\mathcal{F}_F$ associated to a smooth Hitchin fiber is isomorphic to the cohomology of $F$:

$$\operatorname{End}(\mathcal{F}_F)\cong CF^*(F,F)\cong H^*(F;\mathbf{k})$$

This reflects the formality and unobstructedness of self-Floer cohomology in the hyperkähler setting, as proved by Solomon–Verbitsky. The microsupport $ss(\mathcal{F}_F)$ is contained in the conical Lagrangian $\Lambda_{nilp}$—this "conic" property (invariance under the natural $\mathbb{C}^*$-action) ensures that $\mathcal{F}_F$ is locally constant along positive ray orbits in each cotangent fiber (Shende, 2021).

5. Gappedness, Specialization, and Group Actions

A central technical advance is the in-families "gappedness" criterion for composing families of microsheaves: if pairs of conic Lagrangians $$\Lambda_t, \Lambda'_t\subset T^*(M\times \mathbb{R}_{>0})$$ are "positively gapped" (i.e., all Reeb chords from $\Lambda'_t$ to $\Lambda_t$ have length uniformly bounded below), then nearby cycles commute with composition. This ensures compatibility of kernels and convolution operations under family deformations

$$\psi(F_{23}\circ F_{12})\simeq \psi F_{23}\circ \psi F_{12}$$

and applies to functoriality for microlocal bounding cochains and kernel functors in $Sh(M)$ (Li et al., 25 Nov 2025).

When a Lie group $G$ acts by exact symplectomorphisms on $W$, the moment-map correspondence $\Gamma_\mu$ is eventually conic. The associated invertible kernel $K_\mu\in {}_{\Gamma_\mu}(\Gamma_\mu)$ induces a colimit-preserving functor

${}_{\frc_W}(\frc_W)\longrightarrow{}_{\frc_W}(\frc_W)$

yielding a true topological $G$-action on the conic microsheaf category. In cotangent bundles, this reproduces the "microlocal GKS" framework for quantizing contact transformations (Li et al., 25 Nov 2025).

6. Applications: The Global Nilpotent Cone and the Geometric Langlands Program

In the moduli space of Higgs bundles for a smooth projective curve $C$ and reductive group $G$, the global nilpotent cone $$\Lambda_{nilp}=h^{-1}(0)\subset T^*\mathrm{Bun}_G(C)_d$$ forms a conical Lagrangian preserved by the scaling action. Each smooth Hitchin fiber defines a conic microsheaf supported on $\Lambda_{nilp}$:

$$\mathcal{F}_F\in Sh_{\Lambda_{nilp}}(\mathrm{Bun}_G(C)_d)$$

The set of such $\mathcal{F}_F$ forms canonical, orthogonal, rank-one objects with endomorphism algebra $H^*(F)$.

Microsheaves $\mathcal{F}_F$ are natural candidates for Hecke eigensheaves in the Betti or de Rham incarnations of the geometric Langlands correspondence. The convolution action of Hecke correspondences $H_\mu$ preserves the support on $\Lambda_{nilp}$:

$$H_\mu(\mathcal{F}_F)\simeq \mathcal{F}_F\boxtimes \rho^\mu(\chi)$$

where $\chi$ is the local system on $C$ determined by $F$ via nonabelian Hodge theory. Full verification of the Hecke-eigensheaf property in singular, unstable, or noncompact cases remains ongoing, but the Floer-microsheaf construction establishes a conceptual and technical bridge between symplectic topology and geometric representation theory (Shende, 2021).