Microsheaf Quantization Framework

• Microsheaf quantization is a framework that assigns constructible sheaves with prescribed microlocal support to Lagrangian or Legendrian submanifolds, enabling a topological approach to quantization.
• It leverages functorial compositions and sheaf kernels to mirror the geometric composition of Lagrangian correspondences, with real applications in Floer theory and mirror symmetry.
• The method establishes equivalences between wrapped Fukaya categories and microsheaf categories, providing new insights into symplectic geometry and representation theory.

Microsheaf quantization is a categorical framework associating constructible sheaves—with prescribed microlocal support—to Lagrangian or Legendrian submanifolds in symplectic and contact manifolds. It provides a robust, topological incarnation of quantization, interacting intricately with mirror symmetry, Floer theory, and the geometric Langlands program via the extrusion of local systems, functoriality under composition, and compatibility with physical and geometric correspondences. Central advances include the precise identification of wrapped Fukaya categories with microsheaf categories and the establishment of sheaf-theoretic quantization procedures for broad classes of Lagrangians, with extensions to exact WKB analysis and applications to symplectic and representation theory.

1. Microlocal Sheaf Categories and Quantization Objects

Microsheaf quantization begins with the construction of categories of sheaves microlocally supported on conic Lagrangian or Legendrian subsets in cotangent or contact manifolds. Given a manifold $M$ and a closed conic subset $\Lambda\subset T^*M$, the dg or $(\infty,1)$-category $$Sh_\Lambda(M) = \{ F\in Sh(M)\mid ss(F)\subset\Lambda\}$$ encodes sheaves whose microlocal singular support (in the sense of Kashiwara–Schapira) is contained in $\Lambda$ (Li et al., 25 Nov 2025). These categories localize to a sheaf of categories $\mu sh_M$ on $T^*M$.

For an exact symplectic manifold $(W,\lambda)$ with a Weinstein structure and a possibly immersed Legendrian $L\subset W\times\mathbb{R}$ that is eventually conic, one constructs the "core" $\mathfrak{c}_W^L$ and defines the category of conic microsheaf quantizations via global sections with support in $\mathfrak{c}_W^L$, denoted $\mu Sh_L(L) = \Gamma(_W)_{\mathfrak{c}_W^L}$ (Li et al., 25 Nov 2025). These quantizations satisfy natural functoriality and base-change properties, forming the backbone of the categorical approach to quantization.

In parallel, the sheaf-theoretic approach initiated by Tamarkin, Kashiwara, and Schapira constructs quantization objects $\mathcal{F}_\hbar$ for Lagrangian branes in $T^*M$. For a graded, relatively Pin Lagrangian $(L,\alpha,b)$, a simple sheaf quantization $\mathcal{F}_\hbar\in \mu_L(T^*M)$ is an object whose microlocalization yields a rank-one local system in degree zero, and its endomorphism ring is isomorphic to the Novikov ring $\Lambda_0$ (Kuwagaki, 2020).

2. Composition, Functoriality, and Gappedness

A crucial structural property of microsheaf quantization is the compatibility of sheaf-theoretic composition with the geometric composition of Lagrangian correspondences. For eventually conic embedded Legendrians $L_{12}\subset W_1^-\times W_2\times\mathbb{R}$ and $L_{23}\subset W_2^-\times W_3\times\mathbb{R}$, whose contact-reduction composition $$L_{13}=L_{23}\circ L_{12}\subset W_1^-\times W_3\times\mathbb{R}$$ is also embedded, one obtains

$$Q(L_{23}\circ L_{12})\simeq p_{13\,!}\left(p_{12}^!Q(L_{12})\otimes p_{23}^!Q(L_{23})\right)\in_{L_{13}}(L_{13})$$

and the operation is associative up to homotopy (Li et al., 25 Nov 2025). This formalism ensures the categorical parallel between symplectic composition of correspondences and convolution or integral transforms at the level of sheaves.

The gappedness criterion is introduced to guarantee the commutation of nearby-cycle functors $\psi$ with composition and $\otimes$, under the condition of no short Reeb chords between microsupports. For families of kernels with non-characteristic supports and uniformly positive gaps, this yields natural isomorphisms such as

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

and higher coherence for iterated compositions (Li et al., 25 Nov 2025).

3. Floer-Theoretic and Wrapped Fukaya Correspondence

A foundational result is the equivalence established between the wrapped Fukaya category $Ind\,Fuk(W,\omega)$ of an exact Weinstein manifold $W$ and the category of constructible microsheaves with microsupport in the skeleton: $Ind\,Fuk(W,\omega)\cong Sh_{skeleton}(W)$ (Shende, 2021). Exact or unobstructed Lagrangians $L\subset W$ are functorially assigned objects $\mu_L$ in the microsheaf category. This assignment preserves canonical structures: the Floer cohomology $HF^*(L,L)$ is isomorphic to $\mathrm{End}(\mu_L)$, and $$\mathrm{Hom}_{Fuk(W)}(L_1,L_2)\cong \mathrm{Hom}(\mu_{L_1},\mu_{L_2})$$, with orthogonality for disjoint fibers.

In the geometric context of the moduli of Higgs bundles, smooth non-stacky Hitchin fibers $L_b$ yield mutually orthogonal microsheaves on the global nilpotent cone, with endomorphism algebras computing the cohomology of $L_b$ (Shende, 2021). The construction leverages hyperkähler geometry and vanishing-of-disk theorems to ensure all Hitchin fibers are unobstructed Fukaya objects.

4. Sheaf Quantization of Spectral Curves and WKB Analysis

Sheaf quantization techniques have been fruitfully applied to spectral curves arising from $\hbar$-connections on Riemann surfaces. For Schrödinger-type equations

$$(\hbar\partial)^2-Q(z,\hbar)=0,\qquad Q(z,\hbar)=\sum_{k\geq0}Q_k(z)\,\hbar^k$$

under geometric conditions on the quadratic differential (complete GMN, absence of Stokes segments at a chosen phase, WKB-regularity), local systems are constructed on the spectral curve $L=\{(z,\xi)\in T^*C\,|\,\xi^2=Q_0(z)\}$. The sheaf quantization object $$S_\mathcal{M}^\hbar\in Sh^\mathbb{R}_{\tau>0}(C\times \mathbb{R}_t)$$ arises from gluing Borel-summed WKB solutions, with the resulting microlocalization/abelianization on $L$ reproducing exactly the Voros–Iwaki–Nakanishi cluster coordinates (Kuwagaki, 2020).

This quantization realizes an object-level $\hbar$-enhanced Riemann–Hilbert correspondence, connecting the topological sheaf category to classical and irregular holonomic $\mathcal{D}$-modules and to the Legendrian–Stokes knot theory.

5. Applications to Geometric Langlands and Representation Theory

Microsheaf quantization provides candidates for Hecke eigensheaves in the Betti and de Rham versions of the geometric Langlands program. The natural action of Hecke correspondences on $Sh_N(Bun_G(C))$ interacts compatibly with the spectral data, such that the images of $\mu_{L_b}$ under suitable restrictions satisfy the expected Hecke-eigenproperty, with eigenvalues determined by the corresponding spectral local systems (Shende, 2021). The family $\{\mu_{L_b}\}$ thus realizes the anticipated match between eigensheaves on the moduli of bundles and flat $G^\vee$-local systems on $C$.

Lie group actions, transfer functors for subdomains, and contact isotopy quantization (e.g., via GKS kernels) are all compatible with the microsheaf framework, establishing deep connections with symplectic representation theory and topological field theories (Li et al., 25 Nov 2025).

6. Technical Advances, Examples, and Future Directions

Significant technical achievements include the rigorous definition of sheaf kernels for Lagrangian correspondences and the proof of their compatibility under composition. The doubling construction (Guillermou–Nadler–Shende) translates problems for open Lagrangians into the context of closed, compactly microlocalized supports, enabling full microlocalization equivalences (Li et al., 25 Nov 2025). Applications range from the Viterbo restriction via graph kernels to the computation of intersection sheaves and Hom-pairings, providing explicit calculations of Floer complexes as Hom-complexes in microsheaf categories.

A plausible implication is the emergence of a unified language for topological and analytic quantization, harmonizing symplectic geometry, microlocal sheaf theory, and representation theory. Future extensions are anticipated in the categorical paper of Legendrian knots, mirror symmetry, and enhanced Riemann–Hilbert correspondences.