Sheaf Quantization of Hamiltonian Isotopies

• Sheaf quantization of Hamiltonian isotopies is a method that translates symplectic dynamics into derived category transformations using canonical sheaf kernels.
• The approach leverages microsupport and kernel convolution to map Hamiltonian flows into categorical auto-equivalences that mirror Lagrangian correspondences.
• Applications include proving non-displaceability, establishing Morse inequalities, and facilitating mirror symmetry through a unified microlocal sheaf framework.

Sheaf quantization of Hamiltonian isotopies is a geometric and microlocal approach that encodes Hamiltonian symplectomorphisms of cotangent bundles by means of functorial transformations in (derived) categories of sheaves. Originating with the work of Guillermou, Kashiwara, and Schapira (GKS), this formalism constructs, for every homogeneous Hamiltonian isotopy of $T^*M$, a canonical kernel in the derived category of sheaves on $M \times M \times I$, whose microsupport encodes the Lagrangian correspondence dictated by the Hamiltonian flow. This theory unifies symplectic topology, microlocal sheaf theory, and aspects of Floer theory, connecting symplectic invariants with sheaf-theoretic data and providing new tools to study questions of non-displaceability, Morse theory, and mirror symmetry.

1. Fundamental Structures: Sheaves, Microsupport, and Hamiltonian Isotopies

Let $M$ be a real (or complex) manifold and $K$ a field. The objects of interest are complexes of $K$-valued sheaves on $M$, particularly those that are constructible with respect to some stratification; these form the bounded derived category $D^b_c(M)$. For $F \in D^b(M)$, the micro- or singular support $SS(F) \subset T^*M$ is a closed conic coisotropic subset reflecting the non-propagation loci of sections of $F$, with $M \times M \times I$0 locally contained in a union of conormal bundles to the strata for constructible sheaves. The cotangent bundle $M \times M \times I$1 is equipped with the canonical symplectic form, with exact Lagrangian submanifolds $M \times M \times I$2 being those where the Liouville 1-form $M \times M \times I$3 restricts to $M \times M \times I$4 for some $M \times M \times I$5.

A Hamiltonian isotopy is a smooth one-parameter family $M \times M \times I$6 of compactly supported symplectomorphisms of $M \times M \times I$7, generated by a (possibly fiberwise homogeneous) Hamiltonian function $M \times M \times I$8. Each flow $M \times M \times I$9 preserves the symplectic structure and, when homogeneous, commutes with the natural $M$0-rescaling in fibers (Kuwagaki, 2022, Guillermou et al., 2010).

2. The GKS Sheaf Quantization Theorem

The core result is the GKS theorem: Given a homogeneous Hamiltonian isotopy $M$1 of $M$2, there exists a unique object (the “sheaf quantization kernel”)

$M$3

satisfying:

• $M$4, where $M$5 is the conic Lagrangian associated to the isotopy:

$M$6

• The restriction $M$7 is the constant sheaf on the diagonal $M$8.

For each $M$9, the slice $K$0 defines, via the functor

$K$1

an auto-equivalence of $K$2 that transports the microsupport via $K$3. Thus, objects microsupported on a conic Lagrangian $K$4 are sent to objects microsupported on $K$5 (Kuwagaki, 2022, Guillermou et al., 2010).

3. Construction of Sheaf Quantization Kernels

The construction proceeds via microlocal sheaf-gluing and the theory of kernel convolution. The local model is based on conormal deformation to the diagonal, guided by non-characteristic propagation techniques. The conification procedure lifts the Hamiltonian dynamics to $K$6, with the “conified” Hamiltonian

$K$7

yielding flows that preserve the $K$8 region. Tamarkin's category $K$9 provides the setting in which such (possibly non-conic) Lagrangians are analyzed as conic objects. The GKS kernel’s microsupport is then explicitly given by

$K$0

with convolution formulas ensuring compatibility with the symplectic composition of correspondences. The essential uniqueness of this kernel follows from abstract microlocal invertibility and homotopy-theoretic arguments (Kuwagaki, 2022, Guillermou et al., 2010).

4. Explicit Examples: Euclidean, Spherical, and Toric Cases

In $K$1 with $K$2, the sheaf quantization kernel $K$3 is the constant sheaf on $K$4, and its convolution induces symplectic translation on microsupports. On compact symmetric spaces, explicit GKS kernels have been constructed for normalized geodesic flows—for instance, on $K$5 and $K$6 the kernel is built from cones of inclusions of sublevel sets of distance functions, glued along nontrivial extension classes given by the topology of these spaces. The microsupport in each case traces the graph of the geodesic flow and ensures restriction to the diagonal at $K$7 (Arai, 27 Feb 2025).

In the context of mirror symmetry, contact isotopies corresponding to toric Cartier divisors are quantized by families of GKS kernels $K$8 on $K$9, converging (in the sense of nearby cycles) to a limit kernel $M$0 that realizes the mirror of the Picard-group action by convolution with the twisted polytope sheaf $M$1 (Bose et al., 8 May 2025).

5. Applications: Non-Displaceability, Morse Inequalities, Mirror Symmetry

Sheaf quantization provides a purely sheaf-theoretic framework for classical symplectic invariants. Non-displaceability results (such as Tamarkin’s theorem) are realized at the categorical level: if $M$2 in $M$3 satisfy $M$4, then no compactly supported Hamiltonian isotopy can separate their microsupports. For quantizations of exact Lagrangians $M$5, the Hom in the sheaf category matches the Floer cohomology $M$6 in the wrapped Fukaya category (Kuwagaki, 2022). Strong Morse inequalities for constructible sheaves are preserved under Hamiltonian isotopy, as the sheaf kernel auto-equivalences transport both Betti numbers and Morse-theoretic indices. These constructions also underpin sheaf-theoretic approaches to mirror symmetry, with sheaf kernels quantizing mirror group actions (e.g., via Picard group convolution in the toric setting) (Bose et al., 8 May 2025, Nadler et al., 2020).

6. Extensions: Weinstein Manifolds, Non-Smooth Dynamics, and Microlocal Categories

Sheaf quantization extends to Weinstein manifolds via microlocal sheaf-of-category theory. The category $M$7 associated to a Weinstein manifold $M$8 comprises sheaves on its contactization $M$9, microsupported in the skeleton $D^b_c(M)$0. Exact Lagrangians give objects in $D^b_c(M)$1 via microlocal specializations, and Hamiltonian isotopies induce auto-equivalences quantized by convolution with GKS kernels $D^b_c(M)$2 supported on the graph of the isotopy in microlocal phase space (Nadler et al., 2020).**

The theory also encompasses “Hamiltonian homeomorphisms,” establishing that the category $D^b_c(M)$3 of sheaves microsupported in $D^b_c(M)$4 is complete under the interleaving distance. Thus, limits of GKS kernels quantize continuous (not necessarily smooth) Hamiltonian flows, and Lusternik-Schnirelmann invariants can be transferred to the microlocal sheaf context (Asano et al., 2022). This allows for fixed-point and intersection theorems of Arnold type to be proved in entirely sheaf-theoretic settings.

7. Open Problems and Future Directions

Key directions for further research include:

• Sheaf quantization beyond compactly supported or exact Hamiltonian isotopies, incorporating non-exact Lagrangians (requiring the use of Novikov coefficients and $D^b_c(M)$5-equivariant sheaves), and treating time-dependent Hamiltonians with controlled asymptotic growth (Kuwagaki, 2022).
• Global sheaf quantization on Weinstein domains with stops, aiming to establish equivalence between (wrapped) Fukaya categories over the Novikov ring and microlocal sheaf categories, as envisioned in the Nadler–Zaslow–GPS program (Nadler et al., 2020).
• Systematic inclusion of coisotropic branes within microlocal sheaf frameworks, extending beyond existing Lagrangian-centric approaches.
• The development of analytic counterparts to sheaf quantization, such as conjectural $D^b_c(M)$6-Riemann-Hilbert correspondences linking deformation-quantization modules to GKS-type sheaves.

These developments underline the deep unification of microlocal analysis, categorical symplectic geometry, and topological field theories encoded in the sheaf quantization of Hamiltonian isotopies.

---

Key References: (Kuwagaki, 2022, Guillermou et al., 2010, Nadler et al., 2020, Arai, 27 Feb 2025, Bose et al., 8 May 2025, Asano et al., 2022).