Liouville Sectorial Techniques

• Liouville Sectorial Techniques are a framework that fuses geometric data of Liouville sectors with tailored Hamiltonians and almost complex structures to control holomorphic curve behavior.
• They employ maximum principles and confinement properties to ensure compactness and functoriality in Floer theory, enabling robust local-to-global and categorical descent computations.
• This approach underpins critical applications in wrapped Fukaya categories, homological mirror symmetry, and sectorial operator theory through explicit local geometric and analytic structures.

Liouville sectorial techniques form the foundational framework for modern local-to-global computations, categorical descent, and rigidity analyses in symplectic topology, particularly in the context of wrapped Fukaya categories, symplectic cohomology, and homological mirror symmetry. These techniques synthesize geometric input—namely the structure of Liouville sectors and sectorial Hamiltonian/complex data—with categorical localization, pushout, and descent tools. The theoretical apparatus centers on the notion of sectoriality (in both symplectic and analytic contexts) and its maximally compatible ambient Floer-theoretic and categorical structures.

1. Structure of Liouville Sectors and Sectorial Data

A Liouville sector $(M, \lambda)$ is an exact symplectic manifold with boundary composed of two types: a contact-type "ceiling" boundary $\partial_\infty M$ and a "flat-type" (possibly cornered) interior boundary $\partial M$. The Liouville vector field $Z=X_\lambda$ is required to be outward pointing along $\partial_\infty M$ and tangent to $\partial M$. Locally, $M$ is cut out inside a Liouville manifold by a defining function $I$ linear along $Z$, allowing a collar neighborhood $U \cong F \times \{\Re \geq 0\}$, and the Liouville form in this region is expressed as

$$\lambda|_U = \pi_F^* \lambda_F + \pi^* \lambda^\alpha + df, \quad \lambda^\alpha = (1-\alpha)x \, dy - \alpha y \, dx,\; 0 < \alpha \leq 1.$$

A sectorial hypersurface $H \subset M$ is defined by the existence of a "stop function" $I_H$ which is linear at infinity and satisfies controlled Poisson/foliation properties, ensuring $H$ is itself a Liouville sector and intersections occur via "sectorial corners" (Dai, 20 Nov 2025, Oh, 2021).

Sectorial Hamiltonians $H$ are constructed to be functions of an "exhaustion profile" $\psi$, i.e., $H = \rho(\psi)$ near the sectorial boundary, with $\rho' > 0$. Almost complex structures $J$ are constructed to be "sectorial" by solving duality conditions such as $-d\psi \circ J = \lambda_\kappa$ for a suitable cut-off $\lambda_\kappa$, ensuring compatibility with $\lambda$ near all sectorial faces.

2. Maximum Principle and Confinement Properties

A central advance is the identification of sectorial pairs $(H,J)$ for which the analysis of pseudoholomorphic curves is uniformly governed by a maximum principle. Introducing a function $\psi$ with $J$-convexity (i.e., $-d(d\psi \circ J) \geq 0$) leads to pseudoconvex pairs. For $J$–holomorphic curves $u:\Sigma \to M$ solving $\bar{\partial}_J u = 0$, the potential $\psi_k \circ u$ is subharmonic:

$$\Delta(\psi_k \circ u) = -d(d\psi_k \circ J)(du, J du) = |du|^2 \geq 0.$$

This establishes automatic $C^0$ confinement for Floer trajectories: images remain bounded in terms of the levels of $\psi_k$ at boundary punctures, with no need for dissipativity or delicate estimates (Oh, 2021).

For the perturbed Floer equation

$$(du - X_H \otimes dt)^{0,1}_J = 0,\quad H = \rho(\psi_k),$$

the same framework yields:

$$\Delta(\psi_k \circ u) = |\partial_s u - X_H|^2 - \rho'(\psi_k) \partial_s (\psi_k \circ u).$$

These maximum- and strong maximum principles underlie compactness, transversality, and gluing results for all relevant moduli spaces in wrapped Fukaya theory and symplectic cohomology.

3. Sectorial Decomposition and Descent in Symplectic Topology

Sectorial decomposition is a covering of a Liouville manifold by Liouville sectors intersecting in sectorial corners, each induced by sectorial hypersurfaces. For a punctured surface $\Sigma$, a collection of properly embedded arcs $\{\gamma_i\}$ yields sectors $\Sigma_j$ by decomposing along their ascending and descending (stable and unstable) manifolds.

These decompositions are functorial. For example, a sectorial decomposition of $\Sigma$ induces a sectorial decomposition on the symmetric product $\operatorname{Sym}^2(\Sigma)$, with the sectors labeled by pairs of minima and the boundary faces by prescribed "stop hypersurfaces". The result is a refined Mayer–Vietoris formalism at the categorical level, enabling pushout and descent calculations for wrapped Fukaya categories (Dai, 20 Nov 2025, Ganatra et al., 2018).

The key categorical tool is the homotopy colimit

$$\mathcal{W}(X) \simeq \mathrm{colim}\left(\cdots \to \mathcal{W}(X_\alpha \cap X_\beta) \to \mathcal{W}(X_\alpha)\oplus \mathcal{W}(X_\beta) \to \cdots \right),$$

allowing reconstruction of $\mathcal{W}(X)$ from local data. This underpins proofs in homological mirror symmetry for spaces such as the pair of pants (Dai, 20 Nov 2025).

4. Infinity-Categorical Formalism and Monoidal Coherence

Lazarev–Sylvan–Tanaka established that the homotopical category of stabilized Liouville sectors and strict sectorial embeddings, after inverting sectorial equivalences, acquires a natural $\infty$-category structure

$\lioulocal \simeq \lioustrstab[W^{-1}],$

with objects products of Liouville sectors and cotangent sectors $T^*[0,1]^k$, and morphisms strict sectorial embeddings up to isotopy. The $\infty$-category admits a symmetric monoidal structure given by product

$(M, k) \otimes (N, \ell) = (M \times N, k + \ell).$

Any functor from Liouville sectors to an $\infty$-category which is covariantly functorial under strict embeddings, stabilizes under $T^*[0,1]$-product, and inverts sectorial equivalences, extends (uniquely up to contractible choice) to a symmetric monoidal functor out of $\lioulocal^\otimes$. This conceptual advance creates a universal setting for ensuring continuous and multiplicative coherence of symplectic invariants built from sectorial data (Lazarev et al., 2021).

5. Categorical Applications: Wrapped Fukaya Categories and Gluing

Wrapped Fukaya categories $\mathcal{W}(X)$ constitute $A_\infty$-categories attached to Liouville sectors $X$. Liouville sectorial techniques guarantee the following:

• Descent Property: For a sectorial (or Weinstein sectorial) cover $\{X_i\}$,

$$\mathcal{W}(X) \simeq \mathrm{hocolim}_{I \subset \{1, \ldots, n\}} \mathcal{W}(U_I),\quad U_I = \bigcap_{i \in I} X_i.$$

The involved pushforward functors are fully faithful and the image generates, under sectorial and Weinstein assumptions (Ganatra et al., 2018).

• Künneth Formula and Generation: The split wrapped Fukaya category on a product of sectors $X \times Y$ is generated by perturbed products of generators of the $X$ and $Y$ categories, and the Künneth functor is a pre-triangulated equivalence under Weinstein hypotheses (Ganatra et al., 2018).
• Stop Removal and Localization: Enlarging a stop corresponds categorically to localization at a collection of Lagrangian linking disks.
• Homological Mirror Symmetry (HMS): For the complex two-dimensional pair of pants, the sectorial decomposition yields a categorical equivalence between the wrapped Fukaya category of $\operatorname{Sym}^2(\Sigma)$ and the derived category of coherent sheaves on $\{xyz=0\} \subset \mathbb{C}^3$ (Dai, 20 Nov 2025).

These structures rely on sectorial control of holomorphic curve confinement, the strong maximum principle, and functoriality under inclusions and sectorial covers.

6. Analytic Sectoriality: Operators and Fractional Calculus

Sectoriality arises analytically in the paper of densely defined operators $A$ on Hilbert spaces $\mathcal{H}$ with spectrum in a sector of the complex plane. For sectorial angle $\theta$ and vertex $t$,

$$W(A) \subset t + \{\lambda \in \mathbb{C}: |\arg(\lambda - t)| \leq \theta\}.$$

Fractional powers of $A$ (in particular, Liouville fractional integrals and derivatives) can be defined via Dunford-Taylor contour integrals

$$A^\alpha = \frac{1}{2\pi i} \int_{\Gamma} z^\alpha (z - A)^{-1} dz,$$

for suitable contours $\Gamma$. In the semigroup model, these yield expressions for fractional derivatives governed by sectorial calculus.

Spectral theory for such sectorial operators leverages Liouville-sectorial methods—e.g., the “lacunae” technique and power-type contour integrals—to provide convergence and basis results (Abel–Lidskii property), enabling explicit solutions of fractional and integer evolution equations in abstract settings (Kukushkin, 11 Jan 2024).

7. Synthesis and Significance

Liouville sectorial techniques provide a unifying formalism for the control and calculation of symplectic invariants via Floer-theoretic, categorical, and analytic means. The introduction of sectorial data for Hamiltonians and almost complex structures removes the necessity of ad hoc estimates for compactness and confinement, enabling robust, functorial local-to-global constructions in wrapped Fukaya theory, symplectic cohomology, and HMS. The explicit local geometric structure (collar neighborhoods, sectorial corners, stop functions) is reflected in the behavior of moduli spaces, sheaf-theoretic invariants, and operator-theoretic methods for both geometric and analytic problems (Oh, 2021, Dai, 20 Nov 2025, Ganatra et al., 2018, Lazarev et al., 2021, Kukushkin, 11 Jan 2024). By encoding geometric sectoriality into both the analysis of holomorphic curves and the structure of categories and functors, sectorial techniques furnish the foundation for categorical descent, pushout formulas, and the invariance properties necessary for coherent functoriality and mirror symmetry.