Localization and flexibilization in symplectic geometry

TL;DR

The paper develops a symplectic localization framework by constructing the critical Weinstein ∞-category $ ext{Wein}_{ extsf{crit}}^{ullet}$ and introducing $P$-flexibilization endofunctors that invert a finite set of integers in the wrapped Fukaya category. It proves that $X[P^{-1}]$ is an idempotent localization, functorial and symmetric monoidal, and shows two geometrically distinct models—carving Lagrangian disks and forming products with $T^*D^n[P^{-1}]$—are equivalent in the critical category. This equivalence yields a robust localization mechanism compatible with products, stops, and subcritical moves, and it endows $T^*D^n[P^{-1}]$ with an $E_ leftarrow ext{∞}$-algebra structure within the localized category. The results generalize classical flexibilization, relate to Abouzaid–Seidel and Lazarev–Sylvan constructions, and open a pathway to higher algebraic structures arising from symplectic geometry. The framework extends to arbitrary finite CW complexes via regular Lagrangian disks, enabling a universal, homotopy-invariant approach to symplectic localization with potential applications to spectral wrapped Fukaya categories and localization in stable homotopy theory.

Abstract

We introduce the critical Weinstein infinity-category -- the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms -- and for every finite collection P of integers, construct a P-flexibilization endofunctor. Our main result is that P-flexibilization is an idempotent localization functor of the critical Weinstein infinity-category, allowing us to characterize the essential image of the endofunctor by a universal property. This localization has the effect of replacing every Weinstein sector with one in which P is invertible in the wrapped Fukaya category and hence is a symplectic analogue of topological localization of Bousfield and Sullivan, answering a question of Abouzaid and Seidel. When P = {0}, our construction recovers Cieliebak and Eliashberg's flexibilization procedure. Moreover, we show that P-flexibilization is symmetric monoidal as a functor of higher categories, and hence gives rise to a new way of constructing E-infinity-commutative algebra objects from symplectic geometry.

Figures (9)

• Figure 1: Two Morse-Bott Weinstein structures on $T^*D^n$, depicted via their corresponding vector fields on $D^n$; the zero locus of the vector fields is in red. The left figure has sectorial boundary $T^*S^{n-1}$, equipped with a Morse-Bott Weinstein structure having critical locus $S^{n-1}$. The right figure has the same sectorial boundary $T^*S^{n-1}$, but for which the boundary is equipped with a Morse Weinstein structure consisting of two points.
• Figure 2: Converting a stopped domain, with stop in blue, to a sector, with additional critical points in red.
• Figure 3: Homotoping an arbitrary sector to a (sector induced by a) stopped domain, with stop in blue and additional critical point in red.
• Figure 4: Bordered homotopy at time $t$ from Part 1) (top figure) and Part 2) (bottom figure) of Proposition \ref{['prop: bordered_to_interior_homotopy_Weinstein']}. For Part 1), the index $n$ co-cores are $C_{F,t} \times T^*_{-3/4} [-1,0]$, where $C_{F,t}$ are the co-cores of $(F, \lambda_{F,t})$. For Part 2), the index $n$ co-cores are $C_{F,0} \times T^*_{-3/4} [-1,0]$.
• Figure 5: Bordered homotopy at time $t$ from Part 3) of Proposition \ref{['prop: bordered_to_interior_homotopy_Weinstein']}, before cancellation of critical points and after cancellation. The stopped domain $Y_0$ is modified to the stopped domain $Y_0'$, which has the same interior critical points.

Theorems & Definitions (169)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3: A special case of Theorem \ref{['thm: comparison']}
• Definition 1.4
• Corollary 1.5: Porism
• Corollary 1.6
• Theorem 1.7
• Remark 1.8
• Theorem 1.9
• Corollary 1.10