Some memos on Stable Symplectic Structured Space
Eita Haibara, Shelly Miyano
TL;DR
The work builds a framework that fuses symplectic geometry with derived and higher categorical geometry by introducing a symplectic pregeometry T_S^{alg} and a corresponding geometry G_S^{alg}. It defines objects, morphisms, admissibility, and gluing via descent, culminating in the notion of G_S^{alg}-structured ∞-topoi and affine G_S^{alg}-schemes built from Thom spectra and E∞-ring sheaves. A symplectic ∞-category Perf(X) and its graded variants Symp_d(X) are constructed, with K-theory K(Symp_d(X)) capturing higher invariants and hinting at a Calabi-Yau-like structure S_symp ≅ [d]. The framework provides a bridge between algebraic invariants and symplectic/homological properties, enabling new avenues to study symplectic phenomena through higher category theory and derived geometry.
Abstract
In these memos, we define a pregeometry $\mathcal{T}_{\mathbb{S}} ^{alg}$ and a geometry $\mathcal{G}_{\mathbb{S}} ^{alg}$ which integrate symplectic manifolds with $E_{\infty}$-ring sheaves, enabling the construction of $\mathcal{G}_{\mathbb{S}} ^{alg}$-schemes as structured $\infty$-topoi. Our framework and results establish a profound connection between algebraic invariants and homological properties, opening new pathways for exploring symplectic phenomena through the lens of higher category theory and derived geometry.