$L_\infty$-Kuranishi spaces and the moduli space of pseudoholomorphic maps

TL;DR

This paper introduces $L_{ abla}$-Kuranishi spaces, a new categorical framework for moduli spaces in symplectic geometry, utilizing $L_{ abla}[1]$-algebras and homotopy-theoretic compatibilities to generalize classical structures.

Contribution

It develops the theory of $L_{ abla}$-Kuranishi spaces, embedding smooth manifolds into this category and modifying existing notions to incorporate homotopy-theoretic flexibility.

Findings

Moduli space of pseudoholomorphic disks is modeled as an $L_{ abla}$-Kuranishi space.

A homotopy-theoretic model for $L_{ abla}[1]$-morphisms is proposed.

Forgetful and evaluation maps are lifted to morphisms between $L_{ abla}$-Kuranishi spaces.

Abstract

We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which naturally embeds the category of smooth manifolds. Certain notions in \cite{FOOO1} are modified to achieve desired categorical structures; for instance, the tangent bundle condition is interpreted as a quasi-isomorphism condition for the $L_{\infty}$-structures. In this process, the originally strict and rigid cocycle condition for coordinate changes is replaced by more flexible homotopy-theoretic compatibilities. To this end, a model of higher homotopy theory for $L_{\infty}[1]$-morphisms is proposed. Moreover, the moduli space of pseudoholomorphic disks with Lagrangian boundary condition is shown to serve as an example of $L_{\infty}$-Kuranishi spaces, provided that a Whitney stratification with a compatible system of tubular neighborhoods exists on each chart. Finally, the forgetful and evaluation maps for the moduli space are lifted to morphisms between $L_{\infty}$-Kuranishi spaces.