Rel--$C^\infty$ Structures on Gromov-Witten Moduli Spaces

TL;DR

This paper establishes canonical smooth structures on Gromov-Witten moduli spaces of pseudo-holomorphic curves, characterized by a universal property, using gluing analysis and polyfold theory.

Contribution

It introduces a universal characterization of smooth structures on Gromov-Witten moduli spaces, integrating gluing analysis with polyfold theory.

Findings

Canonical relative smooth structures are constructed on moduli spaces.

These structures are characterized by a universal property.

The approach combines gluing analysis with polyfold theory.

Abstract

We show that moduli spaces of transversely cut-out (perturbed) pseudo-holomorphic curves in an almost complex manifold carry canonical relative smooth structures ("relative to the moduli space of domain curves"). The main point is that these structures can be characterized by a universal property. The tools required are ordinary gluing analysis combined with some fundamental results from the polyfold theory of Hofer--Wysocki--Zehnder.