Virtual harmony

TL;DR

This paper reviews and connects various virtual techniques in symplectic geometry, aiming to unify different approaches to structures on moduli spaces and facilitate their application.

Contribution

It provides a comparative analysis of virtual structures across multiple theories, highlighting their compatibility and shared ideas to aid researchers in applying these methods.

Findings

Identifies pairwise correspondences between different virtual structures.

Demonstrates compatibility and unity among various virtual techniques.

Provides accessible explanations to encourage broader application.

Abstract

This article serves a few purposes. First of all, it reviews polyfold--Kuranishi correspondence I (http://arxiv.org/abs/1402.7008) and previews and samples some results from four papers I have been preparing. It is also a written-up and expanded version of a talk I gave at a symplectic conference in Chengdu on June 28, 2015, and it intends to provide bridges and compatibility between various pairs of virtual techniques and to demonstrate some unity among various technical viewpoints in the constructions of structures on moduli spaces in symplectic geometry. More precisely, the abstract perturbative structures (or interchangeably, virtual structures) present in each virtual theory discussed in this paper (and sometimes even the way they essentially originate in applications) are identified pairwise in a way that intertwines the (non-)perturbation mechanisms. To be more helpful to readers and not get them buried under technicalities and notations, we give the ideas and appropriate level of details so that the results will be clear to the relevant experts; meanwhile the ideas of each virtual machinery and how they are related should come through to more application-minded readers so that they might get encouraged to read papers on a given virtual machinery and possibly apply it to remove some technical assumptions in their results. It is meant to be a service to the symplectic community.