it to remove some technical assumptions in their results. It is meant to be a service to the symplectic community.'
author:
- Dingyu Yang
date: 'October 23, 2015'
title: Virtual harmony
---

[^1]

Introduction
============

This article paints a grand unifying picture which illustrates in a precise way the compatibility among various virtual theories in the symplectic geometry literature. Namely, the structures featured in any two virtual theories included in the current discussion can be identified up to R-equivalences in a way that intertwines the perturbation mechanisms and algebraic constructions to extract invariants used in the respective virtual machineries. A couple of reasons for the existence of virtual harmony are (a) each approach is dealing with the same Fredholm problem (with varying domain and analytic limiting phenomena) and all the finite dimensional approaches at least in the differential geometric category are just shadows (outputs of forgetful functors constructed one way or another) of R-equivalent (sc-)Fredholm problems, and (b) in principle[^2] polyfold is a universal analytic framework that other analytic approaches factorize through. To reach wider readership while being directly useful, the article tries to strike a balance between having informal discussions of key ideas, reasons and/or summaries of each approach, 