Real Ruan-Tian Perturbations

TL;DR

This paper introduces a new type of Ruan-Tian perturbation compatible with real Gromov-Witten invariants, simplifying the deformation-obstruction framework for symplectic geometry in arbitrary genera.

Contribution

It develops an analogue of Ruan-Tian deformations tailored for real Gromov-Witten invariants, avoiding embeddings into smooth manifolds and streamlining the construction process.

Findings

Provides a deformation framework compatible with real invariants

Eliminates the need for universal curve embeddings

Systematizes the deformation-obstruction approach

Abstract

Ruan-Tian deformations of the Cauchy-Riemann operator enable a geometric definition of (standard) Gromov-Witten invariants of semi-positive symplectic manifolds in arbitrary genera. We describe an analogue of these deformations compatible with our recent construction of real Gromov-Witten invariants in arbitrary genera. Our approach avoids the need for an embedding of the universal curve into a smooth manifold and systematizes the deformation-obstruction setup behind constructions of Gromov-Witten invariants.