Intersection theory of coassociative submanifolds in G_(2)-manifolds and Seiberg-Witten invariants

TL;DR

This paper explores the counting of instantons with coassociative boundary conditions in G2-manifolds, linking it to Seiberg-Witten invariants, and draws analogies to open Gromov-Witten theory in Calabi-Yau manifolds.

Contribution

It introduces a new approach to counting instantons in G2-manifolds and relates it to Seiberg-Witten invariants, expanding the understanding of coassociative submanifolds.

Findings

Establishes a relationship between instanton counts and Seiberg-Witten invariants.

Provides a framework analogous to open Gromov-Witten theory for G2-manifolds.

Suggests new invariants for coassociative submanifolds in G2-geometry.

Abstract

We study the problem of counting instantons with coassociative boundary condition in (almost) G_(2)-manifolds. This is analog to the open Gromov-Witten theory for counting holomorphic curves with Lagrangian boundary condition in Calabi-Yau manifolds. We explain its relationship with the Seiberg-Witten invariants for coassociative submanifolds.