$G_2$ Holonomy, Taubes' Construction of Seiberg-Witten Invariants and Superconducting Vortices

TL;DR

This paper links superconducting vortices in certain topological quantum field theories to M2 branes in M-theory, providing a physical perspective on Taubes' Seiberg-Witten invariants and extending to various Gaiotto theories.

Contribution

It offers a novel M-theory framework connecting vortices, Seiberg-Witten invariants, and topological QFTs from ${ m N}=2$ theories on arbitrary 4-manifolds.

Findings

Superconducting vortices correspond to M2 branes between M5 branes.

The setup explains Taubes' construction of Seiberg-Witten invariants.

Framework applies to all Gaiotto theories on 4-manifolds.

Abstract

Using a reformulation of topological ${\cal N}=2$ QFT's in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a $G_2$ manifold constructed from the space of self-dual 2-forms over $X^4$, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes. This setup provides a physical explanation of Taubes' construction of the Seiberg-Witten invariants when $X^4$ is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT's arising from ${\cal N}=2$ QFT's from all Gaiotto theories on arbitrary 4-manifolds.