(G_2\) Holonomy, Taubes' Construction of Seiberg-Witten Invariants and Superconducting Vortices

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.