Abstract
Using a reformulation of topological mathcal{N} = 2 QFT’s in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a G2 manifold constructed from the space of self-dual 2-forms over a four-fold X, 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 is symplectic and the superconducting vortices are realized as pseudo-holomorphic curves. This setup is general enough to realize topological QFT’s arising from mathcal{N} = 2 QFT’s from all Gaiotto theories on arbitrary 4-manifolds.
Highlights
Using a reformulation of topological N = 2 QFT’s in M-theory setup, where QFT is realized via M5 branes wrapping co-associative cycles in a G2 manifold constructed from the space of self-dual 2-forms over a four-fold X, we show that superconducting vortices are mapped to M2 branes stretched between M5 branes
In this paper we have shown that embedding the N = 2 topological field theory on 4manifolds into M-theory can be helpful in shedding light on the connection of Taubes’ work and the Seiberg-Witten invariants, as inquired by Taubes at the end of ref
In particular we find that G2 geometry on the space of self-dual 2-forms over the 4-manifold X is necessary for this realization
Summary
Where W is a 7-fold of G2 holonomy with parallel 3-form Φ3 [20, 21], and H a hyperKahler 4fold; neither space is supposed to be compact or complete. For a generic metric on a compact 4-fold X with b+2 (X) ≥ 1, the zero set of a self-dual harmonic 2-form ω is a finite collection of non-intersecting codimension-3 circles αSα1 ⊂ X [35, 36], so that the intersection between the two M5’s takes the form. One shows that for a generic metric one can choose the self-dual harmonic form ω so that it has a single circle of zeros [37]. The symplectic geometry of the manifold Xwith boundary ∂Xcontains a new interesting class of pseudo-holomorphic curves Σ, namely the ones with boundaries on ∂Xwhich have finite area and satisfy some good boundary conditions [19, 39]. One may recover the Seiberg-Witten invariants by a suitable count of such curves [19]
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