This is the sequel to [3] which gives an extension of Taubes' "SW = Gr" theorem to nonsymplectic 4-manifolds. The main result of this paper asserts the following. Whenever the Seiberg-Witten invariants are defined over a closed minimal 4-manifold X, they are equivalent modulo 2 to "near-symplectic" Gromov invariants in the presence of certain self-dual harmonic 2-forms on X. A version for non-minimal 4-manifolds is also proved. A corollary to S 1 -valued Morse theory on 3-manifolds is also announced, recovering a result of Hutchings-Lee-Turaev about the 3-dimensional Seiberg-Witten invariants.

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 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 N = 2 QFT's from all Gaiotto theories on arbitrary 4-manifolds.

We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with deformation index 0. A corollary is that the Gromov‐Witten invariants (without descendants) of symplectic 4‐manifolds can always be computed as a signed and weighted count of honest J‐holomorphic curves for generic tame J: in particular, each such invariant is an integer divided by a weighting factor that depends only on the divisibility of the corresponding homology class. The transversality proof is based on an analytic perturbation technique, originally due to Taubes.© 2016 Wiley Periodicals, Inc.

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