# When can Witten-esque moduli spaces be used to define invariants of geometric structures?

I am trying to understand the big picture around Seiberg-Witten invariants of 4-manifolds. Of course, this points to Taubes work on Gromov-Witten invariants of symplectic manifolds.

It is striking to me that both developments arise, essentially, by studying spaces of functions that satisfy non-linear elliptic equations. For Seiberg-Witten, one studies moduli spaces of sections of bundles over a 4-manifold modulo a gauge group action. In the case of Gromov-Witten, one analyzes spaces of special maps from surfaces to symplectic manifolds.

Perhaps someone has cataloged or systematically studied which elliptic moduli spaces produce "useful" invariants of the underlying geometric structures. Is it a work of genius ala Gromov and Witten to suggest these equations or do we have some methodical process whereby we can define (or guess) such a useful equation and resulting moduli space? For example, might there be a Witten-like equation which helps elucidate contact structures on 7-manifolds, etc?

So my question is, what other Witten-like moduli spaces can be used to define invariants of geometric structures?

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I think, you forget to mention Donaldson, the person who actually made this amazing discovery : "one should study spaces of functions that satisfy non-linear elliptic equations". You might be interested to see what Gromov writes on this topic in the last paragraph of "Soft and hard", ihes.fr/~gromov/PDF/soft_and_hard.pdf . For other theories of this type the must read is Gauge theory in higher dimesnions by Donaldson and Thomas www2.imperial.ac.uk/~rpwt/skd.pdf –  Dmitri Dec 21 '10 at 18:06