Seiberg-Witten

It's an infrared effective theory of 4d $\mathcal{N}=2$ supersymmetric Yang-Mills theory. Its moduli space of vacua parametrizes elliptic curves, namely it's the fundamental domain of $SL(2,\mathbb{C})$; this group is the duality group here, with the modular parameter $\tau$ being the coupling. We obtain this by calculating the monodromy around the strong coupling singularities, where monopoles/dyons (not fundamental fields!) become massless.

After topological twist, we obtain a Witten type TQFT and can construct the SW invariant on the moduli space of monopole configuration connections. This invariant is actually correlator of 0-observables $Tr\varphi$ and classifies the smooth structure on 4d manifolds.

intersection theory