Correspondence between the twisted $N = 2$ super-Yang-Mills and conformal Baulieu-Singer theories

We characterize the correspondence between the twisted $N=2$ super-Yang-Mills theory and the Baulieu-Singer topological theory quantized in the self-dual Landau gauges. While the first is based on an on-shell supersymmetry, the second is based on an off-shell Becchi-Rouet-Stora-Tyutin symmetry. Because of the equivariant cohomology, the twisted $N=2$ in the ultraviolet regime and Baulieu-Singer theories share the same observables, the Donaldson invariants for 4-manifolds. The triviality of the Gribov copies in the Baulieu-Singer theory in these gauges shows that working in the instanton moduli space on the twisted $N=2$ side is equivalent to working in the self-dual gauges on the Baulieu-Singer one. After proving the vanishing of the $\beta$ function in the Baulieu-Singer theory, we conclude that the twisted $N=2$ in the ultraviolet regime, in any Riemannian manifold, is correspondent to the Baulieu-Singer theory in the self-dual Landau gauges -- a conformal gauge theory defined in Euclidean flat space.