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

TL;DR

This paper establishes a correspondence between twisted N=2 super-Yang-Mills theory and Baulieu-Singer topological theory, showing they share the same observables and are equivalent in certain gauges, with implications for understanding their ultraviolet behavior.

Contribution

It demonstrates the equivalence of twisted N=2 super-Yang-Mills and Baulieu-Singer theories in specific gauges and proves the vanishing of the beta function in the latter, linking their ultraviolet regimes.

Findings

Shared observables are the Donaldson invariants for 4-manifolds.

Triviality of Gribov copies in Baulieu-Singer theory aligns with instanton moduli space in twisted N=2.

Beta function vanishes in Baulieu-Singer theory, indicating conformal invariance.

Abstract

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.