Infinitesimal Gribov copies in gauge-fixed topological Yang-Mills theories

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

“…To prove that the algebraic renormalization is in harmony with the Feynman diagram analysis in the self-dual Landau gauges, which shows that the BS model does not receive radiative corrections in this gauge, we must invoke a result recently published in [28]. In this work, it was demonstrated that the Gribov ambiguities [59,60] are inoffensive in the self-dual BS theory 26 .…”

“…However, without recovering the discrete symmetries between the ghosts, such an imposition seems to be artificial. As we will see later, the renormalization ambiguity can be solved in the (A)SDL gauges, i.e., the discrete symmetries can be reconstructed, due to the triviality of the Gribov copies [28], which allows for a non-local transformation with trivial Jacobian, capable of recovering such symmetries.…”

“…In Landau gauge, this operator is Hermitian, and thus, its eigenvalues are real. For values of A µ sufficiently small, the eigenvalues of the FP operator will be positive, as −∂ 2 only has positive eigenvalues 28 . As A µ increases, it will attain a first zero mode (4.12).…”

“…In the self-dual topological BS theory, it was proved in [28] that all topological gauge copies associated to the gauge ambiguities (2.69) and (2.70), are eliminated through the introduction of the usual Gribov restriction, in which the path integral domain is restricted to the region Ω-see eq. (4.13).…”

“…Nevertheless, an entire algebraic proof was still lacking until now. It turns out that, for a complete proof of the vanishing of the β-function of the BS theory in the self-dual Landau gauges, one extra property must be considered: the fact that the Gribov copies are inoffensive to the self-dual BS theory [28]. This property establishes that the self-dual BS theory is conformal, as it allows to recover some discrete symmetries.…”