Abelianization of Wilson loop of non-abelian gauge theory on complex manifold and Polyakov's conjecture. (Q1809930)

The authors consider the gauge theory with the group GL\((2,\mathbb C)\) on a complex 2-dimensional manifold \(M\) which is a compactification of the Euclidean 4-space, associated with a rank two Hermitian holomorphic vector bundle \(\pi : E \rightarrow M\). Using the monoidal transform of \(E\) along the zero section, they construct a complex manifold \(\widetilde M\) which is a \(\mathbb C P^1\)-bundle over \(M\) together with a line bundle \(\widetilde{\pi} : \widetilde E \rightarrow \widetilde M\) such that \(\pi\) and \(\widetilde{\pi}\) have isomorphic spaces of sections. The Abelian gauge theory with the structure group \(\mathbb C^*\) of the bundle \(\widetilde{ \pi}\) is related with the non-Abelian gauge theory of the bundle \(\pi\) and it is called the abelianization. This construction is considered as a realization of the idea of abelianization, proposed by 't Hooft, who used it to explain the confinement in the quantum chromodynamics. The authors calculate the Wilson loop of abelianized gauge theory, starting from the Wilson loop of GL\((2,\mathbb{C})\) gauge theory. They show that their theory in the case of holomorphic gauge theory gives a verification of the Polyakov's conjecture that all gauge theories are equivalent to a certain nonstandard string theory.