Abelianization of Wilson loop of non-Abelian gauge theory on complex manifold and Polyakovʹs conjecture

A gauge theory of GL(2) gauge group is considered on a complex manifold and hermitian connections of holomorphic vector bundles of rank 2 over a compact complex manifold M are abelianized using monoidal transformation. The abelianized connections of the line bundle over a manifold M̃ as a P1 fibre over M are constructed explicitly. The Wilson loop of the GL(2, C) gauge theory is abelianized to that of C∗ gauge theory. By this the Polyakov conjecture is verified for a GL(2) gauge theory on a complex manifold.