Yang-Mills theory as an Abelian theory without gauge fixing

A general procedure to reveal an Abelian structure of Yang-Mills theories by means of a (nonlocal) change of variables, rather than by gauge fixing, in the space of connections is proposed. The Abelian gauge group is isomorphic to the maximal Abelian subgroup of the Yang-Mills gauge group, but not its subgroup. A Maxwell field of the Abelian theory contains topological degrees of freedom of original Yang-Mills fields which generate monopole-like and flux-like defects upon an Abelian projection. ’t Hooft’s conjecture that “monopole” dynamics is projection independent is proved for a special class of Abelian projections. A partial duality and a dynamical regime in which the theory may have massive excitations being knot-like solitons are discussed. 1. General remarks. One of the physical scenarios of the color confinement is based on the idea that the vacuum state of quantum Yang-Mills theory is realized by a condensate of monopole-antimonopole pairs [1]. In such a vacuum the field between two colored sources would be squeezed into a tube whose energy is proportional to its length. The picture is dual to the magnetic monopole confinement in a superconductor of the second kind. Monopoles as classical solutions with finite energy are absent in a pure Yang-Mills theory. To realize the dual scenario of the confinement, ’t Hooft proposed an Abelian projection where the gauge group is broken by a suitable gauge condition to its maximal Abelian subgroup [2]. Since the topology of the SU(N) manifold and that of its maximal Abelian subgroup [U(1)] are different, any such gauge is singular, meaning that a gauge group element which transforms a generic SU(N) connection onto the gauge fixing surface is not regular everywhere in spacetime. The singularities may form worldlines that are usually interpreted as worldlines of magnetic monopoles (whose charges are defined with respect to the unbroken Abelian subgroup). As a result the original YangMills theory turns into electrodynamics with magnetic monopoles. Recent numerical simulations show that the monopole degrees of freedom in the Abelian projection can indeed form a condensate responsible for the confinement [3]. Although the numerical results look rather appealing and stimulating, they still do not provide us with an understanding of the confinement mechanism and a nonperturbative spectrum in Yang-Mills theory. In particular, monopoles seem to emerge as an artifact of gauge fixing. The Abelian group appears as a subgroup of the full Yang-Mills gauge group. However one can easily construct colored states which are singlets with respect to the unbroken (maximal) Abelian subgroup, and, hence, they would not be confined even if the “monopoles” condense. A choice of the gauge may be convenient in practical on leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia