On an effective action of monopoles in Abelian projections of Yang-Mills theory

A path integral description of an effective action of monopoles in Abelian projections of Yang-Mills theories is discussed and used to establish a projection independence of the effective action. A dynamic regime in which the effective dynamics may contain massive solitonic excitations is described. Numerical simulations of lattice Yang-Mills theories show that certain topological defects, which occur upon a partial gauge fixing [1], play an important role in the infrared (nonperturbative) dynamics of Yang-Mills fields. A typical numerical experiment of this kind would involve the following steps. Given a set of Yang-Mills field configurations generated with the probability eW , where SW is the Wilson action, one performs a gauge transformation of each configuration from the Wilson ensemble so that after the gauge transformation the configuration satisfies a certain gauge condition that breaks the original gauge group to its maximal Abelian subgroup. The most popular gauge is the so called maximal Abelian gauge which is achieved by minimizing the L2 norm of non-Abelian components of Yang-Mills connections by means of suitable gauge transformations [2]. Clearly, this procedure would leave the maximal Abelian subgroup of the gauge group unbroken. After the gauge fixing (or the Abelian projection), the non-Abelian components are removed from every configuration and only the Abelian components of the gauge-fixed Wilson ensemble are used to compute expectation values of some gauge invariant operators like, for instance, the Wilson loop. An interesting feature of such a numerical experiment is that, despite a substantial reduction of the degrees of freedom (removing the non-Abelian components of connections), the string tension obtained from the Abelian (gauge-fixed) ensemble is 92% of the full string tension (computed in the original Wilson ensemble) [3]. This is known as the Abelian dominance. Since this phenomenon does not occur if non-Abelian components are removed before the gauge fixing, it is clear that some relevant degrees of freedom have been transferred from non-Abelian components to the Abelian ones upon the gauge transformation which has been used to impose the gauge in the Wilson ensemble. Hence, the effective Abelian theory cannot be a usual Maxwell theory. The topology of the gauge group and its maximal Abelian subgroup are different. Therefore any gauge fixing which breaks SU(N) to U(1) would have singularities. In particular, there are connections for which the gauge transformation that minimizes the L2 norm of their non-Abelian components cannot be regular everywhere in spacetime. It is easy to show that after the gauge transformation the Abelian components of such connections would contain Dirac monopoles as topological defects [1],[2]. Furthermore one can separate numerically monopole (singular) and photon (regular) parts of the Abelian on leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia.