Dynamical Abelian Projection of Gluodynamics 1

Assuming the monopole dominance, that has been proved in the lattice gluodynamics, to hold in the continuum limit, we develop an e ective scalar eld theory for QCD at large distances to describe con nement. The approach is based on a gauge (or projection) independent formulation of the monopole dominance and manifestly Lorentz invariant. 1. The monopole dominance Numerical simulations [1] show that there exist con gurations A of gauge elds which dominate in the path integral for the Wilson loop expectation value, i.e. they give a main contribution to the QCD string tension and, therefore, are the most relevant ones for the con nement in QCD. In fact, if one calculates the Wilson loop average over a subset formed by these speci c con gurations, the di erence between the string tension extracted from such average and the full QCD string tension (extracted from the Wilson loop average over all possible gauge eld con gurations) appears to be about eight per cent. When taken in a speci c gauge (A) = 0 that breaks the gauge group G to its maximal Abelian subgroup GH (e.g. SU(3) to U(1) U(1)), the dominant con gurations are Dirac magnetic monopoles with respect to the unbroken Abelian gauge group [2]. For this reason, the phenomenon is called the monopole dominance. It is should be noted that not in every Abelian projection , the dominant con gurations are monopoles [3]. If the dominant con gurations are selected as those that turn into magnetic monopoles when lifted by a gauge transformation on the surface (A) = 0, the monopole dominance may or may not occur, depending on the choice of . This has been indeed observed in the lattice simulations [4]. In fact, the monopole dominance has been found, up to now, only in the so called maximal Abelian projection. In the continuum limit, this gauge condition has the form D (A )A = @ A off + ig[A H ; A off ] = 0 ; (1) where A = A H + A off and A H;off are Abelian (diagonal) and o -diagonal components of A , respectively. So, not every Abelian projection can be used to select con gurations relevant for the QCD con nement. 1 A talk given at QCD96, Montpellier, France (July, 1996) 2 On leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia