[Paper Review] Abelian Projections and Monopoles

M. N. Chernodub, M.I. Polikarpov|ArXiv.org|Oct 27, 1997

Advanced Differential Equations and Dynamical Systems2 references17 citations

TL;DR

This paper reviews the abelian projection mechanism in lattice gauge theories, demonstrating how abelian monopoles emerge from non-abelian gluodynamics and contribute to confinement via a dual Meissner effect. Using numerical simulations and duality transformations, it shows that monopole condensation in SU(2) and SU(3) gauge theories reproduces flux tube formation between quark-antiquark pairs, supporting the dual superconductor picture of confinement.

ABSTRACT

The monopole confinement mechanism in the abelian projection of lattice gluodynamics is reviewed. The main topics are: the abelian projection on the lattice and in the continuum, a numerical study of the abelian monopoles in the lattice gauge theory. Additionally, we briefly review the notation of differential forms, duality, and the BKT transformation in the lattice gauge theories.

Motivation & Objective

• To establish the abelian projection as a framework for extracting abelian monopoles from non-abelian gauge fields in lattice QCD.
• To investigate whether abelian monopoles condense in the vacuum of SU(2) and SU(3) lattice gauge theories, supporting the dual Meissner effect mechanism.
• To connect the monopole confinement mechanism with duality and the BKT transformation in compact U(1) gauge theories.
• To provide numerical evidence for flux tube formation and monopole current winding in quark-antiquark systems.

Proposed method

• Apply the F12 abelian projection to SU(2) gauge theory, fixing the field strength tensor to be diagonal via gauge transformations.
• Construct abelian monopole currents from the non-abelian gauge fields using the field strength tensor and lattice differential forms.
• Use the BKT transformation to dualize the compact U(1) gauge theory, deriving a monopole action from the original gauge action.
• Perform numerical simulations on the lattice to compute the action density and monopole current distributions in the presence of static quark-antiquark sources.
• Apply the Hodge–de Rham decomposition to separate harmonic, exact, and coexact forms in the dual formulation.
• Use the Villain form of the action to derive a local monopole action, enabling numerical evaluation of monopole condensation.

Experimental results

Research questions

• RQ1 How can abelian monopoles be extracted from non-abelian SU(N) gauge fields using the abelian projection?
• RQ2 Does the vacuum of lattice SU(2) and SU(3) gauge theories exhibit properties of a dual superconductor, as indicated by monopole condensation?
• RQ3 What is the role of duality and the BKT transformation in connecting the original gauge theory to a monopole effective theory?
• RQ4 Can numerical simulations reproduce the flux tube structure between quark-antiquark pairs and show monopole current winding around them?
• RQ5 How does the monopole action derived via duality relate to the confinement mechanism in compact U(1) and non-abelian gauge theories?

Key findings

• Numerical simulations in SU(2) lattice gluodynamics show a clear flux tube structure in the action density between static quark-antiquark sources at β = 2.635.
• Monopole currents are observed to wind around the center of the flux tube, analogous to Cooper pairs in an Abrikosov vortex, indicating monopole condensation.
• The abelian projection successfully isolates abelian degrees of freedom and generates abelian monopoles from non-abelian fields in SU(2) and SU(3) gauge theories.
• The BKT transformation maps the compact U(1) gauge theory to a dual theory with a nonlocal monopole action, which becomes local in the Villain form.
• The monopole action in the Villain formulation is found to be $ S_{\text{mon}}(\ast j) = 4\pi^2\beta(\ast j, \Delta^{-1}\ast j) $, confirming a quadratic form consistent with condensation.
• The results support the dual Meissner effect as the mechanism for confinement, with monopole condensation forming a confining flux tube between quark sources.