Effective Abelian Gauge Theory Research Articles

M2-brane probe dynamics and toric duality

We study the dynamics of a single M2-brane probing toric Calabi–Yau four-fold singularity in the context of the recently proposed M-theory crystal model of AdS 4/CFT 3 dual pairs. We obtain an effective Abelian gauge theory in which the charges of the matter fields are given by the intersection numbers between loops and faces in the crystal. We argue that the probe theory captures certain aspects of the CFT 3 even though the true M2-brane CFT is unlikely to be a usual gauge theory. In particular, the moduli space of vacua of the gauge theory coincides precisely with the Calabi–Yau singularity. Toric duality, partial resolution, and a possibility of new RG flows are also discussed.

Abelian Projection of Massive SU(2) Yang–Mills Theory

We derive an effective Abelian gauge theory (EAGT) of a modified SU(2) Yang–Mills theory. The modification is made by explicitly introducing mass terms of the off-diagonal gluon fields into pure SU(2) Yang–Mills theory, in order that Abelian dominance at a long-distance scale is realized in the modified theory. In deriving the EAGT, the off-diagonal gluon fields involving longitudinal modes are treated as fields that produce quantum effects on the diagonal gluon field and other fields relevant at a long-distance scale. Unlike earlier papers, a necessary gauge fixing is carried out without spoiling the global SU(2) gauge symmetry. We show that the EAGT allows a composite of the Yukawa and the linear potentials which also occurs in an extended dual Abelian Higgs model. This composite potential is understood to be a static potential between color-electric charges. In addition, we point out that the EAGT involves the Skyrme–Faddeev model.

Renormalizable Abelian-Projected Effective Gauge Theory Derived from Quantum Chromodynamics

We show that an effective Abelian gauge theory can be obtained as a renormalizable theory from QCD in the maximal Abelian gauge. The derivation improves in a systematic manner the previous version that was obtained by one of the authors and was referred to as the Abelian-projected effective gauge theory. This result supports the view that we can construct an effective Abelian gauge theory from QCD without losing characteristic features of the original non-Abelian gauge theory. In fact, it is shown that the effective coupling constant in the resulting renormalizable theory has a renormalization-scale dependence governed by the $\beta$-function that is exactly the same as that of the original Yang-Mills theory, irrespective of the choice of gauge fixing parameters of the maximal Abelian gauge and the parameters used for identifying the dual variables. Moreover, we evaluate the anomalous dimensions of the fields and parameters in the resultant theory. By choosing the renormalized parameters appropriately, we can switch the theory into an electric or a magnetic theory.

Quark confinement and deconfinement in QCD from the viewpoint of Abelian-projected effective gauge theory

We give another derivation of quark confinement in QCD from the viewpoint of the low-energy effective Abelian gauge theory of QCD obtained via Abelian projection. It is based on the recently discovered Berezinskii-Kosterlitz-Thouless phase transition in the four-dimensional Abelian gauge theory. Moreover, we show that there exists a critical gauge coupling constant in QCD, above which confinement set in and below which there is no confinement. In a SU( N) gauge theory with N f flavored fermions, we argue that this leads to a critical value of fermion flavors N f c for the confinement as well as the chiral symmetry breaking, which separates the deconfining and chiral symmetric phase from the confining and chiral symmetry breaking phase. A finite-temperature deconfinement transition is also discussed briefly.

Abelian-projected effective gauge theory of QCD with asymptotic freedom and quark confinement

Starting from SU(2) Yang-Mills theory in 3+1 dimensions, we prove that the Abelian-projected effective gauge theories are written in terms of the maximal Abelian gauge field and the dual Abelian gauge field interacting with magnetic monopole current. This is performed by integrating out all the remaining non-Abelian gauge field belonging to SU(2)/U(1). We show that the resulting Abelian gauge theory recovers exactly the same one-loop beta function as the original Yang-Mills theory. Moreover, the dual Abelian gauge field becomes massive if the monopole condensation occurs. This result supports the dual superconductor scenario for quark confinement in QCD. We give a criterion of dual superconductivity and point out that the magnetic monopole condensation may be estimated from the classical instanton configuration. Therefore there can exist an effective Abelian gauge theory which shows both asymptotic freedom and quark confinement based on the dual Meissner mechanism. The inclusion of an arbitrary number of fermion flavors is straightforward in this approach. Some implications to the lower dimensional case will also be discussed.

DYNAMICAL GAUGE SYMMETRY BREAKING AND SUPERCONDUCTIVITY IN THREE-DIMENSIONAL SYSTEMS

We discuss dynamical breaking of non-Abelian gauge groups in three-dimensional (lattice) gauge systems via the formation of fermion condensates. A physically relevant example, motivated by condensed matter physics, is that of a fermionic gauge theory with group SU (2)⊗ U S(1)⊗ U E(1). In the strong U S(1) region, the SU(2) symmetry breaks down to a U(1), due to the formation of a parity-invariant fermion condensate. We conjecture a phase diagram for the theory involving a critical line, which separates the regions of broken SU(2) symmetry from those where the symmetry is restored. In the broken phase, the effective Abelian gauge theory is closely related to an earlier model of two-dimensional parity-invariant superconductivity in doped antiferromagnets. The superconductivity in the model occurs in the Kosterlitz–Thouless mode, since strong phase fluctuations prevent the existence of a local order parameter. Some physical consequences of the SU (2)× U S(1) phase diagram for the (doping-dependent) parameter space of this condensed matter model are briefly discussed.

Abelian-Projected Effective Gauge Theory of QCD with Asymptotic Freedom and Quark Confinement

Starting from SU(2) Yang-Mills theory in 3+1 dimensions, we prove that the Abelian-projected effective gauge theories are written in terms of the maximal Abelian gauge field and the dual Abelian gauge field interacting with magnetic monopole current. This is performed by integrating out all the remaining non-Abelian gauge field belonging to SU(2)/U(1). We show that the resulting Abelian gauge theory recovers exactly the same one-loop beta function as the original Yang-Mills theory. Moreover, the dual Abelian gauge field becomes massive if the monopole condensation occurs. This result supports the dual superconductor scenario for quark confinement in QCD. We give a criterion of dual superconductivity and point out that the magnetic monopole condensation may be estimated from the classical instanton configuration. Therefore there can exist an effective Abelian gauge theory which shows both asymptotic freedom and quark confinement based on the dual Meissner mechanism. The inclusion of an arbitrary number of fermion flavors is straightforward in this approach. Some implications to the lower dimensional case will also be discussed.

Physical basis of effective Abelian gauge theory of QCD

Motivated by recent works of 't Hooft, we propose the nonlinear macroscopic Maxwell-Lorentz system, including magnetic-monopole matter, as a simple phenomenological replacement for a non-Abelian gauge theory system. Electric and magnetic charges may be defined through the Abelian projection for a non-Abelian gauge theory. The phase of the system is dictated by certain characteristic behaviors of the field-strength-dependent dielectric function $\ensuremath{\epsilon}$ at sufficiently weak field strengths. Here, $\ensuremath{\epsilon}$ can be expressed as the product $\ensuremath{\epsilon}={\ensuremath{\epsilon}}_{e}{\ensuremath{\epsilon}}_{m}$, with ${\ensuremath{\epsilon}}_{e}$ resulting from redistribution of electrically charged objects and ${\ensuremath{\epsilon}}_{m}$ from redistribution of magnetic monopoles. Magnetic-monopole matters tend to antiscreen free electric charges, and we relate this to antiscreening of quark colorelectric charges at relatively large-distance scale. Based on these ideas, we provide a theoretical basis of the effective Abelian gauge theory of QCD.