Topological objects in QCD

Topological excitations are prominent candidates for explaining nonperturbative effects in QCD like confinement. In these lectures, I cover both formal treatments and applications of topological objects. The typical phenomena like BPS bounds, topology, the semiclassical approximation and chiral fermions are introduced by virtue of kinks. Then I proceed in higher dimensions with magnetic monopoles and instantons and special emphasis on calorons. Analytical aspects are discussed and an overview over models based on these objects as well as lattice results is given.     

Quantitative comparison of filtering methods in lattice QCD

We systematically compare filtering methods used to extract topological excitations (like instantons, calorons, monopoles and vortices) from lattice gauge configurations, namely APE-smearing and spectral decompositions based on lattice Dirac and Laplace operators. Each of these techniques introduces ambiguities, which can invalidate the interpretation of the results. We show, however, that all these methods, when handled with care, reveal very similar topological structures. Hence, these common structures are free of ambiguities and faithfully represent infrared degrees of freedom in the QCD vacuum. As an application we discuss an interesting power law for the clusters of filtered topological charge.     

Quantum chromodynamics

Quantum Chromodynamics (QCD) is a quantum field theory of the strong interaction with non-abelian gauge fields mediating the interactions between quarks. The experimentally observed strong interactions are to be epiphenomena of these fundamental interactions. The experimental successes of QCD form the basis for our present optimism that a theory of the strong force has been found. However, QCD is still very vulnerable to default on both experimental and theoretical grounds. In this article we offer the reader a review of the properties and attempted solutions of QCD. Each section of this review can be read independently of the others. In the introduction we describe the properties of QCD and the hoped for confinement and PCAC phase transitions. This is followed by a section on the renormalization of non-abelian gauge field theories: the functional methods and the path integral, the BPHZ program, the BRS transformation and proof of renormalizability, the Slavnov-Taylor identities and Schwinger-Dyson equations. The renormalization group equations are derived and applied to physical processes. Two dimensional prototypes of QCD, the abelian Schwinger model and 2-D QCD, are reviewed. An extensive review of the perturbative development of QCD is given with emphasis on infrared divergences, exponentiation of leading logarithms, the Cornwall-Tiktopoulos equation and a non-perturbative approach to QCD. A self contained section on topological solitons follows with discussions of homotopy theory, vortices, monopoles and especially instantons, and the periodic vacuum. Recent results, the attempt to study phase transitions in QCD using the dilute gas approximation and Borel resummations in QCD, are examined. Most of the major areas of interest in QCD are covered in this review; the prime exceptions are lattice gauge theories and phenomenological QCD like the parton and potential models.     

Thermo-field dynamics and quark–hadron phase transition in QCD

Based on the topological structure of gauge theory, an effective dual version of QCD has been reviewed and analyzed for the phase structure and color confining properties of QCD by invoking the dynamical magnetic symmetry breaking. The multi-flux-tube configuration of condensed QCD vacuum has been explored and associated glueball masses and inter-quark potential have been derived. Thermal response of QCD vacuum has been analyzed using path-integral formalism alongwith the mean-field approach and associated thermodynamical potential is used to derive thermal form of glueball masses, monopole condensate, inter-quark potential and monopole density which then lead to an estimate of the critical temperature of QCD phase transition. During its thermal evolution, a smooth transition of hadronic system via a weakly bound QGP phase to the fully deconfined phase is established and the thermal evolution profiles of various parameters are shown to indicate a second-order deconfinement phase transition and the restoration of magnetic symmetry. Monopole density calculations have been shown to lead to gradual evaporation of magnetic condensate into thermal monopoles during QCD phase transition.     

Kertész line and embedded monopoles in QCD

We propose a new class of defects in QCD which can be viewed as embedded monopoles made of quark and gluon fields. These objects are explicitly gauge invariant and they closely resemble the Nambu monopoles in the standard electroweak model. We argue that the "embedded QCD monopoles" are proliferating in the quark-gluon plasma phase while in the low-temperature hadronic phase the spatial proliferation of these objects is suppressed. At realistic quark masses and zero chemical potential the hadronic and quark-gluon phases are generally believed to be connected by a smooth crossover across which all thermodynamic quantities are nonsingular. We argue that these QCD phases are separated by a well-defined boundary-known as the Kertész line in condensed matter systems-associated with the onset of the proliferation of the embedded QCD monopoles in the quark-gluon plasma phase.     

Vortices, tunneling, and deconfinement in bilayer quantum Hall excitonic superfluid

The physics of vortices, instantons, and deconfinement is studied for layered superfluids in connection to bilayer quantum Hall systems at filling fraction nu = 1. We develop an effective gauge theory taking into account both vortices and instantons induced by interlayer tunneling. The renormalization group flow of the gauge charge and the instanton fugacity shows that the coupling of the gauge field to vortex matter produces a continuous transition between the confining phase of free instantons and condensed vortices and a deconfined gapless superfluid where magnetic charges are bound into dipoles. The interlayer tunneling conductance and the layer-imbalance induced inhomogeneous exciton condensate are discussed in connection to experiments.     

The “instanton liquid”

We review the theory of interacting topological fluctuations in the ground state of quantum gauge theories, the so called “instanton liquid”. First we outline some known phenomenological facts, both coming from “real” experiments (for QCD) and from the lattice data. Then we describe interaction of instantons and the statistical mechanics of their ensemble for theSU(2) gauge theory. The very essential role of the light quarks is considered using numerical experiments. One of the main conclusions is that instantons induce chiral symmetry breaking in vacuum, but as they are suppressed (e.g. by the nonzero temperature) this symmetry is restored. Phase transition is found to be very strong and of the first order.     

Dual Meissner effect and magnetic displacement currents

The dual Meissner effect is observed without monopoles in quenched SU(2) QCD with Landau gauge fixing. Magnetic displacement currents that are time-dependent Abelian magnetic fields act as solenoidal currents squeezing Abelian electric fields. Monopoles are not always necessary for the dual Meissner effect. A mean-field calculation suggests that the dual Meissner effect through the mass generation of the Abelian electric field is related to a gluon condensate A(a)(mu)A(a)(mu) not equal 0 of mass dimension 2.     

由正负磁单极对相互作用诱导的孤立狄拉克弦

基于三维旋量Gross-Pitaevskii(GP)方程研究在含时周期性外磁场作用下玻色-爱因斯坦凝聚体的动力学行为.结果显示,在含时周期外磁场的作用下,铁磁态自旋为1的玻色-爱因斯坦凝聚体将发生拓扑形变.当磁场的两个零点进入凝聚体后,自旋向上态的密度布居图在z轴上分别形成向上和向下的凸起.随着磁场的两个零点在凝聚体内逐渐重合,向上和向下的凸起被拉长,最终自旋向上态在z轴上呈线状分布,这与理论分析预测得到的孤立狄拉克弦相对应.最后,通过计算凝聚体的超流涡度给出磁单极的表征图.结果显示,凝聚体在磁场的两个零点处形成正、负磁单极对,分别对应着自旋向上态在z轴上向上和向下的凸起.随着磁场的两个零点重合,正、负磁单极对中的两条狄拉克弦逐渐靠近,之后大约经5 ms,它们完全相连,最终形成孤立的狄拉克弦.     

Various topological excitations in the SO (4) gauge field in higher dimensions

Following the original analysis of Zhang and Hu for the 4-dimensional generalization of Quantum Hall effect, there has been much work from different viewpoints on the higher dimensional condensed matter systems. In this paper, we discuss three kinds of topological excitations in the SO (4) gauge field of condensed matter systems in 4-dimension—the instantons and anti-instantons, the ’t Hooft-Polyakov monopoles, and the 2-membranes. Using the ?-mapping topological theory, it is revealed that there are 4-, 3-, and 2-dimensional topological currents inhering in the SO (4) gauge field, and the above three kinds of excitations can be directly and explicitly derived from these three kinds of currents, respectively. Moreover, it is shown that the topological charges of these excitations are characterized by the Hopf indices and Brouwer degrees of ?-mapping.     

Instantons in the nonperturbative QCD vacuum

The effect of nonperturbative fields on instantons in QCD is investigated. The nonperturbative vacuum is described in terms of nonlocal gauge-invariant vacuum expectation values of gluon fields. An effective action for instantons is obtained in the bilocal approximation, and it is shown that a stochastic background gluon field leads to an infrared stabilization of instantons. The dependence of a characteristic instanton size on the magnitude of the gluon condensate and on the correlation length in the nonperturbative vacuum is found. The size distribution of instantons that is obtained here is compared with the results of lattice calculations.     

A ring of instantons inducing a monopole loop

We consider the superposition of infinitely many instantons on a circle in . The construction yields a self-dual solution of the Yang-Mills equations with action density concentrated on the ring. We show that this configuration is reducible in which case magnetic charge can be defined in a gauge invariant way. Indeed, we find a unit charge monopole (worldline) on the ring. This is an analytic example of the correlation between monopoles and action/topological density, however with infinite action. We show that both the Maximal Abelian Gauge and the Laplacian Abelian Gauge detect the monopole, while the Polyakov gauge does not. We discuss the implications of this configuration.     

Instantons and monopoles in Yang-Mills gauge field theories

This is a survey article on instantons and monopoles and is intended for those who have no prior knowledge of Yang-Mills gauge field theories. With minimal amount of physical motivation and mathematical apparatus, the basic field equations and their solutions, wherever known, are presented. Particular emphasis is put on those problems which are as yet unsolved.     

Central dominance and the confinement mechanism in gluodynamics

New topological objects, which we call center monopoles, naturally arise in the Maximal Center Projection of SU(3) gluodynamics. The condensate of the center monopoles is the order parameter of the theory.     

Confining effective theories based on instantons and merons

An effective theory based on ensembles of either regular gauge instantons or merons is shown to produce confinement in SU(2) Yang-Mills theory. When the scale is set by the string tension, the action density, topological susceptibility and low-lying glueball spectrum are similar to those arising in lattice QCD. The physical mechanism producing confinement is explained, and a number of analytical insights into the effective theory are presented.     

Quark fragmentation in the θ vacuum

QCD vacuum is a superposition of degenerate states with different topological numbers that are connected by tunneling (the θ vacuum). The tunneling events are due to configurations of gauge fields (e.g., the instantons) that induce local P-odd domains in Minkowski space-time. We study the quark fragmentation in this topologically nontrivial QCD background. We find that even though QCD globally conserves P and CP symmetries, two new kinds of P-odd fragmentation functions emerge. We study their experimental manifestations in dihadron production in e(+)e(-) collisions, and find two interesting dihadron correlations: the cos(?(1)+?(2)) correlation usually referred to as the Collins effect, and a P-odd ～sin(?(10+?(2)) correlation that vanishes in the cross section summed over many events, but survives on the event-by-event basis.     

Monopoles and topological field theory

We present a topological quantum field theory for magnetic monopoles in an SU(N) Yang-Mills-Higgs model. This field theory is obtained by gauge fixing the topological action defining the monopole charge. This work extends to the three-dimensional case the quantization of invariant polynomials in four dimensions. We choose the Bogomolny self-duality equations as gauge conditions for the magnetic monopole topological field theory. In this way the geometrical equation discussed e.g. in Atiyah and Hitchin's work are recovered as ghost equations of motion. We give the cocycles of the corresponding topological symmetry. In the N→∞ limit interesting phenomena occur. The functional integration is forced to cover only the moduli space and the role of the ghosts stemming from the gauge fixing process is to provide a smooth semiclassical approximation.     

Supersymmetric instantons and topological quantum field theory

We present an action which generates the supersymmetric self-dual equations corresponding to euclidean super Yang-Mills theory in four dimensions. By adding additional constraint fields with new local symmetries, the classical equations of this system are the usual super self-dual equations when a gauge is chosen for the constraint fields. This construction is a supersymmetric generalization of the Labastida-Pernici action which corresponds to a gauge unfixed version of Witten's topological quantum field theory. We discuss some topological prospects for this model, and the role of supersymmetric instantons in Donaldson theory.     

Quark confinement in 2+1 dimensional pure Yang-Mills theory

We report some progress on the quark confinement problem in 2 + 1 dim. pure Yang-Mills theory, using Euclidean instanton methods. The instantons are regularized Wu-Yang ‘monopoles’, whose long range Coulomb field is screened by collective effects. Such configurations are stable to small perturbations unlike the case of singular, undressed monopoles. Using exact non-perturbative results for the 3-dim. Coulomb gas, where Debye screening holds for arbitrarily low temperatures, we show in a self-consistent way that a mass gap is dynamically generated in the gauge theory. The mass gap also determines the size of the monopoles. We also identify the disorder operator of the model in terms of the Sine-Gordon field of the Coulomb gas and hence obtain a dual representation whose symmetry is the centre ofSU(2).     

Monopoles in gluodynamics and blocking from continuum to lattice

We review the method of blocking of topological defects from continuum to lattice as a nonperturbative tool to construct effective actions for these defects. The actions are formulated in the continuum limit, while the couplings of these actions can be derived from simple observables calculated numerically on lattices with a finite lattice spacing. We demonstrate the success of the method in deriving the effective actions for Abelian monopoles in the pure SU(2) gauge models in an Abelian gauge. In particular, we discuss the gluodynamics in three and four spacetime dimensions at zero and nonzero temperatures. Besides the action, the quantities of our interest are the monopole density, the magnetic Debye mass, and the monopole condensate.