Topological objects in QCD

Abstract

Abstract. 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. AppetiserWhen a lattice gauge configuration is subject to cooling, the result can be as depicted in Fig. 1. Without going into details here, this figure reveals typical aspects of a soliton stabilised by topology. In this lecture I will introduce definitions and properties of these beautiful objects and discuss their relevance for particle physics. Before I come to gauge objects like monopoles and instantons, I will demonstrate the main features by virtue of an example in a scalar theory. The kinkAs the first model let us take one of the simplest quantum field theoretical systems, a real scalar field φ in a 1+1 dimensional Minkowski space with metric η µν = diag(+1, −1). The Lagrangian,shall contain a potential V (φ) that has several minima of same height, set to V = 0. For definiteness I will choose the famous mexican hat potential ,a

Fermionic Boundary Conditions and the Finite Temperature Transition of QCD

“…Dyons (see, e.g., [25]) are natural candidates as their index theorem [26] is sensitive to the boundary conditions: For dyons of given electric and magnetic charge the zero modes exist only in a certain range of boundary angles. These zero modes then mix and form a near-zero mode band.…”

Calorons with non-trivial holonomy

“…There are more recent lectures by Bruckmann [13] and Diakonov [14]. Also, Diakonov and Petrov made some progress on constructing the hyperKähler metric which approximates the metric for an arbitrary number of calorons.…”

Vacuum structure and string tension in Yang-Mills dimeron ensembles

“…Hence, we average over dimeron configurations chosen randomly from a Gibbs distribution with Boltzmann factor expðÀSÞ where S is the Yang-Mills action (15). More specifically, we use the Metropolis algorithm to generate homogeneous Markov sequences of dimeron configurations that visit fields with larger probabilities more often.…”

“…Among the maximally, i.e., in all four (Euclidean) spacetime dimensions, localized candidates for the building blocks of confining field configurations, finally, are regular-gauge instantons [14], calorons with nontrivial [13,[15][16][17] holonomy and merons [14,18,19]. All of these ''pseudoparticles'' carry a nontrivial topological charge, mediate vacuum tunneling events and solve the classical Yang-Mills equation [20].…”