Topological objects in QCD

TL;DR

This work surveys topological excitations in QCD as candidates for nonperturbative phenomena such as confinement and chiral symmetry breaking. It develops core tools (Bogomolnyi bounds, BPS tricks, Nahm and ADHM formalisms) and applies them to kinks, monopoles, instantons, and calorons, highlighting their moduli, zero modes, and the relation to spectral flow and anomalies. The document also connects these continuum constructions to semiclassical QCD models (instantons, calorons) and to lattice studies via cooling and fermionic techniques, arguing that monopoles and calorons especially those with nontrivial holonomy  play key roles in confinement and the QCD vacuum structure. Altogether, it outlines a coherent framework in which topological objects underlie both the qualitative and quantitative features of nonperturbative QCD with implications for effective models and lattice phenomenology.

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.

Figures (17)

• Figure 1: Action density, Polyakov loop and zero mode profile of an $SU(2)$ configuration on an asymmetric lattice after long over-improved cooling in some lattice plane, from bruckmann:04b, cf. Sect. \ref{['sect_cooling']} and Fig. \ref{['fig_cal_higher']}.
• Figure 2: Left: the mexican hat potential. Right: In the particle picture the potential is inverted. The soliton 'rolling' from the hill at $-v$ in the infinite past to the hill at $v$ in the infinite future is depicted.
• Figure 3: The kink solution (left, fat curve, together with an antikink) and its energy density (right).
• Figure 4: A multi-soliton consisting of two kinks and two antikinks.
• Figure 5: The hedgehog as a prototype of a mapping $S^2\to S^2$ with winding number 1.