Several related aspects of the 't Hooft vortex operator are studied. The current picture of the vacuum of quantum chromodynamics, the idea of dual field theories, and the idea of the vortex operator are reviewed first. The Abelian vortex operator written in terms of elementary fields and the calculation of its Green's functions are considered. A two-dimensional solvable model of a Dirac string is presented. The expression of the Green's functions more neatly in terms of Wu and Yang's geometrical idea of sections is addressed. The renormalization of the Green's functions of two kinds of Abelian looplike operators, the Wilson loop and the vortex operator, is studied; for both operators only an overall multiplicative renormalization is needed. In the case of the vortex this involves a surprising cancellation. Next, the dependence of the Green's functions of the Wilson and 't Hooft operators on the nature of the vacuum is discussed. The cluster properties of the Green's functions are emphasized. It is seen that the vortex operator in a massive Abelian theory always has surface-like clustering. The form of Green's functions in terms of Feynman graphs is the same in Higgs and symmetric phases; the difference appears in the sum over all tadpole trees. Finally, systems having fields in the fundamental representation are considered. When these fields enter only weakly into the dynamics, a vortex-like operator is anticipated. Any such operator can no longer be local looplike, but must have commutators at long range. A U(1) lattice gauge theory with two matter fields, one singly charged (fundamental) and one doubly charged (adjoint), is examined. When the fundamental field is weakly coupled, the expected phase transitions are found. When it is strongly coupled, the operator still appears to be a good order parameter, a discontinuous change in its behavior leads to a new phase transition. 18 figures.

Four forces are dominant in physics: gravity, electromagnetism and the weak and strong nuclear forces. Quantum electrodynamics - the highly successful theory of the electromagnetic interaction - is a gauge field theory. In this short book Dr Aitchison gives an introduction to these theories, a knowledge of which is essential in understanding modern particle physics.

This monograph discusses specific examples of selfdual gauge field structures, including the Chern–Simons model, the abelian–Higgs model, and Yang–Mills gauge field theory. The author builds a foundation for gauge theory and selfdual vortices by introducing the basic mathematical language of gauge theory and formulating examples of Chern–Simons–Higgs theories (in both abelian and non-abelian settings). Thereafter, the Electroweak theory and self-gravitating Electroweak strings are examined. The final chapters treat elliptic problems involving Chern–Simmons models, concentration-compactness principles, and Maxwell–Chern–Simons vortices.

Examination of appropriate formulation of quantum gauge invariance covers free fields, causal perturbation theory, spin-1 gauge theories involving both massless and massive gauge fields, spin-2 gauge theories, and non-geometric general relativity. 2001 edition.

We study the Euclidean Green's functions of the 't Hooft vortex operator, primarily for Abelian gauge theories. The operator is written in terms of elementary fields, with emphasis on a form in which it appears as the exponential of a surface integral, We explore the requirement that the Green's functions depend only on the boundary of this surface, The Dirac veto problem appears in a new guise, We present a two dimensional ''solvable model" of a Dirac string, which suggests a new solution of the veto problem. The renormalization of the Green's functions of the Abelian Wilson loop and Abelian vortex operator is studied with the aid of the operator product expansion. In each case. an overall multiplication of the operator makes all Green's functions finite; a surprising cancellation of divergences occurs with the vortex operator. We present a brief discussion of the relation between the nature of the vacuum and the cluster properties of the Green's functions of the Wilson and vortex operators. for a general gauge theory. The surface-like cluster property of the vortex operator in an Abelian Higgs theory is explored in more detail.