Synopses & Reviews
In modern theoretical physics, gauge field theories are of great importance since they keep internal symmetries and account for phenomena such as spontaneous symmetry breaking, the quantum Hall effect, charge fractionalization, superconductivity and supergravity. This monograph discusses specific examples of gauge field theories that exhibit a selfdual structure. The author builds a foundation for gauge theory and selfdual vortices by introducing the basic mathematical language of the subject and formulating examples ranging from the well-known abelian-Higgs and Yang-Mills models to the Chern-Simons-Higgs theories (in both the abelian and non-abelian settings). Thereafter, the electroweak theory and self-gravitating electroweak strings are also examined, followed by the study of the differential problems that have emerged from the analysis of selfdual vortex configurations; in this regard the author treats elliptic problems involving exponential non-linearities, also in relation to concentration-compactness principles and blow-up analysis. Many open questions still remain in the field and are examined in this comprehensive work in connection with Liouville-type equations and systems. The goal of this text is to form an understanding of selfdual solutions arising in a variety of physical contexts. Selfdual Gauge Field Vortices: An Analytical Approach is ideal for graduate students and researchers interested in partial differential equations and mathematical physics.
Review
From the reviews: "This monograph is devoted to a mathematical study of selfdual gauge theories using critical point theory and some analytical and topological techniques. ... The book ends with a substantial bibliography and a useful index. This monograph, dealing in a clear way with some 'hot' topics of theoretical physics and geometry, will be of interest both to mathematicians and physicists." (Jean Mawhin, Zentralblatt MATH, Vol. 1177, 2010)
Synopsis
Elliptic problems arise in the study of significant questions concerning vortices in various selfdual field theories, e.g., Chern--Simons and Electroweak theories. Despite progress in these directions, many open questions still remain and are examined in this work in connection with Liouville-type equations and systems.Key topics unfold systematically beginning with the foundations of gauge theory and examples of selfdual vortices; chapters thereafter treat elliptic problems involving selfdual vortices, Ginzburg--Landau and Chern--Simons models, concentration-compactness principles, and Maxwell--Chern--Simons vortices.To date there is no comprehensive examination of the subject. The primary orientation of the text is partial differential equations, but the reader should have knowledge of some basic tools in field theory. For graduate students and researchers in PDEs and mathematical physics.
Synopsis
This monograph discusses specific examples of selfdual gauge field structures. 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. Many open questions still remain in the field and are examined in this work in connection with Liouville-type equations and systems. The goal of this text is to form an understanding of selfdual solutions arising in a variety of physical contexts.
Synopsis
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.
Table of Contents
Preface.- Selfdual Gauge Field Theories.- Elliptic Problems in the Study of Self-ual Vortex Configurations.- Planar Self-dual Chern-Simons Vortices.- Periodic Selfdual Chern-Simons Vortices.- The Analysis of Liouville-type Equations with Singular Sources.- Mean Field Equations of Liouville-type.- Selfdual Electroweak Vortices and Strings.- References.- Index.