Lectures on Topological Aspects of Theoretical Physics

TL;DR

This series of lectures explores the topological foundations of modern theoretical physics, focusing on fibre bundles, gauge theories, and topological invariants, with applications to anomalies and QCD.

Contribution

It provides a comprehensive overview of topological methods in physics, connecting fibre bundle theory, instantons, monopoles, and index theorems to physical phenomena.

Findings

Application of Pontrjagin's theorem to QCD

Topological classification of gauge fields

Insights into conformal anomalies

Abstract

This series of lectures is planned as a generalization of author's large (more than fifteen years) experience of work in the theoretical physics. The modern theoretical physics is based on the group-theoretical approach which generates the formalism of the principal fibre bundles and the instanton approach. The latter is based on the Pontrjagin's degree of map theorem and this theorem is the original ``bridge'' between homology and cohomology theories. The author plans to devote his two first lectures to fibre bundle theory: this is the foundation on which the modern physics rests -- the theory of gauge groups and the Yang-Mills fields. The idea of connection and curvature (first of all for the principal fibre bundles) will be given also. The lectures are devoted to the Pontrjagin's degree of map theorem, to the theories of monopoles and instantons, to the theory of the topological index of the elliptical operator. The information accumulated to this moment allows us to apply these theories to some questions of conformal anomalies and to the topological aspects of QCD. More comprehensive questions of modern topology (for example the algebraic (co)homology theories, the theory of the spectral sequences) will be expounded in the further lectures.