Lectures on Principal Bundles

04/2014-06/2014 | Lecture Series for Graduate Students, Pennsylvania State University and Massachusetts Institute of Technology

There are currently four fundamental interactions – electromagnetism, weak force, strong force and gravity – that are known and well described by modern physics. While the first three are described by gauge quantum field theories, the general theory of relativity is considered very different in its differential geometric language. However, there exist a unifying mathematical framework that describes all four interactions as gauge theories of a particular gauge group, namely U(1), SU(2), SU(3) and SO(1,3)\subset GL(4).
In this lecture series, we will define principal bundles and their associated bundles that encode gauge and matter fields. Furthermore, we will explain how a connection 1-form on the principle bundle gives rise to the curvature 2-form. Locally, both – the connection 1-form and the curvature form – can be pulled back to the base manifold of spacetime leading to the Lie algebra valued 1-form A^k_\mu and 2-form F^k_{\mu\nu} that are well-known from non-abelian Yang-Mills theories. The same happens in gravity leading to the Christoffel symbol \Gamma^k_\mu and the Riemann tensor R^k_{\mu\nu}. We will see that the ambiguity of performing this pullback gives rise to different gauge choices.
Principal bundles provide therefore an excellent framework to understand all gauge theories including gravity from a very geometric perspective. Moreover, they are used in various settings of mathematical physics: they are the natural language of Chern-Simons theories (tpological QFT), give a better understanding of instantons in non-abelian gauge theories and allow a precise mathematical definitions as holonomies of the connection 1-form.

This lecture series was given for graduate students at the Pennsylvania State University and the Massachusetts Institute of Technology.