Problems and Study Projects in non-Abelian Gauge and String Theory
Harish Parthasarathy · 2022
The notion of a connection on a principal bundle is described in the context of the Yang-mills field equations in curved space-time. A vector bundle on whose fibres a Lie group of transformations act is roughly speaking a fibre bundle. The culminating point of this study is the celebrated Atiyah-Singer index theorem which expresses the index of this operator in terms of an integral of a quantity involving both the Riemann curvature tensor of the base manifold and the curvature tensor of the Yang-Mills field. The Electroweak theory explains how an electron acquires mass, how the gauge Bosons that propagate the nuclear forces acquire mass and it also simultaneously explains why the photon is massless, ie, does not acquire any mass after symmetry breaking occurs when one considers vacuum expectations of the Higgs doublet interacting via a Yukawa coupling to the electrons and leptons.