The mathematical structure of gauge theory is a union of many modern algebraic concepts. The book provides the background to understand gauge theory and use it in computational chemistry. It introduces two powerful methods used in modern computational chemistry: The Diffusion Monte Carlo algorithm and Ring Polymer Molecular Dynamics. Gauge theory is used to derive a convergent version of the ground state probability amplitude approach for the simulation of excited states. Contains some recent advances in theoretical and computational chemistry. Clear, coherently written explanation of the various background mathematical concepts. Accessible to undergraduate students with many exercises for readers to achieve mastery of the subjects.