Topological Gauge Theories

Alessandro Tanzini

Part 1: from classical mechanics to supersymmetry

- review of classical mechanics (Hamilton, Lagrange, Hamilton Jacobi formulations)

- classical field theory: Noether theorem, scalar fields and sigma models, Clifford algebrae and spinor fields, gauge fields, instantons, monopoles

- supersymmetry: supermanifolds, superfields, supersymmetric sigma models, gauge theories with (extended) supersymmetry

Part 2: path integral representation of topological invariants

- path integral formulation of quantum mechanics, Feynman-Kac formula. Gauge fields and BRS symmetry.

- topological quantum field theories: supersymmetric quantum mechanics and the Euler characteristic, topological sigma model and Gromov-Witten invariants, Chern-Simons theory and knot invariants, [BF theory and Batalin-Vilkoviski formalism], topological Yang-Mills theory and Donaldson invariants.