## Introduction to topological field theories

### Prof. Alessandro Tanzini

Number of cycles: 2

First semester: Mon 14:30-16:00, Tue 15:30-17:00 (starting from October 31)

Content:

- First part.
- Introduction to quantum mechanics: recall of Hamiltonian formulation of classical mechanics and Hamilton-Jacobi equations, notion of Hilbert space of quantum states, momentum and coordinate representation, harmonic oscillator, angular momentum, brief introduction to WKB method, path integral formulation.
- Second part.
- Recall of Morse theory. Equivariant cohomology and Localisation Formulae (Duistermaat-Heckman and Atiyah-Bott). Mathai-Quillen formalism for the Euler class of vector bundles. Field theory in zero dimensions: a toy model of Supersymmetry. Euler class of infinite dimensional vector bundles (Atiyah-Jeffrey) Supersymmetric quantum mechanics as the regularized Euler number of loop space. Supersymmetric Quantum Mechanics and Morse theory. Morse inequalities and Morse-Witten complex. Supersymmetry algebra in two dimensions. Topological twist: A model and B model. Mathai-Quillen formalism for Gromov-Witten invariants. Brief introduction to mirror symmetry.

Prerequisites: classical mechanics, differential geometry