Mathematics

The course provides a brief introduction to Topological Field Theories as infinite dimensional generalisation of localisation formulae in equivariant cohomology. It starts with an introduction to Duistermaat-Heckman theorem, equivariant cohomology and Atiyah-Bott formula and their extension on supermanifolds. It then continues with supersymmetric quantum mechanics and its relation with Morse theory, gradient flow lines and Morse-Smale-Witten complex. The second part of the course is devoted to supersymmetric sigma models, their topological twist to the A and B models and the relation of the with Gromov-Witten theory and Landau-Ginzburg models and Hodge variations. In the last part of the course we discuss supersymmetric gauged linear sigma models, vortices and mirror symmetry, and we provide the construction of mirror pairs for toric varieties.