Topological Quantum Field Theories

Annotation

The course discusses the geometric aspects of various topological quantum field theories (TQFT). These theories have a wide range of applications in both physics and mathematics. In particular, certain important invariants of  topological spaces can be described by TQFT. The course focuses on particular examples of TQFTs related to Chern-Simons theory, Frobenius algebras, and representation theory though further  examples such as Poisson sigma models are also reviewed. Other topics covered include relation to graded geometry and the AKSZ construction.

Course plan

1. Classical topological fields theories. Examples. Degrees of freedom
2. TQFT and its origin
3. Atiyah’s axioms
4. 1D TQFT and representation theory.
5. 2D TQFT and Frobenius algebras.
6. Vector bundles with a structure group. Connection, curvature. Gauge transformation.
7. Chern-Weil construction and characteristic classes. Chern-Simons invariant.
8. Example of 3D TQFT relates to Chern-Simons action.
9. Jones polynomial and other invariants.
10. Poisson sigma-model and other topological theories.
11. Graded manifolds and AKSZ construction

Literature

1. F. Quinn, Lectures on axiomatic topological quantum fields theory, в книге  Geometry and quantum field theory, (D.Freed,K.Uhlenbeck eds) pp. 323-459.
2. Koch, J., Frobenius algebras and 2D topological quantum field theories, London Mathematical Society Student Texts, 59. Cambridge University Press, Cambridge, 2004.
3. M. F. Atiyah  Topological quantum field theory, Publications mathématiques de l’I.H.É.S., tome 68 (1988), p. 175-186.
4. A. S. Schwarz, The partition function of degenerate quadratic functional and ray-singer invariants, Lett.Math.Phys. 2 (1978) 247.
5.  Witten  E., Topological quantum field theory, "Commun. Math. Phys.", 1988,  v. 117, p. 353.