Title: Topological quantum field theory: symmetries and defects

2016.04.29 |

Date | Mon 23 May |

Time | 13:15 — 14:15 |

Location | 1531-119 (Aud. D2) |

**Abstract**

A major paradigm of 20th-century science is to understand nature in the language of quantum field theory. Efforts to make mathematical sense of this language have led to successful and ongoing cross-fertilisation between theoretical physics and pure mathematics. In particular, Atiyah and Segal proposed an axiomisation of Feynman's path integral by beautifully linking geometry with algebra.

The talk starts with a lightening review of this functorial approach, and then quickly restricts to the case in which spacetime is two-dimensional and has no geometric structure: two-dimensional topological quantum field theory (TQFT). This seemingly simple situation is still surprisingly rich, and we will see how algebras, categories, and "higher" structures appear naturally; examples of such structures are ubiquitous in many areas of mathematics.

At the end of a mostly expository presentation we turn to the central notion of symmetry, which involves the action of groups on a TQFT. We will see how to interpret symmetries as special kinds of "defects" of the TQFT, which in turn allows for a natural, purely algebraic generalisation of the operation of "modding out by a symmetry". This leads to new equivalences between categories, which we will illustrate with examples from singularity theory and representation theory -- with most details deferred to a more technical companion talk.

**NOTE: This talk is aimed at a general audience of mathematicians.**