Lecture Series on Topological Quantum Field Theory

[Rosanna Sain]

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Institute for Geometry And Physics (IGAP)
Distinguished lecture series

 From Monday 28 May to Thursday 31 May we will have at SISSA
the following course of 4 lectures on Topological Quantum Field Theory:

Prof. Nils Carqueville (U. of Wien)

** Higher structures in topological quantum field theory **

Calendar:
Monday 28 May         14:30 - 16:00     Room 137 at SISSA
Tuesday 29 May        14:30 - 16:00     Room 137
Wednesday 30 May    11:00 - 12:30     Room 136
Thursday 31 May       14:30 - 16:00     Room 137

Abstract:
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These lectures aim to be a gentle introduction to topological quantum
field theory (TQFT) and the categorical structures it naturally gives
rise to, with the aim to make connections to topological phases of
matter and topological quantum computation. No prior knowledge of
category theory will be assumed.

Lecture 1 will motivate the functorial definition of closed TQFTs,
using desired properties of the path integral such as locality. We
will introduce the categories of vector spaces and of n-dimensional
bordisms, and then define a TQFT to be a structure-preserving map
between the two. After a quick study of the case n=1, we will focus on
the case n=2 and argue that
2-dimensional closed TQFTs are equivalent to commutative Frobenius
algebras, which we will illustrate with examples coming from sigma
models and Landau-Ginzburg models. Finally, we will give an overview
of the case n=3 (to which we will return in Lecture 4)

Lectures 2 and 3 deal with refinements of 2-dimensional TQFTs where
one studies a larger class of bordisms (with extra structure):
"open-closed TQFTs" and "defect TQFTs". We will explain how the former
naturally lead to "Calabi-Yau categories" (which will be defined along
the way) that have the interpretation of describing certain D-branes
and open strings in string theory. Similarly, 2-dimensional defect
TQFTs give rise to three-layered structures called 2-categories. We
will introduce them and explain how they neatly encode the totality of
all bulk theories, interfaces and point defects of a given topological
theory. After again discussing the examples of sigma models and
Landau-Ginzburg models, we will illustrate the usefulness of the
higher-categorical language by describing a generalisation of the
orbifold construction which unifies state sum models, equivariant
group actions, and provides new relations between 2-dimensional TQFTs.

Lecture 4 wants to lift all the previous constructions from 2 to 3
dimensions. In particular, we will define 3-dimensional defect TQFTs,
sketch examples, explain what 3-categories are and how they arise
naturally from TQFTs. Time permitting, we will discuss the role of
such categories in the description of some (2+1)-dimensional
topological phases of matter, and how 3-dimensional orbifolds can
provide universal topological quantum computation.
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