Topological Quantum Field Theory

Introduction

Topological quantum field theory (TQFT) is a kind of quantum field theory designed so that its observables depend only on the topology of the underlying spacetime, not on geometric details like distances or a chosen metric, and in that sense it “computes” topological invariants of manifolds. In Atiyah’s axiomatic (functorial) viewpoint, a TQFT assigns a vector space (state space) to each closed $d$-dimensional manifold and a linear map to each $(d+1)$-dimensional cobordism between them, in a way compatible with cutting and gluing. TQFTs have become central in connections between physics and areas of mathematics such as knot theory and low-dimensional topology, with landmark examples including Chern–Simons theory and Witten’s topological twists.

Contents

Cobordisms

Cobordisms
Topological Gluing

Frobenius Algebras

Frobenius Algebra
Comultiplication in a Frobenius Algebra
Frobenius Objects