Contents

Idea

A topological quantum field theory is a quantum field theory which is a functorial quantum field theory is a functor on the (infinity,n)-category of cobordisms ${\mathrm{Bord}}_n$ that do not carry any further structure.

For more on the general idea and its development, see FQFT and extended topological quantum field theory. For the central structural result in TQFT, the cobordism hypothesis see

• Jacob Lurie, On the Classification of Topological Field Theories

Remarks

• often topological quantum field theories are just called topological field theories and accordingly the abbreviation TQFT is reduced to TFT. Strictly speaking this is a misnomer, which is however convenient and very common. It should be noted, however, that TQFTs may have classical counterparts (for instance Chern-Simons theory) which would better deserve to be called TFTs. But they are not usually.

Non-topological QFTs

In constrast to topological QFTs, non-topological quantum field theories in the FQFT description are $n$-functors on $n$-categories ${\mathrm{Bord}}_n^S$ whose morphisms are manifolds with extra $S$-structure, for instance

• $S=$ conformal structure $\to$ conformal field theory

• $S=$ Riemannian structure $\to$ “euclidean QFT”

• $S=$ pseudo-Riemannian structure $\to$ “relativistic QFT”

Examples

• 2d TQFT

• Dijkgraaf-Witten theory

• Chern-Simons theory

• Reshetikhin-Turaev model?

• Turaev-Viro model?

Homotopy QFTs

These somehow lie between the previous two types. There is some simple extra structure in the form of a ‘characteristic map’ from the manifolds and bordisms to a ‘background space’ $X$. In many of the simplest examples, this is taken to be the classifying space of a group, but this is not the only case that can be considered. The topic is explored more fully in HQFT.