nLab
topological quantum field theory

Idea

A topological quantum field theory is a quantum field theory which – as a functorial quantum field theory – is a functor on a flavor of the (∞,n)-category of cobordisms ${\mathrm{Bord}}_n^S$, where the n-morphisms are cobordisms without any non-topological further structure $S$ – for instance no Riemannian metric structure – but possibly some “topological structure”, such as Spin structure or similar.

For more on the general idea and its development, see FQFT and extended topological quantum field theory.

Remark

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 which would better deserve to be called TFTs. But they are not usually.

Non-topological QFTs

In contrast 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.

Related concepts

• FQFT, extended topological quantum field theory

• topological quantum computation

Literature

An introduction to 2D TQFTs can be found in

• Joachim Kock, Frobenius algebras and 2D topological quantum field theories, No. 59 of LMSST, Cambridge University Press, 2003., (full information here).

Further literature includes:

• Anton Kapustin, Topological field theory, higher categories, and their applications, survey for ICM 2010, arxiv/1004.2307
• D. Freed, M. Hopkins, J. Lurie, C. Teleman, Topological Quantum Field Theories from Compact Lie Groups
• Jacob Lurie, On the Classification of Topological Field Theories; TQFT and the Cobordism Hypothesis (video, notes)
• see also cobordism hypothesis and references therein
• Mark Feshbach, Alexander A. Voronov, A higher category of cobordisms and topological quantum field theory, arxiv/1108.3349