topological quantum field theory


Quantum field theory


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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 Bord n SBord_n^S, where the n-morphisms are cobordisms without any non-topological further structure SS – 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.


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 nn-functors on nn-categories Bord n SBord^S_n whose morphisms are manifolds with extra SS-structure, for instance


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’ XX. 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.



The FQFT-axioms for global (i.e. 1-functorial) TQFTs are due to

  • Michael Atiyah, Topological quantum field theories, Publications Mathématiques de l’IHÉS 68 (68): 175–186, (1989) (Numdam)

Lecture notes include

For HQFT see there

An introduction to 2d TQFTs? is in

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


The local FQFT formulation (i.e. n-functorial) together with the cobordism hypothesis was suggested in

and formalized and proven in

Lecture notes include

Othe amplifying the aspects of higher category theory is in

See also

  • Mark Feshbach, Alexander A. Voronov, A higher category of cobordisms and topological quantum field theory, arxiv/1108.3349

Indication of local quantization in the context of infinity-Dijkgraaf-Witten theory is in

Revised on November 17, 2014 22:19:16 by Urs Schreiber (