nLab topological quantum field theory




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.


See also the references at 2d TQFT, 3d TQFT and 4d TQFT.

Discussion of action functionals for topological field theories via equivariant ordinary differential cohomology:

Origin in physics

The concept originates in the guise of cohomological quantum field theory motivated from TQFTs appearing in string theory in

  • Edward Witten, Topological quantum field theory, Comm. Math. Phys. Volume 117, Number 3 (1988), 353-386. (euclid:1104161738)

  • Edward Witten, Introduction to cohomological field theory, International Journal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (pdf)

  • Stefan Cordes, Gregory Moore, Sanjaye Ramgoolam, Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories, Nucl. Phys. Proc. Suppl.41:184-244,1995 (arXiv:hep-th/9411210)

and in the discussion of Chern-Simons theory (“Schwarz-type TQFT”) in

  • Edward Witten, Quantum Field Theory and the Jones Polynomial Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 (project EUCLID)


See also:

Global (1-functorial) TQFT

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

Exposition of the conceptual ingredients:

More technical lecture notes:

An introduction specifically 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).

See also the references at HQFT.

Relation to cut-and-paste-ivariants:

See also:

Local (nn-functorial) TQFT

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

and formalized and proven in

This also shows how TCFT fits in, which formalizes the original proposal of 2d cohomological quantum field theory.

Lecture notes:

A discussion amplifying the aspects of higher category theory is in

See also

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

Last revised on December 14, 2023 at 10:36:59. See the history of this page for a list of all contributions to it.