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# Contents

## 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 $Bord_n^S$, 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 $Bord^S_n$ 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”

## 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.

## References

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

### 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)

### Global (1-functorial) TQFT

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)

Exposition of the conceptual ingrediants includes

and more technical lecture notes include

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).

### Local ($n$-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 include

A discussion amplifying the aspects of higher category theory is in