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 $Bord_n^S$, 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”

Examples

• 2d TQFT

  • TCFT

  • 2d Chern-Simons theory

• 3d TQFT

  • Dijkgraaf-Witten theory

  • Chern-Simons theory

  • Reshetikhin-Turaev model

  • Turaev-Viro model

• 4d TQFT

  • topological Yang-Mills theory

  • topologically twisted D=4 super Yang-Mills theory

  • 4d Chern-Simons theory

  • Yetter 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

• cohomological field theory, TCFT

• FQFT, extended topological quantum field theory

• topological phase of matter

• topological quantum computation

References

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:

• Joe Davighi, Ben Gripaios, Oscar Randal-Williams, Differential cohomology and topological actions in physics (arXiv:2011.05768)

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. [euclid:cmp/1104178138, MR0990772]

Review:

• Danny Birmingham, Matthias Blau, Mark Rakowski, George Thompson, Topological field theory, Physics Reports 209 4–5 (1991) 129-340 [doi:10.1016/0370-1573(91)90117-5]

See also:

• Markus Banagl, Positive topological quantum field theories, Quantum Topology 6 4 (2015) 609-706 [arxiv/1303.4276, doi:10.4171/qt/71]

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 86 (1989) 175-186 [numdam:PMIHES_1988__68__175_0]

Exposition of the conceptual ingredients:

• John Baez, Quantum Quandaries: a Category-Theoretic Perspective (arXiv:quant-ph/0404040)

More technical lecture notes:

• Daniel Freed, Lectures on topological quantum field theory, in: Integrable Systems, Quantum Groups, and Quantum Field Theories, NATO ASI Series 409 (1992) [doi:10.1007/978-94-011-1980-1_5, pdf, pdf]

• Frank Quinn, Lectures on axiomatic topological quantum field theory, in Dan Freed, Karen Uhlenbeck (eds.) Geometry and Quantum Field Theory 1 (1995) [doi:10.1090/pcms/001]

• Kevin Walker, TQFTs, 2006 (pdf)

• Mikhail Khovanov (notes by You Qi), §2 in: Introduction to categorification, lecture notes, Columbia University (2010, 2020) [web, web, full:pdf]

  (with an eye towards link homology)

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:

• Carmen Rovi, Matthew Schoenbauer, Relating Cut and Paste Invariants and TQFTs, The Quarterly Journal of Mathematics 73 2 (2022) 579–607 [arXiv:1803.02939, doi:10.1093/qmath/haab044]

See also:

• Fiona Torzewska, Topological quantum field theories and homotopy cobordisms [arXiv:2208.14504]

• Fiona Torzewska, Topological Quantum Field Theories and Homotopy Cobordisms, talk at CQTS (Dec 2023) [slides:pdf, video:YT]

Local ($n$-functorial) TQFT

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

• John Baez, James Dolan, Higher dimensional algebra and Topological Quantum Field Theory J.Math.Phys. 36 (1995) 6073-6105 (arXiv:q-alg/9503002)

and formalized and proven in

• Jacob Lurie, On the Classification of Topological Field Theories; TQFT and the Cobordism Hypothesis (video, notes)

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

Lecture notes:

• Constantin Teleman, Five lectures on topological field theory, in Geometry and Quantization of Moduli Spaces, CRM Advanced Courses in Mathematics, Birkhäuser (2016) [doi:10.1007/978-3-319-33578-0_3, pdf, pdf]

A discussion amplifying the aspects of higher category theory is in

• Anton Kapustin, Topological field theory, higher categories, and their applications, survey for ICM 2010, (arxiv/1004.2307)

See also

• Mark Feshbach, Alexander 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

• Daniel Freed, Michael Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups