topological quantum field theory in nLab

Contents

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 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.

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

• S=S = conformal structure →\to conformal field theory

• S=S = Riemannian structure →\to “euclidean QFT”

• S=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’ 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.

• 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 (nn-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