Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In metamaterials:
For quantum computation:
A topological quantum field theory (TQFT) is a quantum field theory without dependence on the geometry (notably not on pseudo Riemannian geometry or fixed embeddings) of its spacetime/worldvolume domains.
More precisely, in the (extended) functorial description of QFTs, where a quantum field theory is a (higher) functor on a (higher) cobordism category, a TQFT is such a functor on manifolds and cobordisms that carry no geometric structure (but possibly tangential structure, such as orientation or spin structure, or continuous maps to some classifying space, cf. HQFT).
(In contrast, a conformal field theory depends on conformal structure, a Euclidean field theory depends on Riemannian structure, a relativistic field theory depends on pseudo-Riemannian structure, etc.).
In particular, topological quantum field theories are “generally covariant” in a strong sense (Witten 1988), in that they coherently assign isomorphic (equivalent) spaces of states to diffeomorphic domains.
Historically, TQFTs first arose as theoretical models in the context of high energy physics and string theory (Witten 1988), prominent original examples being 2D TQFTs of topological strings and 3D TQFTs of Chern-Simons theory, the latter remaining a pivotal example (also in guises such as the Reshetikhin-Turaev and the Turaev-Viro model).
With the mathematical axiomatization of TQFTs both established and tractable (Atiyah 1989, following Segal 1988‘s axiomatization of 2D CFT) — this in notorious contrast to the situation for general non-topological QFTs — the topic saw dramatic developments in pure mathematics, where TQFTs (with or without relation to actual physics) have come to serve as standard mathematical tools in fields of low-dimensional topology such as knot theory, knot homology, quantum topology and string topology, also in geometric representation theory, and as a field of study in higher category theory (cf. the cobordism hypothesis).
More recently, TQFTs find relevance in solid state physics as models for the behaviour of gapped ground states of topologically ordered quantum materials. The experimental observation of anyon braiding phases in fractional quantum Hall systems may be the first manifestation of TQFT in observational physics (in this case in the guise of abelian Chern-Simons theory), and the widely anticipated application of such anyon braiding to topological quantum computation could become the first technological application of TQFT.
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]
Romesh K. Kaul: Topological Quantum Field Theories – A Meeting Ground for Physicists and Mathematicians [arXiv:hep-th/9907119]
Ralph Cohen (notes by Eric Malm), p 20 of: MATH 283: Topological Field Theories (2008) [pdf, pdf]
Discussion of action functionals for topological field theories via equivariant ordinary differential cohomology:
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. 117 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
See also:
The FQFT-axioms for global (i.e. 1-functorial) TQFTs are due to:
Exposition of the conceptual ingredients:
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
See also the references at HQFT.
Relation to cut-and-paste-ivariants:
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]
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 September 9, 2025 at 14:21:19. See the history of this page for a list of all contributions to it.