Topological quantum field theory

Topological quantum field theory

A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.

Although TQFTs were invented by physicists (notably Edward Witten), they are primarily of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory.

In condensed matter physics, topological quantum field theories are the low energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, andother strongly correlated quantum liquid states.


In a topological field theory, the correlation functions do not depend on the metric on spacetime. This means that the theory is not sensitive to changes in the shape of spacetime; if the spacetime warps or contracts, the correlation functions do not change. Consequently, they are topological invariants. (Strictly speaking, the argument above only shows that they are diffeomorphism invariants. Showing that they are homotopy invariants takes more effort.)

Topological field theories are not very interesting on the flat Minkowski spacetime used in particle physics. Minkowski space can be contracted to a point, so a TQFT on Minkowski space computes only trivial topological invariants. Consequently, TQFTs are usually studied on curved spacetimes, such as, for example, Riemann surfaces. Most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few higher dimensional theories exist, but they are not very well understood.

Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigation of this class of models.

(Caveat: It is often said that TQFTs have only finitely many degrees of freedom. This is not a fundamental property. It happens to be true in most of the examples that physicists and mathematicians study, but it is not necessary. A topological sigma model with target infinite-dimensional projective space, if such a thing could be defined, would have countably infinitely many degrees of freedom.)

pecific models

The known topological field theories fall into two general classes: Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also sometimes referred to as cohomological field theories.

chwarz-type TQFTs

In Schwarz-type TQFTs, the correlation functions computed by the path integral are topological invariants because the path integral measure and the quantum field observables are explicitly independent of the metric. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action (which determines the path integral) is:S=int_M B F,

The spacetime metric does not appear anywhere in this theory, so the theory is explicitly topologically invariant. Another, more famous example is Chern-Simons theory, which can be used to compute knot invariants.

Witten-type TQFTs

In Witten-type topological field theories, the topological invariance is more subtle. For example the Lagrangian for the WZW model does depend explicitly on the metric, but one shows by calculation that the expectation value of the partition function and a special class of correlation functions are in fact diffeomorphism invariant.

Mathematical formulations

Atiyah-Segal axioms

Atiyah suggested a set of axioms for topological quantum field theory which was inspired by Segal's proposed axioms for conformal field theory. These axioms have been relatively useful for mathematical treatments of Schwarz-type QFTs, although it isn't clear that they capture the whole structure of Witten-type QFTs. The basic idea is that a TQFT is a functor from a certain category of cobordisms to the category of vector spaces.

There are in fact two different sets of axioms which could reasonably be called the Atiyah axioms. These axioms differ basically in whether or not they study a TQFT defined on a single fixed "n"-dimensional Riemannian / Lorentzian spacetime "M" or a TQFT defined on all "n"-dimensional spacetimes at once.

: [ed. "What follows is still in rough draft form and should be regarded suspiciously."]

The case of a fixed spacetime

Let Bord_M be the category whose morphisms are "n"-dimensional submanifolds of "M" and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M, and so form the quotient category hBord_M: The objects in hBord_M are the objects of Bord_M, and the morphisms of hBord_M are homotopy equivalence classes of morphisms in Bord_M. A TQFT on M is a symmetric monoidal functor from hBord_M to the category of vector spaces.

Note that cobordisms can, if their boundaries match up, be sewn together to form a new bordism. We would like the functor to preserve this structure, so that a morph [ed. "unfinished"]

There is an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras.

All n-dimensional spacetimes at once

To consider all spacetimes at once, it is necessary to replace hBord_M by a larger category. So let Bord_n be the category of bordisms, i.e. the category whose morphisms are n-dimensional manifolds with boundary, and whose objects are the connected components of the boundaries of n-dimensional manifolds. (Note that any (n-1)-dimensional manifold may appear as an object in Bord_n.) As above, regard to morphisms in Bord_n as equivalent if they are homotopic, and form the quotient category hBord_n. Bord_n is a monoidal category under the operation which takes two bordisms to the bordism made from their disjoint union. A TQFT on n-dimensional manifolds is then a functor from hBord_n to the category of vector spaces, which takes disjoint unions of bordisms to the tensor product f [ed. "unfinished"]


For some applications, it is convenient to demand extra topological structure on the morphisms, such as a choice of orientation.

ee also

*Topological string theory


*Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | title=Topological quantum field theories | url= | id=MathSciNet | id = 1001453 | year=1988 | journal=Publications Mathématiques de l'IHÉS | issn=1618-1913 | issue=68 | pages=175–186
*Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Topological quantum field theory | url= | id=MathSciNet | id = 953828 | year=1988 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=117 | issue=3 | pages=353–386

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Quantum field theory — In quantum field theory (QFT) the forces between particles are mediated by other particles. For instance, the electromagnetic force between two electrons is caused by an exchange of photons. But quantum field theory applies to all fundamental… …   Wikipedia

  • Noncommutative quantum field theory — In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative… …   Wikipedia

  • List of quantum field theories — List of quantum field theories:*Chern Simons model *Chiral modelComplex Quantum Mechanics*Gross Neveu *Kondo model *Lower dimensional quantum field theory *Minimal model *Nambu Jona… …   Wikipedia

  • Quantum gravity — is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature (electromagnetism, weak interaction, and strong interaction), with general relativity, the theory of the fourth… …   Wikipedia

  • Topological order — In physics, topological order is a new kind of order (a newkind of organization of particles) in a quantum state that is beyond theLandau symmetry breaking description. It cannot be described by local order parameters and long rangecorrelations.… …   Wikipedia

  • Quantum chromodynamics — Standard model of particle physics Standard Model …   Wikipedia

  • Topological string theory — In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry.… …   Wikipedia

  • Topological defect — Also see base concepts: topology, differential equations, quantum theory condensed matter physics .In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a… …   Wikipedia

  • Quantum vortex — In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. The existence of these quantum vortices were independently predicted by Richard Feynman and Alexei Alexeyevich Abrikosov in the 1950s. They were… …   Wikipedia

  • Quantum finite automata — In quantum computing, quantum finite automata or QFA are a quantum analog of probabilistic automata. They are related to quantum computers in a similar fashion as finite automata are related to Turing machines. Several types of automata may be… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”