A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ...

Overview

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.) In quantum field theory, correlation functions generalize the concept of correlation functions in statistics. ... See: International System of Units, colloquially called the Metric System, and also metrication. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

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. In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ... Quantum field theory in curved spacetimes is an extension of the standard quantum field theory to curved spacetimes. ...

Although TQFTs were invented by physicists (notably 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.) Edward Witten at the Institute for Advanced Study Edward Witten (born August 26, 1951) is an American mathematical physicist, Fields Medalist, and professor at the Institute for Advanced Study. ...

Physically speaking, topological field theories are not especially interesting. They are toy models, far simpler in structure than the quantum field theories which describe real-world physics. They can be thought of as warm-up exercises for the much harder task of quantizing gravity. Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. Unfortunately, they are only background independent in a fairly trivial sense: they are independent of the spacetime metric (thought of as a background gravitational field), but they do not admit any local degrees of freedom. There is no radiation in a TQFT: no propagating waves, no gluons, no gravitons.

(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.) In quantum field theory, a nonlinear σ model is describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The target manifold is equipped with a Riemannian metric g. ...

Specific 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.

Schwarz-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 The BF model is a topological field theory, which when quantized, becomes a topological quantum field theory. ...

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. Chern-Simons theory is a topological gauge theory in three dimensions which describes knot and three-manifold invariants. ... A knot invariant is a useful tool in knot theory. ...

Witten-type TQFTs

In Witten-type topological field theories, the topological invariance is more subtle. [ed. NEEDS FINISHING]

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. Sir Michael Francis Atiyah, OM (born 22 April 1929) is a mathematician who was born in London. ... Segal (and its variants) can refer to the following: // People Segal Daniel Scott Segal, pseudonym Dancing Eagle, drummer for The Supersuckers David HaLevi Segal (1586-1667), Polish rabbi and Halakhist David Segal, Green Party councilman for Providence, Rhode Island Erich Segal (b. ... A conformal field theory is a quantum field theory (or statistical mechanics model) that is invariant under the conformal group. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, cobordism is a relation between manifolds, based on the idea of boundary. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

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 Euclidean 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 casses of morphisms in Bord_M. A TQFT on M is a functor from hBord_M to the category of vector spaces. A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. ... The term connection has various uses, including: An act of connecting two or more physical entities in a physical sense or connecting concepts in memory or imagination, see below Telecommunications circuit switching That which connects, relates or joins: An electrical connection A telecommunication circuit such as a fiber-optic connection... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...

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]

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] In mathematics, a monoidal category (or tensor category) is a category equipped with a binary tensor functor and a unit object . ...

Generalizations

For some applications, it is convenient to demand extra topological structure on the morphisms, such as a choice of orientation. In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...