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| A '''topological quantum field theory''' (or '''topological field theory''' or '''TQFT''') is a [[quantum field theory]] which computes [[topological invariant]]s.
| | Latoria happens when she's called but people always misspell the item. Lacemaking is mysterious cure he loves most. Her day job is a courier. Michigan is our birth situate. You can always find his website here: http://euroseonews.wordpress.com/ |
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| Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, [[knot theory]] and the theory of [[four-manifold]]s in [[algebraic topology]], and to the theory of [[moduli spaces]] in [[algebraic geometry]]. [[Simon Donaldson|Donaldson]], [[Vaughan Jones|Jones]], [[Edward Witten|Witten]], and [[Maxim Kontsevich|Kontsevich]] have all won [[Fields Medal]]s for work related to topological field theory.
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| In [[condensed matter physics]], topological quantum field theories are the low energy effective theories of [[topological order|topologically ordered]] states, such as [[Quantum Hall Effect|fractional quantum Hall]] states, [[string-net]] condensed states, and other [[Strongly correlated quantum spin liquid|strongly correlated quantum liquid]] states.
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| ==Overview==
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| In a topological field theory, the [[Correlation function (quantum field theory)|correlation functions]] do not depend on the [[Metric tensor (general relativity)|metric]] of 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.
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| Topological field theories are not very interesting on the flat [[Minkowski spacetime]] used in particle physics. Minkowski space can be [[contractible space|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 [[quantum field theory in curved spacetime|defined on spacetimes]] of dimension less than five. It seems that a few higher dimensional theories exist, but they are not very well understood.
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| Quantum gravity is believed to be [[background independence|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.
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| (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.)
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| ==Specific models==
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| 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.
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| ===Schwarz-type TQFTs===
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| In Schwarz-type TQFTs, the [[correlation function (quantum field theory)|correlation function]]s or [[partition function (quantum field theory)|partition function]]s of the system are computed by the path integral of metric independent action functionals. 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
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| :<math>S=\int_M B F\,</math>
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| The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. The first example appeared in 1977 and is due to A. Schwarz, its action functional is:
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| :<math>\int_M A\wedge dA.</math>
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| Another more famous example is [[Chern-Simons theory]], which can be used to compute [[knot invariant]]s. In general partition functions depend on metric but the above examples are shown to be independent on metric.
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| ===Witten-type TQFTs===
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| The first example of the topological field theories of Witten-type appeared in Witten's paper in 1988 {{Harv|Witten|1988a}}, i.e. topological Yang–Mills theory in four dimensions. Though its action functional contains metric ''g''<sub>αβ</sub>, after [[Topological_string_theory#The topological twist|topological twist]] it turns out to be metric independent. The independence of Stress-energy tensor ''T''<sup>αβ</sup> of the system on metric depends on whether [[BRST quantization|BRST-operator]] is closed. Following Witten's example a lot of examples are found in [[Topological_string_theory|string theory]].
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| ==Mathematical formulations==
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| ===The original Atiyah-Segal axioms===
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| [[Michael Atiyah|Atiyah]] suggested a set of axioms for topological quantum field theory {{Harv|Atiyah|1988a}} which was inspired by [[Graeme Segal|Segal]]'s proposed axioms for [[conformal field theory]], and Witten's idea of the geometric meaning of supersymmetry, {{Harv|Witten|1982}}. Atiyah's axioms are constructed on gluing the boundary with differentiable (topological or continuous) transformation, while Segal's are with conformal transformation. 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 (mathematics)|category]] of [[cobordism]]s to the category of [[vector space]]s.
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| 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.
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| Let Λ be a [[commutative ring]] with 1 (for almost all real-world purposes we will have Λ = '''Z''', '''R''' or '''C'''). Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimension ''d'' defined over a ground ring Λ as following:
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| * A finitely generated Λ-module ''Z''(Σ) associated to each oriented closed smooth d-dimensional manifold Σ (corresponding to the ''[[homotopy]]'' axiom),
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| * An element ''Z''(''M'') ∈ ''Z''(∂''M'') associated to each oriented smooth (''d''+1)-dimensional manifold (with boundary) ''M'' (corresponding to an ''additive'' axiom).
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| These data are subject to the following axioms (4 and 5 were added by Atiyah):
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| #''Z'' is ''functorial'' with respect to orientation preserving [[diffeomorphisms]] of Σ and ''M'',
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| #''Z'' is ''involutory'', i.e. ''Z''(Σ*) = ''Z''(Σ)* where Σ* is Σ with opposite orientation and ''Z''(Σ)* denotes the dual module,
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| #''Z'' is ''multiplicative''.
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| #''Z''(φ) = Λ for the d-dimensional empty manifold and ''Z''(φ) = 1 for the (''d''+1)-dimensional empty manifold.
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| #''Z''(''M*'') = {{overline|''Z''(''M'')}} (the ''[[hermitian]]'' axiom). Equivalently, ''Z''(''M*'') is the disjoint of ''Z''(''M'')
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| '''Remark.''' If for a closed manifold ''M'' we view ''Z''(''M'') as a numerical invariant, then for a manifold with boundary we should think of ''Z''(''M'') ∈ ''Z''(∂''M'') as a "relative" invariant. Let ''f'' : Σ × ''I'' → Σ × ''I'' be an orientation preserving diffeomorphism, and identify opposite ends of Σ × ''I'' by ''f''. This gives a manifold Σ<sub>''f''</sub> and our axioms imply
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| :<math>Z(\Sigma_f)=\text{Trace}\ \Sigma(f)</math>
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| where Σ(''f'') is the induced automorphism of ''Z''(Σ).
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| '''Remark.''' For a manifold ''M'' with boundary Σ we can always form the double <math>M\cup_\Sigma M^*</math> which is a closed manifold. The fifth shows that
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| :<math>Z(M\cup_\Sigma M^*)=|Z(M)|^2</math>
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| where on the right we compute the norm in the hermitian (possibly indefinite) metric.
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| ===The relation to physics===
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| Physically (2)+(4) is related to relativistic invariance while (3)+(5) is indicative of the quantum nature of the theory.
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| Σ is meant to indicate the physical space (usually, ''d'' = 3 for standard physics) and the extra dimension in Σ × ''I'' is "imaginary" time. The space ''Z''(''M'') is the [[Hilbert space]] of the quantum theory and a physical theory, with a [[Hamiltonian (quantum_mechanics)|Hamiltonian]] ''H'', will have an time evolution operator ''e<sup>itH</sup>'' or an "imaginary time" operator ''e<sup>−tH</sup>''. The main feature of ''topological'' QFTs is that ''H'' = 0, which implies that there is no real dynamics or propagation, along the cylinder Σ × ''I''. However, there can be non-trivial "propagation" (or tunneling amplitudes) from Σ<sub>0</sub> to Σ<sub>1</sub> through an intervening manifold ''M'' with <math>\partial M=\Sigma^*_0\cup\Sigma_1</math>; this reflects the topology of ''M''.
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| If ∂''M'' = Σ, then the distinguished vector ''Z''(''M'') in the Hilbert space ''Z''(Σ) is thought of as the ''vacuum state'' defined by ''M''. For a closed manifold ''M'' the number ''Z''(''M'') is the [[vacuum expectation value]]. In analogy with [[statistical mechanics]] it is also called the [[partition function]].
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| The reason why a theory with zero Hamiltonian can be sensibly formulated in the [[Feynman path integral]] approach to QFT. This incorporates relativistic invariance (which caters for general (''d''+1)-dimensional "spacetimes") and the theory is formally defined by writing down a suitable [[Lagrangian]] - a functional of the classical fields of the theory. A Lagrangian which involves only first derivatives in time formally leads to a zero Hamiltonian, but the Lagrangian itself may have non-trivial features which relate it to the topology of ''M''.
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| ===Atiyah's examples===
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| In 1988, M. Atiyah published a paper in which he described many new examples of topological quantum field theory that were considered at that time. {{Harv|Atiyah|1988}} It contains some new [[topological invariant]]s and the new ideas, which are [[Casson invariant]], [[Donaldson invariant]], [[Geometric group theory|Gromov's theory]], [[Floer homology]] and [[Jones polynomial|Jones-Witten's theory]].
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| ====''d'' = 0====
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| In this case Σ consists of finitely many points. To single point we associate a vector space ''V'' = ''Z''(point) and to ''n''-points the ''n''-fold tensor product: ''V''<sup>⊗''n''</sup> = ''V'' ⊗ ... ⊗ ''V''. The [[symmetric group]] ''S<sub>n</sub>'' acts on ''V''<sup>⊗''n''</sup>. A standard way to get the quantum Hilbert space is to give a classical [[symplectic manifold]] (or [[phase space]]) and then quantize it. Let us extend ''S<sub>n</sub>'' to compact Lie group ''G'' and consider "integrable" orbits for which the symplectic structure comes from a [[line bundle]] then quantization leads to the irreducible representations ''V'' of ''G''. This is the physical interpretation of the [[Borel-Weil theorem]] or the [[Borel-Weil-Bott theorem]]. The Lagrangian of these theories is the classical action ([[holonomy]] of the line bundle). Thus topological QFT's with ''d'' = 0 relate naturally to the classical [[representation theory]] of [[Lie group]]s and [[symmetric group]]s.
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| ====''d'' = 1====
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| We should consider periodic boundary conditions given by closed loops in a compact symplectic manifold ''X''. Along to {{Harv|Witten|1982}} holonomy round such loops used in the case of ''d'' = 0 as a Lagrangian is used to modify the Hamiltonian. For a closed surface ''M'' the invariant ''Z''(''M'') of the theory is the number of [[pseudoholomorphic curve|pseudo holomorphic map]]s ''f'' : ''M'' → ''X'' in the sense of Gromov (they are ordinary [[holomorphic map]]s if ''X'' is a [[Kaehler manifold]]). If this number becomes to infinite i.e. if there are "moduli", then we must fix further data on ''M''. This can be done by picking some points ''P<sub>i</sub>'' and then looking at holomorphic maps ''f'' : ''M'' → ''X'' with ''f''(''P<sub>i</sub>'') constrained to lie on a fixed hyperplane. {{Harv|Witten|1988b}} has written down the relevant Lagrangian for this theory. Floer has given a rigorous treatment, i.e. [[Floer homology]], based on {{Harv|Witten|1982}}'s [[Morse theory]] ideas, for the case when the boundary conditions are the interval instead of periodic, the initial and end-points of paths lie on two fixed [[Lagrangian submanifold]]s. This theory has been developed as [[Gromov-Witten invariant]] theory.
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| Another example is [[Holomorphic]] [[Conformal Field Theory]]. This might not be strictly topological quantum field theory at that time because Hilbert spaces are infinite dimensional. The conformal field theories are also related to compact Lie group ''G'' in which the classical phase consists of a central extension of the [[loop group]] ''LG''. Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations of ''LG''. The group Diff<sub>+</sub>('''S'''<sup>1</sup>) now substitutes for the symmetric group and play an important role. The partition function in such theories depends on [[complex structure]]{{disambiguation needed|date=September 2012}}: it is not purely topological.
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| ====''d'' = 2====
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| Jones-Witten theory is the most important theory in this case. Here the classical phase space, associated to a closed surface Σ is the moduli space of flat ''G''-bundle over Σ. The Lagrangian is an integer multiple of the [[Chern-Simons theory|Chern-Simons function]] of a ''G''-connection on a 3-manifold (which has to be "framed"). The integer multiple ''k'', called the level, is a parameter of the theory and ''k'' → ∞ gives the classical limit. This theory can be naturally coupled with the ''d'' = 0 theory to produce a "relative" theory. The details have been described by Witten who shows that partition function for a (framed) link in the 3-sphere is just the value of the [[Jones polynomial]] for a suitable root of unity. The theory can be defined over the relevant [[cyclotomic field]]. By considering [[Riemann surface]] with boundary, we can couple it to the ''d'' = 1 conformal theory instead of coupling ''d'' = 2 theory to ''d'' = 0. This theory has been developed as the Jones-Witten theory and turned out to be the trigger binding the [[knot theory]] and the quantum theory.
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| ====''d'' = 3====
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| Donaldson has defined integer invariant of smooth 4-manifolds by using moduli spaces of SU(2)-instantons. These invariants are polynomials on the second homology. Thus 4-manifolds should have extra data consisting of the symmetric algebra of ''H''<sub>2</sub>. {{Harv|Witten|1988a}} has produced a super-symmetric Lagrangian which formally reproduces the Donaldson theory. Witten's formula might be understood as an infinite-dimensional analogue of the [[Gauss-Bonnet theorem]]. At a later date, this theory is further developed and become the [[Seiberg-Witten theory|Seiberg-Witten gauge theory]] which reduces SU(2) to U(1) in ''N'' = 2, ''d'' = 4 gauge theory. The Hamiltonian version of the theory has been developed by Floer in terms of the space of connections on a 3-manifold. Floer uses the [[Chern-Simons theory|Chern-Simons function]], which is the Lagrangian of the Jones-Witten theory to modify the Hamiltonian. In detail see {{Harv|Atiyah|1988}}. {{Harv|Witten|1988a}} has also shown how one can couple the ''d'' = 3 and ''d'' = 1 theories together: this is quite analogous to the coupling between ''d'' = 2 and ''d'' = 0 in the Jones-Witten theory.
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| Now, it isn't considered on a fixed dimension but on all the dimensions at the same time, namely, topological field theory is viewed as a [[functor]].
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| ===The case of a fixed spacetime===
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| Let ''Bord<sub>M</sub>'' be the category whose morphisms are ''n''-dimensional [[submanifold]]s of ''M'' and whose objects are [[connected space|connected]] components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are [[homotopy|homotopic]] via submanifolds of ''M'', and so form the quotient category ''hBord<sub>M</sub>'': The objects in ''hBord<sub>M</sub>'' are the objects of ''Bord<sub>M</sub>'', and the morphisms of ''hBord<sub>M</sub>'' are homotopy equivalence classes of morphisms in ''Bord<sub>M</sub>''. A TQFT on ''M'' is a [[symmetric monoidal functor]] from ''hBord<sub>M</sub>'' to the category of vector spaces.
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| Note that cobordisms can, if their boundaries match up, be sewn together to form a new bordism. This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece.
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| There is an [[equivalence of categories]] between the category of 2-dimensional topological quantum field theories and the category of commutative [[Frobenius algebra]]s.
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| ===All ''n''-dimensional spacetimes at once===
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| [[File:Pair of pants cobordism (pantslike).svg|thumb|The [[pair of pants (mathematics)|pair of pants]] is a (1+1)-dimensional bordism, which corresponds to a product or coproduct in a 2-dimensional TQFT.]]
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| To consider all spacetimes at once, it is necessary to replace ''hBord<sub>M</sub>'' by a larger category. So let ''Bord<sub>n</sub>'' 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<sub>n</sub>''.) As above, regard two morphisms in ''Bord<sub>n</sub>'' as equivalent if they are homotopic, and form the quotient category ''hBord<sub>n</sub>''. ''Bord<sub>n</sub>'' 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<sub>n</sub>'' to the category of vector spaces, which takes disjoint unions of bordisms to the tensor product of them.
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| For example, for (1+1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with a [[pair of pants (mathematics)|pair of pants]] gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative, while the map associated with a disk gives a counit (trace) or unit (scalars), depending on grouping of boundary, and thus (1+1)-dimension TQFTs correspond to [[Frobenius algebra]]s.
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| Furthermore, we consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds that are related by the above bordisms, then obtain ample and important examples.
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| ===Development at a later time===
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| Looking at the development of topological quantum field theory we should consider that it has many applications to [[Seiberg-Witten theory|Seiberg-Witten gauge theory]], [[topological string theory]], the relationship between [[knot theory]] and quantum theory, and [[quantum knot invariant]]s. Furthermore it has provided objects of great interest to both mathematics and physics. Also of important recent interest is non-local operators in TQFT.({{harvtxt|Gukov|Kapustin|2013}}) If string theory is viewed as the fundamental, then non-local TQFTs can be viewed as non-physical models that provide a computationally efficient approximation to local string theory.
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| ==See also==
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| *[[Quantum topology]]
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| *[[Topological defect]]
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| *[[Topological entropy in physics]]
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| *[[Topological order]]
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| *[[Topological quantum number]]
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| *[[Topological string theory]]
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| *[[Arithmetic topology]]
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| *[[Cobordism hypothesis]]
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| ==References==
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| <references />
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| *{{Citation | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | title=Topological quantum field theories | url=http://www.numdam.org/item?id=PMIHES_1988__68__175_0 | id={{MathSciNet | id = 1001453}} | year=1989 | journal=[[Publications Mathématiques de l'IHÉS]] | issue=68 | pages=175–186 | doi=10.1007/BF02698547 | volume=68}}
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| *{{Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Super-symmetry and Morse Theory | url=http://intlpress.com/JDG/archive/1982/17-4-661.pdf | year=1982 | journal=[[J. Diff Geom.]] | pages=661–692 | volume=17}}
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| * {{Citation | last=Lurie | first=Jacob | title = On the Classification of Topological Field Theories
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| |url = http://www-math.mit.edu/~lurie/papers/cobordism.pdf }}
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| *{{Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Topological quantum field theory | url=http://projecteuclid.org/euclid.cmp/1104161738 | id={{MathSciNet | id = 953828}} | year=1988a | journal=Communications in Mathematical Physics | volume=117 | issue=3 | pages=353–386 | doi=10.1007/BF01223371}}
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| *{{Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Topological sigma models | url=http://prjecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104162092 | year=1988b | journal=Communications in Mathematical Physics | volume=118 | issue=3 | pages=411–449 }}
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| *{{Cite journal | last1=Atiyah | first1=Michael | author1-link=Michael Atiyah | title=New invariants of three and four dimensional manifolds | year=1988 | journal=Proc. Symp. Pure Math., 48, American Math. Soc. | pages=285–299 | volume=48}}
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| *{{Cite journal |first1=Sergei |last1=Gukov |first2=Anton |last2=Kapustin | title=Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories |year=2013 |journal=[[JHEP]] |url=http://inspirehep.net/search?ln=en&ln=en&p=find+a+gukov&of=hb&action_search=Search&sf=&so=d&rm=&rg=25&sc=0 |ref=harv}}
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| {{Quantum field theories}}
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| [[Category:Quantum field theory]]
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| [[Category:Topology]]
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