Ingauge theory andmathematical physics, atopological quantum field theory (ortopological field theory orTQFT) is aquantum field theory that computestopological invariants.

While TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things,knot theory and the theory offour-manifolds inalgebraic topology, and to the theory ofmoduli spaces inalgebraic geometry.Donaldson,Jones,Witten, andKontsevich have all wonFields Medals for mathematical work related to topological field theory.

Incondensed matter physics, topological quantum field theories are the low-energy effective theories oftopologically ordered states, such asfractional quantum Hall states,string-net condensed states, and otherstrongly correlated quantum liquid states.

In a topological field theory,correlation functions aremetric-independent, so they remain unchanged under any deformation ofspacetime and are thereforetopological invariants.

Topological field theories are not very interesting on flatMinkowski spacetime used in particle physics. Minkowski space can becontracted to a point, so a TQFT applied to Minkowski space results in trivial topological invariants. Consequently, TQFTs are usually applied to curved spacetimes, such as, for example,Riemann surfaces. Most of the known topological field theories aredefined on spacetimes of dimension less than five. It seems that a few higher-dimensional theories exist, but they are not very well understood^{[citation needed]}.

Quantum gravity is believed to bebackground-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. This has prompted ongoing theoretical investigations into 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 topologicalsigma model targets infinite-dimensional projective space, and if such a thing could be defined it would havecountably infinitely many degrees of freedom.)

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. See (Schwarz 2000).

InSchwarz-type TQFTs, thecorrelation functions orpartition functions of the system are computed by the path integral of metric-independent action functionals. For instance, in theBF 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

${\displaystyle S=\underset{M}{\int }BF}$

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 toA. Schwarz; its action functional is:

${\displaystyle S=\underset{M}{\int }A\wedge dA.}$

Another more famous example isChern–Simons theory, which can be applied toknot invariants. In general, partition functions depend on a metric but the above examples are metric-independent.

The first example ofWitten-type TQFTs appeared in Witten's paper in 1988 (Witten 1988a), i.e.topological Yang–Mills theory in four dimensions. Though its action functional contains the spacetime metricg_αβ, after atopological twist it turns out to be metric independent. The independence of the stress-energy tensorT^αβ of the system from the metric depends on whether theBRST-operator is closed. Following Witten's example many other examples can be found instring theory.

Witten-type TQFTs arise if the following conditions are satisfied:

1. The action${\displaystyle S}$ of the TQFT has a symmetry, i.e. if${\displaystyle \delta }$ denotes a symmetry transformation (e.g. aLie derivative) then${\displaystyle \delta S=0}$ holds.
2. The symmetry transformation isexact, i.e.${\displaystyle {\delta }^2=0}$
3. There are existingobservables${\displaystyle O_1,\dots ,O_n}$ which satisfy${\displaystyle \delta O_i=0}$ for all${\displaystyle i\in \{1,\dots ,n\}}$.
4. The stress-energy-tensor (or similar physical quantities) is of the form${\displaystyle T^{\alpha \beta }=\delta G^{\alpha \beta }}$ for an arbitrary tensor${\displaystyle G^{\alpha \beta }}$.

As an example (Linker 2015): Given a 2-form field${\displaystyle B}$ with the differential operator${\displaystyle \delta }$ which satisfies${\displaystyle {\delta }^2=0}$, then the action${\displaystyle S=\underset{M}{\int }B\wedge \delta B}$ has a symmetry if${\displaystyle \delta B\wedge \delta B=0}$ since

${\displaystyle \delta S=\underset{M}{\int }\delta (B\wedge \delta B)=\underset{M}{\int }\delta B\wedge \delta B+\underset{M}{\int }B\wedge {\delta }^{2}B=0.}$

Further, the following holds (under the condition that${\displaystyle \delta }$ is independent on${\displaystyle B}$ and acts similarly to afunctional derivative):

${\displaystyle \frac{\delta }{\delta B^{\alpha \beta }}S=\underset{M}{\int }\frac{\delta }{\delta B^{\alpha \beta }}B\wedge \delta B+\underset{M}{\int }B\wedge \delta \frac{\delta }{\delta B^{\alpha \beta }}B=\underset{M}{\int }\frac{\delta }{\delta B^{\alpha \beta }}B\wedge \delta B-\underset{M}{\int }\delta B\wedge \frac{\delta }{\delta B^{\alpha \beta }}B=-2\underset{M}{\int }\delta B\wedge \frac{\delta }{\delta B^{\alpha \beta }}B.}$

The expression${\displaystyle \frac{\delta }{\delta B^{\alpha \beta }}S}$ is proportional to${\displaystyle \delta G}$ with another 2-form${\displaystyle G}$.

Now any averages of observables${\displaystyle ⟨O_i⟩:=\int d\mu O_i{e}^{iS}}$ for the correspondingHaar measure${\displaystyle \mu }$ are independent on the "geometric" field${\displaystyle B}$ and are therefore topological:

${\displaystyle \frac{\delta }{\delta B}⟨O_i⟩=\int d\mu O_{i}i\frac{\delta }{\delta B}S{e}^{iS}\propto \int d\mu O_i\delta G{e}^{iS}=\delta \left(\int d\mu O_{i}G{e}^{iS}\right)=0}$.

The third equality uses the fact that${\displaystyle \delta O_i=\delta S=0}$ and the invariance of the Haar measure under symmetry transformations. Since${\displaystyle \int d\mu O_{i}G{e}^{iS}}$ is only a number, its Lie derivative vanishes.

Atiyah suggested a set of axioms for topological quantum field theory, inspired bySegal's proposed axioms forconformal field theory (subsequently, Segal's idea was summarized inSegal (2001)), and Witten's geometric meaning of supersymmetry inWitten (1982). Atiyah's axioms are constructed by gluing the boundary with a differentiable (topological or continuous) transformation, while Segal's axioms are for conformal transformations. 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 afunctor from a certaincategory ofcobordisms to the category ofvector 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 apply to a TQFT defined on a single fixedn-dimensional Riemannian / Lorentzian spacetimeM or a TQFT defined on alln-dimensional spacetimes at once.

Let Λ be acommutative ring with 1 (for almost all real-world purposes we will have Λ =Z,R orC). Atiyah originally proposed the axioms of a topological quantum field theory (TQFT) in dimensiond defined over a ground ring Λ as following:

• A finitely generated Λ-moduleZ(Σ) associated to each oriented closed smooth d-dimensional manifold Σ (corresponding to thehomotopy axiom),
• An elementZ(M) ∈Z(∂M) associated to each oriented smooth (d + 1)-dimensional manifold (with boundary)M (corresponding to anadditive axiom).

These data are subject to the following axioms (4 and 5 were added by Atiyah):

1. Z isfunctorial with respect to orientation preservingdiffeomorphisms of Σ andM,
2. Z isinvolutory, i.e.Z(Σ*) =Z(Σ)* where Σ* is Σ with opposite orientation andZ(Σ)* denotes the dual module,
3. Z ismultiplicative.
4. Z(${\displaystyle \mathrm{\varnothing }}$) = Λ for the d-dimensional empty manifold andZ(${\displaystyle \mathrm{\varnothing }}$) = 1 for the (d + 1)-dimensional empty manifold.
5. Z(M*) =Z(M) (thehermitian axiom). If${\displaystyle \mathrm{\partial }M={\mathrm{\Sigma }}_0^{\ast }\cup {\mathrm{\Sigma }}_1}$ so thatZ(M) can be viewed as a linear transformation between hermitian vector spaces, then this is equivalent toZ(M*) being theadjoint ofZ(M).

Remark. If for a closed manifoldM we viewZ(M) as a numerical invariant, then for a manifold with a boundary we should think ofZ(M) ∈Z(∂M) as a "relative" invariant. Letf : Σ → Σ be an orientation-preserving diffeomorphism, and identify opposite ends of Σ ×I byf. This gives a manifold Σ_f and our axioms imply

${\displaystyle Z({\mathrm{\Sigma }}_f)=\mathrm{Trace}\mathrm{\Sigma }(f)}$

where Σ(f) is the induced automorphism ofZ(Σ).

Remark. For a manifoldM with boundary Σ we can always form the double${\displaystyle M{\cup }_{\mathrm{\Sigma }}{M}^{\ast }}$ which is a closed manifold. The fifth axiom shows that

${\displaystyle Z\left(M{\cup }_{\mathrm{\Sigma }}{M}^{\ast }\right)=|Z(M){|}^2}$

where on the right we compute the norm in the hermitian (possibly indefinite) metric.

Physically (2) + (4) are related to relativistic invariance while (3) + (5) are indicative of the quantum nature of the theory.

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 spaceZ(Σ) is theHilbert space of the quantum theory and a physical theory, with aHamiltonianH, will have a time evolution operatore^itH or an "imaginary time" operatore^−tH. The main feature oftopological QFTs is thatH = 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 Σ₀ to Σ₁ through an intervening manifoldM with${\displaystyle \mathrm{\partial }M={\mathrm{\Sigma }}_0^{\ast }\cup {\mathrm{\Sigma }}_1}$; this reflects the topology ofM.

If ∂M = Σ, then the distinguished vectorZ(M) in the Hilbert spaceZ(Σ) is thought of as thevacuum state defined byM. For a closed manifoldM the numberZ(M) is thevacuum expectation value. In analogy withstatistical mechanics it is also called thepartition function.

The reason why a theory with a zero Hamiltonian can be sensibly formulated resides in theFeynman path integral approach to QFT. This incorporates relativistic invariance (which applies to general (d + 1)-dimensional "spacetimes") and the theory is formally defined by a suitableLagrangian—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 to the topology ofM.

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 (Atiyah 1988a)(Atiyah 1988b). It contains some newtopological invariants along with some new ideas:Casson invariant,Donaldson invariant,Gromov's theory,Floer homology andJones–Witten theory.

In this case Σ consists of finitely many points. To a single point we associate a vector spaceV =Z(point) and ton-points then-fold tensor product:V^⊗n =V ⊗ … ⊗ V. Thesymmetric group S_n acts onV^⊗n. A standard way to get the quantum Hilbert space is to start with a classicalsymplectic manifold (orphase space) and then quantize it. Let us extendS_n to a compact Lie groupG and consider "integrable" orbits for which the symplectic structure comes from aline bundle, then quantization leads to the irreducible representationsV ofG. This is the physical interpretation of theBorel–Weil theorem or theBorel–Weil–Bott theorem. The Lagrangian of these theories is the classical action (holonomy of the line bundle). Thus topological QFT's withd = 0 relate naturally to the classicalrepresentation theory ofLie groups and thesymmetric group.

We should consider periodic boundary conditions given by closed loops in a compact symplectic manifoldX. Along withWitten (1982) holonomy such loops as used in the case ofd = 0 as a Lagrangian are then used to modify the Hamiltonian. For a closed surfaceM the invariantZ(M) of the theory is the number ofpseudo holomorphic mapsf :M →X in the sense of Gromov (they are ordinaryholomorphic maps ifX is aKähler manifold). If this number becomes infinite i.e. if there are "moduli", then we must fix further data onM. This can be done by picking some pointsP_i and then looking at holomorphic mapsf :M →X withf(P_i) constrained to lie on a fixed hyperplane.Witten (1988b) has written down the relevant Lagrangian for this theory. Floer has given a rigorous treatment, i.e.Floer homology, based on Witten'sMorse theory ideas; for the case when the boundary conditions are over the interval instead of being periodic, the path initial and end-points lie on two fixedLagrangian submanifolds. This theory has been developed asGromov–Witten invariant theory.

Another example isHolomorphicConformal Field Theory. This might not have been considered strictly topological quantum field theory at the time because Hilbert spaces are infinite dimensional. The conformal field theories are also related to the compact Lie groupG in which the classical phase consists of a central extension of theloop group(LG). Quantizing these produces the Hilbert spaces of the theory of irreducible (projective) representations ofLG. The group Diff₊(S¹) now substitutes for the symmetric group and plays an important role. As a result, the partition function in such theories depends oncomplex structure, thus it is not purely topological.

Jones–Witten theory is the most important theory in this case. Here the classical phase space, associated with a closed surface Σ is the moduli space of a flatG-bundle over Σ. The Lagrangian is an integer multiple of theChern–Simons function of aG-connection on a 3-manifold (which has to be "framed"). The integer multiplek, called the level, is a parameter of the theory andk → ∞ gives the classical limit. This theory can be naturally coupled with thed = 0 theory to produce a "relative" theory. The details have been described by Witten who shows that the partition function for a (framed) link in the 3-sphere is just the value of theJones polynomial for a suitable root of unity. The theory can be defined over the relevantcyclotomic field, seeAtiyah (1988b). By considering aRiemann surface with boundary, we can couple it to thed = 1 conformal theory instead of couplingd = 2 theory tod = 0. This has developed into Jones–Witten theory and has led to the discovery of deep connections betweenknot theory and quantum field theory.

Donaldson has defined the 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 ofH₂.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 theGauss–Bonnet theorem. At a later date, this theory was further developed and became theSeiberg–Witten gauge theory which reduces SU(2) to U(1) inN = 2,d = 4 gauge theory. The Hamiltonian version of the theory has been developed byAndreas Floer in terms of the space of connections on a 3-manifold. Floer uses theChern–Simons function, which is the Lagrangian of Jones–Witten theory to modify the Hamiltonian. For details, seeAtiyah (1988b).Witten (1988a) has also shown how one can couple thed = 3 andd = 1 theories together: this is quite analogous to the coupling betweend = 2 andd = 0 in Jones–Witten theory.

Now, topological field theory is viewed as afunctor, not on a fixed dimension but on all dimensions at the same time.

LetBord_M be the category whose morphisms aren-dimensionalsubmanifolds ofM and whose objects areconnected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they arehomotopic via submanifolds ofM, and so form the quotient categoryhBord_M: The objects inhBord_M are the objects ofBord_M, and the morphisms ofhBord_M are homotopy equivalence classes of morphisms inBord_M. A TQFT onM is asymmetric monoidal functor fromhBord_M to the category of vector spaces.

Note that cobordisms can, if their boundaries match, 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.

There is anequivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutativeFrobenius algebras.

To consider all spacetimes at once, it is necessary to replacehBord_M by a larger category. So letBord_n be the category of bordisms, i.e. the category whose morphisms aren-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 inBord_n.) As above, regard two morphisms inBord_n as equivalent if they are homotopic, and form the quotient categoryhBord_n.Bord_n is amonoidal category under the operation which maps two bordisms to the bordism made from their disjoint union. A TQFT onn-dimensional manifolds is then a functor fromhBord_n to the category of vector spaces, which maps disjoint unions of bordisms to their tensor product.

For example, for (1 + 1)-dimensional bordisms (2-dimensional bordisms between 1-dimensional manifolds), the map associated with apair 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 the grouping of boundary components, and thus (1+1)-dimension TQFTs correspond toFrobenius algebras.

Furthermore, we can consider simultaneously 4-dimensional, 3-dimensional and 2-dimensional manifolds related by the above bordisms, and from them we can obtain ample and important examples.

Looking at the development of topological quantum field theory, we should consider its many applications toSeiberg–Witten gauge theory,topological string theory, the relationship betweenknot theory and quantum field theory, andquantum knot invariants. Furthermore, it has generated topics of great interest in both mathematics and physics. Also of important recent interest are non-local operators in TQFT (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.

Stochastic (partial) differential equations (SDEs) are the foundation for models of everything in nature above the scale of quantum degeneracy and coherence and are essentially Witten-type TQFTs. All SDEs possess topological orBRST supersymmetry,${\displaystyle \delta }$, and in the operator representation of stochastic dynamics is theexterior derivative, which is commutative with the stochastic evolution operator. This supersymmetry preserves the continuity of phase space by continuous flows, and the phenomenon of supersymmetric spontaneous breakdown by a global non-supersymmetric ground state encompasses such well-established physical concepts aschaos,turbulence,1/f andcrackling noises,self-organized criticality etc. The topological sector of the theory for any SDE can be recognized as a Witten-type TQFT.

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• Atiyah, Michael (1988b)."Topological quantum field theories"(PDF).Publications Mathématiques de l'IHÉS.68 (68):175–186.doi:10.1007/BF02698547.MR 1001453.S2CID 121647908.
• Gukov, Sergei; Kapustin, Anton (2013). "Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories".arXiv:1307.4793 [hep-th].
• Linker, Patrick (2015)."Topological Dipole Field Theory"(PDF).The Winnower.2: e144311.19292.doi:10.15200/winn.144311.19292.
• Lurie, Jacob (2009). "On the Classification of Topological Field Theories".arXiv:0905.0465 [math.CT].
• Schwarz, Albert (2000). "Topological quantum field theories".arXiv:hep-th/0011260.
• Segal, Graeme (2001). "Topological structures in string theory".Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.359 (1784):1389–1398.Bibcode:2001RSPTA.359.1389S.doi:10.1098/rsta.2001.0841.S2CID 120834154.
• Witten, Edward (1982)."Super-symmetry and Morse Theory".Journal of Differential Geometry.17 (4):661–692.doi:10.4310/jdg/1214437492.
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• Witten, Edward (1988b)."Topological sigma models".Communications in Mathematical Physics.118 (3):411–449.Bibcode:1988CMaPh.118..411W.doi:10.1007/bf01466725.S2CID 34042140.

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