TheChern–Simons theory is a 3-dimensionaltopological quantum field theory ofSchwarz type. It was discovered first by mathematical physicistAlbert Schwarz. It is named after mathematiciansShiing-Shen Chern andJames Harris Simons, who introduced theChern–Simons 3-form. In the Chern–Simons theory, theaction is proportional to the integral of the Chern–Simons 3-form.

Incondensed-matter physics, Chern–Simons theory describescomposite fermions and thetopological order infractional quantum Hall effect states. In mathematics, it has been used to calculateknot invariants andthree-manifold invariants such as theJones polynomial.^[1]

Particularly, Chern–Simons theory is specified by a choice of simpleLie group G known as the gauge group of the theory and also a number referred to as thelevel of the theory, which is a constant that multiplies the action. The action is gauge dependent, however thepartition function of thequantum theory iswell-defined when the level is an integer and the gaugefield strength vanishes on allboundaries of the 3-dimensional spacetime.

It is also the central mathematical object in theoretical models fortopological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abeliananyonic model of a TQC, the Yang–Lee–Fibonacci model.^[2][3]

The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related tofusion rules andconformal blocks inconformal field theory, and in particularWZW theory.^[1][4]

In the 1940sS. S. Chern andA. Weil studied the global curvature properties of smooth manifoldsM asde Rham cohomology (Chern–Weil theory), which is an important step in the theory ofcharacteristic classes indifferential geometry. Given a flatG-principal bundleP onM there exists a unique homomorphism, called theChern–Weil homomorphism, from the algebra ofG-adjoint invariant polynomials ong (Lie algebra ofG) to the cohomology${\displaystyle H^{\ast }(M,\mathbb{R})}$. If the invariant polynomial is homogeneous one can write down concretely anyk-form of the closed connectionω as some 2k-form of the associated curvature form Ω ofω.

In 1974 S. S. Chern andJ. H. Simons had concretely constructed a (2k − 1)-formdf(ω) such that

${\displaystyle dTf(\omega )=f({\mathrm{\Omega }}^k),}$

whereT is the Chern–Weil homomorphism. This form is calledChern–Simons form. Ifdf(ω) is closed one can integrate the above formula

${\displaystyle Tf(\omega )={\int }_{C}f({\mathrm{\Omega }}^k),}$

whereC is a (2k − 1)-dimensional cycle onM. This invariant is calledChern–Simons invariant. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(M) is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as

${\displaystyle \mathrm{CS}(M)={\int }_{s(M)}\frac{1}{2}T{p}_1\in \mathbb{R}/\mathbb{Z},}$

where${\displaystyle p_1}$ is the first Pontryagin number ands(M) is the section of the normal orthogonal bundleP. Moreover, the Chern–Simons term is described as theeta invariant defined by Atiyah, Patodi and Singer.

The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. Theaction integral (path integral) of thefield theory in physics is viewed as theLagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle onM. These explain why the Chern–Simons theory is closely related totopological field theory.

Chern–Simons theories can be defined on anytopological3-manifoldM, with or without boundary. As these theories are Schwarz-type topological theories, nometric needs to be introduced onM.

Chern–Simons theory is agauge theory, which means that aclassical configuration in the Chern–Simons theory onM withgauge groupG is described by aprincipalG-bundle onM. Theconnection of this bundle is characterized by aconnection one-formA which isvalued in theLie algebrag of theLie groupG. In general the connectionA is only defined on individualcoordinate patches, and the values ofA on different patches are related by maps known asgauge transformations. These are characterized by the assertion that thecovariant derivative, which is the sum of theexterior derivative operatord and the connectionA, transforms in theadjoint representation of the gauge groupG. The square of the covariant derivative with itself can be interpreted as ag-valued 2-formF called thecurvature form orfield strength. It also transforms in the adjoint representation.

TheactionS of Chern–Simons theory is proportional to the integral of theChern–Simons 3-form

${\displaystyle S=\frac{k}{4\pi }{\int }_M\text{tr}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A).}$

The constantk is called thelevel of the theory. The classical physics of Chern–Simons theory is independent of the choice of levelk.

Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the fieldA. In terms of the field curvature

${\displaystyle F=dA+A\wedge A}$

thefield equation is explicitly

${\displaystyle 0=\frac{\delta S}{\delta A}=\frac{k}{2\pi }F.}$

The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to beflat. Thus the classical solutions toG Chern–Simons theory are the flat connections of principalG-bundles onM. Flat connections are determined entirely by holonomies around noncontractible cycles on the baseM. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from thefundamental group ofM to the gauge groupG up to conjugation.

IfM has a boundaryN then there is additional data which describes a choice of trivialization of the principalG-bundle onN. Such a choice characterizes a map fromN toG. The dynamics of this map is described by theWess–Zumino–Witten (WZW) model onN at levelk.

Tocanonically quantize Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in aHilbert space. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be aCauchy surface, in fact, a state can be defined on any surface.

Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model.Witten has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space ofconformal blocks of the G WZW model at level k.

For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrablerepresentations of theaffine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.

Theobservables of Chern–Simons theory are then-pointcorrelation functions of gauge-invariant operators. The most often studied class of gauge invariant operators areWilson loops. A Wilson loop is the holonomy around a loop inM, traced in a givenrepresentationR ofG. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention toirreducible representationsR.

More concretely, given an irreducible representationR and a loopK inM, one may define the Wilson loop${\displaystyle W_R(K)}$ by

${\displaystyle W_R(K)={\mathrm{Tr}}_R\mathcal{P}\exp \left(i{\oint }_{K}A\right)}$

whereA is the connection 1-form and we take theCauchy principal value of thecontour integral and${\displaystyle \mathcal{P}\exp }$ is thepath-ordered exponential.

Consider a linkL inM, which is a collection ofℓ disjoint loops. A particularly interesting observable is theℓ-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in thefundamental representation ofG. One may form a normalized correlation function by dividing this observable by thepartition functionZ(M), which is just the 0-point correlation function.

In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to knownknot polynomials. For example, inG = U(N) Chern–Simons theory at levelk the normalized correlation function is, up to a phase, equal to

${\displaystyle \frac{\sin (\pi /(k+N))}{\sin (\pi N/(k+N))}}$

times theHOMFLY polynomial. In particular whenN = 2 the HOMFLY polynomial reduces to theJones polynomial. In the SO(N) case, one finds a similar expression with theKauffman polynomial.

The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. Thelinking number of a loop with itself enters into the calculation of the partition function, but this number is not invariant under small deformations and in particular, is not a topological invariant. This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzeronormal vector at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of thepoint-splittingregularization procedure introduced byPaul Dirac andRudolf Peierls to define apparently divergent quantities inquantum field theory in 1934.

Sir Michael Atiyah has shown that there exists a canonical choice of 2-framing,^[5] which is generally used in the literature today and leads to a well-defined linking number. With the canonical framing the above phase is the exponential of 2πi/(k + N) times the linking number ofL with itself.

Problem (Extension of Jones polynomial to general 3-manifolds)　

"The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?"

See section 1.1 of this paper^[6] for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots.^[7] It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-spaceR³). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.

In the context ofstring theory, aU(N) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifoldX arises as thestring field theory of open strings ending on aD-brane wrappingX in theA-model topological string theory onX. TheB-model topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory.

Chern–Simons theories are related to many other field theories. For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving atwo-dimensional conformal field theory known as a GWess–Zumino–Witten model on the boundary. In addition theU(N) and SO(N) Chern–Simons theories at largeN are well approximated bymatrix models.

In 1982,S. Deser,R. Jackiw and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which theEinstein–Hilbert action in gravity theory is modified by adding the Chern–Simons term. (Deser, Jackiw & Templeton (1982))

In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions (Jackiw & Pi (2003)) and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.

The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is

${\displaystyle \mathrm{CS}(\mathrm{\Gamma })=\frac{1}{2{\pi }^2}\int d^{3}x{\epsilon }^{ijk}({\mathrm{\Gamma }}_{iq}^p{\mathrm{\partial }}_j{\mathrm{\Gamma }}_{kp}^q+\frac{2}{3}{\mathrm{\Gamma }}_{iq}^p{\mathrm{\Gamma }}_{jr}^q{\mathrm{\Gamma }}_{kp}^r).}$

This variation gives theCotton tensor

${\displaystyle =-\frac{1}{2\sqrt{g}}({\epsilon }^{mij}{D}_i{R}_j^n+{\epsilon }^{nij}{D}_i{R}_j^m).}$

Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action.

In 2013 Kenneth A. Intriligator andNathan Seiberg solved these 3d Chern–Simons gauge theories and their phases usingmonopoles carrying extra degrees of freedom. TheWitten index of the manyvacua discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua,supersymmetry was computed to be broken. These monopoles were related tocondensed mattervortices. (Intriligator & Seiberg (2013))

TheN = 6 Chern–Simons matter theory is theholographic dual of M-theory on${\displaystyle Ad{S}_4\times S_7}$.

In 2013Kevin Costello defined a closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve.^[8] He later studied the theory in more detail together with Witten and Masahito Yamazaki,^[9][10][11] demonstrating how the gauge theory could be related to many notions inintegrable systems theory, including exactly solvable lattice models (like thesix-vertex model or theXXZ spin chain), integrable quantum field theories (such as theGross–Neveu model,principal chiral model and symmetric space cosetsigma models), theYang–Baxter equation andquantum groups such as theYangian which describe symmetries underpinning the integrability of the aforementioned systems.

The action on the 4-manifold${\displaystyle M=\mathrm{\Sigma }\times C}$ where${\displaystyle \mathrm{\Sigma }}$ is a two-dimensional manifold and${\displaystyle C}$ is a complex curve is$${\displaystyle S={\int }_M\omega \wedge CS(A)}$$where${\displaystyle \omega }$ is ameromorphicone-form on${\displaystyle C}$.

The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massivephoton if this term is added to the action of Maxwell's theory ofelectrodynamics. This term can be induced by integrating over a massive chargedDirac field. It also appears for example in thequantum Hall effect. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions^[12][13] Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensionalsupergravity theories.

If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds nMajorana fermions then, due to theparity anomaly, when integrated out they lead to a pure Chern–Simons theory with a one-looprenormalization of the Chern–Simons level by −n/2, in other words the level k theory with n fermions is equivalent to the levelk − n/2 theory without fermions.

• Gauge theory (mathematics)
• Chern–Simons form
• Topological quantum field theory
• Alexander polynomial
• Jones polynomial
• 2+1D topological gravity
• Skyrmion
• ∞-Chern–Simons theory

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Specific

• "Chern-Simons functional".Encyclopedia of Mathematics.EMS Press. 2001 [1994].

• Quantum field theory

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