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The Chern–Simons theory is a 3-dimensional
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...
of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians
Shiing-Shen Chern Shiing-Shen Chern (; , ; October 26, 1911 – December 3, 2004) was a Chinese American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...
and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form. In condensed-matter physics, Chern–Simons theory describes composite fermions and the
topological order In physics, topological order describes a state or phase of matter that arises system with non-local interactions, such as entanglement in quantum mechanics, and floppy modes in elastic systems. Whereas classical phases of matter such as gases an ...
in
fractional quantum Hall effect The fractional quantum Hall effect (fractional QHE or FQHE) is the observation of precisely quantized plateaus in the Hall conductance of 2-dimensional (2D) electrons at fractional values of e^2/h, where ''e'' is the electron charge and ''h'' i ...
states. In mathematics, it has been used to calculate
knot invariants In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...
and three-manifold invariants such as the
Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...
. Particularly, Chern–Simons theory is specified by a choice of simple
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
G known as the gauge group of the theory and also a number referred to as the ''level'' of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the
quantum In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ...
theory is
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression (mathematics), expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined ...
when the level is an integer and the gauge
field strength In physics, field strength refers to a value in a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E''). For example, an electromagnetic field has both electric field strength and magnetic field strength. Field str ...
vanishes on all boundaries of the 3-dimensional spacetime. It is also the central mathematical object in theoretical models for
topological quantum computer A topological quantum computer is a type of quantum computer. It utilizes anyons, a type of quasiparticle that occurs in two-dimensional systems. The anyons' world lines intertwine to form braids in a three-dimensional spacetime (one temporal ...
s (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian
anyon In physics, an anyon is a type of quasiparticle so far observed only in two-dimensional physical system, systems. In three-dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical proper ...
ic model of a TQC, the Yang–Lee–Fibonacci model. The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in
conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...
, and in particular WZW theory.


The classical theory


Mathematical origin

In the 1940s
S. S. Chern Shiing-Shen Chern (; , ; October 26, 1911 – December 3, 2004) was a Chinese American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...
and A. Weil studied the global curvature properties of smooth manifolds ''M'' as
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
( Chern–Weil theory), which is an important step in the theory of characteristic classes in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
. Given a flat ''G''-
principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
''P'' on ''M'' there exists a unique homomorphism, called the
Chern–Weil homomorphism In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing ...
, from the algebra of ''G''-adjoint invariant polynomials on ''g'' (Lie algebra of ''G'') to the cohomology H^*(M,\mathbb). If the invariant polynomial is homogeneous one can write down concretely any ''k''-form of the closed connection ''ω'' as some 2''k''-form of the associated curvature form Ω of ''ω''. In 1974 S. S. Chern and J. H. Simons had concretely constructed a (2''k'' − 1)-form ''df''(''ω'') such that :dTf(\omega)=f(\Omega^k), where ''T'' is the Chern–Weil homomorphism. This form is called
Chern–Simons form In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from whic ...
. If ''df''(''ω'') is closed one can integrate the above formula :Tf(\omega)=\int_C f(\Omega^k), where ''C'' is a (2''k'' − 1)-dimensional cycle on ''M''. This invariant is called Chern–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 :\operatorname(M)=\int_\tfracTp_1\in\mathbb/\mathbb, where p_1 is the first Pontryagin number and ''s''(''M'') is the section of the normal orthogonal bundle ''P''. Moreover, the Chern–Simons term is described as the
eta invariant In mathematics, the eta invariant of a self-adjoint elliptic differential operator on a compact manifold is formally the number of positive eigenvalues minus the number of negative eigenvalues. In practice both numbers are often infinite so are de ...
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. The action integral ( path integral) of the field theory in physics is viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on ''M''. These explain why the Chern–Simons theory is closely related to topological field theory.


Configurations

Chern–Simons theories can be defined on any
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
3-manifold In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
''M'', with or without boundary. As these theories are Schwarz-type topological theories, no
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
needs to be introduced on ''M''. Chern–Simons theory is a
gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, which means that a classical configuration in the Chern–Simons theory on ''M'' with
gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...
''G'' is described by a principal ''G''-bundle on ''M''. The connection of this bundle is characterized by a connection one-form ''A'' which is valued in the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
g of the
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
''G''. In general the connection ''A'' is only defined on individual
coordinate patch In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies th ...
es, and the values of ''A'' on different patches are related by maps known as
gauge transformations In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
. These are characterized by the assertion that the
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
, which is the sum of the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
operator ''d'' and the connection ''A'', transforms in the adjoint representation of the gauge group ''G''. The square of the covariant derivative with itself can be interpreted as a g-valued 2-form ''F'' called the curvature form or
field strength In physics, field strength refers to a value in a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E''). For example, an electromagnetic field has both electric field strength and magnetic field strength. Field str ...
. It also transforms in the adjoint representation.


Dynamics

The action ''S'' of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form :S=\frac\int_M \text\,(A\wedge dA+\tfracA\wedge A\wedge A). The constant ''k'' is called the ''level'' of the theory. The classical physics of Chern–Simons theory is independent of the choice of level ''k''. Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the field ''A''. In terms of the field curvature :F = dA + A \wedge A \, the field equation is explicitly :0=\frac=\frac 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 be ''flat''. Thus the classical solutions to ''G'' Chern–Simons theory are the flat connections of principal ''G''-bundles on ''M''. Flat connections are determined entirely by holonomies around noncontractible cycles on the base ''M''. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
of ''M'' to the gauge group ''G'' up to conjugation. If ''M'' has a boundary ''N'' then there is additional data which describes a choice of trivialization of the principal ''G''-bundle on ''N''. Such a choice characterizes a map from ''N'' to ''G''. The dynamics of this map is described by the Wess–Zumino–Witten (WZW) model on ''N'' at level ''k''.


Quantization

To canonically 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 a
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a Cauchy 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 of conformal 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 integrable representations of the
affine Lie algebra In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody ...
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.


Observables


Wilson loops

The
observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
s of Chern–Simons theory are the ''n''-point correlation functions of gauge-invariant operators. The most often studied class of gauge invariant operators are Wilson loops. A Wilson loop is the holonomy around a loop in ''M'', traced in a given representation ''R'' of ''G''. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to
irreducible representation In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s ''R''. More concretely, given an irreducible representation ''R'' and a loop ''K'' in ''M'', one may define the Wilson loop W_R(K) by : W_R(K) =\operatorname_R \, \mathcal \exp\left(i \oint_K A\right) where ''A'' is the connection 1-form and we take the Cauchy principal value of the
contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
and \mathcal \exp is the path-ordered exponential.


HOMFLY and Jones polynomials

Consider a link ''L'' in ''M'', which is a collection of ''â„“'' disjoint loops. A particularly interesting observable is the ''â„“''-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the fundamental representation of ''G''. One may form a normalized correlation function by dividing this observable by the partition function ''Z''(''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 known knot polynomials. For example, in ''G'' = ''U''(''N'') Chern–Simons theory at level ''k'' the normalized correlation function is, up to a phase, equal to :\frac times the
HOMFLY polynomial In the mathematics, mathematical field of knot theory, the HOMFLY polynomial or HOMFLYPT polynomial, sometimes called the generalized Jones polynomial, is a 2-variable knot polynomial, i.e. a knot invariant in the form of a polynomial of variables ...
. In particular when ''N'' = 2 the HOMFLY polynomial reduces to the
Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...
. In the SO(''N'') case, one finds a similar expression with the Kauffman polynomial. The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The linking 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 nonzero
normal vector In geometry, a normal is an object (e.g. a line, ray, or vector) that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the infinite straight line perpendicular to the tangent line to the cu ...
at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of the point-splitting regularization procedure introduced by
Paul Dirac Paul Adrien Maurice Dirac ( ; 8 August 1902 – 20 October 1984) was an English mathematician and Theoretical physics, theoretical physicist who is considered to be one of the founders of quantum mechanics. Dirac laid the foundations for bot ...
and
Rudolf Peierls Sir Rudolf Ernst Peierls, (; ; 5 June 1907 – 19 September 1995) was a German-born British physicist who played a major role in Tube Alloys, Britain's nuclear weapon programme, as well as the subsequent Manhattan Project, the combined Allied ...
to define apparently divergent quantities in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
in 1934. Sir Michael Atiyah has shown that there exists a canonical choice of 2-framing, 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 2Ï€''i''/(''k'' + ''N'') times the linking number of ''L'' 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 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. 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-space R3). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.


Relationships with other theories


Topological string theories

In the context of
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
, a ''U''(''N'') Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold ''X'' arises as the
string field theory String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of ...
of open strings ending on a
D-brane In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes are typically classified by their spatial dimensi ...
wrapping ''X'' in the A-model topological string theory on ''X''. The B-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.


WZW and matrix models

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 a
two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fi ...
known as a G
Wess–Zumino–Witten model In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Ed ...
on the boundary. In addition the ''U''(''N'') and SO(''N'') Chern–Simons theories at large ''N'' are well approximated by matrix models.


Chern–Simons gravity theory

In 1982, S. Deser, R. Jackiw and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the
Einstein–Hilbert action The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the action is given as :S = \int R \sqrt ...
in gravity theory is modified by adding the Chern–Simons term. () In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions () 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 :\operatorname(\Gamma)=\frac\int d^3x\varepsilon^\biggl(\Gamma^p_\partial_j\Gamma^q_+\frac\Gamma^p_\Gamma^q_\Gamma^r_\biggr). This variation gives the
Cotton tensor In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension ''n'' is a third-order tensor concomitant of the metric. The vanishing of the Cotton tensor for is necessary and sufficient condition for the manifold to ...
:=-\frac\bigl(\varepsilon^D_i R^n_j+\varepsilon^D_i R^m_j). 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.


Chern–Simons matter theories

In 2013 Kenneth A. Intriligator and Nathan Seiberg solved these 3d Chern–Simons gauge theories and their phases using monopoles carrying extra degrees of freedom. The Witten index of the many vacua discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua,
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
was computed to be broken. These monopoles were related to condensed matter
vortices In fluid dynamics, a vortex (: vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
. () The ''N'' = 6 Chern–Simons matter theory is the holographic dual of M-theory on AdS_4\times S_7.


Four-dimensional Chern–Simons theory

In 2013 Kevin 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. He later studied the theory in more detail together with Witten and Masahito Yamazaki, demonstrating how the gauge theory could be related to many notions in integrable systems theory, including exactly solvable lattice models (like the six-vertex model or the XXZ spin chain), integrable quantum field theories (such as the Gross–Neveu model, principal chiral model and symmetric space coset
sigma model In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold, although it is most commonly taken to be either a Lie group or ...
s), the Yang–Baxter equation and quantum groups such as the Yangian which describe symmetries underpinning the integrability of the aforementioned systems. The action on the 4-manifold M = \Sigma \times C where \Sigma is a two-dimensional manifold and C is a complex curve is S = \int_M \omega \wedge CS(A) where \omega is a meromorphic
one-form In differential geometry, a one-form (or covector field) on a differentiable manifold is a differential form of degree one, that is, a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the to ...
on C.


Chern–Simons terms in other theories

The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massive
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
if this term is added to the action of Maxwell's theory of
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. This term can be induced by integrating over a massive charged
Dirac field In quantum field theory, a fermionic field is a quantum field whose Quantum, quanta are fermions; that is, they obey Fermi–Dirac statistics. Fermionic fields obey canonical anticommutation relations rather than the canonical commutation relation ...
. It also appears for example in the
quantum Hall effect The quantum Hall effect (or integer quantum Hall effect) is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance exhi ...
. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
theories.


One-loop renormalization of the level

If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds n Majorana fermions then, due to the
parity anomaly In theoretical physics a quantum field theory is said to have a parity anomaly if its classical action is invariant under a change of parity of the universe, but the quantum theory is not invariant. This kind of anomaly can occur in odd-dimen ...
, when integrated out they lead to a pure Chern–Simons theory with a one-loop
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
of the Chern–Simons level by −''n''/2, in other words the level k theory with n fermions is equivalent to the level ''k'' − ''n''/2 theory without fermions.


See also

*
Gauge theory (mathematics) In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of Connection (mathematics), connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should ...
*
Chern–Simons form In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from whic ...
*
Topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...
* Alexander polynomial *
Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...
* 2+1D topological gravity *
Skyrmion In particle theory, the skyrmion () is a topologically stable field configuration of a certain class of non-linear sigma models. It was originally proposed as a model of the nucleon by (and named after) Tony Skyrme in 1961. As a topological solito ...
* ∞-Chern–Simons theory


References

* * * * * * * * * * * * ;Specific


External links

* {{DEFAULTSORT:Chern-Simons theory Quantum field theory