Chern-Simons theory
Encyclopedia
The Chern–Simons theory is a 3-dimensional topological quantum field theory
of Schwarz type, introduced by Edward Witten
. It is so named because its action
is proportional to the integral of the Chern–Simons 3-form.
In condensed matter physics
, Chern–Simons theory describes the topological order
in fractional quantum Hall effect
states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial.
A particular Chern–Simons theory is specified by a choice of simple Lie group
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
theory is well-defined
when the level is an integer and the gauge field strength
vanishes on all boundaries
of the 3-dimensional spacetime.
3-manifold
M, with or without boundary. As these theories are Schwarz-type topological theories, no metric
needs to be introduced on M.
Chern–Simons theory is a gauge theory
, which means that a classical
configuration in the Chern–Simons theory on M with gauge group 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
g of the Lie group
G. In general the connection A is only defined on individual coordinate patches, and the values of A on different patches are related by maps known as gauge transformations. These are characterized by the assertion that the covariant derivative
, which is the sum of the exterior derivative
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
. It also transforms in the adjoint representation.
S of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form
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
the field equation
is explicitly
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
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.
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
. There is no preferred notion of time in a Schwarz-type topological field theory and so one can impose that Σ be Cauchy surface
s, in fact a state can be defined on any surface.
Σ is 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. Conformal blocks are locally holomorphic and antiholomorphic factors whose products sum to the correlation function
s of a 2-dimensional conformal field theory.
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 representation
s of the affine 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.
s of Chern–Simons theory are the n-point correlation function
s 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 representions R.
More concretely, given an irreducible representation R and a loop K in M one may define the Wilson loop
by
where A is the connection 1-form and we take the Cauchy principal value
of the contour integral
and
is the path-ordered exponential.
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
times the HOMFLY polynomial. In particular when N = 2 the HOMFLY polynomial reduces to the Jones polynomial. 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 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
and Rudolf Peierls
to define apparently divergent quantities in quantum field theory
in 1934.
Sir Michael Atiyah has shown that there exists a canonical choice of 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 2πi/(k + N) times the linking number of L with itself.
, a U(N) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold X arises as the string field theory
of open strings ending on a D-brane
wrapping X in the A-model topological string
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.
known as a G Wess–Zumino–Witten model on the boundary. In addition the U(N) and SO(N) Chern–Simons theories at large N are well approximated by matrix model
s.
is unphysical due to an analogy to Chern–Simons state resulting in negative helicity
and energy. There are disagreements to Witten's conclusions.
if this term is added to the action of Maxwell's theory of electrodynamics. This term can be induced by integrating over a massive charged Dirac field. It also appears for example in the quantum Hall effect. Ten and eleven dimensional generalizations of Chern–Simons terms appear in the actions of all ten and eleven dimensional supergravity
theories.
s then, due to the parity anomaly
, when integrated out they lead to a pure Chern–Simons theory with a one-loop renormalization
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.
Topological quantum field theory
A topological quantum field theory is a quantum field theory which computes topological invariants....
of Schwarz type, introduced by Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
. It is so named because its action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
is proportional to the integral of the Chern–Simons 3-form.
In condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
, Chern–Simons theory describes the topological order
Topological order
In physics, topological order is a new kind of order in a quantum state that is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations...
in fractional quantum Hall effect
Fractional quantum Hall effect
The fractional quantum Hall effect is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of e^2/h. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations...
states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial.
A particular Chern–Simons theory is specified by a choice of simple Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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
Partition function (quantum field theory)
In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
of the quantum
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
theory is well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
when the level is an integer and the gauge field strength
Field strength
In physics, the field strength of a field is the magnitude of its vector value.In theoretical physics, field strength is another name for the curvature form...
vanishes on all boundaries
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of the 3-dimensional spacetime.
Configurations
Chern–Simons theories can be defined on any topologicalTopological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
M, with or without boundary. As these theories are Schwarz-type topological theories, no metric
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
needs to be introduced on M.
Chern–Simons theory is a gauge theory
Gauge theory
In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
, which means that a classical
Classical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
configuration in the Chern–Simons theory on M with gauge group G is described by a principal G-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
on M. The connection
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
of this bundle is characterized by a connection one-form A which is valued in the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
g of the Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
G. In general the connection A is only defined on individual coordinate patches, and the values of A on different patches are related by maps known as gauge transformations. These are characterized by the assertion that the covariant derivative
Gauge covariant derivative
The gauge covariant derivative is like a generalization of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations...
, which is the sum of the exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
operator d and the connection A, transforms in the adjoint representation
Adjoint representation
In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
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
Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry.-Definition:...
or field strength
Field strength
In physics, the field strength of a field is the magnitude of its vector value.In theoretical physics, field strength is another name for the curvature form...
. It also transforms in the adjoint representation.
Dynamics
The actionAction (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
S of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form
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
the field equation
Field equation
A field equation is an equation in a physical theory that describes how a fundamental force interacts with matter...
is explicitly
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
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
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 quantizeCanonical quantization
In physics, canonical quantization is a procedure for quantizing a classical theory while attempting to preserve the formal structure of the classical theory, to the extent possible. Historically, this was Werner Heisenberg's route to obtaining quantum mechanics...
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
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. There is no preferred notion of time in a Schwarz-type topological field theory and so one can impose that Σ be Cauchy surface
Cauchy surface
Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future uniquely....
s, in fact a state can be defined on any surface.
Σ is 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
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
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. Conformal blocks are locally holomorphic and antiholomorphic factors whose products sum to the correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
s of a 2-dimensional conformal field theory.
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 representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
s of the affine Lie algebra
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. It is a Kac–Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1...
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.
Wilson loops
The observableObservable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
s of Chern–Simons theory are the n-point correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
s 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
Representation of a Lie group
In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie...
R of G. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to irreducible representions R.
More concretely, given an irreducible representation R and a loop K in M one may define the Wilson loop
bywhere A is the connection 1-form and we take the Cauchy principal value
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.-Formulation:...
of the contour integral
and
is the path-ordered exponential.HOMFLY and Jones polynomials
Consider a link L in M, which is a collection of l disjoint loops. A particularly interesting observable is the l-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 functionPartition function (quantum field theory)
In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
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
times the HOMFLY polynomial. In particular when N = 2 the HOMFLY polynomial reduces to the Jones polynomial. 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
Linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...
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
Framing
Framing or enframing may refer to:* Framing , the most common carpentry work* Framing or Framing effect , terminology used in communication theory, sociology, and other disciplines where it relates to the construction and presentation of a fact or issue "framed" from a particular perspective*...
for each loop, which is a choice of preferred nonzero normal vector 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
Regularization (physics)
-Introduction:In physics, especially quantum field theory, regularization is a method of dealing with infinite, divergent, and non-sensical expressions by introducing an auxiliary concept of a regulator...
procedure introduced by Paul Dirac
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
and Rudolf Peierls
Rudolf Peierls
Sir Rudolf Ernst Peierls, CBE was a German-born British physicist. Rudolf Peierls had a major role in Britain's nuclear program, but he also had a role in many modern sciences...
to define apparently divergent quantities in quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
in 1934.
Sir Michael Atiyah has shown that there exists a canonical choice of 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 2πi/(k + N) times the linking number of L with itself.
Topological string theories
In the context of string theoryString theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
, 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
String field theory is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory...
of open strings ending on a D-brane
D-brane
In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Hořava in 1989...
wrapping X in the A-model topological string
Topological string theory
In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry...
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 2-dimensional conformal field theoryConformal field theory
A conformal field theory is a quantum field theory that is invariant under conformal transformations...
known as a G Wess–Zumino–Witten model on the boundary. In addition the U(N) and SO(N) Chern–Simons theories at large N are well approximated by matrix model
Matrix model
The term matrix model may refer to one of several concepts:* In theoretical physics, a matrix model is a system with matrix-valued physical quantities. See, for example, Lax pair....
s.
Chern–Simons, the Kodama wavefunction and loop quantum gravity
Edward Witten argued that the Kodama state in loop quantum gravityLoop quantum gravity
Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
is unphysical due to an analogy to Chern–Simons state resulting in negative helicity
Helicity
The term helicity has several meanings. In physics, all referring to a phenomenon that resembles a helix. See:*helicity , the extent to which corkscrew-like motion occurs...
and energy. There are disagreements to Witten's conclusions.
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 photonPhoton
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
if this term is added to the action of Maxwell's theory of electrodynamics. This term can be induced by integrating over a massive charged Dirac field. It also appears for example in the quantum Hall effect. Ten and eleven dimensional generalizations of Chern–Simons terms appear in the actions of all ten and eleven dimensional supergravity
Supergravity
In theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity. Together, these imply that, in supergravity, the supersymmetry is a local symmetry...
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 fermionMajorana fermion
In physics, a Majorana fermion is a fermion which is its own anti-particle. The term is used in opposition to Dirac fermion, which describes particles that differ from their antiparticles...
s then, due to the parity anomaly
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....
, when integrated out they lead to a pure Chern–Simons theory with a one-loop renormalization
Renormalization
In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....
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
- Chern–Simons form
- Topological quantum field theoryTopological quantum field theoryA topological quantum field theory is a quantum field theory which computes topological invariants....
- Alexander polynomialAlexander polynomialIn mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
- 2+1D topological gravity