In
mathematics, the Chern–Simons forms are certain secondary
characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic class ...
es. The theory is named for
Shiing-Shen Chern
Shiing-Shen Chern (; , ; October 28, 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 geom ...
and
James Harris Simons
James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hedge fund manager, and philanthropist. He is the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his f ...
, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.
Definition
Given a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
and a
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 iden ...
valued
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction ...
over it, we can define a family of
''p''-forms:
In one dimension, the Chern–Simons
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction ...
is given by
:
In three dimensions, the Chern–Simons 3-form is given by
:
In five dimensions, the Chern–Simons 5-form is given by
:
where the curvature F is defined as
:
The general Chern–Simons form
is defined in such a way that
:
where the
wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
is used to define ''F
k''. The right-hand side of this equation is proportional to the ''k''-th
Chern character
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Ya ...
of the connection
.
In general, the Chern–Simons
''p''-form is defined for any odd ''p''.
Application to physics
In 1978,
Albert Schwarz
Albert Solomonovich Schwarz (; russian: А. С. Шварц; born June 24, 1934) is a Soviet and American mathematician and a theoretical physicist educated in the Soviet Union and now a professor at the University of California, Davis.
Early li ...
formulated
Chern–Simons theory
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and Ja ...
, early
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 which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
, using Chern-Simons form.
In the
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 (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
, the
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of Chern-Simons form is a global geometric invariant, and is typically
gauge invariant
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 (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
modulo addition of an integer.
See also
*
Chern–Weil homomorphism
*
Chiral anomaly
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is equivalent to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have mor ...
*
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 which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
*
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 ...
References
Further reading
*
*
{{DEFAULTSORT:Chern-Simons form
Homology theory
Algebraic topology
Differential geometry
String theory