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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 to which the bundle is "twisted" and whether it possesses sections. Characterist ...
es. The theory is named for
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 James Harris Simons (April 25, 1938 – May 10, 2024) was an American hedge fund manager, investor, mathematician, and philanthropist. At the time of his death, Simons's net worth was estimated to be $31.4 billion, making him the 55th-richest ...
, 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 N ...
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 ident ...
valued
1-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 t ...
\mathbf over it, we can define a family of ''p''-forms: In one dimension, the Chern–Simons
1-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 t ...
is given by :\operatorname \mathbf In three dimensions, the Chern–Simons 3-form is given by :\operatorname \left \mathbf \wedge \mathbf-\frac \mathbf \wedge \mathbf \wedge \mathbf \right= \operatorname \left d\mathbf \wedge \mathbf + \frac \mathbf \wedge \mathbf \wedge \mathbf\right In five dimensions, the Chern–Simons 5-form is given by : \begin & \operatorname \left \mathbf\wedge\mathbf \wedge \mathbf-\frac \mathbf \wedge\mathbf\wedge\mathbf\wedge\mathbf +\frac \mathbf \wedge \mathbf \wedge \mathbf \wedge \mathbf \wedge\mathbf \right\\ pt= & \operatorname \left d\mathbf\wedge d\mathbf \wedge \mathbf + \frac d\mathbf \wedge \mathbf \wedge \mathbf \wedge \mathbf +\frac \mathbf \wedge \mathbf \wedge \mathbf\wedge\mathbf\wedge\mathbf \right\end where the curvature F is defined as :\mathbf = d\mathbf+\mathbf\wedge\mathbf. The general Chern–Simons form \omega_ is defined in such a way that :d\omega_= \operatorname(F^k), where the
wedge product A wedge is a triangular shaped tool, 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 converting a fo ...
is used to define ''Fk''. The right-hand side of this equation is proportional to the ''k''-th Chern character of the connection \mathbf. In general, the Chern–Simons ''p''-form is defined for any odd ''p''.


Application to physics

In 1978, Albert Schwarz formulated
Chern–Simons theory The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who intr ...
, 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 that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...
, using Chern-Simons forms. 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 under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
, the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.


See also

*
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 ...
*
Chiral anomaly In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is analogous 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 that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...
*
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