In mathematics, the ChernâSimons forms are certain secondary characteristic classes. [1] 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 which the theory arose. [2]
Given a manifold and a Lie algebra valued 1-form over it, we can define a family of p-forms: [3]
In one dimension, the ChernâSimons 1-form is given by
In three dimensions, the ChernâSimons 3-form is given by
In five dimensions, the ChernâSimons 5-form is given by
where the curvature F is defined as
The general ChernâSimons form is defined in such a way that
where the wedge product is used to define Fk. The right-hand side of this equation is proportional to the k-th Chern character of the connection .
In general, the ChernâSimons p-form is defined for any odd p. [4]
In 1978, Albert Schwarz formulated ChernâSimons theory, early topological quantum field theory, using Chern-Simons forms. [5]
In the gauge theory, the integral of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.
In mathematics, the ChernâSimons forms are certain secondary characteristic classes. [1] 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 which the theory arose. [2]
Given a manifold and a Lie algebra valued 1-form over it, we can define a family of p-forms: [3]
In one dimension, the ChernâSimons 1-form is given by
In three dimensions, the ChernâSimons 3-form is given by
In five dimensions, the ChernâSimons 5-form is given by
where the curvature F is defined as
The general ChernâSimons form is defined in such a way that
where the wedge product is used to define Fk. The right-hand side of this equation is proportional to the k-th Chern character of the connection .
In general, the ChernâSimons p-form is defined for any odd p. [4]
In 1978, Albert Schwarz formulated ChernâSimons theory, early topological quantum field theory, using Chern-Simons forms. [5]
In the gauge theory, the integral of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.