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 ...
, infinite-dimensional Chern–Simons theory (not to be confused with
∞-Chern–Simons theory
In mathematics, ∞-Chern–Simons theory (not to be confused with infinite-dimensional Chern–Simons theory) is a generalized formulation of Chern–Simons theory from differential geometry using the formalism of higher category theory, which in ...
) is a generalization of
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 ...
to manifolds with infinite dimensions. These are not modeled with finite-dimensional
Euclidean spaces
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
, but infinite-dimensional
topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
, for example
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
,
Banach
Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase "Son of man (Judaism), son of man", combining the Hebrew language, Hebrew word ' ...
and
Fréchet spaces, which lead to
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad ...
,
Banach
Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase "Son of man (Judaism), son of man", combining the Hebrew language, Hebrew word ' ...
and
Fréchet manifolds respectively.
Principal bundles
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 equi ...
, which in finite-dimensional Chern–Simons theory are considered with (
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
)
Lie groups
In mathematics, a Lie group (pronounced ) is a 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 Euclidean space, whereas ...
as gauge groups, are then fittingly considered with Hilbert Lie, Banach Lie and Fréchet Lie groups as gauge groups respectively, which also makes their total spaces into a Hilbert, Banach and Fréchet manifold respectively. These are called Hilbert, Banach and Fréchet principal bundles respectively. The theory is named after
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 Simons, who first described
Chern–Simons forms in 1974,
although the generalization was not developed by them.
See also
*
Four-dimensional Chern–Simons theory
*
Six-dimensional holomorphic Chern–Simons theory
Literature
*
*
*
*
*
* {{cite web , author=Vozzo , first=Raymond , date=2010 , title=Loop Groups and Characteristic Classes , url=https://www.mpim-bonn.mpg.de/webfm_send/24 , access-date=2025-03-09
References
External links
*
infinite-dimensional Chern-Simons theory on
''n''Lab
Differential geometry
Higher category theory