In
mathematics, ∞-Chern–Simons theory (not to be confused with
infinite-dimensional Chern–Simons theory) is a generalized formulation of
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 ...
from
differential geometry using the formalism of
higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher ca ...
, which in particular studies
∞-categories
In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ...
. It is obtained by taking
general abstract analogs of all involved concepts defined in any cohesive
∞-topos
In mathematics, an ∞-topos is, roughly, an ∞-category such that its objects behave like sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external sp ...
, for example that of smooth
∞-groupoids.
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 ...
on which
Lie groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
act are for example replaced by ∞-principal bundles on with
group objects in ∞-topoi act.
[Definition in Schreiber 2013, 1.2.6.5.2] The theory is named after
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 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 ...
, who first described
Chern–Simons forms in 1974,
although the generalization was not developed by them.
See also
*
∞-Chern–Weil theory
Literature
*
*
*
* {{cite arXiv , arxiv=1301.2580 , author1=Domenico Fiorenza , author2=Hisham Sati , author3=
Urs Schreiber Urs Schreiber (born 1974) is a mathematician specializing in the connection between mathematics and theoretical physics (especially string theory) and currently working as a researcher at New York University Abu Dhabi. He was previously a researche ...
, title=A higher stacky perspective on Chern-Simons theory , date=2011-12-07
References
External links
infinity-Chern-Simons theoryon
''n''Lab
Differential geometry
Higher category theory