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 ...
, ∞-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. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who intr ...
from
differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
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 morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
, which in particular studies
∞-categories. It is obtained by taking
general abstract analogs of all involved concepts defined in any cohesive
∞-topos
In mathematics, an ∞-topos (infinity-topos) is, roughly, an ∞-category such that its objects behave like sheaf (mathematics), sheaves of spaces with some choice of Grothendieck topology; in other words, it gives an intrinsic notion of sheaves ...
, 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, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Euclidean space, whereas ...
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 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
*
∞-Chern–Weil theory
Literature
*
*
*
* {{cite book , 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 research ...
, title=Mathematical Aspects of Quantum Field Theories , chapter=A Higher Stacky Perspective on Chern–Simons Theory , series=Mathematical Physics Studies , date=2011-12-07, pages=153–211 , doi=10.1007/978-3-319-09949-1_6 , isbn=978-3-319-09948-4
References
External links
infinity-Chern-Simons theoryon
''n''Lab
Differential geometry
Higher category theory