Infinite-dimensional Chern–Simons theory
{{Short description|Chern–Simons theory on infinite-dimensional manifolds}}
In mathematics, infinite-dimensional Chern–Simons theory (not to be confused with ∞-Chern–Simons theory) is a generalization of Chern–Simons theory to manifolds with infinite dimensions. These are not modeled with finite-dimensional Euclidean spaces, but infinite-dimensional topological vector spaces, for example Hilbert, Banach and Fréchet spaces, which lead to Hilbert, Banach and Fréchet manifolds respectively. Principal bundles, which in finite-dimensional Chern–Simons theory are considered with (compact) Lie groups 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 and James Simons, who first described Chern–Simons forms in 1974,{{cite journal |last1=Chern |first1=Shiing-Shen |author-link1=Shiing-Shen Chern |last2=Simons |first2=James |author-link2=Jim Simons |date=September 1996 |title=Characteristic forms and geometric invariants |journal=World Scientific Series in 20th Century Mathematics |language=en |volume=4 |pages=363–384 |doi=10.1142/9789812812834_0026|isbn=978-981-02-2385-4 }} although the generalization was not developed by them.
See also
Literature
- {{cite arXiv |eprint=math/0301185 |first1=Sylvie |last1=Paycha |first2=Steven |last2=Rosenberg |title=Chern-Weil Constructions on ΨDO Bundles |date=2003-02-21}}
- {{cite arXiv |eprint=math/0411161 |first1=Steven |last1=Rosenberg |first2=Fabian |last2=Torres-Ardila |title=Infinite Dimensional Chern-Simons Theory |date=2004-11-08}}
- {{cite arXiv |eprint=1003.0067 |author=Andrés Larrain-Hubach, Steven Rosenberg, Simon Scott, Fabián Torres-Ardila |title=Characteristic Classes and Zeroth Order Pseudodifferential Operators |date=2010-05-26|class=math.DG }}
- {{cite web |author=Maeda |first1=Yoshiaki |last2=Rosenberg |first2=Steven |last3=Torres-Ardila |first3=Fabián |title=Riemannian geometry on loop spaces, part II: characteristic classes on LM |url=https://math.bu.edu/people/sr/articles/CAGv2.pdf}}
- {{cite book |author=Paycha |first1=Sylvie |last2=Scott |first2=Simon |title=Analysis, Geometry and Topology of Elliptic Operators |date=2006 |chapter=Chern-Weil forms associated with superconnections |pages=79–104 |chapter-url=https://www.researchgate.net/publication/253606665 |access-date=2025-03-09 |doi=10.1142/9789812773609_0005|isbn=978-981-256-805-2 }}
- {{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}}