Gauge invariance of the Chern-Simons action in noncommutative geometry
@article{Krajewski1998GaugeIO, title={Gauge invariance of the Chern-Simons action in noncommutative geometry}, author={Thomas Krajewski}, journal={arXiv: Mathematical Physics}, year={1998} }
In complete analogy with the classical case, we define the Chern-Simons action functional in noncommutative geometry and study its properties under gauge transformations. As usual, the latter are related to the connectedness of the group of gauge transformations. We establish this result by making use of the coupling between cyclic cohomology and K-theory and prove, using an index theorem, that this coupling is quantized in the case of the noncommutative torus.
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References
SHOWING 1-10 OF 23 REFERENCES
Gravity coupled with matter and the foundation of non-commutative geometry
- Mathematics
- 1996
We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond…
Quantum field theory and the Jones polynomial
- Mathematics
- 1989
It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones…
The Local Index Formula in Noncommutative Geometry
- Mathematics
- 1995
In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded…
Noncommutative differential geometry of matrix algebras
- Mathematics
- 1990
The noncommutative differential geometry of the algebra Mn (C) of complex n×n matrices is investigated. The role of the algebra of differential forms is played by the graded differential algebra…
The quantum structure of spacetime at the Planck scale and quantum fields
- Physics
- 1995
We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is…
Invariance Theory Heat Equation and Atiyah Singer Index Theorem
- Mathematics
- 1995
Pseudo-Differential Operators Introduction Fourier Transform and Sobolev Spaces Pseudo-Differential Operators on Rm Pseudo-Differential Operators on Manifolds Index of Fredholm Operators Elliptic…
C∗-algebras associated with irrational rotations
- Mathematics
- 1981
For any irrational number a let Aa be the transformation group C*-algebra for the action of the integers on the circle by powers of the rotation by angle 2πa. It is known that Aa is simple and has a…
K-Theory and C*-Algebras: A Friendly Approach
- Mathematics
- 1993
PART I: C*-ALGEBRAS PART II: FUNDAMENTALS OF K-THEORY PART III: HILBERT MODULES AND A GENERALIZED INDEX THEORY PART IV: APPENDICES
A remark on trace properties of K-cycles
- Mathematics
- 1995
In this paper we discuss trace properties of $d^+$-summable $K$-cycles considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a trace theorem on the algebra $\A$ of a $K$--cycle…