# 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.

## 16 Citations

Absence of higher order corrections to noncommutative Chern-Simons coupling

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- 2001

We analyze the structure of non-commutative pure Chern-Simons theory systematically in the axial gauge. We show that there is no IR/UV mixing in this theory in this gauge. In fact, we show, using the…

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I argue that the gauge group of noncommutative gauge theory consists of maps into unitary operators on Hilbert space of the form $u=1+K$ with $K$ compact. Some implications of this proposal are…

A Chern-Simons action for noncommutative spaces in general with the example SU_q(2)

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Witten constructed a topological quantum field theory with the Chern-Simons action as Lagrangian. We define a Chern-Simons action for 3-dimensional spectral triples. We prove gauge invariance of the…

Noncommutative massive Thirring model in three-dimensional spacetime

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We evaluate the noncommutative Chern-Simons action induced by fermions interacting with an Abelian gauge field in a noncommutative massive Thirring model in (2+1)-dimensional spacetime. This…

A NOTE ON THE TOPOLOGICAL ORDER OF NONCOMMUTATIVE HALL FLUIDS

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We evaluate the ground state degeneracy of noncommutative Chern–Simons models on the two-torus, a quantity that is interpreted as the "topological order" of associated phases of Hall fluids. We…

Noncommutative Chern – Simons theory on the quantum 3-sphere S 3 θ

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- 2020

We consider the θ-deformed quantum 3-sphere S3 θ and study its Chern–Simons theory from a spectral point of view. We first construct a spectral triple on S3 θ as a generalization of the Dirac…

Computing noncommutative Chern–Simons theory radiative corrections on the back of an envelope

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Computing Noncommutative Chern-simons Theory Radiative Corrections on the Back of an Envelope

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We show that the renormalized U(N) noncommutative Chern-Simons theory can be defined in perturbation theory so that there are no loop corrections to the 1PI functional of the theory in an arbitrary…

Some Properties of Non-linear $\sigma$-Models in Noncommutative Geometry

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- 2000

We introduce non-linear $\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space…

## 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…