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## A Chern-Simons action for noncommutative spaces in general with the example SU q(2). 2012. arXiv

Citations: | 1 - 0 self |

### Citations

2981 | Noncommutative Geometry,
- Connes
- 1994
(Show Context)
Citation Context ...nbounded, self-adjoint operator D on H with compact resolvent, such that the commutators [a,D] for a ∈ A are bounded. Spectral triples are the starting point for the study of noncommutative manifolds =-=[9]-=-. One can think of them being a generalization of the notion of ordinary differential manifolds because every spin manifold without boundary can be encoded uniquely by a spectral triple [11]. In this ... |

765 |
Topological quantum field theory,
- WITTEN
- 1988
(Show Context)
Citation Context ...ed to be interesting in its own right [7]. This classical Chern-Simons invariant has found numerous applications in differential geometry, global analysis, topology and theoretical physics. E. Witten =-=[28]-=- used the Chern-Simons invariant to derive a 3-dimensional quantum field theory in order to give an intrinsic definition of the Jones Polynomial and its generalizations dealing with the mysteries of k... |

709 | Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem,
- Gilkey
- 1984
(Show Context)
Citation Context ...lynomial in the Pontrjagin forms pj(TM) of the tangent bundle TM with the Levi-Civita connection. These Pontrjagin forms pj(TM) are in the 4j-th cohomology group H4j(M) of M . For further details see =-=[19]-=-, chapter 2. Since we are only interested in the case p = 3, all Pontrajagin forms pj vanish for j ≥ 1 and we get Â = 1. Hence the formulas for the cyclic cocycle (φ3, φ1) adopt the following form: φ... |

652 |
Non-commutative geometry and string field theory,”
- Witten
- 1986
(Show Context)
Citation Context ...tensoring the algebra of smooth functions on the manifold with a finite dimensional matrix-algebra. The very first consideration about a Chern-Simons action in noncommutative geometry can be found in =-=[27]-=- by E. Witten himself. In the book [13] of A. Connes and M. Marcolli these considerations were generalised (based on a joint work [6] of the first author with A. H. Chamseddine, where a Chern-Simons t... |

278 |
Cyclic Homology,
- Loday
- 1998
(Show Context)
Citation Context ...ples (C∞(M), L2(S),D) consisting of the Dirac operator acting on the Hilbert space L2(S) of L2-spinors over a closed spin manifold M . Next we introduce the noncommutative replacement of n-forms. See =-=[21]-=-. Definition 2.2. For n ≥ 0 let Ωn(A) = A⊗A⊗n the vector space of noncommutative n-forms over A, where A = A−C1 denotes the vector space we obtain from the algebra A if one takes away the unital eleme... |

151 |
Characteristic forms and geometric invariants,
- Chern, Simons
- 1974
(Show Context)
Citation Context ...rst Pontrjagin number of a 4-manifold. This Chern-Simons invariant turned out to be a higher dimensional analogue of the 1-dimensional geodesic curvature and seemed to be interesting in its own right =-=[7]-=-. This classical Chern-Simons invariant has found numerous applications in differential geometry, global analysis, topology and theoretical physics. E. Witten [28] used the Chern-Simons invariant to d... |

144 | Noncommutative Geometry, Quantum Fields, and Motives, - Connes, Marcolli - 2008 |

60 | Equivariant spectral triples on the quantum SU(2) group. K-Theory 28
- Chakraborty, Pal
(Show Context)
Citation Context ... one (1.1) using the Wodzicki residue instead of the integral, fails to be gauge invariant in general. For the quantum group SUq(2) and its spectral triple constructed by P. S. Chakraborty and A. Pal =-=[4]-=- the Chern-Simons action of [20] is not gauge invariant in general. In [5] a definition of a Chern-Simons action is given, using a Chern-Simons form constructed in cyclic cohomology by D. Quillen [23]... |

16 | Spectral action on noncommutative torus
- Essouabri, Iochum, et al.
(Show Context)
Citation Context .... As noncommutative generalization of the 3-torus T3 – whose tangent bundle TT3 is globally trivial, i. e. TT3 ∼= T3 × R3 – 1-forms A ∈ MN(Ω1(C∞(T3Θ)) always decompose as in the commutative case, see =-=[14]-=- for details. Therefore, Lorenz gauge can be chosen and is used in [22] to compute the path integral for the noncommutative 3-torus. For SUq(2) such a decomposition of 1-forms is not available at all ... |

13 | The Chern-Simons action in noncommutative geometry - Chamseddine, Frohlich - 1994 |

11 | The Local Index Formula - Connes, Moscovici - 1995 |

11 |
Gauge invariance of the Chern-Simons action in noncommutative geometry,” arXiv:math-ph/9810015
- Krajewski
(Show Context)
Citation Context ...ectral triple coming from a 3-dimensional spin manifold. Our definition of a noncommutative Chern-Simons action is not the first one. Several proposals were given before: for instance by T. Krajewski =-=[20]-=- and A. H. Chamseddine, J. Fröhlich [5]. The Chern-Simons action in [20], which is defined as the classical one (1.1) using the Wodzicki residue instead of the integral, fails to be gauge invariant i... |

10 | Chern-Simons Forms and Cyclic Cohomology, The interface of mathematics and particle physics - Quillen - 1988 |

7 |
Quantum Field Theory, A Tourist Guide for
- Folland
- 2008
(Show Context)
Citation Context ...-valued N ×N matrices, modulo gauge transformations. Firstly, we give an outline of the gauge breaking mechanism by Faddeev-Popov. For a detailed discussion we refer the reader to G. B. Follands book =-=[15]-=-. Then we have to make sense of the path integral in the noncommutative setting. Theorems 5.3 and 5.4 are crucial tools in this section, not only for the definition of the path integral on SUq(2) but ... |

7 | Spectral geometry of the Moyal plane with harmonic propagation,” arXiv:1108.2184 [math.OA
- Gayral, Wulkenhaar
(Show Context)
Citation Context ...additional linear term: the cocycle φ1 of the local index formula. Such an additional linear term in the definition of a noncommutative Chern-Simons action is new but not necessarily unreasonable. In =-=[18]-=-, V. Gayal and R. Wulkenhaar investigate the Yang-Mills action for a triple constructed on the noncommutative d-dimensional Moyal space and discovered also an additional linear part of the Yang-Mills ... |

4 | The local index formula for SUq(2
- Suijlekom, Da̧browski, et al.
- 2005
(Show Context)
Citation Context ...e spaces are H↑ and H↓ respectively. The precise form of the representation of the generators α and β is not necessary for our purposes. Hence we skip the definitions and refer the reader to [24]. In =-=[25]-=- the authors proceed by constructing a cosphere bundle C∞(S∗q ) for (C∞(SUq(2)),H,D) in an analogous way as in [12], and end up with a similar result for the local index formula. The maps ρ, τ1, τ ↑ 0... |

2 |
On the Spectral Characterization
- Connes
(Show Context)
Citation Context ... manifolds [9]. One can think of them being a generalization of the notion of ordinary differential manifolds because every spin manifold without boundary can be encoded uniquely by a spectral triple =-=[11]-=-. In this framework some core structures of topology and geometry such as the index theorem [10] were extended far beyond the classical scope. By means of the local index theorem proven by A. Connes a... |

2 | Chern-Simons theory for the noncommutative 3-torus
- Pfante
(Show Context)
Citation Context ...in [24] instead, and deduce the non-topological nature of the Chern-Simons action defined in the first part. This paper is one of two papers about the content of my Ph. D. thesis. An additional paper =-=[22]-=- will appear, where we compute the Chern-Simons action explicitly for the noncommutative 3-torus. Also there we make sense of the path integral for this noncommutative space, i. e. we give a definitio... |

2 |
The Isospectral Dirac Operator on
- D’Andrea, Dabrowski
(Show Context)
Citation Context ...h is isomorphic to SU(2). Spectral triples for SUq(2) were constructed by P. S. Chakraborty and A. Pal [4] on the one hand, and L. Da̧browski, G. Landi, A. Sitarz, W. van Suijlekom and J. C. Várilly =-=[24]-=- on the other hand. The index theoretical aspects of the spectral triple in [4] were carefully studied by A. Connes [12]. The same was done for the spectral triple in [24] by the same authors in [25].... |

1 |
Freed Classical Chern-Simons Theory Part
- S
- 1992
(Show Context)
Citation Context ...tive case, where no additional linear terms emerge in the formula of the action functional, the Chern-Simons action is extremal iff the curvature of the connection 1-form is zero (see proposition 3.1 =-=[16]-=- for a proof of this statement). Connection 1-forms with vanishing curvature are called flat. For SUq(2) things do not work so easily any longer because the appearance of the non vanishing linear part... |

1 |
Dirac Operators and Spectral Geometry, http://toknotes.mimuw.edu.pl/sem3/ files/Varilly dosg.pdf
- Várilly
(Show Context)
Citation Context ... σ3 (| D |3) (x, ξ)−1 dν(x)dξ. The integral term on the right hand side of the equality is called the Wodzicki residue Wres (| D |−3) of the pseudo differential operator | D |−3. From example 5.16 in =-=[26]-=- we obtain Wres (| D |−3) = 8πVol(M). Combining all these identities we obtain τ0 (| D |−3) = lim z→0 z Trace (| D |−3−2z) = 1 2 lim s→3+ s Trace (| D |−s) = 3 2 τω (| D |−3) = 1 2(2π3) Wres (| D |−3)... |