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## Noncommutative manifolds, the instanton algebra and isospectral deformations

Venue: | Comm. Math. Phys |

Citations: | 166 - 28 self |

### Citations

2956 | Noncommutative geometry
- Connes
- 1994
(Show Context)
Citation Context ... Dirac operator [19] in ordinary Riemannian geometry. It specifies both the metric on the state space of A by d(ϕ, ψ) = Sup {|ϕ(a) − ψ(a)|; ‖[D, a]‖ ≤ 1} (3) and the K-homology fundamental class (cf. =-=[6]-=-). What holds things together in this spectral point of view on NCG is the nontriviality of the pairing between the K-theory of the algebra A and the K-homology class of D, given in the even case by [... |

873 | Calibrated geometries
- Harvey, Lawson
- 1982
(Show Context)
Citation Context ..., acting in the Hilbert space H while D is a self-adjoint operator with compact resolvent and such that, [D, a] is bounded ∀ a ∈ A . (2) The operator D plays in general the role of the Dirac operator =-=[19]-=- in ordinary Riemannian geometry. It specifies both the metric on the state space of A by d(ϕ, ψ) = Sup {|ϕ(a) − ψ(a)|; ‖[D, a]‖ ≤ 1} (3) and the K-homology fundamental class (cf. [6]). What holds thi... |

789 | String theory and noncommutative geometry
- Seiberg, Witten
- 1999
(Show Context)
Citation Context ...oyal deformation of R n whose algebra is boring. This is particularly so in view of the upsurge of activity in the interaction between string theory and noncommutative geometry started in [11], [22], =-=[25]-=-. The new examples should arise naturally, have non-trivial global features (and also pass the test of noncommutative manifolds as defined in [8]). This paper will provide and analyse very natural suc... |

367 | Noncommutative geometry and matrix theory: Compactification on tori,” JHEP 9802 - Connes, Douglas, et al. - 1998 |

342 | Gravity coupled with matter and the foundation of non commutative geometry
- Connes
- 1996
(Show Context)
Citation Context ...ry and noncommutative geometry started in [11], [22], [25]. The new examples should arise naturally, have non-trivial global features (and also pass the test of noncommutative manifolds as defined in =-=[8]-=-). This paper will provide and analyse very natural such new examples, including the instanton algebra and the NC-4-spheres S4 θ , obtained from basic considerations of noncommutative differential top... |

270 |
Cyclic homology
- Loday
- 1992
(Show Context)
Citation Context ...sual Kronecker symbol and only the class ˜eikik+1 ∈ A is used in the formula. The crucial property of the components chn(e) is that they define a cycle in the (b, B) bicomplex of cyclic homology [5], =-=[20]-=-, B chn(e) = b chn+1(e) . (4) For any pair of integers m, r we let Am,r be the universal algebra associated to the relations, chj(e) = 0 ∀ j < m . (5) More precisely we let Am,r be generated by the r ... |

255 |
Twisted SU(2) group, An example of a noncommutative differential calculus
- Woronowicz
- 1987
(Show Context)
Citation Context ...alytic continuation of the parameter to the quantum group SU(2)q but the usual theory requires q to be real whereas we need a complex number of modulus one which spoils the unitarity of the coproduct =-=[27]-=-. Had we taken the deformation parameter to be real, λ = q ∈ R, like in [14] we would 8have obtained a different deformation S 4 q of the commutative sphere S 4 , whose algebra is different from the ... |

237 | Elements of Noncommutative Geometry
- Gracia-Bond́ıa, Várilly, et al.
- 2000
(Show Context)
Citation Context ...e , (6) 2 ) is the restriction of D to the range H+ of 1+γ 2 [ 0 ∗ D+ D+ 0 [ 1 0 0 −1 and γ is the Z/2 ] . (7) The corner stone of the general theory is an operator theoretic index formula [6], [12], =-=[16]-=- which expresses the above index pairing (4) by explicit local cyclic cocycles on the algebra A. These local formulas become extremely simple in the special case where only the top component of the Ch... |

229 |
The local index formula in noncommutative geometry
- Connes
- 1995
(Show Context)
Citation Context ...e D + e , (6) 2 ) is the restriction of D to the range H+ of 1+γ 2 [ 0 ∗ D+ D+ 0 [ 1 0 0 −1 and γ is the Z/2 ] . (7) The corner stone of the general theory is an operator theoretic index formula [6], =-=[12]-=-, [16] which expresses the above index pairing (4) by explicit local cyclic cocycles on the algebra A. These local formulas become extremely simple in the special case where only the top component of ... |

226 |
Non commutative differential geometry
- CONNES
- 1985
(Show Context)
Citation Context ...the usual Kronecker symbol and only the class ˜eikik+1 ∈ A is used in the formula. The crucial property of the components chn(e) is that they define a cycle in the (b, B) bicomplex of cyclic homology =-=[5]-=-, [20], B chn(e) = b chn+1(e) . (4) For any pair of integers m, r we let Am,r be the universal algebra associated to the relations, chj(e) = 0 ∀ j < m . (5) More precisely we let Am,r be generated by ... |

191 |
Noncommutative geometry and reality
- Connes
- 1995
(Show Context)
Citation Context ...anonical idempotent given in (III.8). The Dirac operator D fulfills 〈( e − 1 ) [D, e] 2 4 〉 = γ where 〈 〉 is the projection on the commutant of M4(C) and γ is the grading operator. The real structure =-=[7]-=- is given by the charge conjugation operator J, which involves in the noncommutative case the Tomita-Takesaki antilinear involution. The order one condition, [[D, a], b 0 ] = 0 ∀ a, b ∈ C ∞ (S 4 θ) . ... |

167 |
C ∗ -algèbres et geométrie differentielle
- Connes
- 1980
(Show Context)
Citation Context ...nk r ≥ 2 admits isospectral deformations to noncommutative geometries. 1I Introduction It is important to have available examples of noncommutative manifolds that are less standard than the NC-torus =-=[4]-=-, [13] or the old Moyal deformation of R n whose algebra is boring. This is particularly so in view of the upsurge of activity in the interaction between string theory and noncommutative geometry star... |

98 |
Instantons on noncommutative R 4 and (2, 0) superconformal six dimensional theory
- Nekrasov, Schwarz
- 1998
(Show Context)
Citation Context ... old Moyal deformation of R n whose algebra is boring. This is particularly so in view of the upsurge of activity in the interaction between string theory and noncommutative geometry started in [11], =-=[22]-=-, [25]. The new examples should arise naturally, have non-trivial global features (and also pass the test of noncommutative manifolds as defined in [8]). This paper will provide and analyse very natur... |

92 |
Deformation quantization for actions of R d
- Rieffel
- 1993
(Show Context)
Citation Context ...anifold M whose isometry group has rank ≥ 2 admits a natural one-parameter isospectral deformation to noncommutative geometries Mθ. The deformation of the algebra will be performed along the lines of =-=[23]-=-. We let (A, H, D) be the canonical spectral triple associated with a compact Riemannian spin manifold M. We recall that A = C ∞ (M) is the algebra of smooth functions on M, H = L 2 (M, S) is the Hilb... |

84 |
Existence de traces non normales
- Dixmier
- 1966
(Show Context)
Citation Context ... grading of H as 1 above, the resolvent of D is of order 2m (i.e. its characteristic values µk 1 − are 0(k 2m)) and ∫ − is the coefficient of the logarithmic divergency in the ordinary operator trace =-=[15]-=- [26]. We began in [10] to investigate the algebraic relations implied by the vanishing, (8) chj(e) = 0 j < m , (9) of the Chern character of e in the cyclic homology of A. Note that this vanishing at... |

69 |
Universal formula for noncommutative geometry actions: unification of gravity and the
- Chamseddine, Connes
- 1996
(Show Context)
Citation Context ...equation (11) with the index formula gives a quantization of the volume, ∫ − ds 4 ∈ N ds = D −1 (12) and fixes (in a given K-homology class for the operator D) the leading term of the spectral action =-=[3]-=-, ( ( )) D Trace f = Λ Λ4 ∫ − ds 2 4 + · · · (13) Since the next term is the Hilbert-Einstein action in the usual Riemannian case [3], [18], [17], it is very natural to compare various solutions (comm... |

55 | The Dirac Operator and Gravitation - Kastler - 1995 |

49 | A short survey of noncommutative geometry
- Connes
(Show Context)
Citation Context ...e, the resolvent of D is of order 2m (i.e. its characteristic values µk 1 − are 0(k 2m)) and ∫ − is the coefficient of the logarithmic divergency in the ordinary operator trace [15] [26]. We began in =-=[10]-=- to investigate the algebraic relations implied by the vanishing, (8) chj(e) = 0 j < m , (9) of the Chern character of e in the cyclic homology of A. Note that this vanishing at the chain level is a m... |

48 |
non-commutative geometry, and the Wodzicki residue
- Kalau, Walze, et al.
- 1995
(Show Context)
Citation Context ...the operator D) the leading term of the spectral action [3], ( ( )) D Trace f = Λ Λ4 ∫ − ds 2 4 + · · · (13) Since the next term is the Hilbert-Einstein action in the usual Riemannian case [3], [18], =-=[17]-=-, it is very natural to compare various solutions (commutative or not) of (11) using this action. II Components of the Chern character and the Instanton algebra Let A be an algebra (over C) and e ∈ Mr... |

35 |
Yang-Mills for noncommutative two-tori, Operator algebras and mathematical physics (Iowa City
- Connes, Rieffel
- 1985
(Show Context)
Citation Context ...≥ 2 admits isospectral deformations to noncommutative geometries. 1I Introduction It is important to have available examples of noncommutative manifolds that are less standard than the NC-torus [4], =-=[13]-=- or the old Moyal deformation of R n whose algebra is boring. This is particularly so in view of the upsurge of activity in the interaction between string theory and noncommutative geometry started in... |

29 | Riemannian geometry of quantum groups and finite groups with nonuniversal differentials
- Majid
- 2002
(Show Context)
Citation Context ...e suspension of the corresponding NC spaces are contained in the Grm,2r. The Dirac operator and quantum groups There exists formulas for q-analogues of the Dirac operator on quantum groups, (cf. [2], =-=[21]-=-); let us call Q these “naive” Dirac operators. Now the fundamental equation to define the sought for true Dirac operator D which we used above implicitly on the deformed 3-sphere (after suspension to... |

26 |
Geometry of Yang-Mills Fields
- Atiyah
- 1979
(Show Context)
Citation Context ...hat Yang-Mills gauge theory on noncommutative R 4 gives a conceptual understanding of the nonzero B-field desingularization of the moduli space of instantons obtained by perturbing the ADHM equations =-=[1]-=-. In [25], Seiberg and Witten exhibited the unexpected relation between the standard gauge theory and the noncommutative one. The above work raises the specific question for NC-spheres S4 θ whether on... |

11 |
P.P.: Dirac operators on quantum SU(2) group and quantum sphere
- Bibikov, Kulish
- 2000
(Show Context)
Citation Context ...at the suspension of the corresponding NC spaces are contained in the Grm,2r. The Dirac operator and quantum groups There exists formulas for q-analogues of the Dirac operator on quantum groups, (cf. =-=[2]-=-, [21]); let us call Q these “naive” Dirac operators. Now the fundamental equation to define the sought for true Dirac operator D which we used above implicitly on the deformed 3-sphere (after suspens... |

8 |
K-groups of C ∗ -algebras deformed by actions of R d
- Rieffel
- 1993
(Show Context)
Citation Context ...a of smooth functions on M with product deformed to the ∗-product defined in (11). Moreover, the real structure is given by the twisted involution ˜ J defined in (16). One checks using the results of =-=[24]-=- and [8] that Poincaré duality continues to hold for the deformed spectral triple. We showed in [8] that the Dirac operator for the Levi-Civita connection minimizes the action functional ∫ −D 2−n (whe... |

7 |
Noncommutative Geometry: The Spectral Aspect
- Connes
- 1998
(Show Context)
Citation Context ...hains to operators given by π(a0 ⊗ a1 ⊗ ... ⊗ an) = a0[D, a1]...[D, an] . (13) One can then check the various conditions which in the commutative case suffice to characterize Riemannian geometry [8], =-=[9]-=-. Theorem 3 a) The spectral triple (C∞(S 4 θ ), H, D) fulfills all axioms of noncommutative manifolds. b) Let e ∈ C∞ (S4 θ , M4(C)) be the canonical idempotent given in (III.8). The Dirac operator D f... |

7 |
Noncommutative residue, part I. Fundamentals
- Wodzicki
- 1987
(Show Context)
Citation Context ...ing of H as 1 above, the resolvent of D is of order 2m (i.e. its characteristic values µk 1 − are 0(k 2m)) and ∫ − is the coefficient of the logarithmic divergency in the ordinary operator trace [15] =-=[26]-=-. We began in [10] to investigate the algebraic relations implied by the vanishing, (8) chj(e) = 0 j < m , (9) of the Chern character of e in the cyclic homology of A. Note that this vanishing at the ... |

1 | A quantum instanton on the spheres S4 q - Landi |