Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

By Alain Connes, Giovanni Landi. 2000

407 citations recorded by the source corpus.

Abstract: We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝn. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.The new examples include the instanton algebra and the NC-4-spheres S4θ. We construct the noncommutative algebras ?=C∞ (S4θ) of functions on NC-spheres as solutions to the vanishing, chj (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M4 (?), e2=e, e=e*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S3θ distinct from quantum group deformations SUq (2) of SU (2).We then construct the noncommutative geometry of Sθ4 as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric gμν on S4 whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation, where <␣> is the projection on the commutant of 4 × 4 matrices.We shall show how to construct the Dirac operator D on the noncommutative 4-spheres Sθ4 so that the previous equation continues to hold without any change.Finally, we show that any compact Riemannian spin manifold whose isometry group has rank r≥ 2 admits isospectral deformations to noncommutative geometries.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ff460ddf5cdded98c0a9f0a4fd51fd97fcb24c02

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Citations in the corpus

Moyal Planes are Spectral Triples (2003, methodology): “on any decent noncompact manifold (without recourse to “lifting” devices). 3.4 On the Connes–Landi spaces example An interesting family of compact spectral triples was constructed by Connes and Landi [25], by isospectral deformation of a commutative spectral triple wherein the Dirac operator is kept”
Dynamical Systems on Spectral Metric Spaces (2010, methodology): “eal multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes-Landi isospectral deformations [22], a spectral triple for the noncommutative tori obtained by deforming the ber tori to noncommuta”
Quantum Isometries and Noncommutative Spheres (2009, background): “ systematic study of the associated spectral triples, with the explicit computation of a number of related Riemannian geometric invariants. See Connes and Dubois-Violette [14], [15], Connes and Landi [16], Dabrowski, D’Andrea, Landi and Wagner [18]. A useful, alternative point of view comes from the”
Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces (2011, background): “The ﬁrst result to suggest otherwise was obtained by Chakraborty and Pal [5], who constructed a spectral triple on Cq[SU2] following the isospectral deformation approach suggested by Connes and Landi [9]. A large number of examples would follow, including spectral 1 triples on general Cq[SUN], on th”
Quantum Spheres and Projective Spaces as Graph Algebras (2002, background): “In December 2000 Connes and Landi gave a general procedure which yields quantizations of even dimensional quantum spheres [7].”
Gromov-Hausdorff Distance for Quantum Metric Spaces (2000, background): “ght mesh with our quantum metric-space theory. Very recently, building on the present paper, Hanfeng Li has shown [46] for the Connes-Landi-DuboisViolette spheres {Sθ} (and related quantum manifolds) [17] [16] with their Dirac operators, that they form compact quantum metric spaces, and that if a se”
Noncommutative geometry and gravity (2005, background): “In the context of Connes noncommutative geometry, the noncommutative torus, the noncommutative spheres [33] and further noncommtative manifolds (so-called isospectral deformations) considered in [33], and in [34], [35], are noncommutative manifolds whose deformed algebra of functions is along the li”
The Quantum Gromov-Hausdorff Propinquity (2013, background): “Noncommutative examples include the (self-adjoint part of) quantum tori and other quantum homogeneous spaces [38], hyperbolic group C*-algebras [36, 40], ConnesLandi-Dubois-Violette quantum spheres [10, 9, 32], among others.”
Singular integrals in quantum Euclidean spaces (2017, background): “An important instance in string theory is given by the noncommutative deformation of R(4) associated to an invertible symbol Θ, which leads to instantons on a noncommutative space in the influential papers [21, 55, 67].”