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Noncommutative manifolds, the instanton algebra and isospectral deformations
 Comm. Math. Phys
"... We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the ..."
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Cited by 167 (29 self)
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We give new examples of noncommutative manifolds that are less standard than the NCtorus or Moyal deformations of R n. They arise naturally from basic considerations of noncommutative differential topology and have nontrivial global features. The new examples include the instanton algebra and the NC4spheres S4 θ. We construct the noncommutative algebras A = C ∞ (S4 θ) of functions on NCspheres as solutions to the vanishing, chj(e) = 0,j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M4(A), e2 = e, e = e ∗. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC3sphere intimately related to quantum group deformations SUq(2) of SU(2) but for unusual values (complex values of modulus one) of the parameter q of qanalogues, q = exp(2πiθ). We then construct the noncommutative geometry of S4 θ as given by a spectral triple (A, 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 √ g d4x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation, e − 1
The Coarse BaumConnes Conjecture for Spaces Which Admit a Uniform Embedding into Hilbert Space
, 1998
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Dbranes and the noncommutative torus
 JHEP
, 1998
"... We show that in certain superstring compactifications, gauge theories on noncommutative tori will naturally appear as Dbrane worldvolume theories. This gives strong evidence that they are welldefined quantum theories. It also gives a physical derivation of the identification proposed by Connes, D ..."
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Cited by 151 (3 self)
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We show that in certain superstring compactifications, gauge theories on noncommutative tori will naturally appear as Dbrane worldvolume theories. This gives strong evidence that they are welldefined quantum theories. It also gives a physical derivation of the identification proposed by Connes, Douglas and Schwarz of Matrix theory compactification on the noncommutative torus with M theory compactification with constant background threeform tensor field. November 1997It is often stated that in toroidal compactification in superstring theory, there is a minimum radius for the torus, the string length. Tori with smaller radii can always be related to tori with larger radii by using Tduality. However, this picture is not correct in the presence of other background fields, as is clear from the following simple example. Consider compactification on T 2 with a constant NeveuSchwarz twoform field B. We quote the standard result [1]: under simultaneous Tduality of all coordinates, the combination (G + B)ij is inverted, where G is the metric expressed in string units. Consider a square torus and take the limit R1 = R2 → 0; B ̸ = 0 fixed. (1) The Tdual torus has G + B =
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Quantum symmetry groups of finite spaces
 COMM. MATH. PHYS
, 1998
"... We determine the quantum automorphism groups of finite spaces. These are compact matrix quantum groups in the sense of Woronowicz. ..."
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Cited by 142 (6 self)
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We determine the quantum automorphism groups of finite spaces. These are compact matrix quantum groups in the sense of Woronowicz.
Dbranes and deformation quantization
 JHEP
, 1999
"... In this note we explain how worldvolume geometries of Dbranes can be reconstructed within the microscopic framework where Dbranes are described through boundary conformal field theory. We extract the (noncommutative) worldvolume algebras from the operator product expansions of open string verte ..."
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Cited by 140 (17 self)
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In this note we explain how worldvolume geometries of Dbranes can be reconstructed within the microscopic framework where Dbranes are described through boundary conformal field theory. We extract the (noncommutative) worldvolume algebras from the operator product expansions of open string vertex operators. For branes in a flat background with constant nonvanishing Bfield, the operator products are computed perturbatively to all orders in the field strength. The resulting series coincides with Kontsevich’s presentation of the Moyal product. After extending these considerations to fermionic fields we conclude with some remarks on the generalization of our approach to curved backgrounds.
Boundary deformation theory and moduli spaces of Dbranes
, 1999
"... Boundary conformal field theory is the suitable framework for a microscopic treatment of Dbranes in arbitrary CFT backgrounds. In this work, we develop boundary deformation theory in order to study the changes of boundary conditions generated by marginal boundary fields. The deformation parameters ..."
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Cited by 140 (26 self)
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Boundary conformal field theory is the suitable framework for a microscopic treatment of Dbranes in arbitrary CFT backgrounds. In this work, we develop boundary deformation theory in order to study the changes of boundary conditions generated by marginal boundary fields. The deformation parameters may be regarded as continuous moduli of Dbranes. We identify a large class of boundary fields which are shown to be truly marginal, and we derive closed formulas describing the associated deformations to all orders in perturbation theory. This allows us to study the global topology properties of the moduli space rather than local aspects only. As an example, we analyse in detail the moduli space of c = 1 theories, which displays various stringy phenomena.
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 125 (15 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Nonassociative star product deformations for Dbrane . . .
, 2001
"... We investigate the deformation of D–brane world–volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world ..."
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Cited by 114 (3 self)
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We investigate the deformation of D–brane world–volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world–volume deformation, identifying the open string metric and the noncommutative deformation parameter. The picture that unfolds is the following: when the gauge invariant combination ω = B + F is constant one obtains the standard Moyal deformation of the brane world–volume. Similarly, when dω = 0 one obtains the noncommutative Kontsevich deformation, physically corresponding to a curved brane in a flat background. When the background is curved, H = dω ̸ = 0, we find that the relevant algebraic structure is still based on the Kontsevich expansion, which now defines a nonassociative star product. We then recover, within this formalism, some known results of Matrix theory in curved backgrounds. In particular, we show how the effective action obtained in this framework describes, as expected, the dielectric effect of D–branes. The polarized branes