Results 1  10
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49
On conformal field theories
 in fourdimensions,” Nucl. Phys. B533
, 1998
"... We review the generalization of field theory to spacetime with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last ..."
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Cited by 365 (0 self)
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We review the generalization of field theory to spacetime with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. Submitted to Reviews of Modern Physics.
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
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
Combinatorics of (perturbative) quantum field theory
, 2000
"... We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic oper ..."
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Cited by 34 (7 self)
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We review the structures imposed on perturbative QFT by the fact that its Feynman diagrams provide Hopf and Lie algebras. We emphasize the role which the Hopf algebra plays in renormalization by providing the forest formulas. We exhibit how the associated Lie algebra originates from an operadic operation of graph insertions. Particular emphasis is given to the connection with the Riemann–Hilbert problem. Finally, we outline how these structures relate to the numbers which we see in Feynman diagrams.
Quantum symmetry groups of noncommutative spheres
 Commun. Math. Phys
, 2001
"... We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. 1 ..."
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Cited by 32 (2 self)
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We show that the noncommutative spheres of Connes and Landi are quantum homogeneous spaces for certain compact quantum groups. We give a general construction of homogeneous spaces which support noncommutative spin geometries. 1
Noncommutative geometry, dynamics and ∞adic Arakelov geometry
"... In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handl ..."
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Cited by 25 (12 self)
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In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g ≥ 2. We use Connes ’ theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger’s Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we consider derived (cohomological) spectral data (A, H · (X ∗),Φ), where the algebra is obtained from the SL(2, R) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the Archimedean factors from a zeta function of a spectral triple. Then, we introduce a different construction, which is related to Manin’s description of the dual graph of the fiber at infinity. We
Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology)
"... This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where ..."
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Cited by 20 (8 self)
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This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But cubical sets can also express topological facts missed by ordinary topology. This happens, for instance, in the study of group actions or foliations, where a topologicallytrivial quotient (the orbit set or the set of leaves) can be enriched with a natural cubical structure whose directed cohomology agrees with Connes ' analysis in noncommutative geometry. Thus, cubical sets can provide a sort of 'noncommutative topology', without the metric information of C*algebras.
The superfield formalism applied to the noncommutative WessZumino model,” JHEP 0010
, 2000
"... Abstract. We introduce the notion of superoperators on noncommutative R 4 and reinvestigate in the framework of superfields the noncommutative WessZumino model as a quantum field theory. In a highly efficient manner we are able to confirm the result that this model is renormalizable to all orders. ..."
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Cited by 19 (2 self)
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Abstract. We introduce the notion of superoperators on noncommutative R 4 and reinvestigate in the framework of superfields the noncommutative WessZumino model as a quantum field theory. In a highly efficient manner we are able to confirm the result that this model is renormalizable to all orders. 2 Work supported by The Danish Research Agency.
Exercising in Ktheory: Brane condensation without tachyon,” hepth/0012262
"... Abstract. We show that the pdimensional noncommutative Yang–Mills model corresponding to a (p − 1)brane allows solutions which correspond to lower branes. This may be interpreted as the Morita equivalence of noncommutative planes of various dimensions. 1. ..."
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Cited by 15 (13 self)
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Abstract. We show that the pdimensional noncommutative Yang–Mills model corresponding to a (p − 1)brane allows solutions which correspond to lower branes. This may be interpreted as the Morita equivalence of noncommutative planes of various dimensions. 1.
Gauge invariance and noncommutativity
, 2002
"... The role of the gauge invariance in noncommutative field theory is discussed. A basic introduction to noncommutative geometry and noncommutative field theory is given. Background invariant formulation of Wilson lines is proposed. Duality symmetries relating various ..."
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Cited by 13 (11 self)
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The role of the gauge invariance in noncommutative field theory is discussed. A basic introduction to noncommutative geometry and noncommutative field theory is given. Background invariant formulation of Wilson lines is proposed. Duality symmetries relating various