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Quantum field theory on noncommutative spaces
"... A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommuta ..."
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Cited by 273 (15 self)
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A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative YangMills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an indepth study of the gauge group of noncommutative YangMills theory. Some of the more mathematical ideas and
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 268 (1 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 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 89 (12 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
A.: Morita equivalence of multidimensional noncommutative tori
 Internat. J. Math
, 1999
"... One can describe an ndimensional noncommutative torus by means of an antisymmetric n × n matrix θ. We construct an action of the group SO(n, nZ) on the space of n × n antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. ..."
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Cited by 58 (9 self)
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One can describe an ndimensional noncommutative torus by means of an antisymmetric n × n matrix θ. We construct an action of the group SO(n, nZ) on the space of n × n antisymmetric matrices and show that, generically, matrices belonging to the same orbit of this group give Morita equivalent tori. Some applications to physics are sketched. By definition [R5], an ndimensional noncommutative torus is an associative algebra with involution having unitary generators U1,..., Un obeying the relations UkUj = e(θkj)UjUk, (1) where e(t) = e 2πit and θ is an antisymmetric matrix. The same name is used for different completions of this algebra. In particular, we can consider the noncommutative torus as a C ⋆algebra Aθ (the universal C ⋆algebra generated by n unitary operators satisfying (1)). Noncommutative tori are important in many problems of mathematics and physics. It was shown recently that they are essential in consideration of compactifications of M(atrix) theory ([CDS]; for further development see [T]). The results of the present paper also have application to physics. If two algebras A and Â are Morita equivalent (see the definition below), then for every Amodule R one can construct an Âmodule ˆ R in such a way that the correspondence R → ˆ R is an equivalence of categories of Amodules
GromovHausdorff distance for quantum metric spaces
 Mem. Amer. Math. Soc
"... Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distanc ..."
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Cited by 35 (5 self)
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Abstract. By a quantum metric space we mean a C ∗algebra (or more generally an orderunit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov–Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, Aθ. We show, for consistently defined “metrics”, that if a sequence {θn} of parameters converges to a parameter θ, then the sequence {Aθn} of quantum tori converges in quantum Gromov–Hausdorff distance to Aθ. 1.
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 30 (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
Differential and complex geometry of two–dimensional noncommutative tori
, 2002
"... We analyze in detail projective modules over twodimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of thetafunctions. The paper is selfcontai ..."
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Cited by 25 (0 self)
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We analyze in detail projective modules over twodimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of thetafunctions. The paper is selfcontained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.
Instanton expansion of noncommutative gauge theory in two dimensions
 Commun. Math. Phys
"... We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive ..."
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Cited by 23 (5 self)
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We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of YangMills theory defined on a projective module for arbitrary noncommutativity parameter θ which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of θ. The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary twotorus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of θ and computes the symplectic volume of the moduli space
Gauge bundles and BornInfeld on the noncommutative torus,” Nucl. Phys. B547
, 1999
"... In this paper, we describe nonabelian gauge bundles with magnetic and electric fluxes on higher dimensional noncommutative tori. We give an explicit construction of a large class of bundles with nonzero magnetic ’t Hooft fluxes. We discuss Morita equivalence between these bundles. The action of the ..."
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Cited by 23 (0 self)
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In this paper, we describe nonabelian gauge bundles with magnetic and electric fluxes on higher dimensional noncommutative tori. We give an explicit construction of a large class of bundles with nonzero magnetic ’t Hooft fluxes. We discuss Morita equivalence between these bundles. The action of the duality is worked out in detail for the fourtorus. As an application, we discuss BornInfeld on this torus, as a description of compactified string theory. We show that the resulting theory, including the fluctuations, is manifestly invariant under the Tduality group SO(4, 4; Z). The Uduality invariant BPS massformula is discussed shortly. We comment on a discrepancy of this result with that of a recent calculation. 1
Crossed products by finite cyclic group actions with the tracial Rokhlin property
, 2003
"... We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*algebras. We prove that the crossed product of a stably finite simple unital C*algebra with tracial rank zero by an action with this property again has tracial rank zero. Under a kind of w ..."
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Cited by 22 (9 self)
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We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*algebras. We prove that the crossed product of a stably finite simple unital C*algebra with tracial rank zero by an action with this property again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Izumi. We use these results, together with work of ElliottEvans and Kishimoto, to give an inductive proof that every simple higher dimensional noncommutative torus is an AT algebra. We further prove that the crossed product of every simple higher dimensional noncommutative torus by the flip is an AF algebra, and that the crossed products of irrational rotation algebras by the standard actions of Z3, Z4, and Z6 are simple AH algebras with real rank zero. In the