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Noncommutative Geometry and Gauge Theory on Fuzzy Sphere
 Comm. Math. Phys
"... The differential algebra on the fuzzy sphere is constructed by applying Connes ’ scheme. The U(1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U(1) gauge transformation on the fuzzy sphere is identified with the left U(N +1) transformation of the field, w ..."
Abstract

Cited by 42 (1 self)
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The differential algebra on the fuzzy sphere is constructed by applying Connes ’ scheme. The U(1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U(1) gauge transformation on the fuzzy sphere is identified with the left U(N +1) transformation of the field, where a field is a bimodule over the quantized algebra AN. The interaction with a complex scalar field is also given. 1
Strong connections on quantum principal bundles
 Commun. Math. Phys
, 1996
"... A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and nonstrong ..."
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Cited by 37 (7 self)
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A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and nonstrong connections are provided. In particular, such connections are constructed on a quantum deformation of the Hopf fibration S 2 → RP 2. A certain class of strong Uq(2)connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the qdependent hermitian metric. A particular form of the Yang–Mills action on a trivial Uq(2)bundle is investigated. It is proved to coincide with the Yang–Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.
Chirality and Dirac Operator on Noncommutative Sphere’, Commun.Math.Phys. 183
 365 andhepth/9605003; ‘Noncommutative Geometry and Gauge Theory on Fuzzy Sphere’, Commun.Math.Phys. 212 (2000) 395 and hepth/9801195
, 1997
"... We give a derivation of the Dirac operator on the noncommutative 2sphere within the framework of the bosonic fuzzy sphere and define Connes ’ triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As ..."
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Cited by 29 (0 self)
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We give a derivation of the Dirac operator on the noncommutative 2sphere within the framework of the bosonic fuzzy sphere and define Connes ’ triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin’s quantization. The map to the local frame in noncommutative geometry is also discussed. 1Fellow of the Japan Society for Promotion of Science, email:
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 22 (3 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
The Fuzzy Supersphere
, 1998
"... We introduce the fuzzy supersphere as sequence of finitedimensional, noncommutative Z2graded algebras tending in a suitable limit to a dense subalgebra of the Z2graded algebra of H1functions on the (22)dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) ..."
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Cited by 18 (2 self)
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We introduce the fuzzy supersphere as sequence of finitedimensional, noncommutative Z2graded algebras tending in a suitable limit to a dense subalgebra of the Z2graded algebra of H1functions on the (22)dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the superdeRham complex are introduced. In particular we reproduce the equality of the superdeRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level".
Duality Symmetries and Noncommutative Geometry of String Spacetime
 COMMUN. MATH. PHYS
, 1998
"... We examine the structure of spacetime symmetries of toroidally compactified string theory within the framework of noncommutative geometry. Following a proposal of Fröhlich and Gawedzki, we describe the noncommutative string spacetime using a detailed algebraic construction of the vertex operator alg ..."
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Cited by 15 (11 self)
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We examine the structure of spacetime symmetries of toroidally compactified string theory within the framework of noncommutative geometry. Following a proposal of Fröhlich and Gawedzki, we describe the noncommutative string spacetime using a detailed algebraic construction of the vertex operator algebra. We show that the spacetime duality and discrete worldsheet symmetries of the string theory are a consequence of the existence of two independent Dirac operators, arising from the chiral structure of the conformal field theory. We demonstrate that these Dirac operators are also responsible for the emergence of ordinary classical spacetime as a lowenergy limit of the string spacetime, and from this we establish a relationship between Tduality and changes of spin structure of the target space manifold. We study the automorphism group of the vertex operator algebra and show that spacetime duality is naturally a gauge symmetry in this formalism. We show that classical general covariance also becomes a gauge symmetry of the string spacetime. We explore some larger symmetries of the algebra in the context of a universal gauge group for string theory, and connect these symmetry groups with some of the algebraic structures which arise in the mathematical theory of vertex operator algebras, such as the Monster group. We also briefly describe how the classical topology of spacetime is modified by the string theory, and calculate the cohomology groups of the noncommutative spacetime. A selfcontained, pedagogical introduction to the techniques of noncommmutative geometry is also included.
On the noncommutative geometry of the endomorphism algebra of a vector bundle
 Masson T., Submanifolds and quotient manifolds in noncommutative
, 1999
"... In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of ..."
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Cited by 13 (4 self)
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In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of noncommutative differential forms.
Some Aspects of Noncommutative Differential Geometry
"... We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finall ..."
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Cited by 13 (2 self)
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We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1
SU(n)connections and noncommutative differential geometry
 J. Geom. Phys
, 1998
"... We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)vector bundle. We show that ordinary connections on such SU(n)vector bundle can be interpreted in a natural way as a noncommutative 1form on this algebra for the differential calculus based on derivation ..."
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Cited by 12 (4 self)
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We study the noncommutative differential geometry of the algebra of endomorphisms of any SU(n)vector bundle. We show that ordinary connections on such SU(n)vector bundle can be interpreted in a natural way as a noncommutative 1form on this algebra for the differential calculus based on derivations. We interpret the Lie algebra of derivations of the algebra of endomorphisms as a Lie algebroid. Then we look at noncommutative connections as generalizations of these usual connections.
Differential Calculus on Fuzzy Sphere and Scalar Field. preprint TU525
 International Journal of Modern Physics A
, 1997
"... We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes ’ spectral triple on the fuzzy sphere and the differential ..."
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Cited by 8 (1 self)
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We find that there is an alternative possibility to define the chirality operator on the fuzzy sphere, due to the ambiguity of the operator ordering. Adopting this new chirality operator and the corresponding Dirac operator, we define Connes ’ spectral triple on the fuzzy sphere and the differential calculus. The differential calculus based on this new spectral triple is simplified considerably. Using this formulation the action of the scalar field is derived. 1