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262
Gauge Theory on Noncommutative Spaces
 Eur. Phys. J. C16
"... We introduce a formulation of gauge theory on noncommutative spaces based on the concept of covariant coordinates. Some important examples are discussed in detail. A SeibergWitten map is established We introduce a natural method to formulate a gauge theory on more or less arbitrary noncommutative s ..."
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Cited by 138 (18 self)
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We introduce a formulation of gauge theory on noncommutative spaces based on the concept of covariant coordinates. Some important examples are discussed in detail. A SeibergWitten map is established We introduce a natural method to formulate a gauge theory on more or less arbitrary noncommutative spaces. The starting point is the observation that multiplication of a (covariant) field by a coordinate can in general not be a covariant operation in noncommutative geometry, because the coordinates
Braided Quantum Field Theory
"... We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for npoint functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have n ..."
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Cited by 46 (6 self)
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We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for npoint functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have nontrivial over and undercrossings. We demonstrate the power of our approach by applying it to φ 4theory on the quantum 2sphere. We find that the basic divergent diagram of the theory is regularised.
S.Koshelev, “A Note on UV/IR for Noncommutative Complex Scalar Field,” hepth/0001215
"... Noncommutative quantum field theory of a complex scalar field is considered. There is a twocoupling noncommutative analogue of U(1)invariant quartic interaction (φ ∗ φ) 2, namely Aφ ∗ ⋆ φ ⋆ φ ∗ ⋆ φ + Bφ ∗ ⋆ φ ∗ ⋆ φ ⋆ φ. For arbitrary values of A and B the model is nonrenormalizable. However, i ..."
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Cited by 38 (1 self)
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Noncommutative quantum field theory of a complex scalar field is considered. There is a twocoupling noncommutative analogue of U(1)invariant quartic interaction (φ ∗ φ) 2, namely Aφ ∗ ⋆ φ ⋆ φ ∗ ⋆ φ + Bφ ∗ ⋆ φ ∗ ⋆ φ ⋆ φ. For arbitrary values of A and B the model is nonrenormalizable. However, it is oneloop renormalizable in two special cases: B = 0 and A = B. Furthermore, in the case B = 0 the model does not suffer from IR divergencies at Recently, there is a renovation of the interest in noncommutative quantum field theories (or field theories on noncommutative spacetime [1, 2]). As emphasized in [3], the important question is whether or not the noncommutative quantum field theory is welldefined. Note that one of earlier motivations to consider noncommutative field theories is a hope that it would be possible
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.
The spectral action for Moyal planes
 J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 31 (6 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the ConnesLott action [15] previously computed by Gayral [23] for symplectic Θ.
Monopoles and solitons in fuzzy physics
 Commun. Math. Phys
, 2000
"... Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of di ..."
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Cited by 24 (0 self)
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Monopoles and solitons have important topological aspects like quantized fluxes, winding numbers and curved target spaces. Naive discretizations which substitute a lattice of points for the underlying manifolds are incapable of retaining these features in a precise way. We study these problems of discrete physics and matrix models and discuss mathematically coherent discretizations of monopoles and solitons using fuzzy physics and noncommutative geometry. A fuzzy σmodel action for the twosphere fulfilling a fuzzy BelavinPolyakov bound is also put forth. 1 A fuzzy space ( [1–8]) is obtained by quantizing a manifold, treating it as a phase space. An example is the fuzzy twosphere S2 F. It is described by operators xi subject to the relations ∑ i x2i = 1 and [xi, xj] = (i / √ l(l + 1))ǫijkxk. Thus Li = √ l(l + 1)xi are (2l+1)dimensional angular momentum operators
Noncommutative chiral anomaly and the DiracGinspargWilson operator
 JHEP 0308 (2003) 046 [arXiv:hepth/0211209
"... It is shown that local axial anomaly in 2−dimensions emerges naturally in the gaugeinvariant quantized GinspargWilson relation if one postulates an underlying noncommutative structure of spacetime. Indeed if one first regularizes the 2−d plane with a fuzzy sphere, i.e with a (2l + 1)×(2l + 1) matr ..."
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Cited by 22 (0 self)
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It is shown that local axial anomaly in 2−dimensions emerges naturally in the gaugeinvariant quantized GinspargWilson relation if one postulates an underlying noncommutative structure of spacetime. Indeed if one first regularizes the 2−d plane with a fuzzy sphere, i.e with a (2l + 1)×(2l + 1) matrix model, then one immediately finds that the gaugeinvariant Dirac operator DGF is different from the chiralinvariant Dirac operator DCF. The fact that gauge states are different from chiral states stems from the free noncommutative Dirac operator DF which is found to be inconsistent with chiral symmetry at high frequencies. Furthermore it splits such that is the
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.
Matrix algebras converge to the sphere for quantum GromovHausdorff distance
 Mem. Amer. Math. Soc
"... Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how t ..."
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Cited by 20 (3 self)
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Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups. On perusing the theoretical physics literature which deals with string theory and related parts of quantum field theory, one finds in many scattered places assertions that the complex matrix algebras, Mn, converge to the twosphere, S 2, (or to related spaces) as n goes to infinity. Here S 2 is viewed as synonymous with the algebra C(S 2) of continuous complexvalued functions on S 2 (of which S 2 is the maximalideal space). Approximating the sphere by matrix algebras is attractive for the following reason. In trying to carry out quantum field theory on S 2 it is natural to try to proceed by approximating S 2 by finite spaces. But “lattice ” approximations coming from choosing a finite set of points in S 2 break the very important symmetry of the action of SU(2) on S 2 (via SO(3)). But SU(2) acts naturally on the matrix algebras, in a way coherent with its action on S 2, as we will recall below. So it is natural to use them to approximate C(S 2). In this setting the matrix algebras are often referred to as “fuzzy spheres”. (See [33], [34], [17], [22], [24] and references therein.) When using the approximation of S 2 by matrix algebras, the precise sense of convergence is usually not explicitly specified in the literature. Much of the literature is at a largely algebraic level, with indications that the notion of convergence which is intended involves how structure constants and important formulas change as n grows. See, for