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19
Noncommutative geometry and gravity
"... We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a starproduct. The class of noncommutative spaces studied is very rich. Nonanti ..."
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Cited by 77 (18 self)
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We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a starproduct. The class of noncommutative spaces studied is very rich. Nonanticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein’s equations for gravity on noncommutative manifolds.
Heatkernel approach to UV/IR mixing on isospectral deformation manifolds
"... We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions o ..."
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Cited by 27 (3 self)
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We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of R l. Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and noncompact spaces, as well as with periodic and nonperiodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the nonplanar parts of the Green functions is understood simply in terms of offdiagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the nonplanar part of the oneloop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Keywords: noncommutative field theory, isospectral deformation, UV/IR mixing, heat kernel, Diophantine approximation.
Heat Kernel and Number Theory on NCTorus
 Commun. Math. Phys
, 2007
"... The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the numbertheoretical aspect of the deformation parameters. The central condition we ..."
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Cited by 18 (11 self)
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The heat trace asymptotics on the noncommutative torus, where generalized Laplacians are made out of left and right regular representations, is fully determined. It turns out that this question is very sensitive to the numbertheoretical aspect of the deformation parameters. The central condition we use is of a Diophantine type. More generally, the importance of number theory is made explicit on a few examples. We apply the results to the spectral action computation and revisit the UV/IR mixing phenomenon for a scalar theory. Although we find nonlocal counterterms in the NC φ 4 theory on T 4, we show that this theory can be made renormalizable at least at one loop, and may be even beyond.
Integration on locally compact noncommutative spaces
 Journal of Functional Analysis
, 2012
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Noncommutative Gravity Solutions
, 2009
"... We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented. Inspired by [1, 2], we obtain solutions of noncommutative Einstein equations by considering twists that a ..."
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Cited by 10 (0 self)
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We consider noncommutative geometries obtained from a triangular Drinfeld twist and review the formulation of noncommutative gravity. A detailed study of the abelian twist geometry is presented. Inspired by [1, 2], we obtain solutions of noncommutative Einstein equations by considering twists that are compatible with
Sukochev F., Measures from Dixmier traces and zeta functions
 J. Funct. Anal
"... For L∞functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, L2functions. To do bette ..."
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Cited by 8 (6 self)
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For L∞functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, L2functions. To do better than L2functions, symmetrised noncommutative residue and Dixmier trace formulas are introduced, for which the identifications are shown to continue to L1+ǫfunctions,ǫ> 0. However, a failure is shown for the Dixmier trace formulation at L1functions. The (symmetrised) noncommutative residue and Dixmier trace formulas diverge at this point. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show that a claim in the monograph J. M. GraciaBondía, J. C. Várilly and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001, that the identification on C∞functions obtained using Connes ’ Trace Theorem can be extended to any integrable function, is false. The results of this paper are obtained from a general presentation for finitely generated von Neumann algebras of commuting bounded operators, including a bounded Borel
On the hermiticity of qdifferential operators and forms on the quantum Euclidean spaces R N q
"... We show that the complicated ⋆structure characterizing for positive q the Uqso(N)covariant differential calculus on the noncommutative manifold RN q boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ˜v. Subare made ..."
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Cited by 7 (7 self)
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We show that the complicated ⋆structure characterizing for positive q the Uqso(N)covariant differential calculus on the noncommutative manifold RN q boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ˜v. Subare made into Hilbert spaces of the spaces of functions and of pforms on RN q spaces by introducing nonconventional “weights ” in the integrals defining the corresponding scalar products, namely suitable positivedefinite qpseudodifferential operators ˜v ′±1 realizing the action of ˜v ±1; this serves to make the partial qderivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the ‘radial coordinate ’ r. Unless we choose a constant m, then the squareintegrables functions/forms must fulfill an additional condition, namely their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are qspecial functions with ‘quantized ’ free parameters.
Spectral action on SUq(2)
, 803
"... The spectral action on the equivariant real spectral triple over A ( SUq(2) ) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S 3. ..."
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Cited by 6 (4 self)
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The spectral action on the equivariant real spectral triple over A ( SUq(2) ) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S 3.
Heat Trace Asymptotics on Noncommutative Spaces
, 2007
"... This is a minireview of the heat kernel expansion for generalized Laplacians on various noncommutative spaces. Applications to the spectral action principle, renormalization of noncommutative theories and anomalies are also considered. ..."
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Cited by 6 (4 self)
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This is a minireview of the heat kernel expansion for generalized Laplacians on various noncommutative spaces. Applications to the spectral action principle, renormalization of noncommutative theories and anomalies are also considered.