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92
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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From Physics to Number theory via Noncommutative Geometry, II  Chapter 2: Renormalization, The RiemannHilbert correspondence, and . . .
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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 50 (14 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.
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 37 (6 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 33 (7 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.
A family of elliptic algebras
 Internat. Math. Res. Notices
, 1997
"... The survey is devoted to associative Z≥0graded algebras presented by n generators and n(n−1) 2 quadratic relations and satisfying the socalled PoincareBirkhoffWitt condition (PBWalgebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve ..."
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Cited by 28 (4 self)
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The survey is devoted to associative Z≥0graded algebras presented by n generators and n(n−1) 2 quadratic relations and satisfying the socalled PoincareBirkhoffWitt condition (PBWalgebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras are described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.
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 26 (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.
MODULI SPACE AND STRUCTURE OF NONCOMMUTATIVE 3SPHERES
, 2003
"... We analyse the moduli space and the structure of noncommutative 3spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a crossproduct algebra associated ..."
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Cited by 26 (8 self)
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We analyse the moduli space and the structure of noncommutative 3spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a crossproduct algebra associated to the characteristic variety and lands in a richer crossproduct. It allows to control the C ∗norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with 8 points in common. We show
Noncommutative twotori with real multiplication as noncommutative projective varieties
 J. Geom. Phys
"... Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate a ..."
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Cited by 21 (4 self)
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Abstract. We define analogues of homogeneous coordinate algebras for noncommutative twotori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative twotorus with real multiplication. These results are based on the techniques of [14] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.