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259
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 50 (5 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.
Projective module description of the qmonopole
 COMUN.MATH.PHYS
, 1999
"... The Dirac qmonopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The ChernConnes pairing of cyclic cohomology and Ktheory is computed for the winding number −1. The nontriviality of this pairing is used to conclude that the quantum p ..."
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Cited by 49 (19 self)
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The Dirac qmonopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The ChernConnes pairing of cyclic cohomology and Ktheory is computed for the winding number −1. The nontriviality of this pairing is used to conclude that the quantum principal Hopf fibration is noncleft. Among general results, we provide a leftright symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (HopfGalois extensions) their associated covariant derivatives on projective modules.
On Groupoid C∗Algebras, Persistent Homology and TimeFrequency Analysis
"... We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in tim ..."
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Cited by 44 (1 self)
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We study some topological aspects in timefrequency analysis in the context of dimensionality reduction using C ∗algebras and noncommutative topology. Our main objective is to propose and analyze new conceptual and algorithmic strategies for computing topological features of datasets arising in timefrequency analysis. The main result of our work is to illustrate how noncommutative C ∗algebras and the concept of Morita equivalence can be applied as a new type of analysis layer in signal processing. From a conceptual point of view, we use groupoid C∗algebras constructed with timefrequency data in order to study a given signal. From a computational point of view, we consider persistent homology as an algorithmic tool for estimating topological properties in timefrequency analysis. The usage of C∗algebras in our environment, together with the problem of designing computational algorithms, naturally leads to our proposal of using AFalgebras in the persistent homology setting. Finally, a computational toy example is presented, illustrating some elementary aspects of our framework. Due to the interdisciplinary nature
String geometry and the noncommutative torus
 Commun. Math. Phys
, 1999
"... We describe an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra A and the noncommutative torus. We show that the tachyon subalgebra of A is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra o ..."
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Cited by 30 (8 self)
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We describe an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra A and the noncommutative torus. We show that the tachyon subalgebra of A is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding even real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d, d; Z) Morita equivalences between ddimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly dualitysymmetric. The dualityinvariant gauge theory is manifestly covariant but contains highly nonlocal interactions. We show that it also admits a new sort of particleantiparticle duality which enables the construction of instanton field configurations in any dimension. The duality nonsymmetric onshell projection of the field theory is shown to coincide with the standard nonabelian YangMills gauge theory minimally coupled to massive Dirac fermion fields. 1
Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review,” arXiv:0807.1939 [hepth
"... We present the state of the art regarding the relation between the physics of Quantum Black Holes and Noncommutative Geometry. We start with a review of models proposed in the literature for describing deformations of General Relativity in the presence of noncommutativity, seen as an effective theor ..."
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Cited by 30 (2 self)
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We present the state of the art regarding the relation between the physics of Quantum Black Holes and Noncommutative Geometry. We start with a review of models proposed in the literature for describing deformations of General Relativity in the presence of noncommutativity, seen as an effective theory of Quantum Gravity. We study the resulting metrics, proposed to replace or at least to improve the conventional black hole solutions of Einstein’s equation. In particular, we analyze noncommutativeinspired solutions obtained in terms of quasiclassical noncommutative coordinates: indeed because of their surprising new features, these solutions enable us to circumvent long standing problems with Quantum Field Theory in Curved Space and to cure the singular behavior of gravity at the centers of black holes. As a consequence, for the first time, we get a complete description of what we may call the black hole SCRAM, the shut down of the emission of thermal radiation from the black hole: in place of the conventional scenario of runaway evaporation in the Planck phase, we find a zero temperature final state, a stable black hole remnant, whose size and mass are determined uniquely in terms of the noncommutative parameter θ. This result turns out to be of vital importance for the physics of the forthcoming experiments at the LHC, where mini black hole production is foreseen in extreme energy hadron collisions. Because of this, we devote the final part of this review to higher dimensional solutions and their phenomenological implications for TeV Gravity.
Twisting all the Way: from Classical Mechanics to Quantum Fields
, 2007
"... We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and sym ..."
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Cited by 29 (6 self)
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We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from?Poisson brackets to?commutators. In particular commutation relations among creation and annihilation operators are deduced.
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 28 (1 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 geometry, dynamics and ∞adic Arakelov geometry
"... In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handl ..."
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Cited by 25 (12 self)
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In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g ≥ 2. We use Connes ’ theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger’s Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we consider derived (cohomological) spectral data (A, H · (X ∗),Φ), where the algebra is obtained from the SL(2, R) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the Archimedean factors from a zeta function of a spectral triple. Then, we introduce a different construction, which is related to Manin’s description of the dual graph of the fiber at infinity. We