55 research outputs found

    Heat trace for Laplacian type operators with non-scalar symbols

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    For an elliptic selfadjoint operator P=−[uμν∂μ∂ν+vν∂ν+w]P =-[u^{\mu\nu}\partial_\mu \partial_\nu +v^\nu \partial_\nu +w] acting on a fiber bundle over a Riemannian manifold, where u,vμ,wu,v^\mu,w are N×NN\times N-matrices, we develop a method to compute the heat-trace coefficients ara_r which allows to get them by a pure computational machinery. It is exemplified in dimension 4 by the value of a1a_1 written both in terms of u,vμ,wu,v^\mu,w or diffeomorphic and gauge invariants. We also answer to the question: when is it possible to get explicit formulae for ara_r?Comment: 37 pages. v2: misprints corrected, references added, section 5.4 adde

    Crossed product extensions of spectral triples

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    Given a spectral triple (A,H,D)(A,H,D) and a C∗C^*-dynamical system (A,G,α)(\mathbf{A}, G, \alpha) where AA is dense in A\mathbf{A} and GG is a locally compact group, we extend the triple to a triplet (B,H,D)(\mathcal{B},\mathcal{H},\mathcal{D}) on the crossed product G⋉α,redAG \ltimes_{\alpha, \text{red}} \mathbf{A} which can be promoted to a modular-type twisted spectral triple within a general procedure exemplified by two cases: the C∗C^*-algebra of the affine group and the conformal group acting on a complete Riemannian spin manifold.Comment: Version 3: version to appear in Journal of Noncommutative Geometr

    Spectral Geometry

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    International audienceThe goal of these lectures is to present some fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea here is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein--Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one. The basics ingredients such as Dirac operators, heat equation asymptotics, zeta functions, noncommutative residues, pseudodifferential operators, or Dixmier traces will be presented and studied within the framework of operators on Hilbert spaces. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus

    Heat trace and spectral action on the standard Podles sphere

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    We give a new definition of dimension spectrum for non-regular spectral triples and compute the exact (i.e. non only the asymptotics) heat-trace of standard Podles spheres Sq2S^2_q for 0<q<10<q<1, study its behavior when q→1q\to 1 and fully compute its exact spectral action for an explicit class of cut-off functions.Comment: 44 pages, 1 figur

    On spectral Actions

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    Oberwolfach conference on "Dirac Operators in Differential and Noncommutative Geometry" November 26th -- December 2nd, 2006 Organizers: Christian Bär (Potsdam) and Andrzej Sitarz (Krakow)}During the talk, based on collaborations with V. Gayral, C. Levy, A. Sitarz and D. Vassilevich, different aspects of the Connes--Lott action and Chamseddine--Connes spectral action for spectral triples are developed. New results concerning the second action for the noncommutative torus and the triple associated to SUq(2) are presented

    Fuzzy Mass Relations in the Standard Model

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    Recently Connes has proposed a new geometric version of the standard model including a non-commutative charge conjugation. We present a systematic analysis of the relations among masses and coupling constants in this approach. In particular, for a given top mass, the Higgs mass is constrained to lie in an interval. Therefore this constraint is locally stable under renormalization flow.Comment: 14 pages LaTeX, one figure postscrip

    Global and local aspects of spectral actions

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    The principal object in noncommutatve geometry is the spectral triple consisting of an algebra A, a Hilbert space H, and a Dirac operator D. Field theories are incorporated in this approach by the spectral action principle, that sets the field theory action to Tr f(D^2/\Lambda^2), where f is a real function such that the trace exists, and \Lambda is a cutoff scale. In the low-energy (weak-field) limit the spectral action reproduces reasonably well the known physics including the standard model. However, not much is known about the spectral action beyond the low-energy approximation. In this paper, after an extensive introduction to spectral triples and spectral actions, we study various expansions of the spectral actions (exemplified by the heat kernel). We derive the convergence criteria. For a commutative spectral triple, we compute the heat kernel on the torus up the second order in gauge connection and consider limiting cases.Comment: 22 pages, dedicated to Stuart Dowker on his 75th birthda
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