01079nas a2200133 4500008004100000245006300041210006300104260003200167520063100199100002000830700002200850700002200872856005100894 2014 en d00aDirac operators on noncommutative principal circle bundles0 aDirac operators on noncommutative principal circle bundles bWorld Scientific Publishing3 aWe study spectral triples over noncommutative principal U(1)-bundles of arbitrary dimension and a compatibility condition between the connection and the Dirac operator on the total space and on the base space of the bundle. Examples of low-dimensional noncommutative tori are analyzed in more detail and all connections found that are compatible with an admissible Dirac operator. Conversely, a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection is exhibited. These examples are extended to the theta-deformed principal U(1)-bundle S 3 θ → S2.1 aSitarz, Andrzej1 aZucca, Alessandro1 aDabrowski, Ludwik uhttp://urania.sissa.it/xmlui/handle/1963/3512501462nas a2200145 4500008004100000245005600041210005600097260005100153520096300204100002501167700002401192700002701216700002201243856005101265 2014 en d00aQuantum gauge symmetries in noncommutative geometry0 aQuantum gauge symmetries in noncommutative geometry bEuropean Mathematical Society Publishing House3 aWe discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M n(R), Mn(C) and Mn(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C H M3 (C) and Aev = H H M4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp.n/ (quaternionic unitary group).1 aBhowmick, Jyotishman1 aD'Andrea, Francesco1 aDas, Biswarup, Krishna1 aDabrowski, Ludwik uhttp://urania.sissa.it/xmlui/handle/1963/3489700967nas a2200133 4500008004100000245005000041210004800091260003400139520056900173653001300742100002200755700002000777856003600797 2013 en d00aCurved noncommutative torus and Gauss--Bonnet0 aCurved noncommutative torus and GaussBonnet bAmerican Institute of Physics3 aWe study perturbations of the flat geometry of the noncommutative
two-dimensional torus T^2_\theta (with irrational \theta). They are described
by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a
differential operator with coefficients in the commutant of the (smooth)
algebra A_\theta of T_\theta. We show, up to the second order in perturbation,
that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We
also calculate first two terms of the perturbative expansion of the
corresponding local scalar curvature.10aGeometry1 aDabrowski, Ludwik1 aSitarz, Andrzej uhttp://hdl.handle.net/1963/737601202nas a2200133 4500008004100000245005000041210005000091260001900141520081700160653001200977100002200989700002101011856003601032 2013 en d00aDirac operator on spinors and diffeomorphisms0 aDirac operator on spinors and diffeomorphisms bIOP Publishing3 aThe issue of general covariance of spinors and related objects is
reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$
and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor
fields on $M$ and a Hilbert space $\HH_{\sigma, g}= L^2(S_{\sigma,
g},\vol{M}{g})$ of $L^2$-spinors of $S_{\sigma, g}$. The group $\diff{M}$ of
orientation-preserving diffeomorphisms of $M$ acts both on $g$ (by pullback)
and on $[\sigma]$ (by a suitably defined pullback $f^*\sigma$). Any $f\in
\diff{M}$ lifts in exactly two ways to a unitary operator $U$ from
$\HH_{\sigma, g} $ to $\HH_{f^*\sigma,f^*g}$. The canonically defined Dirac
operator is shown to be equivariant with respect to the action of $U$, so in
particular its spectrum is invariant under the diffeomorphisms.10agravity1 aDabrowski, Ludwik1 aDossena, Giacomo uhttp://hdl.handle.net/1963/737701090nas a2200133 4500008004100000245005800041210005800099260001300157520067800170653003000848100002200878700002000900856003600920 2013 en d00aNoncommutative circle bundles and new Dirac operators0 aNoncommutative circle bundles and new Dirac operators bSpringer3 aWe study spectral triples over noncommutative principal U(1) bundles. Basing
on the classical situation and the abstract algebraic approach, we propose an
operatorial definition for a connection and compatibility between the
connection and the Dirac operator on the total space and on the base space of
the bundle. We analyze in details the example of the noncommutative three-torus
viewed as a U(1) bundle over the noncommutative two-torus and find all
connections compatible with an admissible Dirac operator. Conversely, we find a
family of new Dirac operators on the noncommutative tori, which arise from the
base-space Dirac operator and a suitable connection.10aQuantum principal bundles1 aDabrowski, Ludwik1 aSitarz, Andrzej uhttp://hdl.handle.net/1963/738400961nas a2200121 4500008004300000245005200043210005100095260001300146520059700159100002200756700002500778856003600803 2011 en_Ud 00aPoincaré covariance and κ-Minkowski spacetime0 aPoincaré covariance and κMinkowski spacetime bElsevier3 aA fully Poincaré covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincaré group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincaré covariance is realised á la Wigner in the presence of two characteristic dimensionful parameters: the light speed and the Planck length. In other words, a Doubly Special Relativity (DSR) framework may well be realised without deforming the meaning of \\\"Poincaré covariance\\\".1 aDabrowski, Ludwik1 aPiacitelli, Gherardo uhttp://hdl.handle.net/1963/389300741nas a2200121 4500008004100000245003700041210003700078260002100115520040400136100002200540700002100562856003600583 2011 en d00aProduct of real spectral triples0 aProduct of real spectral triples bWorld Scientific3 aWe construct the product of real spectral triples of arbitrary finite dimension (and arbitrary parity) taking into account the fact that in the even case there are two possible real structures, in the odd case there are two inequivalent representations of the gamma matrices (Clifford algebra), and in the even-even case there are two natural candidates for the Dirac operator of the product triple.1 aDabrowski, Ludwik1 aDossena, Giacomo uhttp://hdl.handle.net/1963/551000860nas a2200133 4500008004100000245008400041210006900125260001300194520041200207100002500619700002400644700002200668856003600690 2011 en d00aQuantum Isometries of the finite noncommutative geometry of the Standard Model0 aQuantum Isometries of the finite noncommutative geometry of the bSpringer3 aWe compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries.1 aBhowmick, Jyotishman1 aD'Andrea, Francesco1 aDabrowski, Ludwik uhttp://hdl.handle.net/1963/490601351nas a2200109 4500008004300000245003600043210003500079520104400114100002501158700002201183856003601205 2010 en_Ud 00aCanonical k-Minkowski Spacetime0 aCanonical kMinkowski Spacetime3 aA complete classification of the regular representations of the relations [T,X_j] = (i/k)X_j, j=1,...,d, is given. The quantisation of RxR^d canonically (in the sense of Weyl) associated with the universal representation of the above relations is intrinsically \\\"radial\\\", this meaning that it only involves the time variable and the distance from the origin; angle variables remain classical. The time axis through the origin is a spectral singularity of the model: in the large scale limit it is topologically disjoint from the rest. The symbolic calculus is developed; in particular there is a trace functional on symbols. For suitable choices of states localised very close to the origin, the uncertainties of all spacetime coordinates can be made simultaneously small at wish. On the contrary, uncertainty relations become important at \\\"large\\\" distances: Planck scale effects should be visible at LHC energies, if processes are spread in a region of size 1mm (order of peak nominal beam size) around the origin of spacetime.1 aPiacitelli, Gherardo1 aDabrowski, Ludwik uhttp://hdl.handle.net/1963/386300681nas a2200109 4500008004300000245004900043210004900092520034800141100002400489700002200513856003600535 2010 en_Ud 00aDirac Operators on Quantum Projective Spaces0 aDirac Operators on Quantum Projective Spaces3 aWe construct a family of self-adjoint operators D_N which have compact resolvent and bounded commutators with the coordinate algebra of the quantum projective space CP_q(l), for any l>1 and 0