@article {2014,
title = {Dirac operators on noncommutative principal circle bundles},
number = {International Journal of Geometric Methods in Modern Physics;volume 11; issue 1; article number 1450012;},
year = {2014},
publisher = {World Scientific Publishing},
abstract = {We 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 {\textrightarrow} S2.},
doi = {10.1142/S0219887814500121},
url = {http://urania.sissa.it/xmlui/handle/1963/35125},
author = {Andrzej Sitarz and Alessandro Zucca and Ludwik Dabrowski}
}
@article {2014,
title = {Quantum gauge symmetries in noncommutative geometry},
number = {Journal of Noncommutative Geometry;volume 8; issue 2; pages 433-471;},
year = {2014},
publisher = {European Mathematical Society Publishing House},
abstract = {We 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).},
doi = {10.4171/JNCG/161},
url = {http://urania.sissa.it/xmlui/handle/1963/34897},
author = {Jyotishman Bhowmick and Francesco D{\textquoteright}Andrea and Biswarup Krishna Das and Ludwik Dabrowski}
}
@article {2013,
title = {Curved noncommutative torus and Gauss--Bonnet},
journal = {Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518},
number = {arXiv:1204.0420v1;},
year = {2013},
note = {The article is composed of 13 pages and is recorded in PDF format},
publisher = {American Institute of Physics},
abstract = {We 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.},
keywords = {Geometry},
doi = {10.1063/1.4776202},
url = {http://hdl.handle.net/1963/7376},
author = {Ludwik Dabrowski and Andrzej Sitarz}
}
@article {2013,
title = {Dirac operator on spinors and diffeomorphisms},
journal = {Classical and Quantum Gravity. Volume 30, Issue 1, 7 January 2013, Article number 015006},
number = {arXiv:1209.2021v1;},
year = {2013},
note = {This article is composed of 13 pages and is recorded in PDF format},
publisher = {IOP Publishing},
abstract = {The 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.},
keywords = {gravity},
doi = {10.1088/0264-9381/30/1/015006},
url = {http://hdl.handle.net/1963/7377},
author = {Ludwik Dabrowski and Giacomo Dossena}
}
@article {2013,
title = {Noncommutative circle bundles and new Dirac operators},
journal = {Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130},
number = {arXiv:1012.3055v2;},
year = {2013},
note = {This article is composed of 25 pages and is recorded in PDF format},
publisher = {Springer},
abstract = {We 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.},
keywords = {Quantum principal bundles},
doi = {10.1007/s00220-012-1550-8},
url = {http://hdl.handle.net/1963/7384},
author = {Ludwik Dabrowski and Andrzej Sitarz}
}
@article {2011,
title = {Poincar{\'e} covariance and κ-Minkowski spacetime},
journal = {Physics Letters A 375 (2011) 3496-3498},
number = {SISSA;43/2010/FM},
year = {2011},
publisher = {Elsevier},
abstract = {A fully Poincar{\'e} covariant model is constructed out of the k-Minkowski spacetime. Covariance is implemented by a unitary representation of the Poincar{\'e} group, and thus complies with the original Wigner approach to quantum symmetries. This provides yet another example (besides the DFR model), where Poincar{\'e} covariance is realised {\'a} 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 \\\"Poincar{\'e} covariance\\\".},
doi = {10.1016/j.physleta.2011.08.011},
url = {http://hdl.handle.net/1963/3893},
author = {Ludwik Dabrowski and Gherardo Piacitelli}
}
@article {2011,
title = {Product of real spectral triples},
journal = {International Journal of Geometric Methods in Modern Physics 8 (2011) 1833-1848},
number = {arXiv:1011.4456;},
year = {2011},
note = {Based on the talk given at the conference \\\"Noncommutative Geometry and Quantum Physics, Vietri sul Mare, Aug 31 - Sept 5, 2009\\\"},
publisher = {World Scientific},
abstract = {We 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.},
doi = {10.1142/S021988781100597X},
url = {http://hdl.handle.net/1963/5510},
author = {Ludwik Dabrowski and Giacomo Dossena}
}
@article {2011,
title = {Quantum Isometries of the finite noncommutative geometry of the Standard Model},
journal = {Commun. Math. Phys. 307:101-131, 2011},
number = {arXiv:1009.2850;},
year = {2011},
publisher = {Springer},
abstract = {We 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.},
doi = {10.1007/s00220-011-1301-2},
url = {http://hdl.handle.net/1963/4906},
author = {Jyotishman Bhowmick and Francesco D{\textquoteright}Andrea and Ludwik Dabrowski}
}
@article {2010,
title = {Canonical k-Minkowski Spacetime},
number = {SISSA;26/2010/FM},
year = {2010},
abstract = {A 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.},
url = {http://hdl.handle.net/1963/3863},
author = {Gherardo Piacitelli and Ludwik Dabrowski}
}
@article {2010,
title = {Dirac Operators on Quantum Projective Spaces},
journal = {Comm. Math. Phys. 295 (2010) 731-790},
number = {SISSA;17/2009/FM},
year = {2010},
note = {Quantum Algebra},
abstract = {We 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