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Ludwik Dabrowski

Publications and source records attributed to Ludwik Dabrowski.

At least 55 records · Page 3Linked to original sources
arXiv2008

The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere

Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the orthogonal quantum 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one on the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton' projection. We also introduce a real structure which satisfies all required properties modulo smoothing operators.

math.QA↗
arXiv2007

Dirac operators on all Podles quantum spheres

We construct spectral triples on all Podles quantum spheres. These noncommutative geometries are equivariant for a left action of $U_q(su(2))$ and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the 2-sphere. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.

math.QA↗
arXiv2006

The local index formula for quantum SU(2)

The local index formula of Connes--Moscovici for the isospectral noncommutative geometry recently constructed on quantum SU(2) is discussed. The cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula, are presented.

math.QA↗
arXiv2006

Local Index Formula on the Equatorial Podles Sphere

We discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the `degenerate' (i.e. $q=0$) case (related to the quantum disk). We consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one. After the identification of the smooth pre-$C^*$-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the $C^*$-algebra (the nontrivial generator of the $K_0$-group) as well as the pairing with the $q$-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.

math.QA↗
arXiv2005

Local index formula for SU_q(2)

We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.

math.QA↗
arXiv2005

Carbon diffusion in alpha-iron: Evidence for quantum mechanical tunneling

Recent experimental data on the diffusion coefficient of carbon in alpha-iron below liquid nitrogen temperature (LNT) question the classical approach to the observed temperature dependence. As the temperature is lowered below LNT, the diffusion constant tends to a nearly temperature-independent value rather than continuing its activated trend. The low temperature branch is apparently characteristic of a quantum mechanical process dominated by tunneling in ground state. Concommitantly, we apply an occurrence probability approach to describing the overall temperature dependence as a single continuous rate. Within the adiabatic approximation the electronic eigenvalue depending parametrically on the nuclear coordinates is taken to be the potential energy to control the motion of the nuclei. The resulting rate involves all horizontal-tunneling energy-conserving elastic transitions at the quantized energy levels of the migrating atom. A small though not negligible slope of in the temperature dependence as the temperature is raised below 100 K is dealt with by complementing for the rate of a parallel one-phonon inelastic tunneling process in excess to the basic elastic-tunneling rate. Our combined approach agrees well with the experimental data. In particular, the frequency of the coupled vibration is obtained virtually identical to the carbon vibrational frequency from inelastic neutron-scattering data. The migrational barrier is also found to be within the limits expected for alpha-iron.

cond-mat.mtrl-sci↗
arXiv2005

The Spectral Geometry of the Equatorial Podles Sphere

We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.

math.QA↗
arXiv2005

The Dirac operator on SU_q(2)

We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

math.QA↗
arXiv2003

Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces

We study sigma-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space $CP^{q-1}$.

math.QA↗
arXiv2003

Hopf fibration and monopole connection over the contact quantum spheres

Noncommutative geometry of quantised contact spheres introduced by Omori, Maeda, Miyazaki and Yoshioka is studied. In particular it is proven that these spheres form a noncommutative Hopf fibration in the sense of Hopf-Galois extensions. The monopole (strong) connection is constructed, and projectors describing projective modules of all monopole charges are computed.

math.QA↗
arXiv2002

Dirac Operator on the Standard Podles Quantum Sphere

Using principles of quantum symmetries we derive the algebraic part of the real spectral triple data for the standard Podleś quantum sphere: equivariant representation, chiral grading $γ$, reality structure $J$ and the Dirac operator $D$, which has bounded commutators with the elements of the algebra and satisfies the first order condition.

math.QA↗
arXiv2002

Quantum spin coverings and statistics

SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail.

math.QA↗
arXiv2002

Instanton algebras and quantum 4-spheres

We study some generalized instanton algebras which are required to describe `instantonic complex rank 2 bundles'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\in\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.

math.QA↗
arXiv2001

Instantons on the Quantum 4-Spheres S^4_q

We introduce noncommutative algebras $A_q$ of quantum 4-spheres $S^4_q$, with $q\in\IR$, defined via a suspension of the quantum group $SU_q(2)$, and a quantum instanton bundle described by a selfadjoint idempotent $e\in \Mat_4(A_q)$, $e^2=e=e^*$. Contrary to what happens for the classical case or for the noncommutative instanton constructed in Connes-Landi, the first Chern-Connes class $ch_1(e)$ does not vanish thus signaling a dimension drop. The second Chern-Connes class $ch_2(e)$ does not vanish as well and the couple $(ch_1(e), ch_2(e))$ defines a cycle in the $(b,B)$ bicomplex of cyclic homology.

math.QA↗
arXiv2000

Some Properties of Non-linear $σ$-Models in Noncommutative Geometry

We introduce non-linear $σ$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $σ$-models. In particular we construct a $σ$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model.

hep-th↗