TY - JOUR
T1 - Dirac operators on noncommutative principal circle bundles
Y1 - 2014
A1 - Andrzej Sitarz
A1 - Alessandro Zucca
A1 - Ludwik Dabrowski
AB - 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 → S2.
PB - World Scientific Publishing
UR - http://urania.sissa.it/xmlui/handle/1963/35125
U1 - 35363
U2 - Mathematics
U4 - 1
ER -
TY - JOUR
T1 - Curved noncommutative torus and Gauss--Bonnet
JF - Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518
Y1 - 2013
A1 - Ludwik Dabrowski
A1 - Andrzej Sitarz
KW - Geometry
AB - 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.
PB - American Institute of Physics
UR - http://hdl.handle.net/1963/7376
N1 - The article is composed of 13 pages and is recorded in PDF format
U1 - 7424
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - Noncommutative circle bundles and new Dirac operators
JF - Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130
Y1 - 2013
A1 - Ludwik Dabrowski
A1 - Andrzej Sitarz
KW - Quantum principal bundles
AB - 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.
PB - Springer
UR - http://hdl.handle.net/1963/7384
N1 - This article is composed of 25 pages and is recorded in PDF format
U1 - 7432
U2 - Mathematics
U4 - 1
U5 - MAT/07 FISICA MATEMATICA
ER -
TY - JOUR
T1 - The Dirac operator on SU_q(2)
JF - Commun. Math. Phys. 259 (2005) 729-759
Y1 - 2005
A1 - Ludwik Dabrowski
A1 - Giovanni Landi
A1 - Andrzej Sitarz
A1 - Walter van Suijlekom
A1 - Joseph C. Varilly
AB - 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.
PB - Springer
UR - http://hdl.handle.net/1963/4425
N1 - v2: minor changes
U1 - 4175
U2 - Mathematics
U3 - Mathematical Physics
U4 - -1
ER -
TY - JOUR
T1 - The local index formula for SUq(2)
JF - K-Theory 35 (2005) 375-394
Y1 - 2005
A1 - Walter van Suijlekom
A1 - Ludwik Dabrowski
A1 - Giovanni Landi
A1 - Andrzej Sitarz
A1 - Joseph C. Varilly
AB - 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.
UR - http://hdl.handle.net/1963/1713
U1 - 2438
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - The spectral geometry of the equatorial Podles sphere
JF - C. R. Math. 340 (2005) 819-822
Y1 - 2005
A1 - Ludwik Dabrowski
A1 - Giovanni Landi
A1 - Mario Paschke
A1 - Andrzej Sitarz
AB - 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.
UR - http://hdl.handle.net/1963/2275
U1 - 1972
U2 - Mathematics
U3 - Mathematical Physics
ER -
TY - JOUR
T1 - Dirac operator on the standard Podles quantum sphere
JF - Noncommutative geometry and quantum groups (Warsaw 2001),49,Banach Center Publ., 61, Polish Acad.Sci., Warsaw,2003
Y1 - 2001
A1 - Ludwik Dabrowski
A1 - Andrzej Sitarz
PB - SISSA Library
UR - http://hdl.handle.net/1963/1668
U1 - 2450
U2 - Mathematics
U3 - Mathematical Physics
ER -