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 -