%0 Journal Article %D 2014 %T Dirac operators on noncommutative principal circle bundles %A Andrzej Sitarz %A Alessandro Zucca %A Ludwik Dabrowski %X 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. %I World Scientific Publishing %G en %U http://urania.sissa.it/xmlui/handle/1963/35125 %1 35363 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-12-02T15:58:16Z No. of bitstreams: 1 preprint2014.pdf: 282573 bytes, checksum: 862ea930b67b00c451a0c6f093be973d (MD5) %R 10.1142/S0219887814500121 %0 Journal Article %J Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518 %D 2013 %T Curved noncommutative torus and Gauss--Bonnet %A Ludwik Dabrowski %A Andrzej Sitarz %K Geometry %X 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. %B Journal of Mathematical Physics. Volume 54, Issue 1, 22 January 2013, Article number 013518 %I American Institute of Physics %G en %U http://hdl.handle.net/1963/7376 %1 7424 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-06-17T13:10:21Z No. of bitstreams: 1 1204.0420v1.pdf: 126064 bytes, checksum: ffa850e50f99585d61676b1d2398af2f (MD5) %R 10.1063/1.4776202 %0 Journal Article %J Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130 %D 2013 %T Noncommutative circle bundles and new Dirac operators %A Ludwik Dabrowski %A Andrzej Sitarz %K Quantum principal bundles %X 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. %B Communications in Mathematical Physics. Volume 318, Issue 1, 2013, Pages 111-130 %I Springer %G en %U http://hdl.handle.net/1963/7384 %1 7432 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2014-06-18T10:06:43Z No. of bitstreams: 1 1012.3055v2.pdf: 181206 bytes, checksum: be182e0f568384847efe0f656a70634b (MD5) %R 10.1007/s00220-012-1550-8 %0 Journal Article %J Commun. Math. Phys. 259 (2005) 729-759 %D 2005 %T The Dirac operator on SU_q(2) %A Ludwik Dabrowski %A Giovanni Landi %A Andrzej Sitarz %A Walter van Suijlekom %A Joseph C. Varilly %X 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. %B Commun. Math. Phys. 259 (2005) 729-759 %I Springer %G en %U http://hdl.handle.net/1963/4425 %1 4175 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-04T08:01:47Z No. of bitstreams: 1 math_0411609v2.pdf: 293099 bytes, checksum: cfa2846ded2ecf161e83f4269b65e9b2 (MD5) %R 10.1007/s00220-005-1383-9 %0 Journal Article %J K-Theory 35 (2005) 375-394 %D 2005 %T The local index formula for SUq(2) %A Walter van Suijlekom %A Ludwik Dabrowski %A Giovanni Landi %A Andrzej Sitarz %A Joseph C. Varilly %X 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. %B K-Theory 35 (2005) 375-394 %G en_US %U http://hdl.handle.net/1963/1713 %1 2438 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2006-01-18T10:14:50Z\\nNo. of bitstreams: 1\\nmath.QA0501287.pdf: 189281 bytes, checksum: 75a780cbe958f6093e340102ad9bf176 (MD5) %R 10.1007/s10977-005-3116-4 %0 Journal Article %J C. R. Math. 340 (2005) 819-822 %D 2005 %T The spectral geometry of the equatorial Podles sphere %A Ludwik Dabrowski %A Giovanni Landi %A Mario Paschke %A Andrzej Sitarz %X 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. %B C. R. Math. 340 (2005) 819-822 %G en_US %U http://hdl.handle.net/1963/2275 %1 1972 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-22T09:37:10Z\\nNo. of bitstreams: 1\\n0408034v2.pdf: 95565 bytes, checksum: 1c0e7836d4006a796eb943f128938773 (MD5) %R 10.1016/j.crma.2005.04.003 %0 Journal Article %J Noncommutative geometry and quantum groups (Warsaw 2001),49,Banach Center Publ., 61, Polish Acad.Sci., Warsaw,2003 %D 2001 %T Dirac operator on the standard Podles quantum sphere %A Ludwik Dabrowski %A Andrzej Sitarz %B Noncommutative geometry and quantum groups (Warsaw 2001),49,Banach Center Publ., 61, Polish Acad.Sci., Warsaw,2003 %I SISSA Library %G en %U http://hdl.handle.net/1963/1668 %1 2450 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T13:06:10Z (GMT). No. of bitstreams: 1\\nmath.QA0209048.pdf: 89749 bytes, checksum: e1bc2f04a36a957d4fb3131860296c18 (MD5)\\n Previous issue date: 2002