Dirac operator on the standard Podleś quantum sphere (original) (raw)

Dirac operators on all Podleś quantum spheres

Journal of Noncommutative Geometry, 2000

We construct spectral triples on all Podleś quantum spheres S 2 qt . 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 sphere S 2 . 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.

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

Communications in Mathematical Physics, 2008

Equivariance under the action of U q (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 S 4 q with a Dirac operator which is isospectral to the canonical one on the round sphere S 4 and which then gives 4 +summability. 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.

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

2006

Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 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 of the spin structure of 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. A real structure which satisfies all required properties modulo a suitable ideal of `infinitesimals' is also introduced.

The spectral geometry of the equatorial Podleś sphere

Comptes Rendus Mathematique, 2005

We propose a slight modification of the properties of a spectral geometry a la Connes, allowing for some of the algebraic relations to be satisfied modulo compact operators. On the equatorial Podleś sphere we construct U q (su(2))-equivariant Dirac operator and real structure which satisfy these modified properties.

On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces

Journal of Geometry and Physics, 1996

The eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces of arbitrary dimension are computed by separating variables in geodesic polar coordinates. These eigenfunctions are then used to derive the heat kernel of the iterated Dirac operator on these spaces. They are then studied as cross sections of homogeneous vector bundles, and a group-theoretic derivation of the spinor spherical functions and heat kernel is given based on Harish-Chandra's formula for the radial part of the Casimir operator.

A note on spectral triple with real structure on fuzzy sphere

arXiv: High Energy Physics - Theory, 2021

Here we have illustrated the construction of a real structure on fuzzy sphere S 2 * in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on S 2 * given by Watamura et. al. in [U. C

Spectral triple with real structure on fuzzy sphere

Journal of Mathematical Physics, 2022

Here we have illustrated the construction of a real structure on fuzzy sphere S ∗ in its spin-1/2 representation. Considering the SU(2) coviariant Dirac and chirality operator on S ∗ given by Wattamura et. al. in [1], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition, where we find it necessary to enlarge the symmetry group from SO(3) to the full orthogonal group O(3). However the first order condition is violated thus paving the way to construct a toy model for an SU(2) gauge theory to capture some features of physics beyong standard model following Connes et.al.[2].

The quantization of the symplectic groupoid of the standard Podles sphere

2010

We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.

The quantum 2-sphere as a complex quantum manifold

Zeitschrift f�r Physik C Particles and Fields, 1996

We describe the quantum sphere of Podleś for c = 0 by means of a stereographic projection which is analogous to that which exibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with SU q (2) coefficients as well as under the action of SU q (2) vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the *