Dirac Operators on Quantum Flag Manifolds

A Dirac operator D on quantized irreducible generalized flag manifolds is defined. This yields a Hilbert space realization of the covariant first-order differential calculi constructed by I. Heckenberger and S. Kolb. All differentials df=i[D,f] are bounded operators. In the simplest case of Podle' quantum sphere one obtains the spectral triple found by L. Dabrowski and A. Sitarz.     

Quantum Bundle Description of Quantum Projective Spaces

We realise Heckenberger and Kolb??s canonical calculus on quantum projective (N ? 1)-space C _q [C p ^N?1] as the restriction of a distinguished quotient of the standard bicovariant calculus for the quantum special unitary group C _q [SU _N ]. We introduce a calculus on the quantum sphere C _q [S ^2N?1] in the same way. With respect to these choices of calculi, we present C _q [C p ^N?1] as the base space of two different quantum principal bundles, one with total space C _q [SU _N ], and the other with total space C _q [S ^2N?1]. We go on to give C _q [C p ^N?1] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb??s calculus as an associated vector bundle to the principal bundle with total space C _q [SU _N ]. Finally, we construct strong connections for both bundles.     

Quantum Spheres and Projective Spaces as Graph Algebras

 The C ^* -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively. Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation. RID="**" ID="**" Partially supported by the Research Management Committee of the University of Newcastle.     

Stokes Matrices and Monodromy of the Quantum Cohomology of Projective Spaces

In this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a “classical” framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Received: 24 October 1998 / Accepted: 27 April 1999     

Higher Spin Dirac Operators Between Spaces of Simplicial Monogenics in Two Vector Variables

The higher spin Dirac operator \(\mathcal{Q}_{k,l}\) acting on functions taking values in an irreducible representation space for \(\mathfrak{so}(m)\) with highest weight \((k+\frac{1}{2},l+\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2})\), with k, l?∈?\(\mathbb{N}\) and \(k\geqslant l\), is constructed. The structure of the kernel space containing homogeneous polynomial solutions is then also studied.     

Dirac Quantum Geometrodynamics

Russian Physics Journal - With the help of the Dirac procedure for extracting the root from the Hamiltonian operator of the free gravitational field in flat superspace-time, we have constructed a...     

The Dirac Operator on Generalized Taub-NUT Spaces

We find sufficient conditions for the absence of harmonic L ² spinors on spin manifolds constructed as cone bundles over a compact Kähler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi?inescu and the second author.     

Conformally and Projective Covariant Differential Operators

Motivated by the structure of conformal anomalies in two-dimensional gravity and its generalizations, the projective and conformal covariance properties of linear, bilinear and trilinear differential operators are investigated in some detail and the triviality of the covariant trilinear operators is demonstrated.     

Basic Properties of Symplectic Dirac Operators

Symplectic Dirac operators, acting on symplectic spinor fields introduced by B.~Kostant in geometric quantization, are canonically defined in a similar way as the Dirac operator on Riemannian manifolds. These operators depend on a choice of a metaplectic structure as well as on a choice of a symplectic covariant derivative on the tangent bundle of the underlying manifold. This paper performs a complete study of these relations and shows further basic properties of the symplectic Dirac operators. Various examples are given for illustration. Received: 1 July 1996 / Accepted: 24 September 1996     

Twisted Reality Condition for Dirac Operators

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones, we propose a new twisted reality condition for the Dirac operator.     

D-bar Operators on Quantum Domains

We study the index problem for the d-bar operators subject to Atiyah- Patodi-Singer boundary conditions on noncommutative disk and annulus.     

Noncommutative Circle Bundles and New Dirac Operators

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.     

Random Dirac Operators with Time Reversal Symmetry

Quasi-one-dimensional stochastic Dirac operators with an odd number of channels, time reversal symmetry but otherwise efficiently coupled randomness, are shown to have one conducting channel and absolutely continuous spectrum of multiplicity two. This follows by adapting the criteria of Guivarch-Raugi and Goldsheid-Margulis to the analysis of random products of matrices in the group SO^*(2L), and then a version of Kotani theory for these operators. Absence of singular spectrum can be shown by adapting an argument of Jaksic-Last if the potential contains random Dirac peaks with absolutely continuous distribution.     

Quantum Spinor Structures for Quantum Spaces

A general theory of quantum spinor structures on quantum spaces is presented within the formalism of quantum principal bundles. Quantum analogs of basic objects of the classical theory are constructed: Laplace and Dirac operators, quantum versions of Clifford and spinor bundles, a Hodge *-operator, integration operators. Quantum phenomena are discussed, including an example of the Dirac operator associated to a quantum Hopf fibration.