Laplacians and gauged Laplacians on a quantum Hopf bundle
The author focusses on the analysis of a class of Hall Hamiltonians in the noncommutative set up. It is intended as a survey of the general formulation of quantum principal bundles, and as a description of a specific procedure to introduce, on both the total space and the base space of a quantum Hopf bundle, a set of Laplacian operators and to couple them with gauge connections. It presents a detailed formulation of the classical Hopf bundle.NEWLINENEWLINEClassical Hall Hamiltonians are gauged Laplace operators acting on the space of sections of the vector bundles associated to the principal bundles \(\pi : G \to G/K\) over homogeneous spaces with \(G\) semisimple and \(K\) compact and can be constructed in terms of the Casimir operators of \(G\) and \(K\). With \((\rho, V)\) a representation of \(K\), one has the identification of sections of the associated vector bundle \(E = G \times_{\rho(K)} V\) with equivariant maps from \(G\) to \(V\), \(\Gamma(G/K, E) \simeq C^{\infty} (G, V)_{\rho(K)} \subset C^{\infty} (G) \otimes V\). Given a connection on \(G\), one has a covariant derivative \(\nabla\) on \(\Gamma(G/K, E)\), so that the gauged Laplacian operator is \(\Delta^E = (\nabla \nabla^* + \nabla^* \nabla )\), where the dual \(\nabla^*\) is defined from the metric induced on the homogeneous space basis \(G/K\) by the Cartan-Killing metric on \(G\).NEWLINENEWLINEThe natural evolution is to develop models of the Hall effect on noncommutative spaces whose symmetries are described in terms of quantum groups. In [\textit{G. Landi, C. Reina} and \textit{A. Zampini}, Commun. Math. Phys. 287, No. 1, 179--209 (2009; Zbl 1180.58004)], the first model of excitations moving on a quantum 2-sphere in the field of a magnetic monopole has been studied. It is described by a quantum principal \(U(1)\)-bundle over a quantum sphere \(S_q^2\) having as a total space the manifold of the quantum group \(SU_q(2)\) [\textit{T. Brzeziński} and \textit{S. Majid}, Commun. Math. Phys. 157, No. 3, 591--638 (1993; Zbl 0817.58003)]. The natural associated line bundles are classified by the winding number \(n \in \mathbb{Z}\). Equipped \(SU_q(2)\) with the three dimensional left covariant calculus from \textit{S. L. Woronowicz} [Commun. Math. Phys. 122, No. 1, 125--170 (1989; Zbl 0751.58042)], the gauge monopole connection is studied and a gauged Laplacian acting on sections of the associated bundle is completely diagonalized.NEWLINENEWLINEThe author presents an analysis of the set of connections and covariant derivatives on a \(U(1)\)-quantum Hopf bundle on the standard quantum sphere \(S_q^2\), whose total space algebra \(SU_q(2)\) is equipped with the three dimensional left covariant differential calculus by Woronowicz. The introduction of a Hodge duality on both \(\Omega(SU_q(2))\) and \(\Omega(S_q^2)\) allows for the study of Laplacians and of gauged Laplacians. The author develops the analysis started in [Zbl 1180.58004], and describes another generalization of a relation given in [\textit{N. Berline, E. Getzler} and \textit{M. Vergne}, Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften. 298. Berlin etc.: Springer-Verlag. (1992; Zbl 0744.58001)] to the setting of the same Hopf bundle.NEWLINENEWLINEFor the entire collection see [Zbl 1222.58002].
scientific article; zbMATH DE number 6059391