arXiv subjects
Publications and source records attributed to Pei-Ming Ho.
An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.
We consider the problem of constructing the fully gauged effective action in $2n$-dimensional space-time for Nambu-Goldstone bosons valued in a homogeneous space $G/H$, with the requirement that the action be a solution of the anomalous Ward identity and be invariant under the gauge transformations of $H$. We show that this can be done whenever the homotopy group $π_{2n}(G/H)$ is trivial, $G/H$ is reductive and $H$ is embedded in $G$ so as to be anomaly free, in particular if $H$ is an anomaly safe group. We construct the necessary generalization of the Bardeen counterterm and give explicit forms for the anomaly and the effective action. When $G/H$ is a symmetric space the counterterm and the anomaly decompose into a parity even and a parity odd part. In this case, for the parity even part of the action, one does not need the anomaly free embedding of $H$.
The quantum deformation $CP_q(N)$ of complex projective space is discussed. Many of the features present in the case of the quantum sphere can be extended. The differential and integral calculus is studied and $CP_q(N)$ appears as a quantum Kähler manifold. The braiding of several copies of $CP_q(N)$ is introduced and the anharmonic ratios of four collinear points are shown to be invariant under quantum projective transformations. They provide the building blocks of all projective invariants. The Poisson limit is also described.
In a recent paper the quantum 2-sphere $S^2_q$ was described as a quantum complex manifold. Here we consider several copies of $S^2_q$ and derive their braiding commutation relations. The braiding is extended to the differential and to the integral calculus on the spheres. A quantum analogue of the classical anharmonic ratio of four points on the sphere is given, which is invariant under the coaction of $SU_q(2)$.
We describe the quantum sphere of Podleś for $c=0$ by means of a stereographic projection which is analogous to that which exhibits 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 coefficients as well as under the action of \SU vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the complex quantum plane.
We construct the deformed Dirac monopole on the quantum sphere for arbitrary charge using two different methods and show that it is a quantum principal bundle in the sense of Brzezinski and Majid. We also give a connection and calculate the analog of its Chern number by integrating the curvature over $S^2_q$.