Conformal Algebra and Physical States in a Non-Critical 3-Brane on R xS3

Abstract

A world-volume model of a non-critical 3-brane is quantized in a strong coupling phase in which fluctuations of the conformal mode become dominant. This phase, called the conformal-mode dominant phase, is realized at very high energies far beyond the Planck mass scale. We separately treat the conformal mode and the traceless mode and quantize the conformal mode non-perturbatively, while the traceless mode is treated in a perturbative method that is renormalizable and asymptotically free. In the conformal-mode dominant phase, the coupling of the traceless mode vanishes, and the world-volume dynamics are described by a four-dimensional conformal field theory (CFT4). We canonically quantize this model on R ×S3, where the dynamical fields are expanded in spherical tensor harmonics on S3, which include both positive-metric and negative-metric modes. Conformal charges and a conformal algebra are constructed. They yield strong constraints on physical states. We find that all negative-metric modes are related to positive-metric modes through the charges, and thus negative-metric modes are themselves not independent physical modes. Physical states satisfying the conformal invariance conditions are given by particular combinations of positive-metric and negative-metric modes. An infinite number of such physical states are constructed. In the Appendices, we construct spherical vector and tensor harmonics on S3 in practical forms using the Wigner D functions and the Clebsch-Gordan coefficients and calculate the integrals of three and four products of these harmonics over S3.