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 (CFT 4 ). We canonically quantize this model on R × S 3 , where the dynamical fields are expanded in spherical tensor harmonics on S 3 , 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 negativemetric modes. An infinite number of such physical states are constructed. In the appendices, we construct spherical vector and tensor harmonics on S 3 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 S 3 . † J− 1 2 M 2 .(5.3)Thus, only ϕ † 00 commutes with Q M . Consider the operator with the level H = 2J + 2,The triangular condition for the standard Clebsch-Gordan coefficients implies that this coefficient vanishes unless the triangular conditions |J 1 − with integer J + J 1 + J 2 , and the requirement M = M 1 + M 2 are satisfied. Also, the D coefficients satisfy the relations

Physical cosmological constant in asymptotically background-free quantum gravity

“…This anomalous dimension vanishes at b → ∞, which is consistent with the classical limit of gravity. Here, note that the α t -independent terms inγ EH (2.14) andγ Λ (2.15) agree with the exact solutions of these anomalous dimensions derived using BRST conformal symmetry at α t = 0 [22,24,26,27].…”

Mesons from global anti-de Sitter space

“…are non vanishing only forl 1 +l 2 = 2n + 1 +l [23] for non-negative integer values of n. Therefore the final expression for the spectrum of ω is given by equation (5.19) again.…”

Vertex Operators in 4D Quantum Gravity Formulated as CFT

“…The coefficient b 1 has the correct sign of positive-definite such that the action is bounded from below. 2 In this way, the quantized Riegert-Wess-Zumino model [14,15,17,18] originally studied based on the analogy of 2D quantum gravity is now incorporated into the renormalizable quantum gravity with only one coupling constant t that indicates the asymptotic freedom [11,12,19,21]. As shown in [21], quantum diffemorphism algebra in four dimensions becomes complete in the combined system of the Weyl action and the Riegert-Wess-Zumino action.…”

“…Conformal symmetry is realized in the UV limit as the residual gauge symmetry of the diffeomorphism in the radiation + gauge [19,21]. 4 The gauge parameter ξ μ = ζ μ is the conformal Killing vector satisfying∇ μ ζ ν +∇ ν ζ μ − g μν∇λ ζ λ /2 = 0 and the gauge transformation is given under the decomposition (2.1) by…”

“…Recently, we proposed renormalizable 4D quantum theory of gravity formulated as a perturbed theory from such a CFT [11,12], whose dynamics is described by the combined four-derivative system of the Wess-Zumino action found by Riegert [13][14][15][16][17][18][19][20][21] and the Weyl action. Conformal symmetry is realized as diffeomorphism symmetry in the UV limit when we quantize these actions.…”