Abstract
We study the fuzzy hyperboloids AdS^2 and dS^2 as brane solutions in matrix models. The unitary representations of SO(2,1) required for quantum field theory are identified, and explicit formulae for their realization in terms of fuzzy wavefunctions are given. In a second part, we study the (A)dS^2 brane geometry and its dynamics, as governed by a suitable matrix model. In particular, we show that trace of the energy-momentum tensor of matter induces transversal perturbations of the brane and of the Ricci scalar. This leads to a linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism of emergent gravity in matrix models.
Highlights
Geometry and isometry groupThere are three types of two-dimensional non-compact spaces with constant curvature, given by the Anti-de Sitter space AdS2, de Sitter space dS2 and the hyperbolic or Lobachevsky plane H2
For the principal continuous representations our results are new
We studied the fuzzy version of 2-dimensional de Sitter and Anti-de Sitter space, and some of the associated physics
Summary
There are three types of two-dimensional non-compact spaces with constant curvature, given by the Anti-de Sitter space AdS2, de Sitter space dS2 and the hyperbolic or Lobachevsky plane H2. The circles x3 = const are space-like, and there are no closed time-like curves Both AdS2 and dS2 admit the group SO(2, 1) or its cover SU(1, 1) as isometries, generated by vector fields Ka, a = 1, 2, 3. Unitary irreducible representations of SO(2, 1) are spanned by a basis |j, m of weight states, where j is related to the eigenvalue of the Casimir C, and m is the eigenvalue of K3 and the action of K± on |j, m produces a state with weight m ± 1: K3K±|j, m = (m ± 1)K±|j, m ∼ |j, m ± 1. A chain of states obtained by the successive action of K− operator terminates if there exist state such that Denoting this lowest weight by j = m0, it follows that 0 = K+K−|j, j = −C + K3(K3 − 1) |j, j ⇒ C = j(j − 1).
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