Abstract
We compute boundary correlation functions for scalar fields on tessellations of two- and three-dimensional hyperbolic geometries. We present evidence that the continuum relation between the scalar bulk mass and the scaling dimension associated with boundary-to-boundary correlation functions survives the truncation of approximating the continuum hyperbolic space with a lattice.
Highlights
The holographic principle posits that the physical content of a gravitational system, with spacetime dimension d þ 1, can be understood entirely in terms of a dual quantum field theory living at the d-dimensional boundary of that space [1]
The earliest checks establishing the dictionary for this duality were performed by studying free, massive scalar fields, propagating on pure anti–de Sitter space [2,3]
II we describe the class of tessellations we use in two dimensions and the construction of the discrete Laplacian operator needed to study the boundary correlation functions
Summary
The holographic principle posits that the physical content of a gravitational system, with spacetime dimension d þ 1, can be understood entirely in terms of a dual quantum field theory living at the d-dimensional boundary of that space [1]. The earliest checks establishing the dictionary for this duality were performed by studying free, massive scalar fields, propagating on pure anti–de Sitter space [2,3] These established that the boundary-boundary two-point correlation function of such fields has a power law dependence on the boundary separation, where the magnitude of the scaling exponent, Δ, is related to the bulk scalar mass, m0, via the relation. Reference [9] performed a thorough investigation of the scalar field bulk and boundary propagators in two-dimensional hyperbolic space using a triangulated manifold We extend this discussion to other two dimensional tessellations and to boundary-boundary correlators in three dimensions.
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