Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 1 / b / i

Table of contents
- Solution i

Solution

For a scalar operator of dimension

Δ = 2

in a boundary theory of dimension

d = 3

, the AdS scalar mass--dimension relation gives

m^{2} = Δ (Δ - d) = - 2

in unit-radius

AdS_{4}

. In Poincare coordinates,

d s^{2} = \frac{1}{z ^{2}} (η_{μν} d x^{μ} d x^{ν} + d z^{2}),

so the free Klein-Gordon equation is

[z^{2} (\partial_{z}^{2} + □_{2 + 1}) - 2 z \partial_{z} + 2] ϕ = 0 .

This mass is exactly that of a conformally coupled scalar field in four-dimensional anti-de Sitter spacetime. The Weyl transformation

ϕ (x, z) = z ψ (x, z)

therefore maps the equation to the flat half-space wave equation

(\partial_{z}^{2} + □_{2 + 1}) ψ = 0, z > 0 .

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