fields on AdS space

From Physics wiki

1 Wave equation
2 Scalar field
3 Breitenlohner-Freedman bound
4 Further reading

For an overview, see anti de Sitter space (mathematics). Consider the $n+1\,$ -dimensional Minkowski space $\mathbb{R}^{2,n-1}\,$ with metric

$ds^2 = -(dx^0)^2 - (dx^{n})^2 + (dx^1)^2 + ... + (dx^{n-1})^2\,$

and embed into it the one-sheeted quadric defined by

$-(x^0)^2 - (x^n)^2 + (x^1)^2 + ... (x^{n-1})^2 = -R^2\,$ .

This submanifold has codimension $1\,$ and is known as anti de Sitter space or $\mathrm{AdS}_n\,$ . $\mathrm{AdS}_n\,$ can be described by the coordinates where $\tau \in [0,2\pi]\,$ , $\rho \in [0,\infty)\,$ , and $\Omega^i\,$ , which parameterize the unit n-2 sphere $\mathbb{S}^{n-2}\,$ , along with the metric

$ds^2 = R^2(-\cosh^2\rho\,\,d\tau^2 + d\rho^2 +\sinh^2\rho\,\,d\Omega_{n-2}^2)\,$ .

Since the timelike coordinate $\tau\,$ is periodic, we run into problems with causality, namely closed timelike curves. Therefore in practice one works with the universal cover of $\mathrm{AdS}_n\,$ , denoted $\mathrm{CAdS}_n\,$ by letting $\tau \in (-\infty,\infty)\,$ with no identification between points, though in the context of AdS/CFT $\mathrm{AdS}_n\,$ is almost always used and taken to mean $\mathrm{CAdS}_n\,$ .

Wick rotation of the timelike coordinate $\tau\,$ to $\tau_E = -i \tau\,$ is accomplished by letting $x^n \to x^n_E = -i X^n\,$ , which corresponds to the embedding of