This is in continuation of my previous question and is in reference to this paper.

I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of $dS^n$ and $AdS^n$ respectively. But I am not clear about the physical motivations of the quotients that they take.

They take the following quotients,

  • The sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$. Here they want to identify $\phi \sim \phi + \frac{2\pi}{n}$ for some $n \in \mathbb{N}$ where $\phi$ is the phase of $z_1$. And this quotient is given by $\mathbb{Z}_n\backslash SO(2n+2)/SO(2n+1)$

  • On $\mathbb{H}^{2n+1} = SO_0(2n,1)/SO(2n)$ they take the metric as, $cosh^2\rho dt^2 + d\rho^2 + sinh^2 \rho d\Omega_{2n-1}$ and take the quotient as $t \sim t + \beta$ and then the quotient space is given as $\mathbb{Z} \backslash SO_0(2n,1)/SO(2n)$ . Here $\frac{2\pi}{n}$ and $\beta$ are connected by the same analytic continuation as what takes one from the metric on $S^{2n+1}$ to the metric on $\mathbb{H}^{2n+1}$

Clearly in the paper they keep referring to \beta as the inverse temperature but I can't clearly see what is the thermal picture which will correspond to evaluating the partition function on the these quotients as above.

If there were a particle at inverse temperature $\beta$ on spatial $\mathbb{H}^{2n}$ then I would think that one would calculate the partition function on the manifold $\mathbb{H}^{2n}\times S^1$ such that the $S^1$ is of radius $\beta$.

Is this somehow related to the quotient of $\mathbb{H}^{2n+1}$ described above in the second bulled point? (...the same could have been asked about $S^{2n} \times S^1$ also..)

  • 1
    $\begingroup$ If you identify $t$ with $t+\beta$, you may introduce a new angular variable $\theta$, with $d\theta = \frac{2 \pi}{\beta}dt$. The metrics of the new manifold is : $ds^2 = \frac{\beta^2}{4 \pi^2}(cosh^2\rho) ~d\theta^2 + d\rho^2 + (sinh^2 \rho)~ d\Omega_{2n-1}$ So, the "radius" of "$S_1$" :($\frac{\beta }{2\pi}cosh \rho$) does not depends on $\beta$ only, it depends on $\rho$, so you cannot say it is a direct product between $H_{2n}$ and $S^1$, because $\rho$ is a parametrization of $H_{2n}$ too. $\endgroup$ – Trimok Aug 30 '13 at 9:13
  • $\begingroup$ @Trimok Then in what sense is the identification $ t \sim t+\beta$ setting the inverse temperature to $\beta$? I thought that a basic tenet of this paper is the claim that this identification (after Wick rotation) is the same as setting a particle on $AdS_{2n+1}$ to inverse temperature $\beta$. [...and is there a way to convince oneself that this said metric $cosh^2\rho dt^2 + d\rho^2 + sinh^2\rho d\Omega_{2n-1}$ is a metric on $\mathbb{H}^{2n+1}$ - I can't match this to any other standard form of the hyperbolic metric I know of...] $\endgroup$ – user6818 Aug 30 '13 at 22:15
  • $\begingroup$ There is a natural correspondance (with Wick rotation ) between a quantum field theory in $D$ dimensional space-time and a classical statistical mechanics in $D$ dimensional space, and between a quantum field theory in $D+1$ dimensional space-time, and a quantum statistical mechanics in $D$ dimensional space (see : Zee - QFT in an nutshell, Chapter V.2, p262). So, it would mean that a classical statistical mechanics in $H_{2n+1}$ is equivalent to a quantum field theory in $AdS_{2n+1}$. I only found a reference for the $AdS_{2n+1}$ metrics (see last penultimate comment of previous question) $\endgroup$ – Trimok Aug 31 '13 at 6:17

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