# How do Aharony et. al conclude that all scalar fields in the supergravity multiplet are periodic?

This question is for anyone who has read/gone through the paper above or knows anything about AdS/CFT. The paper can be found here.

On page 46, eq. (2.33), the author finds solutions to the scalar field equation $(\Delta- m^2 ) \phi = 0$ in $AdS_{p+2}$ background as $\phi = e^{i \omega \tau} G(\theta) Y_l(\Omega_p)$, with the functions $G$ and $Y$ defined just below this equation.

On page 51, in equation (2.54) the author shows that when $p=3$, $\omega$ is quantized in multiples of $\frac{1}{R}$, i.e. $\omega R \in {\mathbb Z}$.

He then goes on to state the following

This means that all the scalar fields in the supergravity multiplet are periodic in $\tau$ with the period $2\pi$, ...

I do not see how he comes to that conclusion. Am I missing something?

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It looks like a typo. –  Trimok Jun 15 at 7:04
Where do you think the typo is? I think it is pretty important that the period be $2\pi$ since he says that the field theory of the supergravity multiplet can be defined on the original AdS space (that has periodicity $2\pi$ in $\tau$). On the other hand, he says describing the field theory of other fields requires the maximal extension to $-\infty < \tau < \infty$. Can you comment on where exactly you think the typo is? –  Prahar Jun 20 at 3:40
My (very basic) thinking, was that, with $\omega = \frac {n}{R}$, a periodicity of $\tau$ for $\Phi$, means that $\omega \tau = 2\pi$, so the periodicity should be $2\pi \frac{R}{n}$. –  Trimok Jun 20 at 7:08