# 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?

• It looks like a typo. – Trimok Jun 15 '13 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 '13 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 '13 at 7:08

In (2.54), they finish the proof that $\omega R$ is integer, as you correctly noticed. $\omega$ is the dual (momentum) variable to the time-like coordinate $\tau$ so that the wave functions are proportional to $$\exp(-i\cdot\omega\cdot R\tau)$$ so if $\omega R$ is integer, all such wave functions and their combinations are periodic in $\tau$ with periodicity $2\pi$. One may check in eqn (2.23) that $\tau$ is defined as the dimensionless timelike coordinate along the hyperboloid (2.20), so $2\pi$ is the "normal" periodicity meaning that the functions on the universal cover may be reduced to the original hyperboloid (2.20) again.
• Dear @Dilaton, the dimensions of generic (=almost all) operators, like the very simple "Konishi operator", ${\rm tr}(\phi^i\phi^i)$, and its superpartners etc. (and all generic operators in the CFT), are functions of $g$, the Yang-Mills coupling, of the kind $\Delta = 2 + K\cdot g^2+\cdots$, so they're continuous functions of $g$. They have no reasons to be integers, so we say that they're fractional in general. I don't mean that they're rational (non-integer) numbers. I just mean that they're not integers. – Luboš Motl Jan 31 '14 at 8:38