# 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 '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

## 1 Answer

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.

Let me emphasize that this conclusion "the hyperboloid is enough" is only valid in SUGRA, i.e. a small subset of the operators in the CFT. General stringy states have fractional, continuous frequencies.

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Nice answer :-). Can you extend the comment about the fractional frequencies a bit, how and why they can appear in string theory? – Dilaton Jan 31 '14 at 8:27
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
It shouldn't be shocking that the generic dimensions are not integers. The integrality is equivalent, as explained in this very question, to periodicity in time. You surely don't expect general evolution of a complicated enough physical system to be periodic in time (with a small period), do you? The real question should really be the opposite one - why some special theories or subsets of operators do seem to evolve periodically in time (i.e. respect the integrality of all energies/dimensions). In this case, for SUGRA operators, it's proved in 2.54 and the previous equations. – Luboš Motl Jan 31 '14 at 8:40
I must add that this is a particular example of a recent discussion, AdS/CFT is inevitably stringy, see motls.blogspot.com/2014/01/… - The point here is that most operators in the CFT, starting from the very simple Konishi operator above, simply do not correspond to SUGRA (supergravity) fields, they are some "string stuff". So whatever conclusion one derives just from SUGRA (in this case, the integrality of dimensions or periodicity in time, the same thing) is likely to be wrong in the whole CFT, and in this case, it is wrong. – Luboš Motl Jan 31 '14 at 8:45