Boundary conditions in AdS/CFT

Question

This question is in reference to this very famous paper of Witten.

• In general through the whole paper why is the author able to just focus on the scalar field propagating in the bulk and not need to take into account all the other fields and the complicated Lagrangian in the bulk (a Type IIB superstrings?)

• To construct the example in equation 4.1 (middle of page 6) why did the author choose half-BPS operators and is there a simple way to see that an example of ${\cal O}$ written down is a half-BPS operator? (..what are other such?..is there a classification?..)

• How generic is the argument in equation 4.8 (top of page 8) to get the RG flow equation? Or is this a special case which works here for some special reason?

  With a change in the mass/renormalization scale/cut-off one usually asks for the connected n-point functions or the effective potential to be invariant - but here the author seems to want to have the scalar field's boundary asymptotics to be invariant - I found this renormalization condition very new and mysterious.

• I guess the most exciting analysis in this paper is the argument in the first paragraph on the top of page 9. Can someone help understand that?

  • To start off how one know that the operators ${\cal O}_1$ and ${\cal O}_2'$ related to the boundary values of the two scalar fields are actually (super?)conformal primaries of the boundary (S?)CFT?

  • I did not understand how one sees that the deformation as stated in equation 4.12 (and the line before it) preserves quantum conformal invariance.

  • and the main point about the structure of equation 4.12 and the conformal invariance of the boundary being maintainable for $f \neq 0$..

Answer 1

First, the paper is relatively famous but among Witten's papers, at less than 200 citations, it is his average paper.

• Witten focuses on multi-trace operators. It's important that in these operators, there are several "Tr" symbols multiplied. For the sake of concreteness, he considered a CFT with a scalar field and this particular operator but the spin of more complicated operators isn't the real novelty here; the multi-traceness is. He hasn't claimed to have solved all theories with multi-trace operators; he is showing a new thing that the multi-trace operators may do.

• In the $N=4$ gauge theory, all operators of the type "traceless product of the scalar fields" are half-BPS. It's because they carry $U(1)^3\subseteq SO(6)$ R-symmetry charges that match the dimension. One may also interpret them in the BMN way. Among the polynomial operators, these are the only half-BPS operators as can be seen from the BPS bound.

• The equation (4.8) isn't an argument. It is a condition saying that when the "subscript zero" is added to the bare coupling $f$ and the field $\beta$, and one scale $\mu$ is replaced by another $\Lambda$, the $\beta$ field multiplied by the rest – the boundary condition for $\phi$ – agree. The reason why this condition has this modified logarithmic form is explained in equation (4.7) for the boundary condition. So of course, the condition only has the form (4.8) in this particular theory. Other theories would have different couplings and they would behave somewhat differently at infinity.

I don't understand why you think that the boundary conditions considerations are strange. He is using the standard AdS/CFT dictionary to extract an answer to a question about the bulk, namely the boundary conditions for fields, from a reasoning rooted in the boundary CFT, namely its renormalization. Note that changing of the renormalization scale is mapped to moving a "cutoff boundary of the AdS" closer to the boundary or further from the boundary.

Now,

• It's a general fact in AdS/CFT that the fields are dual to conformal primaries on the boundary. They don't have to be chiral primaries, however.

• The condition (4.12) preserves the scale invariance because it has no explicit dependence on dimensionless parameters. In such cases, it's mostly guaranteed in high enough dimension that the whole conformal invariance follows.

• On the top of page 9, he is just saying, concerning your question, that massive fields must sit at the minimum, i.e. value equal to zero, if the equations of motion are obeyed and if the interaction is turned off, i.e. $f=0$.

Sorry for being a bit telegraphic but you're asking too many things.