While computing the conformal partial waves, it seems to me that

$$\int d^dx |O\rangle\langle\tilde O| = \mathcal{N}^{-1}\sum_{n}|P^n O\rangle\langle P^n O|$$ where $\tilde O$ is the shadow dual of $O$, and the RHS is the projector. Can anyone provide a proof of the above statement ?

  • $\begingroup$ You should probably add some references to put your problem in context, e.g. arxiv.org/abs/1204.3894. For instance it is not clear what P is on the RHS. Is it related to the generator of translations? $\endgroup$ – M.Jo Apr 23 at 14:54

No, because this statement is wrong. You cannot make sense of the left hand side as an operator on a Hilbert space, since $|O(x)\rangle$ is not a normalizable state in radial quantization if $x$ is not inside unit sphere. Furthermore, $|\tilde O(x)\rangle$ cannot be defined as a state either, for the same reason.

The left hand side makes sense as an operation you can do to three-point functions. However, it yields Euclidean conformal partial waves, while the right hand-side (which needs to be written more carefully) yields conformal blocks. Refer to this paper.

You can write and prove something like this in Lorentzian signature, in several ways, but it is more subtle.

  • $\begingroup$ @PeterKravchuck What is the difference between a conformal block and a conformal partial wave ? $\endgroup$ – user44690 Apr 24 at 4:40
  • $\begingroup$ @PeterKravchuck What is the difference between a conformal block and a conformal partial wave ? Also, if you look at David Simons' TASI review on bootstrap, he uses the same notation for the projector (which is $|\mathcal{O}|$) and which he uses for the shadow representation. Even in the paper you cited, he says in section 2.6 LHS can be used to compute conformal blocks as well. $\endgroup$ – user44690 Apr 24 at 4:51
  • $\begingroup$ @user44690 There are different conventions in the literature what to call conformal partial waves and what to call conformal blocks. What I mean by a Euclidean conformal partial wave is a single-valued eigenfunction of conformal Casimir operators. Alternatively, it is the object which appears in harmonic analysis see e.g. 1809.05111. It is equal to a particular linear combination of conformal block and shadow block. See section 2.5 of 1204.3894 or 2.7.1 of 1809.05111. $\endgroup$ – Peter Kravchuk Apr 24 at 15:48
  • $\begingroup$ @PeterKravchuck In 2.5 of 1204.3894, they are anyway removing the shadow block using monodromy projection which leaves behind the conformal block, which only reinforces the fact that the above claim is more or less true and therefore, requires a proof. Right ? $\endgroup$ – user44690 Apr 24 at 16:21
  • $\begingroup$ @user44690 As I explained, the above claim does not make sense as a Hilbert space statement. Shadow representation is a valid way of computing things related to conformal blocks, from which conformal blocks can in principle be extracted. This is explained in e.g. 1204.3894 (parts of this story date back to 70s). I am not sure what else is there to prove. $\endgroup$ – Peter Kravchuk Apr 25 at 4:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.