Sparsity in the numerical six-point bootstrap

There are several answers to the question of why we had to resort to discretization in the first paper. The weakest reason would be that polynomial SDP is not directly implemented in SDPA as it is in SDPB. Of course this is no real obstacle, since we can write a small piece of code that chooses a basis of polynomials and translates polynomial SDPs into a problem that can directly be fed into SDPA. In fact, we did exactly that in the old paper in order to impose asymptotic conditions that are sufficiently strong to ensure convergence of the truncation scheme. Another stronger reason is that, without the matrix decompositions presented in the new paper, it is not clear to me how one would conjugate the matrices appearing in the SDP in such a way that you directly get polynomial entries with a low degree bounded by $\Lambda$ as in eq.~(3.27). I would therefore expect that in the old setup we would have been forced to consider polynomials of much higher degree, increasing the computational cost. This ties together with what I consider to be the strongest reason for why we had to use discretization: The matrix size was a limiting factor in the old implementation. To ensure convergence of the truncation scheme we had to work with matrices that had several hundreds of rows and columns. If we had decided to treat the Delta dependence not by discretization but by polynomial SDP this would have further increased the matrix size by roughly an order of magnitude. Anticipating this issue, we worked with Delta discretization. I hope that the reasons that I have just provided sufficiently answer your question. Personally, I do not think that there is anything interesting to learn for the future from what I just wrote. This is why I decided not to further expand on this issue in the draft. If after reading my answer you have a different opinion, I am of course open to reconsidering this choice.

I agree. To address this point, I replaced the vague formulation "the relevant matrices" before eq.(3.26) by the more precise formulation "the matrices $M^{(a,b,c)}(\Delta)$ defining the sum rules (2.12)" and then replaced $M$ by $M^\alpha$ in the equation. Also, this made me realise that there was a inconsistency in the usage of square and curved brackets between section 2 and 3, which I now also fixed. Thank you for your comment.

Actually, the explanation for the bound on the degree in n given in (3.27) is the same as the explanation for the bound on the degree in Delta. The (2-j-k) part just arises from the (2\Delta)_n n! in the projectors onto conformal multiplets. The extra \Lambda arises because each derivative just generates at most one power of Delta or n (see eq. (2.9)). Apart from this general argument, one can of course also check this polynomial dependence by simply looking at explicit expressions for the functionals that are computed in the Generate_Functionals.nb notebook.

You are completely right to compare the discrete parameter n with the role that spin plays in the higher dimensional four-point bootstrap. In response to comment 1 of referee 1, I added some remarks on this in the paragraph "C matrices", which hopefully also address the point that you are making.

That $X^{(n_0,I,J,d_\Delta)}$ matrices are included only for $n_0$ only up to $\Lambda-1$ is indeed a choice that I made and in principle one could include more of these matrices. I commented on this to some degree at the end of page 14. However, I agree that this was not sufficient. I now added a footnote with further comments on page 15.

Thank you, this is a good observation. I corrected the notational inconsistency. As a side effect, the notation in (3.45) is now also less cluttered.

I understand your question as: "Why is it desirable to use X matrices exploiting the polynomial dependence on n rather than truncating in n and then take X matrices of the type constructed in step 2". My answer to that would be two fold. First, for large enough nTrunc the procedure I chose leads to a smaller number of X matrices (which is desirable for numerical efficiency). Secondly, which I consider much more relevant, there is an important qualitative difference between the kind of SDP suggested by the question and the SDP used in my paper. As nTrunc is increased, both the number of parameters and the number of constraints grow in the former. In the latter, the parameters stay the same and only the number of constraints grow. Therefore it is easier to formulate a convergent algorithm (and also to check its convergence) in the latter case. This is the main reason why I chose to exploit the polynomial n dependence in the way described in the paper.

In (3.37) the parameter $n_0$ runs over its full range. In (3.48) its range got restricted and I only pick a small subset of $X^{n_0,I,J,d_\Delta}$ matrices. The missing $X^{n_0,I,J,d_\Delta}(\Delta)$ are replaced by the $X^{(I,J,d_\Delta,d_n)}(\Delta,n)$ matrices. For further comments on this, see my answer to your remark 5.

Good point. It now reads "On the other hand, \texttt{SDPB} -- the highly optimised solver designed specifically to address the needs of the bootstrap community -- is [...]".

& 11. Thank you for noticing this. The spelling mistakes and grammatical errors that you pointed out are fixed now.

Thank you for pointing out that this was not addressed in the previous draft. I comment on the issue in a revised version of the paragraph "C matrices" of Section 3.2. This should also make the motivation for picking the specific conjugation matrices described in that paragraph clearer. I certainly regard this as an improvement of the text.

Of course you could take any other basis of X matrices, i.e. work with appropriate linear combinations of the X matrices that I describe without changing anything. On the other hand, if you decide to not work with a full basis but rather restrict to a subset, this will generically influence your results. By first computing bounds using the full set of X matrices, you can develop some heuristics on which matrices are important and which can be omitted. I comment on this below eq. (3.48). Finally, there is the question of how many of the $X^{(n_0,I,J,d_\Delta)}$ matrices to include, which is brought up at the end of page 14. I added further comments on that in a new footnote on page 15, as well as in my reply to the Report of Referee 2.

I rather prefer the current layout of Section 3.2. The referee is correct in his assessment that the X matrices constructed in step 1, 2 and 3 are not directly used. However, I consider them as necessary intermediate results for the construction of the final version presented in step 4. Of course one could instead just directly present the result of step 4 without any motivation and move step 1,2,3 into an appendix. But my impression is that the flow of the argument is more coherent in the way it is laid out right now. I will therefore not adapt the draft in this point. If the referee however has a strong opinion on this, I may of course reconsider my choice of presentation.