Navigator Function for the Conformal Bootstrap

With this resubmission we have addressed most of the concerns of the referees. A full list is given below. One significant change is the addition of a new short subsection 3.1 where we provide evidence that the navigator function is not everywhere C^2.

In addition we would like to provide the following replies to some of the remarks in the reports.

• Report 1 asked for more details of the Sigma navigator functions. However in this work we did not fully explore Sigma navigators. We added footnote 34 on page 42 to stress that at present both navigators are on equal footing. In our computations with a Sigma navigator some really simple minded choices worked well, like summing about 100 conformal blocks with small scaling dimensions spacing and equal coefficients. There was nothing particularly smart about those choices, so we preferred not to report them.

• Report 1 also asked whether one would find a flat plane in the case of a conformal manifold. We do not expect this to be true, and would also like to point out that this does not logically follow from our claim that the minimum of the navigator corresponds to a CFT.

• In report 1 a question was also raised about using the navigator for OPE space islands. In our view it makes no conceptual difference whether the parameter space in which one navigates consists of scaling dimensions, OPE coefficients, or both (as we do in the paper). Therefore we did not supply any further comments. Of course the authors would be happy to offer practical guidance in specific cases.

• We expect the request in report 2 to "add a comment or plot" about the navigator inside the Ising island to be addressed by the new subsection 3.1.

In response to report 1:
- Added footnote 34 on page 42;
- Changed "a good predictor for the location of the true CFT" -> "close to the true CFT" on page 25.

In response to report 2:
- In abstract: replace "everywhere well-defined" -> "well-defined in large regions of parameter space";
- Added footnote 33 on page 41 to compare the number of SDPB iteration between the navigator method and Delaunay triangulation method;
- Added and the value of the navigator at the minima in equations (5.8) on p31 and (5.9) on p35.

In reponse to report 3:
- We increased the size of black and red dots of figure 13 (previously figure 12) to make them more visible.

Other changes:

- Added "We used the same conformal block normalization as [1]." in appendix D on page 55;
- Corrected caption of figure 13 (previously figure 12): changed "red/black" -> "black/red", changed "allowed/disallowed" to "allowed/excluded";
- On page 23, in the acknowledgments on page 44, around and in footnote 41 on page 51: added information that the announced program is now available under the name "approx_objective" as a part of the SDPB package. Also changed "we than" -> "we thank" in footnote 41;
- Added subsection 3.1 and figure 5 to provide evidence for some non-C^2-differentiability of the navigator. Relatedly:
-- on p14: "The navigator is observed to be smooth" -> "On this scale the navigator is observed to be smooth (see however below)";
-- on p15: "smooth" -> "smooth at this scale";
-- on p33: added footnote 30 on plateaus;
-- on p42: "smooth" -> "(C^1) smooth".