Monodromies of CFT correlators on the Lorentzian Cylinder

1. Referee 1 comment 1: We have modified the file accordingly. In particular, we have incorporated most of the footnotes (from the Introduction) to the main text. In particular we have merged two of the larger footnotes as a separate subsection 1.5.

2. Referee 1 comment 2: We have added a paragraph in the Conclusion section (see the second last paragraph), and have thanked the referee for making this point.

3. Referee 1 comment 3: We have added a paragraph at the end of appendix D: "We parenthetically note that the $D$ function is the result of a correlator computed using contact diagram in the bulk of AdS (via the AdS/CFT correspondence). It would be interesting to perform an analysis similar to this appendix, on the more complicated analytic structures that arise out of exchange or loop computations in the bulk."

4. Referee 1 comment 4: We agree with the Referee that the comments on this footnote 48 are potentially confusing. In response to the referees comments, we have truncated the footnote, omitting the statement the comment about factorization.

5. Referee 1 comment 5: We have added a short paragraph (third last paragraph in the conclusions section) making this point and thanking the referee for the suggestion.

6. Referee 2 comment 1: We have added the following lines at the end of the second and third paragraphs of section 1.3 (in the introduction) to highlight some of the physical implications of our results.

Line added at the end of the second para in section 1.3: {As correlation functions capture the response of a theory to sources, our results are needed (together, of course, with knowledge of the correlators as analytic function of $z$ and ${\bar z}$), to determine the physical response of a CFT on $S^1 \times $ time to arbitrary sources as a function of angle and time. }

Line added at the end of the third para in section 1.3:

1. Referee 2 comment 2: In response to this question, we have added the following new statement at the end of subsection 1.2: `It is also a disappointment because it tells us that the requirement ofpath independence' is automatic, and imposes no new general constraints on four-point functions of operators with integer spins.''

2. Referee 3 comment 1: We have added the following paragraph to the conclusion section as a tentative suggestion for future work:

{The braiding, and fusions matrices that characterize holomorphic conformal blocks of rational CFTs are well known to obey nontrivial pentagon and hexagon identities [see G. W. Moore and N. Seiberg, Lectures on RCFT]. While the discussion of subsection 3.3 below touched on (analogues of) these matrices, we never had occasion to make nontrivial use of the identities these objects obey. It would be interesting to explore the interplay (if any) of these identities with the study of Lorentzian correlators, along the lines of this paper. }

1. Referee 3 comment 2: We feel this could be an interesting topic for further work, and have added a line at the end of Appendix D, making this point: {We parenthetically note that the $Li_2(z)$ shares some of the properties of the function ($Log^2 z$), as a single monodromy operation around each of these functions produces a $Log \;z$.}