On spectrally flowed local vertex operators in AdS$_3$

Here is the list of chages, addressing the comments and suggestions from both referees:

1.1) Page 1, footnote added "Throughout this paper, we follow \cite{Maldacena:2001km} and refer to $x$-basis vertex operators as \textit{local}. However, we note that computing their spacetime OPEs in full generality remains an interesting open problem."

1.3) We agree with the referee that a definition of $x$-basis spectrally flowed operators can be formally given in terms of the OPEs with the currents. There are, however, two issues with this definition. First, it does not fix the normalization. Second, these OPEs contain a large number of terms that, unfortunately, remain unknown. In this paper we have, in a sense, bypassed both issues by providing a point-splitting definition involving auxiliary degenerate fields --- the "generalized" spectral flow operators --- which give an additional handle for the computation of correlation functions.
For clarity, we have changed the text from "there is no $x$-basis definition of spectrally flowed operators beyond the $\w=1$ case described in [1]" to "there is no $x$-basis definition of spectrally flowed operators generalizing the $\w=1$ case described in [1]".

1.4) We agree with the referee about this point, although we do consider that holomorphic covering maps are an important ingredient even for the computation of three-point functions. Nevertheless, we have replaced "relies heavily" by "can be understood in terms of" in the corresponding sentence.

1.5) We thank the referee for his comment about Eq.(34). Here and in Eq.(33) there was a typo: Eq.(33) comes with a "~" sign because the parafermionic decomposition is written as shown up to an (irrelevant) k-dependent factor, while Eq.(34) can be taken as a formal definition, which fixes the normalization, so that we have now written it with a "=" sign. As for Eq.(54), we have added a footnote at the end of page 12, where we clarify that the overall factor we ignore here is fixed by the discussion around Eq.(66).

1.6) We thank the referee for his thorough reading. However, there is no typo in Eq.(39). We have added a line of text and a reference below this equation.

1.8) We thank the referee for pointing this out. We have added a line below Eq.(45) explaining our notation.

1.9) We thank the referee for their comment. We have modified the corresponding text, which now reads as follows: "The identities (45) imply that the insertion of the operators (...) into the correlator on the LHS of (46) gives zero."

1.11) We thank the referee for their comment. In Eq.(65) there is no need for an extra "+" or "-" superscript since the values of the (unflowed) projections "m" uniquely determine the representations that the different states belong to. As for Eq.(73), we have followed the referee's suggestion and added the following comment in the text below: "While we have derived this result for highest/lowest-weight flowed representations, it is expected it to hold in general by analytic continuation in the spins $j$ and $j'$."

1.13) We thank the referee for their comment. We have added a comment above Eq.(74) clarifying the procedure. On the other hand, the subscript "bulk" refers to the discussion below Eq.(61) and singles out the contribution to the two-point function that is non-zero when j1=j2 (as opposed to the one relevant for j1=1-j2).

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2.1) We have changed "SL(2,R) series identifications" for "certain SL(2,R) discrete module isomorphisms" in the abstract, and also included a clarification at the end of the first paragraph of section 4.

2.2) We have clarified the sense in which V_{k/2 k/2} is a "pure exponential" in the text above Eq.(34) and removed the word "trivial" while discussing Eq.(37), whose second line is explicitly the extension of Eq.(34) to w>1.

2.3 and 2.5) We thank the referee for his comments. We have moved the footnote to the text, and included a small discussion of Eq.(36), including Ref.[41].

2.4) We thank the referee for his suggestion. We have reformulated the text and removed the term "frame" from the manuscript.

2.6) We thank the referee for his suggestion. We have added Eq.(55).

2.7) We have highlighted the importance of Eqs.(35) and (86) as suggested by the referee. Regarding their interpretation in terms of the fusion rules of degenerate fields, this can be understood by looking at Eqs.(34), (37) and (74). We have decided to leave a more extensive discussion on this topic for future work.

2.8) We have included a footnote in page 10 addressing the referee's comment on this subject.

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We have also updated Refs.[17] and [44], which are now published.