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OffShell Strings I: Smatrix and Action
by Amr Ahmadain, Aron C. Wall
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Submission summary
Authors (as registered SciPost users):  Amr Ahmadain 
Submission information  

Preprint Link:  scipost_202308_00047v1 (pdf) 
Date submitted:  20230831 12:27 
Submitted by:  Ahmadain, Amr 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We explain why Tseytlin's offshell formulation of string theory is welldefined. Although quantizing strings on an offshell background requires an arbitrary choice of Weyl frame, this choice is not physically significant since it can be absorbed into a field redefinition of the target space fields. The offshell formalism is particularly subtle at treelevel, due to the treatment of the noncompact conformal Killing group SL(2, $\mathbb{C})$ of the sphere. We prove that Tseytlin's sphere prescriptions recover the standard treelevel Lorentzian Smatrix, and show how to extract the stringy $i \varepsilon$ prescription from the UV cutoff on the worldsheet. We also demonstrate that the correct treelevel equations of motion are obtained to all orders in perturbation theory in $g_s$ and $\alpha^{\prime}$, and illuminate the close connection between the string action and the ctheorem.
Current status:
Reports on this Submission
Report 2 by Matthew Headrick on 20231214 (Invited Report)
 Cite as: Matthew Headrick, Report on arXiv:scipost_202308_00047v1, delivered 20231214, doi: 10.21468/SciPost.Report.8272
Strengths
See report.
Weaknesses
See report.
Report
This paper takes up the longdormant mantle of Tseytlin’s nonlinear sigma model approach to string theory. Many technical advances are made, and the whole theory is put on a somewhat more secure foundation. In addition, the theory is explained in a more transparent way than in Tseytlin’s many papers on the subject, which unfortunately suffered from leaps in logic, hidden assumptions, etc. Applications of the theory, in particular to black hole entropy, come in a second paper, which I am not reviewing here.
This paper is long and highly technical, and addresses many subtle and confusing issues. While I think I understand the gist, and did not find any suspicious or outright false claims, I cannot claim to have checked each derivation carefully. Nonetheless, based on what I do understand, I believe the paper easily clears the bar for publication in SciPost. The results are of great importance for our understanding of string theory, and, with some exceptions detailed below, the presentation is generally clear.
Requested changes
Before publication, I would like the authors to address the presentational issues listed below. Some of these are minor or cosmetic, while others are more substantive. In the cases where I suggest a fix, based on my understanding, the authors don’t have to follow my suggestion; but in all cases they need to address the issue. From p. 18 onward, where my list ends, the authors may want to follow the spirit of the suggestions and try to identify and clean up any further presentational infelicities.
p. 2 R column, a few lines below (T1), “super(string) theory”, why is “string” in parentheses?
p. 2 R column, near the bottom, first bullet: What does “the limit where $\log\epsilon^{1}$ is small” mean? $\epsilon$ is dimensionful, so I don’t think you mean the limit $\epsilon\to1$. I think you just mean “at finite $\epsilon$”, i.e. not taking the limit of the next bullet.
p. 3 R column, near bottom: “The sigma model approach is most successful only when the characteristic length of the background spacetime is much less than the string scale”. Don’t you mean “greater”?
p. 3 4 R column, just below (3): “Unfortunately, this method does not give the correct entropy unless perhaps (following Dabholkar [82]) we allow tachyons to condense on the orbifold.” Perhaps the authors did not intend it this way, but to my reading this is a weirdly derogatory and dismissive throwaway comment, toward what many of us believe is an interesting and wellgrounded line of research. Why “perhaps”? Why would we not allow tachyons to condense? Obviously this is not the place for a full discussion of these issues, which presumably comes in paper II. I would suggest just deleting this sentence (and maybe citing Dabholkar in the previous one).
p. 5 L column, top of page: “For products over $n$…” This really confused me. I think you don’t mean products “over $n$”, you mean products over the vertex operators at fixed $n$. The notation strongly suggests a product over $n$, making equations like (22), (30), etc needlessly hard to understand. I realize you don’t want to include yet another index, but some change of notation would be helpful. Maybe put the $n$ over (rather than under) the $\Pi$, since it is a product “up to $n$”?
p. 7 L column, bottom of page: “i.e. is proportional to some $E_A$” Shouldn’t that be $E_a$?
p. 9 R column: Eq (31) is impossible to understand. What does the colon mean? What is on the LHS of the equation? Please rewrite using standard notation.
p. 12 R column: I didn’t understand in what sense the Smatrix emerges in the limit that the effective action becomes nonlocal. Usual QFTs have a local action and an Smatrix. Related to this, my understanding was that the worldsheet cutoff $\epsilon$ is related to the size of the string: in the limit the cutoff is small, the string gets large and the effective action becomes nonlocal in the target space. However, here it seems to be related instead to the distance over which the string can propagate. What is the relation between these things?
p. 13 R column: Eq (40) is missing a minus sign in the exponent.
p. 14 L column: The measure factor in parentheses is confusing, with the $n$ subscript. Maybe just write $d^{2n}z$?
p. 15 caption to fig 5(i): “the hyperbolic volume of the regulated gauge orbit is noncompact” I think you mean “is infinite”.
p. 17 L column: The notation $ij\ldots z$ is confusing, given the other role of $z$ here. I would recommend instead $i_1\ldots i_n$ (particularly since the number $n$ of them is fixed).
p. 17 R column: On the LHS of (60), I believe that $I_0^{eff}$ should be $I_{(\chi)}$.
p. 17: Eq (61) follows directly from (57) and (58). I didn’t understand what was supposed to be gained by the detour through (59) and (60).
Anonymous Report 1 on 2023922 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202308_00047v1, delivered 20230922, doi: 10.21468/SciPost.Report.7854
Strengths
See acompanying file
Weaknesses
See accompanying file
Report
See accompanying file
Requested changes
See accompanying file
Author: Amr Ahmadain on 20240517 [id 4490]
(in reply to Report 1 on 20230922)
Dear Referee,
We have addressed all of your comments, questions and remarks to the best of our ability. We have made significant changes to section III.C and section VII especially VII.A, VII.B and VII.C. This is in addition to several other minor changes to the whole text.
The attached PDF file contains a detailed 6page exposition of the changes made to the text and responses to all of your questions and remarks. Our replies are the bluecolored text in the PDF file.
If you still have any further questions or remarks, we'll be happy to address them.
The Authors
Attachment:
Off_shell_Strings_I_SciPost_Response_to_Anonymous_Referee.pdf
Anonymous on 20240522 [id 4505]
(in reply to Amr Ahmadain on 20240517 [id 4490])I am satisfied with the corrections that have been made and do not think a further round of refereeing would be useful.
Author: Amr Ahmadain on 20240520 [id 4496]
(in reply to Report 2 by Matthew Headrick on 20231214)Dear Matthew,
We have addressed all of your comments, questions and remarks to the best of our ability.
The attached PDF file contains a detailed 3page exposition of the changes made to the text and responses to all of your questions and remarks. Our replies are the bluecolored text in the PDF file.
If you still have any further questions or remarks, we'll be happy to address them.
The Authors
Attachment:
Off_shell_Strings_I_SciPost_Response_to_Matt_Headrick.pdf