Off-Shell Strings I: S-matrix and Action

See report.

See report.

This paper takes up the long-dormant 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.

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 throw-away comment, toward what many of us believe is an interesting and well-grounded 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 S-matrix emerges in the limit that the effective action becomes non-local. Usual QFTs have a local action and an S-matrix. 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 non-local 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).

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