SL(2, $\mathbb C$) quartic vertex for closed string field theory

1. A construction of of the quartic vertex of closed string field theory.

2. Use of SL(2,C) coordinates.

3. Explicit analytic parametrization of the different regions of the moduli space.

1. Too technical and cryptic, except for specialized readers.

2. Notation used is cumbersome.

3. Some statements need improvement.

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In the paper arXiv:2311.07367 submitted to SciPost, the authors, H.\ Erbin and S.\ Majumder, deals with the construction of the
interaction vertices in bosonic closed string field theory. Specifically, they deal with the construction of the quartic vertex at tree
level. Their approach to use local coordinates that transform under SL$(2,\mathbf{C})$. The most studied approach to these
vertices uses the coordinates related to the minimal area metric associated with quadratic differentials. Perhaps due to an
oversight, the authors did not quote one of the pioneering papers in this area, namely the paper by M.\ Saadi \&\ B.\ Zwiebach,
{\em Closed string field theory from polyhedra}, Annals of Physics 192, 213--227 (1989).

The authors succeeded to an extent in that they determine a one-para\-meter family of vertices, for which they obtain analytic
expressions for the boundaries and volumes of the regions of the 4-point amplitude obtained by the cubic vertex and propagators
using Feynman rules. Consequently they determined the region in the moduli space of the 4-punctured sphere that corresponds
to the quartic vertex, parametrized its boundary and computed its volume. They have claimed this to be the main advantage of
using these coordinates over other choices, in particular, the minimal area metric. An explicit expression of the local coordinates
at each puncture, however, remains undetermined.

The same approach has been advocated for other vertices, even beyond the tree level. The results presented are no doubt of
interest and may help with computations in string field theory. In fact, they authors mentioned the problem of mass renormalization
at one loop, although the complexity involved in that is expected to be much higher.

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\noindent{\bf Therefore, I think that the results deserve to be published. However, I cannot recommend publication in its present
form.}

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\begin{enumerate}
\item
The present version of the paper may be understandable only by experts in string field theory, which, in spite of its recent resurgence,
remains somewhat of a niche subject. It will be in the authors' interest to make their results accessible to a broader audience of
string theorists. Perhaps even beyond, since many quantum field theorists are working to understand the space of (effective) field
theories, where some of the recent developments seems rather suggestive of structures seen in SFT.
\item
In particular, my suggestion is to relegate detailed technical calculations to appendices and explain the background and approach
in more details. Sections 3 and 4, which form the main body of the paper, consist of a succession of formulas with very little to
motivate a non-SFT reader going. The notation, loaded with subscripts and superscripts are cumbersome and difficult to read and
keep track of. Admittedly some of it is unavoidable, however, some improvement may not be difficult to achieve.
\item
A few other points:
\begin{quote}
Eq.(3.58) shows that there is a value of $\beta$ for which the $\mathrm{Vol}(\mathcal{V}_{0,4})$ vanishes. This is however, less
than the lower bound allowed for $\beta$. An elaboration of this for those not initiated in SFT may be useful. The authors may
want to identify more such points and provide explanation.

In Eq.(2.1) (and similarly elsewhere) it is presumably meant that the global coordinate $z$ is restricted to the local patch, for this
equation to make sense.

The measure used to compute the volume of the regions of the moduli space may be mentioned.

Finally, the presentation may benefit from a stricter editing by the authors. For example, in the opening sentence of Subsec.~2.1,
``{\em At tree level, Riemann surfaces are $n$-punctured sphere}", the phrase {\em Riemann surfaces} should be replaced {\em
string worldsheets} (and {\em sphere} by {\em spheres}). Likewise, it is not clear what the {\em This} refers to in the first sentence
of Subsec.~3.1.
\end{quote}
\end{enumerate}

Ask for major revision

1. Very well written. Self-contained, well-referenced, and clear presentation of the materials.

2. The analysis is easy to follow. The main result and its significance are appropriately highlighted.

3. The main result, albeit a bit technical, has important consequences for research in string theory.

None.

In my opinion, the current manuscript satisfies all the criteria for being published in this reputed journal. But there are a few minor improvements that can be potentially achieved. For a detailed report, please see the attached file.

See the attached file.

Ask for minor revision