Riemann surfaces for KPZ with periodic boundaries

Referee 1:
I thank the referee for the positive feedback on the paper.

Referee 2:
I thank the referee for the detailed review of the paper.

General comments:
1 - Analytic continuation of polylogarithms is indeed a well known subject. The parts of section 4 which are (presumably) new are the explicit analytic continuation of the functions J and K in terms of which KPZ fluctuations are expressed. This is now made more precise in the first paragraph of section 4.

2 - The fact that the equivalence between the old formulas and the new formulas depends heavily on analytic continuation results from section 4 was indeed mostly emphasized for flat initial condition in the previous version of the paper. I added some text around equation (10) and before equation (13) to make this explicit also for sharp wedge initial condition.

3 - While the relation of the Riemann surfaces with large deviations (section 2.6.1) and particle-hole excitations (section 2.6.2) is rather clear (the full time dynamics is in some sense given by large deviations ``made analytic'' by summing over all the sheets of the Riemann surfaces, which are in one to one correspondence with particle-hole excitations), I agree that the relation with KdV solitons is less clear (but also probably much deeper). In the classical theory of the KdV equation (which excludes the kind of initial conditions needed for KPZ, e.g. a Heaviside step function for KPZ with flat initial condition), any solution of KdV periodic in space is given in terms of a theta function built from a hyperelliptic compact Riemann surface, and soliton solutions correspond to specific degenerations of these Riemann surfaces. The soliton solution (23) can then be interpreted in terms of the theta function for a specific class of degenerate hyperelliptic Riemann surfaces with $\nu$ parametrizing the position of some branch points. This suggests that the correct interpretation of the parameter $\nu$ should not be a coordinate on a Riemann surface but rather a parameter in the space of Riemann surfaces (the moduli space). This might be crucial in order to understand in a natural way the factors $\Xi$ in (6) and (16). Additionally, integrating tau functions for KdV / KP with respect to moduli parameters seems to occur as well in several other contexts. I added a sentence in the first paragraph of section 2.6.3 to emphasize the moduli interpretation.

4 - For the example provided by the referee, appendix B is actually referred to in the sentence just before ``One finds'' (now: ``There, one finds''). However, references to appendices were slightly broken in the previous version (I had changed from a style file that automatically added ``appendix'' when citing an appendix to the scipost one that does not). This has been corrected in the new version. In principle, formulas proved in the appendices have been indicated consistently throughout the paper.

Other comments:
0 - see above.
1 - I replaced ``Ito discretization'' by ``Ito prescription in the time variable'' in the new version. As usual for Langevin equations with multiplicative noise, the Stratonovich prescription (where an infinitesimal increment of Z depends both on Z before and after the increment) and the Ito prescription (where an infinitesimal increment of Z depends only on Z before the increment) lead to distinct solutions of the stochastic partial differential equation. This is the Ito prescription which leads to the proper physical object described by KPZ universality.
2 - References to sections 3.8.2, 3.8.2 have been added to the first paragraph of page 5.
3 - Done, references have been added to the first paragraph of section 2.6.3.
4 - Footnote 4 has been rewritten. Since the term in the exponential is only linear in $x$, its contribution to $u=2\partial_{x}^{2}\log\tau$ does vanish.
5 - KP is the ``natural'' generalization of KdV from 1 dimension of space to 2 dimensions of space from the point of view of classical integrability: modern books about classical non-linear integral equations such as [55-57] treat both together. Initially, since the dependency in x, y, t of the exponential factors match with those of KP solitons, I was quite convinced that tau functions for KP would appear for KPZ with sharp wedge initial condition. The fact that the prefactors do not match (the Cauchy matrix is squared in M_{sw} below (25)) was a bit disappointing. In view of the recent paper [63] by Quastel and Remenik, I am again enclined to believe that there should also be a proper connection to KP for KPZ with periodic boundaries and sharp wedge initial condition (at least; [63] seems to indicate that it should be much more general). I have not managed to find it so far. Maybe one should instead try to compare (25) to a N-soliton solution to a matrix KP equation, like the one appearing in [63], or maybe the full n-point function (16) has to be considered. I added a sentence about this in the last paragraph of section 2.6.3.
6 - Corrected.
7 - Corrected.
8 - $\simeq$ is the standard notation for the ratio going to 1 in the limit. I added ``when $z\to z_{*}$'' in order to specify the limit considered.
9 - Corrected.
10 - Corrected.
11 - Corrected.
12 - Corrected.
13 - Corrected.
14 - Corrected.
15 - The path $\theta_{j}^{2}\!\cdot\!\mathcal{G}_{P}$ corresponds in figure 3 left to the concatenation of the blue and green dashed paths, and in figure 3 right to the green/blue dashed circle. While on the right, it is clear that this path is homotopic to an empty loop, this is less obvious with the representation on the left.
16 - Yes, corrected.

Referee 3:
I thank the referee for the positive feedback on the paper. Both misprints have been corrected.

Additional changes:
Added $t_0=u_0=0$ after equation (13) and $x_0=0$ after equation (14).