Integrable Deformations from Twistor Space

Please see attached report.

This article studies the higher-dimensional origins of a familly of 2-dimensional integrable field theories (IFT$_2$), belonging to the class of $\lambda$-deformed sigma-models. In particular, it shows that these IFT$_2$ are part of a diamond of theories related by localisations and symmetry reductions, including a 6d holomorphic Chern-Simons theory on twistor space (hCS$_6$), a 4d semi-holomorphic Chern-Simons theory (CS$_4$) and a 4d integrable theory (IFT$_4$) whose equation of motion takes the form of an anti-self-dual Yang-Mills equation. This fits into a general scheme aiming at understanding IFT$_2$ from higher-dimensional gauge theories and in the context of the Penrose-Ward conjecture, which received a lot of attention, including in the recent years. In particular, the inclusion of $\lambda$-deformed sigma-models in this framework was a quite natural open question following various recent works (for instance the references [8,9,11,15] in the article).

The paper presents a quite thorough and systematic analysis of this diamond of theories and its underlying geometric structures. Doing so, the authors reveal new aspects which have been quite less explored so far in the literature, including the possibility of boundary conditions in CS$_4$ which are not based on isotropic subalgebras of the defect algebra. This opens up new avenues and raises interesting questions, for instance towards a complete understanding of the Lax connections and conserved charges of these models. In view of these points, it is my opinion that the results of this submission fit the criteria of originality and significance for publication in SciPost Physics.

Overall, I think the paper is quite clear and well-written. I have a few more specific questions and comments that I would like the authors to address. I explain the two main ones below. The remaining ones are listed in the section "Requested changes" and are essentially typos or small suggestions (that the authors should feel free to implement or not). I am happy to recommend this submission for publication once these minor revisions are addressed.

1) My first comment concerns the passage from CS$_4$ to IFT$_2$ in section 6. Indeed, it seems to me that there is a small inconsistency in the treatment of $\mathbf{h}$ (in this report, $\mathbf{h}$ denotes the field valued in the defect group $\mathbb{D}$). Relating equations (4.19), (6.1) and (6.2), one finds $\mathbf{h} = (\hat{h}|_\alpha, \hat{h}|_\tilde{\alpha})$, which according to (3.3) or (B.8) should then be $\mathbf{h} = (h, \tilde{h})$, rather than $\mathbf{h} = (h^{-1}, \tilde{h}^{-1})$ used in section 6. However, this should not affect the final results of the section, as I think $\mathbf{h}$ should also be replaced by $\mathbf{h}^{-1}$ in the rest of the E-model formulation. For instance, from (6.2), one gets $\mathbf{h}\partial_w \mathbf{h}^{-1} = \mathbb{L}_w - \text{Ad}_{\mathbf{h}} \mathbb{A}_w$, with $\mathbb{A}_w$ in $\mathfrak{l}_t$. One then wants to construct the projector $W^+_{\mathbf{h}}$ such that $\mathbb{L}_w \in \text{Im}(W^+_{\mathbf{h}})$ and $\text{Ad}_{\mathbf{h}} \mathbb{A}_w \in \text{Ker}(W^+_{\mathbf{h}})$. This requires $\text{Ker}(W^+_{\mathbf{h}}) = \text{Ad}_{\mathbf{h}} \mathfrak{l}_t$, instead of $\text{Ad}_{\mathbf{h}}^{-1} \mathfrak{l}_t $ as considered in equation (6.4). Let me also note that this seems consistent with the reference [29] as well. Indeed, comparing the constraint (6.3) of this paper with the constraint on $B_\pm$ given above (3.28) in [29], one sees that the field $\mathbf{h}$ considered here corresponds to $h^{-1}$ in [29], so that one should shift $\mathbf{h}$ to $\mathbf{h}^{-1}$ throughout the E-model formulation.

2) My second point concerns the appendix B and the following two related questions :
i) how does the term $\mathcal{M}_w / \langle \pi \gamma \rangle$ in the expression (B.6) of $\mathcal{L}_w$ descend from $\mathbb{C}^2$ to $\mathbb{CP}^1$, as it is of degree -1 in $\pi$ ?
ii) why does this term not contribute at $\pi = \beta$ in $\mathcal{L}_w|_\beta=0$?
It seems to me that a consistent form for $\mathcal{L}_w$ should instead be
$$ \mathcal{L}_w = \frac{\langle\pi\beta\rangle}{\langle\pi\gamma\rangle}\mathcal{M}_w + \mathcal{N}_w, $$
which would then ensure that $\pi$ can be seen in $\mathbb{CP}^1$ and that $\mathcal{L}_w |_\beta = \mathcal{N}_w=0$. Note that in that case, one would have an additional term $\langle \tilde{\alpha}\beta\rangle/\langle \alpha\beta \rangle$ in the coefficient of $\text{Ad}_{\tilde{h}}^{-1} \text{Ad}_h$ in (B.9): in fact , it seems this is necessary to match this coefficient with $\sigma$, using (4.8). Similar comments apply to $\mathcal{L}_{\bar{w}}$. Here also, this should not change the final results of the appendix.

1- First, I would like to ask the authors to address the comments 1) and 2) in the report above and, if relevant, implement the corresponding revisions in the manuscript.

The following points are small typos.

2- In (2.5), I think the $\wedge$ should be removed between $\mathcal{A}_A$ and $\delta\mathcal{A}_B$ as these are components of forms rather than forms. Also, there should be a vol$_4$.

3- After (3.27), there is a "be" missing in "to holomorphic"

4- After (3.41), there is a repetion of "that".

5- In the second line of (3.50), should the $\langle\tilde{\alpha}\beta\rangle$ be $\langle\alpha\beta\rangle$?

6- Before (4.6), hC6$_6$ instead of hCS$_6$

7- Should there be an overall minus sign in (4.14)?

8- Before (5.10), should it be "equations of motion of (5.3)"?

9- After (5.11), "contains implies"

10- In (5.15), should there be derivatives $\partial_\pm$ in addition to $\mathcal{L}^i_\pm$?

11- First paragraph of section 6: should "choice of isotropic subspace" be "choice of two mutually orthogonal susbpaces"?

12- In (6.4), $\xi$ should be $\alpha$

13- $c_G$ in (6.16) is never defined (although I imagine it is the dual Coxeter number)

14- Should there be $\pm$ signs in (6.23)?

15- Before (A.15), should $\mathcal{J}_1\mathcal{J}_2\mathcal{J}_3$ be $-\text{id}$?

Finally, the following are small suggestions which in my opinion could help the reader. The authors should feel free to implement them or not.

16- After (2.2), it could be useful to mention that $\alpha$, $\tilde{\alpha}$ and $\beta$ are constant spinors entering the definition of the model

17- Following footnote 9, the double pole of $\Omega$ at $\pi=\beta$ in principle creates terms in equations (3.7) and (3.16) which involve $\partial_z \hat{h}|_\beta$ or $\partial_z \mathcal{A}'|_\beta$, in addition to $\hat{h}|_\beta=\text{id}$ or $\mathcal{A}'|_\beta=0$ . It could be worth explaining why these do not contribute.

18- Two remarks concerning the passage from (3.34) to (3.35). As far as I understand, this is not completely direct as it uses the constraint (2.13) on $\ell$: it could be helpful to mention that. Also, it might be useful to recall that $A_{AA'}=\beta_{A'}\,B_A$ is not a new object introduced here and came from the analysis of $\mathcal{A}'$ before (3.9).

19- Around (3.47) and (3.49), it can be worth mentioning that $\gamma$ is the spinor that will appear later in the symmetry reduction.

20- Since there are many different parameterisations introduced throughout the paper to fit the various formulations / components of the diamond, it can be helpful to summarise around the equation (5.3) which ones can be taken as a set of independent physical parameters for the IFT$_2$, as for instance $(r_+,r_-,t,\sigma)$.

21- Around (6.2), it might be worth stating explicitly that $\mathbf{h}=(h|_\alpha,h|_{\tilde{\alpha}})$ and $\mathbb{L}=(\mathcal{L}|_\alpha,\mathcal{L}|_{\tilde{\alpha}})$.