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Integrable Deformations from Twistor Space
by Lewis T. Cole, Ryan A. Cullinan, Ben Hoare, Joaquin Liniado, Daniel C. Thompson
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Submission summary
| Authors (as registered SciPost users): | Lewis Cole · Ben Hoare · Joaquin Liniado |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202404_00042v1 (pdf) |
| Date accepted: | 2024-05-22 |
| Date submitted: | 2024-04-25 23:35 |
| Submitted by: | Liniado, Joaquin |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons in 6 dimensions. We provide the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. Starting from 6d holomorphic Chern-Simons theory on twistor space with a particular meromorphic 3-form $\Omega$, we construct the defect theory to find a novel 4d integrable field theory, whose equations of motion can be recast as the 4d anti-self-dual Yang-Mills equations. Symmetry reducing, we find a multi-parameter 2d integrable model, which specialises to the $\lambda$-deformation at a certain point in parameter space. The same model is recovered by first symmetry reducing, to give 4d Chern-Simons with generalised boundary conditions, and then constructing the defect theory.
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Author comments upon resubmission
With kind regards,
Lewis Cole, Ryan Cullinan, Ben Hoare, Joaquin Liniado & Daniel Thompson
List of changes
Report 1
Main questions
• 1 - We agree that in section 6 the correct definition is h = (h, ˜h). The rest of the E-model
formulation was correspondingly amended.
• 2 - We have redone the calculation in appendix B with the correct factor of ⟨πβ⟩ in the numerator
as pointed out by the referee. As expected, the final action is the same, but the intermediate steps
are altered.
Minor changes
• 2 to 6 - Corrected.
• 7 - Changed sign in eq. (4.10) and clarified CS4 conventions to ensure signs are consistent throughout.
• 8 to 12 - Corrected.
• 13 - Added definition of cG.
• 14 - Corrected.
• 15 - We have added explicit expressions in a coordinate basis for Ji in appendix A.3, for which
J1J2J3 = +id. This specification is correlated to the stated Kähler forms being self-dual and our
field theory being ASDYM with our choice of orientation, as opposed to the other way round.
• 16 - Amended.
• 17 - Added footnote 10.
• 18 - Amended.
• 19 - Added footnote 12.
• 20 - Added comment.
• 21 - Amended.
Report 2
• 1 - We agree that this is an important subtlety that is not fully resolved. After some initial
investigations we believe that a more thorough analysis is required and have left it as an open
question. We have added a paragraph highlighting this point below eq.(3.46).
• 2 - Added comment.
• 3 - The referee’s comment motivated us to look in more detail at reality conditions and their
twistorial origin. Our original approach was direct inspection of the 4d IFT. We have retained
this analysis in section 3.3, but we have rewritten it to make the key points clearer and to better
prepare for a discussion from the perspective 6d hCS and twistors, which we have added after. This
highlighted one new reality condition, which we have also included.
• 4 and 5 - Amended.
Published as SciPost Phys. 17, 008 (2024)
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2024-5-17 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202404_00042v1, delivered 2024-05-17, doi: 10.21468/SciPost.Report.9073
Report
The authors have answered my comments and implemented the relevant changes in the manuscript. I am thus quite happy to recommend it for publication.
I just have one small technical question concerning the conventions for the Hyper-Kähler structure in Appendix A.3. This is extremely minor and does not require an additional reviewing round: it can simply be corrected in the final version of the paper if the authors reach the same conclusion as what follows. Using the expression (A.18) of $\mathcal{J}_\pi$ and the relation $\bigl(\mathcal{J}_1,\mathcal{J}_2,\mathcal{J}_3\bigr) = \bigl(\mathcal{J}_\pi \bigl|_{\zeta=0}, \mathcal{J}_\pi \bigl|_{\zeta=-i}, \mathcal{J}_\pi \bigl|_{\zeta=1}\bigr)$, I naively find the opposite of (A.16) for the matrix representation of the $\mathcal{J}_i$'s. In particular, this would mean $\mathcal{J}_1 \mathcal{J}_2 \mathcal{J}_3 = -\text{id}$, which was the reason behind the question 15 in my previous report.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Author: Joaquin Liniado on 2024-05-31 [id 4525]
(in reply to Report 2 on 2024-05-17)We believe that the apparent disagreement is related to the space on which the complex structure is acting. Namely, one can view the complex structure acting on the tangent bundle or on the co-tangent bundle. The corresponding matrix presentations are the transpose of each other. The presentation we have chosen acts on 1-forms, where we have the identity $\mathcal{J}_1 \mathcal{J}_2\mathcal{J}_3 = 1$. Given that all of these matrices are antisymmetric, the presentation acting on vector fields satisfies $\mathcal{J}_1^t \mathcal{J}_2^t \mathcal{J}_3^t = -1$, which corresponds to the presentation considered by the referee. We have thus clarified in our manuscript on which space the complex structure is acting.