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Fusion of irreducible modules in the periodic Temperley--Lieb algebra
by Yacine Ikhlef, Alexi Morin-Duchesne
Submission summary
| Authors (as registered SciPost users): | Yacine Ikhlef |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2312.14837v2 (pdf) |
| Date submitted: | 2024-09-05 11:26 |
| Submitted by: | Ikhlef, Yacine |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We propose a new family ${\sf Y}_{k,\ell,x,y,[z,w]}$ of modules over the enlarged periodic Temperley--Lieb algebra ${\sf{\cal E}PTL}_N(\beta)$. These modules are built from link states with two marked points, similarly to the modules ${\sf X}_{k,\ell,x,y,z}$ that we constructed in a previous paper. They however differ in the way that defects connect pairwise. We analyse the decomposition of ${\sf Y}_{k,\ell,x,y,[z,w]}$ over the irreducible standard modules ${\sf W}_{k,x}$ for generic values of the parameters $z$ and $w$, and use it to deduce the fusion rules for the fusion $\sf W \times W$ of standard modules. These turn out to be more symmetric than those obtained previously using the modules ${\sf X}_{k,\ell,x,y,z}$. From the work of Graham and Lehrer, it is known that, for $\beta=-q-q^{-1}$ where $q$ is not a root of unity, there exists a set of non-generic values of the twist $y$ for which the standard module ${\sf W}_{\ell,y}$ is indecomposable yet reducible with two composition factors: a radical submodule ${\sf R}_{\ell,y}$ and a quotient module ${\sf Q}_{\ell,y}$. Here, we construct the fusion products $\sf W\times R$, $\sf W\times Q$ and $\sf Q\times Q$, and analyse their decomposition over indecomposable modules. For the fusions involving the quotient modules ${\sf Q}$, we find very simple results reminiscent of $\mathfrak{sl}(2)$ fusion rules. This construction with modules ${\sf Y}_{k,\ell,x,y,[z,w]}$ is a good lattice regularization of the operator product expansion in the underlying logarithmic bulk conformal field theory. Indeed, it fits with the correspondence between standard modules and connectivity operators, and is useful for the calculation of their correlation functions. Remarkably, we show that the fusion rules $\sf W\times Q$ and $\sf Q\times Q$ are consistent with the known fusion rules of degenerate primary fields.
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- Provide a novel and synergetic link between different research areas.
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Author comments upon resubmission
Dear editors,
We thank the referees for their careful reading and comments on our manuscript. The version we are now resubmitting contains changes addressing the issues that they raised. You will find below our answers to their requests.
Best regards, Yacine Ikhlef and Alexi Morin-Duchesne
Answer to Referee 1:
We wrote a new Appendix A describing the arguments leading to the dimension counting and to the decomposition of the modules Y. In working out these proofs, we found that the characterisation of the modules Y given in Section 3.2 wasn't quite clear or correct. We therefore reworked a large part of this section so that it is correct and clearer, and adapted the rest of the text accordingly.
We removed the mention of composition factors in Section 3.3, as this indeed only starts to make sense once the decomposition of the modules Y is given in Section 3.4.
In Appendix A, we added an explanation of the homomorphisms from standard modules to quotients of the modules Y.
At the beginning of Section 6.3, we added a remark about using the same notation for fusion products over Vir and over Vir x Vir.
Answer to Referee 2:
We modified the terminology of singular and null states according to the referee's comment.
We agree with the referee that obtaining a functorial description of our construction and understanding its exactness are interesting questions -- we are currently working on this problem. This is beyond the scope of the current paper, as it would require that we understand how to define more generally the fusion of two arbitrary modules. We added a comment at the end of the conclusion stating that these questions are interesting open problems.
We stress that none of the fusion products that we compute actually vanish. Some of them however vanish in certain channels [z,w]. In general, we express the "full" fusion product of two fields in terms of the fusion rules. If some fusion channels vanish, then the corresponding fusion rules contains finitely many channels, which is usual in rational CFT. We added a paragraph at the end of Section 4.2 to clarify this point.
We fixed the typo in (3.8).
We replaced "unwinded" by "unwound".
List of changes
- Added Appendix A, with details on the dimension counting and the decomposition of the modules Y.
- Modified the description of link states in Section 3.2.
- Added a comment at the end of the Conclusion, about the functorial description of our fusion scheme.
- Added a paragraph at the end of Section 4.2 to clarify a point on vanishing fusion channels.
- Implemented minor changes requested by the Referees (see resubmission letter).