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Fusion of irreducible modules in the periodic TemperleyLieb algebra
by Yacine Ikhlef, Alexi MorinDuchesne
Submission summary
Authors (as registered SciPost users):  Yacine Ikhlef 
Submission information  

Preprint Link:  https://arxiv.org/abs/2312.14837v1 (pdf) 
Date submitted:  20240116 17:49 
Submitted by:  Ikhlef, Yacine 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Abstract
We propose a new family ${\sf Y}_{k,\ell,x,y,[z,w]}$ of modules over the enlarged periodic TemperleyLieb 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 $\beta=qq^{1}$ where $q$ is not a root of unity, there exists a set of nongeneric 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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Report
This paper is the latest in a series of works that study constructions in appropriate statistical lattice models that mimic fusion products in their, presumably conformally invariant, scaling limits. Here, the authors are concerned with bulk fusion, rather than the chiral fusion typically studied by conformal field theorists, and their lattice models are built from the periodic TemperleyLieb algebras.
In a previous paper, the authors proposed a definition for the bulk lattice fusion product of irreducible modules over these algebras. This paper notes that this definition does not correspond to an associative product and proposes an improved definition that does. The improvement is noted by the authors to be a little peculiar as it requires the introduction of one or two auxiliary parameters, depending on the specifics of the lattice modules. They also state that the improvement is an even better candidate for bulk lattice fusion, leaving the door open to further tweaks.
Irrespective of whether further tweaks are required, this paper reports an impressive amount of work, even if this work ultimately relies upon some very reasonable conjectures (Claims 3.1, 4.1 and 4.2; Conjectures 4.1 and 4.2). I expect that these claims are extremely difficult to prove, but it is nevertheless laudable that the authors indicate them so prominently, instead of burying them in dense prose.
Aside from the associativity result mentioned, they give the complete decomposition of the improved bulk lattice fusion product of two standard modules when the parameters are suitably generic. They also describe some results involving the radical submodules of the standard modules and the corresponding quotients. Finally, they also explore a few examples with nongeneric parameters, illustrating that in this case the product involves reducible but indecomposable direct summands (as expected for a logarithmic scaling limit).
This is a remarkable, if extremely technical, improvement on the state of the art and so I warmly recommend publication. I have only a few small suggestions that the authors may wish to implement in a revision.
First, I would like to ask the authors to correct their terminology concerning singular and null vectors of the Virasoro algebra, eg. in between (1.6) and (1.7). A singular vector is a (nongenerating) highestweight state and a null vector is a state that is orthogonal to the entire module. So a radical submodule corresponds to null vectors, at least one of which is singular.
Second, I suggest that the authors think of adding some remarks about the "exactness" of their fusion definition. Assuming that bulk fusion in the scaling limit is modelled as a tensor product, one expects that fusing with a given fixed module is rightexact but perhaps not leftexact. Given the authors results about radicals and quotients, it should be possible to say if their results are consistent with rightexactness. It would be very interesting to know if they are consistent with leftexactness or not, because examples where leftexactness fails are still considered quite exotic and hard to understand in the CFT community.
[I actually wonder if this has something to do with the example (4.19) in which fusion is found to be incompatible with isomorphisms...]
I'd also like to see a little discussion around the fact that some fusion products are found to be 0, eg. (4.73) and (4.77), also QxQ in Section 5. What does this mean? Does it have any bearing of the existence of conjugate fields in the scaling limit?
Is there a typo in (3.8)? The w seems out of place.
Finally, I'd like to suggest replacing the term "unwinded" throughout by "unwound", just for grammar pedants... I don't like worrying about whether defects are running out of breath or not... :)
Recommendation
Ask for minor revision
Strengths
1Original approach to bulk lattice fusion
2Clear and pedagogical presentation
Report
This paper introduces a new class of representations of the affine TemperleyLieb (aTL) algebra. These representations are close in spirit, but not the same, as a set of aTL representations defined by the authors in a previous paper. The introduction of these sets of representations are motivated by the search for a lattice analog of the bulk fusion rules in the logartihmic conformal field theory (CFT) describing loop models. The authors clearly indicate why the representations introduced in this paper are better candidates than the previous ones. It is an important open problem to describe fusion rules in logarithmic CFT. Although important results have been found in the case of boundary CFT through an algebraic lattice fusion procedure, a bulk lattice counterpart is yet to be found. I believe this work is an important step in this direction. The pedestrian approach it proposes is something to rely on to find new ideas for a putative systematic algebraic approach.
The paper is well written and structured. It contains in general a good amount of details and pedagogical explanations. I have only a few suggestions that could potentially improve the readability.
Requested changes
1In page 22, it could be good to give details on how to derive the dimensions of modules. The reader is referred to the previous paper where the analogous proof is not very long. Maybe a similar, possibly shorter, proof could be included.
2In page 23, in the description of non generic parameters, composition factors of the modules are given but never discussed before. It would be good to say more on them beforehand.
3For the proof of the decomposition in the generic case in page 24, the reader is referred to the techniques of the previous paper. In the previous paper, explicit homomorphisms from standard modules were given. It would be nice to give explicit homomorphisms in this new case as well, if not too long. Or it could possibly be put in an appendix.
4 From page 52, the notation of tensor product with a square is defined as the fusion product of Vir. But it is used to denote the fusion product of both Vir and Vir x Vir. It would be good to emphasize this.