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Admissible-level \(\mathfrak {sl}_3\) minimal models

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Abstract

The first part of this work uses the algorithm recently detailed in Kawasetsu and Ridout (Commun Contemp Math 24:2150037, 2022. arXiv:1906.02935 [math.RT]) to classify the irreducible weight modules of the minimal model vertex operator algebra \({\textsf {L} }_{{\textsf {k} }}(\mathfrak {sl}_{3})\), when the level \({\textsf {k} }\) is admissible. These are naturally described in terms of families parametrised by up to two complex numbers. We also determine the action of the relevant group of automorphisms of \(\widehat{\mathfrak {sl}}_{3}\) on their isomorphism classes and compute explicitly the decomposition into irreducibles when a given family’s parameters are permitted to take certain limiting values. Along with certain character formulae, previously established in Kawasetsu (Adv Math 393:108079, 2021. arXiv:2003.10148 [math.RT]), these results form the input data required by the standard module formalism to consistently compute modular transformations and, assuming the validity of a natural conjecture, the Grothendieck fusion coefficients of the admissible-level \(\mathfrak {sl}_{3}\) minimal models. The second part of this work applies the standard module formalism to compute these explicitly when \({\textsf {k} }=-\frac{3}{2}\). This gives the first nontrivial test of this formalism for a nonrational vertex operator algebra of rank greater than 1 and confirms the expectation that the methodology developed here will apply in much greater generality.

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Notes

  1. If we were considering coherent families of \(\mathrm {d}(\mathfrak {l})\)-modules instead of \(\mathfrak {l}\)-modules, then the Weyl reflection appearing here would be \(\mathrm {w}_{2}\).

  2. We remark that under the general definition in [51], all of these modules would be examples of relaxed highest-weight modules. However, in this case the nomenclature introduced here is convenient and so we shall adopt it, hoping that no confusion will arise.

  3. There is a well-known exception to this statement when \({\textsf {u} }\geqslant 3\) and \({\textsf {v} }=1\), so \({\textsf {k} }\in \mathbb {Z}_{\geqslant 0}\). Then, the positive-energy category coincides with \(\widehat{\mathscr {W}}_{{\textsf {u} },1}\) and the category of integrable highest-weight \(\widehat{\mathfrak {sl}}_{3}\)-modules; the latter is of course preserved by spectral flow.

  4. We shall generally drop the hat from affine Dynkin labels, trusting that this will not cause confusion.

  5. As we shall see, this arbitrary choice is convenient because the subsets of admissible weights classifying the relaxed, semirelaxed and highest-weight \({\textsf {A} }_{2} \left({\textsf {u} }, {\textsf {v} } \right)\)-modules then satisfy \(C_{{\textsf {u} },{\textsf {v} }}^2 \subseteq B_{{\textsf {u} },{\textsf {v} }}^1 \subset A_{{\textsf {u} },{\textsf {v} }}\).

  6. It is easy to check that \(\widehat{\lambda } \in C_{{\textsf {u} },{\textsf {v} }}^2\) implies that there are no triple intersections.

  7. We recall that the “−-type” quantum Hamiltonian reduction was introduced by Frenkel, Kac and Wakimoto in [40] for regular (principal) nilpotent elements. It differs from the usual “\(+\)-type” regular reduction in that it gauges the negative root vectors instead of the positive ones. Although both reductions give isomorphic W-algebras, the corresponding functors on modules are different. The reduction functor used in Theorem 5.1 is a generalisation of this −-type functor to all nilpotents due to Kac–Wakimoto [59] and Arakawa [60].

  8. We remark that this name, while quite standard, may be a little misleading. It does not refer to a 1-point correlation function of genus 1 in an appropriate conformal field theory, but rather to a chiral version where the trace is taken over a fixed module. In other words, this concept generalises the definition (5.1) of a character by inserting some fixed zero mode (usually unexponentiated!) inside the trace.

  9. The reader will no doubt recognise the action of \({\textsf {S} }\) and \({\textsf {T} }\) on \(\theta \) as a somewhat strange-looking generalisation of the usual formulae familiar from rational models. The terms involving complex arguments seem to be necessary to deal with the unusual automorphy factor that results from transforming Dirac combs.

  10. This second identity is well known for delta functions with real arguments. Here, as in many other applications of the standard module formalism, we assume that it may be extended to complex arguments. We expect that this formula can be established rigorously by finding the correct space of test functions to pair with and hope to pursue this in future work.

  11. This rigidity conjecture is very natural as all rational vertex operator algebras are known to produce rigid module categories (modular tensor categories even) [8] and a growing number of nonrational vertex operator algebras are also known to admit rigid module categories [32, 68,69,70,71,72]. There is, however, a known counterexample [73]. A modified Grothendieck fusion ring for this counterexample was studied in detail in [74].

  12. We thank an anonymous referee for this suggestion.

  13. That \(\widehat{\mathcal {L}}_{0}\) is the Grothendieck fusion unit follows directly from the standard Verlinde formula and the fact that the standard S-matrix is unitary.

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Acknowledgements

KK’s research is partially supported by the Australian Research Council Discovery Project DP160101520, MEXT Japan “Leading Initiative for Excellent Young Researchers (LEADER)”, JSPS Kakenhi Grant numbers 19KK0065 and 21K3775. DR’s research is supported by the Australian Research Council Discovery Projects DP160101520 and DP210101502, as well as an Australian Research Council Future Fellowship FT200100431. SW’s research is supported by the Australian Research Council Discovery Project DP160101520 and the Humboldt Fellowship for Experience Researchers GBR-1212053-HFST-E.

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  1. Priority Organization for Innovation and Excellence, Kumamoto University, Kumamoto, 860-8555, Japan

    Kazuya Kawasetsu

  2. School of Mathematics and Statistics, University of Melbourne, Parkville, 3010, Australia

    David Ridout

  3. School of Mathematics, Cardiff University, Cardiff, CF24 4AG, UK

    Simon Wood

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Kawasetsu, K., Ridout, D. & Wood, S. Admissible-level \(\mathfrak {sl}_3\) minimal models. Lett Math Phys 112, 96 (2022). https://doi.org/10.1007/s11005-022-01580-9

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