A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$

Abstract

The abelian and monoidal structure of the category of smooth weight modules over a non-integrable affine vertex algebra of rank greater than one is an interesting, difficult and essentially wide open problem. Even conjectures are lacking. This work details and tests such a conjecture for L − 3 2 (sl3) via a logarithmic Kazhdan-Lusztig correspondence.We first investigate the representation theory of U H i (sl3), the unrolled restricted quantum group of sl3 at fourth root of unity. In particular, we analyse its finite-dimensional weight category, determining Loewy diagrams for all projective indecomposables and decomposing all tensor products of irreducibles. Our motivation is that this category is conjecturally braided tensor equivalent to a category of W 0A 2 (2)modules. Here, W 0 A 2 (2) is an orbifold of the octuplet vertex algebra WA 2 (2) of Semikhatov, the latter being the natural sl3-analogue of the well known triplet algebra. Moreover, W 0 A 2 (2) is the parafermionic coset of the affine vertex algebra LWe formulate an explicit conjecture relating the representation theory of W 0 A 2 (2) and U H i (sl3) and work out the resulting structures of the corresponding L − 3 2 (sl3)-modules. In particular, we obtain conjectural Loewy diagrams for the latter's projective indecomposables and decompositions for the fusion products of its irreducibles. These products coincide with those recently computed via Verlinde's formula. Finally, we give analogous results for WA 2 (2).

Admissible-level $$\mathfrak {sl}_3$$ minimal models

“…Unfortunately, the structure of the known logarithmic L k (𝔰𝔩 3 )-modules remains mysterious -even their composition factors are unknown. However, a companion paper to this one [48] combines the results presented here with a conjectural logarithmic Kazhdan-Lusztig correspondence to posit not only composition factors but also complete Loewy diagrams and fusion rules (but only in the special case k = − 3 2 ). In this context, the work reported here represents an important part of a rapidly advancing program to understand the representation theory and modularity of higher-rank affine vertex operator algebras and W-algebras.…”

“…A closely related remark is that the Grothendieck fusion ring presented here has a one-dimensional representation which is constant on the D 6 -and spectral flow orbits, but otherwise takes the values This is also addressed in the companion paper [48] where the existence of a Kazhdan-Lusztig-type correspondence is discussed. This correspondence takes the form of a (conjectural) tensor equivalence between the category 𝒲 3,2 of weight A 2 (3, 2)-modules (with finite-dimensional weight spaces) and an appropriate modification of the category of ADMISSIBLE-LEVEL 𝔰𝔩 3 MINIMAL MODELS 35 finite-dimensional modules over a certain quantum group 𝑈 𝐻 𝑞 (𝔰𝔩 3 ) at 𝑞 = 𝔦.…”

“…These 𝐵 𝑄 (𝑝)-algebras are related to (higher-rank) singlet algebras by a parafermionic coset construction [55,76] and thence to triplet algebras. (This is discussed in more detail in the companion paper [48].) We mention that the automorphism groups of the corresponding triplet algebras are PSL 2 (ℂ) and PSL 3 (ℂ), respectively, which may be ultimately responsible for the appearance of adjoint weights in (5.54).…”

“…In fact, the 𝔰𝔩 3 minimal model A 2 (3, 2) is closely related to several other interesting logarithmic vertex operator algebras including the 𝑁 = 4 superconformal minimal model with 𝑐 = −9 [54], the Feigin-Tipunin algebra 𝑊 0 𝐴 2 (2) [55] and Semikhatov's octuplet algebra 𝑊 𝐴 2 (2) [56]. Further details concerning these relations may be found in the companion paper [48].…”

“…We shall not pursue the structures of these reducible but indecomposable A 2 (3, 2)-module here, though their existence is clear. Instead, we refer to a companion paper [48] in which conjectures for these structures are presented.…”