Tensor Structure on the Kazhdan–Lusztig Category for Affine 𝔤𝔩(1|1)

Abstract

We show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in ${\mathcal{O}}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in ${\mathcal{O}}_k^{fin}$. Then using Knizhnik–Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

“…If Y is an intertwining operator of type (which applies because tensoring with M 2mn+1,1 , n = 0, does not fix any simple object in O M(p) ). Moreover, the argument of [CMY3,Proposition 5.0.4] shows that every simple object of Rep W(p) Am is isomorphic to the induction of a simple M(p)-module in O M(p) , and that F W(p) Am (M 1 ) ∼ = F W(p) Am (M 2 ) for simple modules M 1 and M 2 if and only if M 2 ∼ = M 2mn+1,1 ⊠ M 1 for some n ∈ Z. This discussion shows that we can use induction to classify all irreducible W(p) Am -modules (see also [AM4,Theorem 2.5]):…”

“…Since we already proved in [CMY2] that the atypical category C M(p) is rigid, it is then enough to prove that the typical modules F λ , λ ∈ C \ L • are rigid (as M(p)-modules). Our rigidity proof for F λ is new and completely different from the explicit calculational proofs of rigidity for typical modules of the βγ-vertex algebra [AW] and of affine gl 1|1 [CMY3]. The idea is to choose evaluations e λ : F ′ λ ⊠ F λ → M(p) and coevaluations i λ : M(p) → F λ ⊠ F ′ λ (where F ′ λ is the M(p)-module contragredient of F λ , also a typical Fock module) in such a way that the rigidity composition…”

“…By now, it is understood that affine vertex algebras (and their W -algebras) at almost all levels admit uncountably many inequivalent simple modules [KR] and should also admit logarithmic modules (which are indecomposable but reducible modules on which the Virasoro zeromode L(0) acts non-semisimply). Even the construction of logarithmic modules is a difficult task: for affine vertex operator (super)algebras, this is achieved only for those associated to sl 2 , osp 1|2 , sl 3 , and gl 1|1 [Ad3,ACG,CMY3]. Among these, the complete ribbon (super)category of modules is currently understood only in the case of gl 1|1 [CMY3], and the only additional completely-understood example of a non-finite, non-semisimple tensor category of representations for a vertex operator algebra is the βγ-system [AW].…”

“…Even the construction of logarithmic modules is a difficult task: for affine vertex operator (super)algebras, this is achieved only for those associated to sl 2 , osp 1|2 , sl 3 , and gl 1|1 [Ad3,ACG,CMY3]. Among these, the complete ribbon (super)category of modules is currently understood only in the case of gl 1|1 [CMY3], and the only additional completely-understood example of a non-finite, non-semisimple tensor category of representations for a vertex operator algebra is the βγ-system [AW].…”

3d Mirror Symmetry and the $βγ$ VOA

“…We can now introduce the candidate for L B . In [CMY20a], the authors studied the Kazhdan-Lusztig category KL of V ( gl(1|1)) and showed that it is a rigid braided tensor supercategory defined by P (z)-intertwining operators. In Section 4, we will introduce a tensor subcategory KL 0 .…”