Tensor categories of affine Lie algebras beyond admissible levels
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Cited by 25 publications
References 69 publications
On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
We study the representation theory of non-admissible simple affine vertex algebra L −5/2 (sl(4)).We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra V −5/2 (sl(4)), and show that it generates the maximal ideal in V −5/2 (sl(4)). We classify irreducible L −5/2 (sl(4))-modules in the category O, and determine the fusion rules between irreducible modules in the category of ordinary modules KL −5/2 . It turns out that this fusion algebra is isomorphic to the fusion algebra of KL−1. We also prove that KL −5/2 is a semi-simple, rigid braided tensor category.In our proofs we use the notion of collapsing level for the affine W-algebra, and the properties of conformal embedding gl(4) ֒→ sl(5) at level k = −5/2 from [2]. We show that k = −5/2 is a collapsing level with respect to the subregular nilpotent element f subreg , meaning that the simple quotient of the affine W-algebra W −5/2 (sl(4), f subreg ) is isomorphic to the Heisenberg vertex algebra MJ (1). We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor H f subreg . It turns out that this case is much more complicated than the case of minimal reduction.
On the representation theory of the vertex algebra $L_{-5/2}(sl(4))$
We study the representation theory of non-admissible simple affine vertex algebra L −5/2 (sl(4)).We determine an explicit formula for the singular vector of conformal weight four in the universal affine vertex algebra V −5/2 (sl(4)), and show that it generates the maximal ideal in V −5/2 (sl(4)). We classify irreducible L −5/2 (sl(4))-modules in the category O, and determine the fusion rules between irreducible modules in the category of ordinary modules KL −5/2 . It turns out that this fusion algebra is isomorphic to the fusion algebra of KL−1. We also prove that KL −5/2 is a semi-simple, rigid braided tensor category.In our proofs we use the notion of collapsing level for the affine W-algebra, and the properties of conformal embedding gl(4) ֒→ sl(5) at level k = −5/2 from [2]. We show that k = −5/2 is a collapsing level with respect to the subregular nilpotent element f subreg , meaning that the simple quotient of the affine W-algebra W −5/2 (sl(4), f subreg ) is isomorphic to the Heisenberg vertex algebra MJ (1). We prove certain results on vanishing and non-vanishing of cohomology for the quantum Hamiltonian reduction functor H f subreg . It turns out that this case is much more complicated than the case of minimal reduction.
In Adamović (Commun Math Phys 366:1025–1067, 2019), the affine vertex algebra $$L_k({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ L k ( s l 2 ) is realized as a subalgebra of the vertex algebra $$Vir_c \otimes \Pi (0)$$ V i r c ⊗ Π ( 0 ) , where $$Vir_c $$ V i r c is a simple Virasoro vertex algebra and $$\Pi (0)$$ Π ( 0 ) is a half-lattice vertex algebra. Moreover, all $$L_k({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ L k ( s l 2 ) -modules (including, modules in the category $$KL_k$$ K L k , relaxed highest weight modules and logarithmic modules) are realized as $$Vir_c \otimes \Pi (0)$$ V i r c ⊗ Π ( 0 ) -modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case $${\mathfrak {g}}= {{\mathfrak {s}}}{{\mathfrak {l}}}_3$$ g = s l 3 and present realization of the VOA $$L_k({\mathfrak {g}})$$ L k ( g ) for $$k \notin {\mathbb {Z}}_{\ge 0}$$ k ∉ Z ≥ 0 as a vertex subalgebra of $${\mathcal {W}}_ k \otimes {\mathcal {S}} \otimes \Pi (0)$$ W k ⊗ S ⊗ Π ( 0 ) , where $${\mathcal {W}}_ k $$ W k is a simple Bershadsky–Polyakov vertex algebra and $${\mathcal {S}}$$ S is the $$\beta \gamma $$ β γ vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmic modules. We prove the irreducibility of all our relaxed highest weight modules having finite-dimensional weight spaces (whose top components are Gelfand–Tsetlin modules). The irreducibility of relaxed highest weight modules with infinite-dimensional weight spaces is proved up to a conjecture on the irreducibility of certain $${\mathfrak {g}}$$ g -modules which are not Gelfand–Tsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of $${\mathcal {W}}_ k $$ W k from Adamović et al. (Lett Math Phys 111(2), Paper No. 38, arXiv:2007.00396 [math.QA], 2021) and obtain a realization of logarithmic modules for $${\mathcal {W}}_ k $$ W k of nilpotent rank two at most admissible levels. Beyond admissible levels, we get realization of logarithmic modules up to a existence of certain $${\mathcal {W}}_k({{\mathfrak {s}}}{{\mathfrak {l}}}_3, f_{pr})$$ W k ( s l 3 , f pr ) -modules. Using logarithmic modules for the $$\beta \gamma $$ β γ VOA, we are able to construct logarithmic $$L_k({\mathfrak {g}})$$ L k ( g ) -modules of rank three.
Suppose V G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V . We show that if all irreducible V G -modules contained in V live in some braided tensor category of V G -modules, then they generate a tensor subcategory equivalent to the category Rep G of finitedimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V . Additionally, we show that if the fusion rules for the irreducible V G -modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V G -modules. These results do not require rigidity on any tensor category of V G -modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V G is C2-cofinite but not necessarily rational. When V G is both C2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V G -modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU (2), up to modification by an abelian 3-cocycle.
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