Relaxed and logarithmic modules of $$\widehat{{{\mathfrak {s}}}{{\mathfrak {l}}}_3}$$

Abstract

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.