Relaxed and logarithmic modules of (sl3)over-cap

In Adamovic (Commun Math Phys 366:1025-1067, 2019), the affine vertex algebra Lk (sl(2)) is realized as a subalgebra of the vertex algebra Vir(c) circle times (0), where Virc is a simple Virasoro vertex algebra and Pi(0) is a half-lattice vertex algebra. Moreover, all Lk (sl(2))-modules (including, modules in the category KLk, relaxed highest weight modules and logarithmic modules) are realized as Virc. Pi(0)-modules. A natural question is the generalization of this construction in higher rank. In the current paper, we study the case g = sl(3) and present realization of the VOA Lk (g) for k is not an element of Z(>= 0) as a vertex subalgebra of W-k circle times S circle times Pi(0), whereWk is a simple Bershadsky-Polyakov vertex algebra and S is the beta(gamma) vertex algebra. We use this realization to study ordinary modules, relaxed highest weight modules and logarithmicmodules. 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 g-modules which are not GelfandTsetlin modules. The next problem that we consider is the realization of logarithmic modules. We first analyse the free-field realization of W-k from Adamovi ' c et al. (Lett Math Phys 111(2), Paper No. 38, arXiv:2007.00396 [math.QA], 2021) and obtain a realization of logarithmic modules for 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 Wk (sl(3), f (pr))-modules. Using logarithmic modules for the beta gamma VOA, we are able to construct logarithmic L-k (g)-modules of rank three.