# On relaxed and logarithmic modules for affine vertex algebras

• A+

We will first review certain general methods for constructing logarithmic (projective) modules. Then we will show how these methods can be applied on affine vertex algebras by using recent free field realizations, which are motivated by finding inverses of the Quantum Hamiltonian Reductions. A particular emphasis will be put on the realization of the affine vertex algebra $L_k(sl(3))$ as a subalgebra of $BP_k \otimes \mathcal S \otimes \Pi(0)$ where $BP_k$ is the Breshadsky-Polykov   algebra, $\mathcal S$ the $\beta-\gamma$ system   and $\Pi(0)$ is a half-lattice vertex algebra. We present a realization of logarithmic modules for $BP_k$ and logarithmic $L_k(sl_3)$ at (almost) arbitrary non-integral level k. Next we analyse the relaxed modules for $L_k(sl_3)$  with finite-dimensional and infinite-dimensional weight spaces. Finally, we discuss an affine analogs of the Feigin-Semikhatov algebra $W_{A_2}(p)$ and possible applications in a construction on new tensor categories and fusion rules. This talk is based on a recent joint work with T. Creutzig and N. Genra.