Quasi-integrable modules over affine Lie superalgebras (Critical level)

Representation theory of Lie (super)algebras has attracted significant research interest for many years, especially due to its applications in theoretical physics; in this regard, the representation theory of affine Lie (super)algebras is of central importance. To characterize simple modules over affine Lie (super)algebras, it is necessary to study the cases of nonzero and critical levels separately. Although a vast amount of research has been done on the representation theory of affine Lie (super)algebras $\mathcal{L}$, investigations concerning general modules at the critical level remain limited. In all existing studies, the characterization of the modules under consideration is reduced to the characterization of modules over some subalgebras of $\mathcal{L}$. Depending on the structure of the original modules, these subalgebras -- and the corresponding modules -- have different natures some of which are already known, while others need to be studied separately. In this paper, we give a complete characterization of the modules over specific subalgebras $\mathcal{G}$ of a twisted affine Lie superalgebra $\mathcal{L}$ that arise in the study of general zero level simple finite weight $\mathcal{L}$-modules. In particular, in the special case that $\\mathcal{G} = \mathcal{L}$, we obtain a complete characterization of quasi-integrable $\mathcal{L}$-modules of level zero.