Dirac Cohomology and Unitarizable Supermodules over Lie Superalgebras of Type $A(m\vert n)$

Abstract: Dirac operators and Dirac cohomology for Lie superalgebras of Riemannian type, as introduced by Huang and Pand\v{z}i\'{c}, serve as a powerful framework for studying unitarizable supermodules. This paper explores the relationships among the Dirac operator, Dirac cohomology, and unitarizable supermodules specifically within the context of the basic classical Lie superalgebras of type $A(m\vert n)$. The first part examines the structural properties of Dirac cohomology and unitarizable supermodules, including how the Dirac operator captures unitarity, a Dirac inequality, and the uniqueness of the supermodule determined by its Dirac cohomology. Additionally, we calculate the Dirac cohomology for unitarizable simple supermodules. The second part focuses on applications: we give a novel characterization of unitarity, relate Dirac cohomology to nilpotent Lie superalgebra cohomology, derive a decomposition of formal characters, and introduce a Dirac index.