Dirac Operators, Dirac Cohomology and Unitarity for $A(m\vert n)$

TL;DR

This paper explores Dirac operators and cohomology in Lie superalgebras of type A, establishing their connection to unitarity, and provides explicit characterizations and applications for unitarizable supermodules.

Contribution

It systematically relates Dirac cohomology to unitarity in Lie superalgebras of type A, including explicit calculations and new characterizations.

Findings

Dirac cohomology determines unitarizable supermodules.

Established a Dirac inequality linking unitarity and Dirac operators.

Derived formulas for formal characters and introduced a Dirac index.

Abstract

Dirac operators and Dirac cohomology for Lie superalgebras of Riemannian type, introduced by Huang and Pand\v{z}i\'{c}, provide an effective tool for the study of unitarizable supermodules. In this article, we study these objects for Lie superalgebras of type $A$ and relate them systematically to unitarity. In the first part, we establish the basic structure of the theory in this setting. We relate unitarity to the Dirac operator, derive the corresponding Dirac inequality, and show that Dirac cohomology determines unitarizable supermodules. We also determine explicitly the Dirac cohomology of unitarizable simple supermodules. In the second part, we turn to applications. We obtain a new characterization of unitarity, establish a relation with Kostant's cohomology, derive a formula for formal characters, and introduce a Dirac index.