Perturbations of Dirac Operators

TL;DR

This paper explores various perturbations of Dirac operators associated with Lie superalgebras, introducing new invariants and cohomology theories through a unified algebraic framework.

Contribution

It develops three classes of Dirac operator perturbations within the colour quantum Weil algebra, leading to novel invariants and cohomology theories for Lie superalgebra modules.

Findings

Defined semisimple perturbations and associated invariants.

Introduced nilpotent perturbations linking Dirac and Duflo--Serganova cohomologies.

Constructed a Chern-type invariant via Weil algebra deformations.

Abstract

We study perturbations of relative cubic Dirac operators for basic classical Lie superalgebras within the uniform formalism of the colour quantum Weil algebra. This perspective leads to three complementary classes of perturbations and resulting invariants. First, we define semisimple perturbations that assign to each finite-dimensional simple supermodule a finite collection of semisimple orbits, together with canonically defined vector spaces measuring the degree of atypicality. Second, we introduce nilpotent perturbations parametrized by the self-commuting variety of a quadratic Lie subsuperalgebra; the resulting family of cohomology theories combines Dirac cohomology and Duflo--Serganova cohomology. Third, we deform the cubic Dirac operator by a Weil-covariant differential built from the universal $1$-form in the colour quantum Weil algebra and the Weil differential, producing a Chern-type invariant that assigns to each finite-dimensional module a natural class in the cohomology of the Weil complex.