Perturbations of Dirac Operators

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.