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Perturbations of Dirac Operators

Steffen Schmidt

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.

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 -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.
Paper Structure (42 sections, 68 theorems, 278 equations, 1 table)

This paper contains 42 sections, 68 theorems, 278 equations, 1 table.

Table of Contents

  1. Introduction
  2. Vue d'Ensemble
  3. Dirac Operators and Lie Superalgebras
  4. Results
  5. Semisimple perturbations
  6. Nilpotent Perturbations
  7. Bismut--Quillen Superconnection
  8. Conventions and Notation
  9. Leitfaden
  10. Preliminaries
  11. Basic Classical Lie Superalgebra
  12. Highest Weight Supermodules
  13. Infinitesimal Characters and Atypicality
  14. Unitarizable Supermodules
  15. The Cubic Dirac Operator
  16. ...and 27 more sections

Key Result

Theorem 1

Let $L(\Lambda)$ be a finite-dimensional simple ${\mathfrak{g}}$-supermodule with highest weight $\Lambda$ such that $L(\Lambda)\vert_{{\mathfrak{g}_{\bar{0}}}}=\bigoplus_{\mu} L_{0}(\mu)^{n(\mu)}$. Then If $\xi=- \mu-\rho_{\bar{0}}$, then the kernel is $L(\Lambda)^{\mu}\otimes S^{\rho_{\bar{0}}} \otimes \overline{M}({\mathfrak{n}}_{\bar{1}}^{-})$ with $S := S^{{\mathfrak{g}}, {\mathfrak{n}}_{\ba

Theorems & Definitions (123)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 4
  • Theorem 5
  • Definition 6
  • Lemma 7
  • Definition 8
  • Definition 9
  • Remark 10
  • ...and 113 more