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Generalised Killing Spinors on Three-Dimensional Lie Groups

Diego Artacho

TL;DR

This work addresses the problem of classifying invariant generalised Killing spinors on connected three-dimensional Lie groups endowed with left-invariant metrics. It develops a differential-forms realization of the spin representation and uses the Nomizu map to translate the GK condition into an algebraic endomorphism $A$ on the Lie algebra, revealing that the GK property for invariant spinors is governed by a single $A$. The authors prove a striking rigidity: the existence of a nontrivial invariant GK spinor implies that all invariant spinors are GK with the same endomorphism, and this conclusion is independent of the chosen left-invariant metric. They provide a complete Bianchi-type–based classification, give an explicit expression for $A$ in terms of structure constants, and show that $A$ commutes with the Ricci endomorphism when symmetric; they also present the first homogeneous examples of invariant GK spinors with arbitrarily many distinct eigenvalues on Heisenberg groups, illustrating the method and diversity of GK spectra on homogeneous spaces.

Abstract

We present a complete classification of invariant generalised Killing spinors on three-dimensional Lie groups. We show that, in this context, the existence of a non-trivial invariant generalised Killing spinor implies that all invariant spinors are generalised Killing with the same endomorphism. Notably, this classification is independent of the choice of left-invariant metric. To illustrate the computational methods underlying this classification, we also provide the first known examples of homogeneous manifolds admitting invariant generalised Killing spinors with $n$ distinct eigenvalues for each $n > 4$.

Generalised Killing Spinors on Three-Dimensional Lie Groups

TL;DR

This work addresses the problem of classifying invariant generalised Killing spinors on connected three-dimensional Lie groups endowed with left-invariant metrics. It develops a differential-forms realization of the spin representation and uses the Nomizu map to translate the GK condition into an algebraic endomorphism on the Lie algebra, revealing that the GK property for invariant spinors is governed by a single . The authors prove a striking rigidity: the existence of a nontrivial invariant GK spinor implies that all invariant spinors are GK with the same endomorphism, and this conclusion is independent of the chosen left-invariant metric. They provide a complete Bianchi-type–based classification, give an explicit expression for in terms of structure constants, and show that commutes with the Ricci endomorphism when symmetric; they also present the first homogeneous examples of invariant GK spinors with arbitrarily many distinct eigenvalues on Heisenberg groups, illustrating the method and diversity of GK spectra on homogeneous spaces.

Abstract

We present a complete classification of invariant generalised Killing spinors on three-dimensional Lie groups. We show that, in this context, the existence of a non-trivial invariant generalised Killing spinor implies that all invariant spinors are generalised Killing with the same endomorphism. Notably, this classification is independent of the choice of left-invariant metric. To illustrate the computational methods underlying this classification, we also provide the first known examples of homogeneous manifolds admitting invariant generalised Killing spinors with distinct eigenvalues for each .
Paper Structure (16 sections, 7 theorems, 62 equations, 2 tables)

This paper contains 16 sections, 7 theorems, 62 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential forms approach to the spin representation
  4. Invariant connections and Nomizu maps
  5. The Heisenberg groups
  6. Three-dimensional Lie groups
  7. Explicit matrices
  8. $\mathrm{Bian}(\text{I})$
  9. $\mathrm{Bian}(\text{II})$
  10. $\mathrm{Bian}(\text{III})$
  11. $\mathrm{Bian}(\text{IV})$
  12. $\mathrm{Bian}(\text{V})$
  13. $\mathrm{Bian}(\text{VI}_{x})$, $-1 \leq x < 1$, $x \neq 0$
  14. $\mathrm{Bian}(\text{VII}_{y})$, $y \geq 0$
  15. $\mathrm{Bian}(\text{VIII})$
  16. ...and 1 more sections

Key Result

Theorem 1

For each $n \in \mathbb{N}$, the $n$-th real Heisenberg group $H_{2n+1}$ has a left-invariant metric which carries an invariant generalised Killing spinor with $(n+1)$ distinct eigenvalues.

Theorems & Definitions (15)

  • Theorem
  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 5 more