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Higher spin Killing spinors on 3-dimensional manifolds

Yasushi Homma, Natsuki Imada, Soma Ohno

Abstract

We define higher spin Killing spinors on Riemannian spin manifolds in arbitrary dimension and study them in detail in dimension three. We prove a rigidity result for 3-dimensional manifolds admitting higher spin Killing spinors and give expressions for higher spin Killing spinors on the 3-sphere and the 3-hyperbolic space explicitly. We also investigate the Killing spinor type equation on integral spin bundles.

Higher spin Killing spinors on 3-dimensional manifolds

Abstract

We define higher spin Killing spinors on Riemannian spin manifolds in arbitrary dimension and study them in detail in dimension three. We prove a rigidity result for 3-dimensional manifolds admitting higher spin Killing spinors and give expressions for higher spin Killing spinors on the 3-sphere and the 3-hyperbolic space explicitly. We also investigate the Killing spinor type equation on integral spin bundles.
Paper Structure (15 sections, 33 theorems, 158 equations)

This paper contains 15 sections, 33 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Spin geometry with higher spin
  3. Higher spin Killing spinors
  4. Spin geometry with higher spin on 3-dimensional manifolds
  5. Weitzenböck formulas
  6. The basics on higher spin Killing spinors on 3-dimensional manifolds
  7. Cone construction
  8. Higher spin twistor spinors
  9. Eigenvalue estimate for the higher spin Dirac operator
  10. Higher spin Killing spinors on $\mathbb{S}^{3}$
  11. Higher spin Killing spinors on $\mathbb{H}^{3}$
  12. Killing spinor-type equation with integral spin
  13. Ingredients of the integral spin bundles
  14. Killing spinor equation on integral spin bundle
  15. Killing tensors on $\mathbb{S}^{3}$

Key Result

Proposition 2.2

Let $(\rho, W_{\rho})$ be an irreducible representation of $\mathrm{Spin}(n)$ and $W_{\rho} \otimes \mathbb{C}^n \cong \bigoplus_{\lambda} W_{\lambda}$ be the irreducible decomposition. The Clifford homomorphisms $p^{\rho}_{\lambda}(X) \colon W_{\rho} \to W_{\lambda}$ are defined by the orthogonal p Then, the Clifford homomorphisms satisfy the following relations: where $\{e_i\}_i$ is an orthonor

Theorems & Definitions (67)

  • Remark 2.1
  • Proposition 2.2: HT
  • proof
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • ...and 57 more