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A Fueter operator for 3/2-spinors

Ahmad Reza Haj Saeedi Sadegh, Minh Lam Nguyen

TL;DR

This work investigates the non-compactness of moduli spaces for generalized Seiberg–Witten-type equations with $3/2$-spinors on closed 3-manifolds by introducing a 3/2-Fueter operator $\mathfrak{Q}$. The authors construct an aquaternionic moduli space $W_0$, a 4-dimensional fiber over the base, via hyperkähler reduction, and define $\mathfrak{Q}$ as the projection of the Fueter operator to a natural normal bundle. A Haydys-type correspondence is established: solutions to the degenerate RS-SW equations are in one-to-one correspondence with solutions of $\mathfrak{Q}$, yielding a nonlinear, overdetermined elliptic operator. The overdetermination implies potential obstructions to compactness in certain geometries and points toward new 3-manifold invariants derived from RS-type operators, with broader implications for gauged-sigma models and hyperkähler geometry.

Abstract

We show the non-compactness of moduli space of solutions of the monopole equations for $3/2$-spinors on a closed 3-manifold is equivalent to the existence of `3/2-Fueter sections' that are solutions of an overdetermined non-linear elliptic differential equation. These are sections of a fiber bundle whose fiber is a special 4-dimensional submanifold of the hyperkähler manifold of center-framed charged one $SU(2)$-instantons on $\mathbf{R}^4$. This fiber bundle does not inherit a hyperkähler structure.

A Fueter operator for 3/2-spinors

TL;DR

This work investigates the non-compactness of moduli spaces for generalized Seiberg–Witten-type equations with -spinors on closed 3-manifolds by introducing a 3/2-Fueter operator . The authors construct an aquaternionic moduli space , a 4-dimensional fiber over the base, via hyperkähler reduction, and define as the projection of the Fueter operator to a natural normal bundle. A Haydys-type correspondence is established: solutions to the degenerate RS-SW equations are in one-to-one correspondence with solutions of , yielding a nonlinear, overdetermined elliptic operator. The overdetermination implies potential obstructions to compactness in certain geometries and points toward new 3-manifold invariants derived from RS-type operators, with broader implications for gauged-sigma models and hyperkähler geometry.

Abstract

We show the non-compactness of moduli space of solutions of the monopole equations for -spinors on a closed 3-manifold is equivalent to the existence of `3/2-Fueter sections' that are solutions of an overdetermined non-linear elliptic differential equation. These are sections of a fiber bundle whose fiber is a special 4-dimensional submanifold of the hyperkähler manifold of center-framed charged one -instantons on . This fiber bundle does not inherit a hyperkähler structure.
Paper Structure (17 sections, 10 theorems, 113 equations)

This paper contains 17 sections, 10 theorems, 113 equations.

Table of Contents

  1. Preliminaries
  2. Connections on fiber bundles
  3. 3/2-spinors
  4. Hyperkähler manifolds
  5. Hyperkähler reduction
  6. Hyperkähler bundles
  7. The Fueter operator
  8. An aquaternionic moduli space
  9. $\text{SO}(3)\times\text{SO}(3)$ action
  10. An aquaternionic quotient
  11. The tangent bundle $TW^{\mu}$
  12. A normal line bundle on $W^{\mu}$
  13. Two examples of hyperkähler bundles
  14. Special subbundles of hyperkähler bundles
  15. The 3/2-Fueter operator
  16. ...and 2 more sections

Key Result

Theorem 1

(ref. Theorem th:maintheorem and Theorem th:overdetermination) Any solution $(A,\Phi)$ of eq:degenerate gives a solution $\Phi_0\in\Gamma(\mathbb{W}_0)$ of the $3/2$-Fueter equation Conversely, for any solution $\Phi_0$ of eq:3:2Fueterequation, there exists a $\text{Spin}^{c}(3)$-structure $P_{\text{Spin}^{c}}\to Y$ with a connection $A$ on its determinant line bundle, a section $\Phi\in \Gamma(\

Theorems & Definitions (31)

  • Remark 1
  • Theorem
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 1.1
  • Proposition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • ...and 21 more