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$\text{Spin}^h$ Structure, Scalar and Charged Spinor Eigenfunctions on the $SU(3)/SO(3)$ Wu Manifold

Cameron Gibson, Okan Günel, Gabriel Larios, C. N. Pope

TL;DR

<3-5 sentence high-level summary>This paper investigates the Wu manifold SU(3)/SO(3) as a five-dimensional space that does not admit a conventional spin or spin^c structure but supports a spin^h structure. By coupling fermions to an SO(3) Yang–Mills gauge field and using a gauge-covariantly constant spinor, the authors show how the usual holonomy obstruction can be cancelled, enabling well-defined spin^h spinors. They construct explicit gauge-covariantly constant spinors, derive the full spectrum of scalar and spin^h harmonics, and present a detailed analysis of the Dirac operator spectrum in terms of scalar harmonics, including the flux quantisation on RP^2 and global holonomy arguments. The work parallels the CP^2 spin^c construction, provides concrete spin^h realizations, and lays groundwork for subsequent dimensional reductions on Wu, including non-compact duals and connections to toroidal supergravity reductions.

Abstract

Generalised spin structures are necessary for placing fermions on manifolds that do not admit a standard spin structure. This is especially relevant in a dimensional reduction on such a manifold, which can then be compensated by using fermions that are appropriately charged under some Maxwell or Yang-Mills field defined on the internal manifold. A well known example in the physics literature is $\mathbb{CP}^2$, which has four real dimensions and is the coset $SU(3)/U(2)$. In this paper we focus on a five-dimensional coset space, namely the Wu manifold $SU(3)/SO(3)_{\rm max}$, where $SO(3)_{\rm max}$ is maximal in $SU(3)$. Intriguingly, the Wu manifold does not admit a spin structure or spin$^c$ structure, it does admit a spin$^h$ structure. We provide a physical interpretation of the spin$^h$ structure by considering spinors that are coupled to an $SO(3)$ Yang-Mills field defined on the Wu manifold, but which carry half-integer "isospin," thereby canceling the minus sign in the holonomy for uncharged spinors that provides the original obstruction to an ordinary spin structure. We also construct a gauge-covariantly constant spinor in the Wu manifold, and we show how this can be employed in order to construct spin$^h$ spinor harmonics from scalar harmonics. We provide a very explicit construction of all the scalar and spin$^h$ harmonics. In a follow-up paper, we shall employ the results we obtain here in order to discuss dimensional reductions and consistent reductions on the Wu manifold.

$\text{Spin}^h$ Structure, Scalar and Charged Spinor Eigenfunctions on the $SU(3)/SO(3)$ Wu Manifold

TL;DR

<3-5 sentence high-level summary>This paper investigates the Wu manifold SU(3)/SO(3) as a five-dimensional space that does not admit a conventional spin or spin^c structure but supports a spin^h structure. By coupling fermions to an SO(3) Yang–Mills gauge field and using a gauge-covariantly constant spinor, the authors show how the usual holonomy obstruction can be cancelled, enabling well-defined spin^h spinors. They construct explicit gauge-covariantly constant spinors, derive the full spectrum of scalar and spin^h harmonics, and present a detailed analysis of the Dirac operator spectrum in terms of scalar harmonics, including the flux quantisation on RP^2 and global holonomy arguments. The work parallels the CP^2 spin^c construction, provides concrete spin^h realizations, and lays groundwork for subsequent dimensional reductions on Wu, including non-compact duals and connections to toroidal supergravity reductions.

Abstract

Generalised spin structures are necessary for placing fermions on manifolds that do not admit a standard spin structure. This is especially relevant in a dimensional reduction on such a manifold, which can then be compensated by using fermions that are appropriately charged under some Maxwell or Yang-Mills field defined on the internal manifold. A well known example in the physics literature is , which has four real dimensions and is the coset . In this paper we focus on a five-dimensional coset space, namely the Wu manifold , where is maximal in . Intriguingly, the Wu manifold does not admit a spin structure or spin structure, it does admit a spin structure. We provide a physical interpretation of the spin structure by considering spinors that are coupled to an Yang-Mills field defined on the Wu manifold, but which carry half-integer "isospin," thereby canceling the minus sign in the holonomy for uncharged spinors that provides the original obstruction to an ordinary spin structure. We also construct a gauge-covariantly constant spinor in the Wu manifold, and we show how this can be employed in order to construct spin spinor harmonics from scalar harmonics. We provide a very explicit construction of all the scalar and spin harmonics. In a follow-up paper, we shall employ the results we obtain here in order to discuss dimensional reductions and consistent reductions on the Wu manifold.
Paper Structure (56 sections, 294 equations, 1 figure)

This paper contains 56 sections, 294 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Wu manifold
  3. A parameterisation of $SU(3)$
  4. Local properties of the Wu metric
  5. Global aspects of the Wu manifold
  6. $\text{Spin}^c$ and spin$^h$ structures on ${{\mathbb C}{\mathbb P}}^2$
  7. The Fubini-Study metric on ${{\mathbb C}{\mathbb P}}^2$ in real coordinates
  8. Spinors in ${{\mathbb C}{\mathbb P}}^2$
  9. $U(1)$ gauge-covariantly constant spinors on ${{\mathbb C}{\mathbb P}}^2$
  10. $SU(2)$ gauge-covariantly constant spinors on ${{\mathbb C}{\mathbb P}}^2$
  11. Spin$^h$ spinors on ${{\mathbb C}{\mathbb P}}^2$
  12. Spin h structure on the Wu manifold
  13. Holonomy
  14. Vectors
  15. Uncharged spinors
  16. ...and 41 more sections

Figures (1)

  • Figure 1: The fundamental domains in the $(\mu,\nu)$ plane for (a) $SU(3)$ and (b) $SL(3,{\mathbb R})$ are highlighted in bold. The images of these domains under conjugation of a central element by $\omega_i$ are then plotted.