# Spinor bundle

In differential geometry, given a spin structure on a $n$-dimensional Riemannian manifold $(M, g),\,$ one defines the spinor bundle to be the complex vector bundle $\pi_{\mathbf S}\colon{\mathbf S}\to M\,$ associated to the corresponding principal bundle $\pi_{\mathbf P}\colon{\mathbf P}\to M\,$ of spin frames over $M$ and the spin representation of its structure group ${\mathrm {Spin}}(n)\,$ on the space of spinors $\Delta_n.\,$.

A section of the spinor bundle ${\mathbf S}\,$ is called a spinor field.

## Formal definition

Let $({\mathbf P},F_{\mathbf P})$ be a spin structure on a Riemannian manifold $(M, g),\,$ that is, an equivariant lift of the oriented orthonormal frame bundle $\mathrm F_{SO}(M)\to M$ with respect to the double covering $\rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n).\,$

The spinor bundle ${\mathbf S}\,$ is defined [1] to be the complex vector bundle

${\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\,$

associated to the spin structure ${\mathbf P}$ via the spin representation $\kappa\colon {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\,$ where ${\mathrm U}({\mathbf W})\,$ denotes the group of unitary operators acting on a Hilbert space ${\mathbf W}.\,$ It is worth noting that the spin representation $\kappa$ is a faithful and unitary representation of the group ${\mathrm {Spin}}(n)$.[2]