Spinor bundle

In differential geometry, given a spin structure on an $$n$$-dimensional orientable 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 $$\rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n)$$ of the special orthogonal group by the spin group.

The spinor bundle $${\mathbf S}\,$$ is defined 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 $${\mathrm {Spin}}(n).$$