I understand the question as requesting applications of the geometry of spin manifolds outside of pure mathematics. There are many, starting with the Dirac equation for the spinor of a relativistic electron (1920). Recent applications center on topological states of matter and on quantum theories of gravity.

One recent overview is given in Spin Geometry and Some Applications.

For application beyond physics, see Spin Geometry and Image Processing.

I understand the question as requesting applications of the geometry of spin manifolds outside of pure mathematics. There are many, starting with the Dirac equation for the spinor of a relativistic electron (1920). Recent applications center on topological states of matter and on quantum theories of gravity.

One recent overview is given in Spin Geometry and Some Applications.

For an application beyond physics, see Spin Geometry and Image Processing.