Spinors are mathematical objects that are used to describe the state of particles with spin in quantum mechanics. They provide a way to represent half-integer spin representations of the rotation group, allowing physicists to work with particles like electrons and quarks that exhibit unique behaviors under rotation. In the context of the Atiyah-Singer index theorem, spinors are important as they relate to the analysis of differential operators on manifolds and can be crucial for understanding topological properties of vector bundles.
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