Spinorvalued

Spinorvalued refers to mathematical objects whose values lie in a spinor representation of the spin group, rather than in the real numbers or standard complex vector spaces. In differential geometry, a spinorvalued field on a manifold M is a section of the spinor bundle S(M) associated with a spin structure. Equivalently, at each point p, it assigns a spinor in the fiber S_p, a complex vector space carrying the spin representation of Spin(n) (or the appropriate Spin group for the metric).

Spinor representations arise from the Clifford algebra Cl_n and provide the fundamental carrier spaces for fermionic fields in physics. A spinorvalued function or field is thus a map f: M -> S that varies smoothly with p. The Dirac operator D acts on spinorvalued fields, producing another spinorvalued field; D is a first-order elliptic operator defined using the spin connection and the Clifford multiplication.

Spinorvalued forms can be defined by tensoring the spinor bundle with differential forms: S ⊗ Λ^k T^*M. This yields spinor-valued differential forms, useful in index theory, local index formulas, and supersymmetric theories. On Euclidean space R^n with its standard spin structure, spinor-valued fields reduce to functions from R^n to a fixed spinor representation space C^N.

Spinorvalued objects appear in both mathematics—global analysis, representation theory, and geometry—and physics, where they model fermionic fields. Related concepts include the spinor bundle, Dirac and Rarita–Schwinger operators, and Clifford algebras.