spinorielle

Spinorielle is a term used in a hypothetical mathematical framework to denote a class of spinor-like fields that carry additional internal indices or grading. It generalizes the notion of a spinor field by coupling it to an auxiliary vector or gauge bundle, yielding a section of a twisted spinor bundle. In this setting, a spinorielle field combines the transformation properties of a spinor under the local Clifford action with an extra representation of an internal symmetry group.

Formally, let M be a smooth, oriented Riemannian or pseudo-Riemannian manifold with a spin structure. A spinorielle bundle E is a vector bundle associated to the principal Spin(n) bundle via a representation that pairs the spin representation with an internal representation ρI of a group G. A spinorielle field is a smooth section ψ of E. Local expressions transform under Spin(n) as ordinary spinors, together with the action of G on the internal indices.

Differential structure on spinorielle fields is provided by a compatible connection. A spinorielle connection ∇ combines the Levi-Civita (or spin) connection with an internal gauge connection, allowing one to define a Dirac-type operator D acting on sections of E. One has a Weitzenböck-type formula D² = ∇*∇ + 1/4 R + F, where R is the scalar curvature and F encodes the curvature of the internal bundle. This framework enables twisting of spinor fields by additional bundles, affecting indices and spectral properties.

Applications of the spinorielle concept appear in theoretical explorations of fermions in curved spaces with extra internal symmetries and in twisting techniques within index theory. When the internal representation is trivial, spinorielle reduces to the ordinary spinor field. See also spinor, Dirac operator, twisting, and Clifford algebra.