Clifford Algebroids and Nonholonomic Einstein–Dirac Structures

We propose a new framework for constructing geometric and physical models on spacetimes provided with Lie algebroid symmetry, i.e. manifolds provided with additional anchor and generalized Lie algebra commutator structures. The approach is related to the geometry of moving nonholonomic frames with associated nonlinear connections. A strict application of such geometric methods to spinor fields and Dirac operators motivates the theory of Clifford algebroids defined as Clifford bundles, in general, enabled with nonintegrable distributions. We elaborate the algebroid spinor differential geometry and formulate the (scalar, Proca, graviton, spinor and gauge) field equations on Lie algebroids.