Spin Cohomology

We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ = 0, and the algebraic operators D(p) are constructed from skew-products of p gamma matrices. We exhibit a large number of spin cohomology operators and we investigate the spin cohomologies associated with connections whose holonomy is a subgroup of SU(m), G2, Spin(7) and Sp(2). In the SU(m) case, we find that the spin cohomology of complex spin and spinc manifolds is related to a twisted Dolbeault cohomology. On Calabi-Yau type of manifolds of dimension 8k + 6, a spin cohomology can be defined on a twisted complex with operator d+D which is the sum of a differential and algebraic one. We compute this cohomology on six-dimensional Calabi-Yau manifolds using a spectral sequence. In the G2 and Spin(7) cases, the spin cohomology is related to the de Rham cohomology.