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 $\nabla$ and a parallel spinor $\zeta$, $\nabla\zeta=0$, 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 $SU(m)$, $G_2$, $Spin(7)$ and $Sp(2)$. In the $SU(m)$ case, we findthat the spin cohomology of complex spin and spin$_c$ 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 $G_2$ and $Spin(7)$ cases, the spin cohomology is related to the de Rham cohomology.