Spin cohomology

Abstract

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 S U ( m ) , G 2 , S p i n ( 7 ) and S p ( 2 ) . In the S U ( m ) case, we find that the spin cohomology of complex spin and s p i n c manifolds is related to a twisted Dolbeault cohomology. On Calabi-Yau type of manifolds of dimension 8 k + 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 S p i n ( 7 ) cases, the spin cohomology is related to the de Rham cohomology.