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.Comment: 30 pages, late

The super-correlator/super-amplitude duality: Part I

We extend the recently discovered duality between MHV amplitudes and the light-cone limit of correlation functions of a particular type of local scalar operators to generic non-MHV amplitudes in planar N=4 SYM theory. We consider the natural generalization of the bosonic correlators to super-correlators of stress-tensor multiplets and show, in a number of examples, that their light-cone limit exactly reproduces the square of the matching super-amplitudes. Our correlators are computed at Born level. If all of their points form a light-like polygon, the correlator is dual to the tree-level amplitude. If a subset of points are not on the polygon but are integrated over, they become Lagrangian insertions generating the loop corrections to the correlator. In this case the duality with amplitudes holds at the level of the integrand. We build up the superspace formalism needed to formulate the duality and present the explicit example of the n-point NMHV tree amplitude as the dual of the lowest nilpotent level in the correlator.Comment: 56 page