Split string field theory I

We describe projection operators in the matter sector of Witten's cubic string field theory using modes on the right and left halves of the string. These projection operators represent a step towards an analytic solution of the equations of motion of the full string field theory, and can be used to construct Dp-brane solutions of the string field theory when the BRST operator Q is taken to be pure ghost, as suggested in the recent conjecture by Rastelli, Sen and Zwiebach. We show that a family of solutions related to the sliver state are rank one projection operators on the appropriate space of half-string functionals, and we construct higher rank projection operators corresponding to configurations of multiple D-branes.