Half-strings, projectors, and multiple D-branes in vacuum string field theory

A sliver state is a classical solution of the string field theory of the tachyon vacuum that represents a background with a single D25-brane. We show that the sliver wavefunctional factors into functionals of the left and right halves of the string, and hence can be naturally regarded as a rank-one projector in a space of half-string functionals. By developing an algebraic oscillator approach we are able to construct higher rank projectors that describe configurations of multiple D-branes of various dimensionalities and located at arbitrary positions. The results bear remarkable similarities with non-commutative solitons.