Half-strings, Projectors, and Multiple D-branes in Vacuum String Field Theory

TL;DR

The paper demonstrates that the sliver state in vacuum string field theory acts as a rank-one projector in half-string space, with its left and right halves factorizing into a product of Gaussian functionals. Using an oscillator-based algebraic framework, it constructs higher-rank projectors that describe multiple D-$25$-branes and extends the construction to lower-dimensional branes, separated and intersecting brane configurations, and branes with relative motion. The results reveal a close parallel with non-commutative solitons and suggest a gauge-like structure for $*$-rotations, offering a concrete, algebraic route to multi-brane solutions in VSFT and insights into the role of half-strings in open string theory. Together, these findings provide a robust method for encoding complex D-brane configurations in the star-algebra language and open avenues for further exploration of Lorentz invariance, Chan-Paton factors, and K-theoretic interpretations in vacuum string field theory.

Abstract

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 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.