Vacuum String Field Theory

TL;DR

Vacuum string field theory (VSFT) provides a simplified open string field theory expanded around the tachyon vacuum, using a universal ghost-dominated kinetic operator ${\cal Q}$ and a star product that enables analytic construction of D-brane solutions. The matter sector equations reduce to a projector condition under $*$, with the sliver state playing a central role as a rank-one projector; D-branes arise as deformations of the sliver either in a geometric/boundary-CFT sense or in an algebraic half-string formalism. The two complementary approaches—boundary CFT deformations and half-string projector construction—yield consistent D-brane tensions via disk partition functions and enable the explicit assembly of multi-brane configurations, including coincident and distinct branes, through orthogonal projectors. The work demonstrates a remarkably tractable framework for nonperturbative open-string physics and offers insights toward background independence, gauge structure, and potential links to closed strings and K-theory, while leaving key questions about the ghost sector and full quantum consistency open.

Abstract

This is a brief review of vacuum string field theory, a new approach to open string field theory based on the stable vacuum of the tachyon. We discuss the sliver state explaining its role as a projector in the space of half-string functionals. We review the construction of D-brane solutions in vacuum string field theory, both in the algebraic approach and in the more general geometrical approach that emphasizes the role of boundary CFT. -- To appear in the Proceedings of Strings 2001, Mumbai, India.

Figures (3)

• Figure 1: A punctured disk $D$ with a local coordinate around the puncture $P$. The coordinate is defined via a map from the half-disk $H_U$ to $D$. The arcs $AM$ and $MB$ in $D$ represent the left half and the right half of the open string respectively.
• Figure 2: (a) The surface state corresponding to the identity string field $\langle{\cal I}|$. Here the image of $H_U$ covers the full disk, except for a cut in the negative real axis. (b) The surfaces state corresponding to the sliver $\langle \Xi|$. Here the image of $H_U$ covers an infinitesimally thin sliver around the positive real axis.
• Figure 3: The finite $n$ approximation to the sliver presented in the $\widehat{z}_n$ coordinate.