Ghost Sector of Vacuum String Field Theory and the Projection Equation

TL;DR

This paper analyzes the ghost sector of vacuum string field theory using a new basis, showing it reduces to a projection equation and can be reformulated as a $U( olinebreak) olinebreak( olinebreak) olinebreak ext{infinity}$ cubic matrix model, with solutions constructed from projection operators.

Contribution

It introduces a basis simplifying the ghost sector's BRST operator and demonstrates the reduction to a projection equation, connecting the ghost sector to a matrix model formulation.

Findings

Ghost sector reduces to a projection equation with a specific constraint.

Vacuum string field theory reformulated as a $U( ext{infinity})$ cubic matrix model.

Constructed ghost sector projectors analogous to matter sector states.

Abstract

We study the ghost sector of vacuum string field theory where the BRST operator Q is given by the midpoint insertion proposed by Gaiotto, Rastelli, Sen and Zwiebach. We introduce a convenient basis of half-string modes in terms of which Q takes a particularly simple form. We show that there exists a field redefinition which reduces the ghost sector field equation to a pure projection equation for string fields satisfying the constraint that the ghost number is equally divided over the left- and right halves of the string. When this constraint is imposed, vacuum string field theory can be reformulated as a $U(\infty)$ cubic matrix model. Ghost sector solutions can be constructed from projection operators on half-string Hilbert space just as in the matter sector. We construct the ghost sector equivalent of various well-known matter sector projectors such as the sliver, butterfly and nothing states.