Solving Open String Field Theory with Special Projectors

TL;DR

This paper introduces a general class of special projectors in open string field theory, enabling exact solutions to string field equations through a novel algebraic and differential equation framework.

Contribution

It generalizes Schnabl's analytic solution by defining special projectors with specific algebraic properties, leading to a new method for solving string field equations.

Findings

Special projectors form abelian subalgebras of string fields.

Exact solutions are obtained by recasting equations as differential equations.

Classification of special projectors relates to a Riemann-Hilbert problem.

Abstract

Schnabl recently found an analytic expression for the string field tachyon condensate using a gauge condition adapted to the conformal frame of the sliver projector. We propose that this construction is more general. The sliver is an example of a special projector, a projector such that the Virasoro operator \L_0 and its BPZ adjoint \L*_0 obey the algebra [\L_0, \L*_0] = s (\L_0 + \L*_0), with s a positive real constant. All special projectors provide abelian subalgebras of string fields, closed under both the *-product and the action of \L_0. This structure guarantees exact solvability of a ghost number zero string field equation. We recast this infinite recursive set of equations as an ordinary differential equation that is easily solved. The classification of special projectors is reduced to a version of the Riemann-Hilbert problem, with piecewise constant data on the boundary of a disk.