BRST Invariance of the Non-Perturbative Vacuum in Bosonic Open String Field Theory

TL;DR

The paper addresses whether the tachyon-condensed non-perturbative vacuum in bosonic open string field theory is BRST invariant, a prerequisite for a consistent quantum theory around the vacuum. It implements a numerical BRST test within the level-truncated framework, computing the BRST variation $\delta_B|\phi_c\rangle$ for the true vacuum and demonstrating systematic cancellations between linear and quadratic terms, with concrete evidence at multiple truncation levels (e.g., $\delta_B\bar{C}_2$ approaching zero at level two and further cancellations at level four). The key contributions are the demonstration that the true non-perturbative vacuum is BRST invariant to high accuracy, the identification of a BRST-symmetry breaking for a fake vacuum, and the observation of a zero-norm pattern in the truncated solution that may guide analytic reconstruction. Collectively, these results support the physical consistency of tachyon condensation in this framework and provide structural hints toward an analytic expression for the vacuum state.

Abstract

Tachyon condensation on a bosonic D-brane was recently demonstrated numerically in Witten's open string field theory with level truncation approximation. This non-perturbative vacuum, which is obtained by solving the equation of motion, has to satisfy furthermore the requirement of BRST invariance. This is indispensable in order for the theory around the non-perturbative vacuum to be consistent. We carry out the numerical analysis of the BRST invariance of the solution and find that it holds to a good accuracy. We also mention the zero-norm property of the solution. The observations in this paper are expected to give clues to the analytic expression of the vacuum solution.