Exploring the vacuum structure near identity-based solutions

We explore the vacuum structure in the bosonic open string field theory expanded near an identity-based solution parameterized by $a$ ($\ge-1/2$). Analyzing the expanded theory using the level-truncation approximation up to the level $20$, we find that the theory has the tachyon vacuum solution for $a>-1/2$. We also find that at $a=-1/2$, there exists an unstable vacuum solution in the expanded theory and the solution is expected to be the perturbative open string vacuum. These results reasonably support the hypothesis that the identity-based solution is a trivial pure gauge configuration for $a>-1/2$, but it can be regarded as the tachyon vacuum solution at $a=-1/2$.