Homotopy Operators and One-Loop Vacuum Energy at the Tachyon Vacuum
  We construct the homotopy operators for the BRST operator in the theory around the identity-based solutions, which are believed to represent the tachyon vacuum in cubic bosonic open string field theory. Using the homotopy operators, we find that the one-loop vacuum energy at the tachyon vacuum is independent of moduli such as interbrane distances, which are included in the BRST operator. We also revisit the cohomology problem, which was solved earlier without the homotopy operators. 
Shoko Inatomi, Isao Kishimoto, Tomohiko Takahashi
hep-th