BV Analysis of Tachyon Fluctuation around Multi-brane Solutions in Cubic String Field Theory

We study whether the tachyon mode exists as a physical fluctuation on the 2-brane solution and on the tachyon vacuum solution in cubic open string field theory. Our analysis is based on the Batalin-Vilkovisky formalism. We first construct a set of six string states which corresponds to the set of fields and anti-fields containing the tachyon field. Whether the tachyon field can exist as a physical fluctuation is determined by the 6x6 matrix defining the anti-bracket in the present sector. If the matrix is degenerate/non-degenerate, the tachyon field is physical/unphysical. Calculations for the pure-gauge type solutions in the framework of the KBc algebra and using the Ke-regularization lead to the expected results. Namely, the matrix for the anti-bracket is degenerate/non-degenerate in the case of the 2-brane/tachyon-vacuum solution. Our analysis is not complete, in particular, in that we have not identified the four-fold degeneracy of tachyon fluctuation on the 2-brane solution, and moreover that the present six states do not satisfy the hermiticity condition.