String field theory around the Tachyon vacuum

Assuming that around the tachyon vacuum the kinetic term of cubic open string field theory is made purely of ghost operators we are led to gauge invariant actions which manifestly implement the absence of open string dynamics around this vacuum. We test this proposal by showing the existence of lump solutions of arbitrary codimension in this string field theory. The key ingredients in this analysis are certain assumptions about the analyticity properties of tachyon Green's functions. With the help of some further assumptions about the properties of these Green's functions, we also calculate the ratios of tensions of lump solutions of different dimensions. The result is in perfect agreement with the known answers for the ratios of tensions of D-branes of different dimensions.