Inversion Symmetry of Gravitational Coupling in Cubic String Field Theory

It was found that the canonical energy of multi-brane solutions in CSFT constructed by the KBc algebra has a symmetry under the exchange of K=0 and K=\infty (inversion symmetry). On the other hand, the gauge invariant observable (GIO), which is regarded as the energy defined by the gravitational coupling of open string, cannot count the energy from K=\infty and therefore is not equal to the canonical energy. To resolve this discrepancy, we examine the recent argument of Baba and Ishibashi which directly relates the two energies. We find that the gravitational coupling which is equivalent to the canonical energy consists of the GIO and another new term, and the whole has the inversion symmetry.