  String geometry theory is one of the candidates of the non-perturbative
formulation of string theory. In this paper, in the bosonic sector of string
geometry theory, we identify perturbative vacua in bosonic string theory, which
include general string backgrounds. From fluctuations around these vacua, we
derive the path-integrals of perturbative strings on the string backgrounds up
to any order. Furthermore, the identification of the perturbative vacua enables
us to obtain differential equations that determine an effective potential for
string backgrounds. We solve the differential equations and obtain an effective
potential explicitly up to the second order. In a generic region, we show that
the minimum of the second order potential gives a non-trivial background. This
fact supports that the full effective potential can determine the true vacuum
in string theory.
