String geometry theory is a candidate for a non-perturbative formulation of string theory
based on the principle that not only particles but also spacetime itself are composed of
strings. Although the theory is defined by a path integral over string geometries, the
problem of non-renormalizability does not arise due to a non-renormalization theorem.
When the background is fixed to a perturbative vacuum of string theory, fluctuations around
it allow one to derive, to all orders, the path integrals of perturbative string theories on
arbitrary string backgrounds for all the Type I, Type II, and heterotic strings. On the other
hand, non-perturbative effects of the string coupling induce instanton transitions of the
vacuum from a local minimum of the potential to lower-energy minima, so that any state
eventually reaches the minimum of the potential of string geometry theory. This minimum
will correspond to the true vacuum of string theory, and the potential for string backgrounds
—obtained by restricting the potential of string geometry theory to perturbative vacua—is
expected to represent the string theory landscape.
By substituting a string phenomenological model with free parameters into the potential for
string backgrounds derived from string geometry theory, one obtains a potential for the free
parameters, and the free parameters are determined by the minimum of this potential. The
model with the determined parameters represents the ground state within the model. This
will correspond to a local minimum of the potential in the partial region of the string theory
landscape described by the model. By comparing this minimum with the minima of other
models, one could determine which model is closer to the true vacuum of string theory in
the sense of the value of the potential. Through this line of research, it is expected that the
true vacuum of string theory can be discovered. Since the derivation is based on the first
principle, even models that were previously discarded because of the hierarchy problem or
fine-tuning issues may become viable candidates. Furthermore, precise predictions for
experiments and observations can be made.
In this talk, we carry out this analysis explicitly for a simple non-supersymmetric heterotic
model. In this model, the six-dimensional internal space is given by a direct product of two-
dimensional constant-curvature spaces, and the number of fermion generations is
determined by the flux quantum numbers. As a result of the analysis, a constraint relation
between the compactification scale and the flux quantum numbers is obtained. For the
three-generation model, using this condition one can derive the assumption that is usually
imposed by hand—namely, that the string energy scale is higher than the compactification
energy scale, meaning that the compactification can be analyzed within supergravity.