  The saddle points of a Lagrangian due to Efetov are analyzed. This Lagrangian
was originally proposed as a tool for calculating systematic corrections to the
Bethe approximation, a mean-field approximation which is important in
statistical mechanics, glasses, coding theory, and combinatorial optimization.
Detailed analysis shows that the trivial saddle point generates a sum over
geometries reminiscent of dynamically triangulated quantum gravity, which
suggests new possibilities to design sums over geometries for the specific
purpose of obtaining improved mean field approximations to $D$-dimensional
theories. In the case of the Efetov theory, the dominant geometries are locally
tree-like, and the sum over geometries diverges in a way that is similar to
quantum gravity's divergence when all topologies are included. Expertise from
the field of dynamically triangulated quantum gravity about sums over
geometries may be able to remedy these defects and fulfill the Efetov theory's
original promise. The other saddle points of the Efetov Lagrangian are also
analyzed; the Hessian at these points is nonnormal and pseudo-Hermitian, which
is unusual for bosonic theories. The standard formula for Gaussian integrals is
generalized to nonnormal kernels.
