  Lattice formulations of gravity can be used to study non-perturbative aspects
of quantum gravity. Causal Dynamical Triangulations (CDT) is a lattice model of
gravity that has been used in this way. It has a built-in time foliation but is
coordinate-independent in the spatial directions. The higher-order phase
transitions observed in the model may be used to define a continuum limit of
the lattice theory. Some aspects of the transitions are better studied when the
topology of space is toroidal rather than spherical. In addition, a toroidal
spatial topology allows us to understand more easily the nature of typical
quantum fluctuations of the geometry. In particular, this topology makes it
possible to use massless scalar fields that are solutions to Laplace's equation
with special boundary conditions as coordinates that capture the fractal
structure of the quantum geometry. When such scalar fields are included as
dynamical fields in the path integral, they can have a dramatic effect on the
geometry.
