  For the construction of the Lorentzian path integral for gravity one faces
two main questions: Firstly, what configurations to include, in particular
whether to allow Lorentzian metrics that violate causality conditions. And
secondly, how to evaluate a highly oscillatory path integral over unbounded
domains. Relying on Picard-Lefschetz theory to address the second question for
discrete Regge gravity, we will illustrate that it can also answer the first
question. To this end we will define the Regge action for complexified
variables and study its analytical continuation. Although there have been
previously two different versions defined for the Lorentzian Regge action, we
will show that the complex action is unique. More precisely, starting from the
different definitions for the action one arrives at equivalent analytical
extensions. The difference between the two Lorentzian versions is only realized
along branch cuts which arise for a certain class of causality violating
configurations.
  As an application we discuss the path integral describing a finite evolution
step of the discretized deSitter universe. We will in particular consider an
evolution from vanishing to finite scale factor, for which the path integral
defines the no-boundary wave function.
