  We consider the holographic complexity conjectures in the context of the AdS
soliton, which is the holographic dual of the ground state of a field theory on
a torus with antiperiodic boundary conditions for fermions on one cycle. The
complexity is a non-trivial function of the size of the circle with
antiperiodic boundary conditions, which sets an IR scale in the dual geometry.
We find qualitative differences between the calculations of complexity from
spatial volume and action (CV and CA). In the CV calculation, the complexity
for antiperiodic boundary conditions is smaller than for periodic, and
decreases monotonically with increasing IR scale. In the CA calculation, the
complexity for antiperiodic boundary conditions is larger than for periodic,
and initially increases with increasing IR scale, eventually decreasing to zero
as the IR scale becomes of order the UV cutoff. We compare these results to a
simple calculation for free fermions on a lattice, where we find the complexity
for antiperiodic boundary conditions is larger than for periodic.
