  Finite-density calculations in lattice field theory are typically plagued by
sign problems. A promising way to ameliorate this issue is the holomorphic flow
equations that deform the manifold of integration for the path integral to
manifolds in the complex space where the sign fluctuations are less dramatic.
We discuss some novel features of applying the flow equations to gauge theories
and present results for finite-density $QED_{1+1}$.
