  At finite density, lattice simulations are hindered by the well-known sign
problem: for finite chemical potentials, the QCD action becomes complex and the
Boltzmann weight $e^{-S}$ cannot be interpreted as a probability distribution
to determine expectation values by Monte Carlo techniques. Different
workarounds have been devised to study the QCD phase diagram, but their
application is mostly limited to the region of small chemical potentials. The
Lefschetz thimbles method takes a new approach in which one complexifies the
theory and deforms the integration paths. By integrating over Lefschetz
thimbles, the imaginary part of the action is kept constant and can be factored
out, while $e^{-Re(S)}$ can be interpreted as a probability measure. The method
has been applied in recent years to more or less difficult problems. Here we
report preliminary results on Lefschetz thimbles regularization of heavy-dense
QCD. While still simple, this is a very interesting problem. It is a first look
at thimbles for QCD, although in a simplified, effective version. From an
algorithmic point of view, it is a nice ground to test effectiveness of
techniques we developed for multi thimbles simulations.
