  We propose a novel simulation strategy for Yang-Mills theories with a complex
coupling, based on the Lefschetz thimble decomposition. We envisage, that the
approach developed in the present work, can also be adapted to QCD at finite
density, and real time simulations.
  Simulations with Lefschetz thimbles offer a potential solution to sign
problems in Monte Carlo calculations within many different models with complex
actions. We discuss the structure of Generalized Lefschetz thimbles for pure
Yang-Mills theories with a complex gauge coupling $\beta$ and show how to
incorporate the gauge orbits. We propose to simulate such theories on the union
of the tangential manifolds to the relevant Lefschetz thimbles attached to the
critical manifolds of the Yang-Mills action. We demonstrate our algorithm on a
(1+1)-dimensional U(1) model and discuss how, starting from the main thimble
result, successive subleading thimbles can be taken into account via a
reweighting approach. While we face a residual sign problem, our novel approach
performs exponentially better than the standard reweighting approach.
