Lefschetz thimbles have been proposed recently as a possible solution to the complex action problem (sign problem) in Monte Carlo simulations. Here we discuss pure abelian gauge theory with a complex coupling $\beta$ and apply the concept of Generalized Lefschetz thimbles. We propose to simulate the theory on the union of the tangential manifolds to the thimbles.
We construct a local Metropolis-type algorithm, that is constrained to a specific tangential manifold. We also discuss how, starting from this result, successive subleading tangential manifolds can be taken into account via a reweighting approach. We demonstrate the algorithm on U(1) gauge theory in 1+1 dimensions and investigate the residual sign problem.