PUB - Publikationen an der Universität Bielefeld

In this work we take a look at lattice gauge theories with and without dynamical fermions in dimensions 0+1d and 1+1d, gauge groups U(1) and SU(3), complex coupling β and finite chemical potential. These suffer from a sign problem, which prohibits their simulation with traditional MCMC techniques. We alleviate this problem caused by complex actions using a generalization of Picard-Lefschetz theory for complex Morse functions with degenerate critical points i.e. critical submanifolds. The original integration domain is hereby embedded into complex space and a cell-complex homotopic to the original domain is constructed, where these oscillations are systematically reduced. These cells are called Lefschetz thimbles. We consider and evaluate different techniques for the construction and the sampling of this new integration domain. For the sake of performance and due to the fact, that the theories turn out to have a large number of contributing thimbles, we either triangulate the thimbles or even just approximate them by their tangent spaces at the critical points, which we limit for the sake of homotopy to tangential manifolds(TM). On these, the final simulations with parallel sampling of all TMs are done and evaluated. Dynamical fermions proof to be a challenge, since they add a rich fine structure of additional thimbles to the theory and parallel sampling becomes inpractical with rising volume. Therefore we need to consider methods, where only a subsample of all TMs are taken into account.