Thimble regularization at work for Gauge Theories: from toy models onwards

A final goal for thimble regularization of lattice field theories is the application to lattice QCD and the study of its phase diagram. Gauge theories pose a number of conceptual and algorithmic problems, some of which can be addressed even in the framework of toy models. We report on our progresses in this field, starting in particular from first successes in the study of one link models.