This lecture, part of the "New trends in Teichmüller theory" program at the International Centre for Theoretical Sciences, explores canonical bases for moduli spaces of local systems related to Higher Teichmüller theory. Delivered by Hyun Kyu Kim, the 71-minute talk contributes to a two-week thematic program focusing on recent developments in Teichmüller theory, including Higher Teichmüller theory and its connections with Higgs bundles, algebraic geometry, and Anosov representations. Learn about cutting-edge research that draws from complex analysis, hyperbolic geometry, dynamics, partial differential equations, projective geometry, and other fields, highlighting the richness of the theory through multiple perspectives on the same mathematical objects.