Effective Geometry, Complexity, and Universality

Post-Wilsonian physics views theories not as isolated points but elements of bigger universality classes, with effective theories emerging in the infrared. This paper makes initial attempts to apply this viewpoint to homogeneous geometries on group manifolds, and complexity geometry in particular. We observe that many homogeneous metrics on low-dimensional Lie groups have markedly different short-distance properties, but nearly identical distance functions at longer distances. Using Nielsen's framework of complexity geometry, we argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered in complexity theory. We conjecture that at larger complexities, a new effective metric emerges that describes a broad class of complexity geometries, insensitive to various choices of 'ultraviolet' penalty factors. Finally we lay out a broader mathematical program of classifying the effective geometries of right-invariant group manifolds.