  We investigate the structure of confining and deconfining phases in SU(2)
lattice gauge theory via persistent homology, which gives us access to the
topology of a hierarchy of combinatorial objects constructed from given data.
Specifically, we use filtrations by traced Polyakov loops, topological
densities, holonomy Lie algebra fields, as well as electric and magnetic
fields. This allows for a comprehensive picture of confinement. In particular,
topological densities form spatial lumps which show signatures of the classical
probability distribution of instanton-dyons. Signatures of well-separated dyons
located at random positions are encoded in holonomy Lie algebra fields,
following the semi-classical temperature dependence of the instanton appearance
probability. Debye screening discriminating between electric and magnetic
fields is visible in persistent homology and pronounced at large gauge
coupling. All employed constructions are gauge-invariant without a priori
assumptions on the configurations under study. This work showcases the
versatility of persistent homology for statistical and quantum physics studies,
barely explored to date.
