Topological Foundations of Field Domain Control Science

We propose Field Domain Control Science as a new paradigm for regulating physical fields—electromagnetic, acoustic, elastic, or quantum—by deliberately engineering the topological architecture of the underlying spatial domain. In this framework, topology is not a descriptive accessory but the foundational language of control. We establish three core roles: (i) topological invariants as discrete, robust “control knobs”; (ii) topological phase transitions as non-continuous “switches”; and (iii) certain topological field theories (TFTs) as design principles that link geometric structure to field response. Rigorous definitions of key invariants—Euler characteristic, Betti numbers, Chern number—are provided, along with explicit mappings to observable phenomena such as chiral edge modes, corner-localized states, and backscattering-immune transport. The framework is validated against recent experiments in phononic crystals, magnetoelastic metamaterials, and topological water wave platforms. Crucially, it enables inverse design of topologically structured matter, where desired field functionalities are achieved by prescribing the topology of the field domain.