Interrelation of Topological Matter Phases, Axion Electrodynamics, and the Kuznetsov Tensor

The article investigates topological phases of matter, axion electrodynamics, and the application of the Kuznetsov tensor—a crucial direction in modern condensed-matter physics and quantum field theory. Emphasis is placed on establishing rigorous mathematical connections between different phenomenological aspects of these scientific domains.Fundamental properties of materials exhibiting unusual electrical and magnetic characteristics, conditioned by specific topological structures, are explored. Central attention is given to introducing a new mathematical apparatus—the Kuznetsov tensor—which describes the interrelation between classical physical quantities (such as Berry curvature) and geometric anomalies (space defects) arising in complex systems.The author formulates and proves a key statement: an integral including interaction between Berry curvature and the axion field through the trace of the Kuznetsov tensor is a topologically invariant action functional. This creates a path towards building a unified approach, integrating concepts from diverse branches of theoretical physics.Special focus is put on presenting clear and comprehensible demonstrations, emphasizing clarity in conclusions. Clear implications of the proposed theory concerning the behavior of material media displaying unusual electromagnetic effects are formulated. Particular importance lies in insights into the nature of magnetic flux quanta and peculiarities of quantum-Hall-like effects.The obtained formalism allows effective analysis of systems with crystal lattice defects, materials possessing special topological properties, such as topological insulators, and predicting new physical phenomena. It is expected that this work will make a substantial contribution to developing advanced technologies linked to creating new classes of functional materials and future electronic devices.