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Математика

This work is devoted to the study of interacting higher-spin field theories in flat and anti-de Sitter (AdS) spacetime, focusing on the mathematical structures and physical principles required to construct consistent theories involving fields with spin greater than two. The subject lies at the intersection of quantum field theory, general relativity, gauge theory, and modern mathematical physics, where the description of interacting higher-spin particles presents significant theoretical challenges. The research examines how higher-spin gauge symmetries can be formulated and preserved in the presence of interactions and how the geometry of the underlying spacetime influences the structure of the resulting theories. Particular attention is given to the differences between flat space and AdS space, since the nonzero curvature of anti-de Sitter spacetime can substantially alter the possibilities for constructing consistent interacting systems. The work explores the formulation of higher-spin fields, their gauge transformations, equations of motion, and interaction vertices, as well as the constraints imposed by consistency requirements. It considers how different fields and spin sectors can interact while maintaining the necessary symmetries and avoiding mathematical or physical inconsistencies. The analysis of flat-space theories provides a framework for understanding the difficulties associated with higher-spin interactions in the absence of a cosmological scale, while the AdS setting offers a richer environment in which consistent interactions can be formulated. The study is also relevant to broader questions concerning the unification of gauge interactions and gravity, the structure of quantum field theories, and possible extensions of conventional field-theoretic descriptions of fundamental particles. The mathematical methods involved may include differential geometry, representation theory, gauge-theoretic techniques, perturbative field theory, and the analysis of nonlinear systems. By examining interacting higher-spin models in different spacetime backgrounds, the work contributes to the understanding of the general principles governing consistent higher-spin dynamics and the relationship between spacetime geometry and field interactions. The material is primarily intended for researchers and advanced students in theoretical and mathematical physics, particularly those working in quantum field theory, gravitation, gauge theories, string theory, higher-spin physics, and related areas of fundamental physics.