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A few algebraic surprises in Anti-de Sitter space

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Questions of nonlinear stability in global AdS space have recently received a significant amount of attention, both as an interesting problem in mathematical general relativity and nonlinear dynamics, and in relation to thermalization studies within the AdS/CFT paradigm. Working with nonlinear perturbation theory (the main technique available for analytic studies in this area) requires a thorough understanding of the properties of linearized AdS fields'

mode functions, which are the fundamental building blocks in perturbative treatments of nonlinearities. While complicated explicit expressions for these mode functions are available in the literature, they hide a good deal of the elegant underlying structure dictated by the AdS symmetries. Extending the mode functions in the flat embedding space (in which the AdS space can be realized as a hyperboloid) results in families of homogeneous polynomials, on which the AdS isometries act in a straightforward manner. This suggests a simple proof of important selection rules in nonlinear perturbation theory.

Studies of multiplet structures of the mode functions furthermore reveal a relation to the Higgs oscillator, a well-known quantum-mechanical superintegrable system. This AdS connection leads to an explicit construction of the hidden symmetry generators for the Higgs oscillator, a long-standing problem in mathematical quantum mechanics.