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Non-linear sigma models defined on symmetric target spaces have a wide set ofapplications in modern physics, including the description of string propagationin symmetric spaces, such as AdS or dS, or minimal surfaces in hyperbolicspaces. Although it is difficult to acquire solutions of these models, due totheir non-linear nature, it is well known that they are reducible to integrablesystems of the family of the sine- or sinh-Gordon equation. In this study, wedevelop a method to invert Pohlmeyer reduction for elliptic solutions of thereduced system, implementing a relation between NLSM solutions and theeigenstates of the n = 1 Lame problem. This method is applied to produce afamily of classical string solutions in AdS(3), which includes the spikystrings, as well as hoop string solutions with singular evolution of theirangular velocity and radius, which are interesting in the framework ofholographic dualities. Furthermore, application of this method produces a widefamily of static minimal surfaces in AdS(4), which includes helicoids andcatenoids, and which are interesting in the framework of the Ryu-Takayanagiconjecture and the understanding of the emergence of gravity as an entropicforce related to quantum entanglement statistics. The developed formalismallows the study of the area of the minimal surfaces and geometric phasetransitions between them, which are relevant to confinement-deconfinement phasetransitions.

Elliptic String Solutions in AdS(3) and Elliptic Minimal Surfaces in AdS(4)