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Complexity of four-dimensional hairy Anti-de-Sitter black holes with a rotating string and shear viscosity in generalized scalar-tensor theories

In four dimensions, we consider a generalized scalar-tensor theory where the coupling functions only depend on the kinetic term of the scalar field. For this model, we obtain a set of hairy Anti-de-Sitter black hole solutions, allowing us to calculate the computational complexity, according to the Complexity equals Action conjecture. To perform this, the system contains a particle moving on the boundary, corresponding to the insertion of a fundamental string in the bulk. The effect string is given by the Nambu-Goto term, analyzing the time development of this system. Together with the above, we calculate the shear viscosity, where the viscosity/entropy density ratio can violate the Kovtun-Son-Starinets bound for a suitable choice of coupling functions.