Complexity growth of rotating black holes with a probe string

We study the effect of a probe string to black hole complexity according to the complexity equals action conjecture. Our system contains a particle moving on the boundary of black hole spacetime. In the dual description this corresponds to the insertion of a fundamental string on the bulk spacetime. The total action consists of the Einstein-Hilbert term and the Nambu-Goto term. The effect of this string is expressed by the Nambu-Goto term. Focusing on the Nambu-Goto term, we analyze the time development of this system. Our results show some interesting properties of complexity. This gives us a useful hint for defining complexity in quantum field theories.