Quantum fluctuation on the worldsheet of probe string in BTZ black hole

In this paper, we investigate the second-order normal quantum fluctuation on the world-sheet of a probe string in the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole. These fluctuations is treated as the projection of Hawking radiation on the worldsheet and indeed modify the action growth of the string. Then in the string field theory/boundary conformal field theory framework, via the boundary vertex operator we study the correlation function of the Schr\"odinger functional of excited fields on the world-sheet and further extract the field's formula. Our study could shed light on the potential connection between complexity growth and correlation function.