In this work we use the modified replica trick, proposed in arXiv:2205.01150, to compute the late time behaviour of complexity for JT gravity with ${ \cal N} = 1$ and ${ \cal N} = 2$ supersymmetries. For the ${ \cal N} = 1$ theory, we compute the late time behaviour of complexity defined by the ``quenched geodesic length" and obtain the expected saturation of complexity at time $t \sim e^{ S_0}$, to a constant value with time-independent variance. For the ${ \cal N} = 2$ theory, we explicitly compute complexity at the disk level which yields the late-time linear growth of complexity. However, we comment on the expectation of the late-time saturation by speculating the trumpet partition function and the non-perturbative corrections to the spectral correlation, relevant for the late-time behaviour of complexity. Furthermore, we compute the matter correlation functions for both the theories.
Categories: hep-th gr-qc quant-ph