On the saturation of late-time growth of complexity in supersymmetric JT gravity

TL;DR

This paper investigates the late-time behavior of complexity in supersymmetric JT gravity using a modified replica trick, revealing saturation in the N=1 case and linear growth in the N=2 case, with insights into non-perturbative effects.

Contribution

It introduces a modified replica trick to analyze late-time complexity in supersymmetric JT gravity, providing explicit results for N=1 and N=2 theories and discussing non-perturbative effects.

Findings

Complexity saturates at late times in N=1 supersymmetric JT gravity.

Complexity exhibits linear growth at late times in N=2 supersymmetric JT gravity.

Matter correlation functions are computed for both theories.

Abstract

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.