Complexity of the AdS Soliton

TL;DR

This paper investigates holographic complexity in the AdS soliton background, revealing contrasting behaviors between volume and action conjectures and comparing results with free fermion models.

Contribution

It provides a detailed analysis of holographic complexity for the AdS soliton, highlighting differences between CV and CA calculations and comparing with free fermion results.

Findings

CV complexity decreases with IR scale for antiperiodic conditions

CA complexity initially increases then decreases with IR scale

Antiperiodic boundary conditions lead to higher complexity in free fermions

Abstract

We consider the holographic complexity conjectures in the context of the AdS soliton, which is the holographic dual of the ground state of a field theory on a torus with antiperiodic boundary conditions for fermions on one cycle. The complexity is a non-trivial function of the size of the circle with antiperiodic boundary conditions, which sets an IR scale in the dual geometry. We find qualitative differences between the calculations of complexity from spatial volume and action (CV and CA). In the CV calculation, the complexity for antiperiodic boundary conditions is smaller than for periodic, and decreases monotonically with increasing IR scale. In the CA calculation, the complexity for antiperiodic boundary conditions is larger than for periodic, and initially increases with increasing IR scale, eventually decreasing to zero as the IR scale becomes of order the UV cutoff. We compare these results to a simple calculation for free fermions on a lattice, where we find the complexity for antiperiodic boundary conditions is larger than for periodic.