Complexity of the AdS soliton

Abstract

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