Topological and time dependence of the action-complexity relation

Musema Sinamuli,Robert B Mann

doi:10.1103/physrevd.98.026005

Musema Sinamuli, Robert B Mann

Open Access

https://doi.org/10.1103/physrevd.98.026005

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Abstract

We consider the dependence of the recently proposed action/complexity duality conjecture on time and on the underlying topology of the bulk spacetime. For the former, we compute the dependence of the CFT complexity on a boundary temporal parameter and find it to be commensurate with corresponding computations carried out in terms of the rate of change of the bulk action on a Wheeler deWitt (WDW) patch. For the latter, we compare the action/complexity relation for $(d+1)$-dimensional Schwarzschild AdS black holes to those of their geon counterparts, obtained via topological identification in the bulk spacetime. The complexity/action duality holds in both cases, but with the proportionality changed by a factor of 4, indicating sensitivity to spacetime topology.

Highlights

The importance of dualities between quantum field and gravity theories is difficult to underestimate
The AdS=CFT correspondence [1], the first and most successful, posits the existence of a d-dimensional conformal field theory (CFT) on the boundary of a (d þ 1)-dimensional asymptotically anti-de-Sitter (AdS) spacetime, and has led to several dualities between quantities observed in AdS and those in the CFTs defined on their boundaries
Watanabe et al [2] introduced a duality between a quantum information metric defined in the CFT on the boundary of an AdS black hole, and the volume of a time slice in the AdS

Summary

INTRODUCTION

The importance of dualities between quantum field and gravity theories is difficult to underestimate. Using a recent proposal for circuit complexity [13], it has been shown [14] that complexity growth dynamics has two distinct phases: an early regime whose evolution is approximately linear is followed by a saturation phase characterized by oscillations around a mean value To this end, one goal of the current paper is to compute from the CFT perspective the dependence of complexity on boundary time in the late time limit. The last section will be a conclusion and discussion, in which our results will be summarized in the context of previous work

COMPLEXITY AND COST FUNCTION

GAUSSIAN STATES

CONFORMAL FIELD THEORY IN d DIMENSIONS

GEON AND DIRECT PRODUCTS

RATE OF VARIATION OF THE ACTION

VIII. CONCLUSION