Negative Complexity of Formation: the Compact Dimensions Strike Back

TL;DR

This paper demonstrates that the holographic complexity of formation can be negative and decrease over time when considering compact dimensions, challenging previous assumptions and highlighting the importance of compact directions in holography.

Contribution

The authors explicitly construct solutions with arbitrarily negative and decreasing complexity of formation in asymptotically AdS$_{4}\times S^{7}$ supergravity, emphasizing the role of compact dimensions.

Findings

Complexity of formation can be negative when including compact directions.

Complexity can decrease at late times in certain geometries.

Explicit solutions with negative and decreasing complexity are constructed.

Abstract

We show that the vacuum-subtracted maximal volume, the proposed holographic dual to complexity of formation, can be negative when contributions from compact directions are included. We construct explicit solutions with arbitrarily negative complexity of formation in asymptotically AdS$_{4}\times S^{7}$ SUGRA. These examples rely critically on the compact directions, specifically the fact that the full eleven-dimensional spacetime is not asymptotically AdS$_{11}$. While there is some ambiguity in the extension of the holographic complexity proposal to the compact directions, we show that the two natural candidates can both have arbitrarily negative complexity of formation in SUGRA solutions. We further find examples in which complexity can even decrease at late times, including cases of both single-sided geometries and two-sided wormholes. In particular, we construct a cosmological wormhole with simultaneously negative and decreasing complexity of formation (as computed by volume) at late times. We find a distinguished role for relevant primaries in these constructions and comment on possible interpretations.