Holographic complexity of rotating quantum black holes

We study holographic complexity for the rotating quantum BTZ black holes (quBTZ), the BTZ black holes with corrections from bulk quantum fields. Using double holography, the combined system of backreacted rotating BTZ black holes with conformal matters, can be holographically described by the rotating AdS₄ C-metric with the BTZ black hole living on a codimension-1 brane. We investigate both volume complexity and action complexity of rotating quBTZ, and pay special attention to their late-time behaviors. When the mass of BTZ black hole is not very small compared to $$ {g}_{\textrm{eff}}^{2/3}/{G}_3 $$ g eff 2 / 3 / G 3 and the rotation is not very slow compared to parameter ℓ, where g _eff as well as ℓ both characterize the backreaction of quantum fields and G ₃ is the 3D Newton constant, we show that the late-time rates of the volume complexity and the action complexity agree with each other up to a factor 2 and reduce to the ones of BTZ at the leading classical order, and they both receive subleading quantum corrections. For the volume complexity, the leading quantum correction comes from the backreaction of conformal matter on the geometry, similar to the static quBTZ case. For the action complexity, unlike the static case, the Wheeler-de Witt (WdW) patch in computing the action complexity for the rotating black hole does not touch the black hole singularity such that the leading order result is in good match with the one of classical BTZ. However, when the mass of BTZ black hole is small compared to $$ {g}_{\textrm{eff}}^{2/3}/{G}_3 $$ g eff 2 / 3 / G 3 or the rotation parameter a is small compared to ℓ, the quantum correction to the action complexity could be significant such that the late-time slope of the action complexity of quBTZ deviates very much from the one of classical BTZ. Remarkably, we notice that the nonrotating limit a → 0 is singular and the action complexity of the static quBTZ black holes cannot be reproduced by taking the nonrotating limit a → 0 on the one of the rotating black holes. This discontinuity also appears in higher dimensional rotating black holes, such as Kerr-AdS₄ black hole and rotating AdS₅ black hole.