Holographic Subregion Complexity for Singular Surfaces

TL;DR

This work analyzes the divergence structure of holographic subregion complexity for singular entangling regions within AdS/CFT, using the proposal $\mathcal{C}_{\mathrm{subregion}}=\max\left[ V(\gamma)/(8\pi G_N \ell) \right]$ with $\gamma$ the RT surface. It systematically examines kink, cone, crease, and curved-locus geometries in various dimensions, uncovering new universal terms: a $\log\delta$ contribution for a kink in $(2+1)$ dimensions and $\log\delta$ or $\log^2\delta$ terms for cones $c_n$ depending on the parity of $n$, as well as more intricate divergences like $\frac{1}{\delta}\log\delta$ and $\frac{1}{\delta^2}\log\delta$ in certain conical creases. Creases with flat loci generally produce no new universal terms, while curved-locus cases exhibit selective universal logs (notably $c_1\times S^2$) and additional power-law/logarithmic divergences. These results illuminate how subregion geometry controls complexity divergences and motivate future work on monotonic c-functions in odd dimensions, higher-derivative gravity, and the relation between holographic complexity proposals.

Abstract

Recently holographic prescriptions are proposed to compute quantum complexity of a given state in the boundary theory. A specific proposal known as `holographic subregion complexity' is supposed to calculate the the complexity of a reduced density matrix corresponding to a static subregion. We study different families of singular subregions in the dual field theory and find the divergence structure and universal terms of holographic subregion complexity for these singular surfaces. We find that there are new universal terms, logarithmic in the UV cutoff, due to the singularities of a family of surfaces including a kink in (2+1)-dimension and cones in even dimensional field theories. We find examples of new divergent terms such as square logarithm and negative powers times the logarithm of the UV cut-off parameter.

Figures (1)

• Figure 1: Left: The blue plane represents a constant time slice of a $d=3$ CFT with a kink ($k$) entangling region on it. Middle: Conical entangling region in a $d=4$ CFT. Right: A crease ($k\times R^m$) entangling region as a direct generalization of the kink in higher dimensions.