The Junction Law for Multipartite Entanglement in Confining Holographic Backgrounds

TL;DR

This paper explores the realization of the junction law for multipartite entanglement in confining holographic backgrounds, analyzing phase structures and scaling behaviors across various models using multi-entropy as a diagnostic.

Contribution

It provides a detailed analysis of the junction law in both hard-wall and smooth confining geometries, revealing background-dependent features and phase transition behaviors.

Findings

Genuine multi-entropy is localized near junctions in the hard-wall model.

Phase structures differ between hard-wall and smooth geometries, with the plateau disappearing in smooth cases.

Short-distance scaling of GM varies with background, e.g., L^{-4} in D4-soliton, L^{-2} in D3-soliton, and L^{-2} (log L)^2 in Klebanov--Strassler.

Abstract

We investigate how the junction law for multipartite entanglement is realized in confining holographic backgrounds, using genuine multi-entropy (GM) as our main diagnostic. We first study an AdS$_3$ hard-wall toy model as an analytic benchmark, where multi-way cuts and junction geometries can be analyzed explicitly. In this setup, we classify the relevant saddles, determine the dominant phases, and show that the genuinely multipartite contribution diagnosed by GM is localized near the junction. We also examine how this structure depends on subsystem sizes, asymmetry, and the confinement scale, including phase transitions between competing saddles. We then move beyond the hard-wall benchmark to smooth confining geometries, focusing on the D4-soliton and D3-soliton backgrounds and formulating the corresponding framework also for the Klebanov--Strassler background. In the smooth-cap examples, we find that the junction picture persists, while the detailed phase structure differs from the hard-wall case: in particular, the hard-wall plateau does not survive, and GM instead decreases monotonically and vanishes at a finite critical scale. We also find that the short-distance behavior is background-dependent, with $\mathrm{GM}^{(3)}\sim L^{-4}$ in the D4-soliton background, $\mathrm{GM}^{(3)}\sim L^{-2}$ in the D3-soliton background, and $\mathrm{GM}^{(3)}\sim L^{-2}\cdot (\log L)^{2}$ in the Klebanov--Strassler background. These results clarify which features of the junction-law picture are robust in confining holography and which features of the phase structure and short-distance scaling are background-dependent.