Multi-invariants and Bulk Replica Symmetry

TL;DR

This paper investigates the conditions under which multi-partite entanglement measures in holographic CFTs preserve replica symmetry in the bulk, revealing infinite families of invariants and their geometric interpretations.

Contribution

It introduces a class of multi-invariants with specific replica symmetry properties and characterizes their bulk dual geometries, including conical singularities and handlebody volumes.

Findings

Identifies infinitely many multi-invariants with identical holographic evaluations.

Establishes geometric relations involving volumes of handlebodies and moduli space points.

Provides explicit CFT calculations supporting the bulk geometric analysis.

Abstract

In this paper, we analyze the question of replica symmetry in the bulk for multi-partite entanglement measures in the vacuum state of two dimensional holographic CFTs. We first define a class of multi-partite local unitary invariants, multi-invariants, with a given replica symmetry that acts freely and transitively on the replicas. We look for a subclass of measures such that the dual bulk geometry also preserves replica symmetry. We obtain the most general solution to this problem if we require the bulk to preserve replica symmetry for general configurations of the regions. Orbifolding the bulk solution with the replica symmetry gives us a bulk geometry with a network of conical singularities. Our approach makes it clear that there are infinitely many infinitely large families of multi-invariants such that each family evaluates identically on the holographic state. Geometrically, these are equalities involving volumes of handlebodies, possibly of different genus, at particular points in the moduli space. In certain cases, we check our bulk computation with an explicit calculation in CFT. Finally we comment on the generalization to higher dimension.