Generalized Replica Manifolds I: Surgery and Averaging

TL;DR

This paper introduces a novel framework for constructing replica manifolds through correlated averaging over defects, enabling new calculations of entanglement entropy in gauge theories and holography without explicit manifold modifications.

Contribution

It develops a simple, general method for implementing path integral surgery via operator averaging, extending to gauge theories and large-N holographic contexts.

Findings

Framework successfully constructs replica manifolds via averaging.

Connects gauge-invariant entanglement calculations to quiver gauge theories.

Applicable to large-N theories and holography contexts.

Abstract

We develop a simple framework for implementing a type of path integral "surgery" via correlated averaging over codimension-one defects/extended operators. This technique is used to construct replica manifolds by effectively cutting and gluing the path integral without explicitly modifying the underlying manifold. We argue that restricted forms of this averaging can be used to calculate R\'enyi entanglement entropy corresponding to a wide range of choices of subsystem partitioning. When the entanglement entropy being calculated in this way does not simply correspond to entanglement between subregions, we call the resulting objects from this surgery "generalized replica manifolds". We show how this framework extends to gauge theories and, in particular, how in non-Abelian gauge theories it establishes a connection between replica calculations of a gauge-invariant notion of entanglement between color degrees of freedom and a quiver gauge-theory structure. Finally, we discuss how this framework appears in the context of large-$N$ theories and holography, with a bird's-eye view of potential future directions. This paper focuses on averaging over operators that form a representation of the Heisenberg group; a subsequent paper will extend the framework to more general operator averaging.