Generalized Replica Manifolds I: Surgery and Averaging

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 Rényi 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.

• The paper introduces a new framework that implements path integral surgery via correlated averaging over codimension-one defects.
• It extends the methodology to non-Abelian gauge theories, linking entanglement entropy with quiver gauge theory structures.
• The approach offers fresh insights into holography and large-N theories, suggesting promising directions in emergent gravity.

"Generalized Replica Manifolds I: Surgery and Averaging" (2510.20900)

Introduction to Path Integral Surgery and Replica Manifolds

The paper "Generalized Replica Manifolds I: Surgery and Averaging" introduces a novel framework for implementing path integral surgery via a correlated averaging over codimension-one defects or extended operators. This technique is used to construct replica manifolds by effectively cutting and gluing the path integral without explicitly altering the underlying manifold. This approach is particularly promising for calculating Rényi entanglement entropy across diverse subsystem partitionings, including cases beyond spatial region entanglement, termed "generalized replica manifolds."

Extension to Gauge Theories

The framework is extended to gauge theories. In non-Abelian gauge theories, it relates replica calculations of entanglement between color degrees of freedom to a quiver gauge theory structure. The paper also briefly discusses its application in large-$N$ theories and holography, highlighting potential future directions in studying entanglement structures in boundary theories.

Path Integral Surgery Technique

By focusing on operator insertions forming a representation of the Heisenberg group, this paper establishes a foundation for path integral surgery. This leads to a mechanism for selectively targeting degrees of freedom through a form of path integral "selective surgery," which can construct path integrals akin to those needed for certain entanglement entropy calculations. The correlated averaging of sources, coupled to select operators, effectively manipulates the manifold corresponding to these path integrals.

Implications for Entanglement in Holography

The implications for holography are significant, suggesting avenues to probe entanglement structures beyond spatial subregion entanglement. This may link to examples in gauge-gravity duality where standard spatial partitioning does not apply, such as D0-brane holography. New insights into emergent gravity concepts could arise from understanding these non-spatial entanglement structures.

Conclusion

The paper presents a promising avenue toward generalized replica manifolds, offering potential applications in understanding complex entanglement structures in field theories, particularly in the context of holography. This approach broadens the scope of entanglement entropy calculations and provides a pathway toward understanding how these structures can be mirrored in holographic models, potentially using techniques related to large $N$ limits and quiver gauge theories. Future work, as suggested in the paper, will focus on extending these concepts beyond operator averaging in the Heisenberg group.