Entanglement Renyi Entropies in Conformal Field Theories and Holography

TL;DR

The paper develops a holographic framework for entanglement Renyi entropies in conformal field theories with AdS duals, deriving leading UV terms from weak-coupling 4D ${ m N}=4$ SYM and proposing a bulk, n-independent geometric functional on a codimension-2 surface to capture Renyi data. It expresses the 4D CFT Renyi entropy through area-like and curvature-invariant terms, with explicit polynomials in $ abla_n$ for the $s^{(n)}_2$ and $s^{(n)}_4$ contributions, and identifies the bulk duals $ ilde F_a$, $ ilde F_b$, $ ilde F_c$ that reproduce these structures holographically. A central result is the holographic Renyi entropy functional $S(n, ilde{ ext B})$ which combines a volume term and curvature invariants with shifted coefficients, matching the known $n=1$ entropy (Ryu–Takayanagi) and vanishing remainder in the appropriate limits, while highlighting nonlocal logarithmic contributions and remaining ambiguities (e.g., the coefficient $b( abla_n)$). The work clarifies how holography can encode Renyi entropies beyond the leading entropy and points to future investigation of strong-coupling behavior and quantum gravity corrections, providing a bridge between heat-kernel analysis in field theory and geometric functionals in AdS/CFT.

Abstract

An entanglement Renyi entropy for a spatial partition of a system is studied in conformal theories which admit a dual description in terms of an anti-de Sitter gravity. The divergent part of the Renyi entropy is computed in 4D conformal N=4 super Yang-Mills theory at a weak coupling. This result is used to suggest a holographic formula which reproduces the Renyi entropy at least in the leading approximation. The holographic Renyi entropy is an invariant functional set on a codimension 2 minimal hypersurface in the bulk geometry. The bulk space does not depend on order $n$ of the Renyi entropy. The holographic Renyi entropy is a sum of local and non-local functionals multiplied by polynomials of $1/n$.

Figures (1)

• Figure 1: The figure shows the holographic surface $\tilde{\cal B}$ with normal vectors in a constant time section.