Exact and Numerical Results on Entanglement Entropy in (5+1)-Dimensional CFT

TL;DR

The paper studies how entanglement entropy in a (5+1)-D CFT depends on the geometry of the entangling surface, focusing on extrinsic curvature and conical singularities. It identifies conformally invariant extrinsic-curvature functionals T1,T2,T3 and expresses the logarithmic EE term in terms of anomaly coefficients A,B1,B2,B3, subject to a single constraint, with holographic calculations via Lovelock gravity providing precise coefficients. The authors test the framework by relating 5+1D massless theories to 2+1D massive theories through dimensional reduction, performing lattice computations that confirm the predicted IR expansions and the F-theorem monotonicity, and by examining singular surfaces that yield log^2 ε divergences. These results advance the understanding of higher-dimensional holographic EE and offer nontrivial checks of the holographic prescriptions, with implications for potential six-dimensional a-theorem proofs. The combination of analytic holographic results, exact constraints among anomaly coefficients, and numerical EE studies establishes a coherent cross-dimensional picture of EE shape dependence and renormalized EE behavior.

Abstract

We calculate the shape dependence of entanglement entropy in (5+1)-dimensional conformal field theory in terms of the extrinsic curvature of the entangling surface, the opening angles of possible conical singularities, and the conformal anomaly coefficients, which are required to obey a single constraint. An important special case of this result is given by the interacting (2,0) theory describing a large number of coincident M5-branes. To derive the more general result we rely crucially on the holographic prescription for calculating entanglement entropy using Lovelock gravity. We test the conjecture by relating the entanglement entropy of the free massless (1,0) hypermultiplet in (5+1)-dimensions to the entanglement entropy of the free massive chiral multiplet in (2+1)-dimensions, which we calculate numerically using lattice techniques. We also present a numerical calculation of the (2+1)-dimensional renormalized entanglement entropy for the free massive Dirac fermion, which is shown to be consistent with the F-theorem.

Figures (2)

• Figure 1: The renormalized entanglement entropy ${\cal F}$ for the massive real free scalar ( left) and massive free Dirac fermion ( right). The black curves are the results of the numerical lattice computations. The orange curves are the analytic $1/(mR)$ IR approximations coming from the coefficient $\tilde{c}^{\Sigma}_{-1}$ in \ref{['a2Rulekappa']}. The dotted red lines are the $m = 0$ values $F_S$ and $F_D$ for the scalar and fermion, respectively.
• Figure 2: The function $(mR)^3 \left( {\cal F}_{\text{chiral}} - {\pi \over 24} {1 \over mR} \right)$ plotted at large values of $(mR)$, where ${\cal F}_{\text{Chiral}}$ is the renormalized entanglement entropy of the free massive chiral multiplet, which consists of two real scalar fields and one Dirac fermion. The black points are the results of the numerical lattice computation, which is subject to numerical error at this level of precision. The dotted red line is the predicted asymptotic value ${\pi \over 256}$, which we compute using \ref{['FIRchiral']}.