Entanglement entropy and conformal bounds for $d=5$ CFTs

Abstract

The entanglement entropy of spacetime regions $A$ in odd-dimensional conformal field theories (CFTs) contains a universal constant term, $(-1)^{\frac{d-1}{2}}F(A)$. This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of $A$. In the case of general three-dimensional CFTs, $F(A)$ is positive definite and bounded below by the round disk result, $F(A)\geq F_0\equiv F(\partial A=\mathbb{S}^1)$. Additionally, strong evidence has been provided that for every region $A$, $F(A)/F_0$ is maximized, within the space of CFT$_3$'s, by the free scalar field result. In this paper we show that while $F(A)$ remains a local minimum around $F_0\equiv F(\partial A=\mathbb{S}^3)$ for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT$_5$'s. We argue that an analogous conjecture regarding the extremization of $F(A)/F_0$ for general regions within the space of theories fails in $d=5$. We instead analyze the viability of the weaker bound, $F_ε/F_0\leq \left[F_ε/F_0\right]_{\text{free scalar}}$, $\forall$CFT$_5$ for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function $C_T$ and the Euclidean partition function on the sphere, namely, $C_T/F_0\leq \left[C_T/F_0\right]_{\text{ free scalar}}\approx 0.314$, which we show to hold for all known CFT$_5$'s. We also comment on possible extensions of this result to higher dimensions.

Figures (7)

• Figure 1: A smooth spatial region $A$ in $\mathbb R^4$, and two concentric regions $A^-$ and $\overline{A^+}$, obtained by displacing the entangling surface $\partial A$ along its normal direction at each point by distances $-\varepsilon/2$ and $\varepsilon/2$, respectively.
• Figure 2: We plot the coefficient of the universal contribution to the EE generated by a conical entangling region as a function of the opening angle $\Omega$ for a $d=5$ holographic CFT dual to Einstein gravity. The result is normalized by the stress-tensor two-point function coefficient, $C_T$, which universally governs the leading behavior in the $\Omega \rightarrow \pi/2$ regime. The dashed lines correspond, respectively: to the exact result in that limit, $a_{(5)}(\Omega)= 4\pi^4C_T(\Omega-\pi/2)^2/270+\dots$Bueno:2015lza; and to a fit of the form $a_{(5)}(\Omega)/C_T \approx 1.36/\Omega$. As is evident from the plot, $a_{(5)}(\Omega)$ remains positive for all values of the opening angle and approaches the exact result predicted in Bueno:2015lza as $\Omega \rightarrow \pi/2$. This is in fact the first explicit verification of the general results presented in that paper beyond $d=3$ theories. More details can be found in Appendix \ref{['holoCorner']}.
• Figure 3: We plot the universal term $F(A_1 \cup A_2)$ in the EMI model for the union of two identical round balls, $A_1 = A_2 = \mathbb{B}^4$, each of radius $R$ and separated by a distance $\ell$, as a function of the relative separation $\ell/R$. The result is normalized by the single-ball value. At large separations, $F(A)/F_0$ is positive and approaches $+2$. It remains positive for most separations as the spheres move closer together, but it eventually vanishes for $\ell/R \simeq 0.00194$. As the separation decreases further and the spheres approach a point of contact, $F(A)/F_0$ diverges to $-\infty$.
• Figure 4: From left to right, we display the ratios $F_{\text{strip}}/F_0$ for a variety of CFTs in three, four, and five spacetime dimensions. In all cases, the values are normalized with respect to the free scalar result, which provides a convenient common reference across dimensions. In three and four dimensions, the free scalar sets the upper bound, whereas in five dimensions it appears to act instead as a lower bound. The lower bound is given by the Maxwell theory ratio in $d=3$ (trivially) and in $d=4$, where it provides a nontrivial constraint. Each panel includes the ratios corresponding to a free fermion, the EMI model, and holographic theories dual to Einstein gravity. We observe the same hierarchy in three and four dimensions, while in five dimensions this hierarchy is reversed. Regions corresponding to forbidden values of the ratio are shaded in light red.
• Figure 5: Quotient $C_T/F_0$ for two families of $d=5$ SCFTs corresponding, respectively, to $\#_{N,M}$Aharony:1997bh and $T_N$Benini:2009giMitev:2014isa models, respectively. All data points for $C_T$ and $F_0$---as well as the approximate interpolation shown for the $T_N$ model---are extracted from Fluder:2018chf.