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

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, $F_{\epsilon}/F_0\leq \left[F_{\epsilon}/F_0\right]_{\text{free scalar}}$, $\forall$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.