Entanglement entropy and conformal bounds for five-dimensional CFTs

Abstract: The entanglement entropy of spatial regions in odd-dimensional conformal field theories contains a universal constant contribution that encodes important information about the theory. This quantity can be defined in a robust way using mutual information between slightly deformed versions of a given region. In three-dimensional conformal field theories, it is known that this quantity is always positive and bounded from below by the value corresponding to a spherical region. There is also strong evidence that, for any region, its normalized value is maximized by the free scalar theory. In this work, we show that the situation changes significantly in five dimensions. Although the spherical region remains a local minimum under small shape deformations, more general regions can lead to values that become arbitrarily large in magnitude, with either sign. This implies that, in five-dimensional conformal field theories, the quantity is not bounded from above or below. We also demonstrate that the analogous maximization property observed in three dimensions does not hold in five dimensions when considering general regions. Despite this, we find that existing evidence is consistent with a weaker statement: for small deformations of a spherical region, the normalized quantity remains bounded above by the free scalar result across all five-dimensional conformal field theories. This leads to a new conjectured universal bound relating two key physical quantities—the coefficient governing stress-tensor correlations and the sphere free energy—which appears to hold for all currently known examples.