A new energy bound for Einstein-Scalar theory in AlAdS$_4$ and holographic bound for deformed CFT$_3$

Abstract

In this work, we derive an upper bound on energetic quantities, namely vacuum energy and free energy, for static solutions of Einstein-Scalar theory in four dimensional asymptotically locally Anti-de Sitter(AlAdS) spacetime with a nontrivial scalar potential where the scalar field mass parameter(m 2 ) is equal to 0 or -2. This system is the holographic dual of strongly coupled conformal field theory(CFT) in three dimensions being deformed by a relevant or marginal scalar operator of conformal dimension ∆ = 1, 2, 3. The bound is derived from a purely gravitational perspective regardless of the inhomogeneity of the static conformal boundary of AlAdS and the source of the deformation. We demonstrate the bound in simple settings and check the consistency with the known previous bounds.

Does the Round Sphere Maximize the Free Energy of (2+1)-Dimensional QFTs?

“…Hence in both cases the sign of a (t) is not immediately clear. However, note that the large-behavior of a (t) can be obtained in the flat-space scaling limit discussed above, and is given in (3.36); assuming a r 0 k (t) is 10 Explicitly, consider the deformed torus ds 2 = e 2f [(dx 1 ) 2 + (dx 2 ) 2 ], where x 1 and x 2 both have periodicity ∆x, and we take f = cos(2πx 1 /∆x) + O( 2 ). The perturbative heat kernel for the Dirac fermion on a deformed torus is computed in [14], and in this case comes out to be…”