Codimension-two holography

Abstract

A holographic interpretation for some specific Ricci flat string backgrounds of the form A 6 × C 4 is proposed. The conjecture is that there is a Four-dimensional Euclidean Conformal Field Theory (ECFT) defined on a codimension two submanifold of the manifold A 6 (where one of the two remaining holographic coordinates of A 6 is timelike, and the other one spacelike), with central charge proportional to the radius of curvature of the six-dimensional manifold, c ∼ l 4 .

Holographic entanglement entropy probes (non)locality

“…With the standard definition of the size of A, ℓ = 2|A|/|∂A| we see that the minimal hypersurface within this restricted class degenerates to z * = −∞ for ℓ > R, or to z * = z ε for ℓ < R. In the marginal case ℓ = R there is a degeneracy with respect to z * , corresponding to the continuous degeneracy found in (31), with a slightly renormalized value of the critical length, due to the non-smoothness of the class of hypersurfaces considered here.…”

“…Conversely, models with Lorentz invariance on the boundary and volume law must have a 'pathological' density of states, either because the specific heat is negative, or because the species degeneracy decreases at high energies. For example, we may consider the case of flat space, with λ(u) = µ(u) = 1/R, whose holographic dual, if formally defined, is expected to be a nonlocal theory [31]. Conforming to these expectations, when one calculates the entanglement entropy one finds it satisfies the volume law.…”

Goursat's problem and the holographic principle

“…Remarkably enough, the complete set of isometries of the three-dimensional metric (15) includes the full Lorentz group, SO (1,3). Please note that isometries are well-defined, even for singular metrics, through the vanishing Lie-derivative condition £(k)g µν = 0, reflecting the invariance of the metric under the corresponding one-parametric group of diffeomorphisms, although of course this is not equivalent to ∇ µ k ν + ∇ ν k µ = 0 because the covariant derivative (that is, the Christoffel symbols ) is not well defined owing to the absence of the inverse metric.…”

“…Incidentally, the induced metric on all hyperboloids is what could be called euclidean anti-de Sitter, (cf. [1]) with isometry group SO(1, 3) 3 which in our conventions has all coordinates spacelike…”

“…Recently we have pointed out some indications of holographic behavior in a rather general class of Ricci flat string backgrounds with vanishing cosmological constant ( [1]) of the form…”