Vortex holography

Abstract

We show that the Abelian Higgs field equations in the four dimensional anti de Sitter spacetime have a vortex line solution. This solution, which has cylindrical symmetry in AdS 4 , is a generalization of the flat spacetime Nielsen-Olesen string. We show that the vortex induces a deficit angle in the AdS 4 spacetime that is proportional to its mass density. Using the AdS/CFT correspondence, we show that the mass density of the string is uniform and dual to the discontinuity of a logarithmic derivative of correlation function of the boundary scalar operator. 1

A conical deficit in the AdS 4 /CFT 3 correspondence

“…A similar angular deficit was obtained before in [16] considering a vortex line configuration in the limit of weak gravitational field.…”

Abelian Higgs hair for rotating and charged black holes

“…Throughout the black hole mass is taken to be m = 10, and so the horizon is respectively located at r H = 20, 18.66, 14.36, with r H = m = 10 in the extremal case. The figures in (8), when a = 0 are exactly the same as the results of the Schwarzschild black hole [4], [12]. We note that in changing of the rotation parameter from a value of 0 (corresponding to a Schwarzschild black hole) to the extremal value of 10, both the X and P contours undergo only a small change.…”

“…When a = 0, the vortex solutions of the Abelian Higgs equations in flat spacetime (without a black hole) satisfy the l → ∞ limit of equations up to errors which are proportional to mR 2 r 3 ≈ m r 3 [12]. These errors are very tiny far from the black hole horizon, whereas near the horizon r ≈ r H = 2m, they are of the order of 1 m 2 , which is negligible for large mass black holes.…”

Boson stars with negative cosmological constant

“…However, the same system in the presence of a Higgs scalar presents solutions with very similar properties to the asymptotically-flat space counterparts [6,7,8]. More recently it has been shown that the U (1) Higgs field equations have a vortex solution in four-dimensional AAdS [9]. Static spherically symmetric solutions of general relativity with Λ < 0, coupled to a real scalar field, have been studied in [10].…”