Holographic confinement in inhomogenous backgrounds

TL;DR

This paper explores how inhomogeneities in the confinement scale affect holographic confining theories, revealing flux tube behaviors and energy shifts across dimensions using a derivative expansion approach.

Contribution

It generalizes the AdS soliton solution to include slow variations in the confinement scale and computes response coefficients for dimensions 3 to 8.

Findings

Flux tubes tend to align orthogonal to gradients.

Flux tubes are attracted or repelled by gradients depending on the dimension.

Inhomogeneities increase the confining vacuum energy for dimensions greater than 3.

Abstract

As noted by Witten, compactifying a $d$-dimensional holographic CFT on an $S^1$ gives a class of $(d-1)$-dimensional confining theories with gravity duals. The prototypical bulk solution dual to the ground state is a double Wick rotation of the AdS$_{d+1}$ Schwarzschild black hole known as the AdS soliton. We generalize such examples by allowing slow variations in the size of the $S^1$, and thus in the confinement scale. Coefficients governing the second order response of the system are computed for $3 \le d \le 8$ using a derivative expansion closely related to the fluid-gravity correspondence. The primary physical results are that i) gauge-theory flux tubes tend to align orthogonal to gradients and along the eigenvector of the Hessian with the lowest eigenvalue, ii) flux tubes aligned orthogonal to gradients are attracted to gradients for $d \le 6$ but repelled by gradients for $d \ge 7$, iii) flux tubes are repelled by regions where the second derivative along the tube is large and positive but are attracted to regions where the eigenvalues of the Hessian are large and positive in directions orthogonal to the tube, and iv) for $d > 3$, inhomogeneities act to raise the total energy of the confining vacuum above its zeroth order value.