Holographic Wilson loops, Hamilton-Jacobi equation and regularizations

TL;DR

This paper employs the Hamilton-Jacobi equation to compute minimal areas of surfaces bounded by rectangular and circular loops in asymptotically AdS geometries, introducing a natural regularization method and comparing it with traditional approaches.

Contribution

It introduces a Hamilton-Jacobi based method for calculating minimal surface areas with a new regularization scheme, avoiding explicit surface shape calculations.

Findings

Exact solution for rectangular loop minimal area.

Power series expansion for circular loop minimal area.

Regularization by shifting the contour away from the boundary.

Abstract

The minimal area for surfaces whose border are rectangular and circular loops are calculated using the Hamilton-Jacobi (HJ) equation. This amounts to solve the HJ equation for the value of the minimal area, without calculating the shape of the corresponding surface. This is done for bulk geometries that are asymptotically AdS. For the rectangular countour, the HJ equation, which is separable, can be solved exactly. For the circular countour an expansion in powers of the radius is implemented. The HJ approach naturally leads to a regularization which consists in locating the countour away from the border. The results are compared with other regularization which leaves the countour at the border and calculates the area of the corresponding minimal surface up to a diameter smaller than the one of the countour at the border. The results do not coincide, this is traced back to the fact that in the former case the area of a minimal surface is calculated and in the second the computed area corresponds to a fraction of a different minimal surface whose countour lies at the boundary.