Boundary Ring: a way to construct approximate NG solutions with polygon boundary conditions: I. Z_n-symmetric configurations

TL;DR

This paper introduces an algebraic method using the boundary ring to construct approximate minimal surfaces with polygonal boundaries in AdS_5, focusing on Z_n-symmetric configurations and providing a framework for exploring string/gauge duality.

Contribution

It presents a novel algebro-geometric approach to approximate solutions of minimal surfaces with polygon boundaries, specifically for Z_n-symmetric cases, using power series truncation.

Findings

Constructed approximate NG solutions for Z_6-symmetric hexagon

Represented solutions as power series with initial terms evaluated

Boundary conditions help fix free parameters in the series

Abstract

We describe an algebro-geometric construction of polygon-bounded minimal surfaces in ADS_5, based on consideration of what we call the "boundary ring" of polynomials. The first non-trivial example of the Nambu-Goto (NG) solutions for Z_6-symmetric hexagon is considered in some detail. Solutions are represented as power series, of which only the first terms are evaluated. The NG equations leave a number of free parameters (a free function). Boundary conditions, which fix the free parameters, are imposed on truncated series. It is still unclear if explicit analytic formulas can be found in this way, but even approximate solutions, obtained by truncation of power series, can be sufficient to investigate the Alday-Maldacena -- BDS/BHT version of the string/gauge duality.