Boundary Ring or a Way to Construct Approximate NG Solutions with Polygon Boundary Conditions. II. Polygons which admit an inscribed circle

TL;DR

This paper advances methods for approximately solving Nambu-Goto equations with polygon boundary conditions in AdS spaces, focusing on polygons with inscribed circles, to aid in string/gauge duality studies.

Contribution

It develops a simplified formalism for polygons with inscribed circles, linking NG equations and boundary rings, to improve evaluation of regularized NG actions in the Alday-Maldacena program.

Findings

Preserves inscribed circle condition simplifying the problem

Introduces an extended class of functions for future NG action evaluations

Provides detailed theory connecting NG equations and boundary rings

Abstract

We further develop the formalism of arXiv:0712.0159 for approximate solution of Nambu-Goto (NG) equations with polygon conditions in AdS backgrounds, needed in modern studies of the string/gauge duality. Inscribed circle condition is preserved, which leaves only one unknown function y_0(y_1,y_2) to solve for, what considerably simplifies our presentation. The problem is to find a delicate balance -- if not exact match -- between two different structures: NG equation -- a non-linear deformation of Laplace equation with solutions non-linearly deviating from holomorphic functions, -- and the boundary ring, associated with polygons made from null segments in Minkovski space. We provide more details about the theory of these structures and suggest an extended class of functions to be used at the next stage of Alday-Maldacena program: evaluation of regularized NG actions.