Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space (0904.0663v2)

Abstract: We consider minimal surfaces in three dimensional anti-de-Sitter space that end at the AdS boundary on a polygon given by a sequence of null segments. The problem can be reduced to a certain generalized Sinh-Gordon equation and to SU(2) Hitchin equations. We describe in detail the mathematical problem that needs to be solved. This problem is mathematically the same as the one studied by Gaiotto, Moore and Neitzke in the context of the moduli space of certain supersymmetric theories. Using their results we can find the explicit answer for the area of a surface that ends on an eight-sided polygon. Via the gauge/gravity duality this can also be interpreted as a certain eight-gluon scattering amplitude at strong coupling. In addition, we give fairly explicit solutions for regular polygons.

• The paper demonstrates that applying gauge/gravity duality reduces the study of null polygonal Wilson loops to a generalized Sinh-Gordon equation linked with SU(2) Hitchin systems.
• It computes explicit minimal surface areas for octagonal Wilson loops, directly relating geometrical features to strong-coupling scattering amplitudes in gauge theory.
• The study connects integrable models with the moduli space of gauge theory vacua, offering new avenues for exploring higher-dimensional holographic dualities.

Null Polygonal Wilson Loops and Minimal Surfaces in Anti-De-Sitter Space

The paper authored by Luis F. Alday and Juan Maldacena embarks on an exploration of minimal surfaces in AdS$_3$ space, focusing on surfaces that terminate at the boundary of AdS according to configurations defined by null polygonal Wilson loops. The authors utilize the gauge/gravity duality to paper these Wilson loops in the context of strong coupling, translating the problem into a mathematical framework that combines a generalized Sinh-Gordon equation with SU(2) Hitchin equations.

Central to the paper is the analysis of null polygonal Wilson loops, a simplified class of Wilson loops characterized by light-like segments, notable for their connections to scattering amplitudes in gauge theories. By leveraging integrability, the classical equations for a string worldsheet that conclude on a null polygonal boundary in AdS are tackled. The researchers restrict their focus to AdS$_3$ subspaces, where the problem is treated purely geometrically.

The core mathematical inquiry revolves around a reduction to a generalized Sinh-Gordon equation, which is then aptly linked to Hitchin's systems via a clever use of Pohlmeyer-type reduction. Alday and Maldacena examine classical solutions with an aim to illuminate how such configurations elucidate features of integrable systems. A novel connection is drawn to previous work by Gaiotto, Moore, and Neitzke, facilitating the identification of solutions for regular polygonal structures and analyses for octagonal Wilson loops through the relation between the Hitchin system and the moduli space of gauge theory vacua.

One striking result is the explicit computation of the area for minimal surfaces bounded by eight-sided polygons, translating geometrical attributes into strong-coupling scattering amplitudes of gluons through gauge/gravity duality. Here, an incursion into the field of spectral parameters and their relations to SU(2) Hitchin systems helps establish various cross-ratios pertinent to these surfaces, yielding a comprehensive understanding of the physical cross-ratios as(z) approach infinity.

The implications of this research marry theoretical insight with potential practical applications. By formalizing the process of calculating Wilson loop observables at strong coupling, the work enhances computational techniques, offering pathways to solving more complex gauge/string duality problems. Further, suggestions to extend this work to encompass more general AdS embeddings or higher genus surfaces provide a fertile ground for theoretical advancement.

Future directions, as hinted by the paper, include deeper explorations into the field of integrability within the higher-dimensional settings of AdS/CFT and potential connections to phenomena such as the wall-crossing behavior and spectral parameter evaluations in different brane configurations. Thus, the paper not only provides a cornerstone for understanding the interplay between geometry and physics beyond perturbative computations but also paves the way for subsequent endeavors in theoretical physics to grapple with the complexities of holographic dualities.