Segmented strings, brane tilings, and the Y-system

TL;DR

This paper demonstrates how the dynamics of segmented closed strings in AdS3 can be modeled using brane tilings and Y-systems, linking string motion to algebraic and combinatorial structures.

Contribution

It introduces a novel embedding of string motion into brane tiling mutation dynamics and relates the spectral curve to the Kasteleyn matrix determinant.

Findings

String motion modeled by Y-system with constraints

Spectral curve computed via Kasteleyn matrix

Constraints deformable by background two-form coupling

Abstract

I show that the motion of a closed string consisting of $n$ segments in AdS$_3$ can be embedded into the mutation dynamics of the $Y^{n,0}$ brane tiling. The determinant of the Kasteleyn matrix computes the spectral curve. The dynamics is governed by a Y-system with additional constraints ensuring that the string closes in target space. The constraints can be deformed by coupling the worldsheet to a background two-form whose field strength is proportional to the volume form.