Segmented strings, brane tilings, and the Y-system

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.

40 pages, 22 figures

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