Integrable Systems for Generalized Toric Polygons and Higgsed 5d N=1 Theories

TL;DR

This paper extends the connection between toric Calabi-Yau 3-folds, integrable systems, and 5d N=1 theories to generalized toric polygons, revealing new birational transformations and brane web interpretations.

Contribution

It introduces a framework for integrable systems associated with generalized toric polygons and links them to Hanany-Witten transitions and Higgsed 5d theories.

Findings

Integrable systems for GTPs arise from birational transformations of known dimer systems.

Transformations correspond to Hanany-Witten brane transitions.

Higgsing higher-rank theories yields 5d N=1 theories with GTP-shaped toric diagrams.

Abstract

The interplay between toric Calabi-Yau 3-folds, dimer integrable systems, and 5-dimensional quantum field theories has proved fruitful. We extend this framework to generalized toric polygons (GTPs) and show that their integrable systems arise from refined birational transformations of known dimer integrable systems acting on the Casimirs and Hamiltonians as well as the Poisson structure and spectral curves. We argue that these transformations are realized as Hanany-Witten transitions producing (p,q) 5-brane webs dual to GTPs. We show that the resulting 5d N=1 theory is obtained by Higgsing a higher-rank theory whose associated toric Calabi-Yau has a toric diagram of the same shape as the GTP.