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

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.

Figures (3)

• Figure 1: The toric diagrams $\Delta$ of the $dP_1$ model and $\Delta^\prime$ of the $L^{2,5,1}$ model are related by a birational transformation. Under a Hanany-Witten transition, the $(p,q)$-web dual to $\Delta$ becomes a $(p,q)$-web where two external $5$-branes end on the $7$-brane, making the dual polygon $\Delta^\vee$ a GTP. Because $\Delta^\vee$ is of the same shape as $\Delta^\prime$, we can identify the two $5$-branes terminating on the same $7$-brane with zig-zag paths $w_3$ and $w_4$ in the brane tiling of $L^{2,5,1}$. In the corresponding dimer integrable system, we can impose $w_3 = w_4$ resulting in a reduced (frozen) integrable system that describes the dynamics of the $5d$$\mathcal{N}=1$ theory corresponding to the GTP $\Delta^\vee$. We also note that the reduced integrable system is also birationally equivalent to the dimer integrable system corresponding to $\Delta$.
• Figure 2: The brane tiling and toric diagram for $dP_1$.
• Figure 3: The brane tiling and toric diagram for $L^{2,5,1}$.