New dimer integrable systems and defects in five dimensional gauge theory

We study the relation between the quantum integrable systems derived from the dimer graphs and five dimensional $\mathcal{N}=1$ supersymmetric gauge theories on $S^1 \times \mathbb{R}^4$. We construct integrable systems based on new dimer graphs obtained from modification of hexagon dimer diagram. We study the gauge theories in correspondence to the newly proposed integrable systems. By examining three types of defects -- a line defect, a canonical co-dimensional two defect and a monodromy defect -- in five-dimensional gauge theory with $\mathcal{N}=1$ supersymmetry and $\Omega_{\varepsilon_1,\varepsilon_2}$-background. We identify, in the $\varepsilon_2 \to 0$ limit, the canonical co-dimensional two defect satisfying the Baxter T-Q equation of the generalized $A$-type dimer integrable system, and the monodromy defect as its common eigenstate of the commuting Hamiltonians, with the eigenvalues being the expectation value of the BPS Wilson loop in the anti-symmetric representation of the bulk gauge group.