Surface defects as transfer matrices

TL;DR

This paper connects surface defects in 4d supersymmetric theories to transfer matrices in integrable 2d lattice models, identifying specific L-operators and verifying their correspondence through index computations.

Contribution

It introduces a novel representation of surface defects as transfer matrices using L-operators, linking supersymmetric indices to integrable lattice models.

Findings

Surface defects correspond to transfer matrices in lattice models.

Identified L-operator as Sklyanin's in the eight-vertex model.

Verified the correspondence through index calculations in specific theories.

Abstract

The supersymmetric index of the 4d $\mathcal{N} = 1$ theory realized by a brane tiling coincides with the partition function of an integrable 2d lattice model. We argue that a class of half-BPS surface defects in brane tiling models are represented on the lattice model side by transfer matrices constructed from L-operators. For the simplest surface defects in theories with $\mathrm{SU}(2)$ flavor groups, we identify the relevant L-operator as that discovered by Sklyanin in the context of the eight-vertex model. We verify this identification by computing the indices of class-$\mathcal{S}$ and -$\mathcal{S}_k$ theories in the presence of the surface defects.