Surface defects as transfer matrices

Abstract

The supersymmetric index of the 4d 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 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-S and -S k theories in the presence of the surface defects.A Definitions and useful formulas 53 A.1 Theta functions 53 A.2 Elliptic gamma function 54

4d N = 1 $$ \mathcal{N}=1 $$ from 6d (1, 0)

“…The indices of the four dimensional theories were shown in [9] to be written in terms of eigenfunctions of certain integrable models for any number of branes and k of Z k (there should be also a generalization for any ADE type of singularity). See [38,39] for explicit expressions for the Hamiltonians of these integrable models and [40][41][42] for the k = 1 N = 2 case. Moreover knowing the index of a theory one can in principle extract its anomalies [43][44][45][46].…”

Integrability as duality: The Gauge/YBE correspondence

“…First, the role of a single rapidity line is now played by a pair of zig-zag paths which run parallel with each other in the opposite direction. 24 Second, in the bottom figure of Figure 15 such a pair of zig-zag paths are twisted in four locations. One can therefore say what we obtain here is a "doubled twisted" YBE.…”

“…The actual integrable model has been discussed in many papers in the literature. While it is difficult to precisely summarize the rich literature, it might be useful for readers to loosely classify the literature into: 4d N = 1 on S 1 × S 3 [14,15,16,11,12,17,24,23,28], on S 1 × S 3 Z r [13,25,21,22,29], 3d N = 2 on S 1 × S 2 [18,20,27,31], on S 3 [76,11,12] and S 3 Z r [22], 2d N = (2, 2) on S 2 [26], on T 2 [18,19,30], and 1d N = 4 on S 1 [26].…”

On the correspondence between surface operators in Argyres-Douglas theories and modules of chiral algebra

“…14 Note that this choice of x i is slightly different from the one in[34] 15. See[44,45,46,47,48,49,50] for earlier applications of this prescription.…”