Branes and integrable lattice models

Abstract

This is a brief review of my work on the correspondence between four-dimensional N = 1 supersymmetric field theories realized by brane tilings and two-dimensional integrable lattice models. I explain how to construct integrable lattice models from extended operators in partially topological quantum field theories, and elucidate the correspondence as an application of this construction.

Surface defects and elliptic quantum groups

“…3 In addition to the constraint |n| ≤ 1, we require that for every bounded face, the domain walls bounding it all go upward locally for some choice of the vertical direction. See [15] for a discussion on this point.…”

“…From the fact that the supersymmetric index is invariant under continuous deformations of the checkerboard lattice, one deduces that it is captured by a correlation function of line operators in a TQFT. A general argument in open/closed TQFTs [15] then shows that the correlation function equals the partition function of a lattice model, the Bazhanov-Sergeev model in this case. Finally, the TQFT has a hidden extra dimension that emerges via string dualities, and its existence implies the integrability of the model [19,20].…”

“…The proposal builds on a recent development [11] in the correspondence between N = 1 supersymmetric field theories and integrable lattice models [12][13][14][15]. Therefore, let me first review the relevant results.…”

Integrable Lattice Spin Models from Supersymmetric Dualities

“…Such construction gives an opportunity to obtain integrable lattice models directly using brane construction. We will not discuss this direction in the paper and refer the interested reader to the papers [23,25,32].…”

Entanglement branes, modular flow, and extended topological quantum field theory

“…This is a boundary condition that was identified in [16] as an entanglement brane. In a different context, this boundary condition has been used to obtain integrable lattice models from line operators in TQFT's (see [17] and the references within) We begin in section 2 by reviewing the axioms of "open-closed" TQFT and its diagrammatic notation. We will avoid discussion of the underlying category theory, details of which can be found elsewhere [18].…”