Integrable models from 4d holomorphic BF theory

TL;DR

This paper constructs 2d holomorphic integrable field theories from 4d holomorphic BF theory using defect setups, providing explicit examples and analyzing their classical solutions and symmetries.

Contribution

It introduces a method to derive 2d integrable models from 4d holomorphic BF theory and explores their properties and relation to higher-dimensional integrability.

Findings

Constructed explicit 2d theories from 4d holomorphic BF

Identified an infinite family of solutions to equations of motion

Explained the position of 2d holomorphic integrability between 1d and 2d notions

Abstract

We show how to construct 2d field theories with holomorphic integrability from defect setups in 4d holomorphic BF. In a simple example setup, we explicitly construct the 2d theory and perform an initial classical analysis. Making use of the symmetries, we are able to write down an infinite family of solutions to the equations of motion. Comparing with more typical integrable systems, we explain how 2d holomorphic integrability sits between the standard notions of integrability in one and two dimensions. These 2d theories are designed as toy models for integrable theories in three and four dimensions, many of which can be understood as partially or totally holomorphic. We comment on the implications for higher-dimensional integrability and aspects of quantization in the concluding remarks.