Integrable theories and generalized graded Maillet algebras

TL;DR

This paper develops a formalism to analyze the integrability of non-ultralocal models, enabling their lattice discretization, with applications to string theory subsectors and specific models like WKIS and Dirac fermions.

Contribution

It introduces a modified Maillet formalism to handle second derivatives in Lax algebras, facilitating lattice formulations of complex non-ultralocal integrable models.

Findings

Derived a well-defined algebra of transition matrices for the AAF model.

Explicit forms of r-matrices for WKIS and Dirac fermion models.

Extended Maillet's formalism to include second derivatives of delta functions.

Abstract

We present a general formalism to investigate the integrable properties of a large class of non-ultralocal models which in principle allows the construction of the corresponding lattice versions. Our main motivation comes from the su(1|1) subsector of the string theory on AdS_5 x S^5 in the uniform gauge, where such type of non-ultralocality appears in the resulting Alday-Arutyunov-Frolov (AAF) model. We first show how to account for the second derivative of the delta function in the Lax algebra of the AAF model by modifying Maillet's r- and s-matrices formalism, and derive a well-defined algebra of transition matrices, which allows for the lattice formulation of the theory. We illustrate our formalism on the examples of the bosonic Wadati-Konno-Ichikawa-Shimizu (WKIS) model and the two-dimensional free massive Dirac fermion model, which can be obtained by a consistent reduction of the full AAF model, and give the explicit forms of their corresponding r-matrices.