# On integrable boundaries in the 2 dimensional $O(N)$ $\sigma$-models

## TL;DR

This paper explores integrable boundary conditions in 2D O(N) sigma models through classical and quantum methods, including conserved charges, transfer matrices, spectral curves, and Bethe-Yang equations.

## Contribution

It provides a comprehensive analysis of integrable boundaries in O(N) sigma models at both classical and quantum levels, connecting spectral curves with boundary Bethe-Yang equations.

## Key findings

- Classical integrable boundary conditions identified via conserved charges.
- Quantum solutions derived from boundary Yang-Baxter equation.
- Connection established between thermodynamic limit and spectral curve.

## Abstract

We make an attempt to map the integrable boundary conditions for 2 dimensional non-linear O(N) $\sigma$-models. We do it at various levels: classically, by demanding the existence of infinitely many conserved local charges and also by constructing the double row transfer matrix from the Lax connection, which leads to the spectral curve formulation of the problem; at the quantum level, we describe the solutions of the boundary Yang-Baxter equation and derive the Bethe-Yang equations. We then show how to connect the thermodynamic limit of the boundary Bethe-Yang equations to the spectral curve.