The R-matrix bootstrap for the 2d O(N) bosonic model with a boundary

TL;DR

This paper extends the S-matrix bootstrap to 1+1d theories with $O(N)$ symmetry and a boundary, introducing the R-matrix bootstrap to explore the space of reflection matrices and identify integrable models.

Contribution

It develops the R-matrix bootstrap framework for boundary theories with $O(N)$ symmetry, revealing the structure of integrable reflection matrices and their analytic properties.

Findings

Identified vertices corresponding to integrable R-matrices with no free parameters.

Mapped the boundary of the allowed R-matrix space to integrable theories.

Derived the analytic form of a previously unknown R-matrix for the periodic Yang-Baxter solution.

Abstract

The S-matrix bootstrap is extended to a 1+1d theory with $O(N)$ symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole boundary represents integrable theories. We also consider R-matrices which are analytic in an extended region beyond the physical cuts, thus forbidding poles (resonances) in that region. In certain models, this drastically reduces the allowed space of R-matrices leading to new vertices that again correspond to integrable theories. We also work out the dual problem, in particular in the case of extended analyticity, the dual function has cuts on the physical line whenever unitarity is saturated. For the periodic Yang-Baxter solution that has zero transmission, we computed the R-matrix initially using the bootstrap and then derived its previously unknown analytic form.