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

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