We study one of the simplest integrable two-dimensional quantum field theories with a boundary: N free non-compact scalars in the bulk, constrained non-linearly on the boundary to lie on an (N − 1)-sphere of radius \( 1/\sqrt{g} \) . The N = 1 case reduces to the single-channel Kondo problem, for N = 2 the model describes dissipative Coulomb charging in quantum dots, and larger N is analogous to higher-spin impurity or multi-channel scenarios. Adding a boundary magnetic field — a linear boundary coupling to the scalars — enriches the model’s structure while preserving integrability. Lukyanov and Zamolodchikov (2004) conjectured an expansion for the boundary free energy on the infinite half-cylinder in powers of the magnetic field. Using large-N saddle-point techniques, we confirm their conjecture to next-to-leading order in 1/N. Renormalization of the subleading solution turns out to be highly instructive, and we connect it to the RG running of g studied by Giombi and Khanchandani (2020).
