The exact mass-gap of the supersymmetric CP$^{N-1}$ sigma model

A formula for the mass-gap of the supersymmetric \CP^{n-1} sigma model (n > 1) in two dimensions is derived: m/\Lambda_{\overline{\rm MS}}=\sin(\pi\Delta)/(\pi\Delta) where \Delta=1/n and m is the mass of the fundamental particle multiplet. This result is obtained by comparing two expressions for the free-energy density in the presence of a coupling to a conserved charge; one expression is computed from the exact S-matrix of K\"oberle and Kurak via the thermodynamic Bethe ansatz and the other is computed using conventional perturbation theory. These calculations provide a stringent test of the S-matrix, showing that it correctly reproduces the universal part of the beta-function and resolving the problem of CDD ambiguities.