3-body quantization condition in a unitary formalism

Unitarity identifies all power-law finite-volume effects and is, therefore, the crucial S-matrix principle for a mapping between experimental results and those of Lattice QCD calculations. In this contribution we review how 3-body unitarity constrains the form of the 3-body scattering amplitude parametrized by the tower of isobars. The result is discretized and projected to the irreducible representations of the cubic group, leading to a fully relativistic 3-body quantization condition. The latter is used to deduce the finite-volume excited level spectrum of the $\pi^+\pi^+\pi^+$ system, which agrees nicely with the available lattice results by the NPLQCD collaboration.