Fusion of line operators in conformal sigma-models on supergroups, and the Hirota equation

We study line operators in the two-dimensional sigma-model on PSl(n vertical bar n) using the current-current OPEs. We regularize and renormalize these line operators, and compute their fusion up to second order in perturbation theory. In particular we show that the transfer matrix associated to a one-parameter family of flat connections is free of divergences. Moreover this transfer matrix satisfies the Hirota equation (which can be rewritten as a Y-system, or Thermodynamic Bethe Ansatz equations) for all values of the two parameters defining the sigma-model. This provides a first-principles derivation of the Hirota equation which does not rely on the string hypothesis nor on the assumption of quantum integrability.