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

TL;DR

This paper analyzes line operators in a supergroup sigma-model, demonstrating their fusion properties and deriving the Hirota equation from first principles without relying on string hypotheses or integrability assumptions.

Contribution

It provides a first-principles derivation of the Hirota equation for the sigma-model, showing the divergence-free nature of the transfer matrix and its relation to flat connections.

Findings

Transfer matrix is divergence-free at second order in perturbation theory.

Transfer matrix satisfies the Hirota equation for all parameter values.

Derivation of Hirota equation without string hypothesis or quantum integrability assumptions.

Abstract

We study line operators in the two-dimensional sigma-model on PSl(n|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.