Fusion of Integrable Defects and the Defect $g$-Function

Abstract: We study exact defect $g$-functions for integrable line defects in two-dimensional integrable quantum field theory and use them to probe defect fusion. We consider three settings: fusion of purely transmitting topological defects, fusion of non-topological defects with reflection and transmission, and fusion of a defect with an integrable boundary. For topological defects, the separated logarithmic $g$-function is additive, and the fusion limit is controlled by the multiplicative composition of transmission factors. For non-topological defects, separation-dependent phases in the Bethe-Yang equations produce oscillatory finite-size effects, while the fused defect is described by effective reflection and transmission amplitudes. In the Ising examples studied here, fusion involving non-topological defects lowers the finite localized contribution to the entropy, whereas topological defect-boundary fusion leaves it unchanged.