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arXiv:2605.20688

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

Yang He, Yunfeng Jiang, Yuxiao Liu

Published May 20, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2605.20688

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Fusion of Integrable Defects and the Defect ggg-Function
We study exact defect ggg-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 ggg-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.
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The modern way to understand symmetries of a quantum field theory is via its topological defects in various dimensions. In this contribution to the proceedings we focus on line defects in 2d QFT and we point out that topological defects naturally embed into a larger class, namely translation invariant defects. The latter still allow for non-singular fusion and one obtains a monoidal category of translation invariant defects which contains that of topological defects as a full subcategory. We give a simple perturbative description of translation invariant defects in a perturbed conformal field theory via chiral three-dimensional topological field theory. We show in the example of the Ising CFT and the Lee-Yang CFT that even if no topological defects survive the deformation, some translation invariant defects still do.
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