Fusion of Critical Defect Lines in the 2D Ising Model

TL;DR

We address the universal fusion of parallel defect lines in the critical 2D Ising model by representing defects as labeled pairs $(a,\Lambda)$ with $a\in\{\mathbf{1},\boldsymbol{\epsilon},\boldsymbol{\sigma}\}$ and $\Lambda\in O(1,1)/\mathbb{Z}_2$, and show the fusion rule is $ (a,\Lambda)\star(a',\Lambda')=(a\times a',\Lambda\Lambda')$ combined with the Verlinde algebra for the Ising model and Lorentz-group multiplication. The analysis relies on a regulated fusion construction ${\cal D}\star{\cal D}'=\lim_{\delta\to 0} e^{-C/\delta}{\cal D} e^{-\delta {\cal H}}{\cal D}'$, translated through a folding–unfolding dictionary to boundary states of a $c=1$ orbifold and then back to Ising defects. The main contributions include a concrete, universal fusion rule that incorporates both the Ising fusion rules and $O(1,1)/\mathbb{Z}_2$ structure, plus a precise Ramond-sector normalization. This work connects defect fusion in the critical Ising model to established algebraic structures, informing both defect CFT and integrable defect classifications.

Abstract

Two defect lines separated by a distance delta look from much larger distances like a single defect. In the critical theory, when all scales are large compared to the cutoff scale, this fusion of defect lines is universal. We calculate the universal fusion rule in the critical 2D Ising model and show that it is given by the Verlinde algebra of primary fields, combined with group multiplication in O(1,1)/Z_2. Fusion is in general singular and requires the subtraction of a divergent Casimir energy.

Figures (2)

• Figure 1: The two-defect system discussed in the text. The green dots are arguments of a typical two-point function at a horizontal scale $x\gg \delta$. The fusion product ${\cal D}\star {\cal D}^{\prime}$ gives an effective description of this system in the limit where $\delta$ is very large compared to the lattice spacing $\Delta$. Only in this limit is fusion universal.
• Figure 2: Folding transforms defect lines (in red) of the critical Ising model to boundary conditions of the $c=1$$\mathbb{Z}_2$-orbifold theory.