Fusion of Critical Defect Lines in the 2D Ising Model

TL;DR

This paper investigates how two critical defect lines in the 2D Ising model effectively merge into a single defect at large scales, revealing a universal fusion rule linked to the Verlinde algebra and group multiplication.

Contribution

It derives the universal fusion rule for defect lines in the critical 2D Ising model, connecting it to the Verlinde algebra and group theory, and addresses the singular nature of fusion.

Findings

Fusion of defect lines follows the Verlinde algebra.

Fusion rule involves group multiplication in O(1,1)/Z_2.

Fusion requires subtracting divergent Casimir energy.

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.