Non-invertible duality defect and non-commutative fusion algebra

TL;DR

This paper investigates non-invertible duality symmetries in 2D and 4D gauge theories, revealing non-commutative fusion algebras and emergent symmetries at specific CFT points through diagonal gauging techniques.

Contribution

It introduces a novel approach to studying non-invertible duality symmetries via diagonal gauging, demonstrating non-commutative fusion rules in both 2D CFT and 4D gauge theories.

Findings

Fusion algebra is non-commutative.

Non-invertible symmetry becomes emergent at irrational CFT points.

Similar non-commutative fusion rules are found in 4D gauge theory.

Abstract

We study non-invertible duality symmetries by gauging a diagonal subgroup of a non-anomalous U(1) $\times$ U(1) global symmetry. In particular, we employ the half-space gauging to $c=2$ bosonic torus conformal field theory (CFT) in two dimensions and pure U(1) $\times$ U(1) gauge theory in four dimensions. In $c=2$ bosonic torus CFT, we show that the non-invertible symmetry obtained from the diagonal gauging becomes emergent on an irrational CFT point. We also calculate the fusion rules concerning the duality defect. We find out that the fusion algebra is non-commutative. We also obtain a similar result in pure U(1) $\times$ U(1) gauge theory in four dimensions.