Generalizing fusion rules by shuffle: Symmetry-based classifications of nonlocal systems constructed from similarity transformations

TL;DR

This paper explores the classification of nonlocal systems using symmetry and similarity transformations, revealing a fundamental link between ring isomorphism and physical symmetry properties in quantum field theories.

Contribution

It introduces a novel approach to classify nonlocal topological field theories via Galois shuffle operations and symmetry transformations, connecting local nonunitary and nonlocal unitary models.

Findings

Fusion rings outside NIM-rep are reconstructed from local nonunitary CFTs.

Nonlocal SymTFTs are ring isomorphic to those of local nonunitary CFTs.

Classification of RG flows and boundary phenomena are linked between models.

Abstract

We study fusion rings, or symmetry topological field theories (SymTFTs), which lie outside the non-negative integer matrix representation (NIM-rep), by combining knowledge from generalized symmetry and that from pseudo-Hermitian systems. By applying the Galois shuffle operation to the SymTFTs, we reconstruct fusion rings that correspond to nonlocal CFTs constructed from the corresponding local nonunitary CFTs by applying the similarity transformations. The resultant SymTFTs are outside of NIM-rep, whereas they are ring isomorphic to the NIM-rep of the corresponding local nonunitary CFTs. We study the consequences of this correspondence between the nonlocal unitary model and local nonunitary models. We demonstrate the correspondence between their classifications of massive or massless renormalization group flows and the discrepancies between their boundary or domain wall phenomena. Our work reveals a new connection between ring isomorphism and similarity transformations, providing the fundamental implications of ring-theoretic ideas in the context of symmetry in physics.