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

TL;DR

The paper develops a framework uniting generalized symmetry and pseudo-Hermitian systems to study fusion rings beyond NIM-rep via the Galois shuffle in SymTFTs. It shows that shuffled SymTFTs correspond to nonlocal unitary CFTs built from local nonunitary CFTs by similarity transformations and are ring-isomorphic to the original local theory's NIM-rep, enabling a correspondence between their RG classifications and boundary phenomena. The authors illustrate the construction with explicit examples (M(2,5) and Osp(1,2)_1) and discuss implications for massless and massive RG flows, smeared boundary CFTs, and boundary/defect phenomena. The work reveals a deep link between ring-theoretic structures and similarity transformations, offering a new lens on symmetry in physics and potential ties to dS/CFT.

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.

Figures (1)

• Figure 1: The schematic correlation chart of the massless RG flow from UV theory to IR theory. We use "$'$" symbol for IR rings. Notice that the RG flow is generally complicated in the sense that the simple objects in UV can be transformed to some complicated objects in IR.