Decomposition, Trivially-Acting Symmetries, and Topological Operators

TL;DR

This work develops a topological-operator framework for decomposition in two-dimensional CFTs with trivially-acting symmetries, recasting these symmetries as a mix of topological defect lines and topological point operators. By gauging the mixed structure and tracking TPOs through the resulting universes, the authors refine standard decomposition and reveal how mixed anomalies and non-invertible symmetries modify partition functions and vacuum structure. The formalism uses algebra-object gauging and explicit projectors to compute how universes arise and how TPOs distribute among them, illustrated through detailed finite-group and fusion-category examples (including Rep(S3)). The results offer a systematic method to understand extended operator spectra under decomposition and point toward generalizations to higher dimensions and more intricate mixed gaugings.

Abstract

Trivially-acting symmetries in two-dimensional conformal field theory include twist fields of dimension zero which are local topological operators. We investigate the consequences of regarding these operators as part of the global symmetry of the theory. That is, we regard such a symmetry as a mix of topological defect lines (TDLs) and topological point operators (TPOs). TDLs related by a trivially-acting symmetry can join at a TPO to form non-trivial two-way junctions. Upon gauging, the local operators at those junctions can become vacua in a disjoint union of theories. Examining the behavior of the TPOs under gauging therefore allows us to refine decomposition by tracking the trivially-acting symmetries of each universe. Mixed anomalies between the TDLs and TPOs provide discrete torsion-like phases for the partition functions of these orbifolds, modifying the resulting decomposition. This framework also readily allows for the consideration of trivially-acting non-invertible symmetries.

Figures (16)

• Figure 1: The fusion of two group-like line operators.
• Figure 2: The fusion products for Rep$(S_3)$.
• Figure 3: Two effective symmetry lines joined at a topological junction.
• Figure 4: Two line operators joined by a point operator.
• Figure 5: The fusion of two point operators.