[Paper Review] Decomposition, Trivially-Acting Symmetries, and Topological Operators

Daniel Robbins, Eric Sharpe+1 more|arXiv (Cornell University)|Nov 25, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper develops a framework to classify and track trivially-acting symmetries in 2D conformal field theories by treating them as topological operators—specifically, topological point operators (TPOs) and topological defect lines (TDLs)—and analyzing their fusion and gauging. The key contribution is that mixed anomalies between TDLs and TPOs modify orbifold partition functions and obstruct decomposition into disjoint universes, refining the standard notion of decomposition by encoding symmetry structure via extension classes and discrete torsion phases.

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.

Motivation & Objective

• To systematically study trivially-acting symmetries in 2D CFTs, which act non-effectively on local operators but influence global structure via topological operators.
• To extend the framework of decomposition to include the role of topological point operators (TPOs) and their fusion with topological defect lines (TDLs) under trivial symmetries.
• To analyze how mixed anomalies between TDLs and TPOs modify partition functions in orbifolds, particularly by introducing discrete torsion-like phases.
• To generalize decomposition to non-invertible and non-abelian symmetries by focusing on physical behavior of topological operators rather than algebraic group structures.
• To clarify the interplay between gauging trivially-acting symmetries and the resulting structure of vacua in disjoint union theories.

Proposed method

• The authors model trivially-acting symmetries as group extensions Γ of K by G, where K acts trivially on local operators, and associate TDLs and TPOs to the group elements.
• They use the fusion of TDLs and TPOs to define two-way junctions, with TPOs arising as bound states of TDLs under the extension class δ:K→Γ.
• The paper introduces a mixed anomaly between TDLs and TPOs, encoded by characters χγ(δ(k)), which modifies the partition function of the orbifolded theory.
• By gauging the global symmetry Γ, the authors derive the extension class of the gauged theory, showing that the kernel of the new extension class ˜δ corresponds to the mixed anomaly.
• They use Pontryagin duality to relate the ungauged and gauged theories, showing that the mixed anomaly in one theory determines the extension class in the other.
• The framework is applied to examples with abelian and non-abelian groups, including Z2×Z2, S3, and Q8, to demonstrate obstruction of decomposition by mixed anomalies.

Experimental results

Research questions

• RQ1 How do trivially-acting symmetries, which act non-effectively on local operators, manifest as topological operators in 2D CFTs?
• RQ2 What is the role of topological point operators (TPOs) in the decomposition of theories with non-effective symmetries?
• RQ3 How do mixed anomalies between TDLs and TPOs modify the partition functions of orbifold theories?
• RQ4 In what way does the presence of a mixed anomaly obstruct the standard decomposition into a disjoint union of universes?
• RQ5 How can the structure of trivially-acting symmetries be captured via extension classes and duality in the gauged theory?

Key findings

• Trivially-acting symmetries in 2D CFTs are realized as TPOs bound to TDLs via the extension class δ:K→Γ, with the TPOs arising as junctions of TDLs.
• Upon gauging Γ, the TPOs from the ungauged theory become freestanding local operators in the gauged theory, but their behavior is modified by mixed anomalies.
• The mixed anomaly between TDLs and TPOs is encoded by characters χγ(δ(k)), which act as discrete torsion phases in the orbifold partition function.
• The extension class of the gauged theory, ˜δ, has kernel equal to K, indicating that TPOs remain bound to lines when the anomaly is non-trivial, obstructing full decomposition.
• The framework shows that decomposition is refined by tracking trivially-acting symmetries through TPOs, with the extension class and anomaly data determining the structure of the resulting disjoint universes.
• For abelian Γ, the duality between ungauged and gauged theories ensures that the mixed anomaly in one determines the extension class in the other, confirming consistency via Pontryagin duality.