McKay quivers and decomposition

TL;DR

This paper explores the relationship between decomposition in quantum field theories with higher-form symmetries and McKay quivers, revealing geometric and algebraic insights into their structure and connections to orbifold models and gerbes.

Contribution

It establishes the equivalence between orbifold decomposition and McKay quivers, providing geometric interpretations and a group-theoretic derivation for certain cases.

Findings

Decomposition of orbifold sigma-models corresponds to disconnected McKay quivers.

Each quiver component has a geometric meaning via decomposition formulas.

Group theoretic derivation aligns with the geometric interpretation for central trivially acting groups.

Abstract

When a quantum field theory in $d$-spacetime dimensions possesses a global $(d-1)$-form symmetry, it can decompose into disjoint unions of other theories. This is reflected in the physical quantities of the theory and can be used to study properties of the constituent theories. In this note we highlight the equivalence between the decomposition of orbifold $\sigma$-models and disconnected McKay quivers. Specifically, we show in numerous examples that each component of a McKay quiver can be given definitive geometric meaning through the decomposition formulae. In addition, we give a purely group and representation theoretic derivation of the quivers for the cases where the trivially acting part of the orbifold group is central. As expected, the resulting quivers are compatible with the case of $\sigma$-models on `banded' gerbes.