Symmetries and topological operators, on average

TL;DR

This paper investigates how symmetries in disordered quantum field theories behave after averaging, revealing emergent topological operators, exotic selection rules, and their implications for gauge coupling and holography.

Contribution

It constructs symmetry operators in disordered theories that re-emerge after averaging and analyzes their properties, including coupling and gauging limitations, with implications for LogCFTs and holography.

Findings

Symmetry operators become topological after average.

Emergent symmetries can lead to exotic selection rules.

Obstructions exist to gauging these emergent symmetries in the bulk.

Abstract

We study Ward identities and selection rules for local correlators in disordered theories where a 0-form global symmetry of a QFT is explicitly broken by a random coupling $h$ but it re-emerges after quenched average. We consider $h$ space-dependent or constant. In both cases we construct the symmetry operator implementing the group action, topological after average. In the first case, relevant in statistical systems with random impurities, such symmetries can be coupled to external backgrounds and can be gauged, like ordinary symmetries in QFTs. We also determine exotic selection rules arising when symmetries emerge after average in the IR, explaining the origin of LogCFTs from symmetry considerations. In the second case, relevant in AdS/CFT to describe the dual boundary theory of certain bulk gravitational theories, the charge operator is not purely codimension-1, it can be defined only on homologically trivial cycles and on connected spaces. Selection rules for average correlators exist, yet such symmetries cannot be coupled to background gauge fields in ordinary ways and cannot be gauged. When the space is disconnected, in each connected component charge violation occurs, as expected from Euclidean wormholes in the bulk theory. Our findings show the obstruction to interpret symmetries emergent after average as gauged in the bulk.