Higher Form Symmetries and M-theory

TL;DR

This paper explores the geometric origins of discrete higher form symmetries in quantum field theories via M-theory, revealing how flux non-commutativity leads to anomalies that constrain global structures, with applications to various supersymmetric theories.

Contribution

It introduces a geometric framework in M-theory to analyze higher form symmetries and anomalies, extending the understanding to 4d, 5d, and 7d supersymmetric theories and predicting new phenomena.

Findings

Reproduces known global structure classifications in 4d $ ext{SYM}$.

Identifies mixed 't Hooft anomalies in 7d $ ext{SYM}$ from flux non-commutativity.

Predicts new anomalies and symmetry structures in 5d SCFTs.

Abstract

We discuss the geometric origin of discrete higher form symmetries of quantum field theories in terms of defect groups from geometric engineering in M-theory. The flux non-commutativity in M-theory gives rise to (mixed) 't Hooft anomalies for the defect group which constrains the corresponding global structures of the associated quantum fields. We analyze the example of 4d $\mathcal{N}=1$ SYM gauge theory in four dimensions, and we reproduce the well-known classification of global structures from reading between its lines. We extend this analysis to the case of 7d $\mathcal{N}=1$ SYM theory, where we recover it from a mixed 't Hooft anomaly among the electric 1-form center symmetry and the magnetic 4-form center symmetry in the defect group. The case of five-dimensional SCFTs from M-theory on toric singularities is discussed in detail. In that context we determine the corresponding 1-form and 2-form defect groups and we explain how to determine the corresponding mixed 't Hooft anomalies from flux non-commutativity. Several predictions for non-conventional 5d SCFTs are obtained. The matching of discrete higher-form symmetries and their anomalies provides an interesting consistency check for 5d dualities.