Higher Form Symmetries and M-theory

TL;DR

This paper develops a geometric framework for discrete higher-form symmetries in quantum field theories realized via M-theory geometric engineering. By analyzing defect groups and flux non-commutativity, it derives the mixed 't Hooft anomalies that constrain the global structure of 4d $\mathcal N=1$ theories, 7d $\mathcal N=1$ theories, and a broad range of 5d SCFTs, including toric and non-Lagrangian cases. The approach yields explicit global-structure classifications and predicts how instanton charges and Chern–Simons couplings shape the surviving higher-form symmetries, with consistency checks tied to 5d dualities. It also extends to M-theory on $G_2$ spaces to touch on 4d $\mathcal N=1$ SYM, offering a unified picture of discrete symmetries across dimensions and providing practical tools for analyzing dualities in strongly coupled theories.

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.

Figures (8)

• Figure 1: Schematic topology of $B_3^L$Garcia-Etxebarria:2016bpb
• Figure 2: Toric diagram for $Y^{p,q}$. We have defined $l \equiv p-q$.
• Figure 3: One of the triangles defined in the text.
• Figure 4: Triangulation of $Y^{p,q}$ considered in the text.
• Figure 5: Geometry realizing $\mathfrak{su}(p)$ with one flavor. We show the added triangle, which is of minimal area.