Generalised Symmetries and String Theory

Abstract

In this thesis, we study the geometric origin of discrete higher-form symmetries and associated anomalies of -dimensional quantum field theories in terms of defect groups via geometric engineering in M-theory and type IIB string theory by reduction on non-compact spaces . As a warm-up, we analyze the example of 7d SYM theory, where we recover it from a mixed 't Hooft anomaly among the electric 1-form centre symmetry and the magnetic 4-form centre symmetry in the defect group. The case of 5-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. For these theories, we further determine the dimensional Symmetry TFT, or SymTFT for short, by reducing the topological sector of 11d supergravity on the boundary of the space . Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in the cohomology of the space , which in turn gives rise to the background fields for discrete symmetries.

We further extend our analysis to study the 1-form symmetries of 4-dimensional supersymmetric quantum field theories that arise from IIB on hypersurface singularities. The examples we discuss include a broad class of theories such as Argyres-Douglas and theories. In our computation of the defect groups of hypersurface singularities, we rely on a fundamental result in singularity theory known as Milnor's theorem which establishes a connection between the topology of the hypersurface and the local behaviour of the singularity. For the theories, in the simple case when , we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where , we use a similar Maruyoshi-Song flow to conjecture that the defect groups of theories are given by those of theories. In the cases of we cross-check our result by calculating the BPS quivers of the theories and looking at the cokernel of their intersection matrix.