In this thesis, we study the geometric origin of discrete higher-form symmetries and associated anomalies of $d$-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 $X$. As a warm-up, we analyze the example of 7d $mathcal{N}=1$ 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 $d+1$ dimensional Symmetry TFT, or SymTFT for short, by reducing the topological sector of 11d supergravity on the boundary $partial X$ of the space $X$. 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 $partial X$, 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 $mathcal{N}=2$ supersymmetric quantum field theories that arise from IIB on hypersurface singularities. The examples we discuss include a broad class of$mathcal{N}=2$ theories such as Argyres-Douglas and $D_p^b(G)$ 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 $D_p^b(G)$ theories, in the simple case when $b=h^vee (G)$, 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 $bneq h^vee (G)$, we use a similar Maruyoshi-Song flow to conjecture that the defect groups of $D_p^b(G)$ theories are given by those of $G^{(b)}[k]$ theories. In the cases of $G=A_n, ;E_6, ;E_8$ we cross-check our result by calculating the BPS quivers of the $G^{(b)}[k]$ theories and looking at the cokernel of their intersection matrix.