Non-invertible symmetries and quantum supergravities in 4 and 5 dimensions

In this thesis, we discuss non-invertible symmetries in four and five dimensional supegravities and their corresponding non-invertible topological operators, in the context of the no-global symmetry conjecture and its connection to the completeness of the spectrum. After discussing the simpler examples of four-dimensional axion-Maxwell and five-dimensional Maxwell Chern-Simons theories, we introduce N=2 supergravity in four dimensions and its ªgeneralized theta termsº which seem to break every possible 0-form axionic symmetry in the scalar sector. We argue that there are non-invertible 0-form symmetries that are not explicitly broken by these terms. The no-global symmetry conjecture then implies the necessity of non-perturbative corrections due to fundamental instantons, consistently with what is expected from string theory models. After reviewing N=1 five-dimensional supergravity, we construct its non-invertible operator both from a purely five-dimensional point of view and, in M-theory models, by dimensionally reducing the M-theory non-invertible symmetry operator. Then, by further reducing the topological non-invertible operator obtained for the five-dimensional theory, we construct explicitly a 0-form non-invertible symmetry for N=2 four-dimensional supergravity. In type IIA compactifications on Calabi-Yau three-folds, we show that perturbative corrections to the leading N=2 prepotential do not affect the non-invertible operator that we have obtained.