The $c$-map, Tits Satake subalgebras and the search for $\mathcal{N}=2$ inflaton potentials

In this paper we address the general problem of including inflationary models exhibiting Starobinsky-like potentials into (symmetric) $\mathcal{N}=2$ supergravities. This is done by gauging suitable abelian isometries of the hypermultiplet sector and then truncating the resulting theory to a single scalar field. By using the characteristic properties of the global symmetry groups of the $\mathcal{N}=2$ supergravities we are able to make a general statement on the possible $\alpha$-attractor models which can obtained upon truncation. We find that in symmetric $\mathcal{N}=2$ models group theoretical constraints restrict the allowed values of the parameter $\alpha$ to be $\alpha=1,\,\frac{2}{3},\, \frac{1}{3}$. This confirms and generalizes results recently obtained in the literature. Our analysis heavily relies on the mathematical structure of symmetric $\mathcal{N}=2$ supergravities, in particular on the so called $c$-map connection between Quaternionic K\"ahler manifolds starting from Special K\"ahler ones. A general statement on the possible consistent truncations of the gauged models, leading to Starobinsky-like potentials, requires the essential help of Tits Satake universality classes. The paper is mathematically self-contained and aims at presenting the involved mathematical structures to a public not only of physicists but also of mathematicians. To this end the main mathematical structures and the general gauging procedure of $\mathcal{N}=2$ supergravities is reviewed in some detail.