Generalized two-field $\alpha$-attractors from the hyperbolic triply-punctured sphere

We study generalized two-field $\alpha$-attractor models whose rescaled scalar manifold is the triply-punctured sphere endowed with its complete hyperbolic metric, whose underlying complex manifold is the modular curve $Y(2)$. Using an explicit embedding into the end compactification, we compute solutions of the cosmological evolution equations for a few globally well-behaved scalar potentials, displaying particular trajectories with inflationary behavior as well as more general cosmological trajectories of surprising complexity. In such models, the orientation-preserving isometry group of the scalar manifold is isomorphic with the permutation group on three elements, acting on $Y(2)$ as the group of anharmonic transformations. When the scalar potential is preserved by this action, $\alpha$-attractor models of this type provide a geometric description of two-field `modular invariant $j$-models' in terms of gravity coupled to a non-linear sigma model with topologically non-trivial target and with a finite (as opposed to discrete but infinite) group of symmetries. The precise relation between the two perspectives is provided by the elliptic modular function $\lambda$, which can be viewed as a field redefinition that eliminates almost all of the countably infinite unphysical ambiguity present in the Poincar\'e half-plane description of such models.