Generalized $α$-attractors from the hyperbolic triply-punctured sphere

We study generalized $\alpha$-attractor models whose rescaled scalar manifold is the triply-punctured Riemann sphere $Y(2)$ endowed with its complete hyperbolic metric. 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. 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 relation between the two perspectives is provided by the elliptic modular function $\lambda$, viewed as a field redefinition which eliminates most of the infinite unphysical ambiguity present in the Poincare half-plane description of such models.

Comments: 35 pages

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