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We propose a generic recipe for deforming extremal black holes into non-extremal black holes and we use it to find and study the static non-extremal black-hole solutions of several N = 2, d = 4 supergravity models (SL(2, R)/U (1), CP n and ST U with four charges). In all the cases considered, the non-extremal family of solutions smoothly interpolates between all the different extremal limits, supersymmetric and not supersymmetric. This fact can be used to explicitly find extremal non-supersymmetric solutions also in the cases in which the attractor mechanism does not completely fix the values of the scalars on the event horizon and they still depend on the boundary conditions at spatial infinity. We compare (supersymmetry) Bogomol'nyi bounds with extremality bounds, we find the first-order flow equations for the non-extremal solutions and the corresponding superpotential, which gives in the different extremal limits different superpotentials for extremal black holes. We also compute the entropies (areas) of the inner (Cauchy) and outer (event) horizons, finding in all cases that their product gives the square of the moduli-independent entropy of the extremal solution with the same electric and magnetic charges. a

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface Σ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field α-attractor models, being parameterized by a positive constant α, by the choice of a finitely-generated surface group Γ ⊂ PSL(2, R) (which is isomorphic with the fundamental group of Σ) and by the choice of a scalar potential defined on Σ. The traditional two-field α-attractor models arise when Γ is the trivial group, in which case Σ is the Poincaré disk. We give a general prescription for the study of such models through uniformization in the so-called "non-elementary" case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp ends. When Σ is non-compact and the scalar potential is "well-behaved" at the ends, we show that, in the naive local one-field truncation, our generalized models have the same universal behavior as ordinary one-field α-attractors if inflation happens near any of the ends of Σ where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends; we also discuss spiral trajectories near the ends. Generalized two field α-attractors illustrate interesting consequences of nonlinear sigma models whose scalar manifold is not simply connected. They provide a large class of tractable cosmological models with non-trivial topology of the scalar field space.

We give a global formulation of the coupling of four-dimensional scalar sigma models to Abelian gauge fields for the generalized situation when the "duality structure" of the Abelian gauge theory is described by a flat symplectic vector bundle (S, D, ω) defined over the scalar manifold M. The construction uses a taming of (S, ω), which encodes globally the inverse gauge couplings and theta angles of the "twisted" Abelian gauge theory in a manner that makes no use of duality frames. We show that global solutions of the equations of motion of such models give classical locally geometric U-folds. We also describe the groups of duality transformations and scalar-electromagnetic symmetries arising in such models, which involve lifting isometries of M to a particular class of flat automorphisms the bundle S and hence differ from expectations based on local analysis. The appropriate version of the Dirac quantization condition involves a discrete local system defined over M and gives rise to a smooth bundle of polarized Abelian varieties, endowed with a flat symplectic connection. This shows that a generalization of part of the mathematical structure familiar from N = 2 supergravity is already present in such purely bosonic models, without any coupling to fermions and hence without any supersymmetry. arXiv:1609.05872v1 [hep-th] 19 Sep 2016Notations and conventions. The symbols Gp, Ab and Alg denote respectively the categories of all groups, of Abelian groups and of unital R-algebras. The symbol Vect denotes the category of finitedimensional R-vector spaces and linear maps. For any category C, let C × denote its unit groupoid, i.e. the category having the same objects as C and whose morphisms are the isomorphisms of C. Given an Abelian group A, let Tors(A) denote its torsion subgroup and A tf def.= A/Tors(A) be the corresponding torsion-free group; the latter is free when it is finitely-generated.2 Indeed, the bosonic sector of the N = 2 theory should be a particular case of the bosonic sector of the N = 0 theory. Scalar nonlinear sigma models coupled to gravityWe start by recalling the global construction of four-dimensional scalar nonlinear sigma models coupled to gravity. Let M be an oriented four-manifold and Met 3,1 (M ) be the set of Lorentzian metrics defined on M . We assume that Met 3,1 (M ) is non-empty (this requires χ(M ) = 0 when M is compact).