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Abstract We investigate the global structure of the recently discovered family of SL(2,ℤ)-invariant potentials describing inflationary α-attractors. These potentials have an inflationary plateau consisting of the fundamental domain and its images fully covering the upper part of the Poincaré half-plane. Meanwhile, the lower part of the half-plane is covered by an infinitely large number of ridges, which, at first glance, are too sharp to support inflation. However, we show that this apparent sharpness is just an illusion created by hyperbolic geometry, and each of these ridges is physically equivalent to the inflationary plateau in the upper part of the Poincaré half-plane. 

Abstract We study cosmological theory where the kinetic term and potential have SL(2,ℤ) symmetry. Potentials have a plateau at large values of the inflaton field, where the axion forms a flat direction. Due to the underlying hyperbolic geometry and special features of SL(2,ℤ) potentials, the theory exhibits an α-attractor behavior: its cosmological predictions are stable with respect to significant modifications of the SL(2,ℤ) invariant potentials. We present a supersymmetric version of this theory in the framework ofD3 induced geometric inflation. The choice ofαis determined by underlying string compactification. For example, in a CY compactification withT², one has 3α= 1, the lowest discrete Poincaré disk target for LiteBIRD. 

A<sc>bstract</sc> We discuss known maximalD-dimensional supergravities of two types: type I withG/Hcoset spaces and type II derived by compactification from higher dimensions without dualization, these have less manifest symmetries. In 4Dand 6Din type I models we perform explicit gauge-fixing of localHsymmetries in unitary gauges: symmetric, Iwasawa and partial Iwasawa. In 4Dsupergravity I in symmetric gauge globalH-invariance and nonlinearly realizedG-symmetry are valid on shell, classically. The globalH-symmetry andG-symmetry in Iwasawa-type gauges in type I and in type II supergravities are not manifest, if at all present. This fact raises the issue of the gauge equivalence of the S-matrix of various gauge-fixedD-dimensional supergravities and its relation to the ones computable using superamplitude methods. 

A<sc>bstract</sc> We introduce Ward identities for superamplitudes inD-dimensional$$ \mathcal{N} $$ $N$-extended supergravities. These identities help to clarify the relation between linearized superinvariants and superamplitudes. The solutions of these Ward identities for ann-partice superamplitude take a simple universal form for half BPS and non-BPS amplitudes. These solutions involve arbitrary functions of spinor helicity and Grassmann variables for each of thensuperparticles. The dimension of these functions at a given loop order is exactly the same as the dimension of the relevant superspace Lagrangians depending on half-BPS or non-BPS superfields, given by (D− 2)L+ 2 −$$ \mathcal{N} $$ $N$or (D− 2)L+ 2 −$$ 2\mathcal{N} $$ $2N$, respectively. This explains why soft limits predictions from superamplitudes and from superspace linearized superinvariants agree. 

A<sc>bstract</sc> We studyd= 4,$$ \mathcal{N} $$ $N$≥ 5 supergravities and their deformation via candidate counterterms, with the purpose to absorb UV divergences. We generalize the earlier studies of deformation and twisted self-duality constraint to the case with unbroken local$$ \mathcal{H} $$ $H$-symmetry in presence of fermions. We find that the deformed action breaks nonlinear local supersymmetry. We show that all known cases of enhanced UV divergence cancellations are explained by nonlinear local supersymmetry. This result implies, in particular, that if$$ \mathcal{N} $$ $N$= 5 supergravity at five loop will turn out to be UV divergent, the deformed theory will be BRST inconsistent. If it will be finite, it will be a consequence of nonlinear local supersymmetry and E7-type duality. 

Abstract We investigate the two-stage inflation regime in the theory of hybrid cosmological  α-attractors. The spectrum of inflationary perturbations is compatible with the latest Planck/BICEP/Keck Array results, thanks to the attractor properties of the model. However, at smaller scales, it may have a very high peak of controllable width and position, leading to a copious production of primordial black holes (PBH) and generation of a stochastic background of gravitational waves (SGWB). 

Abstract We discuss two-stage dilaton-axion inflation models [1] and describe α -attractor models with either exponential or polynomial approach to the plateau.We implement one of the models of primordial black hole production proposed in [2] in the α -attractor context, and develop its supergravity version. The predictions of this model following from its polynomial attractor properties are: n s and r are α -independent, r depends on the mass parameter μ defining the approach to the plateau. The tachyonic instability at the transition point between the two stages of inflation is proportional to the negative curvature of the hyperbolic space ℛ K = -2/3 α . Thereforethe masses of primordial black holes (PBHs) and the frequencies of small-scale gravitational waves (GWs) in this model show significant dependence on  α . 

A bstract It was argued in [1] that the Volkov-Akulov (VA) model as well as similar models in supergravity and the related KKLT model in string theory, suffer from tachyonic instabilities due to goldstino condensation. The authors of [1] constructed a specific model with two unconstrained interacting chiral superfields with linearly realized supersymmetry which has an unstable vacuum. They claimed that this model becomes equivalent to the VA model in the UV limit. We show that the UV limit of their model is discontinuous, and the vacuum instability of the model proposed in [1] is not relevant to the VA model, to related models in supergravity, and to the KKLT construction. 

Abstract Inflationary α -attractor models can be naturally implemented in supergravity with hyperbolic geometry. They have stable predictions for observables, such as n s = 1 - 2/ N e , assuming that the potential in terms of the original geometric variables, as well as its derivatives, are not singular at the boundary of the hyperbolic disk, or half-plane. In these models, the potential in the canonically normalized inflaton field φ has a plateau, which is approached exponentially fast at large φ . We call them exponential α-attractors . We present a closely related class of models, where the potential is not singular, but its derivative is singular at the boundary. The resulting inflaton potential is also a plateau potential, but it approaches the plateau polynomially. We call them polynomial α-attractors . Predictions of these two families of attractors completely cover the sweet spot of the Planck/BICEP/Keck data. The exponential ones are on the left, the polynomial are on the right.