The cc-map, Tits Satake subalgebras and the search for N=2\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.Comment: 101 pages, LaTeX sourc

Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras

In this paper we show that the supergravity equations describing both cosmic billiards and a large class of black-holes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on the dual Borel algebra B_N of the simple Lie algebra A_{N-1}. As a by product we derive the explicit integration algorithm associated with all symmetric spaces U/H^{*} relevant to the description of time-like and space-like p-branes. The most important consequence of our approach is the explicit construction of a complete set of conserved involutive hamiltonians h_{\alpha} that are responsible for integrability and provide a new tool to classify flows and orbits. We believe that these will prove a very important new tool in the analysis of supergravity black holes and billiards.Comment: 48 pages, 7 figures, LaTex; V1: misprints corrected, two references adde

ALE manifolds and Conformal Field Theory

We address the problem of constructing the family of (4,4) theories associated with the sigma-model on a parametrized family ${\cal M}_{\zeta}$ of Asymptotically Locally Euclidean (ALE) manifolds. We rely on the ADE classification of these manifolds and on their construction as HyperK\"ahler quotients, due to Kronheimer. So doing we are able to define the family of (4,4) theories corresponding to a ${\cal M}_{\zeta}$ family of ALE manifolds as the deformation of a solvable orbifold ${\bf C}^2 \, / \, \Gamma$ conformal field-theory, $\Gamma$ being a Kleinian group. We discuss the relation among the algebraic structure underlying the topological and metric properties of self-dual 4-manifolds and the algebraic properties of non-rational (4,4)-theories admitting an infinite spectrum of primary fields. In particular, we identify the Hirzebruch signature $\tau$ with the dimension of the local polynomial ring {\cal R}=\o {{\bf C}[x,y,z]}{\partial W} associated with the ADE singularity, with the number of non-trivial conjugacy classes in the corresponding Kleinian group and with the number of short representations of the (4,4)-theory minus four.Comment: 48 pages, Latex, SISSA/44/92/EP, IFUM/443/F

R--R Scalars, U--Duality and Solvable Lie Algebras

We consider the group theoretical properties of R--R scalars of string theories in the low-energy supergravity limit and relate them to the solvable Lie subalgebra \IG_s\subset U of the U--duality algebra that generates the scalar manifold of the theory: \exp[\IG_s]= U/H. Peccei-Quinn symmetries are naturally related with the maximal abelian ideal {\cal A} \subset \IG_s of the solvable Lie algebra. The solvable algebras of maximal rank occurring in maximal supergravities in diverse dimensions are described in some detail. A particular example of a solvable Lie algebra is a rank one, $2(h_{2,1}+2)$--dimensional algebra displayed by the classical quaternionic spaces that are obtained via c-map from the special K\"ahlerian moduli spaces of Calabi-Yau threefolds.Comment: 17 pages, misprints in Table 2 correcte

Supersymmetry and First Order Equations for Extremal States: Monopoles, Hyperinstantons, Black-Holes and p-Branes

In this lecture I review recent results on the first order equations describing BPS extremal states, in particular N=2 extremal black-holes. The role of special geometry is emphasized also in the rigid theory and a comparison is drawn with the supersymmetric derivation of instantons and hyperinstantons in topological field theories. Work in progress on the application of solvable Lie algebras to the discussion of BPS states in maximally extended supergravities is outlined.Comment: LaTeX, {article.sty, espcrc2.sty} 11 twocolumn pages. Invited Seminar given at Santa Margerita Conference on Contrained Dynamics and Quantum Gravity September 1995. In the replaced version misprints have been corrected and some sentences have been modifie

Twisted Elliptic Genera of N=2 SCFTs in Two Dimensions

The elliptic genera of two-dimensional N=2 superconformal field theories can be twisted by the action of the integral Heisenberg group if their U(1) charges are fractional. The basic properties of the resulting twisted elliptic genera and the associated twisted Witten indices are investigated with due attention to their behaviors in orbifoldization. Our findings are illustrated by and applied to several concrete examples. We give a better understanding of the duality phenomenon observed long before for certain Landau-Ginzburg models. We revisit and prove an old conjecture of Witten which states that every ADE Landau-Ginzburg model and the corresponding minimal model share the same elliptic genus. Mathematically, we establish ADE generalizations of the quintuple product identity.Comment: 28 pages; v2 refs adde