Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

Abstract

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N = 2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras ,J, the spectrum generating symmetry groups are the conformal groups Conf (J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N = 2 MESGTs are the quasiconformal groups QConf (J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D , N = 2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results. Table 2: Above we list the orbits of spherically symmetric stationary BPS black hole solutions in 5D MESGTs defined by Euclidean Jordan algebras J of degree three. U-duality and stability groups are given by the Lorentz ( reduced structure ) and rotation ( automorphism) groups of J.SO(9) R ⊕ Γ (1,n−1) SO(n − 1, 1) × SO(1, 1)/SO(n − 2, 1) SO(n − 2)M 5 = M 4 = M 3 = J Str 0 (J) / Aut (J) Conf (J) / Str (J) QConf(F(J))/ Conf (J) × SU (2)

Degeneration of groups of type E 7 and minimal coupling in supergravity

“…In general, G 4 ≡Conf(J 3 ) =Aut(M(J 3 )) (see e.g. [34][35][36] for a recent introduction, and a list of refs.). When considering a consistent reduction to a subgroup, G 4 groups of type E 7 may admit a "degeneration" in which the rank-4 invariant symmetric structure q is reducible, namely it is the product of two symmetric invariant tensors.…”

“…r + s of U(r, s), and Q i x and Q i x are the charge vectors in the complex (manifestly U(r, s)-covariant) symplectic frame. By introducing 34) it is immediate to check the degenerate nature of the quartic invariant q-structure (2.21):…”

“…In turn, the "pure" N = 5 theory admits two maximal "degenerative" truncations, which precisely match the kinematical decomposition of the N = 5 gravity multiplet into matter N = 2 multiplets. 34) and it admits two possible interpretations, due to the fact that N = 3 supergravity coupled to 1 vector multiplet and N = 2 supergravity minimally coupled to 3 vector multiplets share the very same bosonic sector (namely, they are "twin" theories; see the discussion in section 9 of [41]). In the N = 3 interpretation, one gets a theory with 1 vector multiplets, and the SU(2) commuting factor can be interpreted as the part of the R-symmetry truncated away in the supersymmetry reduction N = 5 → N = 3.…”

“…Note that this case, as well as the cases treated at points 1, 3 and 4 of section 3.4, is based on the maximal (symmetric) Jordan algebraic embedding (see e.g. [72]):…”

Iwasawa nilpotency degree of non compact symmetric cosets in \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ \mathcal{N} $\end{document}‐extended supergravity

“…[55]). In (5.1) SL(2, R) is the so-called Ehlers group, related to the reduction of pure Einstein gravity to 3 dimensions, whereas in (5.2) and (5.3) SO(1, 1) is the Kaluza-Klein compactification factor.…”