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

Abstract

Abstract:We study properties of D = 4 N 2 extended supergravities (and related compactifications of superstring theory) and their consistent truncation to the phenomenologically interesting models of N = 1 supergravity. This involves a detailed classification of the "degenerations" of the duality groups of type E 7 , when the corresponding quartic invariant polynomial built from the symplectic irreducible representation of G 4 "degenerates" into a perfect square. With regard to cosmological applications, minimal coupling of vectors in consistent truncation to N = 1 from higher-dimensional or higher-N theory is non-generic. On the other hand, non-minimal coupling involving vectors coupled to scalars and axions is generic. These features of supergravity, following from the electric-magnetic duality, may be useful in other applications, like stabilization of moduli, and in studies of non-perturbative black-hole solutions of supergravity/string theory.

D = 3 unification of curious supergravities

“…This last step is particularly challenging for the consistency with local N = 1 supersymmetry, which is however granted by the results 6 in [23] (also cfr. [24]); see, in particular, the discussion around eq. (6.145) therein.…”

Non-Linear Invariance of Black Hole Entropy

“…Through the attractor mechanism [7], all this enjoys an interesting physical interpretation when evaluated at the (unique) event horizon of the extremal black hole (1.3) (denoted below by the subscript "H"); indeed 17) where S BH and A H respectively denote the Bekenstein-Hawking entropy [8] and the area of the horizon of the extremal black hole, and the matrix horizon value M H is defined as…”

“…by [14,15,16,17,18]. Starting from a pair (G, R) made of a Lie group G and its faithful representation R, the three axioms defining (G, R) as a group of type E 7 read as follows :…”

Branching rules of sp(6) ↓ sp(4) × sp(2) and bases of eigenstates

“…The "external" operators correspond to the Casimir operators of sp(6). As the reduction is not multiplicity free, the previous formula (29) indicates that in general n = 1 missing label operator is sufficient to separate degeneracies. However, as we have seen, using equation (23) we can skip this operator, as repeated IRREPs of the subalgebra are distinguished by the values of Γ 1 3 and Γ 2 3 .…”

“…For N = 2 the chain sp(4, R) ↓ sp(2, R) × sp(2, R) has been used in the study of light nuclei, more specifically in the simplification of the collective excitations of the sp(6, R) ⊃ U(3) model [27,28], albeit for the case of infinite dimensional (unitary) representations, while the chain sp(6, R) ↓ sp(4, R) × sp(2, R) has recently been considered in the context of N = 8 supergravity truncation to N = 2 theories with scalarvector minimal coupling [29] 1 , where branching rules are used to reduce kinematically the supergravity multiplets. These recent applications serve as motivation to the problem of obtaining branching rules in an explicit and consistent way.…”