On Invariant Structures of Black Hole Charges

TL;DR

This paper investigates invariant polynomial structures related to two-centered black hole solutions in specific supergravity models, revealing symmetries and algebraic properties relevant for understanding black hole charge configurations.

Contribution

It provides a detailed analysis of invariant polynomial bases in supergravity models with symmetric scalar manifolds, highlighting new algebraic structures and symmetry properties of black hole charges.

Findings

Identification of minimal degree invariant polynomial bases

Analysis of SL(2,R) and U(r,s) symmetry structures

Insights into special Kaehler geometry formulations

Abstract

We study "minimal degree" complete bases of duality- and "horizontal"- invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D=4 Einstein supergravity models with symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL(2,R) "horizontal" symmetry. The first class encompasses N=2 and N=4 matter-coupled theories, with semi-simple U-duality given by SL(2,R) x SO(m,n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial "horizontal" stabilizer SO(2), includes N=2 minimally coupled and N=3 matter coupled theories, with U-duality given by the pseudo-unitary group U(r,s) (related to complex flux representations). Finally, we comment on the formulation of special Kaehler geometry in terms of "generalized" groups of type E7.