Dualities Near the Horizon

TL;DR

This paper explores the properties of the symplectic matrix in 4D supergravity theories, relating it to black hole attractor mechanisms and duality symmetries, providing universal formulas for field strengths at black hole horizons.

Contribution

It derives universal expressions for the symplectic matrix and field strengths at black hole horizons in supergravity, especially for symmetric scalar manifolds with duality group G of type E7.

Findings

Universal form of M at attractor points with no flat directions

Dependence of M on flat directions via Freudenthal dual Q

Explicit near-horizon expressions for field strengths in Bertotti-Robinson geometry

Abstract

In 4-dimensional supergravity theories, covariant under symplectic electric-magnetic duality rotations, a significant role is played by the symplectic matrix M({\phi}), related to the coupling of scalars {\phi} to vector field-strengths. In particular, this matrix enters the twisted self-duality condition for 2-form field strengths in the symplectic formulation of generalized Maxwell equations in the presence of scalar fields. In this investigation, we compute several properties of this matrix in relation to the attractor mechanism of extremal (asymptotically flat) black holes. At the attractor points with no flat directions (as in the N = 2 BPS case), this matrix enjoys a universal form in terms of the dyonic charge vector Q and the invariants of the corresponding symplectic representation RQ of the duality group G, whenever the scalar manifold is a symmetric space with G simple and non-degenerate of type E7. At attractors with flat directions, M still depends on flat directions, but not MQ, defining the so-called Freudenthal dual of Q itself. This allows for a universal expression of the symplectic vector field strengths in terms of Q, in the near-horizon Bertotti-Robinson black hole geometry.