Dualities Near the Horizon

Understand

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.