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We construct the NHEG phase space, the classical phase space of Near-Horizon Extremal Geometries with fixed angular momenta and entropy, and with the largest symmetry algebra. We focus on vacuum solutions to d-dimensional Einstein gravity. Each element in the phase space is a geometry with SL(2,R)*U(1)^{d-3} Killing isometries which has vanishing SL(2,R) and constant U(1) charges.
We construct an on-shell vanishing symplectic structure, which leads to an infinite set of symplectic symmetries.
In four spacetime dimensions, the phase space is unique and the symmetry algebra consists of the familiar Virasoro algebra, while in d > 4 dimensions the symmetry algebra, the NHEG algebra, contains infinitely many Virasoro subalgebras. The nontrivial central term of the algebra is proportional to the black hole entropy. This phase space and in particular its symmetries might serve as a basis for a semiclassical description of extremal rotating black hole microstates.
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