w_{\infty} Algebras, Conformal Mechanics, and Black Holes

TL;DR

The paper analyzes BPS solitons in gauged $\mathcal{N}=2$, $D=4$ supergravity and their near-horizon dynamics, showing that particle motion reduces to a conformal quantum mechanics with a Virasoro symmetry that extends to the $\mathrm{w}_{\infty}$ algebra of area-preserving diffeomorphisms. It establishes a concrete AdS$_2$/CFT$_1$ realization by linking spacetime asymptotic symmetries to phase-space diffeomorphisms, and computes the central charge via dimensional reduction to a Jackiw–Teitelboim model. The work further generalizes to ${\cal N}=1$ and ${\cal N}=2$ superextensions of $\mathrm{w}_{\infty}$, providing a comprehensive framework for superconformal mechanics in black-hole backgrounds and highlighting the rich symmetry structure of near-horizon geometries. Overall, the results illuminate how AdS$_2$ holography manifests in quantum-mechanical systems and offer tools for exploring supersymmetric extensions of phase-space diffeomorphism algebras.

Abstract

We discuss BPS solitons in gauged ${\cal N}=2$, D=4 supergravity. The solitons represent extremal black holes interpolating between different vacua of anti-de Sitter spaces. The isometry superalgebras are determined and the motion of a superparticle in the extremal black hole background is studied and confronted with superconformal mechanics. We show that the Virasoro symmetry of conformal mechanics, which describes the dynamics of the superparticle near the horizon of the extremal black hole under consideration, extends to a symmetry under the $w_{\infty}$ algebra of area-preserving diffeomorphisms. We find that a Virasoro subalgebra of $w_{\infty}$ can be associated to the Virasoro algebra of the asymptotic symmetries of $AdS_2$. In this way spacetime diffeomorphisms of $AdS_2$ translate into diffeomorphisms in phase space: our system offers an explicit realization of the $AdS_2/CFT_1$ correspondence. Using the dimensionally reduced action, the central charge is computed. Finally, we also present generalizations of superconformal mechanics which are invariant under ${\cal N} =1$ and ${\cal N} =2$ superextensions of $w_{\infty}$.