Near-Horizon Extremal Geometries: Coadjoint Orbits and Quantization

TL;DR

This paper explores the structure and quantization of the NHEG algebra, extending Virasoro symmetries in near-horizon extremal geometries, and classifies its coadjoint orbits and representations.

Contribution

It constructs the NHEG group, classifies its coadjoint orbits, and describes the expected unitary representations arising from their quantization.

Findings

NHEG group consists of maps from an n-torus to the Virasoro group.

Orbits are bundles of Virasoro coadjoint orbits over T^n.

Unitary representations and characters are described for the quantized orbits.

Abstract

The NHEG algebra is an extension of Virasoro introduced in [arXiv:1503.07861]; it describes the symplectic symmetries of $(n+4)$-dimensional Near Horizon Extremal Geometries with $SL(2,R)\times U(1)^{n+1}$ isometry. In this work we construct the NHEG group and classify the (coadjoint) orbits of its action on phase space. As we show, the group consists of maps from an $n$-torus to the Virasoro group, so its orbits are bundles of standard Virasoro coadjoint orbits over $T^n$. We also describe the unitary representations that are expected to follow from the quantization of these orbits, and display their characters. Along the way we show that the NHEG algebra can be built from u(1) currents using a twisted Sugawara construction.