Symplectic leaves of W-algebras from the reduced Kac-Moody point of view

TL;DR

This paper studies the structure of symplectic leaves in W-algebras by examining their relation to Kac-Moody algebras and constraints, providing classification and representatives, with detailed analysis for the Virasoro algebra.

Contribution

It offers a new perspective on classifying symplectic leaves of W-algebras via Kac-Moody algebra intersections and provides explicit representatives and energy positivity analysis.

Findings

Classified symplectic leaves of W-algebras.

Provided explicit representatives for each leaf.

Analyzed energy positivity in the Virasoro case.

Abstract

The symplectic leaves of W-algebras are the intersections of the symplectic leaves of the Kac-Moody algebras and the hypersurface of the second class constraints, which define the W-algebra. This viewpoint enables us to classify the symplectic leaves and also to give a representative for each of them. The case of the (W_{2}) (Virasoro) algebra is investigated in detail, where the positivity of the energy functional is also analyzed.