Geometrical action for $w_\infty$ algebra as a reduced symplectic Chern-Simons theory

TL;DR

This paper demonstrates that a geometric action related to the $w_ abla$ algebra can be derived as a specific reduction of 3D Chern-Simons theory, linking algebraic and topological field theories.

Contribution

It establishes a novel connection between the geometric action for the $w_ abla$ algebra and a reduced form of symplectic Chern-Simons theory, revealing new insights into their relationship.

Findings

The geometric action on a certain orbit matches a reduction of 3D Chern-Simons theory.

Group and space coordinates are identified in the reduction process.

The work provides a geometric interpretation of the $w_ abla$ algebra within topological field theory.

Abstract

The geometric action on a certain orbit of the group of the area-preserving diffeomorphisms is considered, and it is shown, that it coincides with a special reduction of the three-dimensional Chern-Simons theory, under which group and space coordinates are identified.