Gelfand-Dickey Algebra and Higher Spin Symmetries On $T^2=S^1\times S^1$

TL;DR

This paper explores higher spin symmetries on the two-dimensional torus, connecting Gelfand-Dickey algebra with conformal field theories and integrable models, highlighting their geometric and algebraic structures.

Contribution

It explicitly derives higher conformal spin symmetries from the Gelfand-Dickey algebra on $T^2$, extending known symmetries and linking them to Liouville and Toda theories.

Findings

Derived generalized symmetries from Gelfand-Dickey algebra

Connected higher spin symmetries to conformal field theories

Discussed geometric properties of diffeomorphisms on $T^2$

Abstract

We focus in this work to renew the interest in higher conformal spins symmetries and their relations to quantum field theories and integrable models. We consider the extension of the conformal Frappat et al. symmetries containing the Virasoro and the Antoniadis et al. algebras as particular cases describing geometrically special diffeomorphisms of the two dimensional torus $T^2$. We show in a consistent way, and explicitly, how one can extract these generalized symmetries from the Gelfand-Dickey algebra. The link with Liouville and Toda conformal field theories is established and various important properties are discussed.