Swapping algebra, Virasoro algebra and discrete integrable system

TL;DR

This paper constructs and analyzes Poisson algebras on the configuration space of twisted polygons in projective space, revealing their connections to the Virasoro algebra and integrable systems.

Contribution

It introduces new Poisson structures on polygon configuration spaces and demonstrates their asymptotic relation to the Virasoro algebra, establishing their Schouten commutativity.

Findings

Poisson algebra on twisted polygons coincides with known algebra for n=2

Two Poisson algebras are asymptotically related to the dual Virasoro algebra

The two Poisson algebras are Schouten commute

Abstract

We induce a Poisson algebra $\{\cdot,\cdot\}_{\mathcal{C}_{n,N}}$ on the configuration space $\mathcal{C}_{n,N}$ of $N$ twisted polygons in $\mathbb{RP}^{n-1}$ from the swapping algebra \cite{L12}, which is found coincide with Faddeev-Takhtajan-Volkov algebra for $n=2$. There is another Poisson algebra $\{\cdot,\cdot\}_{S2}$ on $\mathcal{C}_{2,N}$ induced from the first Adler-Gelfand-Dickey Poissson algebra by Miura transformation. By observing that these two Poisson algebras are asymptotically related to the dual to the Virasoro algebra, finally, we prove that $\{\cdot,\cdot\}_{\mathcal{C}_{2,N}}$ and $\{\cdot,\cdot\}_{S2}$ are Schouten commute.