Poisson structures on loop spaces of $\mathbb{C} P^n$ and an $r$-matrix associated with the universal elliptic curve

Abstract

We construct a family of Poisson structures of hydrodynamic type on the loop space of $\mathbb{C} P^{n-1}$. This family is parametrized by the moduli space of elliptic curves or, in other words, by the modular parameter $\tau$. This family can be lifted to a homogeneous Poisson structure on the loop space of $\mathbb{C}^n$ but in order to do that we need to upgrade the modular parameter $\tau$ to an additional field $\tau(x)$ with Poisson brackets $\{\tau(x),\tau(y)\}=0,~~\{\tau(x),z_a(y)\}=2\pi i~ z_a(y)~\delta^{\prime}(x-y)$ where $z_1,...,z_n$ are coordinates on $\mathbb{C}^n$. These homogeneous Poisson structures can be written in terms of an elliptic $r$-matrix of hydrodynamic type.