Bosonization of Feigin-Odesskii Poisson varieties

Abstract

The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Projective spaces with Feigin-Odesskii Poisson brackets are examples of such moduli spaces over complex elliptic curves [6,7]. By generalizing several results in our previous work [10–12] we construct a collection of auxiliary Poisson varieties equipped with Poisson morphisms to Feigin-Odesskii varieties. We call them bosonizations of Feigin-Odesskii varieties. These spaces appear as special cases of the moduli spaces of chains, which we introduce. We show that the moduli space of chains admits a shifted Poisson structure when the base is a Calabi-Yau variety of an arbitrary dimension. Using bosonization spaces mapping to the zero loci of the Feigin-Odesskii varieties, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries. We also derive explicit formulas for the Poisson brackets on the bosonizations of the Feigin-Odesskii varieties associated with line bundles in a simplest nontrivial case.