Shifted symplectic structures and Poisson vertex algebra

TL;DR

This paper develops a method to construct Poisson vertex algebra structures from shifted symplectic data, linking geometric structures with integrable hierarchies and deformation theory.

Contribution

It introduces a novel construction of PVAs from 1-shifted symplectic structures and connects classical R-matrices with Maurer-Cartan data in a geometric framework.

Findings

Constructed PVAs on arc spaces from shifted symplectic data.

Linked Hamiltonian structures to integrable hierarchies via R-matrices.

Reinterpreted classical R-matrices as Maurer-Cartan elements in deformation theory.

Abstract

We construct Poisson vertex algebra (PVA) structures on arc spaces from $1$-shifted symplectic (QP) data. A Hamiltonian satisfying the classical master equation induces a canonical PVA $\lambda$-bracket, matching the Hamiltonian-operator formalism for integrable hierarchies. As applications, we find the resulting PVA sheaves on $\mathbb P^1$ and reinterpret our classical $R$-matrix as Maurer-Cartan data in a deformation-theoretic geometric framework, yielding AKS-type integrable hierarchies from the corresponding $R$-deformations.