Shifted symplectic structures and Poisson vertex algebra

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.