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Shifted symplectic structures and Poisson vertex algebra

Wenda Fang

TL;DR

The paper develops a geometric framework tying shifted 1-symplectic structures to Poisson vertex algebras on arc spaces by encoding a degree-2 function $P$ on $Y=T^*[1]X$ into a PVA on $X_ abla$, with the key Correspondence Theorem stating that a Maurer–Cartan condition $igl[ int P, int Pigr]=0$ is equivalent to a consistent PVA structure. It unifies chiralizations of Poisson–Lie theory with Hamiltonian PDE formalisms through the Master Formula, and provides a Maurer–Cartan perspective on classical $R$-matrices as deformations, yielding AKS-type hierarchies from $R$-deformations. The work also delivers concrete PVA sheaves on $ ext{CP}^1$ and explicit classifications of Brackets on arc spaces, including Virasoro–Magri-type structures and sieve-based extendability criteria. Collectively, these results offer a new geometric route to construct and study integrable hierarchies within a chiral, graded, and deformation-theoretic setting, with connections to $W$-algebras and Lax-formalisms.

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 $λ$-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.

Shifted symplectic structures and Poisson vertex algebra

TL;DR

The paper develops a geometric framework tying shifted 1-symplectic structures to Poisson vertex algebras on arc spaces by encoding a degree-2 function on into a PVA on , with the key Correspondence Theorem stating that a Maurer–Cartan condition is equivalent to a consistent PVA structure. It unifies chiralizations of Poisson–Lie theory with Hamiltonian PDE formalisms through the Master Formula, and provides a Maurer–Cartan perspective on classical -matrices as deformations, yielding AKS-type hierarchies from -deformations. The work also delivers concrete PVA sheaves on and explicit classifications of Brackets on arc spaces, including Virasoro–Magri-type structures and sieve-based extendability criteria. Collectively, these results offer a new geometric route to construct and study integrable hierarchies within a chiral, graded, and deformation-theoretic setting, with connections to -algebras and Lax-formalisms.

Abstract

We construct Poisson vertex algebra (PVA) structures on arc spaces from -shifted symplectic (QP) data. A Hamiltonian satisfying the classical master equation induces a canonical PVA -bracket, matching the Hamiltonian-operator formalism for integrable hierarchies. As applications, we find the resulting PVA sheaves on and reinterpret our classical -matrix as Maurer-Cartan data in a deformation-theoretic geometric framework, yielding AKS-type integrable hierarchies from the corresponding -deformations.
Paper Structure (12 sections, 12 theorems, 100 equations)

This paper contains 12 sections, 12 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminary: Graded manifold and graded algebra
  3. Graded algebra
  4. Graded manifold
  5. Graded conformal algebras
  6. Graded Lie conformal algebra
  7. Shifted $1$-symplectic structure and Poisson vertex algebra
  8. Conventions
  9. Normalizing operator and normalized expression
  10. Applications: PVA sheaves and the classical R-matrix
  11. Sheaves of PVAs
  12. Classical R-matrix

Key Result

Theorem 1.1

Let $X$ be a smooth, reduced, finitely generated $\mathbb{C}$-scheme, and set $Y:=T^*[1]X$. Then a degree-$2$ element $P\in\Gamma(Y_\infty,\mathcal{O}_{Y_\infty})_2$ defines a Poisson vertex algebra structure on $X_\infty$ (i.e., it makes $\mathcal{O}_{X_\infty}$ a sheaf of PVAs) if and only if $\bi

Theorems & Definitions (46)

  • Theorem 1.1: Correspondence Theorem
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Remark 2.7
  • Remark 2.8
  • ...and 36 more