Wenda Fang | Searcharxiv

arXiv2026

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 $λ$-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.

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arXiv2023

Generalized AKS scheme of integrability via vertex algebra

In this paper, we define and study the classical $R$-matrix for vertex Lie algebra, based on which we propose to construct a new vertex Lie algebra. We give a systematic way to construct the $R$-matrix for affine Kac-Moody vertex Lie algebra and study the universal vertex algebra associated with the new vertex Lie algebra that we obtained by $R$-matrix. As an application, using the classical $R$-matrix we defined, we give a new scheme to construct infinite-dimensional (Liouville) integrable systems via the Feigin-Frenkel center. The scheme-theoretical explanation of our equations and the classical $\mathcal{W}$-algebra case of our theory will come later.

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