We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA), and we construct a spectral sequence relating them. Since in “good” cases the classical PVA cohomology coincides with the variational PVA cohomology and there are well-developed methods to compute the latter, this enables us to compute the cohomology of vertex algebras in many interesting cases. Finally, we describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs.

我们回顾与手性和古典操作员相对应的同调理论。第一个是顶点代数的同调理论，而第二个是泊松顶点代数（PVA）的经典同调论，我们构造了与它们有关的谱序列。由于在“良好”情况下，经典PVA同调与变分PVA同调一致，并且存在完善的计算后者的方法，因此使我们能够在许多有趣的情况下计算顶点代数的同调。最后，我们描述了通过消除第一个同调性来实现可积性的统一方法，该方法适用于哈密顿PDE的经典系统和量子系统。