Bihamiltonian cohomology of KdV brackets

TL;DR

This paper computes the bihamiltonian cohomology groups of the dispersionless KdV hierarchy's Poisson brackets using spectral sequences, confirming a conjecture about their vanishing.

Contribution

It provides the first complete computation of these cohomology groups, resolving a longstanding conjecture in the field.

Findings

Bihamiltonian cohomology groups of the dispersionless KdV hierarchy vanish.

Spectral sequences are effective for computing bihamiltonian cohomology.

The conjecture of Liu and Zhang is proven.

Abstract

Using spectral sequences techniques we compute the bihamiltonian cohomology groups of the pencil of Poisson brackets of dispersionless KdV hierarchy. In particular this proves a conjecture of Liu and Zhang about the vanishing of such cohomology groups.