Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets

TL;DR

This paper computes the Poisson cohomology for scalar multidimensional Dubrovin-Novikov brackets, revealing non-trivial deformation theory in higher dimensions, contrasting with the one-dimensional case.

Contribution

It provides the first explicit computation of Poisson cohomology for scalar multidimensional Dubrovin-Novikov brackets, showing non-vanishing second and third cohomology groups.

Findings

Second and third cohomology groups are non-zero for D>1

Deformation theory is non-trivial in multiple dimensions

Contrasts with the trivial deformation in D=1 case

Abstract

We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin-Novikov type with $D$ independent variables. We find that the second and third cohomology groups are generically non-vanishing in $D>1$. Hence, in contrast with the $D=1$ case, the deformation theory in the multivariable case is non-trivial.