On the structure of homogeneous local Poisson brackets

TL;DR

This work addresses the structure and deformation theory of homogeneous local Poisson brackets of degree $k$ on loop spaces by embedding them into a differential-geometric framework on $M$ and using homological methods. The authors encode the Jacobi identity as a differential $D_P$ on a graded algebra and analyze it with a spectral sequence to extract geometric data, leading to the construction of $k$ flat connections. The principal result is that the linear combinations $ abla^{[s]} = extstyle extstyleigl( abla^{(0)}, abla^{(1)},\dots, abla^{(k-1)}igr)$ with coefficients $c_s^t$ yield flat connections for all $s$, revealing a hidden uniformity across degrees. This flatness provides a robust geometric handle on the leading homogeneous part of dispersive Poisson brackets and sets the stage for a systematic study of their Poisson cohomology and deformations, with concrete implications for low-degree cases and potential extensions to higher-order structures.

Abstract

We consider an arbitrary Dubrovin-Novikov bracket of degree $k$, namely a homogeneous degree $k$ local Poisson bracket on the loop space of a smooth manifold $M$ of dimension $n$, and show that $k$ connections, defined by explicit linear combinations with constant coefficients of the standard connections associated with the Poisson bracket, are flat.

Theorems & Definitions (23)

• Remark 1
• Remark 2
• Remark 3
• Lemma 4: lz11lz13
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7
• Lemma 8