A Darboux theorem for Hamiltonian operators in the formal calculus of variations
Differential Geometry
2007-05-23 v2 Mathematical Physics
math.MP
Symplectic Geometry
Exactly Solvable and Integrable Systems
Abstract
We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) such as the Schouten algebra - these give a convenient language for describing deformation problems; 2) the Deligne 2-groupoid associated to such a dg Lie algebra, which represents the moduli of formal deformations; 3) a refined version of the Schouten bracket in the formal calculus of variations, due to V. O. Soloviev (hep-th/9305133).
Cite
@article{arxiv.math/0002164,
title = {A Darboux theorem for Hamiltonian operators in the formal calculus of variations},
author = {Ezra Getzler},
journal= {arXiv preprint arXiv:math/0002164},
year = {2007}
}