Symmetries of the Space of Linear Symplectic Connections

TL;DR

This paper explores the symmetries and geometric structures of the space of linear symplectic connections, introducing Lie algebras, moment maps, and Poisson structures that deepen understanding of symplectic geometry.

Contribution

It constructs a family of Lie algebras acting on symplectic connections, analyzes their moment maps, and demonstrates compatibility of Poisson structures on symmetric tensor algebras.

Findings

Constructed Lie algebras act Hamiltonianly on symplectic connections.

Identified a formal sum of moment maps combining Cahen-Gutt, Ricci, and translational terms.

Proved compatibility of two Poisson structures on symmetric tensor algebras.

Abstract

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.