Locally conformal symplectic structures: From standard to line bundle approach

Abstract

Motivated by the recent interest growing in contact structures, this paper is devoted to even-dimensional counterpart of contact geometry, namely the locally conformal symplectic one. In physics, these structures come natural from endowing the phase-space with a 2-form that is non-degenerate but locally closed up to a multiplicative non-vanishing function. In view of this, we initially adopt the trivial line bundle situation and show that a given locally conformal simplectic over an even-dimensional manifold leads to a transitive Jacobi pair and conversely. Then, we introduce the geometric perspective on locally conformal symplectic structures in terms of a line bundle over an even-dimensional manifold, L → M endowed with a flat-connection, ∇ and an L-valued non-degenerate and closed 2-form, ω ∈ Ω(M; L). In this framework we also exhibit the connection between locally conformal symplectic structures and transitive Jacobi ones.Motivated by the recent interest growing in contact structures, this paper is devoted to even-dimensional counterpart of contact geometry, namely the locally conformal symplectic one. In physics, these structures come natural from endowing the phase-space with a 2-form that is non-degenerate but locally closed up to a multiplicative non-vanishing function. In view of this, we initially adopt the trivial line bundle situation and show that a given locally conformal simplectic over an even-dimensional manifold leads to a transitive Jacobi pair and conversely. Then, we introduce the geometric perspective on locally conformal symplectic structures in terms of a line bundle over an even-dimensional manifold, L → M endowed with a flat-connection, ∇ and an L-valued non-degenerate and closed 2-form, ω ∈ Ω(M; L). In this framework we also exhibit the connection between locally conformal symplectic structures and transitive Jacobi ones.