We describe in detail a gluing construction for pseudoholo- morphic maps in symplectic geometry, including in the pres- ence of an obstruction bundle. The main motivation is to try to compare the symplectic and enumerative invariants of algebraic manifolds. These descriptions can also be used to enumerate rational curves with high-order degeneracies of lo- cal nature in projective spaces.

We show that the Hamiltonian systems on Sternberg-Wein- stein phase spaces which yield Wong’s equations of motion for a classical particle in a gravitational and a Yang-Mills field, naturally arise as the first order approximation of generic Hamiltonian systems on Poisson manifolds at a critical La- grangian submanifold. We further define a second order ap- proximated system involving scalar fields which first appeared in Einstein-Mayer theory. Reduction and symplectic realiza- tion of this system are interpreted in terms of dimensional reduction and Kaluza-Klein theory.

We start by describing the relationship between the classi- cal prequantization condition and the integrability of a certain Lie algebroid associated to the problem and use this to give a global construction of the prequantizing bundle in terms of path spaces (Introduction). Next, we rephrase the problem in terms of groupoids (second section), and then we study the more general problem of prequantizing groupoids endowed with multiplicative 2-forms (the rest of the paper).