We set S1 = R/2πZ throughout. Let (M,ω) be a symplectic manifold. A symplectic S1-action on (M,ω) is a smooth family ψt ∈ Symp(M,ω), t ∈ S1, such that ψt+s = ψt ◦ ψs for any t, s ∈ S1. One can easily check that the corresponding vector fields Xt ≡ d dtψt ◦ψ −1 t is time-independent, i.e., Xt = X is constant in t. We call X the associated vector field of the given symplectic S1-action. Note that...

Let G be a finite group acting on a symplectic complex vector space V . Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by “symplectic reflections”, i.e. symplectomorphisms with fixed space of codimension 2 in V . Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakam...

By means of the perturbative renormalization group method, we study a long-time behaviour of some symplectic discrete maps near elliptic and hyperbolic fixed points. It is shown that a naive renormalization group (RG) map breaks the symplectic symmetry and fails to describe a long-time behaviour. In order to preserve the symplectic symmetry, we present a regularization procedure, which gives a ...

For symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action i...

We study symplectic manifolds (M, ω) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇ and admitting a metaplectic structure. Let S be the so called symplectic spinor bundle and let R be the curvature tensor field of the symplectic spinor covariant derivative ∇ associated to the Fedosov connection ∇. It is known that the space of symplectic spinor valued exterior...

A symplectic Lie group is a with left-invariant form. Its algebra structure that of quasi-Frobenius algebra. In this note, we identify the groupoid analogue group. We call aforementioned \textit{$t$-symplectic groupoid}; $t$ motivated by fact each target fiber $t$-symplectic manifold. For $\mathcal{G}\rightrightarrows M$, show there one-to-one correspondence between algebroid structures on $A\m...

Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic nilmanifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic. On a symplectic manifold (M,Ω), a symplectic connection is a torsion-free affine ...