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We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group O 2 (R) . The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series. Communicated by Ellen Kirkman [ABSTRACT FROM AUTHOR]

We provide an analytical construction of the gluing map for stable affine vortices over the upper half plane with the Lagrangian boundary condition. This result is a necessary ingredient in studies of the relation between gauged sigma model and nonlinear sigma model, such as the closed or open quantum Kirwan map. [ABSTRACT FROM AUTHOR]

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden–Weinstein–Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds. [ABSTRACT FROM AUTHOR]

Suppose given a Hamiltonian and holomorphic action of G = U (2) on a compact Kähler manifold M, with nowhere vanishing moment map. Given an integral coadjoint orbit O for G, under transversality assumptions we shall consider two naturally associated 'conic' reductions. One, which will be denoted M ¯ O G , is taken with respect to the action of G and the cone over O ; another, which will be denoted M ¯ ν T , is taken with respect to the action of the standard maximal torus T ⩽ G and the ray R + ı ν along which the cone over O intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some indications in this direction. Furthermore, we give explicit transversality criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatorial data associated to the action and the orbit. [ABSTRACT FROM AUTHOR]

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden–Weinstein–Meyer reduction, Mikami–Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg–Kazhdan construction of Moore–Tachikawa varieties in topological quantum field theory. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$ -space $\mathfrak {M}_{G, S}$ to each pair $(G,S)$ , where $G$ is any Lie group and $S\subseteq \mathrm {Lie}(G)^{*}$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak {M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. Although these Hamiltonian $G$ -spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new. [ABSTRACT FROM AUTHOR]

Let K be a compact group. For a symplectic quotient Mλ of a compact Hamiltonian Kähler K-manifold, we show that the induced complex structure on Mλ is locally invariant when the parameter λ varies in Lie(K)*. To prove such a result, we take two different approaches: (i) use the complex geometry properties of the symplectic implosion construction; (ii) investigate the variation of geometric invariant theory (GIT) quotients. [ABSTRACT FROM AUTHOR]

Let $Q$ be a closed manifold admitting a locally free action of a compact Lie group $G$. In this paper, we study the properties of geodesic flows on $Q$ given by suitable G-invariant Riemannian metrics. In particular, we will be interested in the existence of geodesics that are closed up to the action of some element in the group $G$ , since they project to closed magnetic geodesics on the quotient orbifold $Q/G$. [ABSTRACT FROM AUTHOR]

Recently Ohsawa (Lett Math Phys 105:1301–1320, 2015) has studied the Marsden–Weinstein–Meyer quotient of the manifold T ∗ R n × T ∗ R 2 n 2 under a O (2 n) -symmetry and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics. In this paper, we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle F (T ∗ R n) of T ∗ R n . We outline some advantages of this interpretation and explain how it can be extended to more general symplectic manifolds using the notion of the diagonal lift of a symplectic form due to Cordero and de León (Rend Circ Mat Palermo 32:236–271, 1983). [ABSTRACT FROM AUTHOR]

We discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries. [ABSTRACT FROM AUTHOR]

Let ( G , V ) be a representation with either G a torus or ( G , V ) a locally free stable θ -representation. We study the fiber at 0 of the associated moment map, which is a commuting variety in the latter case. We characterize the cases where this fiber is normal. The quotient ( i.e. the symplectic reduction) turns out to be a specific orbifold when the representation is polar. In the torus case, this confirms a conjecture stated by C. Lehn, M. Lehn, R. Terpereau and the author in a former article. In the θ -case, the conjecture was already known but this approach yield another proof. [ABSTRACT FROM AUTHOR]

We analyse a monotone lagrangian in CP2 that is hamiltonian isotopic to the standard lagrangian RP2, yet exhibits a distinguishing behaviour under reduction by one of the toric circle actions, namely it intersects transversally the reduction level set and it projects one-to-one onto a great circle in CP1. This lagrangian thus provides an example of embedded composition fitting work of Wehrheim–Woodward and Weinstein. [ABSTRACT FROM AUTHOR]

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a $${{\mathrm{{{\mathrm{Spin}}}^c}}}$$ -Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635-664, 2007) and proved by Ma and Zhang in (Acta Math 212:11-57, 2014). [ABSTRACT FROM AUTHOR]

We show that the Siegel upper half space $${\Sigma_{d}}$$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $${T^{*} \mathbb{R}^{2d^{2}}}$$ with O(2 d)-symmetry. The reduced symplectic form on $${\Sigma_{d}}$$ corresponding to the standard symplectic form on $${T^{*} \mathbb{R}^{2d^{2}}}$$ turns out to be a constant multiple of the symplectic form on $${\Sigma_{d}}$$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show that the two formulations are related via the symplectic reduction. [ABSTRACT FROM AUTHOR]

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula. [ABSTRACT FROM AUTHOR]

We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of the 'Kepler problem' on the cone over the rotation group SO( n). As a corollary we derive an elementary formula for the general solution of the MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, | ϕ|/ r, agrees with the rotational part of the cone's kinetic energy. [ABSTRACT FROM AUTHOR]