The space $Ω(G)$ of all based loops in a compact semisimple simply connected Lie group $G$ has an action of the maximal torus $T\subset G$ (by pointwise conjugation) and of the circle $S^1$ (by rotation of loops). Let $μ: Ω(G)\to (\t\times i\mathbb{R})^*$ be a moment map of the resulting $T\times S^1$ action. We show that all levels (that is, pre-images of points) of $μ$ are connected subspaces of $Ω(G)$ (or empty). The result holds if in the definition of $Ω(G)$ loops are of class $C^{\infty}$ or of any Sobolev class $H^s$, with $s\ge 1$ (for loops of class $H^1$, connectedness of regular levels has been proved by Harada, Holm, Jeffrey, and the author).
Consider a closed non-degenerate 3-form $ω$ with an infinitesimal action of a Lie algebra $\mathfrak{g}$. Motivated by the fact that the observables associated to $ω$ form a Lie 2-algebra, we introduce homotopy moment maps defined on a Lie 2-algebra rather than just on the Lie algebra $\mathfrak{g}$. We formulate existence criteria and provide a construction for such homotopy moment maps, by characterizing them in terms of cohomology.
We define an equivariant index of Spin$^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.
In this paper, we construct various examples of Lagrangian mean curvature flows in Calabi-Yau manifolds, using moment maps for actions of abelian Lie groups on them. The examples include Lagrangian self-shrinkers and translating solitons in the Euclidean spaces. Moreover, our method can be applied to construct examples of Lagrangian mean curvature flows in non-flat Calabi-Yau manifolds. In particular, we describe Lagrangian mean curvature flows in 4-dimensional Ricci-flat ALE spaces in detail and investigate their singularities.
We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension $4n$ with a tri-Hamiltonian action of a torus of dimension $n$, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperKähler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connecte…
Convexity package for momentum maps on contact manifolds
设环面T有效地作用在紧致连通共定向切触流形上，Psi是辛化上的自然动量映射。证明了：当dim T > 2时，Psi的原点与象的并是一个凸多面锥，Psi的非零水平集是连通的（而零水平集是不连通的），Psi的动量映射是开的.这回答了Eugene Lerman提出的一个问题，当零水平集为空时，他证明了类似的结果。我们还分析了dim T <= 2的例子。
Multisymplectic geometry admits an operation that has no counterpart in symplectic geometry, namely, taking the product of two multisymplectic manifolds endowed with the wedge product of the multisymplectic forms. We show that there is an L-infinity-embedding of the L-infinity-algebra of observables of the individual factors into the observables of the product, and that homotopy moment maps for the individual factors induce a homotopy moment map for the product. As a by-product, we associate to every multisymplectic form a curved L-infinity-algebra, whose curvature is the multisymplectic form …
We study deformations of symplectic structures on a smooth manifold $M$ via the quasi-Poisson theory. By a fact, we can deform a given symplectic structure $ω$ to a new symplectic structure $ω_t$ parametrized by some element $t$ in $Λ^2\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of a Lie group $G$. Moreover, we can get a lot of concrete examples for the deformations of symplectic structures on the complex projective space and the complex Grassmannian.
We study multi-moment maps induced by a two-torus action on the four homogeneous nearly Kähler six-manifolds. Their explicit expression and stationary orbits are derived. The configuration of fixed-points and one-dimensional orbits is worked out for generic six-manifolds equipped with an $\mathrm{SU}(3)$-structure admitting a two-torus symmetry. Projecting the subspaces obtained to the orbit space yields a trivalent graph. We illustrate this result concretely on the homogeneous nearly Kähler examples.
We study various trivializations of moment maps. First in the general framework of a reductive group $G$ acting on a smooth affine variety. We prove that the moment map is a locally trivial fibration over a regular locus of the center of the Lie algebra of $H$ a maximal compact subgroup of $G$. The construction relies on Kempf-Ness theory and Morse theory of the square norm of the moment map studied by Kirwan, Ness-Mumford and Sjamaar. Then we apply it together with ideas from Nakajima and Kronheimer to trivialize the hyperkaehler moment map for Nakajima quiver varieties. Notice this trivializ…
Deformation quantization and geometric quantization on Kähler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a Kähler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and Kähler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).
The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more general class of group-valued momentum maps which is inspired by the Poisson Lie setting. In this approach, the group-valued momentum map assigns to every section of the symplectic fiber bundle a principal circle-bundle with connection. The power of this general framework is illus…
