Quantum moment map and obstructions to the existence of closed Fedosov star products

Invariant Star Products of Wick Type: Classification and Quantum Momentum Mappings

We extend our investigations on $\mathfrak g$-invariant Fedosov star products and quantum momentum mappings \cite{MN03a} to star products of Wick type on pseudo-Kähler manifolds. Star products of Wick type can be completely characterized by a local description as given by Karabegov in \cite{Kar96} for star products with separation of variables. We separately treat the action of a Lie group $G$ on $\Cinf{M}[[ν]]$ by (pull-backs with) diffeomorphisms and the action of a Lie algebra $\mathfrak g$ on $\Cinf{M}[[ν]]$ by (Lie derivatives with respect to) vector fields. Within Karabegov's framework w…

A cohomological framework for homotopy moment maps

Given a Lie group acting on a manifold $M$ preserving a closed $n+1$-form $ω$, the notion of homotopy moment map for this action was introduced in Callies-Fregier-Rogers-Zambon [6], in terms of $L_{\infty}$-algebra morphisms. In this note we describe homotopy moment maps as coboundaries of a certain complex. This description simplifies greatly computations, and we use it to study various properties of homotopy moment maps: their relation to equivariant cohomology, their obstruction theory, how they induce new ones on mapping spaces, and their equivalences. The results we obtain extend some of …

Global Action-Angle Variables for Non-Commutative Integrable Systems

In this paper we analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (NCI systems) on Poisson manifolds. In contrast with local action-angle variables, which exist as soon as the fibers of the momentum map of such an integrable system are compact, global action-angle variables rarely exist. This fact was first observed and analyzed by Duistermaat in the case of Liouville integrable systems on symplectic manifolds and later by Dazord-Delzant in the case of non-commutative integrable systems on symplectic manifolds. In our mor…

Deformation quantization and homological reduction of a lattice gauge model

For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on $G$. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $G = \operatorname{SU} (2)$ the main hypotheses …

Moment map for coupled equations of Kähler forms and curvature

In this paper we introduce two new systems of equations in Kähler geometry: The coupled p equation and the generalized coupled cscK equation. We motivate the equations from the moment map pictures, prove the uniqueness of solutions and find out the obstructions to the solutions for the second equation. We also point out the connections between the coupled cscK equation, the coupled Kähler Yang-Mills equations and the deformed Hermitian Yang-Mills equation. Moreover, using this moment map, we can show the Mabuchi functional for the generalized coupled cscK equation, and a special case of the co…

Gauged Floer theory of toric moment fibers

We investigate the small area limit of the gauged Lagrangian Floer cohomology of Frauenfelder. The resulting cohomology theory, which we call quasimap Floer cohomology, is an obstruction to displaceability of Lagrangians in the symplectic quotient. We use the theory to reproduce the results of Fukaya-Oh-Ohta-Ono and Cho-Oh on non-displaceability of moment fibers of not-necessarily-Fano toric varieties and extend their results to toric orbifolds, without virtual fundamental chains. Finally we describe a conjectural relationship to Floer cohomology in the quotient.

Almost Homogeneous Poisson Spaces

We prove that any holomorphic Poisson manifold has an open symplectic leaf which is a pseudo-Kähler submanifold, and we define an obstruction to study the equivariance of momentum map for tangential Poisson action. Some properties of almost homogeneous Poisson manifolds are studied and we show that any compact symplectic Poisson homogeneous space is a torus bundle over a dressing orbit.

Real Lagrangian Tori and Versal Deformations

Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called \emph{real}. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not re…

Homotopy moment maps

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L-infinity morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such m…

Rigid subsets of symplectic manifolds

We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-…

相对同伦矩映射

Relative homotopy moment maps

本文针对配备相对\(n\)-辛结构的光滑映射，发展相对同伦矩映射理论。通过李群作用定义该映射，用显式方程刻画，关联相对嘉当模型，证明相关结论，还表明拟哈密顿\(G\)空间有规范相对同伦矩映射并可计算。

Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2π} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspo…