In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M).
In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism, denoted by $\calabi\_{S}$, is defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface $S$ of genus greater than 1. This construction is motivated by a question of M. Entov and L. Polterovich. If $U\subset S$ is a disk or an annulus, the restriction of $\calabi\_{S}$ to the subgroup of diffeomorphisms which are the time one map of a Ha…
We show the existence of Calabi quasimorphisms on the universal covering of the group of Hamiltonian diffeomorphisms of a monotone coadjoint orbit of a compact Lie group. We show that this result follows from positivity results of Gromov-Witten invariants and the fact that the quantum product of Schubert classes can never be zero.
In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such $Z$-semistable morphisms of a fixed type is compact. This gives us a counter moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten theory to the fundamental string. It serves and prepares for us the basis toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string …
We prove that quasi-morphisms and quasi-states on a closed integral symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.
On groups of symplectomorphisms of the disk, we construct two homogeneous quasi-morphisms which relate to the Calabi invariant and the flux homomorphism respectively. We also show the relation between the quasi-morphisms and the translation number introduced by Poincaré.
Relative Calabi-Yau structures and ice quivers with potential
Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG（巴黎西岱大学与索邦大学，法国国家科学研究中心，IMJ-PRG）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.
We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.
In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this notion to give a prototypical definition of supersymmetric D-branes of A-type (i.e. A-branes) -- in an appropriate region of the Wilson's theory-space of string theory -- as special Lagrangian morphisms from such objects with a unitary, minimally flat connection-with-singularities. This merges Donaldson's picture of special Lagrangian submanifolds and the Polchinski-Grothendieck Ansatz …
The main purpose of this note is to exhibit a Hamiltonian diffeomorphism loop undetected by the Seidel morphism of certain 2-point blow-ups of $S^2 \times S^2$, exactly one of which being monotone. As side remarks, we show that Seidel's morphism is injective on all Hirzebruch surfaces and discuss how to adapt the monotone example to the Lagrangian setting.
We prove that the Seidel morphism of $(M \times M', ω\oplus ω')$ is naturally related to the Seidel morphisms of $(M,ω)$ and $(M',ω')$, when these manifolds are monotone. We deduce that any homotopy class of loops of Hamiltonian diffeomorphisms of one component, with non-trivial image via Seidel's morphism, leads to an injection of the fundamental group of the group of Hamiltonian diffeomorphisms of the other component into the fundamental group of the group of Hamiltonian diffeomorphisms of the product. This result was inspired by and extends results obtained by Pedroza [P08].
In a suitable regime of superstring theory, D-branes in a Calabi-Yau space and their most fundamental behaviors can be nicely described mathematically through morphisms from Azumaya spaces with a fundamental module to that Calabi-Yau space. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121 [math.AG], with Si Li and Ruifang Song) from the project, we explored this notion for the case of D1-branes (i.e. D-strings) and laid down some basic ingredients toward understanding the notion of D-string world-sheet instantons in this context. In this continuation, D(10), of D(2), we move on to construc…
We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental's nonlinear Maslov index and a reduction theorem for quasi-morphisms on contactomorphism groups previously established together with M. Strom Borman. We explore applications to metrics on this group and to symplectic and contac…
In this note the interrelations between several natural morphisms on the $π_1$ of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (non-linear) Maslov index of loops of quantomorphisms of prequantizations of $\C P^n$ and the Calabi-Weinstein invariant is shown, settling affirmatively a conjecture by A. Givental. We also prove the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds, as suggested by C. Woodward. Finally, a family of generalized action-Maslov inv…
Let Σ_g be a closed orientable surface let Diff_0(Σ_g; area) be the identity component of the group of area-preserving diffeomorphisms of Σ_g. In this work we present an extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface Σ_g, i.e. we show that every non-trivial homogeneous quasi-morphism on the braid group on n strings of Σ_g defines a non-trivial homogeneous quasi-morphism on the group Diff_0(Σ_g; area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff_0(Σ_g; area) is infinite dimensional. Let Ham(Σ_g) be …
We review and streamline our previous results and the results of Y.Ostrover on the existence of Calabi quasi-morphisms and symplectic quasi-states on symplectic manifolds with semi-simple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4-manifolds. We present also new results due to D.McDuff: she observed that for the existence of quasi-morphisms/quasi-states it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blow-ups of non-uniruled symplectic manifolds.
In this paper we study certain algebraic properties of the quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semi-simplicity of the quantum homology algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily-verified sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non semi-simple quantum homology algebra, and others in which …
In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,ω)$ admitting Hamiltonian torsio…
Let (M,w) be a compact symplectic manifold, and L a compact, embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono construct Lagrangian Floer cohomology for such M,L, yielding groups HF^*(L,b;Λ) for one Lagrangian or HF^*((L,b),(L',b');Λ) for two, where b,b' are choices of bounding cochains, and exist if and only if L,L' have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M,w), and so ar…
