Symplectic Manifold Research Articles

Berezin–Toeplitz Quantization on Symplectic Manifolds of Bounded Geometry

We establish the theory of Berezin-Toeplitz quantization on symplectic manifolds of bounded geometry. The quantum space of this quantization is the spectral subspace of the renormalized Bochner Laplacian associated with some interval near zero. We show that this quantization has the correct semiclassical limit.

Commuting Toeplitz Operators on Cartan Domains of Type IV and Moment Maps

Let us consider, for $$n \ge 3$$ , the Cartan domain $$\mathrm {D}_n^{\mathrm {IV}}$$ of type IV. On the weighted Bergman spaces $$\mathcal {A}^2_\lambda (\mathrm {D}_n^{\mathrm {IV}})$$ we study the problem of the existence of commutative $$C^*$$ -algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $$\mathrm {SO}(n) \times \mathrm {SO}(2)$$ of biholomorphisms of $$\mathrm {D}_n^{\mathrm {IV}}$$ that fix the origin. The $$\mathrm {SO}(n) \times \mathrm {SO}(2)$$ -invariant symbols yield Toeplitz operators that generate commutative $$C^*$$ -algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $$\mathrm {SO}(2)$$ . We compute the moment map $$\mu ^{\mathrm {SO}(2)}$$ for the $$\mathrm {SO}(2)$$ -action on $$\mathrm {D}_n^{\mathrm {IV}}$$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $$a = f \circ \mu ^{\mathrm {SO}(2)}$$ , denoted by $$L^\infty (\mathrm {D}_n^{\mathrm {IV}})^{\mu ^{\mathrm {SO}(2)}}$$ , yield Toeplitz operators that generate commutative $$C^*$$ -algebras. Spectral integral formulas for these Toeplitz operators are also obtained.

Quantization: History and problems

In this work, I explore the concept of quantization as a mapping from classical phase space functions to quantum operators. I discuss the early history of this notion of quantization with emphasis on the works of Schrödinger and Dirac, and how quantization fit into their overall understanding of quantum theory in the 1920's. Dirac, in particular, proposed a quantization map which should satisfy certain properties, including the property that quantum commutators should be related to classical Poisson brackets in a particular way. However, in 1946, Groenewold proved that Dirac's mapping was inconsistent, making the problem of defining a rigorous quantization map more elusive than originally expected. This result, known as the Groenewold-Van Hove theorem, is not often discussed in physics texts, but here I will give an account of the theorem and what it means for potential ``corrections” to Dirac's scheme. Other proposals for quantization have arisen over the years, the first major one being that of Weyl in 1927, which was later developed by many, including Groenewold, and which has since become known as Weyl Quantization in the mathematical literature. Another, known as Geometric Quantization, formulates quantization in differential-geometric terms by appealing to the character of classical phase spaces as symplectic manifolds; this approach began with the work of Souriau, Kostant, and Kirillov in the 1960's. I will describe these proposals for quantization and comment on their relation to Dirac's original program. Along the way, the problem of operator ordering and of quantizing in curvilinear coordinates will be described, since these are natural questions that immediately present themselves when thinking about quantization.

A relative Hofer estimate and the asymptotic Hofer-Lipschitz constant

Let (M,ω) be a symplectic manifold and U⊆M an open subset. We study the natural inclusion of the compactly supported Hamiltonian group of U in the compactly supported Hamiltonian group of M. The main result is an upper bound for this map in terms of the Hofer norms for U and M.Applications are upper bounds on the asymptotic Hofer-Lipschitz constant and the relative Hofer diameter of U. The first bound is often sharp and the second one is often sharp up to a factor of 2.

Boucksom–Zariski and Weyl Chambers on Irreducible Holomorphic Symplectic Manifolds

Abstract Inspired by the work of Bauer, Küronya, and Szemberg, we provide for the big cone of a projective irreducible holomorphic symplectic (IHS) manifold a decomposition into chambers (which we describe in detail), in each of which the support of the negative part of the divisorial Zariski decomposition is constant. We see how the obtained decomposition of the big cone allows to describe the volume function. Moreover, similarly to the case of surfaces, we see that the big cone of a projective IHS manifold admits a decomposition into simple Weyl chambers, which we compare to that induced by the divisorial Zariski decomposition. To conclude, we determine the structure of the pseudo-effective cone.

Operad structures in geometric quantization of the moduli space of spatial polygons

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kähler and real polarizations. In this paper, associated to the Kähler and real polarizations, morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ are constructed by using the quantum Hilbert spaces $\mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}}$ and $\mathscr{H}_\mathrm{re}$, respectively. Moreover, the relationship between the two morphisms of operads $\mathsf{f}_{\mathsf{K}\ddot{\mathsf{a}}\mathsf{h}}$ and $\mathsf{f}_{\mathsf{re}}$ is studied and then the equality $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ is proved in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama (2000) for proving $\dim \mathscr{H}_{\mathrm{K}\ddot{\mathrm{a}}\mathrm{h}} = \dim \mathscr{H}_\mathrm{re}$ in a special case.

Symplectic Flatness and Twisted Primitive Cohomology

We introduce the notion of symplectic flatness for connections and fibre bundles over symplectic manifolds. Given an A_infty -algebra, we present a flatness condition that enables the twisting of the differential complex associated with the A_infty -algebra. The symplectic flatness condition arises from twisting the A_infty -algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang–Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

Floer cohomology and flips

We show that blow-ups or reverse flips (in the sense of the minimal model program) of rational symplectic manifolds with point centers create Floer-non-trivial Lagrangian tori. These results are part of a conjectural decomposition of the Fukaya category of a compact symplectic manifold with a singularity-free running of the minimal model program, analogous to the description of Bondal-Orlov (Derived categories of coherent sheaves, 2002) and Kawamata (Derived categories of toric varieties, 2006) of the bounded derived category of coherent sheaves on a compact complex manifold.

The bijectivity of mirror functors on tori

By the SYZ construction, a mirror pair (X,Xˇ) of a complex torus X and a mirror partner Xˇ of the complex torus X is described as the special Lagrangian torus fibrations X→B and Xˇ→B on the same base space B. Then, by the SYZ transform, we can construct a simple projectively flat bundle on X from each affine Lagrangian multisection of Xˇ→B with a unitary local system along it. However, there are ambiguities of the choices of transition functions of it, and this causes difficulties when we try to construct a functor between the symplectic geometric category and the complex geometric category. In this paper, we prove that there exists a bijection between the set of the isomorphism classes of their objects by solving this problem.

A DERIVED LAGRANGIAN FIBRATION ON THE DERIVED CRITICAL LOCUS

AbstractWe study the symplectic geometry of derived intersections of Lagrangian morphisms. In particular, we show that for a functional $f : X \rightarrow \mathbb {A}_{k}^{1}$ , the derived critical locus has a natural Lagrangian fibration $\textbf {Crit}(f) \rightarrow X$ . In the case where f is nondegenerate and the strict critical locus is smooth, we show that the Lagrangian fibration on the derived critical locus is determined by the Hessian quadratic form.

Singular symplectic spaces and holomorphic membranes

We set up a topological framework for degenerations of symplectic manifolds into singular spaces paying a special attention to the behavior of Lagrangian manifolds and their (holomorphic) membranes. We show that degenerations into singular toric varieties provide a source of exotic Lagrangian tori.

Lagrangian fields, Calabi functions, and local symplectic groupoids

A Lagrangian field on a symplectic manifold M is a family Λ={Λx|x∈M} of pointed Lagrangian submanifolds of M. This notion is a generalization of a real Lagrangian polarization for which each Λx is the leaf containing x. Two Lagrangian fields Λ and Λ˜ are called transversal if Λx intersects Λ˜x transversally at x for every x∈M. Two transversal Lagrangian fields determine an almost para-Kähler structure on M. We construct a local symplectic groupoid on a neighborhood of the zero section of T⁎M from two transversal Lagrangian fields on M. The Lagrangian manifold of n-cycles of this groupoid in (T⁎M)n has a generating function whose germ around the diagonal of Mn is given by the n-point cyclic Calabi function of a closed (1,1)-form on a neighborhood of the diagonal of M2 obtained from the symplectic form on M.

A b$b$‐symplectic slice theorem

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of b $b$ -symplectic manifolds started in Guillemin, Miranda, and Pires Adv. Math. 264 (2014), 864–896, we prove a slice theorem for Lie group actions on b $b$ -symplectic manifolds.

Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector 2λ1+2λ2 in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution M˜, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of M˜ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin [58] in the very general case. We also prove that M˜ is the MRC quotient of the main component of the Hilbert scheme of quintic elliptic curves in Y, confirming a conjecture of Castravet.

Mirror Symmetry for Truncated Cluster Varieties

In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cluster variety, we construct a symplectic manifold and prove homological mirror symmetry for the resulting pair. We also describe how our construction can be obtained from toric geometry, and we relate our construction to various aspects of cluster theory which are known to symplectic geometers.

A Minimal Maslov Number Condition for Displaceability in Certain Weakly Exact Symplectic Manifolds

A Minimal Maslov Number Condition for Displaceability in Certain Weakly Exact Symplectic Manifolds

Eggbeater dynamics on symplectic surfaces of genus 2 and 3

The group $$Ham(M,\omega )$$ of all Hamiltonian diffeomorphisms of a symplectic manifold $$(M,\omega )$$ plays a central role in symplectic geometry. This group is endowed with the Hofer metric. In this paper we study two aspects of the geometry of $$Ham(M,\omega )$$ , in the case where M is a closed surface of genus 2 or 3. First, we prove that there exist diffeomorphisms in $$Ham(M,\omega )$$ arbitrarily far from being a k-th power, with respect to the metric, for any $$k \ge 2$$ . This part generalizes previous work by Polterovich and Shelukhin. Second, we show that the free group on two generators embeds into the asymptotic cone of $$Ham(M,\omega )$$ . This part extends previous work by Alvarez-Gavela et al. Both extensions are based on two results from geometric group theory regarding incompressibility of surface embeddings.

Loops in the fundamental group of which are not represented by circle actions

Abstract We study generators of the fundamental group of the group of symplectomorphisms $\operatorname {\mathrm{Symp}} (\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2, \omega )$ for some particular symplectic forms. It was observed by Kȩdra (2009, Archivum Mathematicum 45) that there are many symplectic 4-manifolds $(M, \omega )$ , where M is neither rational nor ruled, that admit no circle action and $\pi _1 (\operatorname {\mathrm {Ham}} (M,\omega ))$ is nontrivial. On the other hand, it follows from Abreu and McDuff (2000, Journal of the American Mathematical Society 13, 971–1009), Anjos and Eden (2019, Michigan Mathematical Journal 68, 71–126), Anjos and Pinsonnault (2013, Mathematische Zeitschrift 275, 245–292), and Pinsonnault (2008, Compositio Mathematica 144, 787–810) that the fundamental group of the group $ \operatorname {\mathrm{Symp}}_h(\mathbb C\mathbb P^2\#\,k\overline { \mathbb C\mathbb P}\,\!^2,\omega )$ , of symplectomorphisms that act trivially on homology, with $k \leq 4$ , is generated by circle actions on the manifold. We show that, for some particular symplectic forms $\omega $ , the set of all Hamiltonian circle actions generates a proper subgroup in $\pi _1(\operatorname {\mathrm{Symp}}_{h}(\mathbb C\mathbb P^2\#\,5\overline { \mathbb C\mathbb P}\,\!^2,\omega )).$ Our work depends on Delzant classification of toric symplectic manifolds, Karshon’s classification of Hamiltonian $S^1$ -spaces, and the computation of Seidel elements of some circle actions.

Quantum cohomology of symplectic flag manifolds

We compute the quantum cohomology of symplectic flag manifolds. Symplectic flag manifolds can be described by non-abelian GLSMs with superpotential. Although the ring relations cannot be directly read off from the equations of motion on the Coulomb branch due to complication introduced by the non-abelian gauge symmetry, it can be shown that they can be extracted from the localization formula in a gauge-invariant form. Our result is general for all symplectic flag manifolds, which reduces to previously established results on symplectic Grassmannians and complete symplectic flag manifolds derived by other means. We also explain why a (0, 2) deformation of the GLSM does not give rise to a deformation of the quantum cohomology.

Survey on the metric SYZ conjecture and non-Archimedean geometry

We survey the metric aspects of the Strominger–Yau–Zaslow conjecture on the existence of special Lagrangian fibrations on Calabi–Yau manifolds near the large complex structure limit. We will discuss the diverse motivations for the conjectural picture, what the best hopes are, and a number of subtleties. The bulk of the survey highlights the role of pluripotential theory, and non-Archimedean geometry in particular, with a list of open questions.