Symplectic Invariants Research Articles

On symplectic capacities and their blind spots

In this paper, we settle three basic questions concerning the Gutt–Hutchings capacities from [11] which are conjecturally equal to the Ekeland–Hofer capacities from [7, 8]. Our primary result settles a version of the recognition question from [4], in the negative. We prove that the Gutt–Hutchings capacities together with the volume do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. In particular, we construct a smooth family of such domains, all of which have the same Gutt–Hutchings capacities and volume, but no two of which are symplectomorphic. We also establish two independence properties of the Gutt–Hutchings capacities. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume by constructing a family of star-shaped domains with smooth boundaries, whose Gutt–Hutchings capacities all agree but whose volumes all differ. We also prove that the capacities are mutually independent by constructing, for any [Formula: see text], a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the [Formula: see text]th. The constructions underlying these results are not exotic. They are convex and concave toric domains as defined in [11], where the authors also establish beautiful formulae for their capacities. A key to the progress made here is a significant simplification of these formulae under an additional symmetry assumption. This simplification allows us to identify new blind spots of the Gutt–Hutchings capacities which are used to construct the desired examples. This simplification also yields explicit computations of the Gutt–Hutchings capacities in many new examples.

Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form

This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. In the two-mode states, the (ordinary) CM is a real symmetric matrix of order 4; therefore, it depends on 10 real variables. However, there is a very efficient representation of the CM called the standard form (SF) that reduces the degrees of freedom to four real variables, while preserving all the relevant information on the state. The SF can be easily evaluated using a set of symplectic invariants. The paper starts from the SF, introducing an architecture that implements with primitive components the given two-mode Gaussian state having the CM with the SF. The architecture consists of a beam splitter, followed by the parallel set of two single–mode real squeezers, followed by another beam splitter. The advantage of this architecture is that it gives a precise non-redundant physical meaning of the generation of the Gaussian state. Essentially, all the relevant information is contained in this simple architecture.

On symplectic invariants of planar 3-webs

The classical web geometry [1,3,4] studies invariants of foliation families with respect to pseudogroup of diffeomorphisms. Thus for the case of planar 3-webs the basic semi invariant is the Blaschke curvature, [2]. It is also curvature of the Chern connection [4] that are naturally associated with a planar 3-web. In this paper we investigate invariants of planar 3-webs with respect to group of symplectic diffeomorphisms. We found the basic symplectic invariants of planar 3-webs that allow us to solve the symplectic equivalence problem for planar 3-webs in general position. The Lie-Tresse theorem, [4], gives the complete description of the field of rational symplectic differential invariants of planar 3-webs. We also give normal forms for homogeneous 3-webs, i.e. 3-webs having constant basic invariants.

Jacobi fields in optimal control: Morse and Maslov indices

In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalisation of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic invariants of Jacobi curves.

The Height Invariant of a Four-Parameter Semitoric System with Two Focus\u2013Focus Singularities

Semitoric systems are a special class of completely integrable systems with two degrees of freedom that have been symplectically classified by Pelayo and Vũ Ngọc about a decade ago in terms of five symplectic invariants. If a semitoric system has several focus–focus singularities, then some of these invariants have multiple components, one for each focus–focus singularity. Their computation is not at all evident, especially in multi-parameter families. In this paper, we consider a four-parameter family of semitoric systems with two focus–focus singularities. In particular, apart from the polygon invariant, we compute the so-called height invariant. Moreover, we show that the two components of this invariant encode the symmetries of the system in an intricate way.

Mixed partition functions and exponentially bounded edge-connection rank

We study graph parameters whose associated edge-connection matrices have exponentially bounded rank growth. Our main result is an explicit construction of a large class of graph parameters with this property that we call mixed partition functions . Mixed partition functions can be seen as a generalization of partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe and V. F. R. Jones, Graph invariants related to statistical mechanical models: examples and problems, J. Combin. Theory Ser. B 57 (1993), no. 2, 207–227.] and they are related to invariant theory of orthosymplectic supergroup. We moreover show that evaluations of the characteristic polynomial of a simple graph are examples of mixed partition functions, answering a question of de la Harpe and Jones. (NOTE. Some of the results of this paper were announced in an extended abstract: G. Regts and B. Sevenster, Partition functions from orthogonal and symplectic group invariants, Electron. Notes Discrete Math. 61 (2017), 1011–1017. Unfortunately that reference contains a mistake; we will comment on that below).

Squared Dehn twists and deformed symplectic invariants

We establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in \cite{murphysiegel}.

Computable genuine multimode entanglement measure: Gaussian versus non-Gaussian

Genuine multimode entanglement in continuous variable systems can be quantified by exploring the geometry of the state-space, namely via the generalized geometric measure (GGM) which is defined as the shortest distance of a given multimode state from a nongenunely multimode entangled state. For multimode Gaussian states, we derive a closed form expression of GGM in terms of the symplectic invariants of the reduced states. Following that prescription, the characteristics of GGM for typical three- and four-mode Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM for photon-added as well as -subtracted states having three- and four-modes and find that both addition and subtraction of photons lead to enhancement of the genuine multimode entanglement content of the state compared to its Gaussian counterpart. Our analysis reveals that when an initial three-mode vacuum state is evolved according to an interacting Hamiltonian, photon addition is more beneficial in increasing GGM compared to photon subtraction while the picture is opposite for four-mode case. Specifically, subtracting photons from four-mode squeezed vacuum states almost always results in higher multimode entanglement content than that of photon addition with both single as well as multimode and constrained as well as unconstrained operations. Furthermore, we observe a novel freezing feature of GGM during some specific cases of photon subtraction with respect to the number of photons subtracted.

Symplectic invariants of semitoric systems and the inverse problem for quantum systems

Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with labels which are symplectic invariants of the system. We will review the construction of these invariants, and explain how they have been generalized or applied in different contexts. One of these applications concerns quantum integrable systems and the corresponding inverse problem, which asks how much information of the associated classical system can be found in the spectrum. An approach to this problem has been to try to compute invariants in the spectrum. We will explain how this has been recently achieved for some of the invariants of semitoric systems, and discuss an open question in this direction.

Symplectic structure perturbations and continuity of symplectic invariants

This paper studies how symplectic invariants created from Hamiltonian Floer theory change under the perturbations of symplectic structures, not necessarily in the same cohomology class. These symplectic invariants include spectral invariants, boundary depth, and (partial) symplectic quasi-states. This paper can split into two parts. In the first part, we prove some energy estimations that control the shifts of symplectic action functionals. These directly imply positive conclusions on the continuity of spectral invariants and boundary depth, in some important cases including any symplectic surface $\Sigma_{g \geq1}$ and any closed symplectic manifold $M$ with $\dim_{\mathcal K} H^2(M; \mathcal K) = 1$. This follows by applications on some rigidity of the subsets of a symplectic manifold in terms of heaviness and superheaviness, as well as on the continuity property of some symplectic capacities. In the second part, we generalize the construction in the first part to any closed symplectic manifold. In particular, to deal with the change of Novikov rings from symplectic structure perturbations, we construct a family of variant Floer chain complexes over a common Novikov-type ring. In this set-up, we define a new family of spectral invariants called $t$-spectral invariants, and prove that they are upper semicontinuous under the symplectic structure perturbations. This implies a quasi-isometric embedding from $(\mathbb R^{\infty}, |-|_{\infty})$ to $({\widetilde {\rm Ham}}(M, \omega), |-|_{\rm Hofer})$ under some dynamical assumption, imitating a result from Usher.

Geometry on the manifold of Gaussian quantum channels

In the space of quantum channels, we establish the geometry that allows us to make statistical predictions about relative volumes of entanglement breaking channels among all the Gaussian quantum channels. The underlying metric is constructed using the Choi-Jamio{\l}kowski isomorphism between the continuous-variable Gaussian states and channels. This construction involves the Hilbert-Schmidt distance in quantum state space. The volume element of the one-mode Gaussian channels can be expressed in terms of local symplectic invariants. We analytically compute the relative volumes of the one-mode Gaussian entanglement breaking and incompatibility breaking channels. Finally, we show that, when given the purities of the Choi-Jamio{\l}kowski state of the channel, one can determine whether or not such channel is incompatibility breaking.

Symplectic classification of coupled angular momenta

The coupled angular momenta are a family of completely integrable systems that depend on three parameters and have a compact phase space. They correspond to the classical version of the coupling of two quantum angular momenta and they constitute one of the fundamental examples of so-called semitoric systems. Pelayo and Vũ Ngọc have given a classification of semitoric systems in terms of five symplectic invariants. Three of these invariants have already been partially calculated in the literature for a certain parameter range, together with the linear terms of the so-called Taylor series invariant for a fixed choice of parameter values.In the present paper we complete the classification by calculating the polygon invariant, the height invariant, the twisting-index invariant, and the higher-order terms of the Taylor series invariant for the whole family of systems. We also analyse the explicit dependence of the coefficients of the Taylor series with respect to the three parameters of the system, in particular near the Hopf bifurcation where the focus–focus point becomes degenerate.

Bang’s problem and symplectic invariants

We consider the Tarski--Bang problem about covering of convex bodies by planks. The results of this kind give a lower bound on the sum of widths of planks (regions between a pair of parallel hyperplanes) covering a given convex body. Previously we have applied some notions of symplectic geometry to study convex bodies, and here we show that the symplectic techniques may be useful in this problem as well. We are able to handle some particular cases with the symplectic techniques, and show that the general cases would follow from a certain ``subadditivity conjecture'' in symplectic geometry, motivated by the results of K.~Ball. We also prove several related results by more elementary methods.

The sharp energy–capacity inequality on convex symplectic manifolds

In symplectic geometry, symplectic invariants are useful tools in studying symplectic phenomena. One such invariant is the Hofer–Zehnder capacity, which is defined for any subset of a symplectic manifold. On the other hand, we can associate the so-called displacement energy to any subset. Many symplectic geometers tried to relate the Hofer–Zehnder capacity and the displacement energy to study the behaviour of closed orbits of Hamiltonian diffeomorphisms. Usher proved the so-called ( $$\pi _1$$ -sensitive) sharp energy–capacity inequality between the Hofer–Zehnder capacity and the displacement energy for closed symplectic manifolds. In this paper, we consider a certain Floer homology on symplectic manifolds with boundaries (not symplectic homology) and its spectral invariants. Then we extend the $$\pi _1$$ -sensitive sharp energy–capacity inequality to convex symplectic manifolds. As a corollary, we also prove the almost existence theorem of closed characteristics near displaceable hypersurfaces in convex symplectic manifolds. In particular, we prove the existence of closed characteristics on displaceable contact type hypersurfaces in convex symplectic manifolds (the Weinstein conjecture).

Typical Gaussian quantum information

We investigate different geometries and invariant measures on the space of mixed Gaussian quantum states. We show that when the global purity of the state is held fixed, these measures coincide and it is possible, within this constraint, to define a unique notion of volume on the space of mixed Gaussian states. We then use the so defined measure to study typical non-classical correlations of two mode mixed Gaussian quantum states, in particular entanglement and steerability. We show that under the purity constraint alone, typical values for symplectic invariants can be computed very elegantly, irrespectively of the non-compactness of the underlying state space. Then we consider finite volumes by constraining the purity and energy of the Gaussian state and compute typical values of quantum correlations numerically.

Symplectic Geometry and Spectral Properties of Classical and Quantum Coupled Angular Momenta

We give a detailed study of the symplectic geometry of a family of integrable systems obtained by coupling two angular momenta in a non-trivial way. These systems depend on a parameter $$t \in [0,1]$$ and exhibit different behaviors according to its value. For a certain range of values, the system is semitoric, and we compute some of its symplectic invariants. Even though these invariants have been known for almost a decade, this is to our knowledge the first example of their computation in the case of a non-toric semitoric system on a compact manifold. (The only invariant of toric systems is the image of the momentum map.) In the second part of the paper, we quantize this system, compute its joint spectrum and describe how to use this joint spectrum to recover information about the symplectic invariants.

Symplectic invariants for parabolic orbits and cusp singularities of integrable systems.

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

Lie–Hamilton systems on curved spaces: a geometrical approach

A Lie–Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot–Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie–Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.

Koszul duality patterns in Floer theory

We study symplectic invariants of the open symplectic manifolds $X_\Gamma$ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree $\Gamma$. For any tree $\Gamma$, we calculate (DG-)algebra models of the Fukaya category $\mathcal{F}(X_\Gamma)$ of closed exact Lagrangians in $X_\Gamma$ and the wrapped Fukaya category $\mathcal{W}(X_\Gamma)$. When $\Gamma$ is a Dynkin tree of type $A_n$ or $D_n$ (and conjecturally also for $E_6,E_7,E_8$), we prove that these models for the Fukaya category $\mathcal{F}(X_\Gamma)$ and $\mathcal{W}(X_\Gamma)$ are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of $X_\Gamma$ for $\Gamma=A_n,D_n$, based on the Legendrian surgery formula of Bourgeois, Ekholm and Eliashberg.

Partition functions from orthogonal and symplectic group invariants

We introduce a new class of graph parameters coming from invariants of the orthosymplectic group, which we call mixed partition functions. This class strictly contains partition functions of edge-coloring models and we give a characterization of this class by means of rank growth of associated connection matrices.