We settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of such a system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and Poisson commuting functions, up to symplectomorphisms. We also give a full description of the semiclassical spectral theory of quantum toric integrable systems. This type of problem belongs to the realm of classical questions in spectral theory going back to pioneer works of Colin de Verdiere, Guillemin, Sternberg and others in the 1970s and 1980s.
In this article we show how one can use the local models of integrable Hamiltonian systems near critical points to prove a localization theorem for certain singular loci of integrables semi-toric systems for dimension greater than 4.
We show that existence and uniqueness of solutions to transported Monge-Ampere problem on complex compact toric manifold follows easily from the real theory of optimal transportation.
The goal of the paper is to develop a systematic approach to the study of (perhaps degenerate) singularities of integrable systems and their structural stability. As the main tool, we use "hidden" system-preserving torus actions near singular orbits. We give sufficient conditions for the existence of such actions and show that they are persistent under integrable perturbations. We find toric symmetries for several infinite series of singularities and prove, as an application, structural stability of Kalashnikov's parabolic orbits with resonances in the real-analytic case. We also classify all …
I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold $S^2\times S^3$. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, $Y^{p,q}$, discovered by physicists by showing that $Y^{p,q}$ and $Y^{p',q'}$ are inequivalent as contact structures if and only if $p\neq p'$.
We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.
We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, the convergence is uniform. Analogous results hold for non-commuting operators.
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners), similarly to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In particular, near a focus singular point, they are locally convex a…
We characterize completely integrable Hamiltonian systems inducing an effective Hamiltonian torus action as systems with zero transport costs w.r.t. the time-$T$ map where $T \in {\mathbb R}^n$ is the period of the acting $n$-torus.
Let X be a smooth complex projective variety of dimension n equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions H_1, ... H_n on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U. Moreover, we show that in many cases, we c…
This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vu Ngoc and classified by Pelayo and Vu Ngoc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of …
Kostant gave a model for the real geometric quantization associated to polarizations via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed counting integral points inside the associated Delzant polytope. In this article we extend Kostant's geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of t…
The Stark problem is a completely integrable system which describes the motion of an electron in a constant electric field and subject to the attraction of a proton. In this paper we show that in the planar case after Levi-Civita regularization the bounded component of the energy hypersurfaces of the Stark problem for energies below the critical value can be interpreted as boundaries of concave toric domains.
This is a short companion paper to arXiv:0810.3470. We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fibers if the central fiber is a toric Fano variety admitting a small resolution.
We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system.
These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart will also be presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can you recover the classical system?
We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover the classification of two-dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric …
The presence of focus-focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action-Angle coordinates on such systems. At focus-focus critical points, the Liouville-Arnold-Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behaviour and the multi-valuedness of the Action integrals is given by a complex logarithm. This extends a previous result by Vu Ngoc to any dimension. We also calculate the monodromy m…
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 th…
