Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConvexity of singular affine structures and toric-focus integrable Hamiltonian systems
58
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 and the local-global convexity principle still applies. They are also globally convex under some natural additional conditions. However, when the monodromy is sufficiently big then the local-global convexity principle breaks down, and the base spaces can be globally non-convex even for compact manifolds. As one of surprising examples, we construct a 2-dimensional integral affine black hole, which is locally convex but for which a straight ray from the center can never escape.
rate research
Read More
We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(mathbb{Z}_2 times D_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields $X_{f_1}, X_{f_2}$ associated to the components $f_1, f_2 colon (M, omega) to mathbb{R}$ of the integrable system on the symplectic manifold $(M,omega)$ via the differential equation $omega(X_{f_i}, cdot) = mathop{}!mathrm{d} f_i$, near the singular fiber containing the $k$ focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.
About 6 years ago, semitoric systems were classified by Pelayo & Vu Ngoc by means of five invariants. Standard examples are the coupled spin oscillator on $mathbb{S}^2 times mathbb{R}^2$ and coupled angular momenta on $mathbb{S}^2 times mathbb{S}^2$, both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on $mathbb{S}^2 times mathbb{S}^2$ and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.
We show that all compact four-dimensional Hamiltonian $S^1$-spaces can be extended to a completely integrable system on the same manifold such that all singularities are non-degenerate, except possibly for a finite number of degenerate orbits of parabolic (also called cuspidal) type -- we call such systems hypersemitoric. More precisely, given any compact four dimensional Hamiltonian $S^1$-space $(M,omega,J)$ we show that there exists a smooth $Hcolon Mtomathbb{R}$ such that $(M,omega,(J,H))$ is a completely integrable system of hypersemitoric type. Hypersemitoric systems generalize semitoric systems. In addition to elliptic-elliptic, elliptic-regular, and focus-focus singular points which can occur in semitoric systems, hypersemitoric systems may also have hyperbolic-regular and hyperbolic-elliptic singular points (hyperbolic-hyperbolic points cannot appear due to the presence of the global $S^1$-action) and moreover degenerate singular points of a relatively tame type called parabolic. Admitting the existence of degenerate points is necessary since there exist compact four-dimensional Hamiltonian $S^1$-spaces whose extensions must include degenerate singular points of some kind as we show in the present paper. Parabolic points are among the most common and natural degenerate points, and we show that it is sufficient to only admit these degenerate points in order to extend all Hamiltonian $S^1$-spaces. In this sense, hypersemitoric systems are thus the nicest and smallest class of systems to which all Hamiltonian $S^1$-spaces can be extended. Moreover, we prove several foundational results about these systems, such as the non-existence of loops of hyperbolic-regular points and properties about their fibers.
We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].
We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are related by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
