arXiv subjects

Konrad Schöbel

Publications and source records attributed to Konrad Schöbel.

9 recordsLinked to original sources

arXiv2024

Superintegrable systems on conformal surfaces

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of dimension three and higher. The governing equations of non-degenerate second order superintegrability in dimension two are structurally significantly different from those valid in higher dimensions. Specifically, we find conformally covariant structural equations, allowing one to classify the (conformal classes of) non-degenerate second order superintegrable systems on conformal surfaces geometrically. We then specialise to second order properly superintegrable systems on surfaces with a (pseudo-)Riemannian metric and obtain structural equations in accordance with the known equations for Euclidean space. We finally give a single explicit set of purely algebraic equations defining the variety parametrising such systems on all constant curvature surfaces.

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arXiv2022

Algebraic Conditions for Conformal Superintegrability in Arbitrary Dimension

We show that the definition of a second order superintegrable system on a (pseudo-)Riemannian manifold gives rise to a conformally invariant notion of superintegrability. Conformal equivalence is the natural extension of the well-known Stäckel transform, which in turn originates from the classical Maupertuis-Jacobi principle. We extend our recently developed algebraic geometric approach for the classification of second order superintegrable systems in arbitrarily high dimension to conformally superintegrable systems, which are presented via conformal scale choices of second order superintegrable systems defined within a conformal geometry. For superintegrable systems on constant curvature spaces, we find that the conformal scales of Stäckel equivalent systems arise from eigenfunctions of the Laplacian and that their equivalence is characterised by a conformal density of weight two. Our approach yields an algebraic equation that governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems. This class contains all non-degenerate examples known to date, and is given by a simple algebraic constraint of degree two on a general harmonic cubic form. In this way the yet unsolved classification problem is put into the reach of algebraic geometry and geometric invariant theory. In particular, no obstruction exists in dimension three, and thus the known classification of conformally superintegrable systems is reobtained in the guise of an unrestricted univariate sextic. In higher dimensions, the obstruction is new and has never been revealed by traditional approaches.

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arXiv2020

An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension

Second-order superintegrable systems in dimensions two and three are essentially classified. With increasing dimension, however, the non-linear partial differential equations employed in current methods become unmanageable. Here we propose a new, algebraic-geometric approach to the classification problem - based on a proof that the classification space for irreducible non-degenerate second-order superintegrable systems is naturally endowed with the structure of a quasi-projective variety with a linear isometry action. On constant curvature manifolds our approach leads to a single, simple and explicit algebraic equation defining the variety classifying superintegrable Hamiltonians that satisfy all relevant integrability conditions generically. In particular, this includes all non-degenerate superintegrable systems known to date and shows that our approach is manageable in arbitrary dimension. Our work establishes the foundations for a complete classification of second-order superintegrable systems in arbitrary dimension, derived from the geometry of the classification space, with many potential applications to related structures such as quadratic symmetry algebras and special functions.

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arXiv2017

An algebraic geometric classification of superintegrable systems in the Euclidean plane

We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving the corresponding system of homogeneous algebraic equations. We then solve these equations explicitly and give a detailed analysis of the algebraic geometric structure of the corresponding projective variety. This naturally associates a unique planar line triple arrangement to every superintegrable system, providing a geometric realisation of this variety and an intrinsic labelling scheme. In particular, our results confirm the known classification by independent, purely algebraic means.

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arXiv2016

Are Orthogonal Separable Coordinates Really Classified?

We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogonal separable coordinates by studying the structure of this variety. We give an example where this approach reveals unexpected structure in the well known classification and pose a number of problems arising naturally in this context.

math.DG↗

arXiv2016

Nijenhuis Integrability for Killing Tensors

The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial differential equations. We give a simple and completely algebraic proof that for a Killing tensor the third and most complicated of these equations is redundant. This considerably simplifies the classification of orthogonal separation coordinates on arbitrary (pseudo-)Riemannian manifolds.

math.DG↗

arXiv2014

The Variety of Integrable Killing Tensors on the 3-Sphere

Integrable Killing tensors are used to classify orthogonal coordinates in which the classical Hamilton-Jacobi equation can be solved by a separation of variables. We completely solve the Nijenhuis integrability conditions for Killing tensors on the sphere $S^3$ and give a set of isometry invariants for the integrability of a Killing tensor. We describe explicitly the space of solutions as well as its quotient under isometries as projective varieties and interpret their algebro-geometric properties in terms of Killing tensors. Furthermore, we identify all Stäckel systems in these varieties. This allows us to recover the known list of separation coordinates on $S^3$ in a simple and purely algebraic way. In particular, we prove that their moduli space is homeomorphic to the associahedron $K_4$.

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arXiv2014

Separation coordinates, moduli spaces and Stasheff polytopes

We show that the orthogonal separation coordinates on the sphere $S^n$ are naturally parametrised by the real version of the Deligne-Mumford-Knudsen moduli space $\bar M_{0,n+2}(R)$ of stable curves of genus zero with $n+2$ marked points. We use the combinatorics of Stasheff polytopes tessellating $\bar M_{0,n+2}(R)$ to classify the different canonical forms of separation coordinates and deduce an explicit construction of separation coordinates and Stäckel systems from the mosaic operad structure on $\bar M_{0,n+2}(R)$.

math.DG↗

arXiv2007

Moduli Spaces of PU(2)-Instantons on Minimal Class VII Surfaces with b_2=1

We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kaehler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $M^{pst}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when $g$ varies in the space of Gauduchon metrics. $M^{pst}_g(S,E)$ can be identified with a moduli space of PU(2)-instantons. The moduli spaces of simple bundles of the above type leads to interesting examples of non-Hausdorff singular one-dimensional complex spaces.

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