Bers’ pants decomposition theorem

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

Download Full-text

3D B-Rep meshing for real-time data-based geometric parametric analysis

Advanced Modeling and Simulation in Engineering Sciences ◽

10.1186/s40323-021-00194-5 ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Tristan Maquart ◽

Thomas Elguedj ◽

Anthony Gravouil ◽

Michel Rochette

Keyword(s):

Real Time ◽

Surface Topology ◽

Time Data ◽

Standard Data ◽

Input Surface ◽

Cad Models ◽

Pants Decomposition ◽

Parametric Studies ◽

Geometry Feature ◽

Complicated Geometry

AbstractThis paper presents an effective framework to automatically construct 3D quadrilateral meshes of complicated geometry and arbitrary topology adapted for parametric studies. The input is a triangulation of the solid 3D model’s boundary provided from B-Rep CAD models or scanned geometry. The triangulated mesh is decomposed into a set of cuboids in two steps: pants decomposition and cuboid decomposition. This workflow includes an integration of a geometry-feature-aware pants-to-cuboids decomposition algorithm. This set of cuboids perfectly replicates the input surface topology. Using aligned global parameterization, patches are re-positioned on the surface in a way to achieve low overall distortion, and alignment to principal curvature directions and sharp features. Based on the cuboid decomposition and global parameterization, a 3D quadrilateral mesh is extracted. For different parametric instances with the same topology but different geometries, the MEG-IsoQuad method allows to have the same representation: isotopological meshes holding the same connectivity where each point on a mesh has an analogous one into all other meshes. Faithful 3D numerical charts of parametric geometries are then built using standard data-based techniques. Geometries are then evaluated in real-time. The efficiency and the robustness of the proposed approach are illustrated through a few parametric examples.

Download Full-text

Hyperbolic three-string vertex

Journal of High Energy Physics ◽

10.1007/jhep08(2021)035 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Atakan Hilmi Fırat

Keyword(s):

Boundary Value Problem ◽

Conservation Laws ◽

Riemann Surfaces ◽

Higher Order ◽

Local Coordinates ◽

Pants Decomposition ◽

Fuchsian Equation ◽

String Amplitudes ◽

Hyperbolic Riemann Surfaces ◽

Liouville’S Equation

Abstract We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

Download Full-text

Curvature of quaternionic Kähler manifolds with $$S^1$$-symmetry

manuscripta mathematica ◽

10.1007/s00229-021-01294-7 ◽

2021 ◽

Author(s):

V. Cortés ◽

A. Saha ◽

D. Thung

Keyword(s):

Curvature Tensor ◽

Decomposition Theorem ◽

Kähler Manifolds ◽

Kahler Manifolds ◽

Riemann Curvature ◽

Riemann Curvature Tensor ◽

Cohomogeneity One ◽

Civita Connection ◽

Levi Civita Connection ◽

Quaternionic Kähler

AbstractWe study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic Kähler side in terms of the initial hyper-Kähler data. Our curvature formula refines a well-known decomposition theorem due to Alekseevsky. As an application, we compute the norm of the curvature tensor for a series of complete quaternionic Kähler manifolds arising from flat hyper-Kähler manifolds. We use this to deduce that these manifolds are of cohomogeneity one.

Download Full-text