In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining technique…
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48 results for “super-exceptional geometry”
The Hidden M-Grouphep-th
The hidden M-algebra is integrated into a super-Lie group, allowing for compactification of extra dimensions.
problem Integrating the hidden M-algebra into a super-Lie group to model super-exceptional spacetimes.
method Left-invariant extension of the decomposed M-theory 3-form, providing a computer-checked re-derivation and streamlined conception of super-Lie groups.
result Lattice subgroups of the hidden M-group allow toroidal compactification of hidden dimensions, akin to topological T-duality.
Super-exceptional embedding construction of the heterotic M5: Emergence of SU(2)-flavor sectorhep-th
New construction reveals SU(2)-flavor fields in heterotic M5-brane model.
problem Tackles the emergence of SU(2)-flavor fields in heterotic M5-brane models.
method Classifies super-exceptional field content and works out interactions.
result SU(2)xU(1)-valued scalar and vector fields emerge from probe M2- and M5-branes.
Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.
problem Combining symmetries in M-theory to extend exceptional tangent spaces.
method Combining known results to show hyperbolic involutory algebra acts on M-algebra through brane-rotating symmetry.
result Extends the hierarchy of exceptional tangent spaces from n ≤ 7 to n = 11.
Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.
problem Characterize surfaces with constant ratio of principal curvatures in different geometries.
method Differential geometry, line geometry, Lie sphere geometry, ordinary differential equations, algebraic geometry.
result Characterized various types of surfaces like rotational, channel, ruled, helical, and translational.
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
We define hermitian geometry as the target space geometry of the two dimensional supersymmetric sigma model. This includes generalised Kähler geometry for , generalised hyperkähler geometry for , strong Kähler with torsion geometry for and strong hyperkähler with torsion geometry f…
Non-lorentzian spacetimesmath.DG
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
New definition of Born geometry connects to known geometries.
problem Defining and understanding Born geometries.
method Using Künneth structures and recursion operators.
result Born connection derived from Künneth connection for integrable geometries.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
The paper extends group constructions to coset geometries, creating new ways to combine geometries.
problem Combining and gluing incidence geometries in a general framework.
method Extending classical group-theoretic constructions to coset geometries.
result Provides a general framework for combining or gluing incidence geometries.
Convex meets complexmath.CV
Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
Geometry of geodesicsmath.DG
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…
Limits of geometriesmath.GT
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…
Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometrymath.FA
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
A car as parabolic geometrymath.DG
We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type , where is a Borel parabolic subgroup in . We discuss the relations of this geometry of a car with the geometry of circles in the plane (a low dimensional Lie sph…
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
Develops Weyl structures for path geometries, simplifying their study.
problem Complexity in studying path geometries using traditional differential geometry methods.
method Defines distinguished connections and Schouten tensor, proving their dependence on line bundle sections.
result Shows a smaller subclass of Weyl structures for path geometries, with interesting connections to BGG sequences.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
The paper extends Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
problem Extending Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
method Extending the Ruh-Vilms theorem to Weitzenböck geometry, considering the flat geometry with torsion.
result The Laplacian of the Gauss map for hypersurfaces in Weitzenböck geometry is related to the mean curvature vector field.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
Reduction of HKT-Structuresmath.DG
KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequiv…
New framework for noncommutative Carrollian geometry using Lie-Rinehart pairs.
problem Developing a geometric framework for ultra-relativistic physics in noncommutative settings.
method Using ρ-Lie-Rinehart pairs to generalize Carrollian Lie algebroids to almost commutative geometry.
result Foundational principles of Carrollian geometry hold in almost commutative geometry.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
The target space geometry of abelian vector multiplets in theories in four and five space-time dimensions is called special geometry. It can be elegantly formulated in terms of Hessian geometry. In this review, we introduce Hessian geometry, focussing on aspects that are relevant for the special geometrie…
Quantum field theory connects Riemannian geometry to quantum fluctuations.
problem Generating Riemannian structures from quantum fluctuations.
method QFT approach to Riemannian Geometry, focusing on Ricci curvature.
result Ricci curvature is crucial in generating Riemannian structures.
New geometry based on Siegel upper half-space with volume formula.
problem Developing a new 3D geometry based on Siegel upper half-space.
method Constructing a geometry fibered over Siegel upper half-space and providing a volume formula.
result Volume of Siegel-Seifert closed manifolds is the fiber circle length times base manifold's Euler characteristic.
Develops a new algebraic framework for differential geometry of infinite dimensional spaces.
problem Creating a differential geometry for infinite dimensional spaces without topology or local coordinates.
method Introduces a general algebraic framework for lifted geometry applicable to various infinite dimensional spaces.
result Stokes' Theorem appears as a form of differentiability in the lifted geometry of spaces of submanifolds.
Study of Teichmüller space geometry using infinitesimal and global methods.
problem Understanding the geometry of Teichmüller space and its tangent/cotangent spheres.
method Systematic study of Thurston metric's infinitesimal and global properties.
result Rigidity statements for the Thurston metric analogous to Royden theorem.
The study identifies two sources of invariants in 2--nondegenerate CR geometries.
problem Characterizing fundamental invariants of 2--nondegenerate CR geometries.
method Analyzes the harmonic curvature and the difference in complex structures.
result Nontrivial examples of CR geometries can be obtained as deformations of models.
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
Study shows nonexistence of certain geometric structures in complex geometries.
problem Failure of Lichnerowicz-type conjectures in specific parabolic geometries.
method Used techniques from Erickson to establish existence of specific geometries.
result Nonexistence of certain geometric structures in Yamaguchi nonrigid parabolic models.
Machine learning applied to algebraic geometry for physics problems.
problem Reformulating algebraic geometry problems as tensor mappings for machine learning.
method Supervised and unsupervised machine learning techniques applied to algebraic geometry problems.
result Machine learning provides insights into the structure of algebraic geometry data.
Survey of global geometry for double field theory.
problem Global description of double field theory geometry.
method Review of Courant algebroids, metric algebroids, AKSZ construction, para-Hermitian geometry.
result Global description of doubled geometry and topological models.
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
We exploit the spinor description of four-dimensional Walker geometry, and conformal rescalings of such, to describe the local geometry of four-dimensional neutral geometries with algebraically degenerate self-dual Weyl curvature and an integrable distribution of alpha-planes (algebraically special real alpha-geometry)…
Geometry-aware similarity metrics for neural representations on Riemannian and statistical manifoldscs.LG
MSA compares neural representations' intrinsic geometry for better understanding.
problem Existing similarity measures fail to capture subtle distinctions between neural network solutions.
method Metric similarity analysis (MSA) using Riemannian geometry.
result MSA can disentangle features of neural computations and compare nonlinear dynamics.
We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\ma…
Symplectic and Poisson structures proved for information geometry's Frobenius manifold.
problem Connecting disconnected theories in information geometry.
method Proving symplectic and Poisson structures on the Frobenius manifold.
result Established a bridge between Vinberg, Souriau, and Koszul's theories.