The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14) \supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4 \supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot \supset Spin(1) …
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48 results for “Spin_8”
The paper explores spin^h structures and their obstructions.
problem Understanding which manifolds are spin^h.
method Analyzing the fifth Stiefel-Whitney class and constructing generalised spin structures.
result Every compact orientable manifold of dimension 7 or lower is spin^h, and there are higher-dimensional manifolds that are not.
Study -dDT connections on manifolds with -structures.
problem Understanding moduli spaces of -dDT connections.
method Introduced and studied -dDT connections using fully nonlinear PDEs.
result Moduli space has finite expected dimension and smoothness under certain conditions.
New -instantons constructed on Joyce's manifold.
problem Constructing -instantons on Joyce's compact manifold.
method Gluing non-flat connections on local model spaces to a flat connection on the -orbifold.
result More than 20,000 new four-parameter families of -instantons.
Knot spinningmath.GT
This is an introduction to the construction of higher-dimensional knots by spinning methods. Simple spinning of classical knots was introduced by E. Artin in 1926, and several generalizations have followed. These include twist spinning, superspinning or p-spinning, frame spinning, roll spinning, and deform spinning. We…
Spin Manifoldsmath.DG
New -manifolds studied for their properties.
problem Understanding new manifold classifications.
method Exploring -manifolds as a new category.
result Highlights of -manifolds discussed.
Study on harmonic flow of Spin(7)-structures in 8D manifolds.
problem Comparing isometric Spin(7)-structures.
method Developed a notion of harmonicity for Spin(7)-structures and studied the harmonic flow.
result Analytical properties of the harmonic flow of Spin(7)-structures presented.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
The paper characterizes new invariant spin spinors on projective spaces.
problem Characterizing new invariant spin spinors on projective spaces.
method Adapting spin representation via exterior forms to the generalised spin context.
result Complete description of the space of invariant spin spinors for , , and .
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.
We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer requirements, transformations allowed by spin frames are more general tha…
Spin Cohomologymath.DG
We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator is introduced which is associated to a connection and a parallel spinor , , and the algebraic o…
Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
Rigidity of elliptic genera proven for non-spin manifolds with -action.
problem Rigidity of elliptic genera for non-spin manifolds with -action.
method Analysis of universal covering spin condition and for rigidity.
result Rigidity of elliptic genera is proven for spin universal coverings but not for non-spin universal coverings.
First non-trivial examples of deformed Spin(7)-instantons constructed.
problem Constructing deformed Spin(7)-instantons and connections.
method Constructing on cotangent bundles of and cones over 3-Sasakian 7-manifolds.
result First non-trivial examples of deformed Spin(7)-instantons.
Study shows certain spin manifolds can't meet DEC condition.
problem Non-existence of spin fill-ins meeting DEC condition.
method Analyzes spin Riemannian manifolds and generalized mean curvature functions.
result Closed spin manifolds cannot satisfy DEC if curvature is large.
Spin TFTs created by gauging line defects in 3D.
problem Creating spin TFTs from oriented TFTs with framed line defects.
method Constructing a spin TFT from an oriented TFT with framed line defects and a commutative Frobenius algebra.
result Spin TFTs extend earlier classifications and reproduce abelian spin Chern-Simons theories.
The paper explores connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
problem Exploring connections between quaternionic and Cayley calibrations in dimensions 8 and 16.
method Starting from collections of 'Kähler 2-forms', the paper constructs canonical 4-forms and calibrated 4-planes in dimensions 8 and 16.
result Explicit formulas for canonical 4-forms and are derived, and their calibrated 4-planes are characterized.
Joyce constructed examples of compact eight-manifolds with holonomy Spin(7), starting with a Calabi-Yau four-orbifold with isolated singular points of a special kind. That construction can be seen as the gluing of ALE Spin(7)-manifolds to each singular point of the Calabi-Yau four-orbifold divided by an anti-holomorphi…
Real vector bundles are determined by their Dirac indices on specific spin manifolds.
problem Determining real vector bundles using Dirac indices.
method Mapping spin or spin manifolds into a compact smooth manifold and using Dirac indices.
result Real vector bundles are uniquely determined by their Dirac indices on prescribed manifolds.
The paper studies curvature identities and solitons on Spin(7)-manifolds.
problem Curvature identities and solitons on Spin(7)-manifolds.
method Analyzes the curvature and torsion of Spin(7)-manifolds, proving identities and conditions.
result Conditions for closed torsion and implications for Ricci flatness and solitons.
This paper gives a combinatorial description of spin and spin^c-structures on triangulated PL-manifolds of arbitrary dimension. These formulations of spin and spin^c-structures are established primarily for the purpose of aiding in computations. The novelty of the approach is we rely heavily on the naturality of binary…
No spin structures found in a hyperbolic 4D space.
problem Finding hyperbolic 4-manifolds without spin structures.
method Constructed a non-compact, orientable, hyperbolic 4-manifold.
result Demonstrated existence of a hyperbolic 4D space without spin structures.
New invariant for spin 3-manifolds using super 3-cocycles.
problem Constructing an invariant for spin 3-manifolds.
method State sum construction based on super 3-cocycles and combinatorial representation.
result The invariant is sensitive to the spin structure.
Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…
Study invariant spin^r structures on homogeneous spaces.
problem Classify invariant spin^r structures on homogeneous spaces.
method Introduce and study -invariance of spin^r structures on homogeneous spaces.
result Classification of invariant spin^r structures in terms of isotropy representation.
A homotopy classification of -structures with applications to exceptional Riemannian holonomymath.DG
Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.
problem Classifying Spin(7) structures on compact 8-manifolds.
method Obstruction theory applied to Spin(7) structures on compact 8-manifolds with abelian fundamental group.
result Compact Riemannian 8-manifolds with holonomy Spin(7) have exactly two Spin(7) structures extending the induced G2 structure on the boundary.
Based on a pair of cohomology operations on so called -formal spaces, we construct the integral cohomology rings of the classifying spaces of the Lie groups and . As applications, we introduce characteristic classes for the reduced topological theory, determine the ring of integra…
Classifies invariant spin structures on spheres.
problem Determining which Lie groups preserve the unique spin structure.
method Two approaches: differential of actions and representation theory.
result Classification of Lie groups preserving the spin structure.
Simplified proof of spin^c structures using twistor spaces.
problem Proving every compact oriented 4-manifold admits spin^c structures.
method Using twistor spaces to provide a simpler, more geometric proof.
result Simplified and clarified understanding of spin^c structures in 4-manifolds.
Researchers compute mod 2 Seiberg-Witten invariants for spin structures and families.
problem Computing mod 2 Seiberg-Witten invariants for spin structures and families.
method Using Pin(2)-symmetry and localisation in equivariant cohomology, the researchers enhance and compute the invariants.
result The mod 2 Seiberg-Witten invariants are computed for spin structures and families, confirming the simple type conjecture mod 2.
Spin Lefschetz fibrations can represent any group and lattice point.
problem Representing groups and lattice points using Lefschetz fibrations.
method Proving any finitely presented group and admissible lattice point can be realized as a spin Lefschetz fibration.
result Spin Lefschetz fibrations can represent any group and lattice point.
In this paper, we extend the study of generalized Killing spinors on Riemannian Spin manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin Killing spinors or imaginary generalized Spin Killing spinors, providing that the dimension of t…
Spin Link Homologymath.QA
New spin on Khovanov-Rozansky homology categorifies spin link polynomial.
problem Categorify spin link polynomial using colored Khovanov-Rozansky homology.
method Equip Λ^n-colored sl_{2n} Khovanov-Rozansky homology with an involution.
result Categorifies spin-colored so_{2n+1} quantum link polynomial for n=1,2,3.
New rigidity theorems for spin^c manifolds using modular invariance.
problem Establishing rigidity theorems for twisted Dirac and Toeplitz operators.
method Liu's modular invariance method and its odd-dimensional extension.
result New Witten rigidity theorems for even and odd-dimensional spin^c manifolds.
Study proves a conjecture for certain spin manifolds.
problem Proving a conjecture about positive scalar curvature metrics.
method Connected sum construction and Gromov-Lawson area enlargement.
result Connected sum of spin manifolds admits no complete metric of positive scalar curvature.
Constructs explicit solutions to Spin(7)-structures gradient flow.
problem Finding explicit solutions to Spin(7)-structures gradient flow.
method Expressed Spin(7)-torsion tensor and gradient flow in terms of torsion forms; used these formulae to find solutions.
result Found explicit solutions including a shrinking soliton on SU(3) and another on a -bundle over .
We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced …
In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian …
We define spin-c prequantization of a symplectic manifold to be a spin-c structure and a connection which are compatible with the symplectic form. We describe the cutting of an S^1-equivariant spin-c prequantization. The cutting process involves a choice of a spin-c prequantization for the complex plane. We prove that …
Flows of Spin(7)-structuresmath.DG
We consider flows of Spin(7)-structures. We use local coordinates to describe the torsion tensor of a Spin(7)-structure and derive the evolution equations for a general flow of a Spin(7)-structure on an 8-manifold M. Specifically, we compute the evolution of the metric and the torsion tensor. We also give an explicit d…
We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic stru…
Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin(9) both…
Defines a cell complex for even spin mapping class group.
problem No specific problem stated; focuses on definition.
method Defines a cell complex with an action of the even spin mapping class group.
result Obtains a finite presentation of the even spin mapping class group.
Study negative definite spin fillings of knot covers.
problem Existence of negative definite spin fillings in branched double covers.
method Derive obstructions and characterize special knots.
result Characterization of special alternating knots.
The study explores psc-metrics on non-spin manifolds with pin^\pm or spin^c structures.
problem Existence of psc-metrics on non-spin manifolds with specific structures.
method Analysis of Dirac operators and index theory on pin^\pm or spin^c manifolds.
result The index of a Dirac operator controls the existence of psc-metrics on certain manifolds.
New cobordism invariants derived from BPS q-series.
problem Understanding BPS partition functions with different structures.
method Analyzing limits of BPS q-series and summing over Spin structures.
result Derived invariants of homology cobordisms from BPS q-series.
A spin network is a cubic ribbon graph labeled by representations of . Spin networks are important in various areas of Mathematics (3-dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network is an integ…