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.
Study on instantons in G2 and Spin(7) manifolds.
problem Investigate instanton properties in G2 and Spin(7) manifolds. method Analyze the characteristic and torsion connections on G2 and Spin(7) manifolds, focusing on parallel torsion and Lee forms. result Conditions for the curvature of the characteristic connection to be a G2 instanton and for the torsion connection to be a Spin(7) instanton are established. 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 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…
The paper studies quotient spaces of Spin(7) manifolds by S^1 actions and their G2 structures.
problem Understanding quotient spaces of Spin(7) manifolds by S^1 actions and their geometric properties.
method Derives equations relating intrinsic torsion of Spin(7)-structures to G2-structures, focusing on three torsion classes.
result Shows that the quotient space cannot have G2 holonomy unless the original manifold is a Calabi-Yau 4-fold.
The Frölicher-Nijenhuis bracket helps classify G2 and Spin(7) structures.
problem Classifying G2 and Spin(7) structures on manifolds. method Using the Frölicher-Nijenhuis bracket to characterize integrability and torsion.
result Analogous characterization of G2 and Spin(7) structures via the bracket. Study homogeneous 8-manifolds with invariant Spin(7)-structures.
problem Classify and describe invariant Spin(7)-structures on homogeneous 8-manifolds.
method Examine canonical presentations G/H, construct invariant Spin(7)-structures, analyze associated connections.
result Explicit examples and classifications of invariant Spin(7)-structures on homogeneous 8-manifolds.
We study Spin(9)-structures on 16-dimensional Riemannian manifolds and characterize the geometric types admitting a connection with totally skew-symmetric torsion.
The paper calculates a functional for a specific Dirac operator.
problem Computing a spectral Einstein functional for a Dirac operator with torsion.
method Computing the spectral Einstein functional for even-dimensional spin manifolds without boundary.
result The spectral Einstein functional for the Dirac operator with torsion is computed.
Formula derived for Spin(7)-structures on 8D manifolds.
problem Deriving the intrinsic torsion for Spin(7)-structures.
method Explicit formula derivation using Levi-Civita and minimal Spin(7)-connections.
result Formula relating intrinsic torsion to Lee form and codifferential components.
Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive …
The paper explores Einstein warped G2 and Spin(7) manifolds.
problem Understanding Einstein warped G2 and Spin(7) manifolds.
method Warped G2 and Spin(7) structures with Einstein fiber are studied.
result Explicit descriptions of torsion forms for these structures.
We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Rieman…
The paper proves an equivariant Kastler-Kalau-Walze theorem for various spin manifolds.
problem Proving an equivariant Kastler-Kalau-Walze theorem for different types of manifolds.
method Analyzing spin manifolds and their boundaries, proving the theorem for various dimensions and with torsion.
result Generalized Kastler-Kalau-Walze theorem for n-dimensional manifolds. The paper calculates spectral torsion for rescaled Dirac operators on manifolds.
problem Computing spectral torsion for rescaled Dirac operators.
method Using trilinear Clifford multiplication and functional of differential one-forms.
result Computed spectral torsion for four types of rescaled Dirac operators.
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the Dirac operator with torsion on spin manifolds.
method Develops a Lichnerowicz type formula and proves a Kastler-Kalau-Walze type theorem.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator with torsion on 4D and 6D almost product Riemannian spin manifolds. Gradient flow studies Spin(7)-structures on compact 8-manifolds.
problem Formulating and studying the gradient flow of Spin(7)-structures.
method Negative gradient flow of an energy functional of Spin(7)-structures.
result Short-time existence and uniqueness of solutions to the flow.
New formulas for mean curvature of submanifolds in geometries with torsion.
problem Characterizing submanifolds in geometries with intrinsic torsion.
method Deriving formulas for mean curvature of associative, coassociative, and Cayley submanifolds.
result New obstructions to the local existence of coassociative 4-folds in G2-structures with torsion.
Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.
problem Understanding smooth fibrations of compact Spin(7)-manifolds by Cayley submanifolds.
method Geometric and topological constraints from Spin(7)-structure, spinnability criterion, gauge-theoretic input.
result Rules out smooth Cayley fibrations on all known compact torsion-free Spin(7)-manifolds.
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 T7-bundle over S1. Study on deformations of Spin(7)-structures on manifolds.
problem Analyzing deformations of Spin(7)-structures on asymptotically conical manifolds.
method Examined the moduli space of torsion-free, asymptotically conical Spin(7)-structures, showing it is an orbifold for generic decay rates.
result Found that the classical Bryant-Salamon metric on positive spinors on S4 has no continuous deformations as an AC Spin(7)-metric. We give a surgery formula for the torsions and Seiberg-Witten invariants associated with Spinc-structures on 3-manifolds. We use the technique of Reidemeister-type torsions and their refinements.
Study connects spectral and algebraic torsion in geometric contexts.
problem Relating different torsion concepts in geometric settings.
method Example of product geometry, focusing on spin manifolds and two-point space.
result Established connection between spectral and algebraic torsion.
Study topological G₂ and Spin(7) strings at 1-loop using double complexes.
problem Calculate topological string partition functions at 1-loop.
method Define double complexes for supersymmetric backgrounds using generalised geometry, compute partition functions as alternating products of determinants of Laplacians.
result Reproduce known results for G₂ string and predict for Spin(7) string.
Study on deformations of LC Spin(7) instantons simplifies the problem.
problem Deformation theory of instantons on locally conformal Spin(7) manifolds.
method Reformulated linearized deformation equations using a t-parameter family of Dirac operators, demonstrating cancellation of torsion terms.
result The deformation space H^1 is governed by Levi-Civita geometry, reducing the problem to a torsion-free setting.
Unified CR-twistor spaces for G2 and Spin(7) structures.
problem Formally integrable CR-structures on manifolds with G2 or Spin(7) structures. method Generalized LeBrun's, Rossi's, and Verbitsky's construction to CR-twistor spaces for manifolds with VCP structures.
result Torsion tensor properties on CR-twistor spaces for G2 and Spin(7) structures. We present a classification theorem for closed smooth spin 2-connected 7-manifolds M. This builds on the almost-smooth classification from the first author's thesis. The main additional ingredient is an extension of the Eells-Kuiper invariant for any closed spin 7-manifold, regardless of whether the spin characteristic…
We show that a homotopy equivalence between manifolds induces a correspondence between their spin^c-structures, even in the presence of 2-torsion. This is proved by generalizing spin^c-structures to Poincare complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses.
Motivated by the description of N=1 M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a n…
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
Ozsvath and Szabo construct a spectral sequence with E_2 term Λ^*(H^1(Y;Z))\otimes Z[U,U^{-1}] converging to HF^\infty(Y,s) for a torsion Spin^c structure s. They conjecture that the differentials are completely determined by the integral triple cup product form via a proposed formula. In this paper, we prove that HF^\…
Quantum invariant constructed for sutured 3-manifolds using Hopf superalgebra.
problem Quantum invariants for balanced sutured 3-manifolds with Spinc structure. method Involutive Hopf superalgebra H and Fox calculus to compute the invariant. result Invariant is a normalization of Reidemeister torsion when H is Borel subalgebra of Uq(gl(1∣1)). New slicing obstruction from knot surgery yields non-trivial results.
problem Detecting non-trivial elements in the smooth concordance group.
method Spin 4-manifold with boundary as 0-surgery on a knot.
result Detects torsion elements and finds non-smoothly slice knots.
We resolve Spin(7)-orbifolds using algebraic and symplectic techniques.
problem Resolving Spin(7)-orbifolds in exceptional holonomy metrics.
method Combines algebraic and symplectic techniques with analytic gluing methods.
result Existence of torsion-free Spin(7)-structures extending Joyce's theorem.
Torsion in cohomology grows exponentially for certain arithmetic hyperbolic manifolds.
problem Growth of torsion in cohomology of arithmetic hyperbolic manifolds.
method Study of sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding hyperbolic manifolds.
result Torsion in cohomology grows exponentially under natural assumptions.
Study mirror symmetry on manifolds with exceptional holonomy groups.
problem Construct mirrors for Spin(7) and G2 manifolds.
method Apply mirror symmetry to pairs of non-compact manifolds and use CFT analysis.
result Confirm geometric mirror constructions for Spin(7) and G2 manifolds.
We consider the Yang-Mills flow on hyperbolic 3-space. The gauge connection is constructed from the frame-field and (not necessarily compatible) spin connection components. The fixed points of this flow include zero Yang-Mills curvature configurations, for which the spin connection has zero torsion and the associated R…
We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ∇ with skew torsion T∈Λ3M in the situation where the tangent bundle splits under the holonomy of ∇ and the torsion of ∇ is of `split' type. We prove an optimal lower bound for the first eige…
We construct examples of asymptotically cylindrical Riemannian 8-manifolds with holonomy group Spin(7). To our knowledge, these are the first such examples. The construction uses an extension to asymptotically cylindrical setting of Joyce's existence result for torsion-free Spin(7)-structures. One source of examples ar…
We study the Dirac spectrum on compact Riemannian spin manifolds M equipped with a metric connection ∇ with skew torsion T∈Λ3M by means of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac operator with torsion is found that generalizes Friedrich's classical Riemannian estimate.…
Reviews interactions between Spin(9) and octonionic geometries.
problem Understanding the role of Spin(9) in octonionic geometry.
method Analyzes canonical 8-forms, vector fields, Hopf fibrations, and manifolds.
result Discovers new insights into the geometry of octonionic Hopf fibrations.
New invariants from twisted Heegaard Floer homology restrict 4-manifold topology.
problem Defining numerical invariants for arbitrary 3-manifolds with torsion spinc structures. method Using Heegaard Floer homology with twisted coefficients.
result Twisted correction terms provide restrictions on 4-manifold topology.
New spinor fields reveal local or global geometric properties of manifolds.
problem Characterizing spinor fields on manifolds.
method Analyzing generalized imaginary Spin^c-Killing spinors and their associated vector fields.
result Local or global geometric descriptions of manifolds based on spinor fields.
The paper introduces a trilinear functional to recover torsion in spectral triples.
problem Recovering torsion in noncommutative spectral triples.
method Introduces a trilinear functional for spectral triples and demonstrates its application to recover torsion.
result The trilinear functional recovers the torsion of the linear connection in canonical spectral triples.
We prove that the mod Z reduction of the torsion of a rational homology 3-sphere is completely determined by three data: a certain canonical spin^c structure, the linking form and a Q/Z-valued constant c. This constant is a new topological invariant of the rational homology sphere. Experimentations with lens spaces sug…
Combings of oriented compact 3-manifolds are homotopy classes of nowhere zero vector fields in these manifolds. A first known invariant of a combing is its Euler class, that is the Euler class of the normal bundle to a combing representative in the tangent bundle of the 3-manifold M. It only depends on the Spinc-s…
Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second autho…
New flow for G2-structures helps find torsion-free structures.
problem Finding torsion-free G2-structures on compact manifolds.
method Ricci-harmonic flow of G2-structures, analyzing Taylor series expansion.
result Stationary points of the flow are torsion-free G2-structures.