We give a combinatorial model for r-spin surfaces with parametrised boundary based on Novak (2015). The r-spin structure is encoded in terms of -valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state sum construction of two-dimensional topolog…
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48 results for “r-spin structure”
Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold Σare r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle STΣ. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomo…
We introduce the stack of r-spin maps. These are stable maps into a variety V from n-pointed algebraic curves of genus g, with the additional data of an r-spin structure on the curve. We prove that this stack is a Deligne-Mumford stack, and we define analogs of the Gromov-Witten classes associated to these spaces. We s…
Higher spin mapping class groups and strata of Abelian differentials over Teichm{ü}ller spacemath.GT
Classifies components of abelian differentials over Teichmüller space.
problem Classifying components of strata of abelian differentials.
method Computing monodromy groups and determining finite generating sets for -spin stabilizer subgroups.
result Complete classification of strata components for .
We give a review of the quantum singularity theory of Fan-Jarvis-Ruan and the r-spin theory of Jarvis-Kimura-Vaintrob and describe the work of Abramovich-Jarvis showing that for the singularity A_{r-1} = x^r the stack of A_{r-1}-curves of is canonically isomorphic to the stack of r-spin curves. We prove that the A_{r-1…
We give definitions of moduli spaces of framed, r-Spin and Pin surfaces. We apply earlier work of the author to show that each of these moduli spaces exhibits homological stability, and we identify the stable integral homology with that of certain infinite loop spaces in each case. We further show that these moduli spa…
Constructs chiral rational homology spheres with hyperbolic groups.
problem Existence of strongly chiral rational homology spheres with hyperbolic fundamental groups.
method Construction of rational homology spheres using -spins and investigation of self-map degrees.
result Strongly chiral rational homology spheres with hyperbolic fundamental groups constructed.
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdV_…
The purpose of this note is introduce a new axiom (called the Descent Axiom) in the theory of -spin cohomological field theories. This axiom explains the origin of gravitational descendants in this theory. Furthermore, the Descent Axiom immediately implies the Vanishing Axiom, explicating the latter (which has no a …
String-net models explore non-spherical fusion categories, revealing new spin structures and representations.
problem Investigating string-net models in non-spherical fusion categories.
method String-net models associate vector spaces to surfaces in terms of graphs decorated by objects and morphisms of a pivotal fusion category.
result String-net spaces count r-spin structures and carry representations of the mapping class group.
Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfacesmath.AG
Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.
problem Characterizing curves that can be vanishing cycles in degenerations of linear systems.
method Computing mapping class group-valued monodromy and identifying it with r-spin mapping class groups.
result Identifies simple closed curves as vanishing cycles and provides characterizations of discriminants and Lefschetz fibrations.
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new -spin and Theta-class intersection numbers.
Bott and Samuelson constructed explicit cycles representing a basis of the Z_2-homology of the orbits of variationally complete representations of compact Lie groups. As a consequence, all those orbits are taut. We were able to show that an irreducible representation of a compact Lie group, all of whose orbits are taut…
The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.
problem Formulating and proving a constant-curvature, holonomy-valued Lorentzian analogue of Minkowski theorem for tetrahedra.
method Formulated and proved a Lorentzian analogue of Minkowski theorem for tetrahedra in dS3 and AdS3.
result A unique strictly convex tetrahedron can be reconstructed from four non-trivial based SO+(1,2) holonomies.
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, …
We give a notion of compatibility between a Riemannian metric and a Jacobi structure. We prove that in case of Poisson structures, contact structures and locally conformally symplectic structures, fundamental examples of Jacobi structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, $\fr…
Extending Jacobi and Riemannian compatibility to Lie algebroids.
problem Generalizing compatibility between Jacobi and Riemannian structures.
method Generalizing previous work on fundamental examples to Lie algebroids.
result Compatibility results for Lie algebroids.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
Study on structures and almost para-contact structures in 7D.
problem Understanding the relation between structures and almost para-contact structures.
method Calculating projections using properties of structures.
result Determined the class of almost para-contact structures induced by structures.
Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of -family of generalized complex structures and study of twistor spaces.
result Existence of generalized hypercomplex structures on -dimensional tori with non-maximal types.
Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
Corner manifoldsmath.DG
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Study -structures on manifolds leading to complex structures.
problem Understanding -structures and their relation to complex structures.
method Explored -structures on differential manifolds, proving their relation to almost-complex structures and providing a canonical connection.
result Established a twistor-like construction for -geometry.
3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…
Spin-harmonic structures on low-dimensional manifolds.
problem Defining geometric structures on low-dimensional manifolds.
method Introducing spin-harmonic structures defined by harmonic unitary spinors.
result Spin-harmonic structures are equivalent to balanced Spin(7) structures in dimension 8.
Study equivalence between Hessian and Born structures on tangent bundles.
problem Equivalence between Hessian and Born structures on tangent bundles.
method Analyzing conditions for Hessian structures and integrability of induced almost Born structures.
result Conditions for equivalence between Hessian and Born structures are established.
Introduces compatibility between Dirac structures and Nijenhuis tensors.
problem No specific problem stated; focuses on extending Poisson-Nijenhuis structures.
method Introduces compatibility between Dirac structures and (1,1)-tensor fields.
result Properties of Dirac-Nijenhuis structures studied, including connections and integrations.
New G2-structures found on Lie groups with strong structural conditions.
problem Existence and structure of extremally Ricci pinched G2-structures on Lie groups.
method Strong structural conditions on Lie algebra, deformation and rigidity studies.
result Three new examples of extremally Ricci pinched G2-structures, all steady Laplacian solitons.
We introduce generalized almost contact structures which admit the -field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which…
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
Introduces VB-structures for geometric objects on manifolds.
problem Properties of higher tangent lifts of geometric structures.
method Introduces weighted structures for various geometric objects on a manifold with a homogeneity structure.
result Proves interesting properties of various weighted structures.
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined -nilpotent structures for Lie subgroups of related to neutral hyperKähler structures.
result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an -nilpotent structure.
Spin-structures on real Bott manifolds with Kähler structure are characterized.
problem Existence of spin-structures on real Bott manifolds with Kähler structure.
method Ishida characterization and techniques from \cite{PS16} using characteristic classes.
result Necessary and sufficient condition for the existence of spin-structures on .
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural -structure on an Inoue surface is the unique -structure.
Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current ar…
In this paper, we show the existence of (co-oriented) contact structures on certain classes of -manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with -structure) admits an almost contact structure. We…
Two Kähler structures are PCR equivalent in the Siegel domain.
problem Equivalence of two Kähler structures in the Siegel domain.
method Construction of complex hyperbolic and Kähler structures from Sasakian structure.
result PCR Kähler equivalent structures in Siegel domain.
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, …
Introduces holed cone structures to generalize cone structures on 3-manifolds.
problem Generalizing cone structures to 3-manifolds with irreducible holonomy representations.
method Introduces holed cone structures and considers their deformation space.
result The deformation space of holed cone structures is a covering space of the character variety.
The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.
problem Classifying and constructing complex symplectic structures on 4D Lie algebras.
method Interpreting complex symplectic and pseudo-Kähler structures, developing a method for constructing hypersymplectic structures.
result Obtained an example of a hypersymplectic structure on a 4-step nilmanifold.