New contact structures defined on differentiable stacks.
problem Defining contact structures on differentiable stacks.
method Introducing -shifted and -shifted contact structures.
result Shifted contact structures provide new insights into geometry.
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48 results for “shifted symplectic geometry”
The paper integrates quasi-Poisson manifolds into multiplicative D-valued moment maps.
problem Integrating quasi-Poisson manifolds into a broader geometric framework.
method Develops new aspects of shifted symplectic and Poisson geometry, establishing Lie-type correspondences and systematic constructions.
result Identifies multiplicative D-valued moment maps integrating quasi-Poisson manifolds, extending known constructions.
New concept of coisotropic structures for differentiable stacks defined.
problem Defining coisotropic structures for differentiable stacks.
method Using twisted Dirac structures and Morita equivalences.
result 1-shifted coisotropic structures transfer through Morita equivalences.
Introduces derived Lie n-groupoids with shifted symplectic structures.
problem Defines structures for higher groupoids and their symplectic properties.
method Introduced derived Lie n-groupoids and their shifted symplectic structures, defining shifted lagrangian structures and proving composition well-defined.
result Shows that the framework includes various reduction procedures.
This work explores symplectic structures on graded manifolds and higher Lie groupoids.
problem Understanding symplectic structures on graded manifolds and their global counterparts.
method Introduction and study of graded manifolds, symplectic Q-manifolds, higher Lie groupoids, and their symplectic structures.
result Developed a graded analogue of Weinstein's tubular neighborhood theorem and explored its applications.
The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.
problem Identifying relationships between different geometric structures on complex varieties.
method Using perfect complexes and shifted symplectic geometries, the paper establishes a Lagrangian correspondence.
result A Lagrangian correspondence between shifted symplectic geometries of flat and Higgs perfect complexes.
This thesis extends contact structures to differentiable stacks using line bundle-valued 1-forms.
problem Extending classical contact structures to differentiable stacks.
method Introducing and -shifted contact structures on Lie groupoids, using line bundle-valued 1-forms and homotopy kernels.
result Definition and examples of and -shifted contact structures on Lie groupoids.
New models for symplectic structures on classifying stacks.
problem Building models for symplectic structures on classifying stacks.
method Introducing -shifted symplectic Lie -groupoids and constructing explicit symplectic Morita equivalences.
result Explicit symplectic Morita equivalences between models of the 2-shifted symplectic structure on classifying stacks.
Survey of bundle gerbes in geometry, field theory, and quantization.
problem Exploring bundle gerbes and their applications in geometry, field theory, and quantization.
method Definition and classification of bundle gerbes with connection, surface holonomy, transgression line bundles, and geometric quantization.
result Bundle gerbes provide a smooth bordism-type field theory and geometric quantization for 2-plectic and symplectic forms.
Introduces a new method for symplectic reduction along submanifolds.
problem Symplectic reduction in various geometric categories.
method Uniform approach to symplectic reduction in smooth manifolds, complex analytic spaces, and algebraic varieties.
result Generalizes and encompasses various known reduction techniques.
Proof that m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
problem Consistent definition of symplectic structures on higher Lie groupoids under Morita equivalence.
method Rigorous proof of m-shifted symplectic forms preservation.
result m-shifted symplectic forms are preserved under Morita equivalence of Lie n-groupoids.
Shifted symplectic Lie and algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetri…
Lie algebroids as spacesmath.DG
In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid -and the natural generalization to dg Lie algebroids-provides an (essentially unique) space. More precisely, we construct a faithful functor from the category of Lie algebroids …
Continues work on derived manifolds and symplectic schemes, constructing virtual classes.
problem Constructing virtual fundamental classes for derived manifolds and schemes.
method Cosection localization, reduced virtual fundamental classes, and applications to Donaldson-Thomas theory.
result Virtual fundamental classes for -shifted symplectic derived schemes are consistent with algebraic and differential geometric constructions.
A strictification result is proved for isotropic distributions on derived schemes equipped with negatively shifted homotopically closed -forms. It is shown that any derived scheme over equipped with a -shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally …
The BPS decomposition theorem splits cohomology of symmetric stacks into invariant parts.
problem Decomposing the cohomology of smooth symmetric stacks into invariant parts.
method Using cohomological Hall induction and intersection cohomology of moduli spaces.
result Establishes the BPS decomposition theorem for various symplectic stacks.
We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop…
Method resolves 4D symplectic orbifolds using complex geometry.
problem Resolving symplectic orbifolds in 4 dimensions.
method Combining complex geometry techniques with symplectic form gluing.
result Examples of 4D symplectic orbifolds successfully resolved.
Symplectic geometric flowsmath.SG
New flows introduced for symplectic geometry.
problem No specific problem stated; focuses on new flows.
method Introduces several geometric flows on symplectic manifolds.
result Examples include the Hitchin gradient flow and dual Ricci flow.
Abstract collects open problems in billiards and symplectic geometry.
problem Open problems in billiards and symplectic geometry.
method Compilation of open problems from discussions.
result Compilation of open problems.
Criterion found for blowing down in 6D symplectic geometry.
problem Blowing down criterion in 6D symplectic geometry.
method Criterion for blowing down in 6D symplectic geometry.
result Criterion established for blowing down in 6D symplectic geometry.
A dictionary connects symplectic to contact geometry, with applications to complex and G-structures.
problem Formalizing the relationship between symplectic and contact geometry.
method Developing a Symplectic-to-Contact Dictionary.
result The dictionary can be applied to complex and G-structures, revealing new geometries.
Introduces systolic inequalities in Riemannian and symplectic geometry.
problem Exploring systolic inequalities in different geometric settings.
method Comparing classical Riemannian metrics to recent symplectic measurements.
result Illustrates connections between Riemannian and symplectic geometry.
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.
The paper explores symplectic geometry of Cartan-Hartogs domains.
problem Understanding the symplectic geometry of Cartan-Hartogs domains.
method Constructing a dual counterpart and computing symplectic capacity.
result A Cartan-Hartogs domain admits symplectic duality if and only if it reduces to a complex hyperbolic space.
Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.
problem Quantization of symplectic manifolds with bounded geometry.
method Berezin-Toeplitz quantization theory.
result Correct semiclassical limit achieved.
Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
Proves a vanishing property for symplectic manifold cohomology.
problem Generalizing complex geometry results to symplectic geometry.
method Based on Tseng and Zhou's vanishing property under symplectic flatness.
result Establishes necessity of symplectic flatness for certain results.
Symplectic structures simplified for compact manifolds.
problem Locally conformally symplectic structures on compact manifolds.
method Symplectic analogue of Vaisman's theorem.
result Locally conformally symplectic structures become globally symplectic.
We will survey some aspects of the smooth topology, algebraic geometry, symplectic geometry and contact geometry of anti-canonical pairs in complex dimension two.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
Survey and generalization of implosion and contraction in symplectic and hyperkähler geometry.
problem Exploring implosion and contraction in symplectic and hyperkähler geometry.
method Survey and extension of implosion construction to general reductive groups, interpretation in Moore-Tachikawa category, generalization of contraction construction.
result Generalization of implosion and contraction concepts to hyperkähler and complex symplectic situations.
The paper tackles isotropy of symplectic forms using Hodge flows.
problem Whether symplectic forms in a given class are isotropic.
method Introduces nonlinear Hodge heat flows to study isotropy.
result The flow converges to the symplectic form smoothly for any initial symplectic form in the class.
Defines duals of higher vector bundles for Lie 2-groupoids.
problem Constructing duals for higher vector bundles over Lie 2-groupoids.
method Develops theory of n-duals for simplicial vector spaces, defines n-duals for Lie 2-groupoids, and studies their properties.
result Proposes a new construction for VB 2-duals of VB 2-groupoids, showing they are VB 2-groupoids themselves and have nondegenerate canonical dual pairings up to homotopy.
A new method simplifies contact Hamiltonian mechanics.
problem Traditional contact Hamiltonian mechanics is complex.
method Introduces sections of line bundles over contact manifolds.
result Reduces contact Hamiltonian formalism to symplectic.
This paper uses a generalization of symplectic geometry, known as -symplectic geometry and developed by Norris, to find observables on three-dimensional manifolds. It will be seen that for the cases considered, the -symplectic observables are derivable from the symplectic observables of . The quantization of…
Generalized Luttinger surgery and other cut-and-paste constructions in generalized complex geometrymath.DG
The paper extends symplectic techniques to generalized complex geometry.
problem Creating stable generalized complex structures on high-dimensional manifolds.
method Introducing generalized Luttinger surgery and generalized Gluck twist.
result Produced stable generalized complex structures with non-homotopy-equivalent components.
Study contact geometry of symplectic divisors, invariant under specific transformations.
problem Understanding contact structures on symplectic divisors and their boundaries.
method Invariant analysis of contact structures under toric and interior blow-ups/blow-downs, open book decomposition construction.
result Contact structure on divisor boundaries is invariant under specified transformations.
In this paper we define a new category of almost complex riemannian 4- manifolds and discuss some basic properties of such pseudo symplectic manifolds. Some motivation based on the Seiberg - Witten theory is imposed.
This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the ch…
We introduce the symplectic twistor operator in symplectic spin geometry, as a symplectic analogue of the twistor operator in Riemannian spin geometry. We focus on the real dimension 2 and compute the space of its solutions on . Our analysis is based on the techniques of metaplectic Howe duality.
We explain how to translate several recent results in derived algebraic geometry to derived differential geometry. These concern shifted Poisson structures on NQ-manifolds, Lie groupoids, smooth stacks and derived generalisations, and include existence and classification of various deformation quantisations.
While symplectic manifolds have no local invariants, they do admit many global numerical invariants. Prominent among them are the so-called symplectic capacities. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of sympl…
Study reveals new geometric structures for magnetic field Hamiltonian systems.
problem Understanding Hamiltonian systems in magnetic fields.
method Investigation of symplectic-Haantjes geometry.
result Non-trivial symplectic-Haantjes manifolds found.
For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic ana…
The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base…
We study the geometry of a family of Lie groups, which contained the classical affine Lie groups, endowed with an exact left invariant symplectic form. We show that this family is closed by symplectic reduction and symplectic double extension in the sense of Dardié and Medina. We prouve also that these groups are endow…