Let (F,u)\to P\to N be a symplectic fibration in math.SG/0503268 McDuff has defined a subgroup Ham^s(F,u) of the group of symplectic automorphisms of(F,u). She has shown that the cohomology class [u] of u can be extended to P if and only if the symplectic fibration has an Ham^s reduction. To show this result, she const…
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48 results for “symplectic gerbes”
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.
Let be a finite group and $\Y$ a -gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the -gerbe $\Y$ and the geometry of $\hat{…
The paper classifies symplectic Lie and algebroids and their applications.
problem Classifying symplectic Lie and algebroids and their higher gauge symmetries.
method Classifying zero-, one-, and two-shifted symplectic algebroids using classical geometric higher structures.
result New examples of twisted Courant algebroids from codimension-two cycles and symplectic interpretations of higher structures.
This paper studies regular Poisson manifolds of compact types, revealing their geometric and algebraic properties.
problem Understanding the geometric and algebraic properties of Poisson manifolds of compact types.
method Analyzing regular Poisson manifolds, proving properties of their leaf spaces, and introducing symplectic gerbes.
result Regular Poisson manifolds of compact types have rich transverse geometry, with linearly varying cohomology classes and a distinguished polynomial function for leafwise symplectic volume.
Action stabilizing bundle gerbe leads to Lie group extension.
problem Stabilizing bundle gerbe with a Lie group action.
method Obtained abelian extension of Lie group from action.
result Universal central extension of Hamiltonian diffeomorphisms.
This work is based on the talk delivered at Poisson 2008. We review the recent advances in Generalized Kahler geometry while stressing the use of Poisson and symplectic geometry. The derivation of the generalized Kahler potential is sketched and the relevant global issues are discussed.
Affine deformations of cotangent groupoids
problem Constructing affine deformations of cotangent groupoids
method Using one-forms and symplectic reduction
result Identifying reduced symplectic forms as multiplicative magnetic forms
Three perspectives on quantizing magnetic Poisson structures are explored.
problem Quantization of magnetic Poisson structures with non-associativity.
method Deformation quantization, symplectic realization, geometric quantization using bundle gerbes.
result Comparison and contrast of different quantization approaches.
Higher Symplectic Geometrymath-ph
We consider generalizations of symplectic manifolds called n-plectic manifolds. A manifold is n-plectic if it is equipped with a closed, nondegenerate form of degree n+1. We show that higher structures arise on these manifolds which can be understood as the categorified or homotopy analogues of important structures stu…
A new gauge principle for string models emerges from groupoid symmetries.
problem Describing gauge symmetries in 2D non-linear sigma models.
method Introducing principaloid bundles and using twisted equivariant structures.
result Emergence of Poisson sigma models from gauge principles.
Master's thesis reviews bundle gerbe theory and introduces new isomorphic gerbe.
problem Understanding bundle gerbe theory and its applications.
method Introduced cup product bundle gerbe and showed its isomorphism to the pullback of the basic bundle gerbe by the Weyl map.
result Stable isomorphism between cup product bundle gerbe and pullback of basic bundle gerbe by the Weyl map.
Bundle gerbesmath.DG
Just as $\Cstar$ principal bundles provide a geometric realisation of two-dimensional integral cohomology; gerbes or sheaves of groupoids, provide a geometric realisation of three dimensional integral cohomology through their Dixmier-Douady class. I consider an alternative, related, geometric realisation of three dimen…
Study of Poisson manifolds of compact types with symplectic properties.
problem Understanding Poisson manifolds of compact types and their properties.
method Establish fundamental properties and connections with symplectic topology.
result PMCTs have properties similar to compact symplectic manifolds, including Hodge decomposition and non-degenerate Poincaré duality pairing.
Introduces Weyl bundle gerbe and shows it's not D-stably isomorphic.
problem Identifying and distinguishing bundle gerbes.
method Introduces cup product bundle gerbe, defines Weyl bundle gerbe, and uses holonomy to show non-isomorphism.
result Weyl bundle gerbe and basic bundle gerbe are not D-stably isomorphic.
Decomposes bundle gerbes on supermanifolds into simpler components.
problem Classifying bundle gerbes on supermanifolds.
method Proves decomposition theorem and cohomological classification theorems.
result Bundle gerbes on supermanifolds decompose into a gerbe over the manifold and a 2-form.
Gerbes and Lie Groupsmath.DG
We present a review of bundle gerbes, emphasizing their relations to Lie groups. Indeed, compact Lie groups do not only carry the structure of a Riemannian manifold, but also canonical families of bundle gerbes. We recall the construction of these bundle gerbes and their relation to loop groups. We discuss several alge…
Equivariant bundle gerbesmath.DG
Develops equivariant bundle gerbes for sigma models.
problem Descent problems and equivariance for bundle gerbes.
method Theory of simplicial extensions for bundle gerbes.
result Shows equivariant bundle gerbes for conjugation and String 2-group actions.
Nonabelian bundle 2-gerbesmath.DG
We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…
This thesis reviews the theory of bundle gerbes and then examines the higher dimensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced and it is shown that there is a class in associated to any bundle 2-gerbe.
Bundle 2-gerbesmath.DG
We make the category BGrb_M of bundle gerbes on a manifold M into a 2-category by providing 2-cells in the form of transformations of bundle gerbe morphisms. This description of BGrb_M as a 2-category is used to define the notion of a bundle 2-gerbe. To every bundle 2-gerbe on M is associated a class in H^4(M;Z). We de…
Review of non-abelian gerbes and their applications in string theory.
problem Anomaly cancellation and geometric description of T-duals in string theory.
method Systematic construction of non-abelian gerbes via descent.
result Extension of non-abelian gerbes to orientifold sigma models and T-duals.
By means of cohomology groups, we study relationships between equivariant gerbes with connection over a manifold with a Lie group action and gerbes with connection over the quotient space.
The paper studies equivariant gerbes on complex Lie groups.
problem Obstruction to equivariant extensions of basic gerbes.
method Modified construction of Gawȩdzki-Reis [2004]
result Exhibit basic equivariant bundle gerbes over non-simply connected Lie groups.
The paper classifies equivariant gerbe connections on Lie groups.
problem Classifying equivariant gerbe connections on Lie groups.
method Using Meinrenken's G-equivariant bundle gerbe connections and differential quotient stacks.
result Isomorphism classes of G-equivariant gerbe connections are classified by degree three differential equivariant cohomology.
Quantum theory for gerbes connects stack invariants.
problem Quantum theory of gerbes over stacks.
method Proves structure result on Gromov-Witten theory.
result Gromov-Witten invariants of gerbes relate to stack invariants.
Rigorous model for 2-gerbes simplifies calculations in physics.
problem Constructing explicit geometric string structures.
method Defined a rigid model for bundle 2-gerbes and connections, proving equivalence to existing models.
result Chern-Simons bundle 2-gerbe can be rigidified and described via geometric string structures.
We consider the notion of stable isomorphism of bundle gerbes. It has the consequence that the stable isomorphism classes of bundle gerbes over a manifold M are in bijective correspondence with H^3(M, Z). Stable isomorphism sheds light on the local theory of bundle gerbes and enables us to develop a classifying theory …
Multiplicative bundle gerbes are gerbes over a Lie group which are compatible with the group structure. In this article connections on such bundle gerbes are introduced and studied. It is shown that multiplicative bundle gerbes with connection furnish geometrical constructions of the following objects: smooth central e…
Develops 2-Hilbert spaces for bundle gerbes in geometric quantization.
problem Higher geometric quantization of bundle gerbes.
method Introduces 2-Hilbert spaces, rig-categories, and duals for bundle gerbes.
result Constructs a 2-Hilbert space of sections for bundle gerbes.
The paper generalizes bundle gerbes over groupoids and their correspondence with PB groupoids.
problem Generalizing bundle gerbes over groupoids and their properties.
method Developed a functorial correspondence between PB groupoids and bundle gerbes over groupoids.
result Built a correspondence between PB groupoids and bundle gerbes over groupoids, including partial quotients.
In a previous paper we outlined how discrete torsion can be understood geometrically as an analogue of orbifold U(1) Wilson lines. In this paper we shall prove the remaining details. More precisely, in this paper we describe gerbes in terms of objects known as stacks (essentially, sheaves of categories), and develop mu…
We study geometry on real gerbes in the spirit of Cheeger-Simons theory. The concepts of adaptations and holonomy forms are introduced for flat connections on real gerbes. Their relations to complex gerbes with connections are presented, as well as results in loop and map spaces.
Constructs small bundle gerbes and proves index theorems for manifolds.
problem Constructing and analyzing bundle gerbes on manifolds.
method Defines and constructs small bundle gerbes, uses pseudodifferential and semiclassical smoothing operators, proves index theorems.
result Proves the Atiyah-Singer type theorem for small bundle gerbes, showing their relation to twisted K-theory.
In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analyt…
We introduce and study the notion of a biholomorphic gerbe with connection. The biholomorphic gerbe provides a natural geometrical framework for generalized Kahler geometry in a manner analogous to the way a holomorphic line bundle is related to Kahler geometry. The relation between the gerbe and the generalized Kahler…
In this paper we study n-composition series of affine manifolds. One composition series are classified using gerbe theory. It is natural to think that n-composition series must be classified using n-gerbe theory. In the last section of this, we propose a notion of abelian n-gerbe theory
We introduce an axiomatic framework for the parallel transport of connections on gerbes. It incorporates parallel transport along curves and along surfaces, and is formulated in terms of gluing axioms and smoothness conditions. The smoothness conditions are imposed with respect to a strict Lie 2-group, which plays the …
Gerbes encode spectral gaps in topological insulators.
problem Capturing spectral gaps in topological insulators.
method Geometric encoding through 'Real' gerbes.
result Gerbes precisely capture spectral gaps.
From a certain strongly equivariant bundle gerbe with connection and curving over a smooth manifold on which a Lie group acts, we construct under some conditions a bundle gerbe with connection and curving over the quotient space. In general, the construction requires a choice, and we can consequently obtain distinct st…
Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approa…
Let be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on , with equivariant 3-curvature representing a generator of . Technical tools developed in this context include a gluing construction for gerbes and a theory of equivariant bundle ge…
Study fractional structures on bundle gerbe modules using rational homotopy theory.
problem Understanding twisted Chern classes of torsion bundle gerbe modules.
method Sullivan's rational homotopy theory to realize twisted Chern classes at the level of classifying spaces.
result Introduction of fractional U-structures as a universal framework.
New result on finite gerbes simplifies classification of certain bundles.
problem Classifying finite-dimensional gerbes with torsion -class.
method Proves torsion for a wide class of finite gerbes built from principal bundles.
result Finite gerbes built from finite-dimensional fibre bundles have torsion -class.
New theory connects non-abelian bundle gerbes to abelian ones.
problem Challenges in extending higher gauge theory beyond fake-flat sector.
method Developed a comprehensive theory of adjusted connections on non-abelian bundle gerbes.
result Established a new coordinate-independent formulation of lifting theorem.
We construct a gerbe over a complex reductive Lie group G attached to an invariant bilinear form on a maximal diagonalizable subalgebra which is Weyl group invariant and satisfies a parity condition. By restriction to a maximal compact subgroup K, one then gets a gerbe over K. For a simply-connected group, the parity c…
Proves well-posedness of gradient solitons on bundle gerbe.
problem Analyzing gradient generalized Ricci solitons on bundle gerbes.
method Analytic Cauchy problem solved for gradient solitons on abelian bundle gerbes.
result Initial data equations solved on compact Riemann surfaces.
An introduction to the theory of bundle gerbes and their relationship to Hitchin-Chatterjee gerbes is presented. Topics covered are connective structures, triviality and stable isomorphism as well as examples and applications.