Extends T-duality to exotic Courant algebroids.
problem Lack of T-duality for exotic Courant algebroids.
method Extends T-duality isomorphism to exotic exact Courant algebroids.
result Exchange of momentum and winding numbers.
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48 results for “exotic algebroids”
We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,) where B A are regular Lie algebroids, both over the same regular foliated manifold (M,…
The authors define some secondary characteristic homomorphism for the triple (A,B,\bigtriangledown), in which B\subset A is a pair of regular Lie algebroids over the same foliated manifold and \bigtriangledown:L\rightarrow A is a homomorphism of Lie algebroids (i.e. a flat L-connection in A) where L is an arbitrary (no…
We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections, we are able to define holonomy of the orbit foliation of a Lie algebroid and pr…
Using our earlier proposal for Ramond-Ramond fields in an H-flux on loop space, we extend the Hori isomorphism of Bouwknegt-Evslin-Mathai from invariant differential forms, to invariant exotic differential forms such that the momentum and winding numbers are exchanged, filling in a gap in the literature. We also extend…
New 4-manifolds with exotic diffeomorphisms found.
problem Existence of exotic diffeomorphisms in 4-manifolds.
method Proves existence of infinitely many contractible 4-manifolds with exotic diffeomorphisms.
result Infinitely many contractible 4-manifolds with absolutely exotic diffeomorphisms.
Exotic submanifolds in 4-manifolds remain exotic after stabilizations.
problem Constructing exotic codimension-1 submanifolds in 4-manifolds.
method Constructing pairs of exotic codimension-1 submanifolds with diffeomorphic complements and showing they remain exotic after stabilizations.
result Exotic submanifolds remain exotic after any number of stabilizations.
Defines Dirac pairs on Jacobi algebroids, generalizing Lie algebroids.
problem Generalizing Dirac pairs to Jacobi algebroids.
method Introducing Dirac pairs on Jacobi algebroids and showing their relationship to Lie algebroids.
result Dirac pairs on Jacobi algebroids characterize compatible structures.
Paper constructs various algebroids using n-systems and metric n-systems.
problem Creating algebroids from n-systems and metric n-systems.
method Using n-systems and metric n-systems to construct algebroids.
result Explicit computations for all resulting structure maps.
Study exotic Dehn twists in 4-manifolds, producing first known exotic diffeomorphisms.
problem Detecting exotic diffeomorphisms in 4-manifolds.
method 2-parameter families Seiberg-Witten theory over .
result Constructed the smallest closed 4-manifold with exotic diffeomorphisms.
Constructing -Lie algebroids via connections
problem Constructing Lie algebroids and -Lie algebroids
method Using connections generated by finite families of differential operators and dual sections
result Providing sufficient conditions for generating families to determine Lie algebroid and -Lie algebroid structures
New methods distinguish exotic 4-manifolds using Heegaard Floer homology.
problem Distinguishing exotic 4-manifolds from standard ones.
method Using Heegaard Floer homology to define new invariants.
result New invariants remain distinct in covers and can distinguish exotic manifolds.
In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. A…
The paper categorifies Lie and Courant algebroids, establishing correspondences and new constructions.
problem Categorification of Lie and Courant algebroids to better understand geometric structures.
method Introducing and studying new algebraic structures like VB-Lie 2-algebroids and VB-LWX 2-algebroids.
result Established correspondences and new constructions between Lie and Courant algebroids.
Introduces new construction for Courant algebroids and curved structures.
problem Understanding and classifying Courant algebroids and their lifts.
method Introduces Courant algebroid lift and curved Courant algebroids, establishing connections to various geometric structures.
result Established a classification of exact curved Courant algebroids and related connections to various geometric structures.
This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs …
E-Courant algebroidsmath.DG
In this paper, we introduce the notion of -Courant algebroids, where is a vector bundle. It is a kind of generalized Courant algebroid and contains Courant algebroids, Courant-Jacobi algebroids and omni-Lie algebroids as its special cases. We explore novel phenomena exhibited by -Courant algebroids and provid…
Constructs exotic proper 2-knots from open 2-handles.
problem Proving topological equivalence of exotic proper knotted surfaces.
method Flexible construction of exotic open 2-handles, distinguishing through genus functions and end Floer homology.
result Constructs infinite families of irreducible exotic proper knotted surfaces.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
The Free Courant Algebroidmath.DG
We introduce the category of generalized Courant algebroids and show that it admits a free object on any anchored vector bundle. The free Courant algebroid is built from two components: the generalized Courant algebroid associated to a symmetric Leibniz algebroid and the free symmetric Leibniz algebroid on an anchored …
The paper constructs exotic knotted surfaces and curves in 4-manifolds.
problem Understanding exotic knotted surfaces and curves in 4-manifolds.
method Local knotting and construction of surfaces and curves.
result First examples of exotically knotted complex curves and symplectic 2-spheres.
We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providin…
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with .
method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
In this paper, we give the notion of a CLWX 2-algebroid and show that a QP-structure of degree 3 gives rise to a CLWX 2-algebroid. This is the higher analogue of the result that a QP-structure of degree 2 gives rise to a Courant algebroid. A CLWX 2-algebroid can also be viewed as a categorified Courant algebroid. We sh…
Pre-Courant Algebroidsmath-ph
Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive …
Left-symmetric algebroidsmath.DG
In this paper, we introduce a notion of a left-symmetric algebroid, which is a generalization of a left-symmetric algebra from a vector space to a vector bundle. The left multiplication gives rise to a representation of the corresponding sub-adjacent Lie algebroid. We construct left-symmetric algebroids from $\mathcal …
Study exotic phenomena in small 4-manifolds using diffeomorphisms.
problem Understanding exotic smooth structures and embeddings in 4-manifolds.
method Using constraints from Seiberg-Witten theory on diffeomorphisms.
result New methods to detect and bound exotic phenomena.
Notes on relative algebroidsmath.DG
Notes on relative algebroids for geometric problems.
problem Geometric problems and their solutions.
method Explains how relative algebroids arise from geometric problems and introduces their structural theory.
result Relative algebroids unify Lie algebroids with partial differential equations.
We show how to construct absolutely exotic smooth structures on compact 4-manifolds with boundary, including contractible manifolds. In particular, we prove that any compact smooth 4-manifold W with boundary that admits a relatively exotic structure contains a pair of codimension-zero submanifolds homotopy equivalent t…
New invariant real rank identifies constant real Lie algebroids.
problem Characterizing complex Lie algebroids with constant real rank.
method Introducing real rank and minimal complex subalgebroid.
result Local splitting and characterization of complex Lie algebroids.
This work explores higher-order algebroids via vector bundle comorphisms.
problem Generalizing concepts of higher-order tangent bundles and Lie algebroids.
method Introduces a vector bundle comorphism approach to describe higher-order algebroids.
result Establishes a one-to-one correspondence between higher-order Lie algebroids and specific algebraic structures.
Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. In the second paper, we calculate the "smoothness structure" part of the path integral in quantum gravity for the exotic R^4 as non-compact manifold. We discuss the influence of th…
Lie algebroid structures on double vector bundles and representation theory of Lie algebroidsmath.DG
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in…
New exotic structures found on 4-manifolds with specific groups.
problem Finding exotic smooth structures on 4-manifolds with certain fundamental groups.
method Constructing new smooth structures using definite intersection forms and fundamental groups.
result Infinitely many exotic smooth structures on 4-manifolds with fundamental group Z/2Z.
4-manifolds can be exotic after connected sum with S^2 x S^2.
problem Existence of exotic contractible 4-manifolds.
method Constructed a cork that remains exotic after connected sum with S^2 x S^2.
result Existence of exotic pair of contractible 4-manifolds.
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (…
Study examines Lie algebroids with homological sections, generalizing Q-manifolds and Lie superalgebras.
problem Exploring Lie algebroids with homological sections.
method Derived bracket formalism to define an odd Loday-Leibniz bracket on sections.
result Sections of inner Q-algebroids come equipped with an odd Loday-Leibniz bracket.
We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an …
New exotic 4-manifolds found with small trisection genus.
problem Finding exotic 4-manifolds with small trisection genus.
method Constructing genus-3 relative trisections for contractible 4-manifolds.
result Exotic 4-manifolds with boundary have trisection genus of 4.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
Defines connections on parabolic vector bundles for Lie algebroids.
problem Characterizing parabolic vector bundles with Lie algebroid connections.
method Constructs Lie algebroid connections on parabolic vector bundles, uses Atiyah exact sequence.
result Characterizes stable Lie algebroid vector bundles with connections.
Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB - algebroids. In this p…
Inspired by recent works of Zang Liu, Alan Weinstein and Ping Xu, we introduce the notions of CC algebroids and non asymmetric Courant algebroids and study these structures. It is shown that CC algebroids of rank greater than 3 are the same as Courant algebroids up to a constant factor, though the definition of CC alge…
In this paper we introduce exotic twisted -equivariant K-theory of loop space depending on the (typically non-flat) holonomy line bundle on induced from a gerbe with connection on . We also define exotic twisted -equivariant Chern character that maps the exotic tw…
Classifies geodesic planes outside convex core of geometrically finite 3-manifolds.
problem Classifying geodesic planes in geometrically finite 3-manifolds.
method Constructive proof involving exotic rays and roofs.
result Existence of exotic roofs depends on the existence of exotic rays and bending lamination properties.
Exotic diffeomorphisms found on complex surfaces and 4-manifolds.
problem Finding exotic diffeomorphisms on 4-manifolds.
method Minimal complex surfaces and spin 4-manifolds with boundary.
result First known instances of exotic diffeomorphisms of irreducible 4-manifolds.