arXiv research

Mathai, Melrose, and Singer compute the index of projective elliptic operators.

Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.

In this note the fractional analytic index, for a projective elliptic operator associated to an Azumaya bundle, of DG/0402329 is related to the equivariant index of Atiyah and Singer for an associated transversally elliptic operator.

Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid $M$, we show that the corresponding category of gerbe modules, if it is non-trivial, is equivalent to the category of finitely generated projective modules over an Azumaya algebra on $M$. This result can be seen as an …

In this Part II of D(11), we introduce new objects: super-$C^k$-schemes and Azumaya super-$C^k$-manifolds with a fundamental module (or, synonymously, matrix super-$C^k$-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 [math.DG]) to define the notion of `differentiable maps…

Researchers create explicit representations for skein algebras of small surfaces, revealing their Azumaya loci.

In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold ($C^{\infty}$ or complex analytic).

Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.

This research builds foundations for D3-branes in string theory.

Study centers of generalized skein algebras, showing almost Azumaya properties.

We show that for complex analytic K3 surfaces any torsion class in H^2(X,O_X^*) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surfa…

New topological criterion extends arithmetic invariants in hyperbolic 3-manifolds.

Paper extends algebraic geometry results to hyperbolic link complements.

An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable. One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic…

Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.

The paper uses algebraic geometry tools to study knot invariants and Dehn surgeries.

Constructs noncommutative spaces for D-branes on complex algebraic spaces.

We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry and calibrated geometr…

In this lecture I review how a matrix/Azumaya-type noncommutative geometry arises for D-branes in string theory and how such a geometry serves as an origin of the master nature of D-branes; and then highlight an abundance conjecture on D0-brane resolutions of singularities that is extracted and purified from a work of …

We consider D-branes in string theory and address the issue of how to describe them mathematically as a fundamental object (as opposed to a solitonic object) of string theory in the realm in differential and symplectic geometry. The notion of continuous maps, $k$-times differentiable maps, and smooth maps from an Azuma…

In this follow-up of our earlier two works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) in the D-project, we study further the notion of a `differentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture is made that the notion of differentiable maps from Azumaya/matri…

D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.

Survey of stated skein modules/algebras of 3-manifolds/surfaces.

Explores smooth maps from supermanifolds, unifying concepts for D-branes.

Introduces a new D-brane action based on maps from Azumaya manifolds.

Clarifies conditions for non-Abelian Dirac-Born-Infeld action's masslessness and dynamics.

The paper finds a special pants decomposition for certain surface group representations.

In earlier works, D(1) (arXiv:0709.1515 [math.AG]), D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we have explained why a D-brane in string theory, when treated as a fundamental dynamical object, can be described by a map $\varphi$ from an Azumaya/m…

Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.

The study of quotient structures in multi-graded bundles, including double vector bundles.

Defines double principal bundles with applications in geometry.

Study of semi-principal bundles using group actions and wreath products.

Establishes a framework for stringor bundles, proving their canonical isomorphism to Stolz-Teichner's.

Generalizes graded bundles with more flexible transformation laws.

In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by p…

Proves correspondence between harmonic and Higgs bundles.

This paper shows vector bundles and differential bundles are equivalent in smooth manifolds.

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…

Introduces Darboux-Lie derivative for fiber bundles.

A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theor…

Categorical bundles provide a natural framework for gauge theories involving multiple gauge groups. Unlike the case of traditional bundles there are distinct notions of triviality, and hence also of local triviality, for categorical bundles. We study categorical principal bundles that are product bundles in the categor…

Study on singularities of bundle homomorphisms induced by Morin maps.

Study curvature of determinant bundle over Teichmüller space.

Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.

Introduces Weyl bundle gerbe and shows it's not D-stably isomorphic.

Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.

Smooth classifying spaces for groups defined using diffeological spaces.

Bundling of graph edges (node-to-node connections) is a common technique to enhance visibility of overall trends in the edge structure of a large graph layout, and a large variety of bundling algorithms have been proposed. However, with strong bundling, it becomes hard to identify origins and destinations of individual…