In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold ( or complex analytic).
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Mathai, Melrose, and Singer compute the index of projective elliptic operators.
problem Computing the index of projective elliptic operators on manifolds with Azumaya bundles.
method Equivariant index of transversally elliptic operators as pullbacks of projective elliptic operators.
result Comprehensive fractional index formula for projective elliptic operators.
New topological criterion extends arithmetic invariants in hyperbolic 3-manifolds.
problem Arithmetic invariants from character varieties of hyperbolic 3-manifolds.
method Culler-Shalen theory and JSJ decompositions of toroidal Dehn fillings.
result Explicit topological criterion for extending Azumaya algebras over ideal points.
In this Part II of D(11), we introduce new objects: super--schemes and Azumaya super--manifolds with a fundamental module (or, synonymously, matrix super--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.
problem Understanding the structure and properties of skein algebras of small surfaces.
method Constructed finite-dimensional representations at all roots of unity, using explicit formulas and analyzing reducibility.
result Azumaya loci of the surfaces contain the smooth loci of classical shadow varieties, with equality for the one-punctured torus and proper containment for the four-punctured sphere.
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, -algebraic) symplectic geometry and calibrated geometr…
Paper extends algebraic geometry results to hyperbolic link complements.
problem Understanding algebraic and number-theoretic properties of canonical curves.
method Generalizes Chinburg-Reid-Stover's results to hyperbolic link complements.
result Azumaya algebra does not extend to canonical surfaces.
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, -times differentiable maps, and smooth maps from an Azuma…
Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.
problem Determining when degree-2 Azumaya algebras with C2-actions extend to entire curves.
method Criteria for extension of algebra and new condition for extension with action, testable by computer algebra systems.
result New conditions for extending degree-2 Azumaya algebras with C2-actions over curves.
The paper uses algebraic geometry tools to study knot invariants and Dehn surgeries.
problem Understanding algebraic and number theoretic properties of canonical components of character varieties.
method Utilizes quaternion Azumaya algebras and Brauer groups of curves over number fields.
result Constructs new knot invariants using algebraic geometry.
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…
Study centers of generalized skein algebras, showing almost Azumaya properties.
problem Understanding the center of generalized skein algebras.
method Generalized skein algebra generated by loops and arcs, computed center, discussed implications.
result Center of Muller-Roger-Yang skein algebra is almost Azumaya.
Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.
problem Character variety and topological invariants of Dehn fillings.
method Extension problem for quaternion Azumaya algebras.
result New phenomena in extension problem not seen in knot complements.
Survey of stated skein modules/algebras of 3-manifolds/surfaces.
problem Understanding stated skein modules/algebras of 3-manifolds/surfaces.
method Discussion of splitting homomorphism, general structures, Frobenius homomorphism, center, dimension, representation theory.
result Skein algebra of non-closed marked surface at any root of 1 is a maximal order.
D-branes on noncommutative spaces mimic string theory, offering new insights into mirror symmetry.
problem Exploring noncommutative mirror symmetry through D-branes on noncommutative Calabi-Yau spaces.
method Constructing noncommutative ringed spaces from local resolutions, realizing D-branes as morphisms, and defining kinetic energy.
result Dynamical D-branes on noncommutative spaces can be described by a Polyakov-like action, suggesting a bridge between string theory and noncommutative geometry.
Study deformation invariants of projective surfaces using twisted sheaves and Azumaya algebras.
problem Deformation invariants of projective surfaces with specific cohomology conditions.
method Virtual intersection numbers on moduli spaces of stable twisted sheaves and Azumaya modules.
result Invariants do not depend on the choice of Brauer-Severi variety or Azumaya algebra.
Constructs noncommutative spaces for D-branes on complex algebraic spaces.
problem Mathematical model for D-branes on noncommutative spaces.
method Toric geometry, Azumaya schemes, invertible sheaves.
result Embeds algebraic Calabi-Yau spaces into soft noncommutative schemes.
Introduces a new D-brane action based on maps from Azumaya manifolds.
problem Developing a new action for D-branes analogous to Polyakov's for strings.
method Enhanced non-Abelian gauged sigma model with dilaton, gauge-theory, and Chern-Simons/Wess-Zumino terms.
result Developed first and second variations of the standard action and equations of motion.
Given a bundle gerbe on a compact smooth manifold or, more generally, on a compact étale Lie groupoid , 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 . This result can be seen as an …
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 …
This research builds foundations for D3-branes in string theory.
problem Constructing dynamical fermionic D3-branes in string theory.
method Develops algebraic and geometric foundations, hybrid connections, and chiral maps.
result Provides a framework for constructing supersymmetric actions.
Explores smooth maps from supermanifolds, unifying concepts for D-branes.
problem Defines smooth maps from Azumaya/matrix supermanifolds.
method Re-examines and reformulates super -algebraic geometry concepts.
result Unifies the notion of smooth maps, making it a complete super parallel.
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.
Clarifies conditions for non-Abelian Dirac-Born-Infeld action's masslessness and dynamics.
problem Resolving a pull-push issue in constructing the non-Abelian Dirac-Born-Infeld action.
method Examines two admissible conditions (1) and (2) on differentiable maps in detail.
result Condition (1) alone implies masslessness of the connection, while (2) decouples the fuzzy cloud.
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…
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…
The paper finds a special pants decomposition for certain surface group representations.
problem Finding a specific pants decomposition for surface group representations.
method Proves the existence of a pants decomposition with irreducible restrictions and no trace ±2 elements.
result Shows the existence of a special pants decomposition for -representations of surface groups.
Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.
problem Study sliced skein algebras and their properties.
method Quotient of Kauffman bracket skein algebra, center calculation, PI-degree calculation, fully Azumaya point analysis.
result Center and PI-degree calculations for sliced skein algebras, fully Azumaya points, and simple modules.
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 from an Azumaya/m…
New class of metric f-manifolds introduced.
problem No specific problem stated; focuses on introducing new class.
method Definition and properties of new class of metric f-manifolds.
result Properties and examples of new class of metric f-manifolds.
Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.
problem Understanding the structure and properties of selfsimilar Hessian manifolds.
method Characterization and description of selfsimilar manifolds with homothetic vector fields.
result Any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product of radiant Hessian manifolds.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
The paper defines trans-para-Sasakian manifolds and explores their geometric properties.
problem Exploring the geometry of trans-para-Sasakian manifolds.
method Definition and study of curvature properties.
result Conditions for Einstein and Einstein manifolds.
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with -dimensional contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…
Study on a new type of manifolds that generalize almost C-manifolds.
problem Understanding weak nearly C-manifolds and their properties.
method Analyzing conditions for local Riemannian product structures and characterizing specific dimensions.
result Conditions for a weak nearly C-manifold to become locally a Riemannian product and characterization of specific dimensions.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
Determines the type of 3-manifolds from their covers.
problem Identifying the type of 3-manifolds.
method Uses Thurston seminorms of covers.
result Finite covers determine the type of 3-manifolds.
The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.
problem Determining which 3-manifolds can be boundaries of exotic 4-manifolds.
method Provided criteria and examples of 3-manifolds that can be boundaries of 4-manifolds with infinitely many distinct smooth structures.
result Identified specific types of 3-manifolds (weakly fillable contact, non-vanishing Heegaard Floer invariant) that are boundaries of 4-manifolds with infinitely many distinct smooth structures.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.
problem Exploring new types of manifolds in general relativity.
method Generalization of existing manifolds and construction of specific examples.
result Existence and properties of extended quasi-Einstein manifolds with solitons.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
On PNDP-manifoldmath.DG
New manifold type PNDP-manifold defined with Einstein warped product structure.
problem Defining manifolds with non-standard dimensions.
method Einstein warped product manifold with special base and fiber structures.
result PNDP-manifolds are Einstein warped product manifolds with specific base and fiber properties.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified