Related papers: Morphisms from Azumaya prestable curves with a fun…

In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a…

In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a target-space with a fixed $B$-field…

We explain how Polchinski's work on D-branes re-read from a noncommutative version of Grothendieck's equivalence of local geometries and function rings gives rise to an intrinsic prototype definition of D-branes (of B-type) as an…

In a suitable regime of superstring theory, D-branes in a Calabi-Yau space and their most fundamental behaviors can be nicely described mathematically through morphisms from Azumaya spaces with a fundamental module to that Calabi-Yau space.…

We review first Azumaya geometry and D-branes in the realm of algebraic geometry along the line of Polchinski-Grothendieck Ansatz from our earlier work and then use it as background to introduce Azumaya $C^{\infty}$-manifolds with a…

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…

In this continuation of [L-Y1] and [L-L-S-Y], we explain how the Azumaya structure on D-branes together with a netted categorical quotient construction produces the same resolution of ADE orbifold singularities as that arises as the vacuum…

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)…

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…

In this sequel to works 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 re-examine --- and reformulate when in need --- several basic notions in super…

In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this notion to give a prototypical…

A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic) D-branes on such a noncommutative space, following…

In contrast to the world-sheet of a fundamental string, the world-volume of stacked D-branes carries an Azumaya noncommutative structure ([L-Y1: Sec.\ 2] (D(1))), allowing it to directly serve as a probe into noncommutative target-spaces.…

We give a general construction of extended moduli spaces of topological D-branes as non-commutative algebraic varieties. This shows that noncommutative symplectic geometry in the sense of Kontsevich arises naturally in String Theory.

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…

We study moduli spaces and moduli stacks for representations of associative algebras in Azumaya algebras, in rather general settings. We do not impose any stability condition and work over arbitrary ground rings, but restrict attention to…

In this thesis we study the AdS3 Wess-Zumino-Novikov-Witten model. We compute the Operator Product Expansion of primary fields as well as their images under the spectral flow automorphism in all sectors of the model by considering it as a…

We study the topological string on local P2 with O-plane and D-brane at its real locus, using three complementary techniques. In the A-model, we refine localization on the moduli space of maps with respect to the torus action preserved by…

This thesis is concerned with D-branes in topological string theory, focusing on the description of B-type D-branes in topological Landau-Ginzburg models. Such D-branes are characterized by matrix factorizations of the Landau-Ginzburg…

In this paper we describe how Grothendieck groups of coherent sheaves and locally free sheaves can be used to describe type II D-branes, in the case that all D-branes are wrapped on complex varieties and all connections are holomorphic. Our…