arXiv subjects

Chien-Hao Liu

Publications and source records attributed to Chien-Hao Liu.

At least 19 recordsLinked to original sources

arXiv2025

A D-brane fantasy on noncommutative mirror symmetry, prelude: Noncommutative ringed spaces from local noncommutative crepant resolutions of a singular Calabi-Yau space, dynamical D-branes thereupon, and questions beyond

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. This feature leads to a D-brane fantasy: {\it Noncommutative Mirror Symmetry between noncommutative Calabi-Yau spaces may be realized as different realizations of a supersymmetric D-brane world-volume quantum field theory exactly like the string world-sheet aspect for Mirror Symmetry between (commutative) Calabi-Yau manifolds}. Driven by this fantasy, in the current notes a class of noncommutative ringed spaces shadowing over a $C^\infty$-manifold with corners are constructed from gluing local noncommutative crepant resolutions of Gorenstein isolated singularities. Dynamical D-branes on such noncommutative target-spaces are realized as maps/morphisms from an Azumaya manifold with a fundamental module with a connection $\nabla$ thereto. The notion of $\nabla$-adjusted kinetic energy for such a map is given via the basic noncommutative differential calculus developed earlier in [L-Y4] (D(11.1)). This provides an action functional for dynamical D-branes on such noncommutative spaces in parallel to the Polyakov action functional for fundamental bosonic strings on a commutative target-space. This sets up a basic stage to begin with for the realization of the D-brane fantasy on Noncommutative Mirror Symmetry. Questions beyond are sampled along the discussion.

math.AG↗

arXiv2022

Soft noncommutative flag schemes

The construction of soft noncommutative schemes via toric geometry in arXiv:2108.05328 [math.AG] (D(15.1), NCS(1)) can be generalized and applied to a commutative scheme with a distinguished atlas of reasonably good affine local coordinate charts. In the current notes we carry out this exercise for flag varieties.

math.AG↗

arXiv2021

Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space

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 arXiv:0709.1515 [math.AG] (D(1)), is given. Any algebraic Calabi-Yau space that arises from a complete intersection in a smooth toric variety can embed as a commutative closed subscheme of some soft noncommutative scheme. Along the study, the notion of `soft noncommutative toric schemes' associated to a (simplicial, maximal cone of index $1$) fan, `invertible sheaves' on such a noncommutative space, and `twisted sections' of an invertible sheaf are developed and Azumaya schemes with a fundamental module as the world-volumes of D-branes are reviewed. Two guiding questions, Question 3.12 (soft noncommutative Calabi-Yau spaces and their mirror) and Question 4.2.14 (generalized matrix models), are presented.

math.AG↗

arXiv2020

Grothendieck meeting [Wess & Bagger]: [Supersymmetry and supergravity: IV, V, VI, VII, XXII] (2nd ed.) reconstructed in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry, I. Construction under trivialization of spinor bundle

Forty-six years after the birth of supersymmetry in 1973 from works of Julius Wess and Bruno Zumino, the standard quantum-field-theorists and particle physicists' language of `superspaces', `supersymmetry', and `supersymmetric action functionals in superspace formulation' as given in Chapters IV, V, VI, VII, XXII of the classic on supersymmetry and supergravity: Julius Wess & Jonathan Bagger: Supersymmetry and Supergravity (2nd ed.), is finally polished, with only minimal mathematical patches added for consistency and accuracy in dealing with nilpotent objects from the Grassmann algebra involved, to a precise setting in the language of complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry. This is completed after the lesson learned from D(14.1) (arXiv:1808.05011 [math.DG]) and the notion of `$d=3+1$, $N=1$ towered superspaces' as complexified ${\Bbb Z}/2$-graded $C^\infty$-schemes, their distinguished sectors, and purge-evaluation maps first developed in SUSY(1) (= D(14.1.Supp.1)) (arXiv:1902.06246 [hep-th]) and further polished in the current work. While the construction depends on a choice of a trivialization of the spinor bundle by covariantly constant sections, as long as the transformation law and the induced isomorphism under a change of trivialization of the spinor bundle by covariantly constant sections are understood, any object or structure thus defined or constructed is mathematically well-defined. The construction can be generalized to all other space-time dimensions with simple or extended supersymmetries. This is part of the mathematical foundation required to study fermionic D-branes in the Ramond-Neveu-Schwarz formulation.

math.AG↗

arXiv2019

Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting map

The complexified ${\Bbb Z}/2$-graded $C^\infty$-Algebraic Geometry aspect of a superspace(-time) $\widehat{X}$ in Sec.\,1 of D(14.1) (arXiv:1808.05011 [math.DG]) together with the Spin-Statistics Theorem in Quantum Field Theory, which requires fermionic components of a superfield be anticommuting, lead us to the notion of towered superspace(-time) $\widehat{X}^{\widehat{\boxplus}}$ and the built-in purely even physics sector $X^{\mbox{physics}}$ from $\widehat{X}^{\widehat{\boxplus}}$. We use this to reproduce the $d=3+1$, $N=1$ Wess-Zumino model and the $d=3+1$, $N=1$ supersymmetric $U(1)$ gauge theory with matter --- as in, e.g., Chap.\,V and Chap.\,VI \& part of Chap.\,VII of the classical Supersymmetry \& Supergravity textbook by Julius Wess and Jonathan Bagger --- and, hence, recast physicists' two most basic supersymmetric quantum field theories solidly into the realm of (complexified ${\Bbb Z}/2$-graded) $C^\infty$-Algebraic Geometry. Some traditional differential geometers' ways of understanding supersymmetric quantum field theories are incorporated into the notion of a purge-evaluation/index-contracting map ${\cal P}:C^\infty(X^{\mbox{physics}})\rightarrow C^\infty(\widehat{X})$ in the setting. This completes for the current case a $C^\infty$-Algebraic Geometry language we sought for in D(14.1), footnote 2, that can directly link to the study of supersymmetry in particle physics. Once generalized to the nonabelian case in all dimensions and extended $N\ge 2$, this prepares us for a fundamental (as opposed to solitonic) description of super D-branes parallel to Ramond-Neveu-Schwarz fundamental superstrings

hep-th↗

arXiv2018

$N=1$ fermionic D3-branes in RNS formulation I. $C^\infty$-Algebrogeometric foundations of $d=4$, $N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace

As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current notes: (1) basic $C^\infty$-algebrogeometric foundations of $d=4$, $N=1$ supersymmetry and $d=4$, $N=1$ superspace in physics, with emphasis on the partial $C^\infty$-ring structure on the function ring of the superspace, (2) the notion of SUSY-rep compatible hybrid connections on bundles over the superspace to address connections on the Chan-Paton bundle on the world-volume of a fermionic D3-brane, (3) the notion of $\widehat{D}$-chiral maps $\widehatφ$ from a $d=4$ $N=1$ Azumaya/matrix superspace with a fundamental module with a SUSY-rep compatible hybrid connection $\widehat{\nabla}$ to a complex manifold $Y$ as a model for a dynamical D3-branes moving in a target space(-time). Some test computations related to the construction of a supersymmetric action functional for SUSY-rep compatible $(\widehat{\nabla}, \widehatφ)$ are given in the end, whose further study is the focus of a separate work. The current work is a sequel to D(11.4.1) (arXiv:1709.08927 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) and is the first step in the supersymmetric generalization, in the case of D3-branes, of the standard action functional for D-branes constructed in D(13.3) (arXiv:1704.03237 [hep-th]).

math.DG↗

arXiv2017

Further studies of the notion of differentiable maps from Azumaya/matrix supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz meeting Grothendieck

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 $C^{\infty}$-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz fermionic strings and of Green-Schwarz fermionic strings from the viewpoint of Grothendieck on Algebraic Geometry. Two theorems that are the super counterpart of Theorem~3.1.1 and Theorem~3.2.1 of D(11.3.1) are proved. They unify the notion of "smooth maps from an Azumaya/matrix super smooth manifold with a fundamental module to a super smooth manifold" introduced in D(11.2), making it a complete super parallel to the setting for D-branes in the realm of algebraic geometry in D(1) (arXiv:0709.1515 [math.AG]) and D(2) (arXiv:0809.2121 [math.AG]), and in the realm of differential or $C^{\infty}$-algebraic geometry in D(11.1) and D(11.3.1). A prototypical definition of dynamical fermionic stacked D-brane world-volume on a space-time in the same spirit of RNS fermionic strings or GS fermionic strings is thus laid down. Similar to D(11.3.1), which paved the path to the construction of non-Abelian Dirac-Born-Infeld action (D(13.1) (arXiv:1606.08529 [hep-th])) and the standard action (D(13.3) (arXiv:1704.03237 [hep-th])) for fundamental bosonic stacked D-branes, the current notes shall serve the same for the construction of supersymmetric action for fundamental fermionic stacked D-branes of various dimensions --- a theme of another subseries of the D-project. A notion of "noncommutative $C^{\infty}$-rings" and "morphism" between them is introduced at the end as a byproduct.

math.DG↗

arXiv2017

Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes

We introduce a new action $S_{standard}^{(ρ,h; Φ,g,B,C)}$ for D-branes that is to D-branes as the Polyakov action is to fundamental strings. This `standard action' is abstractly a non-Abelian gauged sigma model --- based on maps $φ: (X^{\!A\!z},E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ with a fundamental module $E$ with a connection $\nabla$ to $Y$ --- enhanced by the dilaton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples $(φ,\nabla)$ to Ramond-Ramond field. In a special situation, this new theory merges the theory of harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. With the analysis developed in D(13.1) (arXiv:1606.08529 [hep-th]) for such maps and an improved understanding of the hierarchy of various admissible conditions on the pairs $(φ,\nabla)$ beyond D(13.2.1) (arXiv:1611.09439 [hep-th]) and how they resolve the built-in obstruction to pull-push of covariant tensors under a map from a noncommutative manifold to a commutative manifold, we develop further in this note some covariant differential calculus needed and apply them to work out the first variation --- and hence the corresponding equations of motion for D-branes --- of the standard action and the second variation of the kinetic term for maps and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along the same line, the current note brings the Nambu-Goto-string-to-Polyakov-string analogue to D-branes. The current bosonic setting is the first step toward the dynamics of fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallel to what happened to the theory of fundamental strings during years 1976--1981.

hep-th↗

arXiv2016

More on the admissible condition on differentiable maps $φ: (X^{\!A\!z},E;\nabla)\rightarrow Y$ in the construction of the non-Abelian Dirac-Born-Infeld action $S_{DBI}(φ,\nabla)$

In D(13.1) (arXiv:1606.08529 [hep-th]), we introduced an admissible condition on differentiable maps $φ: (X^{\!A\!z}, E;\nabla)\rightarrow Y$ from an Azumaya/matrix manifold $X^{\!A\!z}$ (with the fundamental module $E$) with a connection $\nabla$ on $E$ to a manifold $Y$ in order to resolve a pull-push issue in the construction of a non-Abelian-Dirac-Infeld action $S_{DBI}$ for $(φ,\nabla)$ and to render $\nabla$ massless from the aspect of open strings. The admissible condition ibidem consists of two parts: Condition (1) and Condition (2). In this brief note, we examine these two conditions in more detail and bring their geometric implications on $(φ,\nabla)$ and the full action $S_{DBI}(φ,\nabla)+S_{CS/WZ}(φ,\nabla)$ more transparent. In particular, we show that Condition (1) alone already implies masslessness of $\nabla$ from open-string aspect; and that the additional Condition (2) implies a decoupling of the nilpotent fuzzy cloud of $φ(X^{\!A\!z})$ to the dynamics of $(φ, \nabla)$. We conclude with a refined definition of admissible $(φ,\nabla)$ and a remark on the anomaly factor in the integrand of the Chern-Simons/Wess-Zumino term $S_{CS/WZ}(φ,\nabla)$ for D-branes based on the current study.

hep-th↗

arXiv2016

Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term

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 $φ$ from an Azumaya/matrix manifold $X^{Az}$ (cf. D-brane world-volume) with a fundamental module with a connection $(E,\nabla)$ (cf. Chan-Paton bundle) to the target space-time $Y$. In this sequel, we construct a non-Abelian Dirac-Born-Infeld action functional $S_{DBI}^{(Φ, g, B)}(φ,\nabla)$ for such pairs $(φ,\nabla)$. We next develop a technical tool needed to study variations of this action and apply it to derive the first variation $δS_{DBI}^{(Φ,g,B)}/δ(φ,\nabla)$ of $S_{DBI}^{(Φ,g,B)}$ with respect to $(φ,\nabla)$. The equations of motion that govern the dynamics of D-branes then follow. A complete action for a D-brane world-volume must include also the Chern-Simons/Wess-Zumino term $S_{CS/WZ}^{(C)}(φ,\nabla)$ that governs how the D-brane world-volume couples with the Ramond-Ramond fields $C$ on $Y$. In the current notes, a version $S^{(C,B)}_{CS/WZ}(φ,\nabla)$ of non-Abelian Chern-Simons/Wess-Zumino action functional for $(φ,\nabla)$ that follows the same guide with which we construct $S^{(Φ,g,B)}_{DBI}(φ,\nabla)$ is constructed for lower-dimensional D-branes (i.e. D(-1)-, D0-, D1-, D2-branes). Its first variation $δS^{(C,B)}_{CS/WZ}(φ,\nabla)/δ(φ,\nabla)$ is derived and its contribution to the equations of motion for $(φ, \nabla)$ follows. The current notes lay down a foundation toward the dynamics of D-branes along the line of this D-project.

hep-th↗

arXiv2015

Quantum fluctuations, conformal deformations, and Gromov's topology --- Wheeler, DeWitt, and Wilson meeting Gromov

The moduli space of isometry classes of Riemannian structures on a smooth manifold was emphasized by J.A.Wheeler in his superspace formalism of quantum gravity. A natural question concerning it is: What is a natural topology on such moduli space that reflects best quantum fluctuations of the geometries within the Planck's scale? This very question has been addressed by B.DeWitt and others. In this article we introduce Gromov's $\varepsilon$-approximation topology on the above moduli space for a closed smooth manifold. After giving readers some feel of this topology, we prove that each conformal class in the moduli space is dense with respect to this topology. Implication of this phenomenon to quantum gravity is yet to be explored. When going further to general metric spaces, Gromov's geometries-at-large-scale based on his topologies remind one of K.Wilson's theory of renormalization group. We discuss some features of both and pose a question on whether both can be merged into a single unified theory.

gr-qc↗

arXiv2015

Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case

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/matrix manifolds as defined in D(11.1) is equivalent to one defined through the contravariant ring-homomorphisms alone. A proof of this conjecture for the smooth (i.e. $C^{\infty}$) case is given in this note. Thus, at least in the smooth case, our setting for D-branes in the realm of differential geometry is completely parallel to that in the realm of algebraic geometry, cf.\ arXiv:0709.1515 [math.AG] and arXiv:0809.2121 [math.AG]. A related conjecture on such maps to ${\Bbb R}^n$, as a $C^k$-manifold, and its proof in the $C^{\infty}$ case is also given. As a by-product, a conjecture on a division lemma in the finitely differentiable case that generalizes the division lemma in the smooth case from Malgrange is given in the end, as well as other comments on the conjectures in the general $C^k$ case. We remark that there are similar conjectures in general and theorems in the smooth case for the fermionic/super generalization of the notion.

math.DG↗

arXiv2015

D-branes and synthetic/$C^{\infty}$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds

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 geometry that are both tailored to and guided by D-brane phenomena in string theory and along the line of our previous works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).

math.SG↗

arXiv2014

D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory

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 from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion of `fermionic D-branes' in two different styles, one parallels Ramond-Neveu-Schwarz fermionic string and the other Green-Schwarz fermionic string. A more detailed discussion on the Higgs mechanism on dynamical D-branes in our setting, taking maps from the D-brane world-volume to the space-time in question and/or sections of the Chan-Paton bundle on the D-brane world-volume as Higgs fields, is also given for the first time in the D-project. Finally note that mathematically string theory begins with the notion of a differentiable map from a string world-sheet (a $2$-manifold) to a target space-time (a real manifold). In comparison to this, D(11.1) and the current D(11.2) together bring us to the same starting point for studying D-branes in string theory as dynamical objects.

hep-th↗

arXiv2014

D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds

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 Azumaya/matrix manifold with a fundamental module to a (commutative) real manifold $Y$ is developed. Such maps are meant to describe D-branes or matrix branes in string theory when these branes are light and soft with only small enough or even zero brane-tension. When $Y$ is a symplectic manifold (resp. a Calabi-Yau manifold; a $7$-manifold with $G_2$-holonomy; a manifold with an almost complex structure $J$), the corresponding notion of Lagrangian maps (resp. special Lagrangian maps; associative maps, coassociative maps; $J$-holomorphic maps) are introduced. Indicative examples linking to symplectic geometry and string theory are given. This provides us with a language and part of the foundation required to study themes, new or old, in symplectic geometry and string theory, including (1) $J$-holomorphic D-curves (with or without boundary), (2) quantization and dynamics of D-branes in string theory, (3) a definition of Fukaya category guided by Lagrangian maps from Azumaya manifolds with a fundamental module with a connection, (4) a theory of fundamental matrix strings or D-strings, and (5) the nature of Ramond-Ramond fields in a space-time. The current note D(11.1) is the symplectic/differential-geometric counterpart of the more algebraic-geometry-oriented first two notes D(1) ([L-Y1]) (arXiv:0709.1515 [math.AG]) and D(2) ([L-L-S-Y], with Si Li and Ruifang Song) (arXiv:0809.2121 [math.AG]) in this project.

math.DG↗

arXiv2013

A mathematical theory of D-string world-sheet instantons, II: Moduli stack of $Z$-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold

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 projective Calabi-Yau 3-fold and show that the moduli stack of such $Z$-semistable morphisms of a fixed type is compact. This gives us a counter moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten theory to the fundamental string. It serves and prepares for us the basis toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string world-sheet instanton numbers in superstring theory. This note is written hand-in-hand with D(10.1) and is to be read side-by-side with ibidem.

math.AG↗

arXiv2013

A mathematical theory of D-string world-sheet instantons, I: Compactness of the stack of $Z$-semistable Fourier-Mukai transforms from a compact family of nodal curves to a projective Calabi-Yau 3-fold

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. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121 [math.AG], with Si Li and Ruifang Song) from the project, we explored this notion for the case of D1-branes (i.e. D-strings) and laid down some basic ingredients toward understanding the notion of D-string world-sheet instantons in this context. In this continuation, D(10), of D(2), we move on to construct a moduli stack of semistable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold $Y$. In this Part I of the note, D(10.1), we define the notion of twisted central charge $Z$ for Fourier-Mukai transforms of dimension 1 and width [0] from nodal curves and the associated stability condition on such transforms and prove that for a given compact stack of nodal curves $C_{\cal M}/{\cal M}$, the stack $FM^{1,[0];Zss}_{C_{\cal M}/{\cal M}}(Y,c)$ of $Z$-semistable Fourier-Mukai transforms of dimension 1 and width [0] from nodal curves in the family $C_{\cal M}/{\cal M}$ to $Y$ of fixed twisted central charge $c$ is compact. For the application in the sequel D(10.2), $C_{\cal M}/{\cal M}$ will contain $C_{\bar{\cal M}_g}/\bar{\cal M}_g$ as a substack and $FM^{1,[0];Zss}_{C_{\cal M}/{\cal M}}(Y,c)$ in this case will play a key role in defining stability condition for morphisms from arbitrary Azumaya nodal curves (with the underlying nodal curves not necessary in the family $C_{\cal M}/{\cal M}$) to $Y$.

math.AG↗

arXiv2011

Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes

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 Douglas and Moore in 1996. A conjectural relation of our setting with `D-geometry' in the sense of Douglas is also given. The lecture is based on a series of works on D-branes with Shing-Tung Yau, and in part with Si Li and Ruifang Song.

math.AG↗