22 citations · 221 across the 45 of their papers we have counts for
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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
Chien-Hao Liu, Shing-Tung Yau
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 di…
Soft noncommutative flag schemes
Chien-Hao Liu, Shing-Tung Yau
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 dis…
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
Chien-Hao Liu, Shing-Tung Yau
A class of noncommutative spaces, named `soft noncommutative schemes via toric geometry', are constructed and the mathematical model for (dynamical/nonsolitonic, complex algebraic)…
Grothendieck meeting [Wess & Bagger]: [Supersymmetry and supergravity: IV, V, VI, VII, XXII] (2nd ed.) reconstructed in complexified -graded -Algebraic Geometry, I. Construction under trivialization of spinor bundle
Chien-Hao Liu, Shing-Tung Yau
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 `supe…
A mathematical theory of D-string world-sheet instantons, II: Moduli stack of -(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold
Chien-Hao Liu, Shing-Tung Yau
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 -semistable morphisms from general Azumaya nodal curves…
A mathematical theory of D-string world-sheet instantons, I: Compactness of the stack of -semistable Fourier-Mukai transforms from a compact family of nodal curves to a projective Calabi-Yau 3-fold
Chien-Hao Liu, Shing-Tung Yau
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 Azumay…
Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes
Chien-Hao Liu
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…
D0-brane realizations of the resolution of a reduced singular curve
Chien-Hao Liu, Shing-Tung Yau
Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space via a D-brane probe, the rich…
Algebraic cobordism of filtered vector bundles on varieties: Notes on a work of Lee and Pandharipande
Chien-Hao Liu, Yu-jong Tzeng, Shing-Tung Yau
The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immed…
Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a -field background à la Polchinski-Grothendieck Ansatz
Chien-Hao Liu, Shing-Tung Yau
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…
Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-Grothendieck
Chien-Hao Liu, Shing-Tung Yau
In this sequel to [L-Y1], [L-L-S-Y], and [L-Y2] (respectively arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], and arXiv:0901.0342 [math.AG]), we study a D-brane probe on a co…
Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck
Chien-Hao Liu, Shing-Tung Yau
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…
Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves
Si Li, Chien-Hao Liu, Ruifang Song +1
This is a continuation of our study of the foundations of D-branes from the viewpoint of Grothendieck in the region of the related Wilson's theory-space where "branes" are still br…
Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint
Chien-Hao Liu, Shing-Tung Yau
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 p…
Transformation of algebraic Gromov-Witten invariants of three-folds under flops and small extremal transitions, with an appendix from the stringy and the symplectic viewpoint
Chien-Hao Liu, Shing-Tung Yau
We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent develop…
Extracting Gromov-Witten invariants of a conifold from semi-stable reduction and relative GW invariants of pairs
Chien-Hao Liu, Shing-Tung Yau
The study of open/closed string duality and large duality suggests a Gromov-Witten theory for conifolds that sits on the border of both a closed Gromov-Witten theory and an ope…
A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups
Chien-Hao Liu, Shing-Tung Yau
In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form that arises…
-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram
Chien-Hao Liu, Kefeng Liu, Shing-Tung Yau
In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral for a flag manifold $X=\Fl_{r_1, ...…
On A-twisted moduli stack for curves from Witten's gauged linear sigma models
Chien-Hao Liu, Kefeng Liu, Shing-Tung Yau
Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been use…
The fixed points in Quot-schemes and mirror principle computations
Bong H. Lian, Chien-Hao Liu, Kefeng Liu +1
We describe the -action on the Quot-scheme $\Quot({\cal E}^n)$ associated to the trivial bundle ${\cal E}^n=CP^1\times{\smallBbb C}^n$. In particlular, the topology of the $S^…
Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces
Yi Hu, Chien-Hao Liu, Shing-Tung Yau
Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we…
A reconstruction of Euler data
Bong H. Lian, Chien-Hao Liu, Shing-Tung Yau
We apply the mirror principle of [L-L-Y] to reconstruct the Euler data $Q=\{Q_d\}_{d\in{\tinyBbb N}\cup\{0\}}$ associated to a vector bundle on ${\smallBbb C}{\rm P}^n$ and a m…
On the Splitting Type of an Equivariant Vector Bundle over a Toric Manifold
Chien-Hao Liu, Shing-Tung Yau
From the work of Lian, Liu, and Yau on "Mirror Principle", in the explicit computation of the Euler data for an equivariant concavex bundle over a…