7,978 research outputs found

### $S^1$-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram

In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to
compute the integral $\int_{X}\tau^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ for a
flag manifold X=\Fl_{r_1, ..., r_I}({\Bbb C}^n) via an extended mirror
principle diagram. This method turns the required localization computation on
the augmented moduli stack \bar{\cal M}_{0,0}(\CP^1\times X) of stable maps
to a localization computation on a hyper-Quot-scheme \HQuot({\cal E}^n). In
this article, the detail of this localization computation on \HQuot({\cal
E}^n) is carried out. The necessary ingredients in the computation, notably,
the $S^1$-fixed-point components and the distinguished ones $E_{(A;0)}$ in
\HQuot({\cal E}^n), the $S^1$-equivariant Euler class of $E_{(A;0)}$ in
\HQuot({\cal E}^n), and a push-forward formula of cohomology classes involved
in the problem from the total space of a restrictive flag manifold bundle to
its base manifold are given. With these, an exact expression of
$\int_{X}\tau^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ is obtained. Comments on
the Hori-Vafa conjecture are given in the end.Comment: 44 pages with 6 figure

### On A-twisted moduli stack for curves from Witten's gauged linear sigma models

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 used in the Mirror Principle [L-L-Y] that relates
Gromov-Witten invariants and mirror symmetry computations. In this paper the
A-twisted moduli stack for higher genus curves is defined and systematically
studied. It is proved that such a moduli stack is an Artin stack. For genus 0,
it has the A-twisted moduli space of [M-P] as the coarse moduli space. The
detailed proof of the regularity of the collapsing morphism by Jun Li in
[L-L-Y: I and II] can be viewed as a natural morphism from the moduli stack of
genus 0 stable maps to the A-twisted moduli stack at genus 0.
Due to the technical demand of stacks to physicists and the conceptual demand
of supersymmetry to mathematicians, a brief introduction of each topic that is
most relevant to the main contents of this paper is given in the beginning and
the appendix respectively. Themes for further study are listed in the end.Comment: 36 page

### Azumaya structure on D-branes and deformations and resolutions of a conifold revisited: Klebanov-Strassler-Witten vs. Polchinski-Grothendieck

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 conifold from the
viewpoint of the Azumaya structure on D-branes and toric geometry. The details
of how deformations and resolutions of the standard toric conifold $Y$ can be
obtained via morphisms from Azumaya points are given. This should be compared
with the quantum-field-theoretic/D-brany picture of deformations and
resolutions of a conifold via a D-brane probe sitting at the conifold
singularity in the work of Klebanov and Witten [K-W] (arXiv:hep-th/9807080) and
Klebanov and Strasser [K-S] (arXiv:hep-th/0007191). A comparison with
resolutions via noncommutative desingularizations is given in the end.Comment: 23+2 pages, 4 figure

### 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.Comment: 47 + 2 pages, 3 figure

### 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]).Comment: 19 pages, 6 figure

### $N=1$ 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
$\widehat{\varphi}$ 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
$(\widehat{\nabla}, \widehat{\varphi})$ 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]).Comment: 85+2 pages, 5 figure

### On the Splitting Type of an Equivariant Vector Bundle over a Toric Manifold

From the work of Lian, Liu, and Yau on "Mirror Principle", in the explicit
computation of the Euler data $Q=\{Q_0, Q_1, ... \}$ for an equivariant
concavex bundle ${\cal E}$ over a toric manifold, there are two places the
structure of the bundle comes into play: (1) the multiplicative characteric
class $Q_0$ of $V$ one starts with, and (2) the splitting type of ${\cal E}$.
Equivariant bundles over a toric manifold has been classified by Kaneyama,
using data related to the linearization of the toric action on the base toric
manifold. In this article, we relate the splitting type of ${\cal E}$ to the
classifying data of Kaneyama. From these relations, we compute the splitting
type of a couple of nonsplittable equivariant vector bundles over toric
manifolds that may be of interest to string theory and mirror symmetry. A code
in Mathematica that carries out the computation of some of these examples is
attached.Comment: 24 page

### Azumaya structure on D-branes and resolution of ADE orbifold singularities revisited: Douglas-Moore vs. Polchinski-Grothendieck

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 manifold/variety of the supersymmetric quantum field theory on the
D-brane probe world-volume, given by Douglas and Moore [D-M] under the
string-theory contents and constructed earlier through hyper-K\"{a}hler
quotients by Kronheimer and Nakajima. This is consistent with the moral behind
this project that Azumaya-type structure on D-branes themselves -- stated as
the Polchinski-Grothendieck Ansatz in [L-Y1] -- gives a mathematical reason for
many originally-open-string-induced properties of D-branes.Comment: 20 pages, 2 figure

### 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

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 development of algebraic relative Gromov-Witten theory and its
applications. This gives an algebro-geometric account of Witten's wall-crossing
formula for correlation functions of the descendant nonlinear sigma model in
adjacent geometric phases of a gauge linear sigma model and of the symplectic
approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem.
A terse account from the stringy and the symplectic viewpoint is given in the
appendix to complement and compare to the discussion in the main text.Comment: 38+2 pages, 5 figure

### Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint

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
Azumaya-type noncommutative space. Several originally open-string induced
properties of D-branes can be reproduced solely by this intrinsic definition.
We study also the moduli space of D0-branes on a commutative target space in
this setup. Some of its features resembles gas of D0-branes in Vafa's work.Comment: 57 page

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