7,978 research outputs found

    S1S^1-fixed-points in hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram

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    In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral ∫XΟ„βˆ—eHβ‹…t∩1d\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 S1S^1-fixed-point components and the distinguished ones E(A;0)E_{(A;0)} in \HQuot({\cal E}^n), the S1S^1-equivariant Euler class of E(A;0)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 ∫XΟ„βˆ—eHβ‹…t∩1d\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

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    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

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    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 YY 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 ZZ-(semi)stable morphisms from Azumaya nodal curves with a fundamental module to a projective Calabi-Yau 3-fold

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    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 ZZ-semistable morphisms from general Azumaya nodal curves, of genus β‰₯2\ge 2, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such ZZ-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∞C^{\infty}-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds

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

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    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∞C^\infty-algebrogeometric foundations of d=4d=4, N=1N=1 supersymmetry and d=4d=4, N=1N=1 superspace in physics, with emphasis on the partial C∞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 D^\widehat{D}-chiral maps Ο†^\widehat{\varphi} from a d=4d=4 N=1N=1 Azumaya/matrix superspace with a fundamental module with a SUSY-rep compatible hybrid connection βˆ‡^\widehat{\nabla} to a complex manifold YY 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 (βˆ‡^,Ο†^)(\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

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    From the work of Lian, Liu, and Yau on "Mirror Principle", in the explicit computation of the Euler data Q={Q0,Q1,...}Q=\{Q_0, Q_1, ... \} for an equivariant concavex bundle E{\cal E} over a toric manifold, there are two places the structure of the bundle comes into play: (1) the multiplicative characteric class Q0Q_0 of VV one starts with, and (2) the splitting type of E{\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 E{\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

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    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

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    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

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    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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