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Morse Theory without Non-Degeneracy 非简并的莫尔斯理论
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmath/haaa064
Frances Kirwan;Geoffrey Penington
We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any non-degeneracy assumptions except that the critical locus must have only finitely many connected components.
我们将莫尔斯理论推广到紧黎曼流形上的光滑函数,除了临界轨迹必须只有有限多个连通分量外,没有任何非简并性假设。
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引用次数: 5
On the Voevodsky Motive of the Moduli Stack of Vector Bundles on a Curve 曲线上向量束模堆的Voevodsky动机
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmathj/haaa023
Victoria Hoskins;Simon Pepin Lehalleur
We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky’s category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localizing tensor subcategory generated by the motive of the curve, using Białynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, inspired by the work of Atiyah and Bott on the topology of the classifying space of the gauge group, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.
在Voevodsky的动机范畴中,我们定义并研究了光滑投影曲线上固定秩和阶的向量束的模堆栈的动机。我们证明了这个动机可以写成曲线上线束和的扭商的光滑投影商格式的动机的一个同伦群。当使用有理系数时,我们使用这些Quot方案的Białynicki-Birula分解,证明了丛堆栈的动机位于由曲线的动机生成的局部化张量子类别中。受Atiyah和Bott关于规范群分类空间拓扑的工作的启发,我们猜想了这个堆栈的动机的一个公式,并在Quot格式的交集理论上证明了这个猜想的模a猜想。
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引用次数: 15
Instantons and Bows for the Classical Groups 古典团体的瞬间和弓
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmath/haaa034
Sergey A Cherkis;Jacques Hurtubise
The construction of Atiyah, Drinfeld, Hitchin and Manin provided complete description of all instantons on Euclidean four-space. It was extended by Kronheimer and Nakajima to instantons on ALE spaces, resolutions of orbifolds $mathbb{R}^4/Gamma$ by a finite subgroup Γ⊂SU(2). We consider a similar classification, in the holomorphic context, of instantons on some of the next spaces in the hierarchy, the ALF multi-Taub-NUT manifolds, showing how they tie in to the bow solutions to Nahm’s equations via the Nahm correspondence. Recently Nakajima and Takayama constructed the Coulomb branch of the moduli space of vacua of a quiver gauge theory, tying them to the same space of bow solutions. One can view our construction as describing the same manifold as the Higgs branch of the mirror gauge theory as described by Cherkis, O’Hara and Saemann. Our construction also yields the monad construction of holomorphic instanton bundles on the multi-Taub-NUT space for any classical compact Lie structure group.
Atiyah、Drinfeld、Hitchin和Manin的构造提供了欧几里得四空间上所有实例的完整描述。Kronheimer和Nakajima将其扩展到ALE空间上的实例,由有限子群Γ∧SU(2)分解出的轨道$mathbb{R}^4/Gamma$。在全纯的背景下,我们考虑了在层次结构的下一个空间上的实例的一个类似的分类,即ALF多taub - nut流形,显示了它们如何通过Nahm对应与Nahm方程的bow解联系在一起。最近Nakajima和Takayama构造了颤振规范理论真空模空间的库仑分支,将它们与弓解的相同空间联系起来。人们可以把我们的构造看作是描述了与Cherkis、O 'Hara和Saemann描述的镜像规范理论的希格斯分支相同的流形。我们的构造也得到了在多taub - nut空间上对于任何经典紧李结构群的全纯瞬子束的单构造。
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引用次数: 1
Wick Rotation and the Positivity of Energy in Quantum Field Theory 灯芯旋转与量子场论中能量的正性
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmath/haab027
Maxim Kontsevich;Graeme Segal
We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian metrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.
通过假设配分函数和相关器解析扩展到复值度量的某一域,提出了弯曲时空背景下的酉量子场论的一个新的公理体系。普通黎曼度量包含在允许域内,而洛伦兹度量位于允许域的边界上。
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引用次数: 67
Coherent Systems on the Projective Line 投影线上的相干系统
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmathj/haaa024
Peter Newstead;Montserrat Teixidor i Bigas
It is well known that there are no stable bundles of rank greater than 1 on the projective line. In this paper, our main purpose is to study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank. We include a review of known results, mostly for a small number of sections.
众所周知,在射影线上不存在秩大于1的稳定丛。本文的主要目的是研究当区间数大于秩时,射影线上稳定相干系统的存在性问题。我们包括对已知结果的审查,主要针对少数章节。
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引用次数: 1
An Application of Wall-Crossing to Noether–Lefschetz Loci Wall-Crossing在Noether-Lefschetz基因座中的应用
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmathj/haaa022
S Feyzbakhsh;R P Thomas;C Voisin
Consider a smooth projective 3-fold $X$ satisfying the Bogomolov–Gieseker conjecture of Bayer–Macrì–Toda (such as ${mathbb{P}}^3$, the quintic 3-fold or an abelian 3-fold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class, then we show it has very negative square.
考虑满足Bogomolov-Gieseker猜想Bayer-Macrì-Toda的光滑投影3-fold $X$(例如${mathbb{P}}^3$,五次3-fold或阿贝尔3-fold)。设$L$是支撑在$X$中一个非常正的曲面上的线束。如果$c_1(L)$是一个原始上同类,那么我们证明它有一个非常负的平方。
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引用次数: 8
Anyon Networks from Geometric Models of Matter 物质几何模型中的任意网络
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmath/haab004
Michael Atiyah;Matilde Marcolli
This paper, completed in its present form by the second author after the first author passed away in 2019, describes an intended continuation of the previous joint work on anyons in geometric models of matter. This part outlines a construction of anyon tensor networks based on four-dimensional orbifold geometries and braid representations associated with surface-braids defined by multisections of the orbifold normal bundle of the surface of orbifold points.
这篇论文是在第一作者于2019年去世后由第二作者以目前的形式完成的,它描述了之前关于物质几何模型中任意子的联合研究的预期延续。这一部分概述了基于四维轨道几何和与由轨道点表面的轨道法线束的多截面定义的表面编织相关的编织表示的任意张量网络的构造。
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引用次数: 0
Volumes And Random Matrices 体积和随机矩阵
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmath/haaa035
Edward Witten
This article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)
本文介绍了新发现的黎曼曲面或超黎曼曲面的模空间体积、二维引力或超引力的简单模型以及随机矩阵系综之间的关系。(本文根据奥克兰大学2020年1月在规范论和弦理论数学会议上的演讲改编)
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引用次数: 12
Total Positivity in Springer Fibres 施普林格纤维的总正性
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmathj/haaa021
G Lusztig
Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.
设u是复还原基的全正部分中的一个单势元素。我们考虑u处的斯普林格纤维与旗流形的完全正部分的交点。我们证明了这个交集有一个自然的单元分解,它是全正标志流形的单元分解(Rietsch)的一部分。
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引用次数: 6
Transverse hilbert schemes, bi-hamiltonian systems and hyperkähler geometry 横向希尔伯特格式,双哈密顿系统和hyperkähler几何
IF 0.7 4区 数学 Q3 MATHEMATICS Pub Date : 2020-12-01 DOI: 10.1093/qmath/haaa059
Roger Bielawski
We give a characterization of Atiyah’s and Hitchin’s transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperkähler manifolds arising from the transverse Hilbert scheme construction, with particular attention paid to the monopole moduli spaces.
我们用双Poisson结构给出了辛表面上点的Atiyah和Hitchin横向Hilbert格式的一个特征。此外,我们描述了由横向Hilbert格式构造引起的超kähler流形的几何,特别注意单极模空间。
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引用次数: 1
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