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JT gravity and the ensembles of random matrix theory JT重力与随机矩阵理论的系综
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-07-07 DOI: 10.4310/atmp.2020.v24.n6.a4
D. Stanford, E. Witten
We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.
我们将最近发现的JT引力与双尺度随机矩阵理论之间的关系推广到边界理论可能具有时间反转对称性和可能具有或不具有超对称性的费米子的情况。JT重力与矩阵系综之间的匹配取决于假设的对称性。边界理论中的时间反转对称性意味着在体中必须考虑不可定向的时空。在这种情况下,JT引力的配分函数仍然与共形结构模空间的体积有关,但这个体积有量子校正,必须使用Reidemeister-Ray-Singer“扭转”来计算。边界理论中费米子的存在(因此具有对称性$(-1)^F$)意味着体具有自旋或销子结构。边界的超对称性意味着体理论与JT超引力有关,并且与超黎曼曲面而非普通黎曼曲面的模空间体积有关。在所有情况下,我们将JT重力或超重力与适当的随机矩阵系综相匹配。所有十个标准随机矩阵合奏都出现了——三个戴森合奏和七个Altland-Zirnbauer合奏。为了便于分析,我们扩展到在厄米矩阵的原始Wigner-Dyson系综中最熟悉的其他系综技术。我们还将Mirzakhani关于普通模空间体积的递推推广到超黎曼曲面的情况。
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引用次数: 222
Heat kernel for the quantum Rabi model 量子拉比模型的热核
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-06-23 DOI: 10.4310/ATMP.2022.v26.n5.a8
Cid Reyes-Bustos, M. Wakayama
The quantum Rabi model (QRM) is widely recognized as a particularly important model in quantum optics. It is considered to be the simplest and most fundamental system describing quantum light-matter interaction. The objective of the paper is to give an analytical formula of the heat kernel of the Hamiltonian explicitly by infinite series of iterated integrals. The derivation of the formula is based on the direct evaluation of the Trotter-Kato product formula without the use of Feynman-Kac path integrals. More precisely, the infinite sum in the expression of the heat kernel arises from the reduction of the Trotter-Kato product formula into sums over the orbits of the action of the infinite symmetric group $mathfrak{S}_infty$ on the group $mathbb{Z}_2^{infty}$, and the iterated integrals are then considered as the orbital integral for each orbit. Here, the groups $ mathbb{Z}_2^{infty} $ and $mathfrak{S}_infty$ are the inductive limit of the families ${mathbb{Z}_2^n}_{ngeq0}$ and ${mathfrak{S}_n}_{ngeq0}$, respectively. In order to complete the reduction, an extensive study of harmonic (Fourier) analysis on the inductive family of abelian groups $mathbb{Z}_2^n, (n geq0)$ together with a graph theoretical investigation is crucial. To the best knowledge of the authors, this is the first explicit computation for obtaining a closed formula of the heat kernel for a non-trivial realistic interacting quantum system. The heat kernel of this model is further given by a two-by-two matrix valued function and is expressed as a direct sum of two respective heat kernels representing the parity ($mathbb{Z}_2$-symmetry) decomposition of the Hamiltonian by parity.
量子拉比模型(QRM)被广泛认为是量子光学中一个特别重要的模型。它被认为是描述量子光-物质相互作用的最简单和最基本的系统。本文的目的是利用无穷级数的迭代积分,给出哈密顿函数的热核的解析表达式。该公式的推导是基于对Trotter-Kato积公式的直接评估,而不使用费曼-卡茨路径积分。更准确地说,热核表达式中的无限和源于将Trotter-Kato积公式简化为无限对称群$mathfrak{S}_infty$对群$mathbb{Z}_2^{infty}$的作用的轨道和,然后将迭代积分视为每个轨道的轨道积分。这里,组$ mathbb{Z}_2^{infty} $和$mathfrak{S}_infty$分别是族${mathbb{Z}_2^n}_{ngeq0}$和族${mathfrak{S}_n}_{ngeq0}$的归纳极限。为了完成约化,对阿贝尔群$mathbb{Z}_2^n, (n geq0)$归纳族的调和(傅立叶)分析的广泛研究与图论研究是至关重要的。据作者所知,这是获得非平凡现实相互作用量子系统的热核封闭公式的第一个显式计算。该模型的热核进一步由一个2乘2的矩阵值函数给出,并表示为两个各自的热核的直接和,表示哈密顿量通过宇称分解的宇称($mathbb{Z}_2$ -对称)。
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引用次数: 6
Blowup rate control for solution of Jang’s equation and its application to Penrose inequality Jang方程解的爆破速率控制及其在Penrose不等式上的应用
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-06-20 DOI: 10.7916/d8-avnq-g588
Wenhuan Yu
We prove that the blowup term of a blowup solution of Jang's equation on an initial data set (M,g,k) near an arbitrary strictly stable MOTS $ Sigma $ is exactly $ -frac{1}{sqrt{lambda}}log tau $, where $ tau $ is the distance from $ Sigma $ and $ lambda $ is the principal eigenvalue of the MOTS stability operator of $ Sigma $. We also prove that the gradient of the solution is of order $ tau^{-1} $. Moreover, we apply these results to get a Penrose-like inequality under additional assumptions.
证明了在任意严格稳定MOTS $ Sigma $附近的初始数据集(M,g,k)上Jang方程的爆破解的爆破项恰好为$ -frac{1}{sqrt{lambda}}log tau $,其中$ tau $为到$ Sigma $的距离,$ lambda $为$ Sigma $的MOTS稳定性算子的主特征值。我们还证明了解的梯度为$ tau^{-1} $阶。此外,我们将这些结果应用于在附加假设下的类penrose不等式。
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引用次数: 0
Heterotic/$F$-theory duality and Narasimhan–Seshadri equivalence 异质性/$F$-理论二象性与Narasimhan-Seshadri等价
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-06-17 DOI: 10.4310/atmp.2021.v25.n5.a2
H. Clemens, S. Raby
Finding the $F$-theory dual of a Heterotic model with Wilson-line symmetry breaking presents the challenge of achieving the dual $mathbb{Z}_{2}$-action on the $F$-theory model in such a way that the $mathbb{Z}_{2}$-quotient is Calabi-Yau with an Enriques $mathrm{GUT}$ surface over which $SUleft(5right)_{gauge}$ symmetry is maintained. We propose a new way to approach this problem by taking advantage of a little-noticed choice in the application of Narasimhan-Seshadri equivalence between real $E_{8}$-bundles with Yang-Mills connection and their associated complex holomorphic $E_{8}^{mathbb{C}}$-bundles, namely the one given by the real outer automorphism of $E_{8}^{mathbb{C}}$ by complex conjugation. The triviality of the restriction on the compact real form $E_{8}$ allows one to introduce it into the $mathbb{Z}_{2}$-action, thereby restoring $E_{8}$- and hence $SUleft(5right)_{gauge}$ -symmetry on which the Wilson line can be wrapped.
寻找具有wilsonline对称破缺的异杂模型的$F$-理论对偶,提出了在$F$-理论模型上实现$mathbb{Z}_{2}$-作用的挑战,使得$mathbb{Z}_{2}$-商是Calabi-Yau,且在$SU左(5右)_{规}$对称是Enriques $ mathm {GUT}$曲面上。我们利用具有Yang-Mills连接的实$E_{8}$-束与其伴生的复全纯$E_{8}^{mathbb{C}}$-束之间的Narasimhan-Seshadri等价的应用中的一个不太引人注意的选择,即由$E_{8}^{mathbb{C}}$的复共轭实外自同构给出的等价,提出了一种新的方法来解决这个问题。紧实形式$E_{8}$上限制的琐碎性允许我们将其引入$mathbb{Z}_{2}$-动作中,从而恢复$E_{8}$-以及$SU左(5右)_{规范}$-对称性,在此对称性上可以包裹威尔逊线。
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引用次数: 6
Anomaly cancellation in the topological string 拓扑字符串中的异常消除
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-05-22 DOI: 10.4310/atmp.2020.v24.n7.a2
K. Costello, Si Li
We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira-Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimensions 3 and 5. In complex dimension 5, we show that it can only be coupled consistently at the quantum level to holomorphic Chern-Simons theory with gauge group SO(32). This is analogous to the Green-Schwarz mechanism for the physical type I string. This coupled system is conjectured to be a supersymmetric localization of type I string theory. In complex dimension 3, the required gauge group is SO(8).
我们描述了大N下全纯chen - simons理论与Kodaira-Spencer引力的耦合。我们解释了开闭拓扑b模型在微扰理论中所有环上的一种新的异常抵消机制。在一个循环中,这种异常抵消类似于格林-施瓦茨机制。作为应用,我们引入了复杂3维和5维的Kodaira-Spencer理论的I型版本。在复维5中,我们证明了它只能在量子水平上一致耦合到具有规范群SO的全纯chen - simons理论(32)。这类似于物理类型I弦的格林-施瓦茨机制。该耦合系统被推测为I型弦理论的超对称局域化。在复杂尺寸3中,所需的量规组为SO(8)。
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引用次数: 24
Fully extended BV-BFV description of General Relativity in three dimensions 广义相对论在三维空间中完全扩展的BV-BFV描述
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-05-22 DOI: 10.4310/ATMP.2022.v26.n3.a2
G. Canepa, M. Schiavina
We compute the extension of the BV theory for three-dimensional General Relativity to all higher-codimension strata - boundaries, corners and vertices - in the BV-BFV framework. Moreover, we show that such extension is strongly equivalent to (nondegenerate) BF theory at all codimensions.
我们计算了三维广义相对论的BV理论在BV- bfv框架下对所有高协维层(边界、角和顶点)的扩展。此外,我们证明了这种扩展在所有余维上都强等价于(非简并)BF理论。
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引用次数: 11
Remarks on intersection numbers and integrable hierarchies, I: Quasi-triviality 交点数与可积层次的注释,I:拟平凡性
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-05-20 DOI: 10.4310/atmp.2020.v24.n5.a1
B. Dubrovin, Di Yang
Explicit expression for quasi-triviality of scalar non-linear PDE is under consideration.
研究标量非线性偏微分方程的拟平凡性的显式表达式。
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引用次数: 9
Extremal isosystolic metrics with multiple bands of crossing geodesics 具有多个交叉测地线带的极端等收缩度量
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-03-28 DOI: 10.4310/atmp.2022.v26.n5.a7
Usman Naseer, B. Zwiebach
We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($ngeq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with the conformal extremal metric on the regular $2n$-gon. The hexagon was considered by Calabi. The region covered by the maximal number $n$ of geodesics bands extends over most of the surface and exhibits positive curvature. As $nto infty$ the metric, away from the boundary, approaches the well-known round extremal metric on $mathbb{RP}_2$. We extend Calabi's isosystolic variational principle to the case of regions with more than three bands of systolic geodesics. The extremal metric on $mathbb{RP}_2$ is a stationary point of this functional applied to a surface with infinite number of systolic bands.
我们应用最近开发的凸规划来寻找$2n$边多边形($ngeq 3$)上的最小面积黎曼度量,曲线上的长度条件连接对边。我们认为黎曼极限度规与正则$2n$ -gon上的共形极限度规重合。六边形是卡拉比考虑的。测地线带的最大数目$n$所覆盖的区域扩展到大部分表面并呈现正曲率。当$nto infty$度规远离边界时,接近于$mathbb{RP}_2$上众所周知的圆形极值度规。我们将Calabi的等收缩变分原理推广到具有三个以上收缩测地线带的区域。$mathbb{RP}_2$上的极值度规是该函数应用于具有无限数量收缩带的表面的一个固定点。
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引用次数: 9
Ramond–Ramond fields and twisted differential K-theory Ramond-Ramond场和扭曲微分k理论
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-03-21 DOI: 10.4310/atmp.2022.v26.n5.a2
Daniel Grady, H. Sati
We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. In addition to providing a new conceptual framework and a mathematically solid setting, this allows us to uncover interesting and novel effects. Explicitly, we use our recently constructed Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential K-theory to characterize the RR fields and their quantization, which involves interesting interplay between geometric and topological data. We illustrate this with the examples of spheres, tori, and Calabi-Yau threefolds.
我们提供了一个系统的方法来描述Ramond-Ramond (RR)场作为扭曲微分k理论中的元素。这建立在作者对扭曲微分k理论的几何和计算方面的一系列构造的基础上,这些构造在很大程度上最初是由这个问题引起的。除了提供一个新的概念框架和数学上可靠的设置,这使我们能够发现有趣和新颖的效果。明确地,我们使用我们最近构造的扭曲微分k理论的Atiyah-Hirzebruch谱序列(AHSS)来表征RR场及其量化,这涉及到几何和拓扑数据之间有趣的相互作用。我们用球体、环面和Calabi-Yau三倍的例子来说明这一点。
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引用次数: 16
Asymptotically flat extensions with charge 带电荷的渐近平面扩展
IF 1.5 4区 物理与天体物理 Q1 Mathematics Pub Date : 2019-03-21 DOI: 10.4310/atmp.2019.v23.n8.a1
Aghil Alaee, Armando J. Cabrera Pacheco, Carla Cederbaum
The Bartnik mass is a notion of quasi-local mass which is remarkably difficult to compute. Mantoulidis and Schoen [2016] developed a novel technique to construct asymptotically flat extensions of minimal Bartnik data in such a way that the ADM mass of these extensions is well-controlled, and thus, they were able to compute the Bartnik mass for minimal spheres satisfying a stability condition. In this work, we develop extensions and gluing tools, a la Mantoulidis and Schoen, for time-symmetric initial data sets for the Einstein-Maxwell equations that allow us to compute the value of an ad-hoc notion of charged Barnik mass for suitable charged minimal Bartnik data.
巴特尼克质量是准局部质量的一个概念,它非常难以计算。Mantoulidis和Schoen[2016]开发了一种新技术,以这种方式构造最小Bartnik数据的渐近平坦扩展,使这些扩展的ADM质量得到很好的控制,因此,他们能够计算满足稳定条件的最小球体的Bartnik质量。在这项工作中,我们为爱因斯坦-麦克斯韦方程的时间对称初始数据集开发了扩展和粘接工具,使我们能够为合适的带电最小巴特尼克数据计算带电巴尼克质量的特别概念的值。
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引用次数: 3
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Advances in Theoretical and Mathematical Physics
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