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- 1 -Finsler 几何统一场与信息物理学叶 鹰浙江大学信息资源管理研究所(310028)yye@摘 要： 将Finsler 几何的数学形式与物理意义相结合，确立了Finsler 几何统一场与信息物理学的数学表示。
用Finsler 几何中的Hilbert 形式作为统一场势A，以Chern 联络的曲率形式作为统一场强F，提出用TrF∧*F 代表的偶数维作用项和Chern-Simons 形式代表的奇数维作用项共同构成统一作用量，将序间、空间、时间对应的Cartan 张量作为真实物理状态，从而获得Finsler 几何中的信息物理统一场。
关键词： Finsler 几何；Finsler 空间；几何场论；统一场论；信息物理学1. 引 言1985年Asanov 曾对Finsler 几何中的相对论、宇宙论进行过探讨[1]，近年的研究揭示了Finsler 空间对物理学的重要意义[2-3]，参考主流数学物理研究的几何统一论、超弦理论和M 理论[4-6]，在笔者前期工作[7]和曹盛林教授研究成果[8-10]基础上赋予Finsler 几何以实在的信息物理意义，结果具有统一场论价值。
数学方法取自[11-12]。
2. Finsler 几何场Finsler 几何是Riemann 几何的自然拓广，或者说是没有二次限制的Riemann 几何，其中去掉齐性条件的Finsler 几何也称Lagrange 几何。
Finsler 几何相应于Riemann 几何的有关参量如下：对于纤维丛(E,p,B,F,G)(E 为丛空间，B 为底空间，p:E→B 是满连续映射，F 即纤维，G 是有效作用于F 上的变换群)，相应于切丛TM 上的Riemann 度规：2/121)(j i ij dx dx g d =τ (1)射影化切丛PTM 或球丛SM 上的Finsler 度规为：),...;,...(11n n dx dx x x f d =τ (2)其中f 是Finsler 函数。
高斯-博内-陈定理的历史发展及其意义陈惠勇【摘要】高斯-博内定理是大范围微分几何学的一个经典定理,它建立了黎曼流形的局部性质和整体性质之间的联系,因而被认为是曲面微分几何学中最深刻的定理.通过考察高斯-博内-陈定理的历史发展,指出高斯-博内-陈定理在黎曼流形、微分流形以及拓扑流形上的表现形式,以此阐明高斯-博内-陈定理与现代数学的深刻联系及其意义.%Gauss-Bonnet theorem, which established a link between the local nature of Riemannian manifold and the overall's, is a classic theorem of differential geometry in the large, and considered to be the most profound theorem in differential geometry.The historical development of the Gauss-Bonnet-Chem theorem are examined, the manifestations of the Gauss-Bonnet-Chem theorem in Riemannian manifolds, differential manifolds, and topology manifolds are discussed, and the deep relations and meaning between the Gauss-Bonnet-Chem theorem and the modem mathematics are clarified as well.【期刊名称】《江西师范大学学报（自然科学版）》【年(卷),期】2011(035)001【总页数】5页(P106-110)【关键词】高斯;高斯-博内-陈定理;整体微分几何【作者】陈惠勇【作者单位】江西师范大学数学与信息科学学院,江西,南昌,330022【正文语种】中文【中图分类】O186.1高斯-博内定理是大范围微分几何学的一个经典定理, 它建立了黎曼流形的局部性质和整体性质之间的联系, 因而被认为是曲面微分几何学中最深刻的定理. 本文考察高斯-博内定理的历史发展, 指出高斯-博内-陈定理在黎曼流形、微分流形以及拓扑流形上的表现形式, 以此阐明高斯-博内-陈定理与现代数学的深刻联系及其意义, 并以此来表达对高斯的敬意, 缅怀陈省身先生在这一历史发展过程中所做出的杰出贡献.经典的高斯-博内定理指出: 设M是 2维定向Riemann流形,D为M上一个由有限条光滑曲线组成的, 并具有边界∂D的单连通区域, 那么这里χ(D)是D的Euler示性数,kg为每一条光滑边界曲线的测地曲率,jα为边界的顶点处的内角,K为曲面M的高斯曲率, 而 2-形式dA是曲面M的面积元素.因此, 左边的各项分别是各顶点的外角和、测地曲率的积分和高斯曲率的积分. 它们分别是点、线、面的曲率, 故此公式是以拓扑不变量Euler示性数来表示全曲率(如图1).测地三角形的情形是高斯在其 1827年的历史性论文《关于曲面的一般研究》中证明的. 而如上所述的定理则归功于Pierre Ossian Bonnet和Jacques Binet于1848年完成的工作. 然而, Binet从未发表过他的证明[1].通常所称的紧曲面的Gauss-Bonnet定理应当归功于Walther von Dyck. 1888年冯·迪克证明了如下结果: 如果M是R3的一个紧的定向曲面, 那么这里χ(M)是曲面M的欧拉示性数.通过组合方法, 即将曲面三角剖分并在每一个2-单形上应用经典的高斯-博内公式, 可以推出上面的高斯-博内定理. Daniel Henry Gottlieb认为冯·迪克可能是第一个意识到高斯-博内定理对非球面的曲面也成立的数学家[2]. 而Hirsch[3]则认为, 冯·迪克是第一个将映射度与 Euler-Poincare示性数这两个概念联系在一起, 证明了被误为高斯-博内定理的定理. 这里, 冯·迪克实际上已经得到了高斯-博内定理的整体结果. 当然, 也有证据表明, Descartes似乎已经得到凸多面体情形的整体高斯-博内定理.因此, 整体高斯-博内定理或许应该称为 Descartes-Dyck定理.2维紧致流形上的高斯-博内公式是经典微分几何的一个高峰. 霍普夫曾经指出:“推广此公式到高维紧致流形上去是几何学中极其重要而困难的问题.”高斯-博内定理在高维的推广开始于霍普夫, 他得到了下列结果：设M是Rm+1中一个紧的可定向的超曲面, 并设G:M→Sm是从M到m维单位球面Sm上的高斯映射. 如果Sm的体积形式用ωm表示, 那么对于曲面的情形(m=2的情形), 拉回映射G*ωm代表KdA,因为当m=2时,对于向量场, Poincare-Hopf指标定理表明: 如果M是Rn+1中的一个区域R的边界, 那么这里“degG”表示G的拓扑度(=mS中的点p的原像集G-1(p)中的原像的代数数).设m=2n. 从M=∂ 知道,χ(M)= 2χ(R),所以在n=1的情形(M是3R中的曲面)下,这给出了3R中曲面的 Gauss-Bonnet定理的一个几何证明.可以看出, Hopf的证明所用的方法是“外蕴”的, 即仍然是假定M可以等距地嵌入于欧氏空间Rn+1中. 而且, 这种外蕴的方法经过 C.B. Allendoerfer[4]、W. Fenchel[5]以及 C.B.Allendoerfer和 A.Weil[6]的工作仍然没有突破. 然而众所周知, 高斯-博内定理本身只涉及黎曼流形M的内蕴不变量, 为什么要将M嵌入欧氏空间才能证明呢？这就是数学家们一直耿耿于怀而又一直没有突破的地方. 这需要全新的思想和方法, 而这种全新的思想和方法在理论上的突破和创造, 需要伟人的出现, 他将站在巨人的肩膀上, 因而他必然地比任何其他的人看得更高更远。
James Harris "Jim" Simons，詹姆斯·哈里斯·西蒙斯，1938年出生，是美国著名数学家、学者、投资家和慈善家。
1982年，西蒙斯创建了文艺复兴科技公司（Renaissance Technologies Corporation），这是一家私有的位于纽约的投资公司，目前管理着150亿美元的资产，西蒙斯作为该公司的CEO掌管大局，目前该公司已经成为世界上最成功的对冲基金之一。
西蒙斯的个人身价大约是85亿美元。
西蒙斯与他的妻子生活在麦哈顿和长岛，并是3个孩子的父亲。
西蒙斯总是有意回避公众并极少接受采访。
2009年10月10日，西蒙斯宣布他将于2010年1月1日退休，但保留文艺复兴科技公司荣誉主席职位。
早期人生和事业西蒙斯是一个马萨诸塞州鞋厂老板的儿子,他于1958年毕业于麻省理工学院数学系,在1961年获得加州理工伯克利大学的数学博士学位,西蒙斯在23岁的时候就获得博士学位绝对是神童之一!1964年至1968年期间,西蒙斯是美国国防研究院的研究人员之一，他同时也在麻省理工学院和哈佛大学教授数学。
1968年，他就被Stony Brook University授予数学学院院长的职位，仅仅30岁。
1976年，西蒙斯赢得了美国数学协会的Oswald Veblen 几何学奖，用来表彰他在多位平面面积最小化研究的成果，这个成果证明了伯恩斯坦猜想中N维的第8维，同时也成为了佛拉明的高原问题猜想的有力证据。
西蒙斯最著名的研究成果是发现并实践了几何学的测量问题，这个研究成果被命名为陈氏-西蒙斯定理（这是一个与我国著名数学家陈省身共同研究的成果）。
1978年，西蒙斯离开了学术界而创建了一家投资基金，主要投资于商品期货和其他金融工具。
文艺复兴科技公司在过去的20年中，西蒙斯的文艺复兴科技公司在全球市场中进行投资。
他们开发了许多数学模型用来进行分析和交易，这些基本上是自动化完成。
10位科学家新鲜热评2016年诺奖物理学奖：物质的拓扑相变和拓扑相2016年诺贝尔奖物理学奖授予三位科学家：戴维·索利斯，邓肯·霍尔丹和迈克尔·科斯特利茨，以表彰他们在理论上发现了物质的拓扑相变和拓扑相。
二维物理体系中的拓扑相变和拓扑量子物态，是三位得奖者能做出这一成就的关键，它解释了某种薄层物质的导电率会以整数倍发生变化。
施郁我的预测就差一点施郁，复旦大学物理学系教授，研究方向：量子纠缠及其在凝聚态物理和粒子物理中的运用。
这三位获奖者实际上是凝聚态里面拓扑物相的开创者。
Thouless和Kosterlitz首先研究了在相变当中的拓扑相变，拓扑绝缘体的前期的方向。
Thouless与合作者指出量子霍尔电导是拓扑的，是陈省身数。
Haldane研究了一维磁体的拓扑态，以及一个理论模型，它给出后来提出的拓扑绝缘体的一部分物理。
2007年我在一篇文章里提到过Thouless和Kosterlitz得奖，但是很可惜，今年预测的时候我只猜到了颁奖方向，但是头脑没有转弯，追溯前期工作，这次就选择了比较热门的具体的拓扑绝缘体里的几个人。
曹则贤这是实至名归毫无争议曹则贤，中国科学院物理所研究员。
此三位物理学家获得本年度的诺贝尔物理奖应该说是实至名归，这一决定应该说不会有什么争议。
对于Haldane的工作我不是很了解，但是Kosterlitz和Thouless 的名字读过一些凝聚态理论的研究生可能都是知道，见于Kosterlitz-Thouless 相变这个概念。
1973年，Kosterlitz与Thouless的关于2维XY模型相变问题的合作研究，发现了自高温无序相向低温准有序相的无穷阶相变，后来被命名为Kosterlitz-Thouless 相变。
(Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability andphase-transitions in 2 dimensional systems. J. Phys. C 6, 1181–-1203 (1973)。
a r X i v :n u c l -t h /0610029v 3 3 D e c 2006Effective Field Theory Calculation of nd Radiative Capture atThermal EnergiesH.Sadeghi a ,∗S.Bayegan b ,†and Harald W.Grießhammer c ‡aDepartment of Physics,University of Arak,P.O.Box 38156-879,Arak,Iran.b Department of Physics,University of Tehran,P.O.Box 14395-547,Tehran,Iran.c Center for Nuclear Studies,Department of Physics,The George Washington University,Washington DC 20052,USA.Abstract The cross section for the thermal neutron capture by the deuteron is calculated with pionless Effective Field Theory(EFT).No new Three-Nucleon forces are needed up to next-to-next-to-leading order in order to achieve cut-offindependent results,besides those fixed by the triton binding energy and Nd scattering length in the triton channel.The cross-section is accurately determined to be σtot =[0.503±0.003]mb .At zero energies,the magnetic M 1-transition gives the dominant contribution and is calculated up to next-to-next-to-leading order (N 2LO).Close agreement between the available experimental data and the calculated cross section is reached.We demonstrate convergence and cutoffindependence order by order in the low-energy expansion.PACS numbers:27.10.+h,21.45.+V,25.40.Lw,11.80.Jy,21.30.-Fe keywords:effective field theory,three-body system,three-body force,Fad-deev equation,radiative captureI.INTRODUCTIONThe study of the three-body nuclear system involving neutron radiative capture by deuteron has been investigated in theoretical and experimental works over the past years. The experimental result of this process has most accurately been measured by Jurney, et.al.[1].The value of0.508±0.015(mb)for the cross section was resulted for2200m/sec neutrons.Rapid progress has been made in the theoretical study of the Nd→3Hγreaction such as the p-d and n-d radiative capture.At such energies a magnetic dipole(M1)transition is almost entirely participated.These reactions were studied in plane wave(Born)approxi-mation by Friar et al.[2].In these investigations the authors employed their configuration-space Faddeev calculations of the helium wave function,with inclusion of three-body forces and pion exchange currents.More recently a rather detailed investigation of such pro-cesses has been performed by Viviani et al.[3,19].In their calculations the quite accurate three-nucleon bound-and continuum states were obtained in the variational pair-correlated hyperspherical method from a realistic Hamiltonian model with two-nucleon and three-nucleon interactions.They obtained in Ref.[3]the cross section from Argonne v14two-nucleon and Urbana VIII three-nucleon interactions(AV14/UVIII),also from Argonne v18two-nucleon and Ur-bana IX three-nucleon interactions(AV18/UIX)and including∆admixtures.Cross section values were found0.600(mb)and0.578(mb)which overestimate the experimental value by 18%and14%value,respectively,see table2.It should be noted,however,that the explicit, non-perturbative inclusion of∆-isobar degrees of freedom in the nuclear wave function are found to be in significantly better agreement with experiment than those obtained from per-turbative(∆P T)estimates.This shows that these results for this very-low energy observable are sensitive to details of the short-range part of the interaction.recent calculation using manifestly gauge-invariant currents reduced the spread[19],but the result including three-body currents,0.558mb,still over-predicts the cross-section by10%.Model-dependent currents associated with the∆(1232)were identified as source of the discrepancy.Thus, the question remains how such details of short-range Physics can so severely influence a very-long-range reaction with maximal energies of less than10MeV.During the last few years,nuclear Effective Field Theory(EFT)has been applied to two-, three-,and four-nucleon systems,see e.g.[4,5,6,7,8,9,10].The pionless Effective Field Theory would be an ideal tool to calculate low-energy cross sections in a model-independent way and to possibly reduce the theoretical errors by a systematic,model-independent cal-culation with an a-priori estimate of the theoretical uncertainties.An example of a precise calculation is the reaction np→γd,which is relevant to big-bang nucleosynthesis(BBN). The cross section for this process was computed to1%error for center of mass energies E 1Mev[11,12,13].We have suggested a method for computation of neutron-deuteron radiative capture for extremely low energy(20≤E≤200Kev)with pionless EFT[15],where with this formalism,we can estimate errors in a perturbative expansion up to N2LO within a few percent of the ENDF values[16].The purpose of the present paper is to study the cross section for radiative capture of2neutrons by deuterons nd→γ3H at zero energies with pionless EFT.At these energies,the magnetic M1-transition gives the dominant contribution.The M1amplitude is calculated up to next-to-next-to-leading order(N2LO)with insertion of three body force.Results show less than1%deviation from the available experimental data at zero energy(0.0253eV).This article is organized as follows.In the next section,a brief description of the formal-ism and its input for total cross section of the neutron-deuteron radiative capture will be presented.We discuss the theoretical errors,tabulation of the calculated cross section in comparison with the other theoretical approaches and the most recent data[1]in section III. Finally,Summary and conclusions follow in Section IV.II.NEUTRON-DEUTERON SCATTERING IN TRITON CHANNEL AND RA-DIATIVE CAPTUREThe2S1Λ2∞n=0H2n(Λ) ME+γ2tΛ2+2H2(Λ)FIG.1:The Faddeev equation for Nd-scattering.Thick solid line is propagator of the two inter-mediate auxiliaryfields D s and D t,denoted by D;K:propagator of the exchanged nucleon;H: three-body force.t s(p,k)=12πΛd q q2[D s(q)[K(p,q)+2H(E,Λ)]t s(q)+D t(q)[3K(p,q)+2H(E,Λ)]t t(q)]t t(p,k)=12πΛd q q2[D t(q)[K(p,q)+2H(E,Λ)]t t(q)+D s(q)[3K(p,q)+2H(E,Λ)]t s(q)],(2) where D s,t(q)=D s,t(E−q2M12+or J P=32+(amplitude g1)and J P=3(amplitude g3)we can write:g1:t†(i D· e∗× k+ σ× D· e∗× k)N,g3:t†(i D· e∗× k+ σ× D· e∗× k)N.(4)The contribution of the electric transition E LSJifor energies of less than60KeV to thetotal cross section is very small.Therefore,the electric quadrupole transition E0(3/2)(3/2)2from the initial quartet state will not be considered at thermal energies.The M1amplitudereceives contributions from the magnetic moments of the nucleon and dibaryon operatorscoupling to the magneticfield,which are described by the Lagrange densityL B=e MN√9α4M2N iLSJ[| χLSJ i|2],(6)whereχLSJ i=√pµN√nucleons between them when the photon kernel is expanded to the same order.This process will be done separately for NLO and N2LO.Finally the wave function renormalization in each order will be done.The diagrams in Fig.2represent contributions of electromagnetic interaction with nu-cleon,deuteron,four-nucleon-magnetic-photon operator described by a coupling between the3S1-dibaryon and1S0-dibaryon and a magnetic photon.As mentioned in the intro-duction,in another paper[15],we have presented detailed schematic of these diagrams in neutron-deuteron radiative capture for(20≤E≤200keV)up to N2LO.The last diagrams in Fig.2with insertion of a photon to the N2LO three-nucleon force H2vertex is not M1and we know that M1contribution is the dominant contribution at very low energy and especially for zero energy.Its contribution should therefore be very tiny.Because the leading three-nucleon force H0has no derivatives,it is not affected by the minimal substitution p→p−eA.But the parameter H2is the strength of the three-nucleon interaction with two derivatives.Naturally for the energy range near zero momentum, insertion of photon to H2vertices for momentum p∼0.025eV and M1transition,could be neglected.H2is necessary in neutron-deuteron scattering to improve cut-offindependence but is defined such that it does not contribute at zero momentum.Contributions of a photon coupling to H2are however indeed negligible at zero energy.III.NEUTRON-DEUTERON RADIATIVE CAPTURE RESULTS AT ZERO EN-ERGYWe numerically solved the Faddeev integral equation up to N2LO.We used c= 197.327MeV fm,a nucleon mass of M=938.918MeV,for the NN triplet channel a deuteron binding energy(momentum)of B=2.225MeV(γd=45.7066MeV),a residue of Z d=1.690(3),for the NN singlet channel an1S0scattering length of a s=−23.714fm, L1∼−4.5fm byfixing at its leading non-vanishing order by the thermal cross section.As in Ref.[20],we can determine which three-body forces are required at any given order, and how they depend on the cutoff.Low-energy observables must be insensitive to the cut-off,namely to any details of short-distance physics in the region above the break-down scale of the pion-less EFT,set approx-imately by the pion-mass.It was found in Ref.[20]that no additional three-nucleon forces are necessary to render a renormalisable amplitude at N2LO in this process,besides those needed already in nucleon-deuteron scattering:H0and H2.At N2LO,where we saw that H2is required,we checked this by varying the cut-offbetween150and500MeV.This is a reasonable estimate of the errors of our calculation due to higher-order effects.As seen in Fig.3,in the thermal energy range the cutoffvariation is very small and decreases steadily as we increase the order of the calculation and it is of the order of(k/Λ)n,(γ/Λ)n,where n is the order of the calculation andΛ=150MeV is the smallest cutoffused(see Table I and Fig.3).Also,errors due to cutoffvariation is decreasing when the order of calculation is increased up to N2LO.We determined the two-nucleon parameters from the deuteron binding energy,triplet effective range(defined by an expansion around the deuteron pole,not at zero momentum), the singlet scattering length,effective range(defined by expanding at zero momentum),and6FIG.3:Curve of the cutoffvariation of cross section up to N2LO is shown betweenΛ=150MeV andΛ=500MeV.The short dashed,long dashed and solid line correspond to LO,NLO and N2LO,respectively.TABLE I:Results for the cutoffvariation of cross section up to N2LO is shown betweenΛ=150 MeV andΛ=500MeV.NLO0.000050.000400.000600.000840.00131two body capture process(obtained with comparison between experimental data and theo-retical results for np→dγprocess at zero energy[12]).Wefix the three-body parameters as follows:because we defined H2such that it does not contribute at zero momentum scatter-ing,one canfirst determine H0from the2S1FIG.4:The cross section for neutron radiative capture by deuteron as function of the center-of-mass kinetic energy E in MeV.The short dashed,long dashed and solid line correspond to the contribution of M1capture cross section up to LO,NLO and N2LO,respectively.Single point shows experimental results for this cross section at0.025eV[1].TABLE II:Comparison between different theoretical results for Neutron radiative capture by deuteron at zero energy(0.0253ev).Last row shows our EFT result.The last line quotes deviation between data[1]and theory,if it is larger than the theoretical or experimental uncertainty.Theoryσ(mb)AV18/IX(IA+MI+MD)[3]0.48929%AV18/IX(IA+MI+MD+∆P T)[3]0.63118%AV18/IX(IA+MI+MD+∆)[3]0.5783%AV18/IX(gauge inv.+3N-current)[19]0.5565%EFT(NLO)0.496Experiment[1]0.508±0.015at zero energy(0.0253ev).The calculations by Viviani et al.[3,19]shows sensitivity to8short-range physics namely to details of including the physics of the Delta and pion-exchange currents.The calculation of Ref.[19]with manifestly gauge-invariant current operators is quite sensitive to including meson-exchange three-nucleon currents.One might therefore have been tempted to conclude that a new three-nucleon force is also needed in the pion-less EFT.As shown above,this is not the case:There are no new three-nucleon forces besides those alreadyfixed in nd scattering at the same order.The contribution from the photon coupling to a three-nucleon force is negligible in our calculation.As our result is model-independent and universal,any model with the same input must–within the ac-curacy of our calculation–lead to the same result.Our inputs are thefirst two terms of the effective-range expansion in the singlet-and triplet-S wave of NN scattering,the proton and neutron magnetic moments,the triton binding energy and nd scattering length in the doublet-S-wave,andfinally the thermal cross section of the reaction np→dγ(determining L1).More work is needed to understand why the potential-model calculations[3,19]have the same input but do not seem to reproduce the same result.Addressing convergence of the EFT calculation,we notice that the contributions which are characterized as higher-order in the power-counting are indeed small:The LO result is 0.485mb,with NLO adding0.011mb,and N2LO another0.007mb.Cut-offdependence is negligible.The typical size of the expansion parameter in the pion-less EFT is about γt/mπ≈1/3.We therefore estimate the uncertainty from leaving out corrections at N3LO and higher as about1/3of the N2LO correction or0.003mb.IV.CONCLUSIONThe cross section for radiative capture of neutrons by deuterons nd→γ3H at zero energies with was calculated pionless Effective Field Theory,the unique,model independent and systematic low-energy version of QCD for processes involving momenta below the pion mass.We applied pionless EFT tofind numerical results for the M1contributions.Incident thermal neutron energies have been considered for this capture process.At these energy our calculation is dominated by only S-wave state and magnetic transition M1contribution. The M1amplitude is calculated up to Next-to-Next to leading order N2LO.Three-Nucleon forces are needed up to N2LO order for cut-offindependent results.The triton binding energy and nd scattering length in the triton channel have been used tofix them.Hence the cross-section is in total determined asσtot=[0.485(LO)+0.011(NLO)+0.007(N2LO)]= [0.503±0.003]mb.It converges order by order in low energy expansion.It is also cut-offindependent at this order.We notice that our calculation has a systematic uncertainty from higher-order terms which is now smaller than the experimental error-bar.9AcknowledgmentWe thank L.Marcucci and A.Kievsky for enlightening discussions which spurred this work.[1] E.T.Jurney,P.J.Bendt and J.C.Browne,Phys.Rev.C25,2810(1982).[2]J.L.Friar,B.F.Gibson and G.L.Payne,Phys.Lett.B251,11(1990).[3]M.Viviani,R.Schiavilla and A.Kievsky,Phys.Rev.C54,534(1996).[4] D.B.Kaplan,M.J.Savage and M.B.Wise,Nucl.Phys.B534,329(1998).[5]S.R.Beane and M.J.Savage,Nucl.Phys.A694,511(2001).[6]P.F.Bedaque,H.-W.Hammer and U.van Kolck,Phys.Rev.Lett.82,463(1999);Nucl.Phys.A646,444(1999).[7]P.F.Bedaque,H.-W.Hammer and U.van Kolck,Nucl.Phys.A676,357(2000).[8]H.-W.Hammer and T.Mehen,Phys.Lett.B516,353(2001).[9]P.F.Bedaque and H.W.Grießhammer,Nucl.Phys.A671,357(2000).[10] F.Gabbiani,P.F.Bedaque and H.W.Grießhammer,Nucl.Phys.A675,601(2000).[11]J.-W.Chen,G.Rupak,M.J.Savage,Phys.Lett.B464,1(1999).[12]G.Rupak,Nucl.Phys.A678,405(2000).[13]S.Ando,R.H.Cyburt,S.W.Hong,C.H.Hyun,Phys.Rev.C74,025809(2006).[14]P.F.Bedaque,G.Rupak,H.W.Grießhammer and H.-W.Hammer,Nucl.Phys.A714,589(2003).[15]H.Sadeghi and S.Bayegan,Nucl.Phys.A753,291(2005).[16]ENDF/B online database at the NNDC Online Data Service,.[17]W.Dilg,L.Koester and W.Nistler,Phys.Lett.B36,208(1971).[18]H.W.Grießhammer,Nucl.Phys.A744,192(2004).[19]L.E.Marcucci,M.Viviani,R.Schiavilla,A.Kievsky and S.Rosati,Phys.Rev.C72(2005),014001[20]H.W.Grießhammer,Nucl.Phys.A760(2005),11010。
轴⼦——粒⼦物理和宇宙学的新前沿中国科学院⾼能物理研究所；；2. 暨南⼤学1. 中国科学院⾼能物理研究所⼀引⾔粒⼦物理、宇宙学和天⽂学的深度结合催⽣了当下粒⼦宇宙学研究的⾼速发展。
继2017 年引⼒波之后，2019 的物理学诺贝尔奖再次光顾了宇宙学领域，并颁给了从事宇宙学理论研究的Peebles 教授。
⽬前，正当宇宙学研究在观测层⾯⼤步前进时，理论家和实验家们近年来将⽬光投向新的宇宙学热点，⼀个长期被理论预⾔的基本粒⼦“轴⼦(Axion)”。
⼆粒⼦物理学中的CP问题轴⼦起源于现代物理中对称性及对称性破缺问题的深⼊研究。
1956 年李政道、杨振宁与吴健雄等⼈提出并在实验上验证了宇称P在弱相互作⽤中不守恒。
后来⼈们发现弱相互作⽤中正反粒⼦共轭(C)与宇称(P)的联合变换CP 也不守恒。
C 变换指的是将⼀个粒⼦变成它的反粒⼦，P 变换即空间坐标反演。
在粒⼦物理的标准模型中，Kobayashi和Maskawa 提出的机制在理论上成功解释了弱相互作⽤中的CP 破坏，并为此荣获了2008 年的诺贝尔物理学奖。
然⽽，强相互作⽤中的CP 对称性是否守恒仍是现代物理学中⼀个⼤问题。
在粒⼦物理标准模型中，强CP破坏效应对应于量⼦⾊动⼒学(QCD)中的Chern-Simons 项, 其中G是QCD规范场的场强，是相应的对偶场强，θ为常数，表征强作⽤CP 破坏⼤⼩。
这⼀项在CP变换下不守恒，并可以贡献到中⼦的电偶极矩。
然⽽实验测量只给出中⼦电偶极矩的上限，这个上限很强，要求“参数”θ必须⼩于。
θ为什么这么⼩？这便是著名的“强CP问题”。
在粒⼦物理标准模型中，除强相互作⽤项之外，对应于SU(2)×U(1)规范对称性，还应有两个θ项。
但这两个θ项⼀般情况下没有效应。
⼀是U(1)规范场的真空是平庸的，所以θ项效应为零。
SU(2)规范场的θ本不为零，但标准模型的经典拉⽒量中存在着整体的重⼦和轻⼦对称性。
⼆者⼜在量⼦层次都是被破缺的，也具有反常性质，故SU(2)的θ项效应也表现不出来。
数学奖章上的数学故事——国际数学四大奖项菲尔兹奖菲尔兹（John Charles Fields,1863-1932 年）菲尔兹奖由加拿大数学家约翰• 查尔斯• 菲尔兹（John Charles Fields,1863-1932 年）建议设立。
菲尔兹早年游学美国、欧洲，与诸多大数学家共事。
其后返回加拿大致力于提升数学的地位。
如在他努力下，1924 年世界数学家大会在加举行。
他自上个世纪20 年代末开始筹备该奖，并遗嘱捐赠$47,000给奖项基金。
菲奖在1936 年首颁；后从1950 年起每隔4 年颁发一次，奖励40 岁以下数学成就杰出者，且旨在鼓励获奖者进一步的研究。
获奖者一般为2 至4 人。
该奖有“数学界中诺贝尔奖”之称，其实它早期并无今日如此声誉，这很大程度上源于历届获奖者给它带来的荣耀。
菲尔兹奖包括一面金质奖章和一笔不算多的奖金（目前为15,000 加元）。
奖章正面有古希腊数学家阿基米德的头像（Archimedes, 前287- 前212 年）和希腊文“ΑΡΧΙΜΗΔΟΥΣ”，意为“阿基米德的（头像）”；头像周边刻拉丁文“TRANSIRE SUUM PECTUS MUNDOQUE POTIRI”，此来源于一世纪罗马诗人马尼利乌斯（Manilius）的著作《天文学》，意为“超越他的心灵，掌握世界”。
此外奖章设计者（Robert Tait McKenzie）名字之缩写RTM 及设计年份MCNXXXIII（即1933 年，第二个M 字母以N 代）也刻在奖章上。
获奖者的名字则会被刻于奖章边轮。
菲奖章背面刻有意为“聚全球数学家，为杰出著作而颁”的拉丁文“CONGREGATI EX TOTO ORBE MATHEMATICI OB A INSIGNIA TRIBUERE”。
文字和树枝的背景为球体嵌进圆柱体（“圆柱容球”）的示意图，这象征着阿基米德的得意之作《论球与圆柱》中最著名的一个结果：球与其外切柱体的面积（体积）之比为2 : 3。
a rX iv:c ond-ma t/971187v1[c ond-m at.m es -hall]11No v1997A Chern-Simons Effective Field Theory for the Pfaffian Quantum Hall State E.Fradkin 1,Chetan Nayak 2,A.Tsvelik 3,Frank Wilczek 41Dept.of Physics,University of Illinois at Urbana-Champaign,Urbana,IL,618012Institute for Theoretical Physics,University of California,Santa Barbara,CA,93106-40303Department of Theoretical Physics,University of Oxford,1Keble Road,OX13NP,U.K.4School of Natural Sciences,Institute for Advanced Study,Olden Lane,Princeton,N.J.08540We present a low-energy effective field theory describing the universality class of the Pfaffian quantum Hall state.To arrive at this theory,we observe that the edge theory of the Pfaffian state of bosons at ν=1is an SU (2)2Kac-Moody algebra.It follows that the corresponding bulk effective field theory is an SU (2)Chern-Simons theory with coupling constant k =2.The effective field theories for other Pfaffian states,such as the fermionic one at ν=1/2are obtained by a flux-attachment procedure.We discuss the non-Abelian statistics of quasiparticles in the context of this effective field theory.I.INTRODUCTION Recently there has been considerable interest in a new class of quantum Hall states,combinining aspects of BCS pairing with Laughlin-type ordering [1–6].The states appear to be incompressible,and to exhibit non-Abelian statistics.Their properties have mainly been inferred by extrapolation in quantum statistics (of electrons)from ordinary superconductors [2],from numerical studies [2,6],and from the existence of attractive trial wave functions [1–5].To assure ourselves of the robustness of these properties,and for calculational purposes,it is important to have a low-energy,long-wavelength effective field theory.In this paper,we shall supply such a theory.Analyses of the ground states of the lowest Landau level Hamiltonian with three-body interactions of the form H = i>j>k δ′(z i −z j )δ′(z i −z k )(1)have unearthed a number of fascinating properties of these states [1–5].In particular,at Landau level filling fraction ν=1/2,the ground state is the so-called Pfaffian state,ΨPf =Pf 14ℓ20 |z i |2(2)(where Pf 1z 1−z 21z i −z j i>j(z i −z j )q e −1z i −z j i>j (z i−z j )2e −1z j −z k →Pf (z j−η1)(z j −η2)(z k −η3)(z k −η4)+(j ↔k )However,this is not the only four-quasihole state with quasiholes atη1,η2,η3,η4.Pf(13)(24)and Pf(14)(23)seem to be equally good.In fact,only two of these three are linearly independent,as a consequence of the identity[3]:Pf(12)(34)−Pf(14)(23)=η14η238+1√z i−z j→Pf θa(z i−z j)z k+1−z k+214ℓ20 |z i|2(10)The arbitrary distinct positive integers p1,...,p k correspond to the creation of neutral fermionic excitations;this is the c=1/2sector of the edge theory.S(z1,...,z N)is a symmetric polynomial;this modification of the ground state corresponds to the creation of chiral bosonic excitations in the c=1charged sector of the edge theory.Atν=1/2, the compactification radius of the boson is R=1/√2+1.The specific form of the trial wavefunctions considered above played a crucial role in the determination of the properties of the Pfaffian state,such as the non-Abelian braiding statistics.Of course,we would like to believe that the Pfaffian state is a representative of an entire universality class of states which have the same‘topological properties’,such as braiding statistics and ground state degeneracy on the torus.To investigate the existence and stability of this universality class,we need a low-energy,long-wavelength effectivefield theory for the Pfaffian state. An effectivefield theory for the Laughlin states atν=1/(2k+1)was obtained by a Landau-Ginzburg construction L eff=ψ∗(i∂0−(a0+A0))ψ+¯h22k+11The fundamental objects in the Landau-Ginzburg theory are auxiliary bosons,ψ,which are electrons withfictitious flux attached via a.The fractionally charged,anyonic quasiparticles are vortices.In a dual theory,which results from integrating out the auxiliary bosons,the quasiparticles are fundamental:L dual=2k+12+1.We will use this edge theory to deduce the effectivefield theory of the bulk(which is dual tothe Landau-Ginzburg theory).Ourfirst step will be to show how this is done in the simplest case(for reasons which will become clear)of a Pfaffian state of bosons atν=1.We will then check that this effectivefield theory reproduces the bulk properties exhibited by the wavefunctions.Finally,we will comment on the stability of the state.II.EFFECTIVE FIELD THEORY OF THE BOSONIC PF AFFIAN STATE ATν=1The edge theory of the bosonic Pfaffian state atν=1has c=1z2+f abc J c(0)16π d2x tr g−1∂µg g−1∂µg +2k+2=3/2;which is the central charge ofa triplet of Majorana fermions which indicates the equivalency between these two models.Hence,we may take the chiral sector of the SU(2)2WZW model as the edge theory of the bosonic Pfaffian state atν=1.Now,we are most of the way home because the bulk theory corresponding to the chiral SU(2)2WZW model is simply the SU(2)Chern-Simons theory with Chern-Simons coefficient k=2[7],S=23f abc a aµa bνa cλ (17)where a aµis an SU(2)gaugefield.An alternative approach begins with the coset construction of the c=1/2theory [9,10],but it is less transparent,so we do not pursue it further.III.TOPOLOGICAL PROPERTIES IN THE EFFECTIVE FIELD THEORYWe can now check that the effectivefield theory(17)predicts the same topological properties as the analysis of trial wavefunctions.Let usfirst consider the ground state degeneracy on a torus,which we know to be three for bosons at ν=1.The Hilbert space of the Chern-Simons theory(17)(since the Chern-Simons Hamiltonian vanishes,all statesare ground states)on a torus can be obtained in the following way[7,8].Consider the path integral of(17)over the three-dimensional region M enclosed by a torus,∂M=T2,Ψ[a]= α|T2=a Dαe2 M d3xǫµνλ(αaµ∂ναaλ+24π M d3xǫµνλ(αaµ∂ναaλ+24π d3xǫµνλ(αaµ∂ναaλ+2Since(21)tells us how amplitudes are modified by braiding operations,it gives us a direct handle on the eigenvalues of the braiding matrix.Consider an arbitrary state|ψ in the two-dimensional space of states with four quasiholes. According to the skein relation,q−1|ψ −q B2|ψ = q1/2−q−1/2 B|ψ (22)where B is the braiding operator for two of the quasiparticles.(22)implies a quadratic equation for the eigenvalues of B which yields the eigenvalues e−3πi/8and eπi/8,again in agreement with(8).We can also obtain the quasiparticle statistics from the braiding matrices of the spin-1/2vertex operators in the SU(2)2WZW model.According to[9],these follow from the fusion rules and anomalous dimensions in that conformal field theory.IV.EFFECTIVE FIELD THEORY OF THEν=1/2PF AFFIAN STATEWe would now like to deform the theory(17)so that it describes the fermionic Pfaffian state atν=1/2.(Of course,by the same procedure,we could also obtain the effectivefield theory for fermionic Pfaffian states at any even denominator or for bosonic Pfaffian states at any odd denominator.)The idea is to use theflux-attachment procedure [11–14]to change the bosons into fermions and simultaneously change thefilling fraction fromν=1toν=1/2.To do this,we introduce a U(1)gaugefield,cµ,which couples to the charge current,ǫµαβF3αβ.Thefield cµwill attach a flux tube to each electron through the Chern-Simons equation:ǫµαβ∂αcβ=−jµ=ǫµαβF3αβ(23) This equation follows from the actionS=23f abc a aµa bνa cλ +14πcµǫµαβ∂αcβ(24)which is our proposed effectivefield theory for the fermionic Pfaffian state atν=1/2.Equation(24)needs some explanation because the term which couples a aµand cµbreaks the SU(2)gauge invariance down to the smaller U(1)subgroup generated by T3(as does the coupling to the electromagneticfield,which has the same form).The action(17)and the equations of motion which follow from it are invariant under the transformationa aµT a→g a aµT a g−1−∂µg g−1(25) where g is an SU(2)-valued function which must become identity at the boundary.As a result of(25),the time-evolution of a aµ(in the bulk)is not well-defined because a given set of initial conditions can lead to infinitely many possible solutions,each related to the others by the gauge transformation(25).To put it differently,the action does not specify a dynamics for the longitudinal part of the gaugefield,which decouples from the transverse part.On the other hand,gauge-invariant quantities,such as F aαβF aµνhave perfectly well-defined time-evolution since they are independent of the longitudinal part of a aµ.For calculational purposes,we can choose a particular gauge,thereby specifying a dynamics for the longitudinal part of a aµ.Of course,gauge-invariant quantities will not depend on this choice.In(24),however,the cµǫµαβF3αβterm breaks gauge-invariance by coupling the transverse part of the gaugefield to the longitudinal part.As a result of its coupling to the longitudinal part of a aµ,the transverse part no longer has a well-defined time-evolution either.We can only make sense of the action(24)if we understand the SU(2)part of the action to be gauge-fixed.One possible choice is Coulomb gauge,a a0=0,or the weaker condition a10=a20=0which preserves the U(1)gauge symmetry generated by T3.In this gauge,there is a constraint,F a12=0(26) which generates time-independent gauge transformations.We can impose another condition to eliminate this residual gauge symmetry,but we do not have to.Once we have imposed Coulomb gauge,a given set of initial conditions leads to a unique solution.The time-independent gauge transformations connect different gauge-equivalent initial conditions, but do not render the time evolution ill-defined.In other words,we have a hugely redundant,but completely well-defined dynamics.Hence,we can take(24)as our effectivefield theory,provided we understand the SU(2)gaugefield to be gauge-fixed in the Coulomb gauge or another suitable gauge.Alternatively,and more profoundly,we can view the cµǫµαβF3αβterm as a gauge-fixed form of the gauge invariant operator L spinL spin≡1∆+0(−i∂µ+cµ)∆aǫµαβF aαβ+c.c.(30)4πNow we need no extra constraints.It is noteworthy that while the ordinary Chern-Simons terms are completely general covariant,potential term(29)is invariant only under volume preserving diffeomorphisms.This should not be disturbing,however,because there is a preferred density–though not a preferred shape–associated with incom-pressible quantum Hall liquid.The non-asymptotic form,withλ1,2finite,allows in principle for the description of localized vortex configurations.Because theflux-attachment only affects the‘trivial’U(1)part of the topological properties of the Pfaffian state, the quasiparticle statistics,degeneracy on the torus,etc.can be inferred from those atν=1.It is straightforward to generalize this construction to include more generalfilling factors.Essentially,all that is required is to attach an even number offlux quanta in addition to the procedure we used to map bosons to fermions. This is the standard procedure that is followed to generate the Jain fractions in the fermionic version of the abelian Chern-Simons theory of the FQHE[14].There is,however,a significant subtlety concerning the global consistency of the implementation offlux attachment commonly used in the condensed matter literature,when it is applied to closed surfaces of non-trivial topology.Indeed,the action for the Abelian gaugefield is ordinarily written with a Chern-Simons term with a coefficientθ=1/(2πm),where m is an even integer.This is inconsistent with the requirement of quantization of the Chern-Simons coupling constant needed for a Chern-Simons gauge theory on a closed manifold [15].We should instead take this coefficient to be1and put the coefficient1/θ=(2πm)in front of the term(27) which couples the two gaugefields.In the Appendix we give a more careful discussion of this point,and derive the proper procedure fromfirst principles.V.DISCUSSION1.A main motivation for our effectivefield theory is to establish the existence of a universality class of quantum Hall states of which the Pfaffian state(3)is a representative.The most salient property of this universality classis a 2n −1-fold degenerate set of 2n -quasihole states which transform as the spinor representation of SO (2n )as the quasiholes wind about each other.This is a bit worrisome since arbitrary perturbations might be expected to break this degeneracy.In particular,impurities or small variations in the inter-electroninteractions could,potentially,do this,thereby spoiling the non-Abelian statistics.With theeffective field theories (17)or (24)in hand,however,it is clear that this degeneracy is,indeed,stable in the long-wavelength limit.The leading perturbations are Maxwell terms of the form F a µνF µνa which are irrelevant by one power of q or ωcompared to the Chern-Simons terms.Perturbations which couple to the charge density,such as a random potential or Coulomb interactions,couple to the field strength of the U (1)gauge field in (24).Hence,(24)is just as stable as an Abelian quantum Hall state,at least as far as such perturbations are concerned.2.A Landau-Ginzburg theory of the Pfaffian state should take a paired order parameter as its starting point.Indeed,this is strongly suggested by the appearance of the fields ∆0,∆a in the gauge-invariant form of the effective field theory (30).This leads one to the Landau-Ginzburg theory of a superconductor,but with an Abelian Chern-Simons field to cancel the magnetic field or,in other words,to a theory almost identical to the theory of a Laughlin state (11).The crucial difference is that the paired order parameter must have a structure which allows the existence of neutral fermionic modes at vortex cores,where the gap to an unpaired fermion vanishes.The reader may recall that Bogoliubov-de Gennes quasiparticles in a superconductor become neutral fermions on the Fermi surface and are therefore natural candidates for the zero modes we are looking for.We could then,as in [4–6],interpret the 2n −1-fold degeneracy in terms of the occupation of fermionic zero modes associated with the vortices.This raises the interesting prospect that the duality transformation between the Landau-Ginzburg theory –in which the electrons are the fundamental objects –and the dual theory of equation (17)or (24)–in which the quasiparticles are fundamental –is a highly non-trivial transformation relating two theories which,at first glance,appear to be radically different.3.The observed plateaus at ν=5/2(in other words,ν=1/2in the second Landau level)in a single-layer system and ν=1/2in a double-layer system are promising hunting grounds for excitations with non-Abelian statistics.One way to experimentally determine whether either one is described by the Pfaffian state (2)is to measure the quasihole statistics.To do this,we would like to observe how the quantum state of the system transforms when quasiholes are braided.An elegant way to do this utilizes the two point-contact interferometer proposed by Chamon,et.al.[16].AFIG.2.In the two point-contact interferometer of Chamon,et.al.,quasiholes can tunnel from the lower edge to the upper edge by one of two interfering paths.The interference is controlled by varying the flux,Φ,and number of quasiholes,N q ,in the central region.In this device (see figure 2),quasiholes injected at point A along the bottom edge of the quantum Hall bar can tunnel to the other edge at either of two point-contacts.A quasihole which tunnels at the second point-contact will follow a path which encircles a central region containing flux Φas well as N q quasiholes.As a result of the flux Φ,it will aquire an Aharonov-Bohm phase which depends on its fractional charge.Its state will also be tranformed because its trajectory braids the N q quasiholes in the central region.Consequently,the interference between the two tunneling paths will depend on the charge and statistics of the quasiholes.As discussed by Chamon,et.al.[16],if we hold the electron number (and therefore the quasihole number)in the central region fixed,then the conductance will oscillate as a function of Φwith period e2|t 1|2+|t 2|2 +Re t ∗1t 2e iα ψ|B N q |ψ (31)where t 1and t 2are the tunneling amplitudes at the two point-contacts.Now,the matrix element ψ|B N q |ψ is given precisely by the expectation value in the effective field theory (17)of the Wilson lines of figure 3or,simply,by theJones polynomial V N(eπi/4)of these loops.qFIG.3.The matrix element ψ|B N|ψ is the expectation value of these Wilson lines.The loop on the left represents aqquasihole which tunnels at the second point contact,thereby encircling the N q quasiholes on the right.In other words,if the Pfaffian state exists in nature,a two point-contact interferometer will measure the Jones polynomial!By studying the dependence on N q–the conductance will exhibit the periodicities of the two eigenvalues –we will be able to extract the non-Abelian statistics.of B NqVI.ACKNOWLEDGEMENTSThis work was supported in part by the NSF,grant numbers NSF DMR94-24511at UIUC,and NSF PHY94-07194 at ITP-UCSB.EF was a participant at the ITP Program on Quantum Field Theory in Low Dimensions,and thanks the Director of ITP for his kind hospitality. C.N.would like to thank ICTP,Trieste for hospitality during the workshop on Open Problems in Strongly Correlated Systems,where part of this work was done.A.M.T.is grateful to I.I.Kogan for interesting discussions.APPENDIX A:We will show how to defineflux attachment in a manner compatible with the requirement of quantization of the abelian Chern-Simons coupling constant or,what is the same,of invariance under large gauge tranformations.This issue does not arise for the non-abelian sector since its coupling constant is already correctly quantized.Consider a theory of particles(infirst quantization)which interact with each other as they evolve in time.We will assume in what follows that the particles are fermions(in two spacial dimensions)and that their worldlines never cross.The actual choice of statistics is not important in what follows but the requirement of no crossing is important and,for bosons,it implies the assumption that there is a hard-core interaction while for fermions the Pauli principle takes care of this issue automatically.For simplicity,we will assume that the time evolution is periodic,with a very long period.The worldlines of the particles can be represented by a conserved current jµ.For a given history of the system,the worldlines form a braid with a well defined linking numberνL[jµ],given byνL[jµ]= d3x jµ(x)Bµ(x)(A1) where jµand Bµare related through Amp`e re’s Lawǫµνλ∂νBλ(x)=jµ(x)(A2) Under the assumption of the absence of crossing of the worldlines of the particles,the linkingνL[jµ]is a topological invariant.Thus,if S[jµ]is the action for a given history,the quantum mechanical amplitudes of all physical observables remain unchanged if the action is modified byS[jµ]→S[jµ]+2πnνL[jµ](A3) where n is an arbitrary integer.The quantum mechanical amplitudes are sums over histories of the particles,and take the formAmplitude∝ [jµ]e iS[jµ]+2πinνL[jµ]e iφ[jµ](A4)whereφ[jµ]is a phase factor which accounts for the statistics of the particles.However,the amplitudes remain unchanged if the integrand of Eq.A4is mulitiplied by1written as the expression 1≡ D bµ xδ(ǫµνλ∂νbλ−jµ)=N D bµD aµexp i2π[ǫµνλ∂νbλ−jµ](A6)We can then compute this amplitude as a path integral of a theory in which the particles whose worldlines are represnted by the currents jµ,interact with the gaugefields aµand bµ.These interactions are encoded in the effective actionS eff[a,b,j]=14πǫµνλbµ∂νbλ(A7)where we have solved the constraint j=∂∧b to write the term of the winding number in the form of a Chern-Simons action for the gaugefield bµ[17,18].Hence,the amplitudes can be written in terms of a path integral over an abelian Chern-Simons gaugefield with a correctly quantized coupling constant equal to2n21[1]G.Moore and N.Read,Nucl.Phys.B360(1991),362.[2]M.Greiter,X.G.Wen,and F.Wilczek,Nucl.Phys.B374(1992),567.[3]C.Nayak and F.Wilczek,Nucl.Phys.B479(1996),529.[4]ovanovic and N.Read,Phys.Rev.B53(1996),13559.[5]E.H.Rezayi and N.Read,Phys.Rev.B56(1996),16864.[6]F.D.M.Haldane,private communication.[7]E.Witten,Commun.Math.Phys.121(1989),351.[8]E.Verlinde,Nucl.Phys.B300(1988),351.[9]G.Moore and N.Seiberg,Comm.Math.Phys.123(1989),77.[10]I.I.Kogan,Phys.Lett.390B(1997),189.[11]D.Arovas,J.R.Schrieffer,F.Wilczek,and A.Zee,Nucl.Phys.B251(1984)917.[12]S.C.Zhang,T.Hansson and S.Kivelson,Phys.Rev.Lett.62(1989)82.[13]A.L.Fetter,C.Hanna and ughlin,Phys.Rev.B39(1989),9679.[14]A.Lopez and E.Fradkin,Phys.Rev.B44(1991),5246.[15]Y.Hosotani,Phys.Rev.Lett.62(1989),2785.[16]C.de C.Chamon,D.S.Freed,S.A.Kivelson,S.L.Sondhi and X-G.Wen,Phys.Rev.B55(1997),2331.[17]F.Wilczek and A.Zee,Phys.Rev.Lett.51(1983),2250.[18]Y.S.Wu and A.Zee,Phys.Lett.147B(1984),325.[19]See,for instance,X-G.Wen,Adv.Phys.44(1995),405.。
