Bosonisation in Three-Dimensional Quantum Field Theory

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Computer-Vision计算机视觉英文ppt

At the same time, AI MIT laboratory has attracted many famous scholars from the world to participate in the research of machine vision,which included the theory of machine vision , algorithm and design of system .
Its mainstream research is divided into three stages:
Stage 1: Research on the visual basic method ,which take the model world as the main object;
Stage 2: Research on visual model ,which is based on the computational theory;
the other is to rebuild the three dimensional object according to the two-dimensional projection images .
History of computer vision
1950s: in this period , statistical pattern recognition is most applied in computer vision , it mainly focuse on the analysis and identification of two-dimensional image,such as: optical character recognition, the surface of the workpiece, the analysis and interpretation of the aerial image.

Three-dimensional visual illusion of graphic painting-文档资料

Three-dimensional visual illusion of graphic painting：Visual illusion is the visual design of a special category.It is a set of technology and art in one of a unique form of artistic expression.Visual illusion can give people a taste of the spirit, with strong cultural content and artistic appeal.So by this way of painting, it has a clever and unique perspective. Aspect If the plane can be realistic paintings to life, then the three-dimensional, two-dimensional space can be called even more powerful by aspects.Keywords：Visual form of communication；visual illusion；anti-perspective；relationships1.Visual form of communication is a set of the perfect combination of technology and art"We seem to be designed in an era of focus. Dismally in the creative arts, the expression is the need to focus on solving the problem, he just wound during the wizard, looming uncertain, get rid of. Art form close to the human and the nature When the desire and skill entanglement art, the form of keeping a distance of secular life, somedetached, some sublimation, which is the real human needs, but also on the illusory magic of compensation of virtual life. "In fact, the feelings of a rich life to convey, in addition to language, "Visual Communication" is also a really good medium.As mentioned in the art, this needs to be emphasized is that art form of habitual thinking is not just people directly associated with the Dot, line, flat and color. In fact, we can also target the unique combination of performance, showing a more comprehensive effect. As intuition can get through all the sense, the feelings involved in the mobilization of additional forms of art experience (such as: you can sense in the art form to the "flavor", "sense of music", etc.). It is perhaps for this reason, the field of art also have density, soft, hard and cold to say. Within a certain range, the picture gives more dots to crowded, dense feel. The other hand gives a opposite feeling.Relative to the wavy lines, arcs, straight line is much more rigid. The similar color as fire can give people a warm feeling. so it called warm color. The color tends to ice can give people the feeling of cold, so it called cold color.2.The two-dimensional drawings of what might be calleda three-dimensional cutting-edge design techniquesImages in recently news showed the market is popular variety of three-dimensional paintings, characterized both eyes glued to watching after a while, it appeared in a trance between the eyes - three-dimensional image. When I first saw this work, they are visual artists that transcend the ability to grasp.Three-dimensional picture of the visual effect is mainly used in general on the basic color, but also added some artistic lighting, which makes the high-impact visual effects. General painting and photographic works, including computer-generated three-dimensional animation is the use of the human eyes on the light and shadow, light and shade, the actual situation in the sense ofthree-dimensional, without use of binocular stereo vision, a sight seeing and two are the same. Full use ofthree-dimensional paintings binocular stereo vision, you will see a very exciting world.3.With the help of visual illusionFinally, three-dimensional picture plane can be set up body painting experience, in addition to drawing on thescreen other than the principles and painting techniques, but also help to achieve the desired illusion. Because three-dimensional picture plane through the planar design of this particular visual symbols - visual illusion of passing information to the viewer.We only need a heart Duan Xu will be able to remember all the little things. "When people see the screen, you can naturally goes on. Is left blank on the paintings, the viewer will naturally cover it up. For example, traditional Chinese painting in the "no eyes and if, as, no ears to listen ... ... and if there are dozens of real per pen can not write those, and this is obtained one or two sides suddenly as nuanced."References:[1]E.H.贡布里希.艺术与错觉[M].林夕,李本正,范景中译.长沙：湖南科学技术出版,2004年.。

钱学森开放巨系统

谨以此文缅怀纪念民族魂魄、科学巨擎、我们永远的钱老——人民科学家钱学森先生！钱学森开放复杂巨系统理论视角下的科技创新体系——以城市管理科技创新体系构建为例宋刚北京大学遥感与地理信息系统研究所,北京100871摘要：本文基于钱学森开放复杂巨系统理论，对知识社会环境下的科技创新体系建构进行了研究。

在技术创新双螺旋基础上，进一步从科学研究、技术进步与应用创新的协同互动入手，并以城市管理科技创新体系建设为例，构建了科技创新体系构成图，分析了由以科学研究为先导的知识创新、以标准化为轴心的技术创新和以信息化为载体的现代科技引领的管理创新构成的科技创新体系。

三个体系相互渗透，互为支撑，互为动力，推动着科学研究、技术研发、管理与制度创新的新形态，共同塑造了面向知识社会的创新2.0形态。

现代城市管理作为现代服务业的重要组成部分，其科技创新体系的构建与实践对面向服务的科技创新体系建设具有直接参考价值，对知识社会环境下的科技创新体系的建构具有重要借鉴意义。

关键词：知识社会，科技创新体系，创新2.0，城市管理，复杂性Science & Technology Innovation System in Perspective of Open Complex Giant System Theory of Qian Xuesen - Example from Construction of Urban ManagementScience and Technology Innovation SystemSONG GangInstitute of Remote Sensing and GIS, Peking UniversityAbstract: Science & Technology Innovation System in knowledge-based society is studied in perspective of complexity science. Starting from the double helix structure of technology innovation, science and technology innovation is analyzed as interaction of scientific research, technology development and application innovation. With science and technology innovation system in urban management as an example, three subsystems, namely, knowledge innovation system lead by scientific research, technology innovation system supported by standardization, and management innovation enabled by modern ICT, are constructed and analyzed from the perspective of Open Complex Giant System Theory by Qian Xuesen. The convergence and interaction of the three subsystems triggered the paradigm shift of scientific research, technologyinnovation and management innovation, which lead to innovation 2.0 in a knowledge-based society. This construction can be borrow directly to other modern service-oriented industry, and is also revelatory to construction of science and technology innovation system in a knowledge-based society.Key words: Knowledge Society, Science & Technology Innovation System, Innovation 2.0, Urban Management, Complexity科技进步推动了复杂性科学的发展，让我们以全新的视野审视现代经济社会的发展及其科技创新支撑[1]。

新视野综合训练3第二版阅读原文翻译

Unit1Passage1Valentine’s Day probably has its origin in the ancient Roman celebration called Lupercalia(牧神节) . It was celebrated on February 15. In the Rom an calendar February was in the spring. The celebration honored the go ds Lupercus and Faunus as well as the twin brothers Romulus and Rem us, the legendary founders of Rome. As part of the ceremony the priest s paired up young man and women. The girls names were placed in a box and each boy drew a girl’s name. The couple was paired then until the next Lupercalia.In 260AD the emperor Claudius II, called Claudius the Cruel, decid ed that young soldiers would only be distracted by marriage and so orde red that young men may not marry. Valentinus( Valentine), a Christian pr iest, defied the emperor and got married in secret. He was caught exec uted(处死)，on February 14, the eve of Lupercalia. His name became as sociated with young love forever after. In 496, Pope Gelasius set aside February 14 to honor him as Saint Valentine and it has been St. Valenti ne’s Day ever since.In the Middle Ages, some of the customs of the Lupercalia still per sisted in spite of the attempts of the Church to put an end to these non -Christian customs and Christianize the holiday. Both men and women dr ew names from a bowl to see who their valentines would be. They woul d wear the names on their sleeves for a week. Today we still sometime s “wear our hearts on our sleeves” when we cannot conceal our feelings.In the 1600s, it became common to give flowers, particularly the r ose, as a sign of lov e as the “language of flowers”. This came to Europ e from Turkey. The color and placement of rose held a special significan ce--- a red rose, for example, meant beauty. Flowers have been part of Valentine’s Day ever since.情人节可能起源于古罗马的牧神节庆祝活动被称为（牧神节）。

On the Global Structure of Some Natural Fibrations of Joyce Manifolds

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1
Essential mathematical background for physicists.
To set up some notations and to provide some essential background for physicists, three key ingredients of the work: orbifolds, Joyce manifolds, and Borcea-Voisin threefolds, are concisely explained in this section. Readers are referred to the listed references for more details. Orbifolds ([Th1], also [B-S], [F-M], [Mo] and [Sc]). (Cf. Remark 2.1.4.) An n-dimensional orbifold Q is a topological space locally modelled by a quotient U = U /ΓU , where U is a connected open set in Rn and ΓU is a ﬁnite group that acts on U eﬀectively. Associated to each point p in Q is a group Γp that is isomorphic to the stabilizer of any preimage point of p in a local model U = U /ΓU around p. The set ΣQ = {p ∈ Q|Γp = {1}} is called the singular locus of Q. In general ΣQ is stratiﬁed by manifolds of various dimensions (cf. Figures 2-1, 3-1-1, 3-6-1). In particular, when a discrete group Γ acts on a manifold X eﬀectively and properly discontinuously but not necessarily freely, then the quotient X/Γ is an orbifold with ΣX/Γ descending from the set of ﬁxed points in X of some element in Γ. The group Γp associated to each point p in X/Γ is the stabilizer Stab (p) of any preimage point p of p in X . A covering orbifold of an orbifold Q is an orbifold Q with a projection φ : XQ → XQ between the underlying spaces such that each x ∈ XQ has a neighborhood, modelled by U = U /ΓU , for which each component Vi of φ−1 (U ) is isomorphic to U /Γi , where Γi is a subgroup in Γ and the isomorphisms are compatible with φ. Fact 1.1 [universal covering orbifold]. ([Th1]: Proposition 13.2.4). Any orbifold Q has a universal covering orbifold Q. In other words, if ∗ ∈ XQ − ΣQ is a base point for Q, then φ : Q → Q is a connected covering orbifold with base point ∗ ∈ φ−1 (∗) such that, for any covering orbifold φ′ : Q′ → Q with base point ∗′ ∈ φ′ −1 (∗), there is an orbifold covering map χ : Q → Q′ such that χ(∗) = ∗′ and φ = φ′ ◦ χ.

Three-dimensional Continuum Radiative Transfer Images of a Molecular Cloud Core Evolution

a r X i v :a s t r o -p h /0410298v 1 12 O c t 20043D Continuum Radiative Transfer Images of a Molecular CloudCore EvolutionJ.SteinackerMax-Planck-Institut f¨u r Astronomie,K¨o nigstuhl 17,D-69117Heidelbergstein@mpia.deandngMax-Planck-Institut f¨u r Astronomie,K¨o nigstuhl 17,D-69117Heidelberglang@mpia.deandA.BurkertUniversit¨a ts-Sternwarte M¨u nchen,Scheinerstr.1,D-81679M¨u nchenburkert@usm.uni-muenchen.deandA.BacmannObservatoire de Bordeaux,2Rue de l’Observatoire,BP 89,33270Floiracbacmann@obs.u-bordeaux1.frandTh.HenningMax-Planck-Institut f¨u r Astronomie,K¨o nigstuhl 17,D-69117Heidelberghenning@mpia.deABSTRACTWe analyze a 3D Smoothed Particle Hydrodynamics simulation of an evolving and later collapsing pre-stellar ing a 3D Continuum Radiative Transfer program,we generate images at 7,15,175µm,and 1.3mm for diﬀerent evolutionary times and viewing angles.We discuss the observability of the properties of pre-stellar cores for the diﬀerent wavelengths.For examples of non-symmetric fragments,it is shown that,misleadingly,the density proﬁles derived from a 1D analysis of the corresponding images are consistent with 1D core evolution models.We conclude that 1D modeling based on column density interpretation of images does not produce reliable structural information and that multi-dimensional modeling is required.Subject headings:infrared radiation —ISM:clouds —ISM:dust,extinction —submillimeter —stars:formation1.IntroductionMolecular cloud cores are thought to be the di-rect progenitors of stars.However,their initial properties and early evolution are still poorly un-derstood.Current observations therefore aim to ﬁnd and study these cores in more detail.Density structure,velocityﬁeld,and temperature distri-bution of the gas and dust are key parameters for the physical interpretation of the long lifetimes of these objects.The continuum radiation spectra of deeply embedded sources contain only ambigu-ous information about the density and tempera-ture distribution of the dust.Column densities can be inferred from the analysis of continuum im-ages in the mm-range(e.g.Ward-Thompson et al. 1994)and the mid-infrared(MIR)(e.g.Bacmann et al.2000).The emission of molecules within the core contains information about the struc-ture,velocityﬁeld(Tafalla et al.2004,and ref-erences therein),and the turbulence(Ossenkopf, Klessen,&Heitsch2001,and references therein). The currently discussed supporting mechanisms against gravitational collapse are magneticﬁelds and turbulence.There are a number of simula-tions treating the full3D structure of collapsing pre-stellar cores(e.g.Bate,Bonnell,&Bromm 2003;Ballesteros-Paredes,Klessen,&V´a zquez-Semadeni2003;Burkert&Bodenheimer1993, 2000;Krumholz et al.2003;Klessen,Heitsch,& Mac Low2000;Heitsch,Mac Low,&Klessen 2001).In some of these papers,the resulting col-umn density along a line of sight was compared with observationally obtained column densities. The question which structures of the distribution can be seen at which wavelength,however,can only be answered by producing images of the core for given dust properties.In turn,the majority of models that have been applied to observational data are mostly based on spherical symmetry(e.g. Andr´e,Ward-Thompson,&Motte1996).They rely on spherically symmetric models of isolated star formation,which describe core formation, gravitational collapse,and proto-stellar accretion. Commonly,averaged radial density proﬁles are de-rived and compared with power-law density dis-tributions(e.g.Andr´e,Ward-Thompson,&Motte 1996),or Bonnor-Ebert spheres(e.g.Alves2004). The major diﬃculties using3D models in observa-tions are i)the information loss due to the projec-tion eﬀect,ii)the complex and unique structure of each individual pre-stellar cores,and iii)the nu-merical eﬀort of multi-dimensional radiative trans-fer.In this letter,we investigate how an evolving cloud core simulated by a3D Smoothed Particle Hydrodynamics(SPH)code would appear at dif-ferent wavelengths,which structures are visible, and which density distributions would be inferred using common1D models.The results from the SPH simulation are described in Sect.2,along with a discussion of underlying assumptions.We present the images from the3D Continuum Ra-diative Transfer(CRT)modeling and discuss the observability of diﬀerent structures and physical eﬀects.The comparison to density structures ob-tained from applying1D models to the images is given in Sect.3,and theﬁndings are summarized and discussed in Sect.4.2.Cloud core evolution model and radia-tive transfer modeling2.1.3D SPH simulation of the evolutionof a cloud coreWe have calculated the evolution of a cloud core using a three-dimensional SPH code(version de-scribed in Bate,Bonnell,&Price1995),originally developed by Benz,Cameron,Press,&Bowers (1990).The smoothing lengths of particles are variable in time and space,following the constraint that the number of neighbors for each particle has to be approximately constant with N neigh=50. The SPH equations are integrated using a second-order Runge-Kutta-Fehlberg integrator with indi-vidual time steps for each particle(Bate,Bonnell, &Price1995).The simulation was initiated with a mass of M=3M⊙,adopting a spherically symmetric non-rotating homogeneous cloud with a temperature of T=10K,a diameter of d=0.12pc(corre-sponding to a density of2×10−16kg/m3),and a mean molecular weightµ=2.36×10−3kg/mol. This conﬁguration is Jeans unstable.A turbulent velocityﬁeld is added only at the beginning of the simulation with a Mach number of M=2 and following an approximate Kolmogorov law P(k)dΩk∼k−2for the diﬀerent modes.The tur-bulence supports the cloud core against collapse for theﬁrst105yrs.This enables the formation of a pre-stellar core-like structure,self-consistentlyas a result of turbulent energy dissipation.A vari-able equation of state is used:isothermal for den-sities less than1016molecules/m3and adiabatic for larger densities.Deviating from earlier work,the initial condi-tions are arranged in a way that the core reaches a dynamical equilibrium of density structures and velocityﬁeld before it evolves into a runaway col-lapse.The resulting structure is visualized by iso-density surfaces shown in the left panels of Fig.1.The top left panel shows the early stage of core formation5.6×104yrs after the initial-ization(iso-density of4×10−16kg/m3).Tur-bulence dominates the structure formation and creates severalﬁlamentary low-mass density max-ima.The duration of this period before the on-set of the collapse and thus the total”age”of the pre-stellar core stage depends on how turbu-lence is injected initially and its dissipation.In the course of time,the local density enhancements merge.After some additional8.5×104yrs just at the edge of gravitational instability,a single core has formed(second left panel,5×10−17kg/m3). The kinetic pressure support breaks down due to rapid dissipation of turbulent energy inside the over-dense region and the core starts to collapse (t=1.69×105yrs,third left panel,5×10−17 kg/m3).A new single hydrostatic core forms when the gas becomes optically thick and the cooling time exceeds the dynamical teron,the central part of the core is replaced by a sink particle.In the bottom left panel,the struc-ture hasﬂattened substantially,20%percent of the total mass is already accreted onto the sink particle,and a massive disk has formed through an instability.It contains additional low mass condensations and independently,a second frag-ment has started to form with a hydrostatic core (5×10−17kg/m3).The right panels give the iso-densities for0.16,0.5,1.6,and5.2×10−18kg/m3 for a time of2.4×105yrs,respectively.With in-creasing density,the second condensation becomes visible(see also animation at http://www.mpia-hd.mpg.de/homes/stein/Ani/animcf.htm).2.2.3D CRT modeling of the coresThe SPH density distributions of the gas were discretized on a3D grid and scaled to dust par-ticle distributions assuming a dust-gas mass ra-tio of1/100and an eﬃcient gas-dust mixing.The dust number densities were processed with a3D CRT code(Steinacker et al.2003,2002a; Steinacker,Bacmann,&Henning2002b;Pas-cucci et al.2004),producing640images of the cloud core at diﬀerent wavelengths,times,and viewing angles,respectively.The temperatures were calculated from the radiative heating.Heat-ing by compression is irrelevant during the pre-stellar core phase due to the fact that the cool-ing timescale is much faster than the dynamical timescale.For the illustrative purpose of this let-ter,we used standard dust opacity data(Drain &Lee1984)and a standard interstellar radia-tionﬁeld(Black1994).Some of the images are shown in Fig.2for the wavelengths7,15,175,and 1300µm(from top to bottom)and evolutionary times of5.6and14.1×104yrs,respectively(left to right).The wavelengths are chosen to cover common observational windows(e.g.ISOCAM, ISOPHOT,IRAM,JCMT,CSO,SPITZER,HER-SCHEL).All images are scaled to have maximal contrast.A10%random background noise repre-senting a mean background variation was added for illustrative purpose only.In the MIR,as ex-pected,the core is visible in absorption and the images reveal much of the outer thin structure es-pecially for the early stages.Detection of the in-ner,at later stagesﬂattened structure is diﬃcult and requires a careful background analysis.For wavelengths larger than90µm,the cold dust can be seen in emission,revealing more of the inner structure at high densities,as the core also starts to get optically thin.This emission is dominating the mm images.Animations showing the images for all view-ing angles at diﬀerent times of the evolution, as well as visualizations of the3D density data cube can be found under http://www.mpia-hd.mpg.de/homes/stein/Ani/animcf.htm.3.1D analysis of the mapsProjection eﬀects are a severe source of mis-interpretation for structures seen in absorption or emission,as pointed out already, e.g.,by Ballesteros-Paredes&Mac Low(2002).In Fig.3, we show as an example two structures seen at7µm.The upper left panel depicts an elongatedﬁl-ament at an early stage of the evolution(5.6×104 yrs),while the upper right panel gives aﬂattenedstructure at a later stage(24.4×104yrs).In the middle panels,the viewing angle was changed un-til we see the structures as a core-like feature.In the lower panel,they are zoomed and re-binned to an ISOCAM resolution typical for a core at 150pc distance.We determined the1D num-ber column density N(R)with the radius in the plane of the sky R by azimuthally averaging over annuli.As the absorption patterns have ellipti-cal shape,we have used elliptical annuli.The resulting number column density was inverted to a1D number density distribution n(r)with the radius r using recursive integration.In Fig.4, we show the results for the early stage-ﬁlament in the main panel and for the later stage-disk in the inlet.The range of proﬁles n(r)which have been transformed from N(R)-proﬁles along individual directions is given by the solid thin lines and the direction-averaged proﬁle is plotted as solid thick line.The dashed line indicates the slope of a den-sity distribution following a r−2-power-law as it was derived from1D core evolution models(Shu 1977).Although these absorption maxima are slightly less extended than commonly observed cores,the1D model seems to provide a reasonable description of the derived distributions.It could be inferred from thisﬁt that the underlying den-sity structure has an elliptical shape with a proﬁle that-transformed to a spherical distribution-is in agreement with1D core evolution models. To compare with the true underlying3D density distribution,we calculated the number density n for a grid of equally-sized cells from the SPH den-sity distribution.For each cell,we determined the distance to the center of the”core”-distribution deﬁning a point in the n(r)-diagram.This point distribution was rebinned to a greyscale image for better clarity,where black refers to maximum number of cells per bin.The advantage of this representation is that a1D core with radial power law appears as a line in the n(r)-diagram with a gradient representing the powerlaw index.The true density distributions overlayed as grayscale-image are far from being lines as the ﬁlament is not1D spherically symmetric.The agreement with the1D evolution models would tend to validate static core formation models,al-though the core formation mechanism modeled here is highly dynamical.This is in agreement with theﬁndings of Ballesteros-Paredes,Klessen,&V´a zquez-Semadeni(2003).We also modelled theﬂattened structure at a later stage.The ra-dial proﬁle within the disk will be visible in our grayscale-representation of n(r)as a line with the slope of the power-law exponent,and indeed the inlet in Fig.4shows a pronounced branch of the disk-like structure.If the core-ﬂattening is con-ﬁrmed by an independent source of information, the column density proﬁles(and the resulting den-sity proﬁles)can be used to determine the radial proﬁle of the disk-like structure.4.ConclusionsWe have presented3D simulations assuming that initially low mass condensations pass through a stage of turbulence dominated condensation where they accumulate mass and merge together to form extended pre-stellar core like objects. The typical density structures in the cores are non-spherical throughout their evolution.The asymmetry is driven by the turbulent motion and causes complex structures from the very begin-ning.This complexity is partially seen in images that have been calculated from the densities ob-tained in the cloud core simulation.However, projection eﬀects can lead to a severe misinterpre-tation of images.We showed that a1D analysis of the vicinity of the density maxima would sug-gest a density proﬁles in agreement with1D core collapse models.The underlying density struc-ture,however,is intrinsically three-dimensional and deviates strongly from the obtained1D model distribution.As the column density also enters the opti-cal thin emission in the mm-range(aside from the Planck function),we expect the same pro-jection ambiguities to occur when interpretating mm-maps of dense molecular cloud regions.This aspect will be discussed in a forth-coming paper.We conclude that1D modeling based on col-umn density interpretation of images does not pro-duce reliable structural information.Forﬂattened structures appearing in later stages of the core evo-lution,a2D modeling might be applicable,but for the general case,multi-dimensional continuum and line radiative transfer modeling is required to derive consistent density and temperature distri-butions of the gas and dust in pre-stellar cores.REFERENCESAlves,J.2004,Ap&SS,289,259Andr´e,P.,Ward-Thompson,D.,Motte,F.1996, A&A,314,625Bacmann,A.,Andr´e,P.,Puget,J.-L.,Abergel,A.,Bontemps,S.,Ward-Thompson,D.2000,A&A,361,555Ballesteros-Paredes,J.&Mac Low,M.2002,ApJ, 570,734Ballesteros-Paredes,J.,Klessen,R.S.,&V´a zquez-Semadeni,E.2003,ApJ,592,188Bate,M.R.,Bonnell,I.A.,&Bromm,V.2003, MNRAS,339,577Bate,M.R.,Bonnell,I.A.,&Price,N.M.1995, MNRAS,277,362Black,J.H.1994,ASP Conference Series,58,355 Benz,W.,Cameron,A.G.W.,Press,W.H.,& Bowers,R.L.1990,ApJ,348,647Burkert,A.&Bodenheimer,P.2000,ApJ,543, 822Burkert,A.&Bodenheimer,P.1993,MNRAS, 264,798Draine,B.T.,Lee,H.M.1984,ApJ,285,89 Heitsch,F.,Mac Low,M.,&Klessen,R.S.2001, ApJ,547,280Klessen,R.S.,Heitsch,F.,&Mac Low,M.2000, ApJ,535,887Krumholz,M.R.,Fisher,R.T.,Klein,R.I.,& McKee,C.F.2003,Revista Mexicana de As-tronomia y Astroﬁsica Conference Series,15, 138Ossenkopf,V.,Klessen,R.S.,&Heitsch,F.2001, A&A,379,1005Pascucci,I.,Wolf,S.,Steinacker,J.,Dullemond,C.P.,Henning,T.,Niccolini,G.,Woitke,P.,&Lopez,B.2004,A&A,417,793Shu,F.H.1977,ApJ,214,488Steinacker,J.,Henning,Th.,Bacmann,A.,&Se-menov,D.2003,A&A,401,405Steinacker,J.,Hackert,R.,Steinacker,A.,&Bac-mann,A.2002a,JQSRT,73,557 Steinacker,J.,Bacmann, A.,&Henning,Th.2002b,JQSRT,75,765Tafalla,M.,Myers,P.C.,Caselli,P.,Walmsley,C.M.2004,A&A,416,191Ward-Thompson, D.,Scott,P.F.,Hills,R.E., Andr´e,P.1994,MNRAS,268,276Fig.1.—Iso-density surfaces of the averaged SPH density distributions of a cloud core fragment. The left panel shows densities of40,5,5,and5×10−17kg/m3at the times5.6,14.1,16.9,and 27.2×104yrs after the start of simulation,respec-tively.The right panel gives iso-density surfaces at the time24.4×104yrs and densities of0.16,0.5,1.6,and5.2×10−18kg/m3,respectively.Fig. 2.—Images of the cloud core fragment at the wavelengths7,15,175,and1300µm(left columns from top to bottom),and at the times 5.6and14.1×104yrs after start of simulation(leftto right),respectively.Fig. 3.—Examples of features in7µm images mimicking dense cores.The top panels show struc-tures which look like a core when choosing an appropriate viewing angle(middle panels).The lower panels zoom into the core-like structures andswitch to ISOCAM resolution.Fig. 4.—1D density proﬁles n(r)obtained from the7µm-image by azimuthally integrating the column density around the absorption maximum along elliptical tracks.The thin solid lines mark the range of proﬁles for individual directions,and the direction-averaged proﬁle is represented by the thick solid line.The dashed line indicates an r−2-dependency.The true3D density distribution n SP H was discretized on a cell grid,and trans-formed to a point distribution in the n(r)-plane. The number of points is shown as gray-scale image where black means maximum number of grids with a certain density.The main picture corresponds to an elongatedﬁlament from the early stage,and the inlet to aﬂattened structure from a later stage of the core evolution,respectively.。

The Optimal Isodual Lattice Quantizer in Three Dimensions

V
x · x dx
(1)
lattice has the largest packing radius and the smallest covering radius. The m.c.c. lattice, denoted here by M3 , has Gram matrix √ −1 −1 1+ 2 √ √ 1 −1 1+ 2 1− 2 2 √ √ −1 1− 2 1+ 2 determinant 1, packing radius
(3)
where α, β , h are any real numbers satisfying 0 ≤ 2h ≤ α ≤ β , αβ − h2 = 1. In the indecomposable case the nonzero conorms are: p01 = αβ 2α(1 − β ) α(2 − β ) , p02 = , p03 = , γ γ βγ 2β (1 − α) 2(1 − α)(1 − β ) β (2 − α) , p13 = , p23 = , αγ γ γ (4)
The Optimal Isodual Lattice Quantizer in Three Dimensions
J. H. Conway Mathematics Department Princeton University Princeton, NJ 08544 N. J. A. Sloane AT&T Shannon Labs 180 Park Avenue Florham Park, NJ 07932-0971 Email: conway@, njas@ Jan 02 2006. Abstract The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and covering among all isodual three-dimensional lattices. In this note we show that it is also the best quantizer. It thus joins the isodual lattices Z, A2 and (presumably) D4 , E8 and the Leech lattice in being simultaneously optimal with respect to all three criteria. Keywords: quantizing, self-dual lattice, isodual lattice, f.c.c. lattice, b.c.c. lattice, m.c.c. lattice AMS 2000 Classiﬁcation: 52C07 (11H55, 94A29)

新世纪大学英语综合教程3课后答案Unit6

Electronic Teaching PortfolioBook ThreeUnit Six: HappinessPart I Get StartedSection A Discussion▇Sit in groups of threes or fours and discuss the following questions.1. Are you happy with your college life? Why or why not?2. What was the happiest moment in your life as far as you can remember?3. Can money alone bring happiness? Why or why not?▇ Answers for reference:1. I feel happy with my college life. Away from my parents, I‟m learning to live independently and I enjoy a lot of freedom that I have never experienced before. My life at college is easy and carefree. The teachers are professional and my classmates are friendly and helpful. I have access to lots of modern facilities and I can take part in many interesting activities. So I regard college life as the most enjoyable period in my life.I‟m not feeling so happy, because I‟m not used to the heavy work load and I feel lonely, homesick or bored with school life that has lasted for over ten years.2. (Open.)3. Yes. Because I think money is just like a magic wand (魔杖) that can change everything in the world today. It can bring you all you want, including happiness.No. We cannot deny the fact that money is important. It can help keep us free from want and ensure a happy life. But happiness is based on both material and spiritual welfare. The sense of achievement and self-fulfillment sometimes brings us even greater satisfaction. Money does contribute to material welfare. Yet, material welfare alone cannot bring us happiness.Section B Quotes▇Study the following quotes about happiness. Which quote(s) do you like best? Why?⊙Happiness is the meaning and the purpose of life, the whole aim and end of human existence.—— Aristotle Interpretation:This quotation reveals the important role of happiness in human life. It is human nature to seek and enjoy happiness. Otherwise, human existence would be aimless and meaningless.AristotleAbout Aristotle:Aristotle (384 BC-322 BC): a student of Plato and a famous ancient Greek philosopher. During his life, he wrote numerous books on logic (逻辑学), natural science, ethics (伦理学), politics, and rhetoric (修辞学), etc. His works include Physics, On the Soul (《论灵魂》),Posterior Analytics(《后分析篇》), History ofAnimals (《动物志》),Politics,Rhetoric, Poetics (《诗学》), etc. Aristotle is considered to be one of the greatest thinkers of Europe and his works are still widely quoted today.⊙We have no more right to consume happiness without producing it than to consume wealth without producing it.—— George Bernard Shaw Interpretation:Shaw reminds us that we have two identities both as producer and as consumer. We should produce wealth first and then enjoy it. It is also true of happiness.George Bernard ShawAbout George Bernard Shaw:George Bernard Shaw (1856-1950): an Irish writer famous especially for his plays, which criticize society and the moral values of the time. His best known works include the historical plays Caesar and Cleopatra (《恺撒和克娄巴特拉》) and St Joan (《圣女贞德》), and the comedy Pygmalion (《皮格马利翁》), which was later turned into the popular musical show My Fair Lady (《窈窕淑女》).⊙Happiness lies not in the mere possession of money; it lies in the joy of achievement, in the thrill of creative effort.—— Franklin D. Roosevelt Interpretation:According to Roosevelt, money alone does not mean happiness. True happiness comes when one has succeeded in doing something or when one has done something that has not been done before. The sense of achievement and job satisfaction will bring you immense happiness.Franklin D. RooseveltAbout Franklin D. Roosevelt:Franklin D. Roosevelt (1882-1945): the thirty-second president of the US, from 1933 to 1945. He helped to end the Great Depression (经济大萧条) by starting the New Deal (新政), a program of social and economic changes. He also tried to give support to the Allies (同盟国) without getting the US involved in World War II, but when Japan attacked the US in 1941, he was forced to get the country to join the war. During his lifetime Roosevelt was elected President of the US four times.⊙The surest way to happiness is to lose yourself in a cause greater than yourself. The secret of happiness is not in doing what one likes, but in liking what one does.—— James M. Barrie Interpretation:Barrie believes that we should indulge ourselves in a right cause—a cause that is greater than ourselves. Once we have chosen the right cause, we should learn to love it and pursue it resolutely. If we are dedicated to what we feel obliged to do, we will surely do it well and the resulting sense of fulfillment will bring about truehappiness.James M. BarrieAbout James M. Barrie:James M. Barrie (1860-1937): a Scottish playwright and novelist. He is best remembered for his play Peter Pan (《彼得·潘》), a supernatural fantasy about a boy who refuses to grow up.Section C Watching and Discussion▇Watch the following video clip “Is Culture a Factor in How We View Happiness?” and do the tasks that follow:1. Lynn Ianni (Psychotherapist) gives expert advice on happiness in the video. Pay attention to what she says and fill in the missing words.Eve rybody‟s philosoph y is going to determine what they feel is their optimum level of happiness and what opportunity they have to actually attain it. So there are cultural factors that probably weigh in and as you independently try to figure out whether or not you are happy and what chance you have to increase your level of happiness, those cultural factors are gonna play a part. The messages that you received from your family, from your environment, from your philosophy or religious belief system or cultural belief system are all going to be relevant. And those things are things that you need to explore, understand, analyze, take apart, and modify.2. Discuss the topic with your group members: Do you agree with the psychologist that cultural factors playa pa rt in one’s happiness?▇Answers for reference:(Open.)Script：Is Culture a Factor in How We View Happiness?I think that different cultures have different expectations about what they believe is the optimum level of happiness. And some cultures really teach, you know, survival because of where that nation is or where that culture is in those moments and in that time. So if they allow a person to really believe that happiness is within their grasp that culture has an opportunity to sort of instill that belief across the board. Some cultures really prize self sacrifice, some cultures prize altruism, some belief systems or religious theories. Everybody‟s philosophy is going to determine what they feel is their optimum level of happiness and what opportunity they have to actually attain it. So there are cultural factors that probably weigh in and as you independently try to figure out whether or not you are happy and what chance you have to increase your level of happiness, those cultural factors are gonna play a part. The messages that you received from your family, from your environment, from your philosophy or religious belief system or cultural belief system are all going to be relevant. And those things are things that you need to explore, understand, analyze, take apart, and modify. Because if you want to change how you feel within and those things are factors that are creating the feelings that you actually have especially if they are your cognitive frame work. You have to change the frame work in order for the picture to be different.Part II Listen and RespondSection A Word Bankcommit suicide kill oneself deliberately 自杀lottery▲ n. [C] a system in which many numbered tickets are sold, some of which are later chosen by chance and prizes given to those who bought them 抽彩给奖法mean a. (of people or their behaviour) unworthy; unkind （指人或人的行为）卑鄙的，不善良的unfulfilled a. 未得到满足的；未完成的Section B Task One: Focusing on the Main IdeasChoose the best answer to complete each of the following sentences according to the information contained in the listening passage.1)According to the speaker, happiness _____________.A)is not easily obtained by poor peopleB)is what movie stars are most eager to obtainC)does not naturally follow wealth or successD)necessarily results from wealth or success2)According to the speaker, happiness lies in the following EXCEPT ___________.A)wealth obtained through honest effortB)wealth obtained by winning lotteriesC)your contribution to others‟ happinessD)your successful work3) Instead of being an end, happiness is a(n) ____________.A) beginningB) processC) unattainable goalD) business of the community4) The passage is mainly about ____________.A) the secret to happinessB) the definition of happinessC) the misunderstanding of happinessD) the relationship between happiness and wealth▇ Answers for Reference:1) C 2) B 3) B 4) ASection C Task Two: Zooming In on the Details▇Listen to the recording again and fill in each of the blanks according to what you have heard.1) The world is full of very rich people who are as ___________ as if they were ______________.2) If you obtain wealth through ________ or______________, you will not be happy with it. You will think you are a mean person.3) Long-term happiness is based on ____________, and ______________, contribution, and self-esteem.4) If your happiness depends on ______________, you will always feel unfulfilled because there will always be something ___________.▆ Answers:1) The world is full of very rich people who are as miserable as if they were living in hell.2) If you obtain wealth through luck or dishonest means, you will not be happy with it. You will think you area mean person.3) Long-term happiness is based on honesty and productive work, contribution, and self-esteem.4) If your happiness depends on external circumstances, you will always feel unfulfilled because there will always be something missing.Script:HappinessMany people think that when they become rich and successful, happiness will naturally follow. Let me tell you this is not true. The world is full of very rich people who are as miserable as if they were living in hell. We have read stories about movie stars who committed suicide or died from drugs. Quite clearly, money is not the only answer to all problems.Wealth obtained through dishonest means does not bring happiness. Lottery winnings do not bring happiness. To my mind, the secret to happiness lies in your successful work, in your contribution to others‟ happiness and in the wealth you have earned through your own honest effort. If you obtain wealth through luck or dishonest means, you will not be happy with it. You will think you are a mean person.Long-term happiness is based on honesty and productive work, contribution, and self-esteem.Happiness is not an end; it is a process. It is a continuous process of honest and productive work which makes a real contribution to others and makes you feel you are a useful, worthy person. As one writer put it, “There is no way to happiness. Happiness is the way.” It‟s no use saying, “Someday when I achieve these goals, when I get a car, build a house and own my own business, then I will be really happy.” Life just does not work that way. If you wait for certain things to happen and depend on external circumstances of life to make you happy, you will always feel unfulfilled. There will always be something missing.Part III Read and ExploreText ASection A Discovering the Main Ideas1. Answer the following questions with the information contained in Text A.1) Why did the author bring the news story about Ted Turner to Morrie?2) What problem did Morrie think Ted Turner actually had?3) Who paid a visit to Morrie the night before? And how did he feel about it?4) What did material things mean to Morrie?5) According to Morrie, what are Americans brainwashed into believing? What do they expect from materialthings?6) Does the author think that Morrie was rich? Why or why not?7) What did Morrie suggest that we should do to find a meaningful life?▆ Answers for Reference:1) Because he wondered how Morrie would react to Ted Turner‟s failure in “snatching up the CBS network”.At the same time he wanted to know if Ted Turner would still lament his failure if he were stricken down by the same terminal disease as Morrie was suffering from.2) His problem was a typical one that Americans all have: Americans tend to value the wrong things.3) A local a cappela group came to visit him. He showed an intense interest in their musical performance andfelt excited.4) They held little or no significance to him, especially at a time when he knew his days were numbered. Heseemed to know the expression “You can‟t take it with you” a long time ago.5) They are brainwashed into believing that it is good to own things. Actually, they are hungry for gentleness,tenderness or for a sense of comradeship and, therefore, they desperately seek after material things assubstitutes.6) Morrie was far from better off in material things, but he was wealthy in spiritual ways. For years, Morriehadn‟t bought anything new— except medical equipment. And his bank account was rapidly depleting.But he was rich in love, friendship, caring and he derived plenty of satisfaction and gratification from teaching, communication, and such simple pleasures as singing, laughing, and dancing.7) He advised us to devote ourselves to loving others, to our community around us, and to creating somethingthat gives us purpose and meaning. In other words, if we want to find a meaningful life, we should be ourselves and never show off either for people at the top or for people at the bottom. Instead, we should be kind and candid and ready to offer others what we have to give.2.Text A can be divided into four parts, with the paragraph number(s) of each part provided as follows. Write down the main idea of each part.Paragraph(s) Main IdeaPart One 1-3 ________________________________________________________________________________________________________________________________________________ Part Two 4-9 ________________________________________________________________________________________________________________________________________________ Part Three 10-14 ________________________________________________________________________________________________________________________________________________ Part Four 15-30 ________________________________________________________________________________________________________________________________________________▆▆ Answers for Reference:Paragraph(s) Main IdeaPart One 1-3 The author brought Ted Turner‟s news story to Morrie forhis opinion.Part Two 4-9 Morrie explained that Ted Turner‟s problem was caused bythe endlessly repeated stress on the significance of materialthings.Part Three 10-14 In order to get happiness, people are trying to substitutematerial things for love or tenderness, and they fail todistinguish what they want and what they really need inlife.Part Four 15-30 The way to get satisfaction is to offer with an open heart toothers what you have to give: devote yourself to lovingothers, devote yourself to your community around you, anddevote yourself to creating something that gives youpurpose and meaning.Section B In-Depth StudyOn his graduation, Mitch Albom, the narrator, told his favorite professor, Morrie Schwartz, that he would keep in touch. However, Mitch didn’t resume the contact with his old professor until one night on TV when he saw Morrie being interviewed in a wheelchair. It turned out that Morrie had developed ALS (重症肌无力), a terminal disease (不治之症). Soon Mitch realized that he still had a lot to learn from his teacher. He visited Morrie every Tuesday until the fourteenth one, when Morrie passed away. On those Tuesdays he had “classes”, where Morrie gave lessons and wisdom to him. The text you are going to read is the eighth Tuesday’s class where Morrie talks about what role money or material things are supposed to play in life.The Eighth Tuesday We Talk About MoneyMitch Albom1 I held up the newspaper so that Morrie could see it:2 “I DON‟T WANT MY TOMBSTONE TO READI NEVER OWNED A NETWORK.”3 Morrie laughed, then shook his head. The morning sun was coming through the window behind him, falling on the pink flowers of the hibiscus plant that sat on the sill. The quote was from Ted Turner, the billionaire media mogul, founder of CNN, who had been lamenting his inability to snatch up the CBS network in a corporate megadeal. I had brought the story to Morrie this morning because I wondered if Turner ever found himsel f in my old professor‟s position, his breath disappearing, his body turning to stone, his days being crossed off the calendar one by one — would he really be crying over owning a network?4 “It‟s all part of the same problem, Mitch,” Morrie said. “We put o ur values in the wrong things. And it leads to very disillusioned lives. I think we should talk about that.”5 Morrie was focused. There were good days and bad days now. He was having a good day. The night before, he had been entertained by a local a cappella group that had come to the house to perform, and he relayed the story excitedly, as if the Ink Spots themselves had dropped by for a visit.Morrie‟s love for music was strong even before he got sick, but now it was so intense that it moved him to tears. He would listen to opera sometimes at night, closing his eyes, riding along with the magnificent voices as they dipped and soared.6 “You should have heard this group last night, Mitch. Such a sound!”7 Morrie had always been taken with simple pleasures, singing, laughing, dancing.Now, more than ever, material things held little or no significance. When people die, you always hear the expression “You can‟t take it with you.” Morrie seemed to know that a long time ago.8 “We‟ve got a form of brainwashing going on in our country,” Morrie sighed. “Do you know how they brainwash people? They repeat something over and over. And that‟s what we do in this country. Owning things is good. More money is good. More property is good. More commercialism is good. More is good. More is good. We repeat it — and have it repeated to us — over and over until nobody bothers to even think otherwise. The average person is so fogged up by all this that he has no perspective on what‟s really important anymore.9 “Wherever I wen t in my life, I met people wanting to gobble up something new. Gobble up a new car. Gobble up a new piece of property. Gobble up the latest toy. And then they wanted to tell you about it. …Guess what I got? Guess what I got?‟10 “You know how I always interpreted that? These were people so hungry for love that they were accepting substitutes. They were embracing material things and expecting a sort of hug back. But it never works. You can‟t substitute material things for love or for gentleness or for tenderness or for a sense of comradeship.11 “Money is not a substitute for tenderness, and power is not a substitute for tenderness. I can tell you, as I‟m sitting here dying, when you most need it, neither money nor power will give you the feeling you‟re looki ng for, no matter how much of them you have.”12 I glanced around Morrie‟s study. It was the same today as it had been the first day I arrived. Thebooks held their same places on the shelves. The papers cluttered the same old desk. The outside rooms had n ot been improved or upgraded. In fact, Morrie really hadn‟t bought anything new—except medical equipment — in a long, long time, maybe years. The day he learned that he was terminally ill was the day he lost interest in his purchasing power.13 So the TV was the same old model, the car that Charlotte drove was the same old model, the dishes and the silverware and the towels — all the same. And yet the house had changed so drastically. It had filled with love and teaching and communication. It had filled with friendship and family and honesty and tears. It had filled with colleagues and students and meditation teachers and therapists and nurses and a cappella groups. It had become, in a very real way, a wealthy home, even though Morrie‟s bank account was ra pidly depleting.14 “There‟s a big confusion in this country over what we want versus what we need,” Morrie said. “You need food, you want a chocolate sundae. You have to be honest with yourself. You don‟t need the latest sports car, you don‟t need the big gest house.15 “The truth is, you don‟t get satisfaction from those things. You know what really gives you satisfaction?” What?16 “Offering others what you have to give.”17 You sound like a Boy Scout.18 “I don‟t mean money, Mitch. I mean your time. Your concern. Your storytelling. It‟s not so hard. There‟s a senior center that opened near here. Dozens of elderly people come there every day. If you‟re a young man or young woman and you have a skill, you are asked to come and teach it. Say you know computers. You come there and teach them computers. You are very welcome there. And they are very grateful. This is how you start to get respect, by offering something that you have.19 “There are plenty of places to do this. You don‟t need to have a big talent. There are lonely people in hospitals and shelters who only want some companionship. You play cards with a lonely older man and you find new respect for yourself, because you are needed.20 “Remember what I said about finding a meaningful life? I wrote it down, but now I can recite it: Devote yourself to loving others, devote yourself to your community around you, and devote yourself to creating something that gives you purpose and meaning.21 “You notice,” he added, grinning, “there‟s nothing in there about a salary.”22 I jotted some of the things Morrie was saying on a yellow pad. I did this mostly because I didn‟t want him to see my eyes, to know what I was thinking, that I had been, for much of my life since graduation, pursuing these very things he had been railing against — bigger toys, nicer house. Because I worked among rich and famous athletes, I convinced myself that my needs were realistic, my greed trivial compared to theirs.23 This was a smokescreen. Morrie made that obvious. “Mitch, if you‟re trying to show off for people at the top, forget it. They will look down on you anyhow. And if you‟re trying to show off for people at the bottom, forget it. They will only envy you. Status will get you nowhere. Only an open heart will allow you to float equally between everyone.”24 He paused, then looked at me. “I‟m dying, right?” Yes.25 “Why do you think it‟s so important for me to hear other people‟s problems? Don‟t I have enough pain and suffering of my own?26 “Of course I do. But giving to other p eople is what makes me feel alive. Not my car or my house. Not what I look like in the mirror. When I give my time, when I can make someone smile after they were feeling sad, it‟s as close to healthy as I ever feel.27 “Do the kinds of things that come from the heart. When you do, you won‟t be dissatisfied, you won‟t be envious, you won‟t be longing for somebody else‟s things. On the contrary, you‟ll be overwhelmed with what comes back.”28 He coughed and reached for the small bell that lay on the chair. He had to poke a few times at it, and I finally picked it up and put it in his hand.29 “Thank you,” he whispered. He shook it weakly, trying to get Connie‟s attention.30 “This Ted Turner guy,” Morrie said, “he couldn‟t think of anything else for his tombstone?”▇课文参考译文相约第八个星期二：关于金钱米奇·阿尔博姆1 我举起报纸，让莫里能看见这句话：2 “我不愿意我的墓碑上刻着‘我不曾拥有一个广播电视公司‟。

Two-dimensional Quantum Field Theory, examples and applications

Abstract The main principles of two-dimensional quantum ﬁeld theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum ﬁeld theory. The dynamics of conﬁnement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the eﬀective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks

BPS Geodesics in N=2 Supersymmetric Yang-Mills Theory

USC/97-001 hep-th/9702012BPS Geodesics in N = 2 Supersymmetric Yang-Mills TheoryarXiv:hep-th/9702012v1 2 Feb 1997J. Schulze and N.P. Warner Physics Department, U.S.C. University Park, Los Angeles, CA 90089We introduce some techniques for making a more global analysis of the existence of geodesics on a Seiberg-Witten Riemann surface with metric ds2 = |λSW |2 . Because the existence of such geodesics implies the existence of BPS states in N = 2 supersymmetric Yang-Mills theory, one can use these methods to study the BPS spectrum in various phases of the Yang-Mills theory. By way of illustration, we show how, using our new methods, one can easily recover the known results for the N = 2 supersymmetric SU (2) pure gauge theory, and we show in detail how it also works for the N = 2, SU (2) theory coupled to a massive adjoint matter multiplet.February, 19971. Introduction One of the many remarkable features of string duality is that one has been able to use it to extract new statements about the strong coupling regime of supersymmetric ﬁeld theories. In particular, one can re-derive the quantum eﬀective actions of Seiberg and Witten [1] from the classical eﬀective actions of type II string compactiﬁcations on K3ﬁbrations [2,3]. In the IIB theory, the Yang-Mills BPS states come from 3-branes, and when these are wrapped around 2-cycles in the ﬁbration, the result can be reinterpreted as a sixdimensional self-dual string [4,5,6] compactiﬁed to four dimensions on the Seiberg-Witten Riemann surface, Σ [2]. The BPS states of Yang-Mills theory then become minimum energy winding conﬁgurations of the self-dual string on Σ, where the (local) tension in the string is given by the Seiberg-Witten diﬀerential, λSW . In [1], the diﬀerential λSW was an object whose period integrals gave the central charges, and hence the masses of BPS states. Since one integrated it around cycles, one was only interested in its cohomology class – one was free to add the derivative of any meromorphic function. The stringy approach led to a sharper statement about the BPS states: it is only a speciﬁc local form of λSW that has the interpretation of a string tension on Σ, and the existence of BPS states with given monopole and electric charges (g, q), is equivalent to the existence of a geodesic with these winding numbers on Σ with the metric ds2 = |λSW | .2(1.1)Thus a statement about the stability and existence of strong quantum BPS states is reduced to a classical computation. There have been several attempts to use the foregoing as a tool to probe the BPS structure of the theory [3,7,8,9], but existence of geodesics can be subtle to establish, particularly if one proceeds numerically. One would also like to see precisely what happens as one crosses inside a curve of marginal stability, where some of the BPS geodesics must “cease to exist”. A proper understanding of this has so far proven elusive. Our purpose in this paper is to introduce some analytical tools by which these issues can be addressed. The key idea is to look for “geodesic horizons”. These are maximal, closed geodesics that surround poles in λSW , and have the property that once crossed by a BPS geodesic, they can never be recrossed (hence the name “horizon”) by a BPS geodesic of ﬁnite energy. We ﬁnd that, at least in the SU (2) pure gauge theory, and in the SU (2) gauge theory with adjoint matter, these geodesic horizons conﬁne the winding states on Σ in a manner that provides a simple geometric understanding of the BPS spectrum. Since 1these geodesic horizons are straighforward to characterize and their behaviour at curves of marginal stability can be easily addressed, we anticipate that they will provide a valuable technique in using geodesic methods to analyze the BPS spectrum of more complex models than the ones discussed here. We start in the next section by introducing geodesic horizons and deriving some of their properties. We then describe an illustrative, but unphysical “toy model”. In section 3 we obtain some relatively simple analytic expressions for the indeﬁnite integrals of the Seiberg-Witten diﬀerential in the softly broken N = 4, SU (2) gauge theory, and in its N = 2 supersymmetric pure gauge limit. In section 4 we analyze the BPS spectra in various phases of these gauge theories by using geodesic horizons, and the “shadows” cast by such horizons.2. BPS geodesics and horizons There are a few implicit issues in the geodesic characterization of BPS states, and these need to be brought into the open. First, the geodesic characterization has only been carefully established for pure gauge and for the N = 4 supersymmetric model. While it is almost certainly true in greater generality, one still needs the stringy derivation in order to get the proper local form of λSW so as to properly describe the local string tension. Such a stringy derivation has been obtained for the softly broken N = 4 supersymmetric, SU (2) gauge theory [10]. More generally one appeals to the underlying integrable hierarchy to posit the proper local form of λSW . That is, one ﬁnds that the indeﬁnite integral of the diﬀerential, λSW , selected by the string theory is the Hamilton–Jacobi function of the integrable hierarchy that underlies the construction of the eﬀective action [11,12,13,14]. It is natural to assume that this remains true in general. The second issue concerns what constitutes a BPS geodesic. The geometrical origin from 3-branes in ten dimensions implies that states with magnetic charge must begin and end at “branch points of the ﬁbration” [3], whereas purely electric states merely wind with no speciﬁc base point. From the point of view of the Riemann surface, the branch point is a mere coordinate artefact – the invariant statement of this comes from the fact that λSW has zeroes at the branch points of the ﬁbration. We therefore take BPS geodesics to be those that begin and end at the zeroes of λSW . Given a set of winding numbers (g, q) for a BPS state there is always a corresponding (minimum length) geodesic: one considers all curves with the same ﬁxed end-points and the same winding numbers, and then minimizes the length within this homotopy class. 2This geodesic is then either reducible or irreducible. A geodesic will be called reducible if it is the concatenation of two other BPS geodesic curves. It is thus reducible if it runs into a zero of λSW at an intermediate point along its length. If the shortest curve in the proper homotopy class is reducible, then the corresponding BPS state is the sum of two others, and there is no fundamental Yang-Mills BPS state with these quantum numbers. The BPS spectrum is thus characterized by irreducible BPS geodesics, and transitions in the spectrum must correspond to irreducible geodesics becoming reducible. The geodesic equation for (1.1) has a trivial implicit solutionzw ≡z0λSW (z) = α t ,(2.1)where z0 is the starting point, t is the parameter and α is a constant of integration. The solution is unique provided that the tangent vector is continuous, and the only way the tangent can fail to be continuous is if the curve runs into a zero of λSW – i.e. if the geodesic is irreducible. Therefore the irreducible geodesics can be obtained by solving the initial value problem having used the winding numbers to ﬁx the constant, α. That is, if a and aD are the periods of λSW , then the irreducible geodesic with charges (g, q), if it exists, must be obtained by taking α = q · a + g · aD and letting t run from 0 to 1. If this method produces a curve that does not terminate (at t = 1) at a zero of λSW , then the BPS geodesic with winding numbers (g, q) must be reducible. The latter method is eﬀective, and was used in [3,7], but it is somewhat implicit. To see transitions in the BPS spectrum it is easier to think in terms of minimizing the lengths of curves within a homotopy class, and ﬁnding that these curves move towards another zero of λSW as one approaches a curve of marginal stability. Finally, we note that since the geodesic equation is solved by (2.1), it follows that the length of a geodesic, Γ, is given by: L(Γ) =irr. segs. γL(γ) =irr. segs. γ γλSW (z).(2.2)2.1. Geodesic horizons We wish to show how the existence of certain stable BPS states precludes the existence of others. To see this most clearly we will henceforth work in the covering space, Σ, of the Riemann surface, Σ, lifting the metric and curves in the obvious manner. In particular we will now take λSW (z) and the integral (2.1) on Σ. We will also be interested in the 3intersections of various geodesic curves, and by this we will mean intersections on Σ. The idea is to see how BPS geodesics partition the covering space, and this is all a consequence of three simple facts: (i) geodesics are straight lines in the w-plane (where w is deﬁned in (2.1)), (ii) geodesics representing BPS states of ﬁnite energy are ﬁnite line segments in the w-plane , and (iii) ﬁnite line segments (in the w-plane) that intersect more than once must coincide along a common segment. An irreducible BPS geodesic must consist of a single line segment in the w-plane, beginning and ending at zeroes of dw/dz = λSW , but not encountering any other zeroes of λSW along its length. A reducible geodesic may still be a single line segment, but generically will be a collection of such segments, and might involve traversing the same segment twice. One is thus tempted to conclude that a pair of irreducible geodesics can only intersect once, but there is a minor subtlety in (iii). Suppose two geodesics on Σ intersect at two points, P1 and P2 , then either (a) the two geodesics coincide between P1 and P2 on Σ, or (b) the geodesics lie on diﬀerent branches of z(w), and these branches meet at P1 and P2 . In the latter instance, P1 and P2 must either be poles or zeroes of λSW . Thus if two irreducible geodesics intersect more than once then they are either identical or they have common endpoints. We will say that a closed curve, Γ, has the horizon property if any ﬁnite irreducible geodesic that crosses Γ can never recross it (or even return to it). We will call a closed curve a geodesic horizon if a) it is a closed geodesic that has the horizon property, and b) surrounds a region that contains no zeroes of λSW . We will, however, allow geodesic horizons to pass through zeroes of λSW , and we further deﬁne a geodesic horizon to be reducible if it passes through two or more distinct zeroes of λSW . The irreducible components of a geodesic horizon are ﬁnite line segments in the w-plane. (In some circumstances there will be inﬁnitely many concentric geodesic horizons, and so we will subsequently reﬁne this deﬁnition to only mean the outermost, or maximal such horizon.) The whole point of excluding zeroes from the interior is that it ensures that any irreducible geodesic that crosses a geodesic horizon can never meet a zero, and thus cannot represent a fundamental BPS state. We now show that every pole, z0 , of λSW is surrounded by at least one geodesic horizon. Consider the family of simple, closed curves (in Σ) that satisfy the following: (i) The only pole of λSW that they contain is z0 itself. (ii) They do not surround any zeroes of λSW (but can pass through such zeroes). 4Within this homotopy class there is at least one curve of minimum length: A candidate is any geodesic polygon connecting zeroes of λSW , and surrounding the pole. One might ﬁnd a shorter curve inside the polygon, but the curve of minimal length cannot be trivial since the length of the curves becomes inﬁnite as they approach the pole. These closed curves of minimum length are geodesics, and the minimality of their length means that they must be simple (i.e. not self-crossing). We now show that any such closed curve, Γ, has the horizon property. If Γ is irreducible then the horizon property is almost obvious. The only ﬁnite irreducible geodesic (apart from Γ itself) that can meet Γ more than once is a geodesic that meets Γ only at the (single) zero of λSW that lies on Γ. Such a geodesic thus cannot be irreducible and cross Γ. Suppose that Γ is reducible and that an irreducible geodesic, γ meets Γ at two points, P1 and P2 . Further suppose the γ actually crosses Γ at one of these points, P1 . It follows that P1 cannot be a zero of λSW , since this would violate the irreducibility of γ. Let Γ1 and Γ2 be segments of Γ between P1 and P2 , and let γ0 be the segment of γ between P1 and P2 . Since w(γ0 ) is a single line segment between w(P1 ) and w(P2 ) in the w-plane, it follows that L(γ0 ) ≤ L(Γi ), i = 1, 2, with equality if and only if w(Γi ) = w(γ0 ). Since P1 is not a zero of λSW , the latter equality would imply that that Γi = γ0 . Now observe that both Γ1 ∪ γ0 and Γ2 ∪ γ0 are closed curves, neither contains a zero of λSW 1 , and one of them contains the pole. Suppose that it is the former. Minimality of Γ then requires L(Γ2 ) ≤ L(γ0 ), and hence there must be equality. From the comment above, we therefore conclude that γ0 = Γ2 , and hence γ cannot cross into the interior of Γ. Thus any irreducible closed geodesic around a pole has the horizon property, but a reducible closed geodesic crucially needs to be the geodesic of globally minimal length in order to have the horizon property. Consider now a closed geodesic Γ of minimum length that surrounds several poles. The foregoing argument fails, but in an interesting way. If the geodesic γ goes between the poles so that the total residue in both Γ1 ∪γ0 and Γ2 ∪γ0 is non-zero, then γ can recross Γ. If however, one of the two residues is zero, then it follows from (2.2) that L(γ0 ) ≤ L(Γi ), and the argument still goes through. In terms of BPS states this means that a BPS geodesic can emerge from Γ, provided that the BPS state picks up a hypermultiplet charge from the interior that is diﬀerent from the total hypermultiplet charge enclosed by Γ. We will refer to closed curves like Γ as selective horizons.1This follows because Γ must be simple.5We now need to reﬁne the deﬁnition of irreducible geodesic horizons since there can be inﬁnitely many of them (see the example below). If γ1 and γ2 are two irreducible geodesic horizons around the same pole, then one geodesic must lie inside the other. (They cannot cross since they are closed, and would thus have to cross twice.) To make the deﬁnition more useful, if there is more than one irreducible horizon around a pole then take the geodesic horizon to be the maximal one, that is, the one that is contained in no other such horizon. Again, we exclude the possibility of zeroes of λSW from the interior of the region surrounded horizon. By deﬁnition, a reducible horizon must encounter at least two zeroes of λSW upon its length. It may thus be thought of as a sum of BPS states whose net electric and magnetic charges are zero. We now show that the (maximal) irreducible geodesic horizons must meet one zero of λSW . First observe that an irreducible horizon can only surround a pole with a non-zero residue, and conversely, the horizon around a pole with no residue must necessarily be reducible. This is a trivial consequence of (2.2) and residue calculus. Let the residue of the pole be m, then closed irreducible geodesics around the pole are given by the straight lines between some point w0 and the point w0 + 2πim. Let γ be the “outermost” such closed geodesic, and let w1 be a point on it. Suppose that λSW is non-zero everywhere on γ, then we can locally invert to get z(w) around γ. Look at all straight lines, Sǫ , running from w1 + ǫ to w1 + ǫ + 2πim, and consider preimages, γǫ , in the z-plane of Sǫ . Since γ is maximal, then no matter how small one chooses |ǫ| there must always be some γǫ that is an open curve. Let z1 be the limit point where the γǫ ’s ﬁrst open out. By considering the integral along γǫ , and upon a small segment δz that closes γǫ to a loop around the pole, one easily sees that λSW = dw/dz must vanish at z1 . To summarize, all geodesic horizons must be closed geodesics loops that run through at least one of the zeroes of λSW . They must consist of irreducible geodesic segments, which can be thought of as BPS states. Irreducible horizons can only surround poles with residues, and horizons around poles with no residue must necessarily be reducible. Most imporantantly, any irreducible geodesic that crosses a horizon can never recross the horizon, and so BPS geodesics, fundamental or composite, can only exist if they lie outside, or tangent to, such horizons. Thus there are “BPS horizon states” whose existence and behaviour determines the existence and behaviour of all the other BPS states. 62.2. A toy exampleTo illustrate these ideas we consider an unphysical example with many of the features of important physical examples.Consider the conformal mapping: w = z + 1 log 2z2 z 2 −1z−1 z+1−iπ . 2(2.3)The corresponding diﬀerential is λ =has a double zero at z = 0, and two simplepoles with residues ±1/2. Since w has a triple zero at z = 0, straight lines in the wplane through w = 0 turn a 60◦ corner at z = 0. The right half-z-plane maps onto the combination of the right half-w-plane and the strip {w : Re(w) < 0, −π ≤ Im(w) < 0} (see Fig. 1). The top and bottom of strip may be periodically identiﬁed, making the cylinder that can be associated with the conformal map z → log(z). The left half-zplane maps in the reﬂected manner: to the left half-w-plane and the strip, or cylinder {w : Re(w) > 0, −π ≤ Im(w) < 0}. The two patches are glued together along two parts of the imaginary w-axis: Im(w) > 0 and Im(w) < −π. The points w = 0 and w = −iπ are to be identiﬁed. As a result, circling by 2π around z = 0 results in a circle of 6π in the w-plane.The w-plane thus looks like the z-plane but with two semi-inﬁnite cylinders sewn into it: one on each side of the line Re(w) = 0, −π < Im(w) < 0. Straight lines parallel to the imaginary w-axis, and lying in the strips are closed geodesic loops around the simple poles in the z-plane. The geodesic horizons are both mapped to the straight line in the w-plane across the necks of the “cylinders,” i.e. running along Re(w) = 0, −π ≤ Im(w) ≤ 0 (see Fig. 1.). Clearly, any straight line in the w-plane that crosses one of these horizons will spiral down the cylinder, never to return. 71.510.50-0.5-iπ-1-1.5 -1.5-1-0.500.511.5Fig. 1: The shaded region shows the section of the w-plane that maps onto Re(z) > 0. The second diagram shows the z-plane, and the curves correspond to straight lines parallel to the imaginary w axis. Note the two lobes that make up the geodesic horizons and the closed orbits inside these horizons.Finally, there is a useful physical model of the BPS geodesics that is valid for any meromorphic map w(z). The real and imaginary parts of w are harmonic functions of z, and so straight lines in the w-plane can be thought of as equipotentials of some twodimensional ﬁeld in the z-plane. In models with logarithmic singularities, this perspective is perhaps most useful when we look at lines that select the real part of the logarithm. The singularities can then be thought of as charges. In the example above, straight lines parallel to the imaginary w-axis can be thought of as equipotentials of a uniform electric ﬁeld in the x-direction that has been perturbed by equal and opposite charges at z = 1 and z = −1 (see Fig. 1).3. N = 2, SU (2) Yang-Mills theory with adjoint matter 3.1. The curve and diﬀerential In the formulation of [14,15,16], the Riemann surface and diﬀerential for N = 2, SU (2) Yang-Mills theory with adjoint matter can be deﬁned as follows. One starts with the Weierstraß torus, y 2 = 4 (˜ − e1 )(˜ − e2 )(˜ − e3 ) , ˜ x x x (3.1) and constructs a genus two double cover via ˜ t2 − u = m2 x . ˜ 8 (3.2)˜ The Weierstraß torus can be uniformized in the familiar manner by taking x = ℘(ξ), ˜ ˜ y = ℘′ (ξ). To avoid confusion later, we will denote the modular parameter of this torus ˜ by τ . The diﬀerential is ˜ x ˜ ˜ ˜ d˜ . λSW = t dξ = t (3.3) y ˜ The genus two curve deﬁned by (3.2), and diﬀerential (3.3) have an involution symmetry ˜ y → −˜, ˜ → −t, and the torus of the SU (2) eﬀective action is obtained by dividing out ˜ y t ˜ this symmetry. That is, one takes z = ty /m, and replaces t2 using (3.2), to obtain ˜ ˜˜ z 2 = (˜ + u/m2 )(˜ − e1 )(˜ − e2 )(˜ − e3 ) . ˜ x x x x (3.4)One can map this to the cubic form of [1] by using a fractional linear transformation [14], however we will map it to diﬀerent cubic form that is better adapted to the study of the m → ∞ limit. To this end, introduce x = m2 (˜ + b)/(˜ + u/m2 ) and y = (x − m2 )2 z /m x x ˜ 1 2 where b = 3e1 (e1 + 2e2 e3 ). After some judicious rescaling of x and y the curve and diﬀerential reduce to the form: y 2 = (x − µ2 )(x2 − Λ4 ) , where Λ2 = m2 (e2 − e3 ) , 3e1 µ2 = m2 u − m2 b , u + m2 e1 c0 = m m4 − Λ4 . m2 − µ2 (3.6) λSW = c0 (x − µ2 ) dx , (x − m2 ) y (3.5)The whole point of this formulation is that the curve is exactly that of the pure gauge theory. Indeed to get this limit one takes m → ∞ along with τ → i∞. One then has ˜ 2πi˜ τ (e2 − e3 )/e1 → 24 e , and one takes the double limit so that Λ remains ﬁnite. One must also make the inﬁnite additive renormalization: u → u + m2 b, in order that the parameter µ remain ﬁnite. We now take Λ, µ and m as the fundamental parameters of the theory, and we will let τ denote the Teichm¨ller parameter of the torus (3.5). One can easily verify u that λSW does indeed have the property that if one diﬀerentiates it with respect to µ one gets the holomorphic diﬀerential. The diﬀerential, λSW , has two simple poles with residues ±m, and a double zero at x = µ2 . In the pure gauge limit the two poles coalesce into a double pole. In the massless, N = 4 limit (m2 → µ2 ) these two poles move onto the double zero and all three annihilate each other. The ﬁnite mass theory thus looks like a lattice repetition of the toy example above. Before proceeding to a more detailed analysis of the horizons we wish to give some rather useful explicit formula for w as a function of z. 93.2.Integrating λSWOne can evaluate the indeﬁnite integral in (2.1),and to do this we go to the isoge-nous double cover and uniformize it.That is,set x =µ2+(Λ2+µ2)t 2,and then take t =cn(ξ,k )/sn(ξ,k ),where sn and cn are Jacobi elliptic functions,and ξis the ﬂat coor-dinate of the torus.One then ﬁnds thatλSW =−c (1−sn 2(ξ,k ))dξθ3(0|τ)4=2Λ22Λ2,c =2m (m 2−µ2)(Λ2+µ2)=2mk 2sn(α,k )cn(α,k )Λ2+µ2cn 2(ξ,k )Θ(ξ+α) ,(3.10)where w 0is a constant of integration and,following the notation of [17],Θ(ξ)≡θ4 ξΘ(ξ).(3.11)For the pure gauge theory one getsw =w 0−2Λ2+µ2 E ξ+πθ1(πξ/2K ) ,(3.12)where,following the usual convention E and K are elliptic periods.Explicitly,one has:K ≡π3 2−k 2−θ′′′1(0|τ)/(θ′1(0|τ)θ3(0|τ)4) K .(3.13)Expressions for the integral of λSW in the pure gauge theory were also given in [18,19].10From(3.10)it is trivial to read oﬀthe periods a and a D.One getsa=2(2m Z(α)−c)K+2πin1m,(3.14)a D=2(2m Z(α)−c)iK′+2πim(α/K+n2),where K′=−iτK,and n1and n2are the winding numbers around the simple poles at ξ=iK′±α.Similarly,for(3.12)one getsa=4 Λ2+µ2 2τE−πgeodesic horizons.To do this one simply needs to look for all geodesics that start and ﬁnish at zeroes and surround poles ofλSW.We start by showing how the geodesic method easily replicates the known results for the pure gauge theory[1,20,18].4.1.The pure gauge theoryThe Seiberg-Witten diﬀerential has a double pole(with vanishing residue)and a dou-ble zero.In the uniformization used in the last section,they are located atξ=K and ξ=0(mod2K and2iK′)respectively.The geodesic horizon is thus reducible,its irre-ducible parts must correspond to a collection of BPS states.For largeµ,the W-boson is the BPS state of lowest mass,and so the geodesic horizon for largeµmust be the pair of geodesics corresponding to a W+and W−connecting two zeroes above and below the double pole.This fact is born out by the manifest horizons in Fig.2.These horizons mean that any BPS state can only cross the W-boson trajectory at the zero ofλSW.It follows immediately that any BPS state of magnetic charge larger than one must be reducible to BPS states of magnetic charge one.Thus the existence of the stable W-boson implies that only the W-boson itself,and states with magnetic charge one can be stable2.At the curve of marginal stability one knows that the W±-bosons become unstable to decay into states of monopole and electric charges±(1,0)and±(1,−1).This is very clearly seen in even the most rudimentary numerical calculation of the geodesic horizons (see,for example,Fig.3).Asτapproaches the curve of marginal stability,the W±-boson geodesic horizons deform to the zeroes ofλSW located atξ=K±2iK′(mod2K and 2K′).At and beyond the curve of marginal stability,the geodesic horizon around the pole is the quadrilateral with edges(±1,0)and(∓1,±1).These quadrilateral horizons in every fundamental region of the torus completely block passage of BPS states in every direction: To get a winding number of more than±1in either the(1,0)or(−1,1)directions,a BPS state can only pass through the zeroes at the quadrilateral corners,and thus must be decomposable.Thus the fact that the W boson is unstable to a monopole-dyon pair inside the curve of marginal stability implies that the monopole and dyon are the only stable BPS states.Fig.3:These two diagrams show several geodesics and a single geodesic horizonaround a double pole at(0,2).Theﬁrst diagram hasτ=0.3+1.0i,and the secondhasτ=0.3+0.654i.The curve of marginal stability passes through the pointτ=0.3+0.651i.As one approaches the curve of marginal stability,the geodesic horizon deformsto the zeroes at(1,2)and(3,−2).On and inside the curve of marginal stability thegeodesic horizon becomes the quadrilateral with vertices(1,0),(1,2),(3,0)and(3,−2).One can,in fact,infer the entire structure of the BPS spectrum without doing any computations or simulations.From the diagrams in[3]one can immediately see that there is a curve(homotopic to a circle)in the parameter space on which the W-boson is marginally stable to a monopole and a dyon.Thus the geodesic horizon must deform and touch another zero ofλSW as described above.The fact that the quadrilateral horizon persists inside the curve of marginal stability is then inferred from the monodromies of the theory.Speciﬁcally,one considers what happens asτof the torus is decreased from i∞to near the real axis.When Im(τ)is decreased far enough,one must get to a modular inversion of the strong coupling region again.This means that the geodesic horizons must skip from surrounding a-cycles of the torus to surrounding some combination of a and b-cycles.Such horizon jumps can only occur at curves of marginal stability,and the only way that it can happen so that the monodromies of the theory is respected is if the quadrilateral forms and persists inside the curve of marginal stability,and then it detaches to leave a horizon along the a or b cycle depending on where one crosses the curve marginal stability considered as a function ofτ.The beauty of the foregoing argument is that it only cares about the modular proper-ties of the theory,and how the zeroes and poles(the divisor)ofλSW moves as a function of the parameters.134.2.The softly broken N=4theoryThis diﬀerentialλSW now has three independent periods,and as we will see,the BPS spectrum is richer,but computable.Considerﬁrst the unbroken N=4theory.As can be seen from(3.5),the diﬀerential λSW collapses to the holomorphic diﬀerential of the torus:dxy2=x(x2−Λ4),λSW=c0。

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a rXiv:h ep-th/95413v125A pr1995HD-THEP-95-17Bosonisation in Three-Dimensional Quantum Field Theory R.Banerjee*Institut f¨u r Theoretische Physik Universit¨a t Heidelberg Philosophenweg 16,D-69120Heidelberg Germany Abstract We show in three dimensions,using functional integral techniques,the equivalence between the partition functions of the massive Thirring model and a gauge theory with two gauge ﬁelds,to all orders in the inverse mass,are derived.Specialisation to the lowest (andnext to lowest)orders reveals that the gauge theory simpliﬁes to the Maxwell-Chern-Simons theory.Some interesting consequences of the mapping are discussed in this case.The mapping of bosonic theories into fermionic ones and vice versa,commonly known as bosonisation(fermionisation),provide a powerful approach to study(in1+1dimen-sions)the non-perturbative behaviour of either quantumﬁeld theories[1]or condensed matter systems[2].Although bosonisation is reasonably well understood in1+1dimen-sional theories[3],the situation is much less clear in higher dimensions.This is because a Mandelstam[1]-like operator construction is non-trivial in higher dimensions.Similarly, Schwinger terms in the current algebra in these cases is also rather involved so that bosoni-sation identities relating the currents in the diﬀerent(bosonic or fermionic)theories cannot be so easily deduced as happens in1+1dimensions.Very recently,however,Fradkin and Schaposnik[4]have shown the equivalence(in2+1dimensions)of the massive Thirring model,to the leading order in the inverse mass,with the Maxwell-Chern-Simons gauge theory.This work may be viewed in the perspective of ideas initiated by Polyakov[5] and elaborated by Deser and Redlich[6],who had revealed a connection between the2+1 dimensional CP1model with a Chern-Simons term and of a charged massive fermion,to lowest order in the inverse fermion mass.The extension of theseﬁndings to higher orders is problematic and even leads to ambiguities[6].In the present paper we discuss bosonisation in2+1dimensions,similar in spirit to [4],within a path-integral framework.The partition function of the massive Thirring model(MTM)is shown to be equivalent to allorders in the inverse fermion mass,mapping the Thirring current and its dual with corresponding(dual)ﬁeld strengths in the gauge theory,are provided.A similar mapping for the free term of the MTM is also given. The results of[4],which are valid only in the leading order,are reproduced when the MTM is identiﬁed with the Maxwell-Chern-Simons(MCS)theory.Furthermore,in the next-to-leading order,the MTM is still shown to be equivalent to the MCS theory but the mass of the gauge boson is renormalised.Application of the bosonisation identities inthis case show that for the MTM;(i)the current satisﬁes a self-duality relation,(ii)there is a Schwinger term in the current algebra and(iii)the energy-momentum tensor can be expressed in a Sugawara[7]form.Consider the following(2+1dimensional)master Lagrangian,L=¯ψi(i/∂−m−λN/f)ψi−1Boseﬁelds and was used in[4]to discuss bosonisation.The Lagrangian(1)is invariant under the independent U(1)gauge transformations,ψi→e iα(x)ψi,fµ→fµ−√λ∂µα,Aµ→Aµ−∂µβ(2)while the equations of motion obtained by varying the diﬀerentﬁelds are(i/∂−m−λN/f)ψi=0(3a)jµ=√λǫµνλ∂νAλ=√λFµ(3b)ǫµνλ∂νfλ=∂νFµν(3c)wherejµ=¯ψiγµψi(4) is the gauge-invariant(conserved)U(1)current and we have introduced the dualﬁeld strength Fµ.The generating functional for(1)in the presence of external sources Jµ,˜Jµcoupled to gauge-invariantﬁelds is given byZ= d[ψ,¯ψ,fµ,Aµ]δ(∂µfµ)δ(∂µAµ)e i d3x[L+FµJµ+˜fµ˜Jµ](5)where˜fµ=λ2ǫµνλ∂νfλis the dual to fµnormalised byλ2so that the sources have identical dimensions,and a covariant gauge has been chosen for both fµand Aµﬁelds. The Gaussian integration over Aµis easily performed by implementing the gauge∂µAµ=0 using’t Hooft’s prescription to yieldZ= d[ψ,¯ψ,fµ]δ(∂µfµ)e i d3x[¯ψi(i/∂−m−λN/f)ψi+12Jµ(gµν−∂µ∂ν√2(fµ+∂µβ)(fµ+∂µβ)−12Jµ(gµν−∂µ∂ν√2hµhµ+hµ(Jµ+λ2ǫµνλ∂ν˜Jλ)+1√√4(∂µ˜Jν−∂ν˜Jµ)2−λ2ǫµνρJµ∂ν˜Jρ](9a)whereL MT M=¯ψi(i/∂−m)ψi−λ2Alternatively,doing the fermionic integration in (5)just amounts to evaluating the fermion determinant in presence of the external ﬁeld f µ.This is a well-known [6,12,14]gauge-invariant expression computed in inverse powers of the fermion mass which,when substituted in (5)yieldsZ = d [f µ,A µ]δ(∂µf µ)δ(∂µA µ)eid 3x [−18πǫµνλf µ∂νf λ+λ2m 2ǫµνλf µ∂ν⊔⊓f λ)+F µJ µ+˜f µ˜J µ](11)In the absence of sources,(11)represents an interacting gauge theory involving two ﬁelds f µand A µ.Since (9)and (11)were derived from a common origin (5),we conclude the equivalence of the partition functions associated with the MTM and a gauge theory (GT),i.e.Z MT M =Z GT (12)where Z GT is the r.h.s.of (11)in the absence of sources.Note that Z GT in the lowest (up to m −1)order is just the master expression considered in [10]to discuss the connection[8,15]between the self-dual model of [11]and the MCS theory.It is now possible to deduce the bosonisation identities by comparing the source terms appearing in (9)and (11).We ﬁnd,modulo non-propagating contact terms,the following mappingsj µ↔√λF µ=√λǫµνρ∂νA ρ(13a )˜j µ↔√λ˜f µ= √λ λ2ǫµνρ∂νf ρ(13b )where j µand ˜j µare deﬁned in (4)and (10),respectively.Thus gauge invariance (and conservation)of the Thirring current and its dual are preserved.The above relations are manifestations of the equations of motion (3b,c).This happens because of the Gaussian nature of the problem.Finally,observe that (13a)leads to the identiﬁcation of the current-current interaction term in (9b)with the conventional Maxwell term in (11).This allowsthe further correspondence¯ψi(i/∂−m)ψi↔ǫµνλfµ∂νAλ−λ224πm(∂µfν−∂νfµ)2+0 1orders in the inverse fermion mass.It is now simple to reproduce the results in[4]valid to the leading order(i.e.up to14F2µν+2πλ2ǫµαβAµ∂αAβ+FµJµ+(4πFµ−8π3mλ2)Fµ(Jµ+4π˜Jµ)−8π4F2µν+Mλ2 1+8πMǫµνλ∂νFλ(17)expressed in terms of the dual(3b).Thus,up to0(m−2),we concludeZ MT M≈Z MCS(18)In the leading(up to m−1)order this equivalence was demonstrated in[4].The mass of the MCS gauge boson found there was4π/λ2which agrees with the leading order term in M given in(16b).Comparing the source terms in(9)and(16)reveals the following bosonisation relations between the gauge-invariant operators in the MTM(written on the l.h.s.)and MCS theory(written on the r.h.s.)jµ↔√λ 1+4πN3mλ2 Fµ−8π√3mλ∂νFνµ(19b)Using(19a)to recast the current-current interaction term in(9)as a Maxwell piece that may be identiﬁed with(16b)leads to the further mapping,¯ψi(i/∂−m)ψi↔2π3mλ2 ǫµνλAµ∂νAλ+2π4π 1−8πMCS theory.Since the latter is known[16],one can immediately compute the relevant algebra,i[j0(x),j0(y)]=0i[j0(x),j i(y)]=−N3mλ2 ∂iδ(x−y)i[j l(x),j m(y)]=4πN3mλ2 ǫlmδ(x−y)(22)We therebyﬁnd the existence of a Schwinger term whose structure is reminiscent of the 1+1dimensional model.It is easy to verify that(22)is compatible with the self-duality relation(21).Let us next compute the energy momentum tensor in the MTM(valid up to0(m−2)) by recalling that the corresponding tensor in MCS theory is given in terms of Fµby[8,16]ΘMCSµν=FµFν−gµνN 1−8π2jβjβ)(24) which,interestingly,has the Sugawara[7]structure.The“quantum”nature of the above construction is manifested by the normalisation which is just the inverse of the normali-sation in the Schwinger term(22).Exactly the same phenomenon occurs in1+1dimen-sions[3]where the equivalence of the Sugawara form with the conventional(Noether) energy momentum tensor has also been explicitly shown[17]using operator product expansion techniques.Furthermore,using(22),the conservation of the current jµ,as well as the self-dual relation(21)follow from(24),as expected.Ourﬁndings show that(19)and(21)may truly be regarded as“analogues”of corresponding mappings jµ∼ǫµν∂νφ,jµ5∼∂µφ,jµ∼ǫµνjν5known to exist in1+1dimensions.To conclude,we have shown(in2+1dimensions)by starting from a novel master Lagrangian containing both Bose and Fermiﬁelds,the equivalence,on the level of partition functions,of the massive Thirring model(MTM)and a gauge theory with twoto allorders,mapping the current(and its dual)in the MTM to dualﬁeld strengths(corresponding to the two gaugeﬁelds)in the gauge theory are deduced.A similar identiﬁcation is also found for the free part of the MTM.We next specialise up to 0(m−2)(i.e.the order up to which the Seeley coeﬃcients in the expansion of the fermion determinant are known[6](see11))computations whereby one of the gaugeﬁelds can be eliminated.To this order the MTM gets identiﬁed with the Maxwell-Chern-Simons(MCS) theory whose spectrum has a vector gauge boson with mass4π3mλ2 .In the leading (i.e.up to m−1)order this reproduces the result derived in[4].Detailed bosonisation identities(up 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