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All Abelian Symmetries of Landau-Ginzburg Potentials

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arXiv:hep-th/9211047v2 20 Nov 1992CERN-TH.6705/92

ITP–UH–10/92

hep-th/9211047

ALLABELIANSYMMETRIESOFLANDAU–GINZBURGPOTENTIALS

MaximilianKREUZER∗

CERN,TheoryDivision

CH–1211Geneva23,SWITZERLAND

and

HaraldSKARKE#

Institutf¨urTheoretischePhysik,Universit¨atHannover

Appelstraße2,D–3000Hannover1,GERMANY

ABSTRACT

Wepresentanalgorithmfordeterminingallinequivalentabeliansymmetriesofnon-

degeneratequasi-homogeneouspolynomialsandapplyittotherecentlyconstructedcomplete

setofLandau–GinzburgpotentialsforN=2superconformalfieldtheorieswithc=9.A

completecalculationoftheresultingorbifoldswithouttorsionincreasesthenumberofknown

spectrabyaboutonethird.Themirrorsymmetryofthesespectra,however,remainsatthe

samelowlevelasforuntwistedLandau–Ginzburgmodels.Thishappensinspiteofthefact

thatthesubclassofpotentialsforwhichtheBerglund–H¨ubschconstructionworksfeatures

perfectmirrorsymmetry.WealsomakefirststepsintothespaceoforbifoldswithZZ2torsions

byincludingextratrivialfields.

CERN-TH.6705/92

ITP–UH–10/92

October19921Introduction

Landau–Ginzburg(LG)models[1,2]representafairlygeneralframeworkforconstructingN=2superconformalfieldtheories,whichareneededforsupersymmetricstringvacua[3].

Theyprovide,forexample,alinkbetweenexactlysolvablemodelsandCalabi–Yaucompact-

ifications[4],andalsocontainlargeclassesofsuchmodelsasspecialcases.Ingeneral,LG

theoriescannotbesolvedexactly.Still,somebasicinformationonthemasslessspectrumof

theresultingstringmodelscanbeextractedinasimplewayfromthesuperpotentialowing

tonon-renormalizationproperties.Asabonus,ontheotherhand,averyefficientalgorithm

forthecalculationofthenumberofnon-singletrepresentationsofE6in(2,2)vacuacanbe

derivedfromtheformulaeforchargedegeneraciesofLGorbifoldsgiveninrefs.[5,6].We

thereforebelievethatitisworthwhiletoinvestsomeeffortintotheclassificationofthestring

vacuathatcanbeobtainedinthisway.

Recentlythebasisforthisworkhasbeenlaidbytheenumerationofall(deformation

classesof)non-degeneratepotentialswithcentralchargec=9[7,8].Inthepresentpaper,

asasecondstep,wecalculateallabelianorbifoldsthatcanresultfrommanifestsymmetries

atnon-singularpointsinthemodulispacesofthesepotentials,disregardinghoweverthe

possibilityofdiscretetorsions[9,10](exceptforamodestprobeintoZZ2torsions).Todoso,

weextendtheresultsofVafaandIntriligator[5,6]forthecalculationofthechiralringin

orbifoldsandproveananalogueofawell-knowntheoremforCalabi–Yaumanifolds[11]inthe

LGcontext.

Theoretically,ourresultsareinterestingforthequestionofmirrorsymmetry[12,13,14]

andfortheclassificationofN=2models.Unfortunately,forbothofthese(related)issues,

ourresultsare,inasense,negative,asevenmanyspectrawithalargeEulernumberremain

withoutmirror.Fromaphenomenologicalpointofview,however,extensionsofourcalcula-

tionstowardsincludingtorsionandnon-abelianorbifoldslookverypromising,becausenew

spectramainlyariseintherealmofsmallparticlecontentintheeffectivefieldtheory.

Insection2weshowhowthe(non-singlet)masslessspectrumofanorbifoldcan,ingeneral,

beobtainedfromtheindexandthedimensionofthechiralringinaveryefficientway.The

factthatthecomputationofneitherofthesequantitiesrequiresexplicitknowledgeofabasis

fortheringisvitalforacompleteinvestigationofallnon-degeneratecases.Onlyifthere

arestateswithleft-rightcharges(qL,qR)=(1,0)or(0,1)doweneedextrainformation.

WeshowhowtoextractthenumberofsuchstatesandthattheycanexistonlyifWitten’s

indexvanishes.Section3isquitetechnicalanddetailshowweconstruct,baseduponthe

classificationofpotentials[15],allpossible(linear)abeliansymmetries.Thereaderwhois

onlyinterestedintheresultsmaywishtoproceedtosection4.Implicationsofourfindings

arethendiscussedinthefinalsection.

2Hodgenumbers

Inthissectionwereviewtheresultsofrefs.[5,6]andusethemtoderiveanefficientalgorithm

forthecalculationofthespectrumincaseofvanishingtorsions.Wedo,however,keepthe

discussiongeneralaslongaspossible.

1Thebasicinformationaboutthemasslessspectrumof(2,2)heteroticstringmodelsis

containedinthechiralring,i.e.thenon-singularoperatorproductalgebraofchiralprimary

fields[16].Thesefieldshavealinearrelationbetweentheirconformaldimensionsandtheir

U(1)charges.TheirchargedegeneraciesareconvenientlysummarizedbythePoincar´epoly-

nomialwhich,forleft-andright-chiralfields,isdefinedas

P(t,¯t)=tr(c,c)tJ0¯t¯J0.(1)

AnalogousgeneratingfunctionscanbedefinedforthechargedegeneraciesofRamondground

statesandanti-chiralfields;theyarerelatedtooneanotherbyspectralflow[2].

ForN=2supersymmetricLGmodels,definedbyanon-degeneratequasi-homogeneous

superpotential

W(λniXi)=λdW(Xi),(2)

thePoincar´epolynomialisgivenby[17]

P(t,¯t)=P(t¯t)=󰀅1−(t¯t)1−qi

27denotethenumbersof

27and

27,h12=p11=n27,andtheEulernumberofthemanifold

isχ=2(h11−h12)=2(p12−p11)=2(n

2−qi±(θhi−1

1Namelythosewhichwouldhaveh11=0,incontradictionwiththerequirementoftheexistenceofaK¨ahlerform.

合集下载

BI@ZIF-8

BI@ZIF-8

表面技术 第53卷 第8期 ·84· SURFACE TECHNOLOGY 2024年4月

收稿日期：2023-05-31；修订日期：2023-09-26 Received：2023-05-31；Revised：2023-09-26 基金项目：国家自然科学基金（51871228）；中国科学院金属研究所创新基金（2023-PY12） Fund：National Natural Science Foundation of China (51871228); Innovation Fund of Institute of Metals Research, Chinese Academy of Sciences (2023-PY12) 引文格式：刘韦辰, 韦博鑫, 尹航, 等. 轴向应变作用下含腐蚀缺陷X80管道氢渗透的有限元分析[J]. 表面技术, 2024, 53(8): 84-92. LIU Weichen, WEI Boxin, YIN Hang, et al. Finite Element Analysis of Hydrogen Permeation in X80 Pipeline with Corrosion Defects under Axial Strain[J]. Surface Technology, 2024, 53(8): 84-92. *通信作者（Corresponding author） 轴向应变作用下含腐蚀缺陷X80管道

氢渗透的有限元分析

刘韦辰1,2，韦博鑫1,2*，尹航1,2，许进1,2，于长坤1，孙成1,2

（1.中国科学院金属研究所 辽宁沈阳土壤大气环境材料腐蚀国家野外科学观测研究站，

沈阳 110016；2.中国科学技术大学 材料科学与工程学院，沈阳 110016）

摘要：目的 研究在役天然气管道掺氢输送条件下氢原子在管道腐蚀缺陷处的扩散和分布情况及应变对腐蚀

克劳修斯克拉伯龙方程

克劳修斯克拉伯龙方程

博学笃行 自强不息

1

克劳修斯克拉伯龙方程

引言

克劳修斯克拉伯龙方程是一种经典的线性偏微分方程，描述了物理系统中的波动现象，包括声波、光波、电磁波等。该方程由奥地利数学家耶纳·克劳修斯 (Joseph Johann von Clausius) 和德国数学家拉尔夫·斯·克拉伯龙 (Ralph Schrödinger) 分别在19世纪和20世纪初提出。克劳修斯克拉伯龙方程在科学研究、工程应用和自然界中广泛应用，对于理解和解释波动现象具有重要意义。

方程形式

克劳修斯克拉伯龙方程的一般形式为：

∂²u/∂t² = c²∇²u

其中，u是波函数，t是时间，c是波速，∇²是拉普拉斯算子。方程右侧表示波函数u相对于空间的二阶导数，左侧表示波函数u相对于时间的二阶导数。该方程描述了波函数u在时间和空间上的演化。

方程解析 博学笃行 自强不息

2

对于克劳修斯克拉伯龙方程的解析解的求解是一项复杂的任务，通常需要采用数值方法进行近似求解。然而，对于一些简单的情况，可以找到其解析解以帮助我们理解波动现象。

克劳修斯克拉伯龙方程的解具有波动性质，可以表示为：

u(x, t) = A*sin(kx - ωt + φ)

其中，A是振幅，k是波数，x是位置，ω是角频率，φ是相位。这个解描述了一个沿着x轴传播的波动，振幅随位置的变化满足正弦函数的关系。

在一些特殊的情况下，克劳修斯克拉伯龙方程的解可以表示为波包形式：

u(x, t) = ∫(-∞, ∞)A(k)*e^(i(kx - ωt))dk

其中，A(k)是波包的频谱分布，k是波数。这个解表示了连续的波动，可以通过频谱分布A(k)来描述。

应用领域 博学笃行 自强不息

3

克劳修斯克拉伯龙方程在物理学、工程学、计算机科学等领域有广泛的应用。下面介绍几个常见的应用领域：

声波传播：利用克劳修斯克拉伯龙方程可以描述声波在空气、水等介质中的传播和反射。这在工程学中的声学设计和物理学中的声学研究中很有价值。

卢柯院士Science文章1

卢柯院士Science文章1

DOI: 10.1126/science.1080216, 686 (2003);299 Science et al.W. P. TongNitriding Iron at Lower Temperatures

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镍的XPS

镍的XPS

ORIGINALPAPER

H.W.NesbittáD.LegrandáG.M.Bancroft

InterpretationofNi2pXPSspectraofNiconductors

andNiinsulators

Received24May1999/Revised,accepted:30June1999

AbstractNi2p3/2X-rayphotoelectronspectralpeak

bindingenergiesofNimetal,NiS,andNiAs(allcon-

ductors)spanarangeofabout0.5eVandare,conse-

quently,insensitivetoformalNioxidationstateandto

thenatureoftheligandtowhichNiisbonded,relative

toothermetals(e.g.,Fe).Ni2p3/2peakstructuresand

bindingenergiesre¯ecttwoenergeticcontributions.The

majorcontributionisthatassociatedwiththeelectro-

static®eldproducedbyejectionoftheNi(2p)photo-

electron,theminorcontributionistherelaxationenergy

associatedwith®llingunoccupied,conductionband3d9

and4sNimetalorbitals.Theseconductionbandorbitals

becomelocalizedontheNiphotoion(andsometimes

®lled)inresponsetothe®eldcreatedbythephoto-

emissionevent.BecauseonlythecoreNi2pelectronand