Compactifications with S-Duality Twists(10)
时间:2025-04-03
时间:2025-04-03
We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D=4, N=4 super-Yang-Mills cou
Themassmatrixgeneratesaone-parametersubgroupofSL(2,IR)andthissubgroupwillbethegaugegroupinthelowerdimensionaltheory.Thuscompacti cationwithMewillgiveacompactgaugingSO(2)whereascompacti cationwithMhandMpwillgiverisetoSO(1,1)-gaugedlowerdimensionaltheories[7].(Notethatinthespecialcaseinwhichn=3,therewillbeextravectorgauge eldsinDdimensionsfromthereductionofthe2-formgauge elds,andstrictlyspeakingthegaugegroupisISO(2),ISO(1,1)ortheHeisenberggroupfortheelliptic,hyperbolicandparaboliccases,respectively[14].)
TheparabolicmassmatrixMpisnotinvertible,andhasaone-dimensionalkernel,i.e.
′r=2,l=1,sothatαandα′bothtakeonlyonevalueandAa=(A1,A1).Inthiscasethe
matrixmαβin(2.39)isthe1×1matrix( m)andfromthegaugetransformation(2.41)it
¯n 11′eatsthe(n 2)-formA1′andbecomesmassive.Thecanbeseenthatthe(n 1)-formAn 2¯n 11andn 2formA1gauge eldsremainmassless.remainingn 1formAn 2′′
3TheDoubledFormalism
TypicallyaD=2ndimensionalsupergravitytheoryhasaglobalsymmetrygroupGwhichcanberealisedatthelevelof eldequationsbutnottheaction,asGactsonn-form eldstrengthsH=dAthroughelectric-magneticdualitytransformations.InsuchcasesitispossibletoconstructamanifestlyG-invariantlagrangianthatdependsonthepotentialsAanddual
.Thedual eldsareregardedasindependent elds,butthe eldequationsarepotentialsA
todA,keepingsupplementedwithaG-covariantconstraintrelatingthen-form eldstrengthsdA
thenumberofindependentdegreesoffreedomcorrect.Thenewlagrangianisequivalenttotheoriginaloneasthetwoyieldequivalent eldequationswhentheconstraintistakenintoaccount.Inthissectionwewillreviewthisformalism,whichwasintroducedin[21]whereitwascalledthe‘doubledformalism’.Wewill rstconsiderthecaseG=SL(2,IR)andthengivethegeneralcaseinthefollowingsubsection.
3.1G=SL(2,IR)Case
Considerthefollowinglagrangianin2ndimensionswithneven
11e2φdχ∧ dχ χFn∧Fn(3.46)L= 22
HereFn=dAn 1andφandχarescalar elds.The eldequationsofthislagrangianhaveanSL(2,IR)S-dualityinvariance(forevenn)actingonFthroughelectromageneticdualitytransformations,aswenowdiscuss.
De ninganewn-formGby
Gn=δL
…… 此处隐藏:267字,全部文档内容请下载后查看。喜欢就下载吧 ……下一篇:进口成人用品常见英语单词