Advances in Lie Superalgebras

281 Pages·2014·2.008 MB·English

by  Maria Gorelik, Paolo Papi (eds.)

Springer INdAM Series
Volume 7
Editor-in-Chief
V.Ancona
SeriesEditors
P.Cannarsa
C.Canuto
G.Coletti
P.Marcellini
G.Patrizio
T.Ruggeri
E.Strickland
A.Verra
Forfurthervolumes:
http://www.springer.com/series/10283
Maria Gorelik • Paolo Papi
Editors
Advances in
Lie Superalgebras
Editors
MariaGorelik PaoloPapi
DepartmentofMathematics DipartimentodiMatematica
TheWeizmannInstituteofScience Sapienza–UniversitàdiRoma
Rehovot,Israel Roma,Italy
ISSN:2281-518X ISSN:2281-5198(electronic)
SpringerINdAMSeries
ISBN978-3-319-02951-1 ISBN978-3-319-02952-8(eBook)
DOI10.1007/978-3-319-02952-8
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Preface
ThefirstexamplesofLiesuperalgebrasappearinalgebraictopologyinthelate40’s
(theWhiteheadproductonhomotopygroupsisaLiesuperalgebrabracket)andin
the context of deformation theory of complex structures (Nijenhuis, Frölicher and
Nijenhuis) in the late 50’s. Shortly after, Gerstenhaber, in a series of fundamental
papers,shednewlightontheroleofLiesuperalgebrasinhistheoryofdeformation
ofringsandalgebras,whileSpencerandhiscollaboratorsdevelopedapplicationsto
pseudogroupstructuresonmanifolds.
A renewed interest in Lie superalgebras came from Physics in the early 70’s:
manyexamplesarisenaturallyas“supersymmetries”forquantumfieldtheories,e.g.
intheWess-Zuminomodel1.ItwashoweverKac’slandmarkAdvancespaper2which
establishedthestudyofLiesuperalgebrasasabranchofAlgebrainitsownright.
Since then the subject has received dramatic developments, so that up to now
morethan900papershaving“Liesuperalgebra”intheirtitlecanbecountedinthe
MathSciNet.
Thisvolumeoriginatesfrom“LieSuperalgebras”,heldinRoma,IstitutoNazio-
nalediAltaMatematica“FrancescoSeveri”,December14-19,2012.
Itconsistsoforiginalpapersand/orextendedexpositionsofthetalksdeliveredat
theconference.
Webelievethatthecontributions,kindlyofferedbytheinvitedspeakers,clearly
illustrate one of the most remarkable features of the theory of Lie superalgebras
whichis,theastonishingrangeofitsconnectionswithotherbranchesofMathemat-
icsandMathematicalPhysics.
Itisourpleasure tothank ProfessorVincenzo Ancona, PresidentofIndam,the
ScientificCommitteeandtheentirestaffofIndam,forallowingustheopportunity
togathersomanyspecialistsinsuchahighlystimulatingmeeting.
1AthoroughdiscussionoftheroleofLiesuperalgebrasuptothemid70’scanbefoundinCorwin-
Ne’eman-Sternberg,Rev.Mod.Ph.,47,575–603(1975).
2Kac,V.G.:Liesuperalgebras.AdvancesinMath.26(1)8–96(1977).
v
vi Preface
Byhappycoincidence,thepublicationofthisvolumecoincideswithVictorKac’s
seventiethbirthday.ItwouldbehardtobelievethatthetheoryofLiesuperalgebras
wouldhaveprogressedsofarwithouthiscontributioninthefield.Withtheconsent
ofallcontributingauthors,wewouldliketodedicatethisvolumetohim.
RehovotandRoma MariaGorelik
September2013 PaoloPapi
Contents
Superbosonisation,Rieszsuperdistributions,andhighestweightmodules 1
AlexanderAlldridgeandZainShaikh
Homologicalalgebraforosp(1/2n) ................................. 19
KevinCoulembier
Finitenessandorbifoldvertexoperatoralgebras ..................... 35
AlessandroD’Andrea
OnclassicalfiniteandaffineW-algebras ............................ 51
AlbertoDeSole
Q-typeLiesuperalgebras ......................................... 67
MariaGorelikandDimitarGrantcharov
WeightmodulesofDDD(((222,,,111,,,ααα))) ..................................... 91
CrystalHoyt
OnSUSYcurves ................................................ 101
RitaFioresiandStephenDiwenKwok
DiracoperatorsandtheverystrangeformulaforLiesuperalgebras..... 121
VictorG.Kac,PierluigiMösenederFrajriaandPaoloPapi
ParaboliccategoryO forclassicalLiesuperalgebras .................. 149
VolodymyrMazorchuk
OnKostant’stheoremforLiesuperalgebras......................... 167
ElenaPoletaeva
ClassicalLiesuperalgebrasatinfinity .............................. 181
VeraSerganova
ClassicalW-algebraswithinthetheoryofPoissonvertexalgebras ....... 203
DanieleValeri
vii
viii Contents
Vertexoperatorsuperalgebrasandoddtracefunctions................ 223
JethrovanEkeren
SerrepresentationsofLiesuperalgebras ............................ 235
RuibinZhang
Superbosonisation, Riesz superdistributions,
and highest weight modules
AlexanderAlldridgeandZainShaikh
Abstract Superbosonisation, introduced by Littelmann–Sommers–Zirnbauer, is a
generalisationofbosonisation,withapplicationsinRandomMatrixTheoryandCon-
densedMatterPhysics.Inthissurvey,welinkthesuperbosonisationidentitytoRep-
resentationTheoryandHarmonicAnalysisandexplaintwonewproofs,oneviathe
Laplacetransformandonebasedonamultiplicityfreenessstatement.
1 Introduction
Supersymmetry (SUSY)has its origins in Quantum Field Theory. It is usually as-
sociated with High Energy Physics, especially with SUGRA,where the fermionic
fields correspond to physical quantities, the mathematical incarnation of a (as yet,
hypothetical)fundamentalphenomenon.However,beyondthisfascinatinganddeep
theory, and its independent mathematical interest, SUSY also has applications in
quitedifferentareasofphysics,notably,inCondensedMatter.
Here,thegeneratorsofsupersymmetrydonotcorrespondtophysicalquantities.
Rather, they appear as effective symmetries of models for low-temperature limits
of the fundamental Quantum Field Theory. This idea goes under the name of the
SupersymmetryMethod,andwasdevelopedbyEfetovandWegner[9].
Itsparticularmeritisthepossibilitytoderive,bytheuseofHarmonicAnalysison
certainsymmetricsuperspaces,preciseclosedformexpressionsforstatisticalquan-
tities–suchasthemomentsoftheconductanceofametalwithimpurities[29,30]–
inaregimewherethesystembecomescritical,forinstance,exhibitingatransition
fromlocalisationtodiffusion,whichisnottractablebyothermethods.
A.Alldridge( )
MathematicalInstitute,UniversityofCologne,Weyertal86–90,50931Köln,Germany
e-mail:[email protected]
Z.Shaikh
MathematicalInstitute,UniversityofCologne,Weyertal86–90,50931Köln,Germany
e-mail:[email protected]
M.Gorelik,P.Papi(eds.):AdvancesinLieSuperalgebras.SpringerINdAMSeries7,
DOI10.1007/978-3-319-02952-8_1,©SpringerInternationalPublishingSwitzerland2014
2 A.AlldridgeandZ.Shaikh
Inconnectionwiththephysicsofthinwires,thesubjectwaswellstudiedinthe
1990s;ithasrecentlygainedsubstantialnewinterest,sincethe‘symmetryclasses’
investigatedinthiscontext[4,14,31]havebeenfoundtooccuras‘edgemodes’of
certain2Dsystemsdubbed‘topologicalinsulators’(resp.superconductors)[12].
Mathematically, several aspects of the method beg justification. One both sub-
tleandsalientpointisthetransformationofcertainintegralsoverflatsuperspacein
highdimensionN→∞,whichoccurasexpressionsforstatisticalGreen’sfunctions,
intointegralsoveracurvedsuperspaceoffixedrankanddimension–thelatterbeing
moreamenabletoasymptoticanalysis(bysteepestdescentorstationaryphase).Tra-
ditionally,thisstepisperformedbytheuseoftheso-calledHubbard–Stratonovich
transformation,whichisbasedonacarefuldeformationoftheintegrationcontour.
Thisposessevereanalyticalproblems,whichtothepresentdayhaveonlybeen
overcomeincasesderivedfromrandommatrixensemblesthatfollowthenormaldis-
tribution[15].ToextendtheSupersymmetryMethod’srangebeyondGaussiandis-
order,forinstancetoestablishuniversalityforinvariantrandommatrixensembles,
acomplementarytoolwasintroduced,basedonideasofFyodorov[13]:theSuper-
bosonisation Identity of Littelmann–Sommers–Zirnbauer [21]. (A more complete
accountofthehistoryofsuperbosonisationistobefoundintheintroductionof[3].)
We now proceed to describe this identity. In general, it holds in the context of
unitary,orthogonal,andunitary-symplecticsymmetry.Werestrictourselvestothe
firstcase(ofunitarysymmetry),althoughourmethodscarryovertotheothercases.
One considers the spaceW :=Cp|q×p|q of square super-matrices and a certain
subsupermanifold Ωof purely even codimension, whose underlying (Riemannian
symmetric) manifold Ω is the product of the positive Hermitian p×p matrices
0
withtheunitaryq×qmatrices.Let f beasuperfunctiondefinedandholomorphic
onthetubedomainbasedonHerm+(p)×Herm(q).Thesuperbosonisationidentity
states (cid:2) (cid:2)
|Dv|f(Q(v))=C |Dy|Ber(y)nf(y), (1)
Cp|q×n⊕Cn×p|q Ω
forsomefinitepositiveconstantC,provided f hassufficientdecayatinfinityalong
themanifoldΩ.Here,QisthequadraticmapQ(v)=vv∗,|Dv|istheflatBerezinian
0
density,and|Dy|isaBereziniandensityonΩ,invariantunderacertainnaturaltran-
sitivesupergroupactionwewillspecifybelow.
Remark that any GL(n,C)-invariant superfunction on Cp|q×n⊕Cn×p|q may be
writtenintheform f(Q(v)).Thus,anotablefeatureoftheformulaisthatitputsthe
‘hidden supersymmetries’ (from GL(p|q,C)) into evidence through the invariant
integral over the homogeneous superspace Ωwhere ‘manifest symmetries’ (from
GL(n,C))enterviasomecharacter(namely,Ber(y)n).
Aremarkablespecialcaseoccurswhen p=0.ThenEq.(1)reducesto
(cid:2) (cid:2)
|Dv|f(Q(v))=C |Dy|det(y)−nf(y),
C0|q×n⊕Cn×0|q U(q)
whichisknownastheBosonisationIdentityinphysics.Noticethattheleft-handside
isapurelyfermionicBerezinintegral,whereastheright-handsideispurelybosonic.