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Modularity of Bershadsky–Polyakov minimal models 2022 Zachary Fehily.pdf

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LettersinMathematicalPhysics(2022)112:46
/s11005-022-01536-z
ModularityofBershadsky–Polyakovminimalmodels
ZacharyFehily1·DavidRidout1
Received:4November2021/Revised:4November2021/Accepted:4April2022/
Publishedonline:14May2022
©TheAuthor(s)2022
Abstract
TheBershadsky–PolyakovalgebrasaretheoriginalexamplesofnonregularW-
algebras,obtainedfromtheafflnevertexoperatoralgebrasassociatedwithsl3by
.(CommMathPhys385:859–904,
2021),weexploredtherepresentationtheoriesofthesimplequotientsofthese
algebraswhenthelevelkisnondegenerate-,wecombinetheseexplo-
rationswithAdamovi´c’sinversequantumHamiltonianreductionfunctorstostudythe
modularpropertiesofBershadsky–Polyakovcharactersanddeducetheassociated

fortheafflnevertexoperatoralgebrasassociatedwithsl2,exceptthattheroleof
theVirasorominimalmodelsinthelatterishereplayedbytheminimalmodelsof
Zamolodchikov’sW3algebras.
KeywordsVertexoperatoralgebras·W-algebras·Relaxedhighest-weightmodules·
Modulartransformations·Fusionrules
MathematicsSubjectClassiflcation17B69·81T40
Contents
1Introduction.............................................2
...........................................2
..............................................3
..............................................5
2Bershadsky–Polyakovminimalmodels...............................6
–Polyakovvertexoperatoralgebras.........................7
–Polyakovweightmodules.............................8
BZacharyFehily
******@
DavidRidout
david.******@
1SchoolofMathematicsandStatistics,UniversityofMelbourne,Parkville3010,Australia
123:.
,
.......................................11
3InversequantumhamiltonianreductionforBershadsky–Polyakovalgebras............13
.......................................14
-latticevertexalgebra.................................16
.............................17
4Charactersandmodularity......................................18
................................19
-pointfunctionsforstandardmodules...........................21
-pointfunctions...........................23
5TheBershadsky–PolyakovminimalmodelsBP(u,3)........................25
...........................................25
...................................29
:BP(4,3)andBP(5,3)................................34
(4,3)..........................................34
(5,3)..........................................36
6GeneralBershadsky–Polyakovminimalmodels...........................38
...........................................38
....................................44
:BP(3,4)........................................49
.....................52
(u,v)one-pointfunctions.....................52
IdentitiesforW3(u,v)S-matrixelements..............................53
FusionrulesforW3(u,v)......................................57
References................................................58
1Introduction

TheBershadsky–PolyakovalgebrasBPk,k∈C,areamongthesimplestandbest
knownnonregularW-algebras[2,3].Theymaybecharacterised[4]asthesubregular
(orminimal)quantumhamiltonianreductionsofthelevel-kuniversalafflnevertex
algebrasVk(sl).Thispaperisasequelto[1]inwhichtherepresentationtheory
3
,weareinterestedinthe
k
characters,modulartransformationsandfusionrulesofthesimplequotientBPkwhen
kisanondegenerateadmissiblelevel.
Whenk+3∈Z≥0,BPisknowntoberationalandC2-coflnite[5,6].Forthese
2k
levels,whichareadmissible,modulartransformationsofcharactersandfusionrules
areinprincipleknown[7].Foradmissiblelevelswithv>2,thesearethenondegen-
erateadmissiblelevels,
wasshown[1]bycombiningtheuntwistedandtwistedZhualgebrasofBPkwith
Arakawa’sresults[8]
ofsimpleweightBPk-modules,withflnite-dimensionalweightspaces,wasobtained
alongwithaconstructionofcertainnonsemisimpleweightBPk-modules.
Nonrationalityisasigniflcantobstacletocomputingmodulartransformationsand
fusionrules,whichisessentialdataforconstructinglogarithmicconformalfleldthe-

vertexoperatoralgebraswasproposedin[9,10]andisusuallyreferredtoasthe
standardmoduleformalism.
123:.
ModularityandfusionrulesfortheBershadsky–PolyakovalgebrasPage3of6146
Thisformalismhasbeenshowntoreproducethe(Grothendieck)fusionrulesin
manyexamples[11–16]thatcanbeindependentlyverifled[17–27].Otherwise,appli-
cationsofthestandardmoduleformalismconsistentlypasstheusualconsistency
tests,forexamplethattheGrothendieckfusioncoefflcientsarenonnegativeintegers
[28–31].
Atnondegenerateadmissiblelevels,therepresentationtheoryofBPksharesmany
featureswiththatofthesimpleafflnevertexoperatoralgebraLk(sl2).Inparticular,a
keyresultinthestandardmoduleanalysisforLk(sl2)isthefactthatVirasorominimal
modelcharactersappearasfactorsofthestandardcharacters[13,32].Consequently,
themodularS-transformsandGrothendieckfusionrulesofLk(sl2)arealsonaturally

explainedbyAdamovi´c[33]usingaconstructionofthestandardmodulesfromsimple

inversequantumhamiltonianreductionintroducedbySemikhatov[34]andisbelieved
togeneralisewidely.
In[35],inversequantumhamiltonianreductionwasgeneralisedtotheBershadsky–
Polyakovalgebras,withtheroleoftheVirasorominimalmodelsbeingplayedby
Zamolodchikov’sW3minimalmodels[36].Thisgeneralisationleadstotheexpectation
thatBershadsky–Polyakovcharacters,modularityandfusioncanbeunderstoodin
termsofW3minimalmodeldata,,
weconflrmthisexpectationandtherebyprovidefurtherevidencefortheclaimthat
Adamovi´c’sinversequantumhamiltonianreductionfunctorsareafundamentaltool
foranalysingtherepresentationtheoryofgeneralW-
bepresentedinaforthcomingarticle[37]thatwillstudyinversequantumhamiltonian
reductionforthesubregularW-algebrasassociatedwithsln,Lk(sl2)andBPkbeingthe
subregularalgebrasforn=2and3,respectively.

Assumethatk∈Cisnondegenerate-admissible,meaningthatitdeflnesparameters
u,v∈Z≥3by().Asshownin[1],theweightBPk-modulesthenincludesimple
highest-weightmodulesHλ,λ∈u,v,andgenericallysimpletwistedrelaxedhighest-
weightmodulesRtw,[j]∈C/Zand[λ]∈/Z,alongwiththeirimagesunder
[j],[λ]u,v3
thespectralflowfunctorsσ,∈,u,vandu,varecertainflnitesetsof
2
sl3-,asistheZ3-actiononu,
+1/2tw1
theR[j],[λ]=σR[j−κ],[λ],whereκ=6(2k+3)and[j]isrestrictedtoliein
R/Z.
MainTheorem1()Letkbenondegenerate-,thechar-
actersofthestandardmodulesareusuallylinearlydependentandonehastoinstead
considerone-

⎪⎪⎪⎪⎪⎪⎪⎪2πiκθ−(+1)τchW[λ](τ;u)
chR[j],[λ](θ⎪ζ⎪τ;u)=e
η(τ)2
123:.
,

e2πim(j+2κ)δ(ζ+τ−m),()
m∈Z
whereW[λ]isasimplemoduleforthelevel-,thereexist
choicesforusuchthatthesestandardone-pointfunctionsarelinearlyindependent.
Asthestandardmodulesareparametrisedbyacontinuouslabel[j]∈R/Z(aswell
asdiscretelabelsand[λ]),theS-transformofagivenstandardone-pointfunction
willnotbeaweightedsumofone-pointfunctions,butratheraweightedintegral.
Again,theW3minimalmodelS-matrixmakesaconspicuousappearance.
MainTheorem2()Letkbenondegenerate-,theS-

transformoftheone-pointfunctionofR[j],[λ]isgivenby
2⎪⎪⎪⎪⎪⎪⎪⎪
ζζ⎪⎪⎪⎪ζ⎪⎪⎪⎪1u
chR[j],[λ]θ−−+ζ⎪⎪⎪⎪⎪⎪⎪⎪−;
ττ⎪⎪⎪⎪τ⎪⎪⎪⎪ττu

|τ|,[j],[λ]⎪⎪⎪⎪⎪⎪⎪⎪
=S,[j],[λ]chR[j],[λ](θ⎪ζ⎪τ;u)d[j],()
−iτR/Z
∈Z[λ]∈u,v/Z3
whereuistheconformalweightofuandtheentriesofthe“S-matrix”(integral
kernel)are

S,[j],[λ]=SW3e−2πi2κ+(j−κ)+(j−κ).()
,[j],[λ][λ],[λ]
ThevacuummoduleHkω0isnotastandardmodule,butlikeallsimpleweightBPk-
modulesitadmitsaninflnite(one-sidedconvergent)resolutionbystandardmodules
().TheEuler–Poincaréprinciplethenallowsustocalculateitsmodular
S-transform.
MainTheorem3()Letkbenondegenerate-,theS-
transformoftheone-pointfunctionofthevacuummoduleisgivenby
2⎪⎪⎪⎪⎪⎪⎪⎪
ζζ⎪⎪⎪⎪ζ⎪⎪⎪⎪1u
chHkω0θ−−+ζ⎪⎪⎪⎪⎪⎪⎪⎪−;
ττ⎪⎪⎪⎪τ⎪⎪⎪⎪ττu

|τ|,[j],[λ]⎪⎪⎪⎪⎪⎪⎪⎪
=R[j],[λ](θ⎪ζ⎪τ;u)d[j],()
−iτ∈ZR/Z[λ]∈/Z
u,v3
wheretheentriesofthe“vacuumS-matrix"aregivenby
2πiκπi(j−κ)
,[j],[λ]W3ee
Svac.=Svac.,[λ]
.()
2cos3π(j−κ)−i∈Z32cosπai(j,λ)
twitw
Here,ai(j,λ)=(j−κ)+2j∇(λ)andjisdeflnedin().
123:.
ModularityandfusionrulesfortheBershadsky–PolyakovalgebrasPage5of6146
HavingestablishedthemodularS-transformsofthestandardmodulesandthevac-
uummodule,onecannowapplythe(conjectural)standardVerlindeformula()to

anontrivialcalculation,requiringseveralobscureidentitiesinvolvingW3minimal
modelfusioncoefflcients,buttheresultisasfollows.
MainTheorem4()Letkbenondegenerate-,the
Grothendieckfusionrulesofthestandardmodulesare
 ++2+−1
RR=NW3[λ]R+R
[j],[λ][j],[λ][λ],[λ][j+j−4κ],[λ][j+j+2κ],[λ]
[λ]∈u,v/Z3

+NW3[λ]R++1
[λ],[(r,s−ωi+ωi+1)][j+j−2κ],[λ]
[λ]∈u,v/Z3i∈Z3
+
+NW3[λ]R,()
[λ],[(r,s+ωi−ωi+1)][j+j],[λ]
whereweparametriseλas(r,s)asin().
AseverysimpleweightBPk-modulemayberesolvedintermsofstandardmodules,
thisresultimpliestheGrothendieckfusionrulesforarbitrarysimpleweightmodules.
Thesegeneralresultsaredoubtlesslyunpleasantandwedonotattempttoderivethem
,wenoteaninterestinggeneralisationofanobservationof
[13]forLk(sl2).
MainTheorem5()Ifkisnondegenerate-admissible,thenthesimple
highest-weightmodulesHλ,withλ=(r,s)ands=[v−2,−1,0],spanasubring
ofthefusionringofBPkthatisisomorphictothefusionringoftherationalafflne
vertexoperatoralgebraLu−3(sl3).

Westartbydescribingvariouspropertiesofthethreefamiliesofvertexoperatoralge-
brasthatareinvolvedintheinversequantumhamiltonianreductionexploitedinthis
–PolyakovalgebrasBPk,reviewed

admissiblelevel,denotedbyBP(u,v).Afterintroducingspectralflowautomorphisms
andappropriatecategoriesofBP(u,v)-modules,werecalltheclassiflcationresultsof
[1]anddetailthestructureofthespectralfloworbitsofthehighest-weightBP(u,v)-
modules.
,
,withanaccountoftherepresentationtheoryoftheW3minimalmodel
vertexoperatoralgebraW3(u,v).AsW3(u,v)isrational[38],ithasflnitelymany
simplemodulesandallarehighest-
thehalf-latticevertexalgebra,wequicklyreviewthe
constructionofthisvertexalgebra,beforechoosingaconformalstructureanddeflning
123:.
,
certain“relaxed”-modulesthatwillprovecrucialforinversequantumhamiltonia