Decomposition of the Moonshine Vertex Operator Algebra as Virasoro Modules

Transcription

1 Joural of Algebra do:6jabr9996 avalable ole at o Decomposto of the Mooshe Vertex Operator Algebra as Vrasoro Modules Masaak Ktazume Departmet of Mathematcs ad Iformatcs Chba Uersty Chba 6- Japa Chg Hug Lam Isttute of Mathematcs Uersty of Tsukuba Tsukuba -7 Japa ad Hromch Yamada Departmet of Mathematcs Htotsubash Uersty Kutach Tokyo 6-6 Japa Commucated by Geoffrey Maso Receved Jue 999 I ths artcle we obta a decomposto of the Mooshe vertex operator algebra V assocated wth the algebra L L 7 L Our method s based o a coset decomposto of the Leech lattce assocated ' wth A usg some codes I fact we costruct a code vertex operator algebra MC k M D t wth a terary code D ad a code C ad use t to obta a decomposto of V Academc Press INTRODUCTION Vertex operator algebras VOAs as modules of a tesor product of ratoal Vrasoro vertex operator algebras were frst studed by Dog et al 7 They showed that the Mooshe VOA V cotas mutually orthogoal elemets called coformal vectors of cetral charge such that each of them wll geerate a copy of the ratoal Vrasoro VOA L sde V ad the sum of these coformal vectors s the Vrasoro elemet of V I other words the Mooshe VOA cotas a 9-69 $ Copyrght by Academc Press All rghts of reproducto ay form reserved

2 9 KITAZUME LAM AND YAMADA subalgebra somorphc to L Ths subalgebra s very mportat the study of the Mooshe VOA V cf 7 6 For example by usg ths fact Dog proved that the Mooshe VOA V s holomorphc; e V s the oly rreducble V -module I fact a very elegat theory has bee developed for VOAs cotag a subalgebra somorphc to a tesor product of the Vrasoro L cf 6 I 6 Dog et al obtaed several other subalgebras of the Mooshe vertex operator algebra V whch are somorphc to the tesor products of Vrasoro VOAs I ther lst there are two ratoal subalgebras L ad L L 7 L As we have metoed the frst subalgebra L s a very mportat tool for studyg the propertes of the Mooshe VOA I ths artcle we shall study the secod subalgebra L L 7 L We shall obta a decomposto of the Mooshe VOA V assocated wth ths subalgebra Our method s based o a embeddg of ' A to the Leech lattce where A deotes the root lattce of ' type A We shall obta a coset decomposto of A usg some obary codes I fact we shall costruct a subalgebra whch we shall deote by MC k M D t sde the lattce VOA V usg a terary code D ad a code C over We shall the obta a decomposto of V ad the twsted module V T by ths code VOA A decomposto of V wll the be obtaed The authors thak Masahko Myamoto for stmulatg dscussos ad the referee for hs helpful suggestos LATTICE VOA V AND VIRASORO ELEMENTS ' A Let be a set of fudametal roots of a lattce of type A wth a er product ² : ad We shall deote L ' A ad ts dual lattce ' ' L L ² L: For coveece we shall wrte x ad y The L has a bass gve by x y 6 x y 6 ad L has exactly cosets L Let V be the lattce VOA costructed from the lattce L ' L A Bya theorem of Dog V has exactly rreducble modules gve by L V L s a coset of L L L

3 DECOMPOSITION OF THE MOONSHINE VOA 9 For detals about the deftos of VL ad V L please refer to Frekel et al 9 or Dog Now let us cosder the sublattce L of L It s easy to show that L L L L x y x y where L L L L ad L L Smlarly we also have L L L L L a b c y x y x where L L L La L Lb L Lc L ad K a b c We shall deote the other cosets of L L usg the followg scheme For K ad j deote L j L L j j Note that L K j gve all the cosets of L L ad partcular we have ad L L For smplcty we shall deote Moreover we deote ad for K L j L j for j V j V j for K j L V V V V V V V L L L L V a V V b V V c V L L L a b c Next we shall recall a mportat result of Dog et al 6 They showed that the Vrasoro elemet of V VL V' A ca be wrtte as a sum of three mutually orthogoal coformal vectors x x x x x x x

4 96 KITAZUME LAM AND YAMADA ' ' where x e e ad the cetral charge of ad s 7 ad respectvely Let T be the subalgebra geerated by ad The 7 T L L L Next we shall dscuss the decomposto of V V V V a V b ad V c as drect sums of rreducble T-modules LEMMA cf As T-modules V a b or c ca be decomposed as follows V s a drect sum of the followg rreducble T-modules ad each of them has multplcty : L L L L L L L L L L L L L L L L L L L L L L L L V ad V are both drect sums of the followg rreducble T-modules ad each of them has multplcty : L L L L L L L L L L L L Both V a ad V b are drect sums of the followg rreducble T-modules ad each of them has a multplcty : L 6 L 6 L L 6 L 6 L L L L L L L 6 6 V c s a drect sum of the followg rreducble T-modules ad each of them has a multplcty : L L L L L L L L L L L L L L L L L L L L L L L L

5 DECOMPOSITION OF THE MOONSHINE VOA 97 Let us deote M V t M j V j j a b c k where the labels t ad k are abbrevatos for terary ad Kle s four-group respectvely THEOREM cf 7 Mt s a ratoal VOA ad has exactly sx equalet rreducble modules amely Mt Mt Mt Wt Wt ad Wt As L -modules 7 Mt L L Wt L L M L W L t t Mt L Wt L The fuso rules are ge by the followg table: Mt Wt g Mt Wt Mt Wt t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t W M W W M W W M W M W M W M W W M W W M W W M W M W M W M W W M W W M W W M W THEOREM Mk s a ratoal VOA ad has exactly eght equa- let rreducble modules amely Mk Mk a Mk b Mk c Wk Wk a Wk b ad Wk c 7 As L L -modules 7 7 Mk L L L L a 7 7 Mk L 6 L 6 b 7 7 Mk L 6 L 6 c 7 7 k 7 7 k a 7 Wk L 6 L b 7 Wk L 6 L c 7 7 k M L L L L W L L L L W L L L L

6 9 KITAZUME LAM AND YAMADA The fuso rules are ge by the followg table: a a b b c c Mk Wk Mk Wk Mk Wk Mk Wk a a a b b b c c c k k k k k k k k k k k k a a c c b b k k k k k k k k a a a c c c b b b k k k k k k k k k k k k b b c c a a k k k k k k k k b b b c c c a a a k k k k k k k k k k k k c c b b a a k k k k k k k k c c c b b b a a a k k k k k k k k k k k k W M W W M W W M W W M W M W M W M W M W W M W W M W W M W W M W M W M W M W M W W M W W M W W M W W M W M W M W M W M W W M W W M W W M W W M W Remark By Lemma the rreducble modules Wt ad Wk j ca also be defed as Wt V j j Wk V j a b c Ths realzato wll be used Secto to obta a decomposto for the Leech lattce VOA TERNARY CODE VOA I ths secto we shall recall the costructo of a terary code VOA from a terary code The structure of ts rreducble modules wll also be dscussed Let D be a terary code of legth ad let L be a orthogoal sum of -copes of the lattce L For ay D we defe ad L L L V V t t M M where L V ad M are defed as the prevous secto t

7 DECOMPOSITION OF THE MOONSHINE VOA 99 Deote L t L L V V ad M t M t D D D D D D D It s kow that L t D LD s a doubly eve lattce f D s a self-orthogoal terary code cf Moreover we have THEOREM If D s a self-orthogoal terary code the M t s a VOA D wth the Vrasoro elemet ge by Ý ˆ where s the elemet wth the th etty beg t the Vrasoro elemet of Mt ad all other ettes beg I fact MD s a subalgebra of V V but the Vrasoro elemets are dfferet D L D Next we shall study the rreducble modules of MD t Frst let us deote t t M M t Let U be a rreducble M-module The u U where U are rreducble Mt -modules We shall defe the -word of U as where U U U t t f U M or W U f U M or W t t f U M or W By the fuso rules cf Theorem t s easy to see that the fuso t t product M U s also rreducble as a M -module Defe Clearly X X M t U U D t U s a M-module Moreover we have the followg theorem t THEOREM X s a rreducble M -module f U D U D Moreoer eery rreducble module of MD t s somorphc to XU for some t rreducble M -module U wth U D t t -CODE VOA Now we shall dscuss the costructo of VOAs from cf codes

8 9 KITAZUME LAM AND YAMADA DEFINITION Let K a b c be the Kle fourgroup A code of legth s smply a subgroup of K A codeword s sad to be eve f the umber of ozero ettes s eve ad a code s called eve f all ts codewords are eve We also defe a er product o K as follows For ay x y K defe f x y ad x y x y ½ otherwse ad for ay K Ý Now let C be a code of legth For ay C defe ad L L L V V k k M M K As the prevous secto we shall defe LC L VC V ad MC k M k C C C k THEOREM If C s a ee code the MC s a VOA I k ths case M s also a subalgebra of V wth dfferet Vrasoro elemets C C k k Let M Mk ad U be a rreducble M -module We shall defe U U U

9 DECOMPOSITION OF THE MOONSHINE VOA 9 where k k f U M or W a a a f U Mk or W k b b U b f U Mk or W k c f U M c or W c k k I addto we defe FU M k U C k THEOREM F s a rreducble M -module f U C U C Moreoer eery rreducble module of MC k s somorphc to FU for some k rreducble M -module U wth U C Remark cf Lemma As a VOA V cotas a subalgebra L wth the same Vrasoro elemet S Mk Mt 7 7 L L L L L L Therefore all V a b c defed Secto ca be regarded as S-modules I fact we have ad V V M M W W L k t k t V j M M j W W j k t k t V M M W W k t k t where a b c ad j V j M M j W W j k t k t LEECH LATTICE I ths secto we shall decompose the Leech lattce as cosets over the ' lattce A I partcular we shall descrbe these cosets usg a terary code ad a code

10 9 THEOREM the ectors KITAZUME LAM AND YAMADA The Leech lattce s a lattce of rak geerated by e ; X s a geerator of the Golay code C X e e e e ' j j where e e Ý e ad X X I partcular cotas ectors of the form ' where the two s ca be at ay posto ad ca be ether or Now let us recall the followg theorem of Dog et al 6 THEOREM Let N be ay Nemeer lattce The there s at least oe ad geeral seeral sometrc embeddg ' N where s the Leech lattce Now let N be a Nemeer lattce such that N A e A ca be sometrcally embedded N The N N A N E6 or E where N R deotes the Nemeer lattce assocated wth the root lattce R I partcular we have ' ' E A Next we shall dscuss the coset structure of E A ad ' E usg certa codes As oted Secto we have LEMMA Let D be the terary tetracode that s D s defed by the geeratg matrx The ž / ' t LD E where L t L t s defed as Secto D D ' '

11 DECOMPOSITION OF THE MOONSHINE VOA 9 Note that the weght dstrbuto of D s as follows: weght: umber: COROLLARY Let D D D D The t t ' D D L L E t I partcular L D are the cosets of E A Now let us cosder E Frst deote ' ' ' a ' a ' b ' b ' c ' c ' a ' a ' b ' b ' c ' c '

12 9 KITAZUME LAM AND YAMADA The we have ' ² a b a b a b a b : A E Now deote aˆ a aˆ a aˆ a ˆb b ˆb b ˆ b b ˆc c ˆc c ˆc c where Note that the lattce ² a ˆb ˆ c : ˆ s somorphc to ' A ad s cotaed the Leech lattce Thus t gves a atural embeddg ' ' A E For ay codeword let us defe Ý ˆ where ˆ a ˆ ˆ b orc ˆ whe a b orc respectvely Now by drect computato t s possble to detfy the coset represe- ' tatves of E by a code C where the geeratg matrx of C s gve by a a a a c c c c a a a a c c c c a a a a c c c c a a a a c c c c a a c b b b a c a b c c b c a a Namely a codeword C ca be detfed wth a coset represetatve ' of E Note that C s eve ad self-dual wth respect to the er product defed Secto Moreover the weght dstrbuto of '

13 DECOMPOSITION OF THE MOONSHINE VOA 9 C s as follows: weght: umber: Note that C has o codeword of weght Next let us cosder the set C D For ay C ad D we deote L L where L s aga defed as Secto Defe L L C D C D Sce C ad D are codes t s clear that L C D PROPOSITION L s a ee self-dual lattce C D s a lattce The above proposto s a cosequece of the followg geeral theorems THEOREM 6 Let C be a ee code of legth ad let D be a self-orthogoal terary code of the same legth The LC D s a ee lattce Proof Frst ote that for ay a L K ad b L j j we have Thus for ay x L ² : a a mod ² : b b mod ² : a b mod we have ² : x x mod where ad are the weghts of ad respectvely Sce C s eve ad D s self-orthogoal mod ad mod for ay C ad D Therefore we have ad L C D s eve ² : x x mod

14 96 KITAZUME LAM AND YAMADA THEOREM 7 Let C be a ee code of legth ad let D be a self-orthogoal terary code of the same legth If both C ad D are self-dual the L s a self-dual lattce C D Proof Frst we shall ote that for ay a L s ad b L t j ² : a b s t j mod where L s ad L t j are defed as Secto s t ad j Sce L L we have C D LC D L Now let x x x L C D L The x L for some codeword ad Let C D ad y y y L The we have However ² : Ý ² : x y x y Ý Ý ² x y : mod relatvely prme Thus we must also have Therefore Ý O the other had ad are Ý Ý ad I other words s orthogoal to C ad s orthogoal to D Sce both C ad D are self-dual we have C D ad hece x L C D LEMMA L has o ector of square legth C D Proof Let I I 6 7 ad I 9 For ay C ad D we deote support of S support of S By the defto of the tetracode D t s easy to see that for ay ad D I S or

15 DECOMPOSITION OF THE MOONSHINE VOA 97 O the other had by the defto of C for ay C there exst j wth j such that ad I S or j I S Thus for ay C ad D Let a L j L The S S S S mod f j mod f j ² a a: mod f j mod f j Now let x L L C DIf or the t s easy to see that ² : ² x x Ý x x : Assume that ad The S S S S Case S S say S S The Moreover Therefore ² : x x ² : ² x x Ý x x : Case S S say j S S The ² x j x j : But ad I S for at least two Therefore ² : ² x x Ý x x : Hece there exsts o vector of square legth

16 9 KITAZUME LAM AND YAMADA By the above lemmas we have THEOREM 9 LC D Leech lattce COROLLARY The Leech lattce VOA ca be decomposed as V V C D where V V V L Remark Note that the lattce ² C ' E : ' geerated by C ad E cotas a set of geerators of the ² ' Leech lattce cf Theorem Therefore C E : By usg ths fact t s straghtforward to show that L C D Leech lattce By the above method oe ca also obta a smlar decomposto of the Leech lattce usg a embeddg ' ' ' ' 6 N E A or N A A I each case we shall obta a self-dual terary code D ad a self-dual code C such that LC D Note that the terary code D s somorphc to the terary Golay code the case that ' As we have remarked before ' N A A V M M W W k t k t Thus we have the decomposto COROLLARY ž / k t k t C D k t C D V M M W W I partcular V cotas M M as a subalgebra wth the same Vrasoro elemet ths case For coveece let us deote U M M U W W k t k t

17 DECOMPOSITION OF THE MOONSHINE VOA 99 For ay we deote ž / C D U U k t By Theorems ad U s a rreducble MC M as a M k M t module C D V U D module Thus I other words V s a drect sum of rreducble M k M t modules C D 6 THE TWISTED MODULE V T I ths secto we shall dscuss the decomposto of V T wth respect to L L 7 L 6 Frst let us recall the defto of V T Let be the Leech lattce ad let Let h h be a bass of ad let h h be a dual bass of h h The twsted affe algebra of s the algebra wth the bracket gve by ž / ˆ t c x t y t x y m c ad c m ² : ˆ m where x y ad m has a tragular decomposto gve by ˆ ˆ ˆ ˆ c ˆ where t Let ˆ be a cetral exteso of by ad let K be a subgroup of ˆ such that K ˆ defes a cetral exteso of by

18 9 KITAZUME LAM AND YAMADA PROPOSITION 6 9 Chap The extra-specal group K ˆ has a uque up to equalece rreducble module T such that Moreoer dm T As a vector space K o T ž / T V S ˆ T where S ˆ s the symmetrc algebra of ˆ The vertex operator s defed as follows For a ˆ ² a a: ² a a: Y a z E z E z az ad for a V k k where ad wth C ad Y z Y exp z z d ž ž /! dz / k d ž! ž dz / k / Y z : z Ý z z k z Y a z : z Ý Ý m m m C h m h z gve by the equato m x y m Ý C m x y log ž / m By defto t s straghtforward to show that cf 9 Chaps 9 ad V T T V T t T T

19 DECOMPOSITION OF THE MOONSHINE VOA 9 Moreover for ay a wth ² a a: ad T we have a Xa e 6 ² : 6 where X e a e a a By the defto of ad t s clear that Thus we obta 6 for ay T Let be a commo hghest weght vector of hghest weght for the Vrasoro algebras geerated by the compoet operators of Y z such that The Thus or 6 or or or 6 Next we shall gve a explct costructo for T Detals are essetally wrtte Chapter of Frekel et al book 9 see Chap 9 also ' Let E The ad Moreover ˆ s a maxmal abela subgroup of ˆ ad K ˆ s a elemetary abela group ˆ Let : K be a character such that K The as a ˆK module where F ˆ ˆ T Id F s the oe-dmesoal -module ˆ affordg the character

20 9 LEMMA 6 KITAZUME LAM AND YAMADA There s a ector T such that aˆ ˆb cˆ e e e where a ˆ ˆb ˆc are defed as Secto Proof By Frobeus recprocty ² Q : ² Q Q : S where Q K ˆ ad S K ˆ Hece T cotas F wth multplcty as a S-represetato Ths meas that there exsts a vector T such that ' e for ay E ad the vector s uque up to a scalar ˆ ' ' Sce a ˆb c ˆ are A E we have the desred results T PROPOSITION 6 There s a hghest weght ector V of hghest weght wth respect to the subalgebra L L 7 L I partcular V T cotas a -submodule somorphc to 7 L L L Proof Let T be the vector obtaed Lemma 6 ad set ' a ' ˆb ad ' ˆ ˆc The by 6 ad 6 we have ž x ž / / x x x ž / x x x Thus we have the desred result

21 DECOMPOSITION OF THE MOONSHINE VOA 9 7 Now by the fuso rules of L L ad L we have 7 Mk Mt L L L L L L L L L L L L 7 L L L a 7 Mk Mt L L L L L L 7 7 L 6 L 6 L b 7 Mk Mt L L L L L L 7 7 L 6 L 6 L c 7 Mk Mt L L L L L L L L L L L L 7 L L L 7 Wk Wt L L L 7 7 L L L L L L 7 L L L 7 L L L a 7 Wk Wt L L L L L L L L L L L L 7 L 6 L L b 7 Wk Wt L L L 7 6 L L L 7 L 6 L L

22 9 KITAZUME LAM AND YAMADA ad c W W L L L 7 k t 7 7 L L L L L L 7 L L L 7 L L L for ay where deotes the fuso product wth respect to the 7 VOA L L L For coveece we deote T 7 M Mk Mt L L L ad T 7 W Wk Wt L L L Remark 6 We shall ote that the modules M T ad W T are fact rreducble -twsted modules of the VOA Mk Mt V ' A where the acto of o Mk Mt s duced from that of V' A ad s duced from the map of the lattce ' A Sce V T s the uque rreducble -twsted module of V oe ca show THEOREM 6 ž / T T T C V M W 7 MOONSHINE VOA V We shall obta a decomposto of V wth respect to the algebra L L 7 L ths secto Frst let us recall the defto of V cf 9 Let be a automorphsm of such that a a for a We shall exted to V as follows h t ht ad e a e a for h a ad Smlarly we ca exted to V T by defg h t ht

23 DECOMPOSITION OF THE MOONSHINE VOA 9 ad for h T ad As a vector space the Mooshe VOA s defed as Note that fxes V V V T L L 7 L V Ufortuately the code VOA MC k M D t s ot fxed by the automorphsm I other words V MC k M D t Therefore a slghtly dfferet method s used to obta the decomposto of V Frst we shall ote that the automorphsm ca be exteded to the Fock space L L V V V By restrctg the acto of to V we kow that L V L L L L L L L L L L L 7 7 V L L L L L L 7 7 L L L L L L a b 7 7 V or V L 6 L 6 L 7 L 6 L L a b 7 7 V or V L 6 L 6 L 7 7 L 6 L L

24 96 KITAZUME LAM AND YAMADA c 7 7 V L L L L L L 7 7 L L L L L L c V L L L L L L ad for ay L L L L L L V j V j ad j Thus j j j j j j V V V V V V 7 as L L L -modules Let C ad D The we also have V V L L Note that s also a elemet of D Therefore we have ž L / ž L L / C C D V V V V Let us deote V V V V for ay The t s easy to see that s / V V L ž s eve where C ad s s s deotes a eve bary codeword Sce V V we kow that ad Therefore L L L L L V V x x x V V V x x x V L L L VL VL VL VL VL V L

25 DECOMPOSITION OF THE MOONSHINE VOA 97 as -modules Thus we have ž ž ž /// L/ s ž C s eve C D s ž ž ž V C s eve /// V V V ž k t k t // ž C D M M W W as -modules Next we shall cosder the decomposto of V T Nevertheless t s clear that V T s the sum of all -submodules whch have tegral weghts Thus ž s ž ž C s eve /// V V ž k t k t // ž C D T T ž ž / / C M M W W M W T T T where M W s the sum of all -submodules of M W T whch have tegral weght Remark 7 We shall ote that the above decomposto s ot uque ' It depeds o the embeddg of A to ad the choce of smple ' roots for A I the above decomposto we use a embeddg ' ' A E There are fact may other choces but our embeddg yelds a very smple terary code D cf Corollary whch smplfes some of the computato Nevertheless the same method wll work for all embeddgs ad there s o essetal dfferece the argumet APPENDIX A Fuso Rules of L L 7 ad L I ths Appedx we shall lst the fuso rules for the VOAs L L 7 ad L whch we have used ths artcle For the detals of the proofs we refer the reader to Dog et al 7 ad Wag

26 9 KITAZUME LAM AND YAMADA The followg tables should be terpreted as follows: f the h h j etry s k k the the fuso rule s gve by Lc h Lc h j Lc k Lc k PROPOSITION A The fuso rules for L are as follows PROPOSITION A The fuso rules for L 7 are as follows: PROPOSITION A The fuso rules for L are as follows:

27 DECOMPOSITION OF THE MOONSHINE VOA 99 REFERENCES J H Coway ad N J A Sloae Sphere Packgs Lattces ad Groups Sprger- Verlag BerlNew York 9 C Dog Vertex algebras assocated wth eve lattces J Algebra C Dog The represetato of Mooshe module vertex operator algebras Cotemporary Math C Dog R L Gress Jr ad G Hoh Framed vertex operator algebras codes ad the mooshe module Comm Math Phys C Dog ad J Lepowsky Geeralzed Vertex Algebras ad Relatve Vertex Operators Progress Mathematcs Vol Brkhauser Bosto 99 6 C Dog H L G Maso ad S P Norto Assocatve subalgebras of Gress algebra ad related topcs Proceedgs of the Coferece o the Moster ad Le algebra at the Oho State Uversty J Ferrar ad K Harada Eds de Gruyter Berl New York C Dog G Maso ad Y Zhu Dscrete seres of the Vrasoro algebra ad the mooshe module Proceedgs of Symposum Pure Mathematcs Vol 6 pp 96 Amer Math Soc Provdece RI 99 I B Frekel Y Huag ad J Lepowsky O axomatc approaches to vertex operator algebras ad modules Mem Amer Math Soc 99 9 I B Frekel J Lepowsky ad A Meurma Vertex Operator Algebras ad the Moster Pure ad Appl Math Vol Academc Press Bosto 9 M Ktazume M Myamoto ad H Yamada Terary codes ad vertex operator algebras J Algebra 799 C Lam Represetatos of terary code vertex operator algebras preprt C Lam ad H Yamada -codes ad vertex operator algebras J Algebra to appear H L Represetato Theory ad Tesor Product Theory of Vertex Operator Algebras PhD dssertato Rutgers Uversty 99 M Myamoto Gress algebras ad coformal vectors vertex operator algebras J Algebra M Myamoto Bary codes ad vertex operator super algebras J Algebra M Myamoto Represetato theory of code VOAs ad costructo of VOAs hepth96 7 M Myamoto -state Potts model ad automorphsm of vertex operator algebra of order q-alg97 W Wag Ratoalty of Vrasoro vertex operator algebras Duke Math J IMRN