On the correspondence between mirror-twisted sectors for N=2 supersymmetric vertex operator superalgebras of the form V ⊗V and N=1 Ramond sectors of V 4 KatrinaBarron 1 0 2 n a J 9 1 AbstractUsingrecentresultsoftheauthoralongwithVanderWerf,wepresentthe classificationandconstructionofmirror-twistedmodulesforN=2supersymmetric ] A vertexoperatorsuperalgebrasoftheformV⊗V forthesignedtranspositionmirror mapautomorphism.Inparticular,weshowthatthecategoryofsuchmirror-twisted Q sectorsforV⊗V isisomorphictothecategoryofN=1RamondsectorsforV. . h t a m 1 Introduction [ 1 In [B4], [B5], the author studied twisted modulesfor N=2 supersymmetricvertex v operator superalgebras(N=2 VOSAs) for finite order VOSA automorphismsaris- 5 3 ing from automorphismsof the N=2 Neveu-Schwarzalgebra of N=2 infinitesimal 6 superconformaltransformations.Amongsuchautomorphismsisthemirrormap.In 4 [B4],mirrormapsweregivenforN=2VOSAsoftheformV⊗V whereV isanN=1 . 1 supersymmetricVOSAoftheformVL⊗Vfer,whereVLisarankd latticeVOSAor 0 the d free boson vertex operatoralgebra andV is the d free fermionVOSA. In fer 4 particular,weshowedthatoneofthemirrormapsforsuchanN=2VOSA,V⊗V, 1 isgivenbythesignedtranspositionmap : v i k˜ =(12) : V⊗V −→V⊗V, u⊗v 7→ (−1)|u||v|v⊗u (1) X r where|v|= jmod2forv∈V(j),withtheZ -gradingofV givenbyV =V(0)⊕V(1). a 2 In[B6]and[BV2],theauthoralongwithVanderWerfconstructedandclassified thecyclicpermutation-twistedV⊗k-modules,whereV isanyVOSAandkisapos- itiveinteger.Fork even,thisclassificationisintermsofparity-twistedV-modules wheretheparityautomorphismofaVOSAisthemap KatrinaBarron UniversityofNotreDame,NotreDame,Indiana,USA,e-mail:[email protected] 1 2 KatrinaBarron s : V −→V, v 7→ (−1)|v|v. (2) In this note, we apply the results of [BV2] to the setting of the mirror map (1) actingonan N=2supersymmetricVOSA oftheformV⊗V, to showthatthecat- egoryofk˜-twisted(V⊗V)-modulesisisomorphictothecategoryofs -twistedV- modules,whichare theN=1Ramondsectorsforthe N=1supersymmetricVOSA, V.Thisclassificationalsoprovidesanexplicitconstructionofthesemodules. In particular, our result shows that if a representation Ms of the N=1 Ramond algebra is also a parity-twisted modules for a VOSAV, whereV ⊗V is N=2 su- persymmetric,thenMs isalsonaturallyarepresentationofthemirror-twistedN=2 Neveu-Schwarzalgebra. These results can be used to calculate the graded dimen- sionsfor one modulein termsof the gradeddimensionsforthe otheras shownin Corollary2 below.Notethatforourresults,we donotneedtomakeanyassump- tionsabout,forinstance,thevaluesofthecentralcharge,thecompletereducibility oftherepresentations,ortherationalityoftheVOSAs. CertainrepresentationsoftheN=1RamondalgebraandrelatedVOSAconstruc- tions have previouslybeen studied in, e.g., [FQS], [GKO], [FFR], [L], [S], [IK1], [Mi],[AM].Certain representationsofthemirror-twistedN=2Neveu-Schwarzal- gebrahavepreviouslybeenstudiedin, e.g.,[BFK], [Dob],[Ma],[K],[DG2], [G], [IK2],[LSZ].Inparticular,the relationshipbetweencharactersofcertainmodules fortheN=1Ramondalgebraandcertainmodulesforthemirror-twistedN=2Neveu- Schwarzalgebrahadpreviouslybeenobserved.Ourexplicitisomorphismbetween mirror-twistedsectorsforV⊗V andN=1RamondsectorsforV,givesaconstructive andoverarchingexplanationofthisphenomenonthroughthetheoryofVOSAs. 2 The notions ofVOSAand twistedmodule Followingthenotationof[B6],[BV2],werecallthenotionofVOSAandthenotions ofweak,weakadmissibleandordinaryg-twistedV-moduleforaVOSA,V,andan automorphismgofV offiniteorder. Let x,x ,x ,x , denote commuting independent formal variables. Let d (x) = 0 1 2 (cid:229) xn. Expressions such as (x −x )n for n∈C are to be understoodas formal n∈Z 1 2 powerseriesexpansionsinnonnegativeintegralpowersofthesecondvariable. Definition1. A vertex operator superalgebra is a 1Z-graded (by weight) vector 2 space V = V , satisfying dimV < ¥ and V = 0 for n sufficiently nega- n∈1Z n n n 2 tive, thatisLalso Z2-gradedbysign,V =V(0)⊕V(1), withV(j) = n∈Z+jVn, and 2 equippedwithalinearmap L V −→(EndV)[[x,x−1]], v7→Y(v,x)= (cid:229) v x−n−1, (3) n n∈Z and with two distinguished vectors 1∈V , (the vacuum vector) and w ∈V (the 0 2 conformalelement) satisfying the followingconditionsfor u,v∈V: u v=0 forn n Mirror-twistedsectorsforN=2VOSAsV⊗V fromN=1RamondsectorsforV 3 sufficientlylarge;Y(1,x)v=v;Y(v,x)1∈V[[x]],andlim Y(v,x)1=v; x→0 x −x x −x x−1d 1 2 Y(u,x )Y(v,x )−(−1)|u||v|x−1d 2 1 Y(v,x )Y(u,x ) 0 x 1 2 0 −x 2 1 (cid:18) 0 (cid:19) (cid:18) 0 (cid:19) x −x =x−1d 1 0 Y(Y(u,x )v,x ) (4) 2 x 0 2 (cid:18) 2 (cid:19) (the Jacobi identity), where |v| = j if v ∈V(j) for j ∈ Z ; writing Y(w ,x) = 2 (cid:229) L(n)x−n−2, i.e., L(n)=w , for n∈Z, then the L(n) give a representation n∈Z n+1 of the Virasoro algebra with central charge c∈C (the central charge of V); for n∈ 1Z and v∈V , then L(0)v=nv=(wtv)v; and the L(−1)-derivative property 2 n holds: dY(v,x)=Y(L(−1)v,x). dx AnautomorphismofaVOSA,V,isalinearmapgfromV toitself,preserving 1andw suchthattheactionsofgandY(v,x)onV arecompatibleinthesensethat gY(v,x)g−1=Y(gv,x),forv∈V.ThengV ⊂V forn∈ 1Z. n n 2 Let Z denote the positive integers. If g has finite order,V is a direct sum of + theeigenspacesVj ofg,i.e.,V = Vj,wherek∈Z isaperiodofg(i.e., j∈Z/kZ + gk=1)andVj={v∈V |gv=h jv},forh afixedprimitivek-throotofunity. L Definition2. Let (V,Y,1,w ) be a VOSA and g an automorphism of V of period k∈Z .Aweakg-twistedV-moduleisavectorspaceMequippedwithalinearmap + V −→(EndM)[[x1/k,x−1/k]], v7→Yg(v,x)= (cid:229) vgx−n−1, (5) n n∈1Z k withvg∈(EndM)(|v|),andsatisfyingthefollowingconditionsforu,v∈V andw∈ n M:vgw=0fornsufficientlylarge;Yg(1,x)w=w; n x −x x −x x−1d 1 2 Yg(u,x )Yg(v,x )−(−1)|u||v|x−1d 2 1 Yg(v,x )Yg(u,x ) 0 x 1 2 0 −x 2 1 (cid:18) 0 (cid:19) (cid:18) 0 (cid:19) =x−11 (cid:229) d h j(x1−x0)1/k Yg(Y(gju,x )v,x ) (6) 2 k j∈Z/kZ x12/k ! 0 2 (thetwistedJacobiidentity)whereh isafixedprimitivek-throotofunity. Asaconsequenceofthedefinition,wehavethatYg(v,x)=(cid:229) vgx−n−1 for n∈Z+j n k j∈Z/kZandv∈Vj,andforv∈V,wehaveY (gv,x)=lim Y (v,x).It g x1/k→h −1x1/k g also follows that writingYg(w ,x)=(cid:229) Lg(n)x−n−2, i.e., setting Lg(n)=w g , n∈Z n+1 forn∈Z,thentheLg(n)satisfytherelationsfortheVirasoroalgebrawithcentral chargecthecentralchargeofV. Ifwetakeg=1,thenweobtainthenotionofweakV-module.Theterm“weak” meanswearemakingnoassumptionsaboutagradingonM. Aweakadmissibleg-twistedV-moduleisaweakg-twistedV-moduleM which carriesa 1N-gradingM= M(n),suchthatvgM(n)⊆M(n+wtv−m−1) 2k n∈1N m 2k L 4 KatrinaBarron for homogeneousv∈V, n∈ 1N, and m∈ 1Z. If g=1, then a weak admissible 2k k g-twistedV-moduleiscalledaweakadmissibleV-module. An (ordinary)g-twistedV-module is a weak g-twistedV-moduleM graded by CinducedbythespectrumofL(0).Thatis,wehaveM= l ∈CMl ,whereMl = {w∈M|L(0)gw=l w},forL(0)g=w 1g.MoreoverwerequLirethatdimMl isfinite andMn/2k+l =0forfixedl andforallsufficientlysmallintegersn.Ifg=1,then ag-twistedV-moduleisaV-module. 3 The construction andclassificationof(12 ··· k)-twisted V⊗k-modules LetV =(V,Y,1,w )be a VOSA, andlet k bea fixedpositiveinteger.ThenV⊗k is also a VOSA, andthe permutationgroupS acts naturallyonV⊗k as signedauto- k morphisms,i.e.,asarightactionwehave (12 ··· k):V⊗V⊗···⊗V −→V⊗V⊗···⊗V (7) v1⊗v2⊗···⊗vk 7→ (−1)|v1|(|v2|+···+|vk|)v2⊗v3⊗···⊗vk⊗v1. Letg=(12 ··· k).Below,wewillrecalltheclassificationandconstructionofg- twistedV⊗k-modulesfrom[B6]and[BV2].Thisconstructionisbasedonacertain operatorD (x)firstdefinedin[BDM],whichwenowrecall. k Considerthepolynomial 1(1+x)k−1 inxQ[x].Following[BDM],fork∈Z , k k + wedefinea ∈Qfor j∈Z ,by j + exp − (cid:229) a xj+1 ¶ ·x= 1(1+x)k−1. (8) j ¶ x k k j∈Z+ ! Forexample,a =(1−k)/2anda =(k2−1)/12.LetV =(V,Y,1,w )beaVOSA. 1 2 In(EndV)[[x1/2k,x−1/2k]],define D k(x)=exp (cid:229) ajx−kjL(j) (k21)−2L(0) x21k(k−1) −2L(0). (9) j∈Z+ ! (cid:16) (cid:17) For v∈V, and k any positive integer,denote by vj ∈V⊗k, for j=1,...,k, the vectorwhose j-thtensorfactorisvandwhoseothertensorfactorsare1.Thenfor g=(12 ··· k),wehavegvj=vj−1for j=1,...,kwhere0isunderstoodtobek. Let(M,YM)beaV-module,and(Ms ,Ys )as -twistedV-module,wheres isthe paritymaponV.Wedefinetheg-twistedvertexoperatorsforV⊗k onM,forkodd, andonMs ,forkeven,asfollows:Set Mirror-twistedsectorsforN=2VOSAsV⊗V fromN=1RamondsectorsforV 5 Y (D (x)v,x1/k) forkodd M k Y (v1,x)= (10) g  Ys (D k(x)v,x1/k) forkeven andfor j=0,...,k−1,define Y (vj+1,x)= lim Y (v1,x). (11) g g x1/k→h jx1/k LetV be an arbitrary VOSA and h an automorphism ofV of finite order. De- notethecategoriesofweak,weakadmissibleandordinaryh-twistedV-modulesby Ch(V),Ch(V)andCh(V),respectively.Ifh=1,wehabituallyremovetheindexh. w a NowagainconsidertheVOSA,V⊗k,andthek-cycleg=(12 ··· k).Forkodd, define Tk : C (V) −→ Cg(V⊗k), (M,Y ) 7→ (Tk(M),Y )=(M,Y ). (12) g w w M g g g Forkeven,define Tgk : Cws (V) −→ Cwg(V⊗k), (Ms ,Ys ) 7→ (Tgk(Ms ),Yg)=(Ms ,Yg). (13) Thefollowingtheoremisprovedin[B6]forkodd,andin[BV2]forkeven. Theorem1.([B6],[BV2]) (1)Forkodd,thefunctorTkisanisomorphismfromthecategoryC (V)ofweak g w V-modulestothecategoryCg(V⊗k)ofweakg=(12 ··· k)-twistedV⊗k-modules. w (2) For k even, the functor Tk is an isomorphism from the category Cs (V) of g w weak parity-twistedV-modulesto the category Cg(V⊗k) of weak g=(12 ··· k)- w twistedV⊗k-modules. (3)Foranyk∈Z , thefunctorTk restricted tothe respectivesubcategoriesof + g weakadmissible,ordinaryorirreduciblemodulesinC (V)orCs (V),respectively, w w isanisomorphismbetweenthesesubcategoriesandthecorrespondingsubcategory ofweakadmissible,ordinaryorirreducibleg-twistedV⊗k-modules. 4 N=2 supersymmetric VOSAs,Ramondsectors, and mirror-twisted sectors Inthissection,werecallthenotionsofN=1orN=2supersymmetricVOSA,follow- ingthenotationandterminologyof,forinstance,[B1],[B2]and[B3].Firstwewill needthenotionofseveralsuperextensionsoftheVirasoroalgebra. TheN=1Neveu-SchwarzalgebraorN=1superconformalalgebraistheLiesu- peralgebrawithbasisconsistingofthecentralelementd,evenelementsL forn∈Z, n andoddelementsG forr∈Z+1,andsupercommutationrelations r 2 6 KatrinaBarron 1 [L ,L ]=(m−n)L + (m3−m)d d, (14) m n m+n m+n,0 12 m 1 1 [L ,G ]= −r G , [G ,G ] = 2L + r2− d d, (15) m r 2 m+r r s r+s 3 4 r+s,0 (cid:16) (cid:17) (cid:18) (cid:19) form,n∈Z,andr,s∈Z+1.TheN=1RamondalgebraistheLiesuperalgebrawith 2 basis the centralelement d, even elements L for n∈Z, and odd elements G for n r r∈Z,andsupercommutationrelationsgivenby(14)–(15),wherenowr,s∈Z. The N=2 Neveu-Schwarz Lie superalgebra or N=2 superconformal algebra is the Lie superalgebrawith basisconsistingofthe centralelementd, evenelements L and J for n∈Z, and odd elements G(j) for j=1,2 and r∈Z+1, and such n n r 2 (j) thatthesupercommutationrelationsaregivenasfollows:L ,d andG satisfythe n r supercommutationrelationsfortheN=1Neveu-Schwarzalgebra(14)–(15)forboth (1) (2) G =G andforG =G ;theremainingrelationsaregivenby r r r r 1 [L ,J ]= −nJ , [J ,J ] = md d (16) m n m+n m n m+n,0 3 (1) (2) (2) (1) (1) (2) J ,G = −iG , J ,G = iG , G ,G = i(s−r)J . (17) m r m+r m r m+r r s r+s h i h i h i TheN=2RamondalgebraistheLiesuperalgebrawithbasisconsistingofthecentral elementd,evenelementsL andJ forn∈Z,andoddelementsG(j) forr∈Zand n n r j=1,2,andsupercommutationrelationsgivenbythoseoftheN=2Neveu-Schwarz algebrabutwithr,s∈Z,insteadofr,s∈Z+1. 2 NotethatthereisanautomorphismoftheN=2Neveu-Schwarzalgebragivenby k : G(1)7→G(1), G(2)7→−G(2), J 7→−J , L 7→L , d7→d, (18) r r r r n n n n calledthemirrormapautomorphismoftheN=2Neveu-Schwarzalgebra. Let (V,Y,1,w ) be a VOSA, and suppose there exists t ∈V such that writ- 3/2 ing Y(t ,z) = (cid:229) t x−n−1 = (cid:229) G(n+1/2)x−n−2, the G(n+1/2) = t ∈ n∈Z n n∈Z n+1 (EndV)(1) generate a representation of the N=1 Neveu-SchwarzLie superalgebra such that the L(n) are the modes of w . Then we call (V,Y,1,t ) an N=1 Neveu- SchwarzVOSA,oranN=1supersymmetricVOSA,orjustanN=1VOSAforshort. Suppose a VOSA, V, has two vectors t (1) and t (2) such that (V,Y,1,t (j)) is an N=1 VOSA for both j =1 and j =2, and the t (j) =G(j)(n+1/2) generate n+1 a representation of the N=2 Neveu-Schwarz Lie superalgebra. Then we call such a VOSA an N=2 Neveu-Schwarz VOSA or an N=2 supersymmetric VOSA, or for short,anN=2VOSA. Forthecaseoftheparitymap,s ,as -twistedV-module,forV anN=1orN=2 VOSA, is naturally a representation of the N=1 or N=2 Ramond algebra, respec- tively.(Seeforinstance[B4],[B5],aswellasreferencestherein). SupposeV isanN=2VOSAsuchthatV hasanautomorphismgk whichisalift ofthemirrormapk fortheN=2Neveu-Schwarzalgebra.Thatislettinggk actby conjugationonEndV,thengk restrictstothemirrormapk ontheelementsL(n), Mirror-twistedsectorsforN=2VOSAsV⊗V fromN=1RamondsectorsforV 7 J(n), and G(j)(r), for n∈Z, j=1,2, and r∈Z+1, which give the N=2 Neveu- 2 SchwarzalgebrarepresentationontheN=2VOSA,V.Following[B4],[B5],wecall such an automorphismgk of an N=2 VOSA,V, a mirror map. Then a gk -twisted V-moduleisnaturallyarepresentationofthe“mirror-twistedN=2Neveu-Schwarz algebra”. The mirror-twisted N=2 Neveu-Schwarzalgebrais the Lie superalgebra with basis consisting of even elements L , and J and central element d, odd el- n r ements G(1) and G(2), for n∈Z and r ∈Z+1, and supercommutation relations r n 2 (1) given as follows: The L and G satisfy the supercommutation relations for the n r (2) N=1Neveu-Schwarzalgebrawithcentralcharged;theL andG satisfythesu- n n percommutationrelationsforthe N=1 Ramondalgebrawith centralcharged;and theremainingsupercommutationrelationsare 1 [L ,J ] = −rJ , [J ,J ]= rd d, G(1),G(2) = −i(r−n)J (19) n r n+r r s r+s,0 r n r+n 3 h i (1) (2) (2) (1) J ,G = −iG , J ,G = iG . (20) r s r+s r n r+n h i h i Notethatthismirror-twistedN=2Neveu-Schwarzalgebraisnotisomorphictothe ordinaryN=2Neveu-Schwarzalgebra[SS]. 5 Mirror-twisted modules forthe classofN=2 VOSAsofthe formV ⊗V There are large classes of N=2 VOSAs of the formV ⊗V such thatV is an N=1 VOSA,andk˜ =(12),thesignedtranspositionmapgivenbyEqn.(1),isa mirror mapforV⊗V.ExamplesofsuchN=2VOSAs,werestudiedin[B4].Theseinclude thefollowingexamples:LetV bearankdpositivedefiniteintegrallatticeVOSAor L thed freebosonvertexoperatoralgebra,andletVd bethed freefermionVOSA. fer Asnotedin[B4],theVOSAV =V ⊗Vd ,isnaturallyanN=1VOSA,andV⊗V L fer is naturallyan N=2VOSA. Thisusesthe constructionofa VOSA froma positive definiteintegrallattice,followingforinstance[DL],[X],[BV1].SuchN=2VOSAs havemorethanonemirrormapaswasshownin[B4],wheretheauthorconstructed mirror-twistedmodulesfortheseVOSAsfortheothermirrormap. ForsuchN=2VOSAsoftheformV⊗V,andforthesignedtranspositionmirror- mapk˜,wehavethefollowingimmediatecorollarytoTheorem1. Corollary1.The categoryofweakmirror-twisted (V⊗V)-modulesforthesigned transposition mirror map automorphism of an N=2 VOSA of the form V ⊗V is isomorphic to the category of weak N=1 Ramond-twistedV-modules (i.e., parity- twistedV-modules).Inaddition,thesubcategoriesofweakadmissible,ordinary,or irreduciblemodulesareisomorphic. Inparticular,itfollowsthatifMs isarepresentationoftheN=1Ramondalgebra suchthatMs isaweakparity-twistedmoduleforanN=1VOSA,V,andsuchthat 8 KatrinaBarron V⊗V is an N=2 VOSA, then Ms is also naturally a representationof the mirror- twistedN=2superconformalalgebraandisaweakk˜-twistedmoduleforV⊗V. Furthermore,fromtheconstructionofsuchmodulesgivenbythefunctorTk for g k=2asin(10),(11),(13),(seealso[BV2]),wehaveasaconsequenceofCorollary 6.5in[BV2],thefollowing: Corollary2.Mk˜ isanordinaryk˜-twisted(V⊗V)-modulewithgradeddimension dimqMk˜ =trMk˜q−2c/24+Lk˜(0)=q−c/12 (cid:229) dim(Ml )ql l ∈C if and only if (Tk˜2)−1(Mk˜)=Mk˜ is an ordinary s -twisted V-module with graded dimension dimq(Tk˜2)−1(Mk˜)=trMk˜q−c/24+Ls (0)=dimq2Mk˜, wherecisthecentralchargeofV. Acknowledgements TheauthorwouldliketothankVladimirDobrev andtheother organizers fortheirkindinvitationtopresentherresearchattheworkshopinVarna,andfortheirgenerous hospitalityduringherstay. References AM. D.Adamovic´andA.Milas,TheN=1tripletvertexoperatorsuperalgebras:twistedsec- tor,SIGMASymmetryIntegrabilityGeom.MethodsAppl.4(2008),Paper087,24pp. B1. 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