On the correspondence between mirror-twisted sectors for N=2 supersymmetric vertex operator superalgebras of the form $V \otimes V$ and N=1 Ramond sectors of $V$

0.09 MB·

by  Katrina Barron

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
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1 AbstractUsingrecentresultsoftheauthoralongwithVanderWerf,wepresentthe
classificationandconstructionofmirror-twistedmodulesforN=2supersymmetric
]
A vertexoperatorsuperalgebrasoftheformV⊗V forthesignedtranspositionmirror
mapautomorphism.Inparticular,weshowthatthecategoryofsuchmirror-twisted
Q
sectorsforV⊗V isisomorphictothecategoryofN=1RamondsectorsforV.
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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.
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