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A spinor-like representation of the contact superconformal algebra K'(4)

a r X i v :h e p -t h /0011100v 2 20 D e c 2000A spinor-like representation of the contact superconformal algebra K ′(4)

Elena Poletaeva

Centre for Mathematical Sciences

Mathematics,Lund University

Box 118,S-22100Lund,Sweden

Electronic mail:elena@maths.lth.se In this work we construct an embedding of a nontrivial central extension of the contact superconformal algebra K ′(4)into the Lie superalgebra of pseudodi?erential symbols on the supercircle S 1|2.Associated with this embedding is a one-parameter family of spinor-like tiny irreducible representations of K ′(4)realized just on 4?elds instead of the usual 16.I.Introduction Recall that a superconformal algebra is a simple complex Lie superalgebra,such that it con-tains the centerless Virasoro algebra (i.e.the Witt algebra)W itt =⊕n ∈Z C L n as a subalgebra,and has growth 1.The Z -graded superconformal algebras are ones for which adL 0is diagonal-izable with ?nite-dimensional eigenspaces;see Ref. 1.In general,a superconformal algebra is a subalgebra of the Lie superalgebra of all derivations of C [t,t ?1]?Λ(N ),where Λ(N )is the Grassmann algebra in N odd variables.The Lie superalgebra K (N )of contact vector ?elds with Laurent polynomials as coe?cients

is characterized by its action on a contact 1-form (Refs.1,2,3,and 25);it is isomorphic to the SO (N )superconformal algebra (Ref.4).K (N )is simple except when N =4.In this case K ′(4)=[K (4),K (4)]is simple.Note that K ′(N )is spanned by 2N ?elds.It was discovered independently in Ref.3and Ref.5that the Lie superalgebra of contact vector ?elds with poly-nomial coe?cients in 1even and 6odd variables contains an exceptional simple Lie superalgebra (see also Ref.2,Refs.6,7,and Refs.23,24).In Ref.3the exceptional superconformal algebra CK 6was discovered as a subalgebra of K (6),and it was shown that the derived Lie superalgebra of divergence-free derivations of C [t,t ?1]?Λ(2),which is spanned by 8?elds,can be realized inside K (4)using the construction of CK 6inside K (6).

Note that a Lie algebra of contact vector ?elds can be realized as a subalgebra of Poisson

algebra;see Ref.8.The Poisson algebra of formal Laurent series on ˙T

?S 1=T ?S 1\S 1has

a well-known deformation,that is the Lie algebra R of pseudodi?erential symbols on the circle. The Poisson algebra can be considered to be the semiclassical limit of R;see Refs.9,10,11,and 12.

In this work we de?ne a family R h(N)of Lie superalgebras of pseudodi?erential symbols on the supercircle S1|N,where h∈]0,1],which contracts to the Poisson superalgebra.

For each h we construct an embedding of a central extension?K′(4)into R h(2).These central extensions are isomorphic to one of3independent central extensions,which are known for K′(4)(Refs.1,2,13and14).The corresponding central element is h∈R h(2).The elements of embeddings of?K′(4)are power series in h;considering their limits as h→0,we obtain an embedding of K′(4)into the Poisson superalgebra.

The idea of our construction is as follows.We consider the Schwimmer-Seiberg’s deformation S(2,α)of the Lie superalgebra of divergence-free derivations of C[t,t?1]?Λ(2)(Refs.15and 1)and observe that the exterior derivations of S′(2,α)form an sl(2)ifα∈Z.The exterior derivations of S′(2,α)for allα∈Z generate a subalgebra of the Poisson superalgebra isomorphic to the loop algebra?sl(2)[sl(2)corresponds toα=1].We prove that the family S′(2,α)for all α∈Z and?sl(2)generate a Lie superalgebra isomorphic to K′(4).The similar construction for each h∈]0,1]gives an embedding of a nontrivial central extension of K′(4):

?K′(4)?R

(2).(1.1)

It is known that the Lie algebra R has two independent central extensions;see Refs.9,10,and11. Accordingly,there exist,up to equivalence,two nontrivial2-cocycles on its superanalog R h=1(N). The2-cocycle on K′(4),which corresponds to the central extension?K′(4)is equivalent to the restriction of one of the2-cocycles on R h=1(2).

Finally,the embedding(1.1)for h=1allows us to de?ne a new one-parameter family of tiny irreducible representations of?K′(4).Recall that there exists a two-parameter family of representations of K′(N)in the superspace spanned by2N?elds.These representations are de?ned by the natural action of K′(N)in the spaces of“densities”;see Ref.1.

We obtain representations of?K′(4),where the value of the central charge is equal to1, realized on just4?elds,instead of the usual16.

II.Superconformal algebras

In this section we review the notion of a superconformal algebra and give the necessary

de?nitions.

A superconformal algebra is a complex Lie superalgebra g such that

1)g is simple,

2)g contains the Witt algebra W itt=der C[t,t?1]=⊕n∈Z C L n with the well-known commu-tation relations

[L n,L m]=(n?m)L n+m(2.1)

as a subalgebra,

3)adL0is diagonalizable with?nite-dimensional eigenspaces:

g=⊕j g j,g j={x∈g|[L0,x]=jx},(2.2)

so that dim g j

The superalgebras W(N).Consider the superalgebra C[t,t?1]?Λ(N),whereΛ(N)is the Grassmann algebra in N variablesθ1,...,θN.Let p be the parity of the homogeneous element. Let p(t)=ˉ0and p(θi)=ˉ1for i=1,...,N.By de?nition W(N)is the Lie superalgebra of all derivations of C[t,t?1]?Λ(N).Let?i stand for?/?θi and?t stand for?/?t.Every D∈W(N) is represented by a di?erential operator,

D=f?t+

i=1

f i?i,(2.3)

where f,f i∈C[t,t?1]?Λ(N).W(N)has no nontrivial2-cocycles if N>2.If N=1or2, then there exists,up to equivalence,one nontrivial2-cocycle on W(N).

The superalgebras S(N,α).The Lie superalgebra W(N)contains a one-parameter family of Lie superalgebras S(N,α);see Refs.15and1.By de?nition

S(N,α)={D∈W(N)|Div(tαD)=0}forα∈C.(2.4)

Recall that

Div(D)=?t(f)+

i=1

(?1)p(f i)?i(f i)(2.5)

and

Div(fD)=Df+fDivD,(2.6)

where f is an even function.Let S′(N,α)=[S(N,α),S(N,α)]be the derived superalgebra. Assume that N>1.Ifα∈Z,then S(N,α)is simple,and ifα∈Z,then S′(N,α)is a simple ideal of S(N,α)of codimension one de?ned from the exact sequence,

0→S′(N,α)→S(N,α)→C t?αθ1···θN?t→0.(2.7) Notice that

S(N,α)～=S(N,α+n)for n∈Z.(2.8) There exists,up to equivalence,one nontrivial2-cocycle on S′(N,α)if and only if N=2;see Ref. 1.Let?S′(2,α)be the corresponding central extension of S′(2,α).Note that S′(2,α)is spanned by4even?elds and4odd?elds.Sometimes the name“N=4superconformal algebra”is used for?S′(2,0);see Refs.4and3.

The superalgebras K(N).By de?nition

K(N)={D∈W(N)|D?=f?for some f∈C[t,t?1]?Λ(N)},(2.9)

where

?=dt?

i=1

θi dθi(2.10)

is a contact1-form;see Refs.1,2,3,and25.(See also Ref.26,where the contact superalgebra K(m,n)was introduced,and Ref.24).Every di?erential operator D∈K(N)can be represented by a single function,

f∈C[t,t?1]?Λ(N):f→D f.(2.11)

Let

?(f)=2f?

i=1

θi?i(f).(2.12)

Then

D f=?(f)?t+?t(f)

i=1

θi?i+(?1)p(f)

i=1

?i(f)?i.(2.13)

Notice that

D f+g=D f+D g,(2.14)

[D f,D g]=D{f,g},

where

{f,g}=?(f)?t(g)??t(f)?(g)+(?1)p(f)

i=1

?i(f)?i(g).(2.15)

The superalgebras K(N)are simple,except when N=4.If N=4,then the derived superalgebra K′(4)=[K(4),K(4)]is a simple ideal in K(4)of codimension one de?ned from the exact sequence

0→K′(4)→K(4)→C D t?1θ

1θ2θ3θ4→0.(2.16) There exists no nontrivial2-cocycles on K(N)if N>4.If N≤3,then there exists,up to equivalence,one nontrivial2-cocycle.Let?K(N)be the corresponding central extension of K(N). Notice that?K(1)is isomorphic to the Neveu-Schwarz algebra(Ref.17),and?K(2)～=?W(1)is isomorphic to the so-called N=2superconformal algebra;see Ref.18.The superalgebra K′(4) has3independent central extensions(Refs.1,2,13and14),which is important for our task.

III.Lie superalgebras of pseudodi?erential symbols

Recall that the ring R of pseudodi?erential symbols is the ring of the formal series

A(t,ξ)=

?∞

a i(t)ξi,(3.1)

where a i(t)∈C[t,t?1],and the variableξcorresponds to?/?t;see Refs.9,10,11,and12.The multiplication rule in R is determined as follows:

A(t,ξ)?B(t,ξ)= n≥01

(Refs.12and19).One can construct the contraction of the Lie algebra R to P using the linear isomorphisms:

?h:R?→R(3.5)

de?ned by

?h(a i(t)ξi)=a i(t)h iξi,where h∈]0,1],(3.6) see Ref.12.The new multiplication in R is de?ned by

A?h B=??1

(?h(A)??h(B)).(3.7) Correspondingly,the commutator is

[A,B]h=A?h B?B?h A.(3.8) Thus

[A,B]h=h{A,B}+hO(h).(3.9) Hence

lim h→0

(A?h B)?(XY),(3.13) where A,B∈R,and X,Y∈Θh(N).The product in R h(N)determines the natural Lie superalgebra structure on this space:

[(A?X),(B?Y)]h=1

(B?h A)?(Y X).(3.14)

For each h∈]0,1]there exists an embedding

W(N)?R h(N),(3.15)

such that the commutation relations in R h(N),when restricted to W(N),coincide with the commutation relations in W(N).In particular,when h=1,we obtain the superanalog R(N):= R h=1(N)of the Lie algebra of pseudodi?erential symbols on the circle.

The Poisson superalgebra P(N)has the underlying vector space P?Θ(N),where

Θ(N):=Θh=0(N)is the Grassman algebra with generatorsθ1,...,θN,ˉθ1,...,ˉθN,whereˉθi=?i for i=1,...,N.The Poisson bracket is de?ned as follows:

{A,B}=?ξA?t B??t A?ξB?(?1)p(A)(

i=1

?θ

A?ˉθ

B+?ˉθ

A?θ

B),(3.16)

where A,B∈P(N);cf.Refs.2,5.Thus

lim h→0[A,B]h={A,B}.(3.17) Correspondingly,we have the embedding

W(N)?P(N).(3.18)

Remark3.1:Recall that there exist,up to equivalence,two nontrivial2-cocycles on R(Refs. 9,10,and11).Analogously,one can de?ne two2-cocycles,cξand c t,on R(N);cf.Ref.20.Let A,B∈R,and X,Y∈Θh=1(N).Then

cξ(A?X,B?Y)=the coe?cient of t?1ξ?1θ1...θN?1...?N(3.19)

in([logξ,A]?B)?(XY),

where

[logξ,A(t,ξ)]=Σk≥1

(?1)k+1

t?k?kξA(t,ξ).(3.22)

IV.The construction of embedding

Let DerS′(2,α)be the Lie superalgebra of all derivations of S′(2,α).

Lemma4.1:The exterior derivations Der ext S′(2,α)for allα∈Z generate the loop algebra

?sl(2)?P(2).(4.1) Proof:In Ref.21we observed that the exterior derivations of S′(2,0)form an sl(2).Let

{Lαn,E n,H n,F n,hαn,p0n,x0n,yαn}n∈Z(4.2) be a basis of S′(2,α)de?ned as follows:

Lαn =?t n(tξ+

Let

Lα0=?Lα0+

Letα∈Z.Then

Der ext S′(2,α)～= F?α+1,H0,Eα?1 .(4.17)

Theorem4.1:The superalgebras S′(2,α)for allα∈Z together with?sl(2)generate a

Lie superalgebra isomorphic to K′(4).

Proof:Let

I0n=t n(θ1?1+θ2?2),

r n=t n?1θ1θ2?1,(4.18)

s n=t n?1θ1θ2?2.

Then according to(4.3)

Lα

=L0n?1

I0n,(4.22)

I n=I0n+H n,

p n=p0n+t n,

x n=x0n?q n.

Set

h n=h0n,y n=y0n.(4.23) Then g=gˉ0⊕gˉ1,where

gˉ0= L n,I n,E n,H n,F n,E n,H n,F n ,(4.24)

gˉ1= h n,p n,x n,y n,r n,s n,q n,t n .

We will describe the nonvanishing commutation relations in g with respect to this basis.

For[gˉ0,gˉ0]the relations are:

[L n,L k]=(n?k)L n+k;(4.25) [H n,E k]=2E n+k,[H n,F k]=?2F n+k,[E n,F k]=H n+k;

[H n,E k]=2E n+k,[H n,F k]=?2F n+k,[E n,F k]=H n+k;

[L n,X k]=?kX n+k,where X k=I k,E k,H k,F k,E k,H k,F k.

For[gˉ0,gˉ1]the relations are:

[L n,X k]=(?k+n

)X n+k,where X k=r k,s k,q k,t k;

[I n,X k]=nY n+k,where X k=h k,p k,x k,y k,and Y k=r k,t k,?q k,?s k,respectively;

[H n,X k]=X n+k,where X k=h k,x k,q k,r k;

[H n,X k]=?X n+k,where X k=y k,p k,s k,t k;

[E n,X k]=Y n+k,[F n,Y k]=X n+k,

where X k=y k,p k,s k,t k,and Y k=h k,x k,?r k,?q k,respectively;

[H n,X k]=X n+k+nY n+k,

where X k=p k,x k,q k,t k,and Y k=t k,?q k,0,0,respectively;

[H n,X k]=?X n+k?nY n+k,

where X k=h k,y k,r k,s k,and Y k=r k,?s k,0,0,respectively;

[E n,X k]=Y n+k?nZ n+k,[F n,Y k]=X n+k?nˉZ n+k,where X k=h k,y k,r k,s k,

Y k=x k,p k,?q k,?t k,Z k=q k,?t k,0,0,andˉZ k=?r k,s k,0,0,respectively.

Finally,for[gˉ1,gˉ1]the relations are:

[h n,x k]=(k?n)E n+k,[p n,y k]=(k?n)F n+k,(4.27) [h n,p k]=L n+k?12(k?n)H n+k,

[h n,q k]=E n+k,[x n,r k]=E n+k,[p n,s k]=F n+k,[y n,t k]=F n+k,

[p n,q k]=?E n+k,[x n,t k]=?E n+k,[h n,s k]=?F n+k,[y n,r k]=?F n+k,

[p n,r k]=1

2(H n+k+H n+k),[x n,s k]=1

(H n+k?H n+k),

[h n,t k]=1

2(H n+k+H n+k),[y n,q k]=1

(H n+k?H n+k).

Recall that the elements of K(4)can be identi?ed with the functions from

C[t,t?1]?Λ(4).Let

ˇθ

=θ2θ3θ4,ˇθ2=θ1θ3θ4,ˇθ3=θ1θ2θ4,ˇθ4=θ1θ2θ3.(4.28) The following16series of functions together with t?1θ1θ2θ3θ4span C[t,t?1]?Λ(4): f1n=2nt n?1θ1θ2θ3θ4,(4.29)

f2n=?1

it n(θ2θ3?θ1θ4)?1

t n?1(±θ1θ2?θ3θ4?iθ1θ3?iθ2θ4),k=3,4, f5n=it n(θ1θ4?θ2θ3),

f k n=

√

Remark4.2:We have obtained an embedding

K′(4)?P(2).(4.31)

In general,a Lie algebra of contact vector?elds can be realized as a subalgebra of Poisson algebra; see Ref.8.We will explain this from the geometrical point of view in application to our case. Recall that the Lie algebra V ect(S1)of smooth vector?elds on the circle has a natural embedding into the Poisson algebra of functions on the cylinder˙T?S1=T?S1\S1with the removed zero

section;see Refs.11,12and19.One can introduce the Darboux coordinates(q,p)=(t,ξ)on this manifold.The symbols of di?erential operators are functions on˙T?S1which are formal Laurent series in p with coe?cients periodic in q.Correspondingly,they de?ne Hamiltonian vector?elds on˙T?S1:

A(q,p)?→H A=?p A?q??q A?p.(4.32) The embedding of V ect(S1)into the Lie algebra of Hamiltonian vector?elds on˙T?S1is given by

f(q)?q?→H f(q)p.(4.33) Notice that we obtain a subalgebra of Hamiltonian vector?elds with Hamiltonians which are homogeneous of degree1.(This condition holds in general,if one considers the symplecti?cation of a contact manifold;see Ref.8.)In other words,we obtain a subalgebra of Hamiltonian vector ?elds,which commute with the(semi-)Euler vector?eld:

[H A,p?p]=0.(4.34) We will show that for N≥0there exists the analogous embedding:

K(2N)?P(N).(4.35) The analog of the formula(4.32)in the supercase is as follows(Refs.2,5):

A(q,p,θi,ˉθi)?→H A=?p A?q??q A?p?(?1)p(A)

i=1

(?θ

A?ˉθ

+?ˉθ

A?θ

).(4.36)

Then K(2N)is de?ned as the set of all(Hamiltonian)functions A(q,p,θi,ˉθi)∈P(N)such that

[H A,p?p+

i=1

ˉθ

?ˉθ

]=0.(4.37)

Equivalently,we have the following characterization of the embedding(4.35).Consider a Z-grading of the(associative)superalgebra P(N)=⊕j∈Z P j(N)de?ned by

deg p=degˉθi=1for i=1,...,N,(4.38)

deg q=degθi=0for i=1,...,N.

Thus with respect to the Poisson bracket,

{P j(N),P k(N)}?P j+k?1(N).(4.39) Then

K(2N)=P1(N).(4.40) Theorem4.2:There exists an embedding,

?K′(4)?R

(2),(4.41)

for each h∈]0,1],such that the central element in?K′(4)is h∈R h(2),and

lim h→0?K′(4)=K′(4)?P(2).(4.42) Proof:For each h∈]0,1]andα∈Z we have an embedding,

DerS′(2,α)?R h(2).(4.43)

The exterior derivations Der ext S′(2,α)for allα∈Z generate the loop algebra,

?sl(2)= F

,H n,E n n∈Z?R h(2),where(4.44)

F n=?t n?1ξθ1θ2,(4.45)

H n=

The basis(4.24)in g is de?ned by Eqs.(4.3),(4.18),(4.22)-(4.23)and(4.45)-(4.46).The commu-tation relations in g with respect to this basis are given by Eqs.(4.25)-(4.27).The Lie superalgebra g is isomorphic to a central extension,

?K′(4)=K′(4)⊕C C(4.47) of K′(4).The corresponding2-cocycle(up to equivalence)is

c(t n+1,t k+1θ1θ2θ3θ4)=δn+k+2,0,(4.48)

c(t n+1θi,t k+1?i(θ1θ2θ3θ4))=

δn,?k,(4.51)

c(x n,s k)=

δn,?k,

c(y n,q k)=

Theorem5.1:There exists a one-parameter family of irreducible representations of?K′(4) depending on parameterμ∈C in the superspace spanned by2even?elds and2odd?elds where the value of the central charge is equal to one.

Proof:Let g∈tμC[t,t?1],whereμ∈R\Z.One can think ofξ?1as the anti-derivative,

ξ?1g(t)= g(t)dt.(5.1) Let f(t)∈C[t,t?1].According to(3.2),

ξ?1?f=

∞

n=0

(?1)n(ξn f)ξ?n?1.(5.2)

Notice that this formula,when applied to a function g,corresponds to the formula of integration by parts.Let

Vμ=tμC[t,t?1]?Λ(2)=tμC[t,t?1]? 1,θ1,θ2,θ1θ2 ,μ∈R\Z.(5.3)

Using the realization of?K′(4)inside R(2)(see Theorem4.2for h=1)we obtain a representation of?K′(4)in Vμ.A central element in?K′(4)is I0=1∈R(2);the2-cocycle is de?ned by Eq.

(4.51).Let{v i m},where m∈Z and i=0,1,2,3,be the following basis in Vμ:

v0m=

n+μ)v i m+n,i=1,2,

L n(v3m)=?(m+n+μ+1)v3m+n,

E n(v1m)=v2m+n,

F n(v2m)=v1m+n,

E n(v3m)=v0m+n+2,

F n(v0m)=?v3m+n?2,

H n(v i m)=?v i m+n,i=1,2,

H n(v i m)=±v i m+n,i=0,3,

h n(v1m)=?(m+n+μ)v3m+n?1,y n(v2m)=(m+n+μ)v3m+n?1,

h n(v0m)=v2m+n?1,y n(v0m)=v1m+n?1,

x n(v1m)=(m+n+μ)v0m+n+1,p n(v2m)=?(m+n+μ)v0m+n+1,

x n(v3m)=v2m+n+1,p n(v3m)=v1m+n+1,

r n(v1m)=v3m+n?1,s n(v2m)=v3m+n?1,

q n(v1m)=v0m+n+1,t n(v2m)=v0m+n+1,

I n(v i m)=v i m+n,i=0,1,2,3.

Note that I0acts by the identity operator.One can then de?ne a one-parameter family of repre-sentations of?K′(4)depending on parameterμ∈C in the superspace V= v0m,v3m,v1m,v2m m∈Z,

where p(v i m)=ˉ0,for i=0,3,and p(v i m)=ˉ1for i=1,2,according to the formulas(5.5).

Remark5.1:The elements{L n,H n,h n,p n}n∈Z span a subalgebra of K′(4)isomorphic to K(2).Note that V decomposes into the direct sum of two submodules over this superalgebra:

V= v0m,v2m m∈Z⊕ v3m,v1m m∈Z.(5.6)

Remark5.2:We conjecture that there exists a two-parameter family of representations of ?K′(4)in the superspace spanned by4?elds.In order to de?ne it,instead of the superspace of functions,Vμ,one should consider the superspace of“densities”.

Acknowledgments

Part of this work was done while I was visiting the Max-Planck-Institut f¨u r Mathematik in Bonn(Ref.27).I wish to thank MPI for the hospitality and support.I am grateful to B.Feigin, A.Givental,I.Kantor,B.Khesin,D.Leites,C.Roger,V.Serganova,and I.Zakharevich for very useful discussions.

When the paper was in print I found out that Refs.23,24,25,and26were missing.I would like to thank V.G.Kac for reading my work and clarifying that some corrections have to be made in regard to the references.

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