q

291

Progress of Theoretical Physics Supplement No. 110, 1992

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Missing Link :between Virasoro and sl(2) Kac-Moody Algebras Mitsuhiro KATO and Yasuhiko YAMADA*

Institute of Physics, University. of Tokyo, Komaba, Tokyo 153 *National Laboratory for High Energy Physics (KEK), Tsukuba 305

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§ 1.

Introduction

Virasoro algebra and sl(2) affine Kac-Moody algebra are the simplest examples of infinite dimensional Lie algebras and have been studied extensively from both mathematical and physical points of view. An important relationship between Virasoro and -------sl(2) Kac-Moody algebra is ·known as hamiltonian reduction. 1>-a> Starting from the latter with level k, k+2=pfq, by imposing a constraint on one of the sl(2) currents, one obtain the Virasoro algebra with central charge c=l-6(p-q)2jpq. So far general framework for the correspondence of these algebras is well established. For the representations, however, there seems to be some uncertainty in the literature. ·Let us consider, for instance, minimal 4 >and admissible 5 >. 6>representations for each

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s

s

q

------------, • • • •

q

I

-·-----------,I

• • • • • • • • • • • •

• • • • • • • • r p

(a) Virasoro

I I I I I I I I.

r

p

(b) sl(2)

Fig. 1. Grids for the minimal representations for the Virasoro algebra (a), and admissible representations for the~ algebra (b).

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We discuss the representations of the fractional level sl(2) affine Kac-Moody algebra and their relation to the Virasoro representations. A practical way of calculation of the free field null vectors and explicit relation between those .for both algebras are given with many examples. For the complete degenerate representations some subtle points about the representations on and beyond the boundaries of the conformal grids are clarified.

M. Kato and Y Yamada

292

algebra. Their highest weights are given as hr,s={(rq-sP) 2 -(q-p) 2 }/4Pq with 1 O) and Lolh>=hlh>. A vector lx>EM '/,~ that satisfies the conditions Lnlx>=O (n >0) and Lolx> =(h+N)lx> is called null vector (or singular vector) of grade N. It is known 9 >-I 2 > the following: THEOREM

2.1.

For the value of central charge c=13-6(t+1/t) and highest weight hr,s ={(rt-s) 2 -(t-1)2}/4t with t(=!=O)EC and r, sEZ, rs>O, there exists unique (up to normalization) null vector lxr,s>EM'/,Yz of grade N=rs (Fig. 2). Notice that hr,s+N =h-r,s. We give some examples


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Missing Link between Virasoro and sl(2) Kac-Moody Algebras

293

ix2,1>=(L~I-: tL-2)ih2,1>,

~ L-2 )lh1,2>,

lx1.2>=( L2_1-

lx3,1>=(L~1-4tL-2L-1 +2t(2t -l)L-3)1hs,1>,

lx4.1>=(L~1-IOtL-2L~1 +2t(12t -5)L-sL-1

+9t 2L~2-6t(6t 2 -4t + l)L-4)1h4,1>,

lx2.2>=( L~1

2 Z(t : I) L-2L2_1

f

6

2

3(t2-~t+I) L-4)1h2,2>'

lx1.4>=( L~1- 1 L-2£2_1 + ; 2 L~2

(t -~t +I) L-sL-1

25 ( tt-; IZ) L-sL-1

(t -t~t+ 6 ) L-4)1h1,4>. 2

(2·2)

The null vectors ixr,s> and lxs,r> are related by the exchange of t and 1/t. To find a complete expression of the null vectors is still open problem. See Ref. 18) for an interesting recent development.

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2) Similar result holds for s/(2) Kac-Moody algebra/ 4 >' 15>

[Hn, Em]= En+m , 8

8

• • • • • • • • • •

• • • • • • • • • • X---+---® •

•

•

X

--+--i....,..@

• • • • •

• • • • • • • • • • • • • • • Fig. 2. Allowed values of ( r, s) for the generic degenerate highest weight hr,s for the Virasoro algebra. For example, a highest weight vector corresponding to the symbol 0 generates a unique null vector with weight corresponding to X.

•

•

• • • • •

• • • • • --------------+-------------~ r

•

--------------+-~~-.--i~+-~

• • • • • • • • • • • •

r

• • • • • • • •

Fig. 3. Allowed values of ( r, s) for the generic degenerate highest weight jr,s for the s/(2) algebra.


+ t4-~;2+1 L~2

2

M. Kato and Y. Yamada

294

[Hn, F m] = - Fn+m , [En, F m] = 2Hn+m + knon+m,o .

(2·3)

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Let Mt.zp> be the Verma module over sl(2), generated by the highest weight vector lj>, such that Enlj>=Hnlj>=Fnlj>=O (n>O), Eolj>=O and Holj)=jlj). This lj> is primary with respect to the Sugawara energy-momentum tensor with ck=3k/(k+2), hj = j(j + 1)/(k+2). A null vector ix>EMt.zp> (of grade Nand charge Q) is defined by Enix>=Hnix> =Fnix>=O (n >O), Eoix>=O, Hoix>=(j + Q)ix> and Loix>=(hj+ N)ix>. We have the following: 16>' 17> THEOREM

2.2.

Examples:

ix-r,-1> = E~1ij-r,-1> ,

Cr 2:0)

ixr,o>=Forljr,o),

(r 2:0)

lx-1,-2>=(E:.1Fo-Ck+ 1)(k+2)E-2+2(k+ 1)E-1H-1)V-1.-2>, lx1,1>=(E-1Fo2-(k+2)(k+3)F-1-2(k+3)H-1Fo)lh,l>,

ix-2,-2> =(E'!_1Fo2+ X1E-2E:_1Fo+ X2H-1E~1Fo+ xaH'!..1E:_1 + X4H-2E:_1 + X5F-1E~1 + X6E-aE-1 + X1E-2H-1E-1 + XsE:_2)1j_2,-2) , ix2,1>=(E:.1Fo4+ Y1F-1E-1Fo 2+ Y2H-1E-1Fo 3 + YaH'!..1Fo 2+ y4H-2Fo2 + y5E-2Fo3 + y6F-2Fo+ Y1H-1F~1Fo+ xsF'!..1)ij2,1>,

(2·4)

where xl=-2k(k+7), x2=4k, xa=4k(k+1), x4=2k(k+7), x5=2k, x6=-2k(k+2) 2 X (k+7), x1= -4k(k+ 1)(k+6), xs=k(k+ 1)(k +9k+22) and Y1 = -2(k+3)(k+4), Y2 =-4(k+4), Ya=4(k+3)(k+1), Y4=-2(k+3)(k+4), y5=2(k+4), y6=2(k+2)(k+3) 2 X ( k + 4), Y1 = 4( k + 2)( k + 3)( k + 4), Ys = ( k + 3)( k + 4 )(k + 3k + 2). The first two examples are familiar in the integrable representation theory (k EZ ~o). The null vectors ix-r.-s> and lxr,s+1) are related by the algebra automorphism a,

a(En) = Fn+1 ,

a(Fn)=En-1,

. k

a(Hn) = - Hn +z-on,O .

For general formulas see Refs. 17) and 19). The above statements of Theorems 2.1 and 2.2 are so similar for both cases that one may ask if there exist any relations between the explicit form of the null vectors. In § 4, we give the answer for this question by using the free field realizations.


For level kE C, k+2=t=O and highest weight j parametrized as 2jr,s+ 1 =r-s(k+2) with r, sEZ, rs>O or r>O, s=O, there exists unique null vector ixr,s> EMg,Zp> of grade N=rs and charge Q=-r; here we have a relation jr,s+Q=j-r,s (Fig. 3). Note that the s=O sequence was missing in the Virasoro case.

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295

Completely degenerate representations

§ 3.

_......_Here we discuss about completely degenerate representations of Virasoro and s/(2) algebras. Some subtle points for the representations on or out of the boundary of the conformal grid are clarified. 1) For the Virasoro algebra, it is known that all the nesting structures are classified into the several types of diagrams in Fig. 4 and subdiagrams of them. 10 >' 23 > More precisely we have: THEOREM

3.1.

The completely degenerate representations fall into four subcases, r=t=O (mod p)

and s=t=O (mod q),

Subcase D for t=qjpE Q>o and

r=O (mod p)

or s=O (mod q),

Subcase E for t= -qjpE Q=Hnlj>=Fn-Pij>=O (n>O), the diagram in Fig. 3 will change as in Fig. 5 (for P= -2). Thus the boundary line s=O moves under the affine W eyl translation. '

§ 4.

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Null vector correspondence via Hamiltonian reduction ,_..---__

Here we give an explicit relation between Virasoro and s/(2) null vectors. Both the algebras (2·1) and (2·3) are realized by free fields [an, am]=non+m,o and [,Bn, Ym] =on+m,o as 1

Ln=~ :an-mam:+(n+ 1)Aan 2 meZ

(4 ·1)

and

En= fin, (4. 2)


Fig. 5. With the twisted highest weight condition, degenerate highest weights appear in different positions.

3.2.

Subcase E for k+2=-pjqEQ in the Fock space such as a+(ao+tl)ihr,s>=+(rt-s)ihr,s>. We pick up the case + ·since otherwise corresponding null vector will vanish in the Fock space (co-singular vector). Similarly, the Sf{JJ null vectors lxr-,s>sz in the Verma module correspond to a co-singular vector in the W akimoto module. So we concentrate on IX-r.-s>st(2). After the above remarks, we give the null vectors as

(4 ·3)

(4 ·4)

From these formulas, it is easy to see the following: THEOREM

4.1.

The coefficient of !3nr in the expression of lx-r.-s>sz is nothing but the Virasoro null vector lxr,s+n> Vir. . We need very nontrivial integral formulas to show that the integral (4·3) is non vanishing. 11)-IS) Once we know that it is so, nevertheless, it is rather simple to get the explicit expression for null vectors. To do this, we introduce a convenient basis Pn (n>O), of oscillators defined as follows. First recall the Schur polynomial Pn(x) = Pn(XI, Xz, Xs, · · ·) defined by 00

00

exp( ~ XnZn) = n=l

~ n=O

Pn(X )zn .

(4 ·5)

For example Po(x)=l, P1(x)=x1, Pz(x)=xz+(l/2)xl 2, Ps(x)=xs+xzXI +(1/6)xi 3 , Then the basis Pn is given by the substitution xn=a+(l/n)a-n. In terms of this basis, the Virasoro null vectors take the form (4 ·6) where

(r,s). = I Vl,···,Vr

f

r

r

IT dz·IT(z·-z·)2ti1z· for the integral I, for n>O

(4. 7)


where f(zi) is multivalued function on {(z1, ···, Zr)E Crizi=i=Zi, Zi=i=O} such as

298


This relation can be derived by partial integration, and is equivalent to the null vector condition. It is interesting to note the similarity to the Virasoro condition for the matrix integrals (see § 6). Using the above formulas, we get the explicit form of the null vectors,

lx1,;> = P2l h1,2> , lx2.1>=( P1 2- t ~ 1 P2)lh2,1>, lx1.s> =Psi h1,s> , 3

6t )1 6 2t+ 1 P2P1+ (2t+ 1)(t+ 1) Ps hs.1>,

lx1.4> = P41 h1,4> , 2 lx2.2>=( P2 - t ~ 1 PsP1 + (t ~i)(; ~

2 ) P4)ih2,2>,

2 4 12t np2+ 12t n2 IX4,1 >= (p1 3t+1 T2 1 (2t+1)(3t+1) T2 6t

2

24t

3 )

(4 ·9)

+ (2t + 1)(3t + 1) PsP1 (t + 1)(2t + 1)(3t + 1) p 4 jh4'1> ·

Note that we can express the null vectors in fewer terms of Pn's than that of an's or Ln's. Let us turn to the s/(2) case. Since the integral in the s/(2) null vector is essentially the same encountered befo~e, we get

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(4 ·10)

where the summation runs over

m1, ... ,

mr>O and

l/1, ... ,

Vr2:0 such that

r

~(vi+ mi)= rs.

i=1

As before, we get the explicit form of the null vectors,

lx-1,-1>= P-1V-1,-1), lx-1.-2>=(,8-2+ P-1P1)Ij-1,-2>, IX-1.-s>=(/3-s+ P-2P1 + P-1P2)Ij-1,-2>, · lx-2,-1>= p:.1V-2.-1>, lx-2.-2>=(,8:.2-

t~ 1 .8-sP-1+

t!l .8-2.8-1P1+,8:.1(P12-

t~l P2))V-2,-2>, (4 ·11)


lxs.1>= ( P1

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§ 5.

299

Correspondence of the representations

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It is known 24 > that the irreducible characters xi!~(q) and x:~~2 >(q, z) inside the grid for Virasoro and s/(2) algebras are related by the relation

(5 ·1) This simple correspondence can be naturally understood since the representations inside the grid have the same structure for both cases and the relation h~~~Z)- csl(Z) /24 = h~~~- cv1r /24-1/12 holds. At this point a question may be raised, namely, are there any relations between the representations on the boundaries of the grids for Virasoro and those for s/(2)? For r-o the answer for the question is simple since the nesting diagrams are the same for both cases and the relation above holds without any modification. . But for s=O and r=t=O, it seems that there is no natural relation between Virasoro and s/(2) representations since the embedding structures are different with each other (type D or F for Virasoro and type C or E for s/(2)). As we discuss shortly, nevertheless, there also exists a relation between these representations. For s · 0 representations, the L.H.S. of (5 ·1) vanishes, because we have extra null vectors Fonlj> which terminate the infinite degeneracy at the ground level (origin of the z= 1 pole). ____ On the diagram for s/(2),

---

}r,O

---

~

1-r,o

~

}r,2q

}-r,-2q

~

}r,-2q

}r,4q

}-r,2q

~

}-r,-4q

~

}r,-4q

all the arrows from SW to NE are of this kind, thus in passing to the Virasoro algebra this diagram splits into two copies of corresponding Virasoro embedding diagram (remind the relation hr,s=h-r,-s),

hr,O

~

h-r,O

~

hr,2q

~

h-r,2q

~

hr,4q

h-r,-2q

~

hr,-2q

~

h-r,-4q

~

hr,-4q

(5· 2)

The upper row has overall minus sign relative to the lower row. This provides another explanation of the vanishing residue in (5·1). ---..... In summarize, the hamiltonian reduction of the s=O representations of the s/(2) are difference of two identical copies of corresponding Virasoro representations. § 6.

Conclusion and discussion

We have analyzed the structure of the completely degenerate representations of ;::..---:---,._ 2 the Virasoro with central charge c=l-6(p-q) /Pq and s/(2) with level k+2=p/q. The relation between these representations is quite interesting from the point of view of the quantum hamiltonian reduction and 2d gravity. 7> The correspondence


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300

between the representations in the conformal grid {(r, s)l1 on the primary state lhr,s> with a+ a-= -1, a++ a= -2).. and a+ 2 =2t, and it is given "'"' - "'"' .L.J .L.J JN=I_ mJ,···,mr liJ,···,llr

(6·5)

4) Equation (4 · 8) can be derived from the following relation (6·6)

For n21 we get Eq. (4·8).

For n=-1, 0 we have

(6· 7) respectively. ing function 1rr(z ':1:' 1,

The last one gives the condition L:r=rvi= rs.

• • ·,

Introducing the generat-

Zr)Z -v1 ... Zr-vr , - "J