Volume 172, number 2 PHYSICS LETTERS B 15 May 1986

O N T H E U N I T A R Y R E P R E S E N T A T I O N S O F N = 2 S U P E R C O N F O R M A L T H E O R Y

P. DI V E C C H I A l

Faehbereich Physik, University of Wuppertal, Gauss-Strasse 20, D-5600 Wuppertal 1, Fed. Rep. Germany

J.L. P E T E R S E N and M. Y U

The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark

Received 25 January 1986

The Kac formula for superconformal dimensions (generalized to N = 2) is further developed (compared to a previous article). A list of discrete values of the central charge for which unitary representations are expected to exist is proposed. For several of these, unitarity is checked by computer. For two values, unitarity is proven by providing explicit fermionic representations. For one of those values, the N = 2 theory coincides with a sub theory of one of the known unitary N = 1 theories, thus extending a similar situation between N = 0 and N = 1.

Two-dimensional conformal [1 ] and superconfor- mal [2,3] field theories have recently received a great deal of renewed [4,5] interest, part ly due to their beautiful applications in two-dimensional statistical mechanics [ 3 , 6 - 8 ] , and partly due to their relevance as tools for analyzing various aspects of (super-)string theory.

I f N is the number of supersymmetry generators, the situation for N = 0 and N = 1 is rather well clari- fied. Kac [4] wrote down possible values of conformal dimension for degenerate representations, and Friedan, Qiu and Shenker completed the list by extending it to the Ramond sector. This list of conformal dimensions has a simple interpretat ion in terms of the vertex operators well known in dual models [2,6,7,9-11 ].

Friedan, Qiu and Shenker fur~:her pointed to the existence of isolated unitary theories for 0 < c N < 1, where c N is the central charge of the Virasoro algebra for N = 0, 1. Their results for these unitary represen- tations may be summarized as follows:

N = 0 : c 0 = l - 6 / m ( m + l ) , m = 2 , 3 . . . . .

h p q = ( [ ( m + l ) p - m q ] 2 - 1 } / a m ( m + l ) , (1)

1 Address after 1 February 1986: NORDITA, Blegdamsvej 17, DK-2100 Copenhagen ~, Denmark.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

p = 1 , 2 . . . . . m - l , q = 1 , 2 . . . . . p . ( l c o n t ' d )

N = I : c l = l - 8 / m ( m + 2 ) , m = 2 , 3 . . . . .

hpq = { [(m + 2)p - mq] 2 _ 4} /8m(m + 2)

+ ~ [1 - ( - ) P - q ] ,

p = l , 2 . . . . , m - l , q = l , 2 , . . . , m + l ,

NS: p - q e v e n , R: p - q o d d . (2)

The analyses of refs. [3,8] were part ly based on com- puter evidence but Goddard and Olive [12], and Goddard, Kent and Olive [13] have provided a meth- od for explicit ly verifying the results by realizations based on free fermions.

In ref. [14] two of us (PDV and JLP) and H.B. Zheng formulated the N = 2 superconformal theories [5] along modern lines. Also the method of Feigin and Fuchs [9] (as described in ref. [6]) was used to derive the analogue of Kac's formula. Surprisingly, however, the result only seemed to contain one pa- rameter, but clearly did not describe all the degenerate representations one may (laboriously) construct by hand. The result of ref. [14] was

hp = ~(1 - c2)(p2 - 1) + ~ [1 + ( - ) P ] ,

p = 1, 2 . . . . . (3)

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Volume 172, number 2 PHYSICS LETTERS B 15 May 1986

At level p/2, degenerate representations with these conformal dimensions were indeed found. However, in the NS sector this was true only for p odd, for p even entirely different values are obtained. Similarly, in the R sector eq. (3) was valid for p even but not for p odd. Tedious calculations were carried out for p = 1 , 2 , 3 , 4 , 5 , 6 , 7 .

In this paper we first show how to extend the for- mula to cope with all cases. A preliminary account of that was given in ref. [10] and essentially the same result has recently been obtained by Nam [11 ] using the method of Thorn [15].

Using the idea of "first intersections" [3,8] we may then propose a discrete list of c 2 values for which unitary representations should exist. Remark- ably we then find that for these c 2 values, the expres- sion eq. (3) becomes complete:

N = 2 : c 2 = 1 - 2 / m , m = 2 , 3 . . . . .

hp = ( p 2 _ 1) /am + ~ [1 + ( - ) P ] ,

p = 1 , 2 ..... m - l , (4)

p even in the NS sector, p odd in the R sector. (This finding contradicts a statement by Nam [11 ] .)

For general c 2 values, however, we shall now argue that one finds [10] for degenerate representations

~ p q = ( ( p - m q ) 2 - 1 } / 4 m , p, q E N , (5)

where "~p,q = hp q in the NS sector and ~p q = hp,_ - ~ in the R sector, and where c 2 is parametrized ~y m as in eq. (4) even for m non-integer. Eq. (5) agrees with Nam's result [11 ] (his R = p and his S = 2q, the possible S values were not specified).

In the Feigin-Fuchs construction one represents (super-)primary operators by vertex operators

• ~ (Z)= : e x p [ i a ~ ( Z ) ] : , Z = ( z , O , O ) , (6)

where ~ ( Z ) is a free boson icN = 2 superfield (see ref. [14] for details). Non-trivial values for c 2 are ob- tained by providing a "vacuum charge" of size - 2 a 0 in terms of which c 2 and the superconformal dimen- sion h a of ~ a become

c 2 = 1 - 8a 2 , h a = o r ( a - 2 a 0 ) .

Let us use rn as a parameter rather than a 0 = 1/2x/-m and write

h a = c~(a - 1/vCm). (7)

Greens functions (correlators) of several vertex- operators wit h different a/values, vanish unless

a i = 1/~V/~. (8)

Although a and 1/x /m-- u represent the same opera- tors, eq. (8) can only be fulfilled provided a i is suit- ably quantized in which case a violation of eq. (8) may be "screened away" using "screening operators" [6, 14]. For N = 9, 1, two possible values for screening charges a+ exist [2,610]. However, for N = 2,

a _ = 0 , a , = l / x / ~ . (9)

This observat:on led to eq. (3). However, as emphasized in ref. [14], it is easy to

construct degenerate representations for which other values than ect. (3) are obtained. The simplest non- trivial case oc:urs at level ~- in the NS-sector where one finds [5,10,11 ]

h l l = l ( m - 2 ) , (10)

corresponding; by eq. (7) to

a l l = ~ x / m or a _ l , _ l = l / x / r m - - ~ x / ' m - . (11)

Clearly, if a vertex operator with this a value exists, we may consi ]er operator-product expansions of that with the vertex operators of ref. [14], giving rise in general to a v tlues of the form

apq = (1/2x/~n)(1 - p + mq) ,

( p , q ) o ( _ p , _ q ) , p, q E N , (12)

where the operators are invariant under a sign change of (p, q). Thi~; generalizes the expression of ref. [14] where q = O. ".?o be sure, Greens functions of such ver- tex operators can usually not be constructed. In fact, the condition that a surplus charge may be screened away is (cf. eqs. (8), (9)) obtained from

M M ) M 1 ( M _ ~ P i + m ~ qi ,

i=1 ~ api'qi = 2.¢rm i=1 t=1

as M

M - 2 + ~ ttPi = negative integer, i=1

M

q i = O . i=1

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Volume 172, number 2 PHYSICS LETTERS B 15 May 1986

But because of the invariance under (Pi, qi) ~ ( -P i , - q i ) these conditions may at least be fulfilled for an infinity of correlators. Thus, Greens functions of op- erators with q ~ 0 satisfy an additional conservation law as expressed. Notice, however, that for m integer, the q terms and the p terms are indistinguishable.

In the argument given above, we cannot exclude that the ct could be quantized in different units still. However, we have explicit ly evaluated all highest weight states at levels ½, 1, ~, 2, ~ in both NS and R sectors, and we have always found agreement with eq. (5). Nam's result [I 1 ] is further support. In these ex- plicit calculations we find that the conformal dimen- sions eq. (15) of highest weight states occur at a level

given by

N = p / 2 f o r q = 0 , p . q f o r q ~ 0 . (13)

In table 1 we list the first few discrete c 2 values together with the possible superconformal dimensions in these theories. It is easy to see, by studying the lowest non-trivial level, that h l l of eq. (10) represents the highest value of h compatible with unitarity.

We have checked all those points up to level ~ (ex- cept for the R sector at m = 7) numerically for unitar- ity (cf. refs. [3,8]). Below we shall give explicit fermi- onic representation along the lines of refs. [12,13] which will prove the assertation for c 2 = ½ and ~ (m = 3 , 6 ) .

First, however, let us compare in table 2 the dis- crete c N values f o r N = 0, 1,2 (eqs. (1), (2), (4)). The conventions are such that the (N = 0) Virasoro-algebra parts for N = 1,2 have c N values such that

c o = ~c 1 = 3c 2 and c 1 = 2c 2 . (14)

As emphasized in ref. [3], the first c 1 value corre-

Table 2 The first few c N values corresponding to unitary theories for N=0, 1,2.

m Co Cl c2

3 1/2 7/15 1/3 4 7/10 2/3 1/2 5 4/5 27/35 3/5 6 6/7 5/6 2/3

sponds to the second c o value, and the second c 1 value to c o = 1. Friedan, Qiu and Shenker substantiated the equivalence of the two theories.

Similarly here, we see that the first c 2 value corre- sponds to the second c I value, and the second c 2 value to c 1 = 1. From eq. (2) we write down the table of hp,q values for c I = 2/3 (table 3). We see that many more h values are encountered than in the c 2 = -~ the- ory (table 1). However, the column with q = 1 consti- tutes a subtheory [2] which seems to correspond to the c 2 = ~ theory.

To see the correspondence, we decompose the N = 1 and N = 2 Verma modules in N = 0 ones. The N = 2 module of dimension zero contains the unit opera- tor as the primary field, and the N = 2 supercurrent

[141

J N = 2 ( z ) = -2 r (z ) + ( i /v~) [e ~(z) + 0 G(z)l

+ OOL(z), (15)

as a descendant field. However, the U(1) K a c - M o o d y current T is an N = 0 primary h = 1 field. Similarly G, t~ are h = ~ p r i m a r y N = 0 fields. This follows from the OPE of two supercurrents [14] written in compo- nents:

Table 1 Values of N = 2 superconformal dimensions according to eq. (4) for the first few unitary theories. Numbers in parenthesis refer to the R sector.

m c 2 h

2 0 3 1/3 4 1/2 5 3/5 6 2/3 7 5/7

0 0,(3/8) 0,(5/16), 112 0,(11/40), 2/5,(7/8) 0, (1/4), 1/3, (3/4), 1 0,(13/56), 2/7,(37/56), 6/7,(11/8)

Table 3 N = 1 superconformal dimensions of cl = ~ theory [3,4] ac- cording to eq. (2). Numbers in parenthesis refer to the R sec- tor.

P q

1 2 3 4 5

1 0 (1/16) 1/6 (9/16) 1 2 (3/8) 1/16 (1/24) 1/16 (3/8) 3 1 (9/16) 1/6 (1/16) 0

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Volume 172, number 2 PHYSICS LETTERS B 15 May 1986

2 L(z') + - - L(z)L(z') = ~ L'(z') + (z - z') 2

L(z)r(z ' ) = ~ T'(z) + - - (Z - Z') 2

r ( z ' ) ,

L(z)G(z') = 1 G'(z') + 3/2 G(z') z - z ' ( z - z ' ) 2 '

(same for G ) ,

T(z) Y(z') = ~ C2/(Z -- Z')2 ,

3c2/2

(z - z ' ) 4 '

algebras using the techniques of refs. [12,13]. We have not yet been able to find representations for all the c 2 values of eq. (4). However, we have noted the fol- lowing: Let ~ba(z), ~](z) , i = 1,2, 3; a = 1 ..... M b e free fermi- ons:

<~a(z)~b(z')) = ~i]~ab(z -- Z')-I . (20)

Define

M

2 Oa=l

T(z) G(z ' ) J ± 2 z - z ' LG(z')J '

G ( z ) O ( z ' ) _ 2 [L(z') -+ r'(z')l ± 4 r ( z ' ) G ( z ) G ( z ' ) z - z ' ( z - Z') 2

2c 2 + - - (16)

( z - z ' ) 3

Thus if we denote by (h)N the Verma module for N stlpersymmetries and corresponding to the primary field having (super-)conformal dimension h, we have

(0)2 ~ (0)0 * (1)0 * (~)0 ~ ( ~ ) 0 ' (17)

Similarly

(0)1 ~ (0)0 * (~)0 , ( 1 )1~ (1)0 (~ (~)~ . (18)

Thus we expect:

(0)2 ~ (0)1 * (1)1. (19)

This is substantiated by working out the OPE's of the modules using the rules of ref. [12]. Agreement with eqs. (16) is obtained provided we identify (~)0 of (0)1 with G 1 and (~)~ of (1)1 with G 2, where G ~ G 1 + iG 2, G ~ G 1 _ iG 2.

Finally [3]

( } ) 2 - •

Notice that in R sectors we have two highest weight states for each h value [3,10]: Ih, +) = G O Ih, - ) . For N = 1, Golh, +) ~ Ih, - ) . F o r N = 2, Golh, +) = 0 = G0lh, - ) , and T O has eigenvalues +~ for [h, -+). We use supersymmetry invariant vacua [3,14].

Finally we give an explicit construction o f N = 2

+ : - 1 ,

M

G Oa=l

M

G(z)= l----~i,'iik ~ ~ba(z)~bT(z)t~(z), 3X/6 " a= 1

M

G(z) = 3 ~ ieilk a~= l - ~ a ( z ) ~ ( Z ) ~ ( z) " (21)

Then it is stn ightforward to check the OPE's eqs.(l 6), giving

' ( 2 2 ) c 2 = ~ M .

Thus we have verified the existence of unitary repre- sentations foJ m = 3, 6 (eq. (4) and table 2), corre- sponding to M = 1,2.

We are grateful to J. Sidenius and H.B. Zheng for useful discussions. We thank S. Nam for sending us his paper which we received during the preparation of this work.

References

[ 1] A.A. Belay n, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys B 241 (1984) 333.

[2] M.A. Bershadsky, V.G. Knizhn~ and M.G. Teitelman, Phys. Lett. B 151 (1985) 31.

[3] D. Friedan Z. Qiu and S.H. Shenker, Phys. Lett. B 151 (1985) 37.

[4l V.G. Kac, in: Proc. Intern. Congress of Mathematicians (Helsinki, : 978), Lecture Notes in Physics, VoL 94 (Springer, Berlin, 1979) p. 441.

[5] M. Ademoilo et al., Phys. I_,¢tt. B 62 (1976) 105; Nucl. Phys. B 11 l (1976)77; B 114 (1976) 297.

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[6] V1.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 [FS12] (1984) 312.

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[9] B.L. Feigin and D.B. Fuchs, Funkt. Anal: Appl. 16 (1982) 114.

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