New modular invariants for N = 2 tensor products and four-dimensional strings

Nuclear Physics B330 (1990) 103-123 North-Holland

NEW MODULAR

INVARIANTS FOR N = 2 TENSOR PRODUCTS

AND FOUR-DIMENSIONAL

STRINGS

A.N. SCHELLEKENS and S. YANKIELOWICZ *+ CERN, 1211 Geneva 23, Switzerland

Received 10 July 1989

The construction of modular invariant partition functions of tensor products of N = 2 superconformal field theories is clarified and extended by means of a recently proposed method using simple currents, i.e. primary fields with simple fusion rules. Apart from providing a conceptually much simpler way of understanding space-time and world-sheet supersymmetry projections in modular invariant string theories, this makes a large class of modular invariant partition functions accessible for investigation. We demonstrate this by constructing thousands of (2,2), (1,2) and (0,2) string theories in four dimensions, including more than 40 new three generation models.

1. Introduction

A n i m p o r t a n t p r o b l e m one faces in constructing f o u r - d i m e n s i o n a l string theories is t h a t of f i n d i n g m o d u l a r invariant p a r t i t i o n functions that satisfy several other c o n s t r a i n t s , the most i m p o r t a n t of which are world-sheet and space-time supersymm e t r y . This p r o b l e m has been solved in various special contexts (see e.g. refs. [1-8]) b u t up to n o w an organizing principle was lacking. In theories built with free b o s o n s o r f e r m i o n s or c o m b i n a t i o n s thereof meeting these requirements is quite simple, b u t it was t h o u g h t to be significantly m o r e c o m p l i c a t e d for m o r e general world-sheet theories, w h i c h are not free (or at least not in an obvious way). H e r e we will show that p r o b a b l y for any theory of interest, and certainly for a n y t h e o r y c o n s i d e r e d so far, solving these p r o b l e m s can always be r e d u c e d to a simple c o n s t r u c t i o n which does not differ significantly from one c o n f o r m a l field theory to a n o t h e r . This construction, described in a previous p a p e r [9], yields for almost all ( r a t i o n a l ) c o n f o r m a l field theories m a n y new m o d u l a r invariants in a d d i t i o n to the d i a g o n a l invariant, and requires only k n o w l e d g e of (some of) the fusion rules of the t h e o r y . It is quite easy to a d a p t these m o d u l a r invariants to o n e ' s needs, a n d in * Work supported in part by the US Israel Binational Science Foundation, and the Israel Academy of Science. + Permanent address: School of Physics and Astronomy, Raymond and Beverley Sackler Faculty of Exact Sciences, Tel-Aviv University, Israel. 0550-3213/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)

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particular we will show that it is straightforward to satisfy the consistency conditions of fermionic string theory in this way. More importantly, the method makes it possible to explore a large class of modular invariant partition functions with special properties in a systematic way, thereby enlarging the set of consistent string theories that is accessible in practice. In this paper we demonstrate the use of this method by applying it to the special case of tensor products of N = 2 minimal models. In addition to a description of these models in a way which, at least in our opinion, is conceptually simpler, we have obtained many new (2, 2) models, as well as (2, 1) and (2, 0) models. In the latter category we find at least 44 models that have a net number of three generations. This p a p e r is organized as follows. We begin with a brief summary of the results of refs. [9] and [10]. Then, in sect. 3, we review the construction of four-dimensional superstrings, in order to demonstrate the uses of our method in this context. In sect. 4 we derive some properties of minimal N = 2 superconformal field theories which are important for our construction. In particular we determine the simple currents and the center. In sect. 5 we present some results of calculations of spectra for tensor products of such theories. In sect. 6 we present some conclusions, and discuss the possibility of extending our results in several ways.

2. Simple currents and modular invariant partition functions Consider a rational conformal field theory which has some primary field J that has simple fusion rules with every other primary field, i.e. J x qbi = ~k- Define N to be the smallest integer so that j N = 1, and define the monodromy parameter r in terms of the conformal spin of the current J [9]:

h(J) -

r ( N - 1) 2N

(modl),

so that r is defined m o d u l o N for N odd and m o d u l o 2 N for N even. In the former case we make use of this freedom by choosing r even. The charge Q of a field with respect to the current J is defined as Q(cb)=h(~)+h(J)-h(JXeb)

(mod 1),

and takes values t / N , t ~ Z. The action of the current J organizes the primary fields of the theory into orbits, consisting of the fields ~b, j × ~/i. . . . . j d ~ = q~, where d is a divisor of N. On each orbit the charge Q takes the values (t + r n ) / N m o d 1, n = 0 . . . . . d. Obviously rd must be a multiple of N. With each current J one can associate a modular invariant partition function, provided that r is even (which can always be arranged if N is odd, but not if N is

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even). This partition function is given by =

k,I

To write down a formula for the matrix M we need a quantity Q, which roughly speaking is equal to ~_Q. More precisely we define 0 on each orbit as

O.(J"~)=Ct+rn)/2N

(modl).

(2.1)

Note that since t is defined modulo N, the charge 0 on each orbit is only defined up to a overall half-integer. The matrix M is now given by [10] N

Mk,= E /s(cbk, JP~,)/}l(0(cbk) + 0 ( ~ , ) ) ,

(2.2)

p=l

where 61(X) = 1 for x ~ 71 and zero otherwise; the first &function is equal to one if ~k can be obtained from ~t by acting p times with the current J, and is zero otherwise. Note that because the second/S-function involves the sum of two charges the overall half-integer ambiguity in the definition of (~ is irrelevant. Clearly the matrix M has positive integer coefficients, and Moo = 1, where "0" denotes the identity. Note that the matrix elements of orbits of length d are N/d, since the sum in eq. (2.2) is over N values of p, even on shorter orbits. The simple formula (2.2) describes all modular invariants discussed in ref. [9], for odd as well as N even, for integer as well as fractional spins of the current, and for any conformal field theory that has simple currents. The abelian group formed under fusion by all simple currents of a theory is called the center. Every simple current J generates a cyclic subgroup of the center, and for every cyclic subgroup eq. (2.2) defines a different modular invariant if the monodromy parameter r of the generating current is even. In the application we study in this paper the center is the product of the centers of the factors in a tensor product, and can be very big (for example (Z12) 9 x Z 4 occurs). The number of cyclic subgroups of such a product is enormous! However, this is still not the complete set of modular invariants that one can construct. Given two modular matrices M(J1) and M(J2) one can always multiply them to obtain a new modular invariant partition function. If one thinks of the currents J as orbifold twists (in the way explained in ref. [9]) this is almost the same as performing two successive twists, but not quite. For example, if one applies the same twist twice in the orbifold procedure, the second twist does not alter the theory obtained with the first one. However, if one squares the matrix M(J) obtained from a fractional spin current J one gets the identity, and not M(J). Nevertheless it seems plausible that multiplying modular matrices covers the same space of solu-

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tions as performing successive twists, and in any case the result is of course modular invariant. The only potential problem is that the product need not have Mdo = 1. We show in the appendix that for all products of matrices obtained by means of simple currents one can always divide the entire matrix by Md0, without producing non-integer matrix elements. In general two matrices M(J1) and M(J2) need not commute. If they do not, their product does not yield a symmetric matrix, so that one gets a truly "heterotic" modular invariant. There are (at least) two exceptions to this. The two matrices commute if J1 and J2 are mutually local (see the appendix), or if the two currents are part of the same cyclic subgroup, as discussed in ref. [9]. The effect of fractional spin and integer spin currents is rather different. Fractional spin currents lead to matrices M that act like a permutation of the characters. Integer spin currents J give matrices M(J) that act like projection operators, removing some characters from the theory and combining the remaining ones into fewer characters of some larger algebraic structure. In particular the identity character is enlarged so that the current J appears as a chiral field in the spectrum. In fact one has considerable control over the chiral currents appearing in the leftand right-moving sectors. If one builds a modular invariant

x M( J1)M( J2)... M( Jk_l)M( Jk)x,

(2.3)

then if J1 has integral spin it appears as a left-moving chiral current; if furthermore ,/2 has integral spin and is local with respect to J2 it appears as well, etc. The same considerations apply to the right-moving sector. Note however that eq. (2.3) always defines a modular invariant partition function, even for currents of fractional spin, and irrespective of the ordering of integral and fractional spin currents. A simplification occurs when one chooses to extend the left- and right-moving algebras by the same integral spin chiral currents, by inserting the corresponding matrices M on the right as well as the left. In between those matrices one can still use any current one wishes, but not all of them are really needed to get all possible modular invariant partition functions, because one is actually constructing modular invariant partition function for a smaller set of characters of an integer spin extended algebra. This has two consequences: (i) Simple currents of the original CFT that become part of the same representation of the new extended algebra will yield the same partition function. (ii) Simple currents of the original CFT that are not part of the new CFT (i.e. that are projected out) need not be considered. It is certainly allowed to use such currents in eq. (2.3), but they will not given anything new, since any effect they might have is projected out on the left and the right. Note that if one avoids using currents that have fractional charge with respect to the integral spin currents J one would like to preserve on left and right, it is not even necessary to include the projection matrices M(J) both on the left and the

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right: the projection matrices commute with all other matrices in the chain and hence act on the left-moving as well as the right-moving characters. Further obvious simplifications are possible since one only needs to consider characters with integral charge, and only one member of each representation of the extended algebra.

3. Four-dimensional heterotic strings In this section we will explain how our method helps in understanding the construction of heterotic strings. For definiteness, we will consider four-dimensional heterotic strings. To build such a theory one considers a conformal field theory consisting of a right-moving set of NSR fermions ~p", # = 1 . . . . . 4 plus the corresponding ghosts, tensored with some internal C F T with central charges c L = 22, c R = 9. We will denote the latter as ~22;9. For purposes of classification it is useful to replace the NSR fermions and their ghosts by something more pleasant with the same conformal and modular properties. This is achieved by the bosonie string map*, which replaces this system by an SO(10) × Es K a c - M o o d y algebra. Thus to construct all heterotic strings in four dimensions, we simply have to construct all compactified bosonic strings with an internal s e c t o r 6~922;9× ( S O ( 1 0 ) x E 8 ) R. To get a consistent fermionic string theory one must have world-sheet supersymmetry in the right-moving sector of the heterotic string. This implies that the right-moving part of ~f22:9 is built out of super-Virasoro representations. The generator of the world-sheet supersymmetry in the internal sector is denoted as G int. The total supercurrent ~p~ OX, + G int which couples to the world-sheet gravitino must have a well-defined spin structure. Thus the Ramond (Neveu-Schwarz) sector of SO(10) must be paired with the Ramond (Neveu-Schwarz) internal sector. Here we encounter the first application of our construction. For our purpose we cannot work with the usual super-Virasoro representations consisting of all descendants created by the world-sheet supercurrent, since (in the NS sector) the corresponding characters are not eigenstates of the modular transformation T. The solution is to split the representations and to regard each of its two members as a primary field of a smaller extended algebra, which includes only even powers of the supercurrent. In this algebra the supercurrent G is itself a primary field, which has simple fusion rules; its orbits are simply the original doublets. To achieve the proper pairing of the NS and R sectors of the tensor product of two superalgebras with supercurrents G 1 and G 2 one simply uses the spin-3 simple current GIG2. The corresponding modular matrix M(GtG2) is inserted before the right-moving characters in eq. (2.3). As explained before, integer spin invariants amount to making projections. The result * This map originates historically from refs. [11,12]. The fact that it preserves modular invariance and changes the spin statistics signs correctly in going from fermionic strings to bosonic strings was first proved in the context of the covariant lattice construction [3]. The same map was later applied by Gepner [7] in order to relate type-II strings to heterotic strings.

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of the G1G2 projection is to leave only states which are local with respect to this current, i.e. to pair the two corresponding Ramond (Neveu-Schwarz) sectors. In the case of interest to us G 1 = G int and G 2 = ~ OX~ (which is effectively equivalent to ~ since 3X, has trivial monodromy). Thus if one has found by any method a modular invariant ~kM°lxl of a ~22;9 × (SO(10)Es)R system, one can turn it into a fermionic string by multiplying M ° from the right by the matrix M ( ~ t ~ G i n t ) . If the internal sector is itself a tensor product of several superconformal field theories one needs one such matrix for each factor. We have now obtained a fermionic string theory which may be consistent up to first order in perturbation theory, but which may run into problems at higher orders unless it has space-time supersymmetry. The modification of the partition function that yields a space-time supersymmetric theory is a second application of our construction. A necessary and sufficient condition for space-time supersymmetry is the presence in the spectrum of a right-moving chiral current of conformal spin-l, transforming as an SO(10) spinor. Hence this current must be equal to the spin operator S ~t of SO(10) times an operator S int from the Ramond sector of the internal conformal field theory, which has conformal spin ~- . This saturates the lower bound on the conformal spin of Ramond primary fields in a super-Virasoro algebra. If such an operator exists and if it is furthermore a simple current we have solved the problem: note that the SO(d) spin operator is a simple current for d even, so that S = S~tSi~t is a simple current as well. Hence we can define a matrix M(S) and insert it on the right of the chain of modular matrices (where it commutes with the matrices for world-sheet supersymmetry). The additional spin-1 current S extends SO(10) t o E 6 [3], and hence implies the existence of a U(1) current in the internal sector. It is not hard to verify that this U(1) current in its turn promotes the N = 1 super-Virasoro algebra to N = 2, which leads to the alternative description of space-time supersymmetry in relation to N = 2 world-sheet supersymmetry [14]. Thus we may assume without loss of generality that the right-moving internal sector is a representation of an N = 2 algebra (which however may consist of building blocks that are not separately N = 2 superconformal field theories). It is well known that in such algebras the NS and R sectors are connected by spectral flow [15]. In particular the vacuum is connected to one of the R a m o n d ground state (with h = c/24). The structure of the fusion rules is preserved under spectral flow. In particular it follows that the Ramond ground state connected to the identity has simple fusion rules, because of course the identity has that property. Hence if one has any modular invariant partition function involving a right-moving N = 2 super-Virasoro algebra with c = 9 in the internal sector, it can * The conformal spin of the SO(N) spin operator is N/16, so that one gets 85 for SO(10). One can understand this value ~ as arising from a spin field of the euclidean space-time rotation group, SO(4), (generated by the NSR fermions) wbSch contributes ¼, and the "spin field" of the superghost system [13], which contributes 3. The map to the bosonic string replaces this entire system (NSR plus ghost spin fields) by the SO(10) spin field.

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N = 2 tensor products

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always be modified to a partition function of a space-time supersymmetric string theory. The advantage of this approach is that there exists a completely general p r o o f of modular invariance [10] of the resulting superstring partition function, independent of the details of the N = 2 theory one uses. A necessary condition for this sort of construction is that one must have at least one modular invariant to start with. In general, the only known invariant is the diagonal one, which one can use at the expense of generality by simply choosing C22;9 = ( S O ( 1 0 ) x E s ) L x C9:9, with the same N = 2 algebras on the left and the right. Note that although part of the left-moving sector is now also built out of N = 2 super-Virasoro representations, there is no reason whatsoever to pair NS and R sectors here, nor is there any reason to extend SO(10) to E 6. This will automatically happen if one includes in the modular chain (2.3) no other matrices than M ( ~ " G ira) and M ( S ) (since those matrices act then also on the left-moving sector), but if any other matrices are included the left-moving world-sheet a n d / o r space-time supersymmetry will in general be broken. We will call theories with a left- and right-moving level-] SO(10) x E 8 K a c - M o o d y algebra, left- and right-moving world-sheet supersymmetry and an extension of both the left- and right-moving SO(10) factor to E6* " ( 2 , 2) theories". Using the bosonic string map, they can be mapped to supersymmetric heterotic strings with a n E 6 gauge group, or a type-II strings with (at least) N = 2 space-time supersymmetry. When the left-moving SO(10) algebra is not extended t o E6, we speak of a (1,2) theory. Such theories can be mapped to heterotic strings with an SO(10) gauge group, or type-II theories with only right-moving space-time supersymmetries. Finally we speak of (0,2) theories if there is a right-moving SO(10)× E 8 K a c M o o d y algebra, right-moving world-sheet supersymmetry, and a right-moving extension of SO(10) to E 6. Such a theory can be mapped to a supersymmetric heterotic string, but not to a type-II string. In principle the right-moving sector is not restricted to have any particular structure, but in the applications we study here it will also have an SO(10) x E 8 K a c - M o o d y algebra, and the " 0 " in (0,2) refers to the absence of world-sheet supersymmetry on the left. As explained above, unless one makes a special effort to preserve those world-sheet supersymmetries, they will be broken. Thus generically our construction yields (0, 2) theories. Note that even for (0, 2) theories it may occasionally happen that SO(10) extends t o E 6. A different class of (0, 2) theories built out of N = 2 minimal models was recently studied in ref. [16]. These authors do not break the N = 2 symmetry of the internal sector, but they use orbifold-like twists to break SO(10)L to a subgroup. This sort of construction can be implemented on top of what we described above (in sect. 6 we will explain briefly how to do this), but we will not pursue this in the present paper.

* For extensions to E 7 and E~ we will adopt the same terminology. This does not properly reflect the complete set of world-sheet supersymmetries in such theories, but neither does the label "(2,2)", when interpreted literally.

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4. Simple currents in N = 2 minimal models

The arguments given above are completely general, but we will consider in the rest of this paper a more specific situation, namely tensor products of N - - 2 minimal models. We begin by determining the center of these models. The N = 2 minimal models are labeled by an integer k, in terms of which the central charge is c = 3 k / ( k + 2) [17]. We have a combination of such models in both the left- and the right-moving sector. In discussing the highest weight states, the characters, the field identification and the fusion rules we concentrate on one of the two sectors, but of course all these algebraic results hold for the left- and right-moving sectors separately and independently. The unitary representations and characters of the minimal series have been discussed in refs. [7,18, 19]. In a basis of T eigenstates, the highest weight states are labeled by three integers (l, q, s), where 0 ~< l ~< k and q and s are defined modulo2(k + 2) and 4 respectively; furthermore these values are restricted by the requirement l + q + s = 0 mod2. This set of integers still covers the set of highest weight states twice: there is an identification of (l, q, s) with ( k - 1, q + k + l, s + 2). Thus to get all highest weight states we can restrict s to the values 0 and 1, corresponding to the NS and R sectors respectively. In the following we will denote these states and the corresponding characters simply by their labels. The fusion rules of these models have been derived in ref. [20], and are very simple min(l 1 + / 2 , k - l I - l 2 )

(11' ql, S1) X (12' qz' S2) =

E /=

(I, ql + q2' St + S2)"

Ill - 1 2 1

with implicit periodicities of q and s as explained above. One should use the field identification to map s back into the range 0,1 if necessary. Note that the fusion rule for the first label is simply that of SU(2), level k. Hence it follows immediately that the simple currents are (0, q, s) and (k, q, s), for any allowed value of q and s. The current (0,1,1) generates an orbit of fields which are manifestly different from the identity until the field (0, 2(k + 2), 2(k + 2)) is reached. This field is the identity if 2(k + 2) = 0 rood4 (i.e. k even) but not if 2(k + 2) = 2 rood4 (i.e. k odd). In the latter case the orbit continues, returning finally to the identity after 4(k + 2) steps. It is easy to see that this orbit includes all simple currents. If k is even, the orbit of the simple current (0, 1, 1) has length 2(k + 2), but it does not include the simple current (k, k + 2,0). The latter generates an additional Z 2 orbit. Thus we find the following answer for the center of the N = 2 minimal models Z4(k+2)

(k odd) ;

Zz(k+z) X Z 2 (k even).

As explained in the previous section, we introduce a separate primary field for each of the two members of an N = 2 supermultiplet. In particular the supercurrent

A . N . S c h e l l e k e n s , S. Y a n k i e l o w i c z

/

N = 2 tensorproducts

111

itself is regarded as a p r i m a r y field, and is labeled by (k, k + 2, 0) (or, equivalently, (0, 0, 2)). It is a simple current of order 2 and conformal spin 3, which m a y be used to pair all p r i m a r y fields into supermultiplets. The two partners in such a multiplet are (l, q, s) and (k - l, q + k + 2, s) - (1, q, s + 2). Note that for fixed s the supercurrent m a p s fields with labels inside the " c o n e " ]q] = l + 1 to fields outside that cone, and vice versa. The fields inside the cone are the usual super-Virasoro primary fields, and the ones outside the cone are super-Virasoro descendants (the fields on the cone itself are discussed below). The characters corresponding to each p r i m a r y field have an expansion of the form ~h-c/24~o~ ,4 ,7, . With one exception, only the tt z..,n=Ot,entl first term in the q-expansion is relevant for the massless spectrum. The exception is the v a c u u m , 10), with h = 0. In that case one has to include descendant states J 110) for each spin-1 field in the chiral algebra. There is just one such field, n a m e l y the U(1) current in the N = 2 algebra. T o proceed, we need to know the h-values of the highest weight states of each representation, and the multiplicity d o of those states. In the N S sector (s = 0) the conformal dimensions of two fields in the same supermultiplet differ by half-integers, and the lowest value within a supermultiplet is o b t a i n e d for Iql ~< l. In that case the conformal spin (weight) is given by

l(l+ 2 ) - q 2 h -

4 ( k + 2)

(4.1)

T h e multiplicity of these states is one (as m a y be deduced for example from the f o r m u l a for the characters.) If we denote the corresponding ground state as ]/, q, 0), then the second state in the supermultiplet is given by G+l/2]l,q,O). Thus in general the resulting state, corresponding to the field ( k - l,q+ k + 2,0), has a c o n f o r m a l dimension given by eq. (4.1) plus ~-. Furthermore, it has multiplicity two, since it can be obtained by G + as well as G . However, it m a y h a p p e n that one or b o t h of the excited states G+-a/21l,q,O) have zero norm. G r o u n d states [l, q , 0 ) for which this h a p p e n s are called chiral states, and they occur for ]ql = I. They are annihilated b y either G + 1/2 or G - x / 2 depending on the sign of q, and by both these o p e r a t o r s if q = 0. Thus for ]ql = l, l :g 0 the difference in the conformal dimension between the two states in the supermultiplet is still ~, but the excited state has multiplicity 1. Finally, for l = q = 0 the first excited state does not occur for h - 3 but for h = 23 The corresponding field has multiplicity two, and the two fields are simply the two supercurrents of the N = 2 algebra. In the R a m o n d sector all highest weight states are annihilated by either Go~ or G o . Some states are annihilated by both of these operators. They are precisely the ones that saturate the lower b o u n d h = c/24, and are usually called the R a m o n d g r o u n d states. T h e highest weight states with h > c/24 come in degenerate pairs, which are m a p p e d into each other by Go-+. The two states in such a pair are (l, q, 1)

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and (k - l, q + k - 2,1) - (l, q, 3). Using the field identification one can always m a k e sure that for one of the two states Iqf ~< l - 1. Then h is given by

h-

l(/+ 2)_q2 4 ( k + 2)

1 + 8'

(4.2)

for both states. This formula covers all R a m o n d highest weight states except those with t q] = l + 1, which are precisely the Ramond ground states. Here one has to m a k e a choice of chirality: for one sign of q, the value of h is correctly given by eq. (4.2), and is equal to c/24. The other sign of q yields then the supersymmetric partners of these states, which have h = c/24 + 1.

5. Conformal zoology In this section we present some results of an application of our method to N = 2 tensor products. We have considered all possible combinations yielding c = 9. For these combinations, we have constructed the complete massless spectra of most (2, 2) and (1, 2) models (the total number of each is a few hundred and a few thousand respectively), but only a few of the (0, 2) models (which are far more numerous, but require more computer time). Our procedure is straightforward. One simply generates all right-moving highest weight states and uses (for some choice of currents ,/1. . . . . Jk) eqs. (2.2) and (2.3) to obtain the combination of left-moving characters and a given right-moving one. Each character is a product of a character of each of the N = 2 models in the tensor product and one of the four characters of SO(10). A computation of this kind seems almost impossible once one realizes how many characters one typically has. For example, the k = 1, N = 2 minimal theory has 12 characters. One needs nine such theories to build a ¢ = 9 internal sector of a heterotic or type-II string, so that one has a total of 4 × (12) 9 characters. Fortunately there are several very important simplifications. - - T o get the massless spectrum one only needs to consider character combinations with a highest weight state with h ~< 1. - - B e c a u s e we always have world-sheet supersymmetry on the right, we need only consider right-moving combinations of NS or R characters, but not mixed ones. In fact, the simplifications described at the end of sect. 2 make it possible to avoid putting in the matrices M(+~'Gi) explicitly, at least if one has world-sheet supersymmerry also on the left. Such a short-cut is unfortunately not possible for (0, 2) theories, which explains why computing their spectrum is more time-consuming. - - I n space-time supersymmetric theories it is sufficient to use either right-moving R a m o n d or Neveu-Schwarz characters. We have actually used both as a consistency check, but settled on using Ramond characters. Because of right-moving world-sheet

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products

113

supersymmetry one only needs to consider Ramond ground states to get the massless modes of the string. For each N = 2 factor there are just k + 1 such states, with I = 0 . . . . . k. On the other hand, in the NS sector one has to consider all chiral p r i m a r y fields, of which there are twice as m a n y for each factor in the tensor product (although further short-cuts are possible in this case as well). - - B e c a u s e of CPT invariance it is sufficient to consider just one of the two SO(10) spinor representations. Thus the number of right-moving characters one has to consider is only lqi(k i + 1), where the product is over all factors in the N = 2 tensor product. For example, in the (k = 1) 9 theory one has to consider finally only 512 of the 4 × (12) 9 characters. - - T h e matrices M(J) are very sparse, and eq. (2.2) allows us to determine which matrix elements in a given row or column are non-zero without having to construct the entire matrix. N o t e that in this procedure we automatically obtain all left-moving representations that are paired with a given right-moving one. In particular getting the SO(10) o r E 6 singlet representations is as easy as getting the massless generations. The extended left- and right-moving algebras of (1, 2) and (2, 2) theories imply a reduction of the number of a priori distinct currents, as explained in sect. 2. First of all one only needs to consider pure NS or pure R currents. Furthermore one can always restrict each simple current to have l = 0 in each factor of the tensor product: a current with l = k i in the ith factor is equivalent to one with l = 0, obtained by acting with the extended algebra generator +"Gi (note that this also changes the SO(10) character). This implies in particular that in such theories it is not necessary to consider D-type invariants of the SU(2) labels (the D-invariants are generated by the currents (k, 0, 0)). In (0, 2) theories one should however allow both l = k and l = 0 . If one is only interested in (2, 2) theories one may restrict the currents further by requiring that they have integral charge with respect to the space-time supercharge S. Then the presence of the projection matrix M(S) on the right is also effective on the left, since M(S) commutes with all other matrices. Some of the modular invariant partition functions we obtain in this case have been discussed before [7,16,21]: they include "fl-projections", Z~+ 2 twists and D-invariants (which however are not needed), and combinations thereof. However, some of the conditions mentioned in the literature (such as integral inner products a m o n g / 3 's) are clearly not necessary for modular invariance. Our method puts all of these constructions on equal footing, and makes it possible to combine them in any way one wishes without ever having to worry about modular invariance. While all partition functions we get are modular invariant, it is much harder to determine a priori which ones are distinct. This problem is solved if one uses just one current: every distinct subgroup of the center gives a distinct modular invariant partition function (if the current satisfies level matching). But two partition functions which are formally different may well yield the same massless spectrum. This

114

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products

sort of degeneracy occurs always if there are discrete symmetries among the characters, e.g. if several N = 2 factors are identical. For example in the (1) 9 theory the center is ZnZ92, and has a huge number of discrete subgroups. Many of the corresponding partition functions are however related to each other by permutations of the factors. The problem of classifying the distinct spectra becomes even more difficult if one uses more than one current. It is obvious that a complete enumeration of all four-dimensional string theories is impossible, even for the (relatively) infinitesimal subclass we are considering here. The results we present here should therefore only be regarded as exploratory. For that reason we have not attempted an exhaustive construction of all distinct partition functions, but instead used randomly generated combinations of simple currents. It is certainly possible, with a considerable effort, to list all (2, 2) models that can be obtained with our method, but this seems quite pointless. One may be slightly more hopeful about a complete enumeration of all "interesting" models (e.g. those with 3 (4?) generations). It should be possible to obtain a complete classification of all three-generation models that can be obtained (with our method) from tensor products of N = 2 minimal models (or more general N = 2 coset models, constructed in ref. [22]). A very helpful empirical observation is the following: the net number of chiral generations for a given tensor product is quantized ! More precisely, given a combination of minimal models with total c = 9, the number of generations that one gets for all its modular invariant partition functions is always a multiple of some integer (not equal to one). This integer is in most cases 12 (or a multiple of 12), less often 6, rarely 4", and only in one case it is equal to 3. We will return to this case below. To get an idea of the possible spectra, consider first the product of nine k = 1 models. We choose this example, because it demonstrates just about every interesting phenomenon that can occur. It is not quite typical in the sense that it can be described in terms of free bosons (since the k = 1 model has a realization in terms of one free bosons [24]), but there is no trace of that in the massless spectra. We have randomly generated combinations of up to nine simple currents (including the space-time supercurrent, but not the world-sheet supersymmetry projections) and computed the massless spectra using eqs. (2.3) and (2.2). Altogether we have found 77 different spectra with at least one space-time supersymmetry and unbroken world-sheet supersymmetry for the left-movers. This is probably the complete list of such models for this tensor product, or nearly so, since even after extensive searches nothing new was found. The results are listed in table 1 (to save space, we have omitted some uninteresting non-chiral models (number 26-43 and 53-64), whose number of generations lies between the values listed just above them and just below them). Note that if one maps these theories to type-II instead of heterotic string * In the notation of ref. [23], we find four generations for numbers 60, 150 and 168. Furthermore, the combinations 146, 148 and 157 have a quantization in units of 40, 32 and 8 respectively.

o ~

i

~

~ o~ o

~

B~

b

Jl o

~1

I I [ I I I~

~1

I I I I I I I I I I I I I I I I I I I I I I

116

A.N. Sckellekens, S. Yankielowicz / N = 2 tensorproducts

TABLE 1 (continued) nr. 65 66 67 68 69 70 71 72 73 74 75 76 77

NR

NL

(S, s)

2 2 2 2 2 2 4 4 4 4 4 4 4

0 2 2 2 2 4 1 0 0 0 2 4 4

0 20 8 8 2 ---------

(S, c)

(S,0)

0

204 140 200 164 176 216 --------

----------

( V, 0)

( V, t,)

19 13 31 13 13 24 32 37 10 1 31 78 24

4 ------4 10 16 ----

(S, v)

order

12

5 2 4 3 5 5 8 6 5 5 5 5 3

------------

TABLE 2 Notation of gauge group representations used in table 1 (0)

(v)

(s)

(c)

(1) (1) (1) (1)

(10)

(16) (27) (56)

(16) (27) --

L

SO(10) E6 E7 E~

---

t h e o r i e s , o n e f i n d s all p o s s i b l e c o m b i n a t i o n s

of left- and right-moving

space-time

supersymmetries. Some

o f t h e 77 m o d e l s h a v e b e e n d i s c u s s e d b e f o r e , a n d s e r v e a s a c h e c k o n t h e

calculations. invariants

T h e f i r s t o n e is d i s c u s s e d i n ref. [7], a n d a p p e a r s in t h e list o f d i a g o n a l

o f ref. [23]. N u m b e r

3 and

4 have been

constructed

l a t t i c e s a n d s h i f t v e c t o r s i n ref. [25], a n d t h e r e s u l t s a g r e e . N u m b e r theory constructed torus; numbers

76 a n d 77 a r e e a s i l y r e c o g n i z a b l e

ten-dimensional radius)

i n ref. [7], w h i c h is c o m p a c t i f i e d

E 8 × E 8 heterotic

respectively. Numbers

compactifications

know only the number

to four dimensions

on the A 2

as torus-compactifications

string on the E 6 and by Narain

of the

(A2) 3 torus (with fixed

72, 73 a n d 74 a r e e x a m p l e s

of the type discussed

using covariant 66 is t h e (1) 6 K 3

of more

[26]. A l t h o u g h

general

torus

in general we

o f e x t r a g a u g e b o s o n s , it is o f t e n e a s y t o d e t e r m i n e t h e g a u g e

group,

because

fusion

rules are known,

the extra gauge bosons and impose

are always

simple

strong constraints

currents.

Hence

their

on the possible structure

constants. The number about

o f ( 1 , 2 ) a n d (2, 2) t h e o r i e s f o r t h e o t h e r t e n s o r p r o d u c t s *

400 (typically, products

varies from

o f k = 1, 2, 4, 6 a n d 10 g i v e v e r y l a r g e n u m b e r s

* Tables of spectra, similar to table 1, for the other tensor products are available on request.

of

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products

117

solutions) to just a few (for example, we did not find any additional solution for (1)(5)(41)(1804) in comparison with ref. [23]). In ref. [23] the complete massless spectrum for the diagonal invariants (with a space-time supersymmetry projection) of all these models was presented. Our results agree completely with those of ref. [23], except for some minor differences for K 3 × T 2 compactifications (see below). The n u m b e r of generations and anti-generations for the diagonal invariants has also been listed in ref. [27]. These results disagree substantially with those of ref. [23], and hence with ours. A formula for the number of generations and anti-generations for L a n d a u - G i n z b u r g models associated with N = 2 superconformal compactification has recently been derived by Vafa [28]. Unfortunately this formula does not apply to most of our results, since it deals only with the projection needed for supersymmetry, and not with any others. Furthermore it does not give the number of scalars and extra gauge bosons, and is not applicable to (1, 2) and (0, 2) models, at least not in its present form. Although the exceptional invariants of the N = 2 minimal models are not generated by our procedure, once they are known there is no problem in using them. It turns out that the unit of quantization of the number of generations is in general different for diagonal and exceptional invariants. On the basis of the results of ref. [29], one expects a relation between the following factors in the tensor product [k = 10]E-- [ k = l l A [ k = 2 I A ,

[ k = 2 8 ] E - - [ k = l l A [ k = 31A ,

where the subscripts " E " and " A " indicate the type of invariant. The classification of critical points of the superpotential of N -- 2 L a n d a u - G i n z b u r g theories predicts that upon replacement of one of the factors on the left by the combination on the right one should get the same spectrum, up to deformations of the manifold. Thus as least for (2, 2) theories one expects to find the same number of generations. We find that the relation goes much further than that, and that the entire list of modular invariant partition functions, (1,2) as well as (2,2), is identical, for all massless states in the spectrum. Here we disagree with the results of ref. [23], where in some cases different answers were found for the number of scalars in compactifications of the form K 3 × T 2 (in all compactifications of this type that we have found T2 can be identified as either the A 2 or the D 2 torus). Hence it appears unnecessary to consider E 6 or Es invariants. The only tensor product for which we have found examples with three generations is (1)(16) 3, with a n E 7 invariant in all three k = 16 factors. One (2,2) theory with three generations is already known for this tensor product [30]. This example is not accessible with our method since it involves a twist that permutes the three k = 16 factors. The corresponding twist field is not part of the field content of the original theory. Our method can of course only produce new modular invariants w i t h i n a given CFT; to get the example of ref. [31] we would first have to set up a new

118

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products TABLE 3 String theories with a net number of three generations obtained from the tensor product (1)(16) 3, with an exceptional invariant in all three k = 16 factors. The notation is as in table 1. The last column denotes the simple current used to obtain the result. The astcrisk in column 3 indicates unbroken left-moving world-sheet supersymmetry.

nr.

NR Nk ( S , s )

(S,c)

(S,O) (V,O) ( V , v ) ( S , v )

1 2 3

1 1 1

0 O* 1

16 16 15

13 13 12

230 204 201

4 4 3

4 5 6 7

1 1 1 1

1 0 0 0

15 15 15 15

12 12 12 12

185 258 226 210

3 4 4 4

8

1

0

14

li

222

4

9

1

0

14

11

220

4

10 11

1 1

0 1

14 13

11 lO

218 209

4 3

--

29

12

1

0

13

10

218

4

--

29

13 14 15

1 1 1

0 1 0

13 12 12

10 9 9

202 183 242

4 3 6

--

23

--

25

16

1

0

12

9

236

4

--

29

i7 I8

1 1

0 0

12 12

9 9

222 212

4 4

27 27

19 20

1 1

0 0

12 12

9 9

210 210

4 4

25 23

21 22

1 1

0 1

12 11

9 8

206 197

4 3

25

23

1

0

11

8

232

4

25

24

1

0

11

8

232

4

23

25

1

0

11

8

222

4

23

26

1

0

11

8

218

4

25

--

35

--

29

--

39 31 31

--

29 29

Currents [ v, (0, 0, 0), (0, 7,1), (16,16, 0), (0, 0, 0)] [ v, (0, 0, 0), (0,14, 0), (0,10, 0), (0, 16,0)1 [v,(0,0,0),(0,0,0),(0,0,0),(0,11,1)] [0,(0,0,0),(16,- 1 , 1 ) , ( 1 6 , - 11, 1),(0,1,1)] [s,(1, 2,1),(0,0,0),(0,18,0),(16, - 16,0)] [c,(0,1,1),(0,-

2,0),(0,-

11, 1), (0,1,1)]

[0, (0, 2, 0), (0, 0, 0), (0, 9,1), (0, 4, 0)] [0, (0, 0, 0), (0, 16,10), (0, 0, 0), (0,1,1)] [v,(0,0,0), (16, 13,1), (0, l, 1), (0, 1,1)] [0, (1, 3,0), (16, 6, 0), (0, 0, 0), (16, - 2, 0)] [s, (1, 2,1),(0, 5,1), (0, 4,0),(16, 14,0)] [0, (1,1,0),(0,1,1), (0, 12,0),(16,1,1)] [0,(1, 1,0),(0, 10,0),(16,12,0),(0,1,1)] [v,(0,0,0),(0,1,1),(16,4,0),(0,1,1)] [v, (0, 2, 0), (0,1,1), (0,1, 1),(16,1,1)] [c, (0,1,1),(16,17,1),(0,0, 0),(0,0, 0)] [0,(1, - 1,0),(0,0,0),(0,1,1),(0, - 10,1)] [0, (0,0,0),(16, 7,1),(16,15,1), (0, 7,1)] [0,(1,1,0),(16, 15,1),(0,0,0),(0,1,1)] [0, (0, 0, 0), (16,13,1),(0,1,1), (0,17,1)] [c,(1, - 2,1),(0,0,0),(0,0,0),(0,1,1)] [0,(0,0,0),(0,0,0),(0,-

15,1),(0,0,0)]

[v,(1,1,0),(O, - 5,1),(0,1,1),(0, 2,0)1 [0,(0,

-

2,0),(0,

8,0),(0,1,1),(0,0,0)]

[0, (1, 3,0), (0,1,1), (0, 0, 0), (16,14, 0)] [c, (0,1,1), (0, 0, 0),(0,0,0), (16, 4,0)] [0, (1,1,0), (0,1,1), (0, 0, 0), (0,4,0) 1 [0, (0, O, 0), (0, 8, 0), (0,1,1), (0, 7,1)] [0, (0, 2, 0), (0, 0, 0), (16,15,1), (0, 0, 0)] [v,(0, 0,0), (0,1,1),(16, 7,1),(0,-2,0)] [v,(0,0,0),(16, - 5,1),(16, 11,1),(16,8,0)] [s,(0,1,1),(0,0,0),(0,0,0),(0,1, 1)] [s,(0,1,1),(16,11,1),(0,1,1),(0, - 12,0)] [ v , ( 0 , 1 , - 1),(0, - 17,1),(0,16,0),(0,1,1)l [c, (0,1,1), (0,13,1), (0,0,0),(16, 8,0)1 [c,(1, - 2,1),(16, - 11,1),(0,0,0),(0,1, 1)1 [s,(0,1,1),(0, 8 , 0 ) , ( 0 , - 15,1),(0,1,1)] [v,(0,0,0),(16,7,1),(0, - 7,1),(16, - 8,0)] [v,(1,1,0),(0,1,1), (0,0,0),(0,1,1)1 [0, (1,1,0), (0, 0, 0), (16, 3,1), (16, 8, 0)]

119

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products

TABEL 3 (continued) nr.

N R N L (S, s)

(S, c) (S,0) (V,0) (V, v) (S, v)

Currents [0,(0,2,0),(0,0,0),(16, - 17,1),(16, 7,1)] [s, (0,1,1), (0, 5,1), (0, 0, 0), (0, - 15,1)] [v,(O,O,O),(O,O,O),(O,- 1,1),(0,16,0)] [s, (0,1,1), (0, O, 0), (16, - 7,1), (16,13,1)] [s,(O, 1 , 1 ) , ( 1 6 , - 16,0),(0,0,0),(0,0,0)]

27

1

0

11

8

216

4

--

25

28

1

0

11

8

216

4

--

23

29

1

0

11

8

206

4

30 31

1 1

0 1

11 10

8 7

206 195

4 3

---

23 --

32

1

0

10

7

234

4

--

31

33

1

0

10

7

220

4

--

21

34 35

1 1

0 0

10 10

7 7

218 212

4 4

--

21 23

36

1

0

10

7

212

4

37

1

0

10

7

210

4

--

25

38

1

0

10

7

208

4

--

23

39

1

0

10

7

204

4

--

21

40 41

1 1

0 0

10 9

7 6

202 206

4 4

---

23 25

42

1

0

8

5

208

4

--

21

43

1

0

8

5

206

4

--

21

44

1

0

8

5

200

4

--

19

25

fv, (0,0, 0), (0, 4, 0), (0,1, i), (0, O, 0)]

conformal

However, space-time the

former

left-moving

21

field theory built out of a Z 3 twisted product

did however generations,

[v,(1,1,0),(16,16,0),(0,0,0), (16,17,1)] [s,(0,1,1),(O,O,O),(O,- 15,1),(0,1,1)] [c, (0,1,1), (0,1,1), (0, O, 0), (0, O, 0)] [c,(0, - 1,1),(0, - 5,1),(0,11,1),(0,13,1)] [v,(0,0,0),(0, - 6,0), (0,0,0),(16, 9,1)] [c, (0,1,1), (0, 0, 0), (0, 0, 0), (0,1,1)] [0, (1,1, 0), (0, O, 0), (0,1,1), (16, O, 0)] [v,(1,1,0),(0,2,0),(16, - 13,1),(0,0,0)] [c,(1,2,1),(0,6,0),(0,1,1),(0,- 12,0)] [0, (0, 2, 0), (0, O, 0), (0,1,1), (0,1,1)] [v,(O, - 2,0),(16,5,1),(0,0,0),(0,3,1)] [c,(0,1,1),(0, 4,0),(16, 9,1),(0,1,1)] [0, (0, O, 0), (0, O, 0), (0,1,1), (0,1,1)] [s,(O, - 1 , 0 ) , ( 1 6 , - 14,0),(16, 6,0),(0,0,0)] [ v, (0, O, 0), (16,16,0),(0,1,1), (0,1,1)] [0,(0,0,0),(16, - 2,0),(0,0,0),(0,5,1)] [ v, (0, O, 0), (16,10, 0), (16,17,1), (0, O, 0)] [c,(1, 2 , 1 ) , ( 1 6 , - 9 , 1 ) , ( 0 , - 7 , 1 ) , ( 0 , 2 , 0 ) ] [o, (0, 2, 0), (16,1,1), (0,16, 0), (0,13,1)] [ v, (0, O, 0), (0,1,1), (0,10, 0), (0,1,1)1 [c,(1,0,1),(0,0,0),(16,10,0),(0,- 14,0)] [ v , ( 1 , 1 , 0 ) , ( 0 , - 9,1),(0,1,1),(0, - 12,0)] [0, (0, O, 0), (16,10, 0), (0,1,1), (0, - 6, 0)1 [v,(O,O,O),(O,- 16,0),(0,1,1),(0,0,0)] [ v , ( 1 , - 1 , 0 ) , ( 0 , - 16,0),(0,1,1),(16,0,0)] [ v , ( 1 , - 1,0),(0,9,1),(0,1,1),(0,0,0)] [s, (0,1,1), (16, - 3,1), (0, O, 0), (0,1,1)]

get the model

o f t h r e e k = 16 t h e o r i e s . W e

t h a t is u s e d a s a n i n t e r m e d i a t e

14 anti-generations,

s t e p i n ref. [30], w i t h 23

173 E 6 scalars and 3 extra U(1) gauge bosons.

t h e r e is a m u c h e a s i e r w a y t o g e t t h r e e g e n e r a t i o n s , and/or we

world-sheet get

one

world-sheet

three-generation

(1,2)

supersymmetry theory

supersymmetry

with

a n d t h a t is t o b r e a k

for the left-movers. By just giving up three

to be

generations.

broken

we

If we

get at least

also

allow

43 m o r e

theories (there are almost certainly several others). The spectra, as

120

A.N. Schellekens, S. Yankielowicz /

N = 2 tensor products

well as the currents we used to get them, are listed in table 3. Note that the currents are only given to make the results reproducible. In general, there are many different choices for the currents that yield the same spectrum. All entries in table 3 are exactly solvable conformal field theories corresponding to healthy supersymmetric string theories with a gauge group containing E 6 (if NI_ = 1) or SO(10) (if N L = 0). None of these theories has gauge anomalies, not even the U(1) anomalies that can occur in certain types of (0, 2) theories (and that are canceled by the Green-Schwarz mechanism at the price of vacuum instability [31]). This follows from modular invariance [32], combined with the presence of an E 8 factor in the gauge group [25]. The string theories listed in table 3 are part of a growing list of such theories which already includes examples based on orbifolds [33], free fermions [34], covariant lattices [35], and the (2, 2) theory of ref. [30] based on the same tensor product we are considering here. Even though all theories in table 3 are closely related to each other, they can be expected to have quite different phenomenological implications. One can only hope that at least some of them are in clear contradiction with experiment.

6. Outlook and conclusions Clearly the method we have advocated in this paper greatly extends the list of four-dimensional string theories accessible to exploration. However, this is by no means all one can do. Up to now we have always kept an unbroken SO(10) × E 8 K a c - M o o d y algebra on the left. However, just as one can break the left-moving " s p a c e - t i m e " and world-sheet supersymmetries, one can break this K M algebra as well. To do so, one simply starts with characters of some conformal subalgebra of SO(10) × Es. Of course one wants to get the full SO(10) x E 8 algebra on the right, in order to be able to map this sector to a fermionic one. But this can always be achieved by putting some projection matrices in front of the right-moving characters to add the missing SO(10) × E 8 roots. This opens the way to constructing string theories whose gauge group is something a bit closer to the standard model than SO(10), perhaps even SU(3) × SU(2) × U(1) n (where n is almost inevitably larger than 1). There is no reason why one could not get three generations in such a model, and in fact there could well be many more models than those listed in table 3, since the center of the conformal field theory one starts with is even larger. We hope to come back to this in the future. The total number of different (0, 2) theories with unbroken SO(10) x E 8 may well be close to a hundred thousand or more, for the minimal N = 2 theories alone (to understand this estimate, note that for the one case where we have studied (0, 2) theories fairly exhaustively, namely the three-generation models, the (0, 2) theories are at least 20 times more numerous than the (1, 2) and (2, 2) theories. The number

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products

121

of (1, 2) and (2, 2) theories is several thousand.) Obviously, breaking SO(10) or E 8 will enlarge this number even more. This seemingly hopeless situation is made slightly more bearable by our observation that the number of generations for each tensor product is quantized. This is certainly true (although not proved) for (1, 2) and (2, 2) models, and appears t o b e true as well for (0, 2) models, but we do not know whether it will continue to hold if one starts breaking the gauge group. If it does, most tensor products are irrelevant, since they can never give three generations. We should add that, from out point of view, twists that interchange several identical N = 2 factors should be regarded as producing new conformal field theories, since they involve twist fields that are not in the original theory. If there is a general quantization rule of the kind discussed above, it should be applied to the new theory. One may hope that there is a simple way to derive the basic unit in terms of which the number of generations is quantized for each product of superconformal field theories, so that one knows in advance where to look for phenomenologically interesting models. Obviously all our results can equally well be applied to tensor products of other N = 2 theories, such as those of ref. [22]. We intend to investigate this in more detail in the future. We are grateful to A. Liitken and L. Ibaflez for discussions and explanations of their work.

Appendix If M 1 and M 2 are two matrices defining a positive modular invariant for some conformal field theory, then their product is obviously a positive modular invariant as well. However, it may happen that the identity appears more than once in the product. This is not a problem if the entire matrix can be divided by the multiplicity of the identity, without the appearance of fractional multiplicities elsewhere in the product matrix. That this question is not completely academic becomes clear if one considers instead of multiplication the addition of two matrices M 1 and M 2. This also produces a new positive modular invariant, but in this case one can usually not normalize Moo to one. It would be quite interesting to know if in general the product of two "good" modular matrices yields another good matrix, but we do not know a general proof. In this appendix we will show however that this statement is true for matrices M generated by simple currents (this conclusion can be extended to the exceptional invariants of N = 2 minimal models, used in sect. 5). Consider the product M(Jk)... M(J1) acting on some field ~, and denote the orders of the currents as N 1. . . . . N~. The matrix M(J1) maps • to a finite set of

122

A.N. Schellekens, S. Yankielowicz / N = 2 tensor products

other fields Jl"lq~, where n 1 is defined modulo N and satisfies n101(J1) = -201(@),

(A.1)

where we have used eq. (2.2) and the identity*

O(

= O( *) + nO(J).

The second matrix maps the fields Jl~'~ to fields J~2j~'~cb, where n 2 satisfies n202(J2) + 2naQ2(J1) = - 2 0 z ( ~ ) ,

(A.2)

and so on for J3,---, Jk- One can solve these inhomogeneous equations by first solving the homogeneous equations that are obtained by choosing • = 1. The solution to the homogeneous equations defines combinations of currents JTk... J ~ that appear in the partition function as left-movers in combination with the right-moving identity. We will call this set of currents J . Clearly J is a group under addition of the labels ( n 1. . . . . n k ) m o d u l o ( N 1. . . . . N~). The identity of the conformal field theory is of course included in J but it may occur p times, where p > 1. This means that there are p sets of labels (n 1. . . . . nk) SO that J l n I . . . Jknk = 1. We call this set d . Given any other current J, one can always make p copies of it by acting with s¢. Those copies are all equal to J, but have different labels. Furthermore it is clear that if two elements of o¢ define the same current in the conformal field theory, then the difference of their labels must be in d . Hence each current in J appears precisely p times, and we may write the set of currents J as o ¢ = d × J ' , where all currents in J ' are different. The general solution of the inhomogeneous equations is obtained by taking one special solution, and acting on it with all solutions to the homogeneous equations. If the solutions to the homogeneous equations have a p-fold degeneracy, then obviously they produce at least p-fold degeneracy when acting on ~, due to the currents in d . One can get a properly normalized product matrix by using instead of o¢ the coset J / ~ ¢ = o¢'. Even if the multiplicity of the identity is thus always one, it may of course happen that for some J ' ~ J ' J ' × • = • even if J ¢ 1. This simply means that some of the orbits are shorter than the identity orbit. It is easy to see that two matrices M(J1) and M(J2) commute if J1 and -/2 are mutually local. Then the second term on the left-hand side of eq. (A.2) vanishes, * Note that this equation and the subsequent ones are invariant if n i is shifted by multiples of N~.The "level matching" condition ri = 0 mod2 is essential here. All equations involving Q are modulo integers.

A.N. Schellekens, S. Yankielowi~z / N = 2 tensor products

123

which obviously implies that the equations for n a and n 2 decouple. Hence one gets the s a m e answer if one interchanges Jl and J2.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

[15] [16] [17]

[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

[32] [33]

[34] [35]

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