New exceptional modular invariant partition functions for simple Kac-Moody algebras

Nuclear Physics B346 (1990) 349—386 North-Holland

NEW EXCEPTIONAL MODULAR INVARIANT PARTITION FUNCTIONS FOR SIMPLE KAC-MOODY ALGEBRAS D. VERSTEGEN * K U Leuven, Instituut voor Theoretische Fysica, Gelestijnenlaan 200 D, 3030 LeuLen, Belgium Received 20 April 1990 (Revised 31 May 1990)

Using numerical and analytical methods, we obtain many new exceptional modular invariant partition functions due to automorphisms of extensions of Kac—Moody algebras. We then conjecture the existence of such an invariant for all Kac—Moody algebras with fixed points under either a simple current of order 2 and spin 4, or currents of order 3 and spins (3,3), or currents of order 5 and spins (2,3,3,2). In particular, the first condition is satisfied fortwo infinite series: B~and D,, at level k = 8, and three finite series involving ô~,and SU(N) algebras. Exceptional invariants of other types are also presented: they occur for G 2, k = 4; F~,k = 3; SU(9), k = 3; etc.

c,,,

1. Introduction Two-dimensional conformal field theories (CFTs) [1] play a central role in both two-dimensional critical phenomena and string theories. The CFTs relevant to string theories have central charge c 2~1 and unlike the unitary minimal models with c < 1 [2—4] have not yet been completely classified. The reason is that modular invariance of the partition function, which is a crucial requirement in string theory, can only be achieved if an infinite number of irreducible Virasoro representations are included in Z [5]. One approach is then to find extensions of the Virasoro algebra such that the partition function involves only a finite number of primary fields of the extended algebra. The theories for which this is possible are called rational conformal field theories (RCFT). Important examples of RCFTs are the Wess—Zumino—Witten models [6,7], which are characterized by the presence of a Kac—Moody symmetry [8,9]. Many modular invariant partition functions for untwisted Kac—Moody algebras are already known, but there is as yet no complete classification, except for SU(2) [4]. *

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—

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The arithmetical method of Cappelli—Itzykson—Zuber is however very difficult to generalize to higher-rank algebras [10, 11]. The WZW models lead to many other CFTs through the Goddard—Kent—Olive coset construction [12]. Examples are the minimal unitary Virasoro models, which can be viewed as SU(2)k ~ SU(2)1/SU(2Y’ +1 (the superscripts are the levels, and the last SU(2) is the diagonal subalgebra), and the N 2 minimal superconformal theories*. In both cases, the modular transformation properties of the characters involve those of the characters of affine SU(2) theories. This allows a complete classification of the minimal (unitary) Virasoro [3, 4] and N 2 superconformal theories [14]. The latter have been used by Gepner as building blocks for the internal sector of four-dimensional heterotic strings [15],because N 2 world-sheet supersymmetry is necessary for N 1 space-time supersymmetry. Similarly, Kazama and Suzuki [16] have introduced a large class of N 2 superconformal theories based on cosets involving e.g. the SU(n), SO(n) and Sp(2n) algebras. They too can be used to formulate many four-dimensional string theories (see ref. [17] for a partial investigation). Finding new modular invariants for Kac—Moody algebras is therefore interesting for string phenomenology. But before elaborating on this, we describe the possible types of modular invariant partition functions. The partition function Z takes the following form: =

=

=

=

=

~ xn(T)MAA,xA,(T)*, A, A

(1.1)

where Xn(T) is the character of a representation of the Kac—Moody (KM) algebra. Primary fields of the CFT are associated to pairs (A, A’) of integrable representations of the KM algebra such that MA 1 0. Because of this physical interpretation, MAA should be a non-negative integer and M 00 should be equal to 1 (“0” stands for the identity representation). Modular invariance requires MA A to vanish if 4~ and /~ have different conformal dimensions (modulo 1); M must also commute with the matrix S which implements T —p 1/T on the characters. At present the following solutions are known: (a) The diagonal invariant, MAA (b) The charge conjugation invariant; MAj. CAA” where C is non-trivial for SU(N), N ~ 3, E6, and D~,n odd. More generally, any automorphism of the finite Dynkin diagram [e.g. triality for SO(8)] gives a modular invariant partition function. (c) Many Kac—Moody algebras have outer automorphisms. These symmetries of the affine Dynkin diagram were used [18, 191 to construct infinite series of modular ~‘

—

=

=

*

Original references can be found in refs. [13, 141.

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351

invariants. They are the generalizations of the so-called complementary series for SU(2). They are either “integer spin invariants” (typically sums of squares) involving only a subset of all primary fields or “automorphism invariants”, which take the form (1.2) M~ ‘=6A~(A)~ i.e. the automorphism ~ permutes all primary fields*. Schellekens and Yankielowicz [20,21] showed that the concept of “simple current” allows a systematic construction of these invariants and gives new ones for SU( N) (N ~ 4) and O,~algebras. A simple current J has unique fusion rules [1, 22,231 with all primary fields q~,J~ The relation between these two approaches is provided by Verlinde’s result [24]: the S-matrix diagonalizes the fusion rules and can be used to compute them (and reciprocally). The multiplicative group generated by the simple currents and called the center of the KM algebra is in general isomorphic to the center of the underlying Lie group (but it is sometimes bigger [25]). The new solutions obtained in ref. [20] are associated to subgroups of the center of the KM algebra. (d) Invariants due to conformal embeddings. H is conformally embedded in G if H c G, if they have equal central charges and if the ratio of the levels of H and G is equal to the index of the embedding; this is only possible if G has level 1. All conformal embeddings have been classified [26]. The finite reducibility theorem [27] ensures that the characters of G can be written as a finite sum of characters of fl. For example, the diagonal solution of G gives rise to a sum of squares of characters of H, but there are sometimes other possibilities. (e) One can also combine these mechanisms by taking products of the Mmatrices. A theorem by Moore and Seiberg [28] states that all modular invariant partition functions of a CFT are either of the form (1.2), with ,ts any automorphism of the fusion rules, or assume this form after the algebra has been suitably extended. This extended algebra is again a Kac—Moody algebra in the case of conformal embeddings (e.g. at k 10, SU(2) extends to 132 [29,30]) or is of a new type for the other integer spin invariants. An example is SU(2), k 0 mod 4, where the KM algebra is extended by a single chiral primary field. The invariants not mentioned above can thus only be of the following types: (i) Invariants due to automorphisms of the fusion rules of the extended algebra. There are three known examples: SU(2), k 16 [3,4]; SU(3), k 9 [28]; and SU(5), k 5 [31]; the number of additional chiral primary fields is respectively 1, 2 and 4. (ii) Invariants due to automorphisms of the fusion rules (of the KPvI algebra) not associated to simple currents. =

~‘.

=

=

=

=

=

*

By abuse of language, the objects cbA(r) pertaining to the left (or the right) sector separately are also called primary fields.

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(iii) Integer spin invariants not generated by simple currents, nor by conformal embeddings. Two examples are known: F4, k 6 [20,32] and SU(10), k 2 [33]. (iv) Invariants due to exceptional simple currents. The purpose of this paper is to find exceptional modular invariant partition functions for untwisted (simple) Kac—Moody algebras. This will not only shed light on the mathematical structure of KlvI algebras but is also important for classifying the partition functions of the N 2 superconformal theories introduced by Kazama and Suzuki, just as the partition functions of SU(2) are important for classifying those of the minimal N 2 superconformal theories used in the Gepner models. The massless spectra and the symmetries of the Gepner models have been analyzed in quite some detail [13,34] and it has been observed [34] that the only known models having a net number of 3 generations (withoutinvariants any orbifoldization) 6. Hence, regular and excepinvolve the exceptional invariant of SU(2)’ tional invariants lead to different physical predictions, and thus it is important to investigate systematically exceptional invariants for the KTvI algebras appearing in the Gepner models and their generalizations. Our approach is as follows. We have written a computer program which looks for modular invariants of any type and we have explored all “small” values of k for algebras of rank 2, 3 and 4. The highest value of k considered is for SU(3) which is shown to have no new invariant for k ~ 26. We found in this way several new solutions of type (i) and (ii). Of direct relevance to four-dimensional string models (cf. table 3 of ref. [13] and tables 3 and 4 of ref. [17]) are the invariants for SU(2)8 ~ SU(2)8 [eq. (4.la)]*, SO(S)8 [eq. (3.1)] and SU(4)8 [eq. (4.lb)]. The main result of this paper is the following observation: for a given number of additional chiral fields, invariants due to automorphisms of the fusion rules of the extended algebra only appear for specific values of the conformal dimensions of these additional fields. This leads us to conjecture many new solutions for algebras not covered by our computer search. Several are explicitly constructed using analytical methods. In sect. 2, we briefly review the modular transformation properties of affine characters and we recall some notions about fusion rules, simple currents, integer spin invariants and automorphisms of extended algebras (in the case of SU(2), k 16). In sect. 3, we discuss exceptional invariants arising in KM algebras with one simple current J. We observe that within the range of our computer search, exceptional invariants due to an automorphism of the extension of the Kivi algebra by J appear iff the conformal dimension h(J) 4. Examples are B 2, B3, B4, all at level 8, and C4, k 4. The simple current of SU(2), k 16 also has h 4. We then conjecture the existence of exceptional solutions of that type for all B,~,k 8 =

=

=

=

=

=

=

=

=

=

*

D2 is not simple but is nevertheless considered here because of its similarities with the other D~ algebras.

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and for C~with nk

/

Simple Kac—Moody algebras

C2). Using numerical methods or deriving the fusion rules of the extended algebra analytically, we verify our conjecture for B5, k 8, C8, k 2 and C16, k 1. In sect. 4, we consider KM algebras with a more involved set of simple currents. We conjecture an exceptional invariant if one of the currents has order 2 and spin 4, and if the theory has fixed points under this current, i.e. for D~,k =8;D~,with nk 32 and k ~ 2; and SU(N) with Nk 32 and k ~ 2 (note that SU(4) D3). We verify this explicitly in several cases. One can often use conformal embeddings to derive these exceptional invariants from known ones. An example is the 2, exceptional invariant of SU(16) =

16 (note that SU(2)

=

=

~

=

and

353

~2

=

=

=

=

=

SU(2)16 e SU(16)2 c SU(32)’.

(1.3)

SU(3) is the simplest theory with simple currents of order 3 (they generate thus a Z 3 group). The only known exceptional invariant (even after an extensive numerical search) arises at k 9, with h(J1) h(J2) 3. We verify that SU(9), k 3, which is the only theory with currents of the same order and spin and with fixed points, also has an exceptional modular invariant. Assuming that a similar conjecture applies to currents of order 5, we find that the exceptional invariant of SU(5), k 5 [31] is the only one of its kind. We also briefly discuss theories with simple currents of higher order. In sect. 5 we present two invariants due to automorphisms of the fusion rules of the KM algebra not associated to simple currents. There is one for G2, k 4 and one for F4, k 3. They actually have the same fusion rules! We also discuss some integer spin invariants not generated by simple currents and of the same type as that found by Walton [33] for SU(10), k 2. We summarize our results in sect. 6. The computer program is described in appendix A and the special case of B4, k 2 is discussed in appendix B. =

=

=

=

=

=

=

=

=

2. Preliminaries We first briefly recall a few well-known facts (see ref. [22] for details). Affine modular invariants are sesquilinear forms [eq. (1.1)] in the affine characters defined by 2~~~1Tu~/24), (2.1) Xn(T) =Tre where the trace is over the representation space generated by the negative modes of the Kac—Moody algebra ~ acting on 4~A(O)I0).The primary field 4A(z) carries an irreducible integrable representation of ~ (also called integrable highest weight module).

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The highest weight A is a linear combination of fundamental weights of an untwisted KM algebra with non-negative coefficients A,. It must satisfy the “integrability” condition, (~p+a0)•A=k,

(2.2)

(a).

where l~(i (a0) is the highest root of g Such A’s are called dominant highest weights. Expanding ~/i in terms of simple roots, ~

~/~2

(2.3)

one finds (2.4) i=O

where k is a non-negative integer if one normalizes the length-squared of ~i to 2. The Dynkin labels (A1,...,Ar) of a representation of the underlying Lie algebra, with ~ m,A, ~ k,

(2.5)

j=1

actually suffice to label the integrable representation A of ~ at level k. In the following all highest weights will implicitly satisfy (2.4) or (2.5). (In particular, we never use the notation in which A is shifted by the sum of all fundamental weights, and is thus “strictly dominant”.) The affine characters transform as follows under the generators of the modular group [22,35]: 2 2(k+hv) (A+p)

TA~=exp ~

=

p2 —

e2’~’~ 1—c/24)

8.

(2.6)

Here A, A’ are weights of the Lie algebra and p is the sum of the fundamental ones. The scalar product is defined with the usual metric tensor in weight space. h’~ is the dual Coxeter number of the algebra, hv= Eme, =

(1

(2.7)

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355

hA is the conformal dimension of 4~t,

hA=

A ~(A +2p) 2(k+hv)

(2.8)

‘

and finally c is the central charge of the Virasoro algebra,

c=

5AA’

1~

ik (k+hvy/2

X

E

k dim g k+h”

(2.9)

.

volume cell of M* volumecellofM

s(w)exp

1/2

—2~iw(A+p) (A’ +p) ,

(2.10)

weW

where ~ + I is the number of positive roots of the underlying Lie algebra, M is the lattice of long roots and M* is its dual. The sum in eq. (2.10) runs over all elements w of the Weyl group W; r(w) is the sign of w. This matrix is easily seen to be symmetric. It is also unitary [22] and it satisfies ~2 C and (ST)3 C, with C the charge conjugation matrix. The relevant numerical factors, together with an explicit form of the Weyl group are given in appendix A. The fusion algebra is a commutative and associative algebra which tells which operators can appear in the operator product expansion of any two primary fields [1,22]. One writes =

=

=

EIV~.Jk4~k,

(2.11)

k

where NIlk is the number of couplings of

çb

1 and ~ to cbk. Note that this only refers to one chiral sector and that the sum only runs over primary fields; the objects in eq. (2.11) also have no z-dependence. Moore and Seiberg [28] have shown that if the algebra is maximally extended, the operator ~k allowed by eq. (2.11) will appear in the OPE. A remarkable observation due to E. Verlinde [24,36] is that the S-matrix diagonalizes the fusion matrices NI (with matrix elements This can be used to express the fusion coefficients in terms of the S-matrix elements,

“c3k).

NIJk

=

E

“~‘~“

.

(2.12)

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Reciprocally, one can compute the S-matrix from the eigenvalues and eigenvectors of the NI~ S,1/501

~

(2.13)

E (N~)IkA~ A~’°A~~~

(2.14)

=

with =

k

In unitary theories, A~°~ is the eigenvector with all components ~ 1, and we further require the S-matrix to be unitary and symmetric. The latter condition yields the relations

A(J)A(~)

(2.15)

=

(without them there would be no intrinsic order among the eigenvectors). An important subset of the primary fields are the simple currents (noted J here). They have “unique fusion rules” with any other field, Jçb

=

(2.16)

~‘.

The smallest N such that j~V ~ (identity) defines the order of J. Schellekens and Yankielowicz [21] demonstrated the existence of non-diagonal modular invariants for any RCFT with simple currents satisfying certain properties, and they gave explicit expressions for MLt. Similar results have been obtained in ref. [37] for KM algebras. In that case, the effect of almost all simple currents on fields is equivalent to that of some automorphism of the extended Dynkin diagram*. The advantage of this approach over that of Bernard [18] and Altschüler, Lacki and Zaugg [19] is that it gives more compact and general formulas, and that it yields some new modular invariants when the center of the Lie group has non-trivial subgroups. For the classical algebras, the (known) simple currents, at level k, are [23,31] A~:J1=(o;o,...,O,k,0 0), with “k” in lth position; the 0th component is separated 4i from the others by semicolon. They are all generated by J1, J1 =4~.The fusion arules off and (J1~? 1 are =

=

J1~(A0A1,...,A~)=(A~A0,A1,...,A~_1). *

The only known exception occurs for E8, k

=

(2.17)

2 [25],where the field associated to the representa-

tion of dimension 3875 is also a simple current, but it does not lead to an exceptional modular invariant.

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357

B~:J=(0;k,0,...,0), with f~(A0..., A~) (A1 A0, A2,..., An).

(2.18)

=

~~:J=(0;0,...,k),

with J.(A0...,A)

=(A~A~1,...,A0).

(2.19)

D~:f~=(0;k,0,...,0),J~=(0;0,...,k) and J~=J~J,, with

(2.20) f~~(Ao;...,A~)=(A~...,Ao), ifniseven, J (A0..., A~) (A~1 A~,A~2, .

=

. . . ,

A~

A1,A1J)

,

if n is odd.

(2.21)

The center of is Z,,~1,that of f3,~and is Z2, and that of O,1 is Z2 X Z2 if n is even and Z4 if n is odd. E6 has two currents of order 3, E7 one of order 2 while G2, F4 and E8 have no simple currents for generic k. In sect. 3 we will discuss invariants arising from an automorphism of an extension of the Kac—Moody algebra. The symmetry algebra extends whenever [28,38,39] the CFT has a partition function of the integer spin type (sum of squares typically). The extended algebra may be another KM algebra (all such conformal embeddings have been classified [26]) or an algebra of a new type. The systematic study of extensions of the Virasoro algebra has just been initiated and very little is known about extensions of KM algebras. To any current J of integer conformal dimension one can associate a modular invariant tensor, ~

N

MAn.

=

E

(2.22)

~

a=l

where N is the order of J and A belongs to an orbit of charge Q~(A) 0 (MAA. is 0 otherwise). The concept of charge of a field cP~ w.r.t. a current J is a generalization of that of conjugacy class. More precisely, it follows from the general formulas of refs. [20, 21] that Q~(A)is a fractional number defined modulo =

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D. Verstegen

1 and given, if h(J) i.e.

=

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Simple Kac—Moody algebras

integer, by the congruence of A divided by the order of f, A1

+

2A2

+

3A3 + inA~,

order of J1

I

A inB~, A1 + A3

Q~(A)

=

Q~(A)

A1+A 2

+

2

A5

+

inCa,

inD~,

2A1+2A3+...+(n—2)A =

1+nA inD~.

“

(2.23)

The orbits {f”A, a 1,.. N) have in general length N, but they are sometimes shorter. A well-known example is SU(2), k 0 (4), 2+~x2+xk2~2+...+2Ixk/2~2, (2.24) Z=Ixo+xkI =

.,

=

where the subscript is the Dynkin-label twice the isospin). In the theory with extended algebra, the new primary fields will have characters Xo + Xk’ X2 + Xk_2’~~and there will be two distinct fields with character Xk/2~ If k 16, the fusion rules of the extended algebra are invariant under 42 ~-‘ [28, 38], where (=

=

~fr

8+

is the field with character X2 X2 + X14 and 48~ is one of the fields with character X8~ Performing this exchange on one of the chiral sectors yields another modular invariant, 1 + iX6~ + Ix z l~I2+ 1X4 8-I ~X2X8~~ c.c., (2.25) ~

=

=

which, when rewritten in terms of SU(2) characters, can be recognized as the exceptional “E7” invariant [3,4], Z=

2+ 1X4+X1 2+ IX6+X1O~2+ X8I2+(X2+X14)X~ 21

Xo+X161

+c.c. (2.26)

Of course, we are only interested in automorphisms which leave the value of h unchanged. Automorphisms which do not commute with T cannot give rise to new modular invariant partition functions.

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359

3. Automorphisms of extensions of Kac—Moody algebras: the Z2 case This section deals with exceptional solutions similar to that of SU(2). k 16: the KM algebra has one simple current and extends if h(f) is integer: there is an exceptional invariant if the fusion rules of the extended algebra have a non-trivial automorphism. Theories with several simple currents are discussed in sect. 4. To find numerically the modular invariants of a given Kac—Moody 4 invariantalgebra, under we T. proceed all as modular follows. invariants We first identify the products Finding is equivalent to findingXnXn all eigenvectors of eigen=

value 1 of the real part of the matrix S ® S* restricted to the products invariant under T. One has then to form linear combinations of these eigenvectors satisfying the integrality and positivity conditions. This is done assuming that the M~ 1.are not larger than some bound b. Most of our results are obtained with b 16. Details can be found in appendix A. After presenting some new exceptional invariants obtained in this way, we formulate our conjecture for KM algebras where the center is Z2. It predicts many new invariants for the B,, and C,, algebras. Their existence is checked numerically in some cases and analytically in some others. To this effect we develop a method to compute the fusion rules of the extended algebra. Exceptional invariants of the Z2-type have been found in the following cases (the largest value of k considered is also indicated): Case (i). B2, k 8 (k ~ 20): =

=

Z=I00+80I~+~20+60/lI+I22+42/ll+I04+44~/Il+j14+34j~Il +

(40,02,62)3/11

+

(32,06,26)4/11

+

(24,12,52)6/11 (3.1)

+ (16,30,50)9/11 + (08,10,70)2/11,

where 00 stands for the character X(o,o) and (40,02,62)

4012+ (40)(02+62)*

+

(02+ 62)(40)*.

(3.2)

The subscript is the conformal dimension h modulo 1. Since (4,0), (3,2), (2,4), (1,6) and (0,8) are the fixed points under f, the similarity with A1, k 16 is quite clear. Case (ii). B3, k 8 (k ~ 10): 2+1004+40412+1104+30412+1200+60012+1112+31212 Z=I000+8001 + 210 + 41012 + 020 + 42012 + 022 + 22212 + {040,002,602) =

=

+{130,206,006}

+

(008, 100,700)

+

(220,102,502)

+

(122,202,402)

+(204,012,412)

+

(016,300,500)

+

(302,030,230)

+

(400,010,610)

+(212,014,214)+{024,110,510)+{114,120,320)+[106,310,032],

(3.3)

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with [106,310,032](106)(310 + 032)* + (310)(106

+

032)* + (032)(106

+

310)*. (3.4)

Under the automorphism of the extended algebra, some fields thus stay invariant, e.g. 4oo4 (with character X004 + X404); others transform as (3.5)

~040~~’~0O2’

and we notice that a 4~b’ new4~ possibility appears: Xb’ x~in the triplet fixed points underthe f, characters give rise to Xa’ fields which transform [Xa, Xb’ xe],under with the 4~a’automorphism: as follows 4~a4~b’

~b~~~tic_,

(3.6)

~

(a quantity with a tilde will always denote a quantity pertaining to the extended algebra). Case (iii). B 4, k = 8 (k ~ 8)* and B5, k = 8: The expressions are rather lengthy and not particularly illuminating. They contain the same type of terms as the invariant of B3, k = 8. Case (iv). C4, k = 4 (k ~ 5): Z= I0000+000412+{0002,0022,2000) + (1011,0220,2200) +

+

(0201,1010, 1012)

(1210,0101,0102)

+

+

(0400,0100,0103)

(2020,0121,2100)

+

regular terms. (3.7)

By regular terms we mean terms which are the same for the integer spin invariant given by eq. (2.22) (which is a sum of squares) and for the associated exceptional invariant. On the other hand, we found no exceptional solution of type Z2 for C3 (k ~ 8). We observe that within the range of our computer search, the extended algebras of the type Z2 have an automorphism if and only if the conformal dimension of the simple current is 4. At level k,

*

h(f)=~k

forB,,,

h(f)=jkn

forC,,.

There is a rich spectrum of modular invariants if k

=

(3.8)

2, but one is not the partition function of any

conformal theory; all the others can be explained by simple currents or conformal embeddings (see appendix B).

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We have thus considered all h ~ 10 for B2, h ~ 5 for B3, h ~ 4 for 134 and h ~ 5 for C4. For C3, only h(J) = 3, 6,... are possible. Furthermore, there is a complete classification for SU(2), and the only exceptional solution occurs for k = 16, with h(f) = 4 again. We conjecture that extended algebras of the Z2 type will have an automorphism commuting with T whenever the conformal dimension h(J) is 4. This condition is probably also necessary. This leads to the prediction of invariants for 6= exceptional A’ 6, C~= B~, C~,C~and all B,,, k = 8, and for C,,, level k, with nk = 16, i.e. C~ 1 C~ 6.On the other hand, the condition h(J) = 4 is never met for E7. We checked numerically that C8, k = 2 has an exceptional invariant given by ~ +{08,11,77}9/11

+

(44,02,68)8/11 + (35,06,82)7/11 (3.9)

+(26,13,75}5/11+ (17,33,55)2/11.

In this formula we have switched to a more convenient notation when the rank is large and k is small: if (separated by a comma if necessary) is the representation with Dynkin labels n1 and n3 equal to 1, and all others equal to 0; if i =j, n, = 2. For example the simple current f = (0. .02) is denoted 88; its effect on other fields is given by .

f. (i,j)

=

(8—i,8

—j)

(3.10)

With this “positional” notation, the representations of congruence 0 have i +j = even. To conclude with this case, we note an 8interesting the fusion rules and C~2property: are actually isomorphic! of the congruence 0 representations of B~ The correspondence takes the simple form (i,j)B 2~(i,i+j)c,.

(3.11)

These fields also have opposite conformal dimensions (modulo 1). In the rest of this section we will discuss a case for which the Weyl group is too large to allow a brute force computation of the S-matrix, but which does not have too many primary fields: C16, k = 1. Our method is as follows. We compute the products of the finite representations associated to the primary fields of the theory, R1R~=EMIJkRk. k

(3.12)

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This gives a set of constraints on the fusion coefficients (3.13)

I’~iJk~
We then write the most general commutative algebra satisfying these constraints and having crossing symmetry. Simple currents lead to further relations between the coefficients. Since our interest lies in the extended algebra, only charge 0 representations need to be considered. Verlinde pointed out that because the S-matrix can be derived from the fusion rules, the formula (ST)3 = C yields relations between the N,.Jk and the conformal dimensions h, [24]. But this would require computing the S-matrix for all the algebra satisfying the above constraints and associativity. We will instead use relations derived by Vafa and based on a study of the representations of the mapping class group [40]. They are expressed directly in terms of the N,Jk and h.. This gives a unique solution for the fusion coefficients in all the cases we considered. We then derive formulas relating the fusion coefficients of the extended algebra to those of the KM algebra. The remaining unknowns are determined using again Vafa’s formulas and associativity. It is then obvious to verify that the extended algebra has an automorphism. We use the same positional notation as before; the primary fields of C 16, k = 1 are thus labeled by a single integer i indicating which Dynkin label n, = 1, with 0 ~ i ~ 16; i is even for the congruence 0 representations. There is an integer spin invariant given by Z

=

0

+

16I~+ 12

+

14I~/9+

14

+

12I~/3+ 6

+

10l~,,3+

2I8I~9, (3.14)

where the subscript is h mod 1. The products of the associated finite-dimensional representations take an especially simple form, 16

R1R1=

E

Rim,

(3.15)

i,m=O

with li—il

~

Il

—

ml ~ min{i +j,32

l+m~i+j, Il

—

—

i—i),

l~m, mI

—

li—il

~ 21.

(3.16)

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363

R0, is interpreted as R,, and R00 as the identity representation. This formula further seems to be valid for any Lie algebra Cr, with 16 and 32 replaced by r and 2r. In the fusion rules, only integrable representations may appear on the r.h.s., i.e. only the terms with 1 = 0 are allowed. The fusion rules are thus of the form min{i +1, 32—i —11

E

(3.17)

NIjmcbm,

m=Ii-/

with m

—

i

—jJ

E

21 and

N,.1,,,

=

0 or 1. q~16is the simple current

4~t64~16= 4~0‘

(3.18)

4~16~J= 4’16—1

It is easy to check that the products ~ with 2 ~ i, j © 6, and 4848 determine all the others. We then impose commutativity and crossing symmetry. We use the simple current to obtain trivial relations as N

(3.19)

881=N88161

(since

4848

is invariant under multiplication with

4~16), and

also a less trivial one, (3.20)

N66 12=N46~0.

This reduces the number of coefficients to 19. We now invoke Vafa’s relations to determine them [40], (h,

+ h. +

hk

+ hI)ENijrNrki

=

Ehr(NijrNI-kl

+NjkrNrji +N,krN,.il)mod

1

(3.21)

(“mod 1” will be implicit hereafter). Setting =f = I and k = i, with i the conjugate representation of i, leads to a first set of equations, (3.22)

4hiEN1~,= Ehr(NI~r+2NI~r),

r

r

1~1ijk= 0 or 1. which simplify further if from the general formula with i =1, Other relations follow I = 1, E/1r(NIirNiir + 2NI~~rNItp) =

k

=

I or with i

= k,

(2h, + 2h 1) ENI1r1’1iir

=(2hj+2hi)~N,jrN,~1~.

(3.23a) (3.23b)

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Simple Kac—Moody algebras

Eqs. (3.23a, b) are not equivalent because we did not yet impose associativity. If all fields are self-conjugate as is the case here, the N,Jk become completely symmetric. Eqs. (3.22) and (3.23) suffice to determine the fusion rules: we find that all the NIjm in eq. (3.17) are equal to 1. The fusion rules of the congruence 0 representations of C16, k = 1 are thus the same as those of the vector representations of SU(2), k

=

16. The isomorphism is simply (3.24)

4~SU(2)

Here too, =

_h(4~I6)mod1.

(3.25)

But in view of other applications, we will not use this coincidence to derive the fusion rules of the extended algebra. Our starting point is the formula [21,41] Sj~~aj$6 =

exp[2~n(/3Q(a)

+

aQ(b)

+ a/3r/N)]Sab.

(3.26)

Here J is any simple current of order N and monodromy parameter r [21], and a and b label arbitrary primary fields. We only need a special case of eq. (3.26), namely when the current has integer spin; r then vanishes and the charges Q are those given in sect. 2. Eq. (3.26) is easy to check in the case of SU(2), where (a+1)(b+1)ir

___

k+2

Sa,byk+25in

(3.27)

4~k—a~

and thedefiniteness effect off we is ~14)~ For assume that N is even and that all orbits under J have length N or length N/2. This covers the cases of C 16, k = 1 (with N = 2) and of some algebras discussed in sect. 4. The generalization is not difficult. We use the labels a, b, c,... (respectively f, g, h,...) for fields of charge 0 and belonging to orbits of length N (respectively N/2). Fields of charge * 0 will be labeled by x, y The characters, fusion rules and S-matrix of the extended theory will be distinguished by a tilde. Thus N

~\Ta=

Xj’~,,’

(3.28)

with a any representative of the orbit. The multiplicity 2 of some terms in the partition function indicates that there are two fields which should be treated as different primary fields of the extended algebra. We distinguish accordingly their

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Simple Kac—Moody algebras

365

characters, N/2 Xf~

(3.29)

XJ~f•

a= 1

For charge 0 representations, eq. (3.26) implies SJ~aJi3b=

and similarly for

f,

g.

(3.30)

5a,b’

We also note another consequence of eq. (3.26), N

N/2

E a=1

E 5J’~f,x°~

5J”a,x

(3.31)

a=l

The matrix S is thus partially known [32], 5af~ ~NSaf,

5a,b~15a,b,

Sf±,g++Sf±,g~ ~NSf,g.

(3.32)

Verlinde’s formula yields

d

5ad5bd5c4d + 50d

L

5a,f5b,f~c’~f~

()

where the sums run over one representative of each original orbit. Using eqs. (3.30), (3.31) and (3.32), we can extend the sum over all fields of the original KM algebra. We find N

Nabc

=

E Nab J~c.

(3.34)

a=1

Proceeding similarly, we obtain N/2

Nab f±~ a~1’~~’~

(3.35)

For the other fusion coefficients, we only obtain constraints, e.g. N/2

Naf±g++Naf_g+

~~ 1NaJ~~ N/2

Nf±g±h++ ~f+g+h~+

Nf+g_h±+ N1+g~_=

(3.36)

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D. Verstegen

366

Simple Kac—Moody algebras

These results turn out to be equivalent to iN

(3.38)

=1

2 N/2 41f±+4~f_ —EJ4f

(3.39)

which should be viewed as a recipe for computing or constraining the fusion coefficients. We apply this to C, 6, k = 1, where the new characters are

~=x~+x~61,

(i

0,2,4,6),

=

(3.40)

X8~X8~

There are first a few immediate products, ~2~2

=

~0 + ~2 + Q~4,

~2~4

=

~2

+ ~4 + &‘

~2~6

=

~4

+ ~6 + ~8++

~4~4

=

~0 + ~2 + ~4 + ~6 + ~

~4~6

=

~2

~6~6

=

~0 + ~

~8’

+ ~4 + 2~6 +

~

~8’

2ç6

From (~8±+

4~8)

=

4’84~8

+

2~4+

6 +~

~

we deduce

48~48~=

a~0+ b~2+ c~4+ d~6+ e~8++(1

48-48-—

a~0+

~

(3.41)

—

+

c~4+ d~6+ (1

—

e)~8*+e~8-, (3.42)

with a, b, c, d, e = 0 or 1. 42~ 4~ and 46 are self-conjugate while ~ and 48- could be self-conjugate (a = 1) or be exchanged under C (a = 0). Eq. (3.23) with i = 2, 1 = 8~and I = 4, 1 = 8~give a = c = 1, b = d = 0. Associativity of the product 48~~8~48- yields the last coefficient, e = 1. It is straightforward to check that these fusion rules are left

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Simple Kac—Moody algebras

367

invariant by 4~2 ~ This gives rise to an exceptional modular invariant in the way explained in sect. 2. As expected, this fusion algebra is isomorphic to that of the extension of SU(2), k = 16. ~

4. Automorphisms of extensions of Kac—Moody algebras with several simple currents We consider here Kac—Moody algebras with more than one simple current. We have explored numerically the cases of D2, k ~ 8 and D3, k ~ 10. For D3, k = 12; D4, k ~ 8 and D5, k = 8 we have restricted our search to invariants involving only sums over orbits of J~of characters of vector representations. For D2, D4 and D5 we took b = 4. SU(3) and SU(5) are discussed below. We found exceptional solutions for D~,D~,D~and D16. The first two are D~:

Z=

00+08+80+8812+04+40+48+8412 + 22 + 66 + 62 + 2612+ 24 + 42 + 46 + 6412 2+(02+06+82+86+20+28+60+68)(44)*+c.c.,

(4.la)

+21441

D~:

Z =

000 +

800 +

080 +

00812 + 2122212

+002+ 215 +520+051+502 + 251 + 120+015)(222)* + c.c.+ 400 + 04412 +(400+044)(106+611+160+011)*+c.c.+regularterms.

(4.lb)

These solutions, as well as those of D and D~,follow from acting with an automorphism of the fusion rules of the extended algebra on (one chiral sector of) the regular invariant given by MA, A’

— —

\ 0,

J8AA.

+

6A.JA

+

8J~,J~

+

~

jf Q~(A)= Q~(A)= 0, otherwise.

( 42 ) .

This is the integer spin invariant tensor of D,, generated by J~if n is odd, and the product of the tensors generated by f~and J~(for example) if n is even. It only exists if both h(J~)= k/2 and h(J 5) = kn/8 are integer. The common feature of all these exceptional solutions is that the identity block contains the character of a simple current of order 2 and spin 4 (.1); on the other hand, the spin of the other characters in the identity block (J5 and ~ seems to play no role. We conjecture an exceptional modular invariant for all KM algebras

368

D. Verstegen

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Simple Kac—Moody algebras

with a simple current of order 2 and spin 4, and with fixed points under this current, (i)AI1D,,, k=8(f=f~). (ii) D,,, with nk = 32 (J = .15), i.e. D~6,D~,D~and D~6. 8, SU(8)4 and SU(16)2. We have (iii) SU(N), with Nk = 32 (i’=i’N/2)’ i.e. SU(4) used h(J 1) =kl(N—l)/2N

for SU(N)”.

(4.3)

This list is partly redundant since D3 = SU(4) and since triality exchangesmodular .J~, and 6 has exceptional for D4. since Furthermore one already knew that D~ invariants, D 2 = SU(2) ~ SU(2). 2 and SU(8)4. D~ We will verify the conjecture analytically for D~6,SU(16) 6has a regular integer spin invariant generated by f,,

.15

Z=I0,0+16,16I~+l1,i+15,15I~+l2+14I~/l6+l4+12I~/4 + l6+10l~/16+2l8I~+2l0,16l~/16+2l1,15I~/16,

(4.4)

with the same notations as for C~6. (16,16), (15, 15) and (1,1) are the simple currents with h = 4,4, 1 respectively. The representations (0, i6) and (1, 15) belong to the lattice w~+ Q and the others to the root lattice 0. There is also the partition function give by (4.2), 2+212+1412+214+1212+216+i012+41812. Z’=l0,0+1,1+15,i5+16,161 (4.5) It is clear from the conformal dimensions that the symmetry algebra of Z’ cannot have an automorphism leading to a new invariant. On the other hand, we will show that the fusion rules of the KlvI algebra extended by (16, 16) have a non-trivial automorphism. The method has been explained in sect. 3 and we will only indicate the main steps. The products R,, R 3 with i, j even, take the same form as those of R. and R1 in C16 [cf. eqs. (3.15) and (3.16)], provided i +j ~ 14; the same rule applies almost literally to R8R8, one only has to replace the representations R115 and R16 (whose weights do not belong to the root lattice) by R,15~6, i

R~R,6=

E

jeven=0

i—i

R.,6+

E

R115,

(i~ 14),

(4.6)

jodd==1

14

R,6R,6=

E

R1+R16,6,

j even = 0

where R0, is interpreted as R,, and R0 and R0,, as the identity.

(4.7)

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D. Verstegen

Simple Kac—Moody algebras

369

The non-trivial fusion rules of the associated primary fields are =

411—J +

+

if i,j=2, 4,6,8 and i+j~ 14 (with the convention 4848

+ ~I3~ +

= ~

(4.8)

8i,j4~1,1’ 4~=4,~);

+ d’t6,16’

~15,15

(4.9)

14

E

~0,16~0,16

I even

(4.10)

4~j+4116,16. =

0

Others can be derived using the known fusion rules of the simple currents (cf. sect. 2). The extended algebra has three simple currents of order 2: ç~ with character ~ ~ +X15 15’ and ~ and 48- both with character X8~ The products read ~

~

1=2,4,6,

~1,1~i=~i’

=

~1,1~0,16~

~8-i’

2,4,6,

=

~1,15~

+ ~54+

~2~2

=

~0,0

~2~4

=

~2

=

4~2+

~6~6

=

~0,0

+ ~4 +

~4~4

=

~0,0

+

=

~4

~0,16~~0,16~

~0,0

4’ll,

+ ~6’

+

d~6’

~

43~ 1’

~

~

~8’

+ ~4 +

~

(4.11)

~0,160,16~2~6~

This algebra is invariant under ~-: ~0,16~

‘

~1,1

:: ~

‘

~6

~-:

~1,t5~ ~

(4.12)

370

D. Verstegen

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Simple Kac—Moody algebras

which commutes with T. We conclude that there is an exceptional invariant for -j-’~2

‘-‘16’

Z= 0,0+ 16, 1612+ 4+ 1212+ ((8), (1, 1), (15, 15))

+((0,16),(2),(14))+{(1,15),(6),(10))

(4.13)

(and of course another one with the last two Dynkin labels exchanged). It is not clear to us why the identity block of the exceptional solution sometimes contains all integer spin currents (D16) and sometimes only the spin 4 current (D~6). 2, both possibilities occur. We It turns first out the thatextension in the next example, consider of SU(16)2 by allSU(16) three currents, with h(J 4) = h(f12) = 3 and h(f8) = 4. Under .14, there are 8 orbits of length 4 and two orbits of length 2: ((0, 8),(4.12)) and ((2, 10),(6, 14)). We used the positional notation, in which

f4~(i,j)=(i+4modl6,j+4modl6).

(4.14)

It is interesting to compute the conformal dimensions: one sees that for each orbit of length 2 there is an orbit of length 4 with the same conformal dimension. The exceptional invariant is, as in many cases above, thus easy to guess, Z=I0,0+4,4+8,8+12,12I~+I2,2+...I~/4+I0,4+...l~/l2+l2,6+...l~/3 +

1,7+

...

1/3+13,9+

+(0,8+4,12)(3,5 +

+

...

l~~+ 0,8+4,12I~/9

...)16/9+

c.c.

l2,10+6,14l~3/36+(2,10+6,14)(1,3+ ...)~3/36+c.c.,

(4.15)

where the dots indicate the other terms of the length 4 orbits. We have verified the modular invariance of this expression in two ways. First, we have explicitly constructed the fusion rules of the extended algebra. Noting (0,8) ±the two fields with character X(o,8) + X(4, 12)’ ~ the field with character X(3,s) + X(7,9) + X(l 1,13) + X(15, t)’~~~’ one verifies that (4.16)

is an automorphism. Secondly, we used a method due to Bouwknegt [42]. If ~he h’ c g is a conformal embedding, the characters of ~ can be expressed as finite combinations of products

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D. Verstegen

of characters of

Simple Kac—Moody algebras

I~tand i~’[27]. Any invariant of z”= E ~ p.,

where ~i, refer to ~‘

i’,

g

371

can thus be written as follows: (4.17)

m, n

i~and m, n to h’. It follows from the unitarity of T and S that if z=

EZPVX?LX:

(4.18)

is an invariant of h, then the contracted tensor =

EZp.pZ,~mpn

(4.19)

will give rise to a modular invariant quantity with positive and integer coefficients. Z~

0 is however generally not equal to 1. We will use the conformal embedding

1, (4.20) ~ SU(qY cSU(pq) with p = 2, q = 16, to obtain the exceptional solutions of SU(16)2 by relating them to the exceptional invariant of SU(2)’6. Walton has shown how to compute the branching rules for this embedding [33]. In particular, if p = 2 and q is even, he finds ~2t = (i) ~i +1, ~i + i). (4.21)

E

i even

(~

=

0

There is a similar relation for w21~1 (i) is the character of the representation of SU(2)9 with Dynkin label i; the positional notation is used for the characters ~2t of SU(2q)’ and (m, n) of SU(q)2 (m and n are defined mod q). The proof uses the branching rules for the representations of the associated Lie algebras and the relations between the outer automorphisms of SU(2q)’ and those of SU(2)~and SU(q)2. First we choose the following invariant for ~ = SU(32)1: Z”

=

Iw°+ ~8 + w16 + w2412 + 1w4 + w12 + w20 + w2812

(4.22)

From eq. (4.21) we deduce 16

E3 (i)(—~i+4j,+i+4j),

i even =0 j=’O

(4.23)

/ Simple Kac—Moody algebras +w28. Contracting Zmp,, with the invariant tensor and similarly for w4 + coming from the regular invariant of SU(2)’6 [eq. (2.24)] yields (after dividing by 4) the regular invariant of SU(16)2 generated by .14. On the other hand, contracting Z,~m,rn with the exceptional invariant tensor of SU(2)16 [eq. (2.26)] leads to the exceptional solution (4.15). One obtains the second exceptional invariant of SU(16)2 by choosing instead for SU(32)1 372

D. Verstegen ...

7

Z”=

Elw41+w4~2l2.

(4.24)

1=0

The computation is straightforward. We proceed in the same way to deduce an exceptional invariant for SU(8)4 from that computed numerically for SU(4)8 [cf. eq. (4.ib)]. We need the following branching rules for SU(32)1 SU(4)8 ~ SU(8)4: —~

41= w

3i +2i

E

(i

2+i3 1,i2,i3)

—i +2i2+i3 +1,

1

I

+1,

i, +i~+i~~8

—i1—2i2+i3

+1,

—i1 —2i2—3i3

+1

,

(4.25)

where the sum involves representations of SU(4) of quadrality 0 only. The 4 is exceptional invariant of SU(8) Z’= I0000+2222+4444+666612+21135712 +(1357)(0116

+

...

+0277

+

..

.)* +

c.c.

+21246012+(2460)(1227+...+1300+...)*+c.c. + 10044 + 226612 + (0044 + 2266)(1223 + +

1155

+ 337712 +

(1155

. ..

)*

+ c.c.

+ 3377)(2334 + . ..)* +

c.c.+ regular terms.

(4.26)

The dots denote the other terms of the orbits under i’2 = (2222). In the last part of this section, we discuss exceptional invariants appearing in algebras with integer spin currents of order * 2. SU(3) has a regular integer spin invariant for k = 0 mod 3. It is due to the simple current f~= (k, 0), with h(J,) = k/3. There is an exceptional invariant at k = 9

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D. Verstegen

Simple Kac—Moody algebras

373

given by [28]

z=

00

+

09

+ 9012 + 213312 + (11 + 17 +

71)(33)*

+

c.c.+ regular terms. (4.27)

It results from an automorphism of the fusion rules of the extended algebra and thus involves only sums of characters over the orbits of triality 0. We checked numerically that there is no other invariant of that form for k ~ 42 and no other exceptional invariant of any form for k ~ 26 (the bound on the coefficients of Z is 16; see appendix A for details). We assume that a variant of our conjecture holds: a Kac—Moody algebra with currents of order 3 and fixed points will have an exceptional invariant of the form (4.27) if these currents have spin 3. The simple currents of order 3 of SU(3p) are and J2~,and they have spin 3 iff 3pk = 27. Since SU(27) has no fixed points 3. Before proceeding we note that the under J9, h(f) the only new possibility condition = 3 is never met forisESU(9) 6. 1 We use the same contraction technique as above, taking for SU(27) Z”

=

1w°+ w9

1w3

+ w1812 +

+ w12 +

w21 12

+

1w6

+

w15

+ w2412,

(4.28)

and for SU(3)9 the tensor of (4.27). We use the branching rule 3’

w

.

=

E

—i

,

1 —2i2

where the sum is restricted to i1 2+ Z’=I000+333+6661

i2—i1 +1,

(‘1,~2)

E 2

+

2i~+i2 +1,

+1

(4.29)

2i2 = 0(3) (triality 0). The final result is

21j,3+j,6+j12+(j,3+j,6+j)

j=’O 2

x

E (2 + 31 +j,3 + 3i +j,4 + 31

+J)*

+

i=0

+

regular terms.

(4.30)

Currents of order 4 appear in the identity block of some of the exceptional invariants discovered above, but they are automatically accompanied by a current of order 2, and their spin seems to play no role in deciding whether or not there is an exceptional invariant. SU(5)5 also presents an exceptional invariant [31], with h(J 1) = h(J4) = 2 and h(J2) = h(f3) = 3, but the same reasoning as above leads to the condition kN = 25

374

D. Verstegen

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Simple Kac—Moody algebras

for SU(NY’, so that we expect no other exceptional invariant of that type. We checked this numerically for SU(5)10. What about KM algebras with integer spin currents of order ~ 6? There are some reasons to believe that there are no exceptional invariants associated with them. If N is odd (even), k = N (respectively 2N) is the smallest level such that SU(N)” has currents of integer dimension and order N. We verified that SU(7)7 has no invariant arising from an automorphism of the extended algebra. If SU(6)12 had an exceptional invariant, a generalization of the above conjecture would predict one for all SU(N) with kN = 72 and N ~ 6, k ~ 2. But inspection of the conformal dimensions suggests that this is impossible for SU(36)2. We find the same negative result for currents of order 8. We also note that the “magic” values of h(J) leading to exceptional invariants are rather small [4, (3,3) and (2,3, 3, 2)] and show no sign of increasing with the order of the current. On the other hand, the smallest possible integer values of h(J) increase with the order of J; they are (3,5,6,6,5,3) for e.g. SU(7)7. It is also interesting to compare our observations with the work of Bouwknegt on extensions of the Virasoro algebra [431: he finds that the spin of the chiral field used to extend the Virasoro algebra is an important factor in the classification, and that small values play a privileged role.

5. Other exceptional modular invariants Two of the exceptional modular invariant partition functions discussed in this section are automorphism invariants, and the others are integer spin invariants. The Lie groups G 2 and F4 have a trivial center. A consequence is that for generic k there is just the diagonal solution. For G2, there are also two invariants due to conformal embeddings, at k = 3 and k = 4. They have been given explicitly in ref. [44]. We have considered all k ~ 25 and we have found one new solution, at k = 4,

2+(10)(04)*+c.c.+(20)(01)*+c.c. Z=

10012+10312+11112+10212+ 1121 (5.1)

Since Z is an automorphism invariant, the fusion rules must be invariant under (10)~-(04),

(20)’~-*(01).

(5.2)

We have computed the fusion rules using Verlinde’s formula, eq. (2.12), and we have verified the invariance, but we also find that this automorphism is not due to a simple current.

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375

We now turn to the case of F4. We have explored all k ~ 10. There are two 3 C Dr” yields solutions due to conformal embeddings. F~ z=I0000+ooloI2+l000l+010012+21001112,

(5.3)

and F~9C D~” gives another such invariant. Then there is the “very exceptional” solution at k = 6 discovered in ref. [201. It is of the integer spin type but is not due to a simple current. But there is also a new solution for k = 3, S=1000012+ 1000212+1001012+1001112+ 1100112 +(0001)(0100)* +c.c.+(0003)(1000)* +c.c.

(5.4)

Computing the fusion rules, we find that there are the same as those of G 2, k

=

4!

Up to the automorphism (5.2), the isomorphism is given by (00) ~(0000),

(03) ~(0010),

(11)

(0011),

(10) ~(0001),(04)~(0100),

(02)

(20) ~(1000),

(01)

(0003),

(12) ~(1001).

(0002),

(5.5)

2, by Walton has an exceptional integer spin invariant SU(10) of contracting the obtained diagonal invariant of SU(20)’ [expressed in terms for of characters SU(10)2 and SU(2)10] with the invariant tensor coming from the conformal embedding SU(2)’°C 50(5)~[cf. eq. (4.19)1. We stress that SU(10)2 is not conformally embedded in some Kac—Moody algebra. This exceptional solution has thus the same status as that of F~.On the other hand, taking for SU(20)’ and SU(2)1°a regular solution (i.e. diagonal or due to simple currents) leads to a regular solution for SU(10)2 too (and similarly in other cases). Walton then suggests that any non-regular invariant of SU(p)9 should give rise to one of SU(q)’3. Several exceptional invariants were computed in this way in sect. 4. Besides the embeddings (4.20) involving the unitary groups, one could also use SO(p)~SO(q)”CSO(pq)’, C~C~’cSO(4pq)’, SO( p)4

e SU(2)~° C C~.

(5.6a) (5.6b) (5.6c)

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Does that mean that any conformal embedding h~.ci, with hN one of the classical Lie algebras, leads to an exceptional integer spin invariant for some other algebra? We think the answer is no. The reason is that most of these embeddings belong to infinite series, with h~equal to SU(N)N2,

(N~4),

SO(N)~’2,

SU(N)”~2,

SO(N)~2,

Cj~1,

SU(N)N, Cr1.

(5.7)

For each algebra h~in this list, there is one with N and k exchanged. We expect the solution due to h~c 1 to lead by the above mechanism to that due to h~c I’. On the other hand, isolated conformal embeddings could give rise to new invariants. SU(2)1°c SO(S)1 is precisely such a case. The others can be found in ref. [26]. The following algebras have thus probably exceptional integer spin invariants too:

SU(9)3,

SU(10)8,

SU(21)3,

SU(28)2,

SO(4)’°,

SO(8)12, SO(i0)~, SO(28)4, C~, C~, C~ 0, C~8.

(5.8)

3. A more systematic treatment will be presented We verify this explicitly SU(9) spin invariant due to the embedding into E~, elsewhere. SU(3)9 has anforinteger Z=I00-i-09+90+14+41+4412+2122+25+5212.

(5.9)

For SU(27)’ we choose the regular integer spin invariant of eq. (4.28). We only need the following piece of the branching rule:

w°+w9+w’8(00+09+90)(000+333+666)+(14+41+44)(126+045+378) +(22+25+52)(027+135+468)+...,

(5.10)

where w’ denotes a character of SU(27)1, mn a character of SU(3)9 (Dynkin labels) and mnp a character of SU(9)3 (positional notation). Acting with the appropriate KfvL automorphisms yields the branching rules of w3~+ w9~3’+ w18~3~ (3 = 1,2).

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Simple Kac—Moody algebras

After dividing the contracted tensor by 9, we obtain 2

z’

=

j=0

2

E

2

[(3i +j,3i +j,3i +1)

+

(1

+

3i +j,2

+

3i +j,6 + 3i

+j)]

1=0

2

+2

E (3i+j,2+3i+j,7+3i+j)

2 .

(5.11)

1=0

6. Conclusion Since few exceptional modular invariant partition functions were known, we developed a computer algorithm that finds all modular invariants Z of a given Kac—Moody algebra g~,provided the rank 1, the level k and the coefficients of Z are not too large. We found exceptional solutions due to: (i) Automorphisms of the fusion rules of extensions of Kac—Moody algebras by integer spin simple currents (“regular extensions”); (ii) Automorphisms of the fusion rules of KM algebras not due to simple currents (G~’and F). We observe that within the range of our computer search, exceptional invariants of type (i) arise iff the KM algebra has fixed points under either a simple current of order 2 and spin 4, or simple currents of order 3 and spins 3,3, or simple currents of order 5 and spins 2,3,3, 2 (other currents may be present too). We conjecture that any simple KM algebra satisfying one of these conditions will have an exceptional invariant of type (i). We also have some reasons to believe that this list exhausts all the possibilities for type (i). To prove this conjecture one would have to explicitly construct and classify all (regular) extensions of KM algebras. In many cases, we have verified analytically that the conjecture holds. We have developed a method to compute the fusion rules of the extended algebra and hence to check the presence of an automorphism. We have also derived several exceptional modular invariants by relating them to known ones through conformal embeddings of the form SU(p)9 ~ SU(q)’~’c SU(pq)’. It is likely that in the same way the exceptional solution of C~(with nk = 16) can be related to that of C~ using (5.6b), and similarly for the finite D~series using (5.6a). It would also be interesting to investigate whether the exceptional solution for SO(N)8 can be explained by some solution due to the triality of SO(8) through SO(8Y~’® S0(N)8 c SO(8N)1. We used the same idea to give a list of algebras which are likely to have exceptional invariants due to extensions of the KM algebra by chiral fields that are not simple currents and such that the extended algebra is not a KM algebra.

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Simple Kac—Moody algebras

Finally we note that the methods developed here could also be applied to other theories, such as sums of SU(2) algebras. I would like to thank Ph. Ruelle and E. Thiran for arousing my interest in this problem. I have also benefited from conversations with J. Figueroa-O’Farrill, J. Fuchs, A. Schellekens, S. Schrans, A. Sevrin, D. Speiser, W. Troost, A. van Proeyen, J. Vermaseren and J. Weyers. Part of this work was done while the author was at NIKHEF-H (Amsterdam); the financial support of the “Stichting FOM” is acknowledged.

Appendix A THE NUMERICAL COMPUTATION

We first give explicit formulas for some of the factors appearing in eqs. (2.3)—(2.10) and we describe the Weyl groups of the simple Lie algebras (except E6, E7, E8). We then describe the main features of the computer program that we have used to find modular invariant partition functions in a systematic way. We also show how simple modifications of the original program give partial results for larger values of k. The dual Kac labels (m1, m2,.. , m,) of eqs. (2.3)—(2.5) are given by .

Ar:

(1,1,...,1),

Br:

(1,2,2,...,2,1),

Cr:

(1,1,...,1),

Dr:

(1,2,...,2,1,1),

F4:

(2,3,2,1),

G2:

(2,1).

(A.1)

Those of E6, E7, E8 can for example be found in ref. [9]. The dual Coxeter number is given by eq. (2.7) with m0 = 1. To compute 5, we need

volume cell cellof ofM* M volume

2{/3, 1/2 =

13~},

(A.2)

det”

where the 13~ are the vectors spanning a fundamental cell of M, the lattice of long roots. If the algebra is simply laced, the matrix of scalar products is the Cartan matrix. As usual, we express the non-vanishing roots in terms of orthonormal

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379

vectors e•, Ar:

±(e,—e’) ,

1 s~i
Br:

±e1±e1,

Cr:

(±e,±e1)/V~,±~/~e1 1~
Dr:

±e,±e1

F4:

roots of B4~the l6vectors ~(±e1 ±e2±e3 ±e4).

±e1

(A.3)

Finally the roots of G2 are ±(a1+a2),

±(a1+ 2a2),

±(a1+3a2),

±(2a1+3a2),

(A.4)

where the simple roots satisfy a~=2,

a~=2/3,

a1~a2= —1.

(A.5)

There are standard choices for the simple roots of the classical algebras, and for F4 one can take (a,, a2, a3, a4)

=

(e2

—

e3, e3

—

e4, e4, ~(e1

—

e2

—

e3

—

e4)).

(A.6)

The Weyl group of Ar (W(Ar)) consists of all permutations of the e,, that of Br and Cr of the semi-direct product of all permutations and of all sign changes of the e,. W(Dr) is like W(Br) or W(Cr), but with the constraint that only even numbers of sign changes are allowed. To get the Weyl group of F4, one takes the semi-direct product of W(D4) with the outer automorphisms of the D4 root system. With the standard choice of simple roots for D4, (a1,a2,a3,a4)=(e,—e2,e2—e3,e3—e4,e3+e4),

(A.7)

these automorphisms are all the permutations of (a1,a3,a4). Finally, W(G2) contains the 6 reflections along the lines orthogonal to the 6 positive roots, and the 6 rotations of angle 2~n/6. A simplification in the cases of Br, C~,Deven, F4, G2 (and also E7 and E8) is that the total reflection 11 is an element of the Weyl group. It satisfies e( ]1) = 1 (rotation) if r is even, and = 1 if r is odd. Remembering the factor ~ one sees that in the above cases, S is not only symmetric but also real. To compute modular invariants, we first impose invariance under T because this gives powerful and easily solvable constraints: x,x$ is invariant under T r + 1 iff —

—

~(—

1)

—

—~

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h(~1)mod1. Under r l/T, the products x~x$ transform with the tensor product S ® 5*~We assume that S is real and we consider the following combination of characters: =

—*

—

Z= EM11x1xj~,

(A.8)

with all x~x7invariant under T. Such an expression can be invariant under T l/T only if it is an eigenvector of eigenvalue 1 of the matrix S ® S restricted to the T-invariant products. The fact that the S-matrix is real, symmetric and that its square is J~implies that this condition is also sufficient. If S is complex, the unitarity of S implies that its real and imaginary parts commute, and thus can be simultaneously diagonalized. An expression like (A.8) is modular invariant iff it is an eigenvector of eigenvalue 1 of the real part of S ® once again restricted to the products invariant under T. The eigenvalues of the real part of the restricted S 0 S~matrix are typically scattered between 1/2 and 1, with rather high degeneracies. Those (approximately) equal to 1 are widely separated from the others, so that there is no ambiguity in identifying them. We now address the issues of positivity and integrality of the coefficients of Z. The eigenvectors obtained above (the qualification “with eigenvalue 1” will be implicit hereafter) have to be combined in an appropriate way in order to give physically meaningful modular invariants: the coefficients in Z will be the multiplicities of the various conformal fields. One also requires the coefficient of IX(0,o 0)1 to be 1 (unicity of the vacuum). Call n the number of eigenvectors v~’~ and n’ the number of their components. We must find real coefficients Xa~ such that —~ —

S~,

—

X,,L)$~’)=

1),

(A.9)

for some positive integers l~.Since the eigenvectors are orthogonal, it is possible to 1).We take find of n linearly among the lines of the matrix fv,~” the linear one them toindependent correspondlines to lx(oo 0)1.n’We thus have to solve problem (A.9) for some vector L = (1, 1,, 12,... 11)T Having found the Xe corresponding to L, we check the n’ n remaining integrality and positivity conditions. Of course, in practice we must assume that the 1. are bounded by some integer b, so that the number of possibilities to be explored is (1 + bY’ In most cases we took b = 16. This is probably sufficient, since the biggest coefficient among all known modular invariant partition functions for KM algebras of rank ~ 4 is 9. —

~.

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381

The accuracy of the computation is surprisingly good: the deviation of the coefficients of Z from integers is smaller than 5 X iO~,even if n’ is as large as 1000. However the number of eigenvalues equal to 1 tends to grow with n’, i.e. with the level k. For example, it is 22 for C3, k = 8, and exploring (1 + 16)22_I possibilities for the vector L is clearly unfeasible. But many lines of the matrix {v~) are equal to each other, and several occur also multiplied by 1. Because of the positivity condition, the coefficients of the x~x7that correspond to lines occurring with both signs must be 0. We call such lines “null lines”. We then construct a maximal set of linearly independent combinations of null lines. In the example mentioned above, n’ (the number of products x~x7invariant under T) is 1095, there are 156 different lines among the 1095; of these 156 lines, 57 are null lines out of which one can take 19 linearly independent combinations. Using the line corresponding to X(0 0)1 and these 19 combinations to formulate the linear problem, we see that the number of free integers in L has been reduced to 22 1 19 = 2! This is by no means exceptional: one observes that in most cases this method dramatically reduces the number of possibilities to be explored. We have used this program to explore systematically all “small” values of k for algebras up to rank 4. All known invariants have been reproduced and a few new ones have been found (see sects. 3—5 and appendix B). But one can obtain partial results for higher values of k by restricting to particular types of solutions. An interesting class of exceptional invariants are those due to an automorphism of an extension of a Kac—Moody algebra by chiral fields. If these chiral fields are simple currents, this invariant will only involve the characters of representations of charge 0 (cf. sect. 2) and more precisely, only the combinations —

—

—

N

Xj’~a,

(A.1o)

a=1

with N the length of the orbit of a under.!. All characters of such orbits have the same transformation properties under the modular group [cf. eq. (3.30)1. We can thus apply the same procedure as above to this restricted and folded S-matrix. It is convenient to render it symmetric by changing the normalization of some characters. For example, for SU(3), k = 0 mod 3, we define a new character x’ for the fixed point (k/3; k/3, k/3) under J=(0;k,0),

X(’k/3;k/3,k/3)

=

~X(k/3;k/3,k/3y

(A.11)

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Simple Kac —Moody algebras

With this new basis, the transformation matrix takes the form 35ab V~5af

~5af

,

(A.12)

‘~‘ff

with a, b representatives of orbits of triality 0, f the fixed point and gab’ 5af’ ‘~‘ff the original matrix elements. One returns to the original normalization of characters after the eigenvectors have been computed. No new exceptional invariant has been found in this way for SU(3), k = 27, 30,. 42, for SU(5), k = 10 and for SU(7), k = 7. Note the following subtlety: although the matrix (A.12) can be used to look for modular invariants, it is not a “good” S-matrix in all respects. For example, in order to get positive and integer fusion coefficients, one has to construct the S-matrix of the extended theory, in which one distinguishes three fields with character Xf• Another interesting possibility is to use the fact that the characters of a representation and of its complex conjugate are equal, so that one can keep only one of them. The transformation matrix under r 1/r can be made symmetric by an appropriate change of normalization. After modular invariants have been found, one verifies whether they can be obtained from known invariants by the above identification. This method allowed us to check that there is no new invariant for SU(3), 17 ~ k ~ 26. . . ,

—~ —

Appendix B THE CASE OF B 4, k =2

has for a generic value of k only two invariants: the diagonal one and one due to the simple current (k000). For k = 2, the numerical search gives 9 modular invariants. We will show that 8 of them follow from standard mechanisms, and that the last one is not the partition function of any conformal field theory. The richness of the spectrum is a consequence of the many conformal embeddings: 2cA’~1cE~, (B.1) B 134

B2cD~’cE~’.

(B.2)

We first have the diagonal invariant

z,= l0000I~+l2000l~,44+ 0010l~,84+l1000I~/9,9 + I0100I~/9,36+ I0002I~o/9,,26+

l0001I~/2,16+ l1001l~,128~

(13.3)

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383

The two subscripts are the value of h and the dimension of the associated Lie algebra representation. The last two representations in (B.3) are the spinor ones. Z2= I0000+200012+21001012+21100012+21010012+21000212

(B.4)

can be explained by the simple current (2000) or by reduction of the diagonal solution of A5: one checks for example that in agreement with 1B5 X(oo. .0) X(0000) X(2000)’ A8

=

B4

+

B4

the adjoint representation of A 8 (of dimension 80) goes to the adjoint of dimension 36) plus the representation (2000) (of dimension 44).

z3= I0000+001012+ 1000112+12000+001012+1100112 2 + sI2 can be obtained from the diagonal solution 1012 + Id (note that there are two possible embeddings of B4 in D8).

+

134

(of

(B.6)

Id2 of D

2+(2000+0010)(1001)*+c.c.

8, k = 1 (B.7)

Z4= l0000+001012+ 100011 follows from the solution 1012+ 1v12+sc* +cs~of D 8. Z5= 10000+0010+ 100112

(B.8) 2 of D can be obtained by reduction of the ordinary integer spin invariant 10 + s1 8, which can itself also be viewed as the reduction of 10... 012 of E8. Z6= 10000+2(0010) + 200012 2 of D can be obtained similarly from 10 + cI 8, or from I0...0l~~-*l0...0+0010...0+0...100I~~ Z7 and Z8 follow from the reduction of (0

+ s)(0 +

(B.9)

(B.10)

c)* and its complex conjugate,

z7=(0000+0010+lOol)(0000+2(oolo)+2000)*,

(B.11)

Z8=(0000+2(0010) +2000)(0000+0010+ 1001)*.

(B.12)

The last modular invariant quantity is Z9= I0000+0010+200012+21001012+ 1000212+1010012+ 1100012. (B.13)

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Since (0010) and (2000) have conformal dimension 1, we expect this solution to be due to some conformal embedding too; but dim(adjoint of B4)

+

dim(0010)

+

dim(2000)

=

164,

(B.14)

which is not the dimension of any simple Lie algebra. Z9 is not a partition function, but just an accident due to the large number of modular invariants,

z9=~(Z2+Z6).

(13.15)

Such a phenomenon has already been observed in ref. [311 in the case of SU(5), k = 5. Note however that a relation of the form (B.15) does not necessarily imply that one of the invariants involved is spurious: here one has also 2Z1—Z2—2Z3+Z6=0,

(B.16)

Z3~Z4~Z5~Z6+Z-,+Z7+Z8=0.

(B.17)

For completeness, we now give another proof of our claim. It uses a result of Moore and Seiberg [28] which asserts that the symmetry algebra of a CFT with a partition function of the type (B.13) can be enlarged in such a way that the partition function becomes diagonal when expressed in terms of the characters of the extended algebra. The characters Xoooo + X0010 + X2000’ X0010’ X0002’ Xoioo’ X,ooo transform among themselves under T l/T. The transformation matrix is obtained by folding the original S-matrix, —~ —

2 ~

5folded

=

1 ~ ~-

4 where x

0.11576 and y

2 3

1 3

1 3

X

—

—4 —4

1 3

3

)‘

1 3

1 3

—X

3

—

y

—x—y

x

—x—y

x

y

)1

,

(B.18)

0.5 1069 are the solution of

x2 +y2 + (x +y)2

=

4,

(x +y)2 —xy

=

4.

(B.19)

Because of the coefficient 2 in eq. (B.13), there are two fields with character Xooio in the extended theory. The new S-matrix which acts on the characters

D. Verstegen

Xoooo’ XOOiO, XII0’ 2

-

X00O2~,~oioo,Riooo 1

4 4

Simple Kac—Moody algebras

1

3

1

385

must be of the form 1

3

4 4(1—A) 4 40+A) 1

/

4(1+A) 4(1—A) 1

—~

—~

—4 —4

—4 —4

Unitarity of the matrix demands A using Verlinde’s formula,

=

1

3

3

3

—4 —4

—4 —4

—4 —4

X

y

—x—y

y

—x—y

—x—y

x

(B.20)

y

3 e’4. We compute the fusion coefficients

~0010)~(0010Y(0010)~

(1

+

e2’4)/4,

(B.21)

which can be an integer only if A = ±31.But then, ~ —1! The difficulty was already apparent before, since the inequality S~o~ S,~[381was not satisfied. We conclude that Z 9 is not the partition function of a conformal field theory.

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