HIGHER LEVEL KAC-MOODY REPRESENTATIONS ... - Science Direct

Nuclear Physics B308 (1988) 477-508 North-Holland, Amsterdam

HIGHER LEVEL KAC-MOODY REPRESENTATIONS AND RANK REDUCTION IN STRING MODELS P. FORGACS Central Research Institute for Physics, H-1525 Budapest 114, P.O. Box 49, Hungary

Z. HORV,~TH and L. PALLA Institute for Theoretical Physics, Roland Ertvgs University, H-1088 Budapest, Puskin u. 5-7, Hungary

P. VECSERNYI~S Central Research Institute for Physics, H-1525 Budapest 114, P.O. Box 49, Hungary

Received 8 February 1988

It is shown that an orbifold-like construction based on an external automorphism yields the E 8 model when applied to several 10-dimensional heterotic strings. The decomposition of the internal space into direct products of level two l~8 and critical Ising representations is given. All characters and string functions of level-two I~8 representations are derived explicitly. The conformal field theory underlying the E8 string is determined in detail. We also elucidate the role of modular invariance in the apparent uniqueness of the E 8 string.

1. Introduction String theories [1] offer a w a y to unify all interactions. H e t e r o t i c strings p r o v i d e us w i t h a large n u m b e r of m o d e l s i n c o r p o r a t i n g the o b s e r v e d low energy gauge i n t e r a c t i o n s together with chiral fermions. If we w a n t to connect directly the gauge g r o u p s of these theories (Gstnng) with that of effective low energy m o d e l s (Geff) then o n e m u s t face the p r o b l e m that the ranks a n d d i m e n s i o n s of the two are r a t h e r different. T h e r e are two ways to o v e r c o m e this difficulty, either to invoke the n o t i o n o f a ' s h a d o w w o r l d ' consisting of fields c o m p l e t e l y d e c o u p l e d f r o m Gel f, or to find ' s t r i n g y ' w a y s to reduce the r a n k of Gstring. T h e a i m of this p a p e r is to m a k e a m o d e s t step in the latter direction. A l l b u t o n e o f the k n o w n 10-dimensional heterotic strings have gauge groups with r a n k 16, the o n l y exception being the E 8 string c o n s t r u c t e d in refs. [2-4]. This string m o d e l was first m e n t i o n e d in ref. [2] where it was suggested that it can be o b t a i n e d 0550-3213/88/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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P. Forg6cs et al. / Kac-Moody representations

from the E 8 × E 8 model by an orbifold-type construction using the outer automorphism that exchanges the two Es's. Almost simultaneously and without any reference to the E 8 x E 8 string the spectrum of the E 8 string was determined by the 'spin structure' construction [3] using 32 real fermions to describe the internal space. More recently the procedure suggested in [2] has been carried out in detail [4] to get the contribution of the untwisted and twisted sectors separately. In four-dimensional models obtained by using the covariant lattice construction (CLC) [5] the rank of the gauge group is necessarily 22. Though there are several methods yielding strings with gauge groups of rank less than 22; (e.g. orbifolds [6] with non-abelian point groups or with discrete torsion and the spin structure construction [7]) it seems worthwhile to develop alternatives connected with the lattice construction. The CLC offers certain advantages such as the easy determination of the gauge group and the representation content, complete classification [8] and the possibility of straightforward computations of scattering amplitudes [9]. In this paper we would like to elucidate in 10 dimensions the connection between the E 8 model and the ones constructed by the CLC. As this is not completely trivial we present our results in detail hoping that this could serve as a basis of deeper understanding of rank reduction in lower dimensions. The outcome of this investigation is that the rank reduction using the outer automorphism applied to the E 8 x E 8 model leads to the appearance of level-two representations of the E8 Kac-Moody algebra (KM). The difference between the central charges of the level-one and level-two representations is carried by a critical Ising model. Having explored the level-two 1~8 and Ising representation content of the internal space in some depth we determined in terms of the characters of these representations the partition functions of the untwisted and twisted sectors of the E 8 string. Furthermore, we were able to derive explicit expressions for the characters of the various level-two 1~8 representations in terms of ordinary theta functions. That made possible the explicit determination of the string functions of these representations, too. We also determined in detail the conformal field theory underlying the E 8 string to show that all scattering processes can be calculated using the level-two E 8 currents and the Ising primary fields. Perhaps surprisingly we found that applying the same procedure to other models (D 8 x D8, (A 1 × E7) 2) we always ended up with the E 8 string. The reason for this seems to be that modular invariance demands the appearance of such combinations of level-two I38, or A 1 x E 7 representations which necessarily add up to level-two E8 representations. This paper is organized as follows: in sect. 2 we present the orbifold-like construction based on the external automorphisms of E 8 × E 8, D 8 x D8, (A 1 × E7) 2. In sect. 3 the structure of the internal space of the E 8 string is worked out in detail. Sect. 4 is devoted to the derivation of the level-two E8 characters. We develop the conformal field theory underlying the E 8 model in sect. 5. The role of modular invariance in the uniqueness of the E 8 string is elaborated in sect. 6. We draw our


479

conclusions in sect. 7. There is also an appendix devoted to the explicit determination of the level-two E8 string functions.

2. Orbifold-like construction and the E s string

In this section we first work out in detail the orbifold-like construction of the E s string starting with the E 8 × E 8 model in a way that emphasizes the role of the modular group and then apply the procedure for the D 8 × D 8 and ( A 1 )< E7) 2 strings, respectively. All orbifold constructions consist of basically two steps. In the first one we truncate the Hilbert space of the original string theory by keeping only those states that are invariant under the action of a suitably chosen discrete group, G, the so-called point (or more generally space) group of the orbifold. In the second step of the construction we enlarge this restricted Hilbert space by adding the twisted sectors, i.e. Hilbert spaces where the boundary conditions of the string are modified. The nature of these modifications and the number of twisted sectors depend on G. The necessity of introducing these twisted sectors follows from the requirement of modular invariance [6]: keeping only the G invariant subspace of the original Hilbert space would lead to a modular non-invariant theory but by adding the twisted sectors modular invariance is restored. Therefore we shall use appropriate modular transformations to generate the contribution to the partition function of the twisted sectors from that of the G invariant subspace of the original "untwisted" Hilbert space, Put. The discrete group, G, used in the construction of the E 8 string from the E 8 X E 8 model is a Z 2 group consisting of the identity element and of g = Ce :~iJ,2 where C is the transformation exchanging the two Es's and the second factor describes a 2~ rotation of the 10-dimensional space time. Since this group is abelian and has only two elements there is just one twisted sector. The partition function on the G invariant subspace of the untwisted string's Hilbert space has the following form:

Put = ½Tr q#°- lZIL'°- x/2 + ½Tr fi,qL'°- XEIL~°-I/2 ----et{tl-I- Put,2

(2.1)

where L0e and L~ are the hamiltonians of the left and right moving modes, q = e 2~rir with ~- being the modular parameter and ~ is the operator representing g. The first term, Pult, is nothing but ½ times the partition function of the E 8 × E 8 string; in the light cone gauge it has the form: 1 ( I m r ) -4 ~_~ p~t= 1 (v~8+v~8+88) 2 (~(~34_u~4)_{7~4) 2 (n~) 8 4~116

(2.2)

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P. Forgdcs et al. / Kac-Moody representations

where o~

"q('r) = e i''/12 I--I (1 - q") n=l

is the Dedekind ~/ function. The first factor in eq. (2.2) is the contribution of the eight transverse bosonic coordinates, the second one is the partition function of the E 8 x E 8 internal space and the two parts of the third factor correspond to the space-time bosonic and fermionic contribution of the right moving, bosonized world sheet fermions, respectively. (Of course on account of the Riemann identity for the theta functions P1 t vanishes identically - this is a manifestation of supersymmetry.) The partition function of the E 8 × E 8 internal space can be written formally as Pz, xE~(r) =

E

(2.3)

~ala21ei~(H~+n~)Laxa2)=(~ e~'~) 2.

(al,a2)~FsOFs

Here F 8 • F 8 is the weight lattice of E 8 × E 8 and we used Hilala2)= a~lala2), i = 1, 2 and e.g. aa denotes a vector of the first F 8 lattice. The second term of Put, Pu2tdiffers from P~at at two places. First as a consequence of exchanging the two E8's the contribution of the internal space is

(ala2lei'(t6+n2)la2al) = (al, a2) E F8 ~ F8

y'

ei~O~(alla2)ei~'?(a2lal)

al, a2EF 8 .

. xl/2

= y" e2i'~2=(Pe~×e,(2r))

.

(2.41

a ~ F8

Using some well known identities of the theta functions [10] (PE8 x Es(2'r)) 1/2 can be written: 1 (pt~xz,(2r)) a/2 _ 2~ig0,~0,4 { 1( v~8 + ~48) +

7 o~4o,4 4v3v' J"

(2.5)

The second difference between P,~2t a n d Pult stems from the presence of the 2~r rotation in P~]. The effect of this rotation amounts to inserting the ( - 1 ) G projection operator (where Fsp is the space-time fermion number) in Pit and this in turn changes the sign of ½~24. Putting all this together we get finally:

Put

( Imr)-4 2(TI~) 8

1~41~4 7 ,q.4 o~4 2"1116 ( 1 ( ~ 8 " ~ - l ~ 8 ) q - ~ v 3 v 4 }

~4

(1(~4--~;)-'}-1a4).

(2.6)

This expression is invariant under the T generator (T: ~"~ ,r + 1) of the modular group but changes when acted upon by the other generator, S, (S: "r ~ - 1/'r); thus

481


we can see explicitly that restricting ourselves to the G invariant subspace of the untwisted E 8 x E 8 string spoils modular invariance. However it is not difficult to see that using S and T we can generate from Put expressions that together with Put form a " m o d u l a r family" in the sense that the modular group transforms them among themselves. Fortunately in the present case this is a rather short procedure since defining the quantities M and N as the S and T S transforms of Put (2.6):

M-

(im,r)-4 4 4 03#2 7 ~q,4q,4 ~ 1 2(~/~) 8 2~/16 { 1 ( 0 8 + 0 2 8 ) + ~ v 3 ~ 2 j ~ ( ½ ( 0 4 - ~ 4 ) + ½ ~ 4 ) ,

(2.7)

N=

( i m T ) - 4 0402 4 4 1 _ 2(7/~) 8 2~/16 { ~ ( 0 8 + 0 8) - _7 4 vq,4o~4 4 ~ 2 } ~ 4__( ~ ( 014 4 q - ~ 4 ) n t - ~ 01--4 3),

(2.8)

_

we already get a modular family, Put, M and N. Furthermore it is easy to show that the s u m Put "}- M + N is modular invariant, thus we can identify M + N with the contribution of the twisted sector, Ptw = M + N. Exploiting various identities among the theta functions we can write Put q- Ptw as"

Put + Ptw =

(Im ~-) -4 8 --4 8 --4 8 64~/24~x2 { 31(08 + 08 + 04 ) ( 0 3 0 3 + 0404 -

-

-

--4 8 0202)

30(~4016 -- v4~4q'X6v4-- 0--42 0 216 ) } ,

(2.9)

which is precisely the form of the partition function of the E 8 string one obtains from the spin structure construction of ref. [3]. This illustrates the correctness of our procedure to calculate the contributions of the twisted sectors to the partition function by applying modular transformations to Put- The advantage of this method is that it makes it unnecessary to construct the twisted sectors of the Hilbert space. Next we investigate what happens if we apply the previous orbifold construction based on a Z 2 group whose nontrivial element is the product of the exchanging operator and of the 2~r rotation to the D 8 x D 8 [2, 3,11] and (A 1 X E7) 2 [2, 3,12] string models. One could naively expect that the gauge groups of the models obtained this way are just identical to the diagonal ones, i.e. D 8 and A 1 x E 7. In both cases we again use the previous method to compute the contribution of the twisted sectors to the partition function. For the D 8 X D 8 string the partition function of the G invariant subspace is given by eq. (2.1) again, but PuXt and P~ look slightly more complicated. Part of this complication is due to the lack of space-time supersymmetry of the model. Here the conjugacy classes of the internal space and of the right moving world sheet fermions are connected; in the light cone gauge only the following (D 8 x Dg)D 4 conjugacy

482


classes are present [12]:

((0,0) + (c, c))~,

((s, v) + (v, s))o,

((o, ~) + (s, s))s,

((c,O) + (o, c))c.

Therefore P1 t now has the following form:

eu,,-

1 ( I m r ) -4

f__l[/~ff__O,8"~2..p_,l/~2161 ~, [\ ] ' 1--4

1q,8/" o~8

,q,8,_1~4

2 0/~)8rt16~ 4

+

1

+

1--4

1,°'8/a8- @48)½(7t~4+ ~4)}.

+ 2°2,°,

(2.10)

We construct p 2 again by implementing explicitly the operators that exchange the two Ds's and the 2~r rotation. The effect of the latter one is again changing the sign of the space-time fermionic contributions (i.e. that of the ~1-4 2 ) relative to the space-time bosons. To compute the contribution of the internal space we note that the relation we found earlier between the contributions of the "exchanged" and "non-exchanged" cases, eqs. (2.4) and (2.3), is not specific to E 8 × E8; in fact it is valid for any case when exchanging two isomorphic factors of a direct sum lattice. The only thing we have to be careful about, when considering the D8 X D 8 string, is that vectors in different conjugacy classes are orthogonal to each other, thus there is no contribution from the (v, s), (s, v), (c,O) and (0, c) sectors. The final form of P~ is: Pu2t- 21 ~12~/8~8(2r)(Imr) -4 {½[@8(2r) + t,}48(2~.) + @8(2r)] ½(~4_ ~4) 8 1--4 }. +1[O~(2r)-v~84(2r) +@2(2r)]~02

(2.11)

Since Pit given by eq. (2.10) is ½ times the partition function of the D 8 x D 8 string it is necessarily modular invariant. Therefore the non-invariance of Put is entirely due to the presence of p2, and it is easy to check that this expression indeed changes under the S transformation while staying invariant under T. It is straightforward to show that we can generate a modular family from P2t (2.11), by taking its S and TS transform; and also that the sum

Put+ S(P~)+ TS(p2)=P

(2.12)

is modular invariant. However, perhaps surprisingly it turns out that P is just identical to the partition function of the E 8 string (2.9). So the orbifold construction based on the external automorphism exchanging the two Ds's reduces the rank of

483


the gauge group from 16 to 8, the resulting model is identical to the E 8 string instead of the naively expected D s. Let us now consider the (A 1 × E7) 2 model. The partition function in the light cone gauge can be written in terms of the (A 1 × Ev) 2 × D 4 conjugacy classes as:

P(A,×E#--

(Imr)-4 {[(0A,,0ET)2+ (1Al, lET)211(~4--~44)

+ [(1A1,0E /2 + (0A,,

I( 4 +

+ 2[(0Al,1E )(1 l,0 t]

(2.13)

where we already expressed the contribution of the D 4 conjugacy classes in terms of theta functions; 1ET,0E~ denote the conjugacy classes of E 7 containing the fundamental (56) and the adjoint (133) representations, respectively, and similarly for A 1. The partition function of these conjugacy classes can be represented in terms of the theta functions as follows: 0n = ~-I (032 + 02) 1/2 ,

1A = ~ - ( 0 ~ - - 042)1/2 '

0E 7 = 1 ~ - [ ( 0 2 q_ ~2)1/2(~6 "b ~6) q._ (1.~2__~:)1/206], 1E~ = 1~-~ [(0~ + 02)1/206 + (02 -- 0:)1/2(~6 + ~6)].

(2.14)

P~t is again I times the original partition function (2.13) whereas P~ is given by: P~

1 (Imr) -4 8 ~12~/8~/8(2r){(02(2r) + t~z(2r))l/2 × [(0~(2r) + 02(2r))1/z(06(2r) + 06(2r)) +(t~2(2r) - 02(2,))1/2~6(2r)]

~2(2r))l/206(2r) +(02(2r)-v~2(2r))1/2(O6(2r)- 0 6 ( 2 r ) ) ] } I ( ~ vaff(2r))l/2[(va2(2r) + vaff(2r))l/2(06(2r)+ va6(2r)) +(~2(2r) -0:(2,))1/2[(~2(2¢) +

+ {( 02(2r) -

~4)

-t- ( ~2(2"r ) -- ~: (2 ~"))1/21}26(2¢)] + ( 0 2 ( 2 r ) + t}42(2r))l/2[(O~(2r) + O42(2r))a/2v~26(2¢)

+(O2(2r)_O2(2r))X/E(o6(2r)_O6(2r))]}½(a4+7~4).

(2.15)

484


After some algebra we get for P~t the following form in terms of theta functions with arguments r: 1 ( I m ¢ ) -n Pu2t -- 64

~12//24

o,16~4 -- ,q,16~4 8 8--4 1~81~8~44..}_15~81~8~4 u4 **4 -- ~2 1~3~3 + { "3 "3 }

(2.16)

As before we get a modular family by applying the S and TS transformations to P2t, however, we obtain the following result: eu, + s e a + v s e 2 = *'/17,12 - 2kn (3.6) rules out all representations containing weights with length squares greater than 4 in the present case. The states eliminated by this condition are the so-called null states [17] which in the basic level-one representation first appear at the second grade [18]. In ~ o ® ~ o at grade-two we have the following types of states: Ta_2108)2 ® 10); 108)2 ®OU_210) and T{_~Tb_}ll08)2® 10). The first two types give altogether 248 + 1 states while for the third one we find from the bound (3.6) that none of the 27000 + 3875 + 1 = 30876 states are excluded. The reason for this is that in the level-two representation we consider here, ~ 0Es~ null states appear only at grade three. Thus the total number of states in ~ o ® ~ o at grade two is 31125 which is smaller than the total number of states in ~ by 3875. From this it is clear that ~ o ® ~ o alone cannot account for all the states in ~ and consequently

In ~ 2 ® ~

at grade two there are two kinds of states: l a ) 2 ® ~ 1 1 ~ )

and

T~-~la)2 ® 16)- The first one gives 248 states while from an analysis of the bound (3.6) and the null vectors we found that the number of states of the second type is 1 + 248 + 3875 + 30380 = 34504. Thus at grade two ~ 2 ® ~ 2 contains precisely as

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P. Forgfics et al. / Kac-Moody representations

many states as J~as- Since there are no other ~ aE8 ~ ~ i b combinations giving integer eigenvalues for La02+)g'0 and ~as, ~ are orthogonal to each other we conclude that ~s_~. ( ~ O 8 ~ 0 )

(~]~( ~ l E s ~ )

~ a s ~ _ ~ 2E8 ~ 2

,

•

Knowing the level-two t~.s representation content of ~ and ~as enables us to keep track of the E 8 quantum numbers when constructing the E 8 string. In the first step of this construction we had to keep the subspaces of ~ = ~ i n t ®J~F staying invariant under G = (Cexp(2rr/J12)>. JF F is the direct sum of Hilbert spaces of space-time bosons, Jt~pB, (being invariant under the 2~r rotation) and space-time fermions, ~pF, (changing sign as the result of the 27r rotation). Thus the subspaces in ~ invariant under G are: ®

=

®

®

E8

(3 7 )

In the next section we show how we can use this correlation between certain level-two E8 representations and space-time fermions or bosons to compute in a simple and explicit way the characters of these representations.

4. Determination of the level-two E8 characters From the previous discussion we have seen how the G invariant subspace of decomposes into different representations of the 1~8 current algebra, the Ising algebra and of the bosonized world sheet fermions. Each of these pieces have their own partition functions of the form [18]: Z ~ = ch~(0") q h~= ch#e('r)qh~-c(~)/24q c(#e)/24 = B~e(,r)q c(~e)/24

(4.1)

where h a is the Virasoro h.w. and c ( ~ ) is the central charge of the representation in question and ch~(~-) = Trq ~ = ~ pnq n,

(4.2)

n=0

where q = exp(20ri~-}, /V is the number operator and correspondingly pn gives the number of states in ~ at grade n. (Of course for the right-moving world sheet fermions one should replace ~" by its complex conjugate.) We shall refer to the functions B~(T) as characters for brevity, they are introduced since they have simple transformation properties under the modular group. Evidently the untwisted contribution to the partition function of the E 8 string, Put, becomes a sum over the appropriate triple products of partition functions of

P. Forgfcset al. / Kac-Moodyrepresentations

489

each three types. Since the sum of central charges on both the left and the right-hand sides is the same as for the E 8 × E 8 string there is no trace anomaly and we get using (4.1):

Put

(],ImT, -4 1 1 1 1 --4 (7/~)8 ~14(-R2R2!~4+(B°B°+BEsBI)~(O3-O4))~Es~I 2~2 E8 I

,

(4.3)

where the first factor is the contribution of the eight transverse bosonic coordinates and the barred 0 's correspond to the D 4 characters of spinors and vectors. (The sign in front of the first one is due to the spin-statistics relation.) Comparing the coefficients of the various D 4 characters in this form of Put and in the one we obtained earlier (2.6) enables us to derive simple and explicit expressions for B~8, (a = 0,1,2) in terms of ordinary theta functions. The functions B E8 a can in principle be obtained from the formulae given by Kac [16], however, those expressions are of little practical use as they give B E, a as a nontrivial sum of level 32 theta functions over the entire Weyl group of E 8. To compare the coefficients of the D 4 characters in the two different expressions for Put correctly we have to rewrite eq. (2.6) by appropriately exhibiting in it the space-time spinorial and vectorial contributions of the original E 8 × E 8 partition function. Defining

E = (~28 --[-L~38-q-~8) 2

4 4 ( ~1(~38 --~~48) --~ 4t,3~4 7~q,4q~4) , el = 03194

eq. (2.6) for Put takes the form:

P u t - (Imp')-4 1 1 1 ('r/~) 8 ~411168((½E-Pl)(-O4)+(tE+Pa)(~4-04)}.

(4.4)

The additional piece of information we make use of is the explicit form of the Ising characters, B~'. From the formulae given by Rocha-Caridi and Dobrev [19] after some algebra we get the somewhat more familiar infinite product representation:

BO=1q-1/48 I--I(l_qn-t/2)+ I-'[(l+qn-1/2) = n=l

n ~ = lq-1/48{n=l ~ (l+qn-1/2)--

B2= qt/2,,=l I~I

(1 + q " ) = ~ -

n=l

~

n=l f i (1--qn 1/2)1 = + (

.

(~3 +~44) ~3-

'

~4),

(4.5)

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P. Forgdzcset aL / Kac-Moody representations

Equating the two expressions for the coefficient of the space-time spinor charac1--4 in eqs. (4.3) and (4.4) for Put makes possible the determination of B~: ter, - 7v~2,

B2

1

f~2~31/2(~E-¼Pa).

(4.6)

Expanding this expression in powers of q around q = 0

~(-~-~2~3,/2 ( ~ E - J P , ) = q7/24(248 + 34504q + 1022752q 2 + ... ),

(4.7)

we see that it indeed has the form required by eqs. (4.1) and (4.2) for B2E. It is also worth remarking that the first two coefficients in the bracket do indeed coincide with the numbers we got in the previous section as multiplicities of states in ~ at the first two grades. The determination of the other level-two E8 characters is slightly more complicated since equating the coefficients of 71(343 - ~44)in (4.3) and (4.4) gives only one equation for two unknowns, B° 8 and B~. We circumvent this difficulty by exploiting the transformation properties of Bi,a B E8 a (a = 0,1, 2) and ~ / - 1 6 ( 1 E - ~P1)l under the modular group. We can compute explicitly the action of the S, T, a and on ~ --16zl generators on the Ising characters, BI, t~z:r, - - ~, P 1 ) using eqs. (4.5), (2.7) and (2.8), respectively. It is known [16] that the 1~8 characters, B~,, form a modular family. The behavior of BOa and B 1 is particularly simple under T transformation; they change to exp( -'31t~r}B °E8 and --exp{--lTgqr}BE8 , . 3 1 1 respectively. Thus our strategy is to start with

21

1

(4.8)

by applying to it S and T transformations until we can build up expressions having the previously mentioned transformation properties under T. As a first step we define a quantity A by taking the S transform of eq. (4.8) ~

__~(1

1

4 4 1(038 + 1~28) "~ 4u3 ~2 J ' e2= 03o2{

(4.9)

where P2 is obtained from PI(-1/~-). Then we introduce another function, B,


491

defined by the T transform of (4.9)

~3 = _ 0 : 0 4 ( 1(0~ + o2") - ~ 4

}.

(4.10)

F r o m eqs. (4.9), (4.10) it is easy to see that the combinations A + B and A - B have the expected transformation properties under T for B° 8 and B 1E 8 ~ respectively. Furthermore, A + B and A - B have the following expansions around q = 0: A + B - q-al/48(1 + 248q + • • • ), A - B -

q-3,/48q3/2(--3875 + " ' " ).

(4.11)

F r o m the above it is clear that we can identify A + B with B° 8 and A - B with B18: -

1[(1 1)1(1 1)] E8 .31j2 EN N xN -N 2.

B1 -

-

--

(4.12)

To give additional support for this identification we checked two further properties: first we showed algebraically that with these B_ _° 8 and B~ 8 the sum BE0 B 0I + ~ , ~ I __ indeed coincides with the coefficient of ½ ( ~ 4 _ ~ 4 ) in eq. (4.4), moreover we checked using R E D U C E that the coefficients of the first n powers of q in the expression of B ° and B 1E, are indeed positive integers: B ° = q-31/48(1 + 248q + 31124q 2 + 871627q 3 + . - - ), B ~ = q41/4831(125 + 5863q + 116899q 3 + - . .

).

(4.13)

In addition we note that the first three coefficients in B ° neatly coincide with the multiplicities of states at the first three grades we got in the previous section for ~ o . This explicit and simple form of the various level-two 1~8 characters is interesting f r o m both the mathematical and the physical point of view. On the mathematical side it makes possible the explicit determination of the so called "string functions" for the three level-two h.w. representations. (See appendix.) The physical interest of


492

these formulae stems from the fact that they enable us to classify the true physical particles of the full E 8 string with level-two l~8 and Ising representations. In order to carry out this classification we have to rewrite the partition function of the full E 8 string, PE~, in terms of B~E, and B b. We do this in the following way: we express first in eqs. (4.9) and (4.10) A and B in terms of B ° and B1E~,"~ and ~44/~ in terms of B ° and B~. Then we multiply eqs. (4.8), (4.9) and (4.10) by ~ ( i m ~.)-4 ~8~12

~4(im ~.) '

~8~12

~4(im ~.)-4

4 '

~8~12

respectively and add together the resulting equations. The terms containing E at the right-hand side cancel while the terms containing/'1,/2, P3 precisely reproduce the form of PEs, (2.9), derived in sect. 2. Thus (Im,r) -4 PE8-

'08~ 12

(,')l/2 /~21~4 [ B o B o B 1E 8B II' ~] 2I ( 1 9-34- - ~ 2 ) ~E8~I2"2 + \ E8 I +

-I-(B1E8B°+ B 0E8Bt

t

-4

44)} .

(4.14)

Because of the left-right level matching condition physical particles appear in the q, ,~ expansion of PE8 as q,~/n with their masses proportional to n. We exhibit only the lowest-lying ones with n = 0 and - 1 arising from the D 4 conjugacy classes in (4.14). 1--4 Since the expansion of ~-8~-a2 starts as (1 + 8q + 12F/)q 1/3q 1/2 while 5#2 8q 1/2 and B E28B I2 - 248q 1/3 we see that from the spinor conjugacy class (which corresponds to space-time fermions) we get 496 massless fermions. This means that these fermions also carry the R vacuum Ising quantum numbers. In the vector conjugacy class, where ~(v~3_ t -4 ~ 4 ) _ 8~1/2, we get the (10-dimensional) gravity multiplet and the gauge bosons from the first two terms of B E° 8B °I ~ q - 2 / 3 + 248ql/3 by appropriately combining them with ~-8~-12. Thus both the gravity multiplet and the gauge bosons carry the NS vacuum Ising quantum numbers. Since B 1E 8B 1I 3875q 4/3 it gives only massive (n >I 1) particles. The only additional particle of the E s string with n ~