Kaluza-Klein approach to the heterotic string II

Volume 171, number 2,3 PHYSICS LETTERS B 24 April 1986

K A L U Z A - K L E I N A P P R O A C H T O T H E H E T E R O T I C S T R I N G II

M.J. D U F F l, B.E.W. NILSSON 2, N.P. W A R N E R 3

CERN, CH-1211 Geneva 23, Switzerland

and

C.N. POPE

Blackett Laboratory, Imperial College, London SW7 2BZ, UK

Received 4 February 1986

Further properties of d=10 heterotic superstrings, including the Lorentz Chern-Simons terms, are obtained from spontaneous compactification of the d = 506 bosonic string on the group manifolds G = SO(32) or E s × E s. These involve generalizing our previous Kaluza-Klein ansatz for the gauge bosons A, of G L to (Az, Bz) of G L × G R, which necessitates the incorporation of Kaluza-Klein scalars to maintain consistency. The heterotic strings are then obtained by identifying Bt,(SO(8)internal C G) with o~. + (SO(8)s-acetime) where ~o. + is the connection with torsion + ½H. ~. In particular we obtain the

~0 a# , t exact result dH = (a'/2)(R+ A R+ - 2 F A F ), non-linear in H. The type II strings are obtained in a similar way by setting A,(SO(8)) = ~z- and Bz(SO(8)) = 0~ +.

The massless sector o f d --- 10 superstrings describes

the interaction o f N = 1 supergravity coupled to N = 1 super-Yang-Mills with gauge groups SO(32) or E 8 X E 8. This runs counter to the traditional Kaluza-Klein philosophy for which the fundamental theory has only spacetime symmetries and for which, in particular, the Yang-Mills fields are components of the gravitational field in a higher dimension. In this traditional Katuza-Ktein approach, the Yang-MiUs gauge group is merely a subgroup of the general coordinate group.

In a previous paper [1 ], however, it was shown that the gauge bosons of the heterotic string [2] i n fact have a traditional Kaluza-Klein origin in the d = 506

bosonic string [1 ]. Specifically (1) The conformal invariance of the closed bosonic

string in d spacetime dimensions places restrictions on

the background fields gMN(X), BMN(X) and q~(x) [3].

i On leave of absence from the Blacker Laboratory, Imperial College, London SW7 2BZ, UK.

2 On leave of absence from the Institute of Theoretical Physics, S-412 96 GOteborg, Sweden.

3 Present address: Mathematics Department, M.I.T., Cam- bridge, MA 02139, USA.

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These restrictions are equivalent to the Einste in-matter field equations with cosmological constant propor- tional to (d - 26)/a ' [4].

(2) The presence of the cosmological constant for- bids fiat vacuum solutions except when d = 26, but for d > 26 permits a spontaneous compactification, i.e., a ground-state solution of the form

Md = Mn X G , (1)

where Mn is n-dimensional spacetime with zero cosmological constant, and G is a compact group manifold of dimension k = d - n.

(3) This gives rise, in a conventional Kaluza-Klein fashion [5], to an effective n-dimensional theory with a finite number of massless states and an infinite tower of massive states, all belonging to various representations of GL X GR. The massless sector describes the n- dimensional fields guy., Buy an.d ¢, massless Y a ng - Mills gauge bosonsAh and B~ corresponding to GL and GR respectively where i = 1 .. . . . k, and massless scalars S i~ in the (adjoint, adjoint) representation of GL X GR. From a string point of view, this corresponds to decomposing the d-dimensional creation

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


operators a~t ) and ~ ) into spacetime and extra dimensional components. In particular, acting on the vacuum with

~M '~N " (- ~ - ~at~ 1)¢~b 1) ~-I)~(S1)

^'(-1) X(-1)= \ a(_ 1)a~/._ 1 ) C~_1)~_1) ) , (2)

we obtain the above-mentioned massless states [6]. (4) In order to focus attention on the massless sec-

tor, one makes a Kaluza-Klein "ansatz" for gMN, BMN and ~. In ref. [1], our aim was to obtain the massless bosonic sector of the heterotic string by choosing n = 10 and G = SO(32) or E8 × E8. Hence our ansatz included only the fields (gu~, Buy, ¢, A i). This ansatz was consistent in the sense described in ref. [5], i.e., all solutions of the resulting ten-dimensional theory were solutions of the original 506-dimensional theory. This is to be contrasted with a generic Kaluza-Klein theory where for consistency one must keep only GR singlets and all the GR singlets, in particular GR singlet scalars which, in the present case, would be massive.

(5) In this way, we reproduced the bosonic sector of the low-energy limit of the heterotic string [2], i.e., the N = 1 supergravity coupled to Yang-Mills of Chapline and Manton including, of course, the correct Yang-Mills Chern-Simons term [7,8].

Although this represents encouraging evidence that all the properties of the superstring may be derived from the bosonic string [9], many important questions remain unanswered. In this paper, we address the questions: (i) Where do the Lorentz Chern-Si- mons [10] terms come from? (ii) Does the derivation of the ten-dimensional t3 functions from d = 506, car- ried out successfully at one loop in ref. [1 ], persist to two loops ? Bear in mind that the coefficients of (Riemann) 2 in the two-loop/3 functions are in the ratio 2 : 1 : 0 for the bosonic, heterotic and type II strings respectively [4], whereas a naive Kaluza-Klein argument would yield the same coefficients. This prompts (iii): Can the equivalence be demonstrated to all orders by deriving the ten-dimensional Fradkin-Tseytlin [3] superstring actions from the corresponding bosonic one in d = 506?

As we shall see, the key to answering all these questions resides in (a) making an improved Kaluza-Klein ansatz involving not only the gauge bosons A~ of GL, but also the B/~ of GR; (b) including Kaluza-Klein scalars S q in the (adjoint, adjoint) representation of

GL X GR in order to maintain consistency; (c) trun- cating to an SO(8) subgroup of GR (in the heterotic case [2]) or of both GL and GR (in the type II case [11 ]) and then identifying the SO(8) gauge potentials with the spin-connection of the transverse spacetime so(8).

Let us begin by deriving the massless sector of the n-dimensional theory which results from the spontaneous compactification of the bosonic string on the group manifold G. Although we are ultimately inter- ested in n = 10 and G = SO(32) or E8 × E8, at this point n = d - k and G will be arbitrary solutions of the critical dimension formula [12-15]

d - 26 = kCA/(2 p + CA), (3)

where CA is the second-order Casimir in the adjoint representation and p is an integer. This follows by noting that if we substitute the ground-state values of gMN, I~MN and q~ quoted in ref. [1] into the Fradkin- Tseytlin action, the result is nothing but a non-linear o model on Mn × G with a Wess-Zumino term, for which the radius of the group manifold m -1 is quan- tized in units of ot '-1/2

pm2a '= 1 . (4)

The appearance of the integer p follows from the quantization of the coefficient of the Wess-Zumino term. In what follows, we restrict our attention to the case p = 1, for which the non-linear tr model is equivalent on the one hand to a theory of free fermions [16], and on the other to a theory of free bosons defined on Mn × T r, where r is the rank of G [15]. We shall re- turn to this point later when we compare our results with those of Casher et al. [17].

Since the isometry group of the group manifold is GL X GR, and since the VEVs of ¢ and HMN P given in ref. [1 ] are also GL × GR-invariant, the n-dimensional theory will contain the Yang-MiUs gauge bosons of GL × GR. For generic Kaluza-Klein theories, however, a consistent Kaluza-Klein ansatz for the massless sector requires that we keep only a subgroup of the full isometry group [5]. We recall that a "consistent" ansatz is one for which all solutions of the n-dimensional theory are solutions of the original d-dimensional theory. For a generic Kaluza-Klein theory with homogeneous extra dimensions, a consistent ansatz re- tains all those fields and only those fields invariant under a transitively-acting subgroup K of the isometry

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group. In particular, for group manifolds the gauge bosons are only those o f GL and not those of the full isometry group GL X GR. Moreover, the GL ansatz is in general consistent only if we include Kaluza-Kiein scalars in tl" ~ symmetfized adjoint × adjoint representation of GL. Experience with d = 11 supergravity, however, teaches us that there may exist certain ex- Ceptional theories where a consistent ansatz can be achieved without demanding this "K-invariance" [5]. This can happen either by including fields which are not K-invariant [e.g., the SO(8) ansatz for the S 7 compactification o f d = 11 supergravity] or by omitting fields which are [e.g., the omission of Kaluza- Klein scalars in the SO(3) ansatz on $7]. In the present context, an example o f the latter phenomenon was provided in ref. [1], where we showed that the GL ansatz for the bosonic string was consistent in spite of omitting Kaluza-Klein scalars. In this paper we go one step further and show that even the full GL X GR ansatz is consistent, provided we pay the price of including scalars S i / in the (adjoint GL, adjoint GR) representation. This somewhat confusing situation is summarized in table 1. In presenting these results, we shall first give the ansatz'for GL X GR without including the scalars and then indicate how their inclusion solves the problem of inconsistency *'

Let us introduce the Killing vectors K I on the group manifold G

K I = (L i, R i ) , (5)

where L i are the generators o f left translations

[1, i, L /] = m c i / k L k , (6)

%' Since there seems to be some confusion on this point, we emphasize that the omission of scalars will not involve any linearization in the gravity or Yang-Mills sector.

Table 1 Gauge groups surviving in consistent truncations of theories compactified on the group manifold G, and the corresponding scalar representations.

With KK scalars Without KK scalars

generic KK theory GL: (adjLXadjL)symm - bosonic

string theory GL × GR: (adjL, adjR) GL

and R i are the generators of fight translations

[Ri, R / ] = - m c i J k R k , (7)

and

I t , aq = 0. (8)

The corresponding Yang-Mills gauge potentials are denoted by

A I = (A i, B i ) , (9)

where A i are the gauge bosons of G L and B i the gauge bosons of GR. The corresponding field strengths are given by

F I = (F i, G i ) , (10)

where

F i = dA i + }cil .kA j ^ A k ' (11)

G i = d B i -- ~ c ~ k B / ^ B k . (12)

The Kilting vector components L / and R / satisfy the Cartan-Maurer equations

V~L~= ,_oi . / . k (13)

V,R~='-o~ o / o k (14) ~,,~c jk~,,a~, b ,

and we also introduce the notation

i i k Cab c = Ci/k L a L b L c

i / k (15) = C i j k R a R b R c •

We are now in a position to state the ansatz for ~, gMN and BMN and to calculate the corresponding cur- vatures and field strengths. For the scalar, we write

~ ( x , y ) = constant + ¢ ( x ) . (16)

For the metric ansatz, we employ the one-forms ~A = c A M dx M, for which

~MN = ~MA &NB ITAB , (17)

and write

~'~(x, y ) = e a(x) , (18)

~"(x,y)

= e~fy) - m -1 [#%v) Ai(x) +R ~"0') # ( x ) ] . (19)

For the antisymmetric tensor ansatz, we write

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1~ I - 2 i i (A]L/a_ BI'R/a) = B - ~m (A L , + BiRia) A

_ m - l ( A i ^ L i _ B i ^ R i ) + m b , (20)

and where

B = ~ B ~ ( x ) e c~ ̂ e ~ (21)

and

b = ~bab(Y ) e a A e b , (22)

such that

1 a e b e c db = c = -geabce A ^ • (23)

It is important to note that (a, a) correspond to tan- gent space indices and ~ , rn) to world indices. The ansatz of ref. [1 ] is recovered by setting B~ = 0.

The quantities ~b(x), gpv = eu~ezflrla[~ and Buy will be interpreted as the scalar, metric and antisymmetric tensors in n-dimensional spacetime, and the quantities Ai~(x) and Bi~(x) will be the Yang-Mills gauge bosom for GL and GR respectively. Eqs. (17) - (19) are just the "standard ansatz" for the metric tensor familiar to Kaluza-Klein theories [5]. The novel feature of the bosonic string is the Kaluza-Klein ansatz for BMN, which also involves the Yang-Mills gauge bosons. In this Kaluza-Klein interpretation, the gauge symmetry GL X GR is just a subgroup of the d-dimensional general coordinate group. To see this in more detail, consider a general coordinate transformation

x M ~ x M _ {M (24)

and the corresponding transformations of ~, gMN and BMN. Then focus one's attention on the very special transformation

gM(x,y) = (0, OJ(X) L i m ( y ) + f i ( x ) R i m ( y ) ) , (25)

with ~i and f i arbitrary. Then from the Kaluza-Klein ans/itze (16) - (23) , we may compute the transformation rules for the n-dimensional fields. We find not only the usual Yang-MiUs transformation rules for 4,

i i guy, A u and Bu, but also that the Buy field transforms as

~Bpu = h p ~ v o i _ h i a p o t i _ B u a z , i+Biaufi. (26)

It is now tedious but straightforward to compute the curva tu re I~ABCD and the field strength I21ABC, which will be published in detail elsewhere [18]. In particular the n-dimensional field strength H is not

just dB but rather

H = dB - m - 2 ( ~ A -- ~2B) , (27)

where

~2 A = F i A A i -- ~cijkA i A A ] A A k , (28)

~'-~B = Gi A B i + ~cijkB i A B / A B k , (29)

i.e., we have acquired n-dimensional Chern-Simons terms, even though there were no such terms in d dimensions where/7/= cl/~. The non-covariance of I2 A and ~2 B under Yang-Mills transformations is exactly cancelled by the unusual transformation rule for Buu of (26), ensuring that H does not transform. Hence, although in d dimensions dH = 0, in n dimensions we have (on " , 2 u s l n g a m =1)

dH + o: (F i A F i - G i A G i) = 0 . (30)

Note that the left- and right-handed gauge fields enter with opposite sign, and that this equation is exact to all orders mt~" ' !

The d-dimensional ~ functions [4,1 ] are now re- lated to the n-dimensional f functions by

~4~(x,y) = f ¢ ( x ) , (31)

/~aga (x, y ) = fga~ (x ) , (32)

~o~%(x,y ) B = f~a(X), (33)

~gb(x , y ) A i f B i ( x ) R ~ ) m - 1 = ( fed(x) L b + , (34)

- 8 ; , (35) flo~b(X,y) = ( ~ i ( x ) L b - f i B ( x ) R ~ ) m -1

"g s i / i j m - 2 L b R a ) f ~ b ( x , y ) = (3~(x)(LaR b + , (36)

~3Bab(X,y) S i / i ] -2 (37) = f i ] (x ) (LaRb - L b R a ) m ,

where

16rr2f ~ = 4(a~b) 2 - 4I-q~b - R + ~ H 2

, _,,,~i ,:,ion# + GiGio~#) + (38) + ~u Lr~Or ... ,

g - lu u ~8 + 2VaV0~ foLl~ - Rag -- g'* a'r6 " '#

MttTi tz, i3, + ~ i rT_ioN + (39) - - ~ k~t~T*'/3 ~Jo~,yu/3/ --- ,

&~ = v ~ / ~ - ~ a ~ 0 + . . . . (40)

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jA i - ch rzif J I l l ~i'y8 i(3 - " ~ " ~ - ~ " ~ 3 " 8 " - 2F~ ~ ¢ + ..., ( 4 1 )

i cn ,.i 8 2G/ a ¢ + ( 4 2 ) = L . f l ,J a ~-.,7a../8 u - - # ....

Hence we appear to obtain sensible n-dimensional equations for the gravity fields ¢(x), gu~,(x), and Bu~,(x). interacting with the Yang-Mills field Ai~(x) and B~(x), in which all cross terms have cancelled. Note that in the above # functions and in the Chern- Simons equation (30), all the y-dependence of the original ansatz has disappeared without imposing any inconsistent constraints on the n-dimensional fields. The problem arises when we consider the/3gb and ~B components which yield

/3(s/3 = F(i°~#Gj) ~ + .... (43)

fl [s/j] = F[ ia#G/ l sO + ' " , (44)

whose vanishing would obviously lead to unacceptable constraints on F and G, not required by the higher- dimensional field equations, but imposed by an inconsistent Kaluza-Klein ansatz.

Fortunately, an inconsistency in the ab components is a rather mild one and can be cured by the introduction of scalar fields Si! in the (adjoint GL, adjoint GR) representation into the Kaluza-Klein ansatz, so that

and/3 B acquire extra terms of the form rqS(ii) and rqS[ij] respectively, where [] = c9 ac/)a. (This is contrasted with inconsistencies in the e/3 components which would require massive spin-2 fields or ab, which would require massive spin-l, and hence give rise to the reintroduction of an infinite tower of massive states [5].) This is reminiscent of the SO(8) ansatz in the S 7 compactification of d = 11 supergravity. A naive K-invariance argument would suggest that all the components of the Einstein equation would be inconsistent, whereas one finds that the problem resides only in the ab components and that this is cured by the introduction of Kaluza-Klein scalars. However, whereas the gravity-Yang-Mills ansatz can easily be written down exactly, it is much more difficult to construct the exact scalar ans~'tz. Similarly, in the bosonic string the exact ansatz involving Sij(x) is rather complicated, and here we restrict ourselves to the linearized results which we now quote without proof. Define the fluctu- ations hMN(X, Y), ]¢MN(X, Y) and 6(x, y) by

gMN = (gMN) + hMN , (45)

t~MN = (BMN) + ]¢MN ,

then

+ kab

(46)

(47)

(48)

=S " , [ , i R j 2(k+2) - • ijtX)[l'b a - m 2 C A ( 3 k _ 4 ) DaDb(LZcRlC)) ,

~= 1 i ja -~Si/(x) L aR , (49)

where

1 c D a v b = V a v b+~Hab V c . (50)

It should be emphasized that the above scalar ansatz is valid only at the linearized level, but one can confirm using standard techniques [5] that the scalars siJ(x) are indeed massless as expected.

So far, we have isolated the massless sector of the bosonic string propagating on Mn X G, and determined the 13-functions to one-loop order, and we now turn our attention to the heterotic string. To go beyond the Chapline-Manton equations obtained in ref. [1], however, it is also necessary to include the two-loop contributions to/~/N and ~d/MN. When [1MNP is neglected, the contribution to ~g/N is given by

Ot' I~MPQR 1 ~ / Q R . (51)

Combined with the Yang-Mills terms of (39), this yields the order a ' contribution to/3ug v

ot'(Ruoo.rR~,O°v - Fu ipFv i° - Giup Gi ° ) . (52)

Note that F 2 and G 2 both enter with a sign opposite toR 2.

In ref. [1 ], the heterotic string/3 functions were simply obtained by choosing n = 10, G = SO(32) or E8 X E 8 and by setting B~ (and the scalars S ij) to zero in (38)-(42), but this naive ansatz breaks down at two loops because the correct contribution to/3gv in the heterotic string is

a ' t l~ . pov F ipFif) (53) ~.2 . . x / z p o . r z x v - -

Comparison of (52) and (53) tells us that what is required is an ansatz for which (with Huv p = O)

ap o Par-c:- i ~ip (54) ~'~lapo~r~'"v - ~ , u p "- 'u "

In order to achieve this, we consider the decomposi-

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tion GR D SO(8)interna 1 and equate the corresponding spin connection Bu[IJI ( I ,J = 1 ..... 8) with the spin connection 6o/i,_ of the transverse SO(8)spacetime .2. It is here that we make contact with the work of Casher et al. [ 17] who consider the derivation of the heterotic here that we make contact with the work of Casher et al. [17] who consider the derivation of the heterotic string from T 16 compactification of the bosonic string in d = 26. In order to account for the fermions in 10 dimensions, they identify the spacetime group of the heterotic string with

SO(8)diagonal C SO(8)interna I X SO(8)spacetime , (55)

and choose the embedding of SO(8)internal C GR to be such that the adjoint representation of GR contains the 8s representation of SO(8)internal. [Since, in con- trast to the bosonic sector of d = 11 supergravity [5 ], the bosonic string admits a sensible euclideanization, it may be possible to interpret the covariant generaliza- tion of (55) as an embedding of the full spacetime SO(10) in GR.]

The crucial point is that (54) now provides us with the desired Lorentz Chern-Simons term when substi- tuted into (30). Before doing so, however, we should restore the dependence on Huv o neglected in (51)-(54). As discussed by Hull [19,20], this is achieved by replacing 6ou by 6o~ in (53), where 6o~ is the spin connection with torsion given by +~Huv o. Noting the sign change in (41) and (42), this corresponds to replacing

+ in (54) ,3 6ou by 6ou . Hence

1^ ,to ab ^ Rab 2F i ^ F i) (56) dH = ~t~ t~+ - .

We note that this equation is now non-linear in H but exact to all orders in a' .

It should be admitted, however, that in common with Casher et al. [17], we have no dynamical under- standing of the decomposition GR D SO(8) and the identification (55). Nor do we see any justification for the truncation of the string spectrum which seems to be entailed in reproducing that of the heterotic string. The idea is that the GR index m on the creation operators ~m n) (m = 1 ..... 496) is truncated to ~n) (I = 1 ..... 8) where I is the 8s spinor of SO(8)internal, and hence

,2 The factors of 1 in (53) and (54) compensate for double- counting.

.3 This implies a different bosonic string ~ from that calcu- lated by Fridling and van der Ven [21] whenH :~ 0. See Hull and Townsend [22].

the particle states with I indices transform as fermion representations of SO(8)diag.. Hence, in some case, the GR Yang-Mills boson Bwr is really the gravitino fly/, and the scalars Si! are really the gauginos X/I. In our group manifold approach, however, the origin of fermion statistics is very obscure.

To obtain the type II strings, we repeat the above procedure for both GR and GL, choosing the embed- dings leading to either (8c, 8s) in the case of type IIA or (8s, 8s) in the case of type liB. The extra 64 bosonic degrees of freedom are then provided by SI ' j or SIj, respectively. Vanishing two-loop t3 functions are obtained from (53) and (54) by setting A u ~ 6o~ and

- 1 - . . .

Bu ~ cou which then yields the Chern-Sxmons relation - - . j

dH-- ^ R; - R ^ R_% *'. Finally we note that the derivation of ten-dimen-

sional superstring properties from the bosonic string in d = 506 holds to all orders in a', as may be seen by substituting the Kaluza-Klein ansatz directly into the string lagrangian. Neglecting the dilation, we have

L =~+xMo_xN(gMN+ BMN) (57)

on using the orthonormal gauge for the world-sheet metric. Substituting (16) to (23), we obtain

L = 8+XuO_XV[guv+ Buy

i m j n+ i m j n+ i m ] n +(AuLi AvL/ BuRi BvR 1 AuLi BvR/)gmn]

u i n m U i n m +~+X A u L i ~ , y g m n + ~ _ X BuR i 3+y gmn

+ (V+y m V_yn)gmn+(~+y m a yn) Bmn, (58)

i m u where V+y m =b+ym + AuL i O+X and V_y m = i m t~ + ~ ym + BuRi 3 _ X • By the identification B u "~ wu,

and invoking non-Abelian fermionization [ 16], we re- cover the corresponding action for the heterotic string. [This differs from the manifestly gauge-invafiant action given by Hull and Witten [19,20], since in our action Buy already has the non-trivial Yang-Mflls and Lorentz transformation rules (26).] The type II string actions may also be derived in this way.

The identification of Yang-Mills gauge potentials with gravitational spin connections, first proposed in a rather different context [23] and employed here in going from 506 to 10 dimensions, has also been employed by Candelas et al. [24] in going from 10 to 4.

.4 Note the departure from the naive result dH = 0.

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This is suggestive of some grander scheme which would combine the two into some purely gravitational phenomenon in d = 506. Indeed, the ultimate ut i l i ty of our Ka luza-Kle in approach may be to throw light on the correct compactification from 10 to 4, and we note in this connection the new exact Chern-S imons relation (56). Note, incidentally, that although the coefficient of the torsion is not fixed uniquely by anomaly cancellations and supersymmetry alone [20], it is fixed by Kaluza-Kle in . I t is ~ times the torsion introduced by Scherk and Schwarz [25] and Nepomechie [26]. Moreover, as discussed in ref. [1 ], there seems to be no compelling reason to insist that Huv o be zero in the true ground state. As emphasized by Hull [20], this might have the phenomenological advantage of replacing E 6 by SO(10) as the effective symmetry in four dimensions. We also note the following practical advan- tages o f the d = 506 approach. Once the higher-loop/~ functions of the bosonic string are known, it is then a straightforward matter of substituting in the K a l u z a - Klein ansatz to obtain the ten-dimensional/3 functions of the heterotic string. Furthermore, i f the tr ick A u "" 6o~ is to solve the heterotic string equations to all orders in cx' (or if the vanishing of the higher-loop type II/~ functions is to hold to all orders in a ' ) , then the Yang-MiUs field strengths coming from the (t~') n terms in the d = 506/3 functions must conspire to can- cel the Riemann terms coming from the (a ' ) n+l terms. Thus the fact that our Kaluza-Kle in ansatz itself involves ~' may help us to deduce (n+ 1)-loop ~ functions.

We are grateful to Chris Hull and Hermann Nicolai for discussions.

Note added. After the complet ion of this work, we became aware o f a paper by Nepomechie [27], who also considers the GL X GR ansatz but sets Bu "" ~o u rather than B u ~ ¢o+u. He does not comment on the en- suing inconsistency of (43), (44) and its cure by Kalu- za -Kle in scalars. Furthermore, he does not discuss our result that the Ruo~Rv °a~ contr ibution to comes not only from the one-loop term GuoiGOv i in /3 g , but also from the two-loop term kMeaag~v eAa in t~tU.

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Kaluza-Klein approach to the heterotic string II