Kaluza-Klein approach to the heterotic string

Volume 163B, number 5,6 PHYSICS LETTERS 28 November 1985

KALUZA-KLEIN APPROACH TO THE HETEROTIC STRING

M.J. D U F F 1, B.E.W. NILSSON 2 CERN, CH 1211 Geneva 23, Switzerland

and

C.N. POPE Blackett Laboratory, Imperial College, London SW7 2BZ, UK

Received 8 July 1985

Both the gravltaUonal and gauge field sectors of the d = 10 superstnng, including the Chern-Slmons terms, may be obtaaned from the purely gravataUonal bosomc stnng m d = 10 + dim G xaa a spontaneous compactlficatlon on the group mamfold G with (Hmn p ) the parallelmng torsion The Kaluza-Klem relaUon between the couphng constants a'g 2 - ~2 is that of the heterotlc stnng Consistency of the Kaluza-Klem ansatz is essentml and reduces the symmetry from GL × Ga to GL

The most appealing feature of Kaluza-Klem theories is that what we perceive to be internal symmetries are really spacetime symmetries in a higher d~menslon In particular, it is not necessary to postulate the separate existence of Yang-Mdls fields, they are an automatic consequence of gravaty .1.

It is ~romc therefore that the recent spectacular successes of superstrings [2-4] seem to Ignore ttus beautiful concept. For although formulated m ten rather than four dimensmns, the Yang-Mills fields of E s × E s or SO(32) are present as primary fields already m the d = 10 lagrangian. Moreover, the favoured compacUficaUon to d = 4 [4] yields no Kaluza-Klein gauge fields since Calabl-Yau mamfolds have no continuous symmetries.

In this paper we show, in fact, that Kaluza-Klem can prowde an explanatxon for the gauge fields of the heterotm string [3], as might have been suspected from the Kaluza-Klein like relation between the Yang-MiUs coupling con-

1 On leave of absence form the Blackett Laboratory, Imperial College, London SW7 2BZ, Umted Kangdom

2 On leave of absence from the Insmute of Theoreucal Physics, S-41296 Goteborg, Sweden

,1 For a recent rewew see ref [1]

0370-2693/85/$ 03 30 © Elsevier Science Publishers B.V. (North-Holland Physms Publisl~ng I~wsmn)

stant g, the gravitational constant r and the slope parameter or', namely a'g 2 - r 2. (This is to be contrasted with the corresponding relation g 2 _

Ka' of the open string [2] which appears to admit no such K - K interpretation.) The Yang-Mills gauge group will, as usual, be a subgroup of the higher dunenslonal general coordinate group. Although the theory compactifies on a group mamfold G, however, our Kaluza-Klein ansatz will revolve only the gauge bosons of G L and not those of the full isometry group G L × G a. As discussed elsewhere [1,5-8] this is essential for the consistency of the Kaluza-Klein ansatz. Surpris- ingly, consistency is achieved without the intro- duction of Kaluza-Klein scalars.

Our starting point is the off-sheU effective action F of Fradkin and Tseytlin [9] for the infinite set of local fields corresponding to the modes of a free bosonic string spectrum. F is the generating functional of all possible off-shell scattering amplitudes on arbitrary backgrounds. In the case of closed oriented strings F is defined (in Euclidean signature) by

r [ ~ ( x ) , ^ ^ gMN( X ) . B,. ,N( X ) . . . ]

= Y'. e°×f[dVab][dxMle -s , (1) X=2,0, - 2

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where o is a dimensionless parameter and where

S = L 2 ~ [ ~ / - ~ ' ~ a b O a X M O b X N g M N ( X )

+ :Oox'% Ng N(x) + . . . ]. (2)

Here x M(Z) defines the embedding of the two- dimensional stnng worldsheet M 2 m a spaceume M a (M, N = I . . . . . d); z a ( a = 1 , 2 ) are the coordinates on M2; R(~,) is the curvature scalar of the metnc Tab. The gravlton ~MN(X), the antlsymmetnc tensor BMN(X ) and the &laton ~ ( x ) correspond to the massless modes of the bosomc stnng spectrum. The dots refer to terms describing the Ingher spin massive modes and the scalar " tachyon". In writing the action (2), and in the remainder of the paper, we revert to Mmkow- ski signature for both the worldsheet and for spaceUme.

For stnng consistency, the two-dimensional theory must be conformally mvanant and hence the two-dimensional worldsheet stress tensor must be traceless. The general structure of the trace is given by

0: _}_ flgNV[-~V abO a X MOb X N

-'[- ~N~,ab c~aXM C~ b XN, ( 3 )

where fl*, fig and fin are local functlonals of ~(x) , ~MN(X) and BMN(x). As noted by Polyakov [10] the first term is just the two-dimensional gravitational trace anomaly [11] while the second and third terms are present even in flat space and arise from the self-interaction. The fl functions in the tree approximation (X = 2) have been calcu- lated by Callan et al. [12]. They find

16~rZfl * = ( d - 2 6 ) / 3 a '

+ [ 4 ( O e O ) 2 - 4 0 ~ - k + ~ [ - I 2 ] + O ( a ' ) , (4)

fl~tN = R MN - ¼ :-IMeQf-INeQ + 2~ MO N~O

(5) fl~tN = ~TPI?-IeMN - 2[-IeMuOeq ' + O( a'), (6)

where [-IMN e = 3 O[MBNp I and satisfies

OtQ~.NPl=O. (7)

The absence of a trace anomaly, i e. the Vanlsinng of fl*, fig and fls, is therefore just equivalent to the Einstein-matter-field equations obtained from the effective action

r - f dax fS~e-2*

× [ ( d - 2 6 ) / 3 a ' - R - 4 ( 0 ~ ) 2 + ~ a 2

(8) One obvious solution of the field equations

corresponds to ( ~ ) = constant, (gMN) = ~IMN and (HMNP) = 0, but tins is valid only for d = 26. For d > 26, the cosmological term in (4) obhges us to look for solutions in which some of the dimensions are compactlfied and we can now follow the traditional Kaluza-Klein interpretation. Accord- lngly we spht the mdices X M = (X p, ym) where x ' (bt = 1 . . . . . n) refers to spacetime and ym (m = 1 . . . . . k) refers to the extra dimensions. One solution which suggests itself corresponds to the case where the extra &menslons are a group manifold G with k = dim G. In this case

( g m n ( X , Y ) ) = g m n ( Y )

= (1/C^)cmpqCf q, (9)

where Cmn p are thd structure constants of the group, converted to world mdmes, and C A is the second-order Casirmr m the adjomt representa- tion. This will indeed be a solution provided

( ~ ( x , y ) ) = constant, (10)

and

( f lmnp(X ,y ) )=Hmnp(y )=mCmnp , (11)

where m Is a constant with &mensions of mass. The left-mvariant Kalhng vector fields K 'm(y) are normalized by

gmnK'mKJ n = 8 'j, (12)

and satisfy

[K' , K:] = mc':kK k, (13)

where K ' = K ' m O m . With these normalizations Hm,,p corresponds to the parallelizing torsion .2.

,2 Setting H,.np - c,.np was first attempted (albeit unsuccess- fully) m ref [13]

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Volume 163B, number 5,6 PHYSCIS LEI"rERS 28 November 1985

The curvature is given by

( Rmnpq) __ I~,2~ t~ 1 4,,~ ~mnt'pql,

-~m CA(gin. ~,

( R k ) = ¼m2CAk,

( 1 4 )

(15)

(16)

where R k is the curvature scalar of the k-dimen- sxonal group mamfold. Substituting (9)-(11) into the field equations (4)-(7) we see that (5)-(7) are ldentmally satisfied while (4) yields

d - 26 = ka'm2CpJ2 + O(a'2). (17)

Our aim is to obtain the bosonic sector of the heterotm stnng m ten dimensions. Neglecting the higher derivative terms, this corresponds to the bosonic sector of the Chapline-Manton [14,15] N = 1 supergravity with fields q~(x), g~(x) , H~,~p(x) coupled to Yang-Mills fields A'~(x) with G = E 8 × E 8 or SO(32). This means that d ffi 506 and k = 496. We shall now write down a Kaluza- Klein ansatz which achieves this goal, but for the sake of generality we allow G and the dimensmns n and d to be arbitrary. For the scalar, we write

~ ( x , y ) -- constant + q,(x). (18)

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

^ A n ( 1 9 ) gMN = eM eN TAB,

and write

M'(x, y ) ffi ea~,(x)dx ~',

~ a ( x , y ) = e a m ( Y ) d y m

- m - l K ' a ( y ) A ' , ( x ) d x ~.

(20)

(21)

The quantifies q~(x), ea~(x) and A'~,(x) will be interpreted as the scalar, vxelbem and Yang-Mdls gauge potential in n-dimensional spacetime. We must also make an ansatz for HMNp(x, y). Experience with antisymmetric tensors in d = 11 supergravtty [1,16] suggests that this must also revolve the Yang-Mills fields as well as the d = 10

three-form H~,~(x). Accordingly, we wnte

H,o~,( x, y ) = H,¢v( x ), (22)

l~I, oc(x, y ) = - m - l F ' , , o ( x ) K ' c ( y ) , (23)

I-l , ,bc(x,y)=O, ttabc(X,y)=mCabc, (24,25)

where F' ,o(x ) is the Yang-Mills field strength

F'~,~ = O~,A'~ - O,A'~, + C~kAJ~Ak,. (26)

It lS important to note that (a, a) are tangent space indices and (#, m) are world indices and that Cab ~ ffi C,jkK~aKJbKk c.

Several comments are now in order. That eqs. (18)-(25) correspond to the correct Kaluza-Klem ansatz can only be justified a posterion. Ftrst they must be mathematically consistent and, as em- phasized elsewhere [1,5-8] this is a highly non-triv- ial requirement. A "consistent" ansatz is one for which all solutions of the n-dimensional theory are solutions of the origanal d-&menslonal theory. For a generic Kaluza-Klein theory with homogeneous extra dimensmns, a consistent truncation is achieved by an ansatz which retains all those fields and only those fields mvariant under a translUvely acting subgroup of the isometry group [6]. In particular, for group manifolds the gauge bosons are only those of G L and not those of the full lsometry group G L x G R [5]. In certain cases, ~t may not be necessary to retain all the G a smglets. Indeed, suprisingly, our ansatz (18)-(25) is consistent m spite of omitting the usual Kaluza-Klein scalars. Secondly, eqs. (18)-(25) should be physi- cally consistent m that we recover the desired n-dtmensxonal theory. In our case that means, for n ffi 10, that we recover the low energy hmat of the heterotic string [3], i.e. the Chapline-Manton N = 1 supergravity coupled to Yang-Mtlls with gauge group G including, of course, the correct Yang-Mil ls Chern-Simons terms [14,15].

Note that we have not specified the ground-state values of the spacetime fields 4,(x), g~,(x), H~,,o(x ) or A'~,(x). Since, by defimtmn, a "consistent" ansatz ~s one for which all solutmns of the n-dimensional field equatmns are solutmns of the original d-dimensional field equatmns, ~t is not necessary at this stage to single out any preferred configuration such as g~ ffi ~1~,~, q' = H~,,p = A'~ ffi O. (For example, there can be other solutions such as Calabx-Yau mamfolds.)

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We now substitute the ans~tze (18) to (25) into the field equations (4)-(7). Most of the relevant calculations may be found m appendix A of ref. [1]. The flg~ equatmn becomes

R~,~ - -114 0oi4 4- 2V~O~,

- m-2F'~,oF' f = 0 (27)

The fir. equation becomes

D o r , f - ~...1 ,~- 214..~o*F,_ 0. - 2r'~°Oo* = 0, (28)

where D o is the Yang-Mills covanant denvatlve Note that since our ansatz xs invanant under the transitive action of G a, all the y dependence of the ansatz disappears from the n-dlmenmonal equations. Note also that half the Yang-Mllls contribution to the energy-momentum tensor comes from the standard metnc ansatz and an equal contribution from the antzsymmetnc tensor ansatz ,3. The ~ , equation is satisfied identically. (This is the equation m which one nught have expected to encounter lnconmstencies by omitting the Kaluza-Kle ln scalars S 'J (x ) since m a genenc theory one obtains an equation of the form r-lS'J - F'~,~F j~'~. In our case, however, the inclu- sion of the Yang-Mills fields m the ansatz (23) for / t , ~ exactly cancels this right hand rode and hence S 'J may consistently be set to zero).

The fl~ equation becomes

von°~, ~ - 2H°~,~Oo, = 0, (29)

the tiff. equation simply reproduces (28), while the fl~. equation is satisfied identically.

Now we consider the 13' equation, which becomes

4O~,,O~', - 4123, - R + ~H~,.oH~"°

' - = 0 , ( 3 0 ) + 5m 2 F ' ~ F '~

after using (17) Finally, eq. (7) becomes

3 -2 , 0, (31) O[~H~oo] + ~m F [ ~ F ' p o ] =

or, in terms of dzfferentlal forms,

d H + m - 2 F ' A F ' = 0. (32)

,3 Smularly m d = 11 supergravtty [5] one quarter of the Yang-Mal ls stress tensor comes from the standard metric ansatz and three quarters from the ant lsymmetnc tensor a n s a t z

In the case n = 10, the reader wall no doubt recognize eqs. (27)-(32) as the bosonic sector of the Chaphne-Manton N = 1 supergrawty w~th couphng constant K coupled to N = 1 Yang-Mdls with coupling constant g, where we make the identification

g2 = r2m2. (33)

It IS perhaps remarkable that m (32) we correctly reproduce the Yang-Mllls Chern-Slmons term m n dimensions even though there was no such term in d dimenmons (The appearance of the F ' A F ' term on reducing from d to n &menslons is known m the mathematical literature as a transgression).

Eq. (33) is of the standard Kaluza-Klem form since m-1 Is just the radius of the extra dimensions As m d = 11 supergravlty, however, the numencal coefficient will differ from that obtained m pure gravity Kaluza-Klein theories [17] owing to the Yang-Mdls content of the antlsymmetrlc tensor ansatz [16].

If, on the other hand, we compare eqs. (27)-(32) with the low energy hnut ( ~< 2 derivatives) of the heterotlc stnng [12], we find

m2ot ' = 1. (34)

Hence we recover the well-known relatxon [3] between K, g and a ' of the heterot~c string

~,g2 = ~2. (35)

One major &fference from genenc Kaluza- Klein compactlfications Is that we can obtain an n-dImenmonal theory with zero cosmological constant. The particular ~ dependence of the action (8) ensures that the Einstein equation (5) contains no cosmological term, while the cosmo- logacal term m the d-&menslonal ~ equatmn cancels out in the n-dImensmnal , equatmn.

By ignonng the O(a ' ) terms m the d-dimenmonal equatmns we have obtained in tins paper the Chapl ine-Manton ten-dimenmonal equations including the Yang-Mills Chern-Stmons term. In a subsequent paper we shall show that including

^ ^

t h e o l t R M N p Q R M N P Q Jr • • • t e r m s i n d-dimensions, we can obtain the modified field equations of Candelas et al. [4] m ten &mensions with their a ' ( R~,~o,,R~'"°" - F'~,~F"") terms. This lagranglan

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Volume 163B, number 5,6 PHYSCIS LETTERS 28 November 1985

will contain not only the Chapline-Manton Yang-Mil ls Chern-Slmons term but also the Green-Schwarz [2] gravitatmnal Chern-Simons term ensuring both Yang-MiUs and grawtatlonal anomaly cancellauons m the case G = Es × E8 or SO(32); i.e. eq. (32) becomes

d H + a ' ( F ' A F ' - R ab A R ab) = O. (36)

One is then m a position to seek a second compactxfication from n = 10 to n = 4 of the kind discussed by Candelas et al. [4]. If so desired, of course, one could descend directly from d = 506 to n = 4 in winch case the embedding of the Yang-Malls potential m the spin connecuon [18] employed by Candelas et al. [4] could be reinter- preted as a purely gravatational phenomenon Since, m our compacUficatlon from d = 506 to n = 10 the H,,n; background was non-tnvaal, moreover, there seems to be no compelhng reason to insist that ~t be zero in going from n = 10 to n = 4. An interesting question IS whether one can attach physical significance to compacUficaUons winch go directly from d = 506 to n = 4 and winch do not admit any mtermedmte n = 10 interpretation. The answer to this question would seem to require a better understanding of how the ferm- ions are incorporated.

It is mterestmg to note that whereas m d dimensions the a ' expansion corresponds to an expansion in numbers of derivatives (Le. zero slope hmit = low energy hrmt), in n dimensions this correspondence breaks down. For example, m the present paper we obtain a ' F ' ~ , , F '~'~

terms in n dimensions even though we ignored ^ ^

a ' R M l v e Q R ~ m ' Q terms in d dimensions. In fact we conjecture that the group manifold with paraUehz- mg torsion solves the field equations to all orders m a ' and that the Kaluza-Klem interpretation apphes to the full string theory and not merely to its field theory lunit.

All tins, of course, is entirely consistent with prexaous work on strings in curved space and non-linear o models [19-24]. The lagrangtan (2) with the fields set equal to their background values, Le. q, = constant;^~,,, the metnc on the group manifold; and H,,,,,p the parallehzang torsion as nothing but the non-linear o model wxth

Wess-Zununo term. According to Witten [20], tins model has vamshing fl functions provaded a ' m 2 = 1, winch is just eq. (34). Furthermore, the critical dimension of these theories is given by [23,24]

d - 26 = k C A / ( 2 + CA) (37)

(e.g. with G = SO(32), k -- 496, C A -- 60 and hence d = 506 and n = 10). On the other hand, our equaUon (17) yields

d - 2 6 = ½ k C A + . . . , (38)

on using (34). Tins suggests that solving the fl* equation to all orders in a ' would yield the exact equation (37).

Note, however, that the ordinary bosonic stnng formulated on the group manifold G would yield states belongang to representations of the full lsometry group G L )< G a m contrast to the heterotlc string where the states belong only to representations of G L. In our Kaluza-Klem formulation tins is taken care of by the consistency of the truncaUon in the Kaluza-Klem ansatz which demands that we retain only G a smglets.

In tins Kaluza-Klem picture, the E 8 × E s or SO(32) symmetry of the heterotlc stnng is a subgroup of the general coordinate group m d = 506. This is to be contrasted with the F renke l -Kac -Godda rd -Ohve [25] approach where the gauge symmetry emerges from com- pacUficauon of the d = 26 string on the maximal torus T 16 [26,3,27]. No doubt the two approaches are ultimately equivalent and it will be interestmg to establish the appropriate dictionary. In particular one can ask whether the n = 10 fermlons can be incorporated m any natural way into the d -- 506 bosomc theory. For example, one has the feeling that the gravitational, Yang-Mills and mixed anomaly cancellations of Green and Schwarz m n = 10 [2] may have a sLmpler pure gravity interpretation m d = 506.

We are grateful to C. Isham, D. Ohve, M. Perry and E. Sezgm for discussions, and to the organizers and participants of the Cambridge Workshop on Supersymmetry and its Applications, June-July 1985.

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Kaluza-Klein approach to the heterotic string