- Email: [email protected]
Superstring σ-models from bosonic ones
Volume 193, number 4
SUPERSTRING
PHYSICS LETTERS B
o-MODELS
23 July 1987
FROM BOSONIC ONES
A. CHAMSEDDINE 1.2,M.J. DUFF 3, B.E.W. NILSSON 4, C.N. POPE and D.A. ROSS 5 CERN, CH-1211 Geneva 23, Switzerland Received S May 1987
We consider the derivation of the background field heterotic u-model starting from the corresponding bosonic a-model compactilied on the group manifold G [G= E8x Es or Spin( 32)/&l. It is necessary not only to identify the SO( 8) c G, gauge potentials with the space-time SO( 8) spin connections, but also to identify the Kaluza-Klein scalars with the G, Yang-Mills field strengths. Similar remarks apply to Type II. This explains the origin of the four-fermion interactions on the world sheet.
It has been suggested that superstrings have their origin in the bosonic string [ l-51. In fact, it has now been proved that the heterotic and Type II superstrings are modular invariant truncations of the bosonic string [ 61. The aim of this paper is to derive the background-field lagrangian for the heterotic string in ten dimensions with background fields g,,, Bpy and A,$ L Heterotic=a+X~a-XY(g~,+B~,)+il”a+~a
G=EsxEs or corresponding to So(32), G= Spin( 32)/Z2 heterotic strings. The derivation of (1) from (2) proceeds as follows. The theory described by eq. (2) is known to be conformally invariant when gMNand BMNare chosen so that the lagrangian reduces to that of a non-linear rr-model on Mdx G with Wess-Zumino term [ 71, where Md is d-dimensional Minkowski space (d= D -dim G) and G is a simply-laced non-abelian group manifold with
+il”(w,“B+5H~ff~)128a+X~+iW’d_V/’ +i$A;Jy/Ja_
.P + 4 FafirJ~“2flv/‘v/J
,
(1)
starting from the lagrangian for the bosonic string in D dimensions with background fields gMNand 8,,
L=a+x”a_XN(g,~N+B,,).
(2)
We shall work in the light-cone formalism so that p, Y= 1, ...) 8 and M, N= 1, .... D-2. In (1) I,J= 1, .... 32, and the background Yang-Mills potentials A: take values in the Lie algebras SO(16) xSO(16) or ’ On leave of absence from the American University of Beirut, Beirut, Lebanon. 2 Present address: Theoretical Physics ETH-Hiinggerberg, CH8093 Zurich, Switzerland. 3 On leave of absence from Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK. 4 Present address: Institute for Theoretical Physics, CTH, S4 1296 Goteborg, Sweden. ’ Permanent address: Department of Physics, The University, Southampton SO9 SNH, UK.
444
D-26=dim
C, G .-2+c,
=dimG
- rankG,
(3)
where C, is the second-order Casimir in the adjoint representation. In the language of Kaluza-Klein, the Einstein-matter field equations for the metric ,&N and the antisymmetric tensor BMN (obtained by required vanishing trace of the world sheet stress tensor) admit a spontaneous compactitication on G from D to d dimensions. The massless states in d dimensions are given by the metric g,,, the antisymmetric tensor B,,, the Yang-Mills gauge potentials (Ah, 1;) of GL x G, and scalars S’j’ in the (adjoint GL, adjoint GR) representation of G,xG, [ 2,4]. Here GL and G, correspond to the left and right actions of G on the group manifold (i, i’ = 1, .... dim G). In principle, we now substitute the Kaluza-Klein ansatz for the above massless states into (2), noting
0370-2693/87/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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that the truncation of the massive Kaluza-Klein modes is, contrary to generic Kaluza-Klein theories [ 81, completely consistent [ 41. Unfortunately, although the ansatz for g,, B,,,,, Al and 2; is known exactly, the ansatz for P’ is known only to linearized order [ 41 in the case where G is non-abelian. To begin with, therefore, we shall see how far we can get by ignoring these Kaluza-Klein scalars and then return to the problem of including them. The ansatz for gMN is then [4] &“(X#) =g,“(X) + [Ah(X) L”‘+&(X) x [A:(X)
L”‘+J$
(X) R”“]g,,,,(y)
Rrnl’]
LETTERS
23 July 1987
B
where we have defined the “current” J$
=CMAak
xM )
(9)
and where O$$& is the spin connection with torsion ? +Ej,,, .
c?& = WABC ? 4Ei,,, ,
(10)
the connection one-form being L;)AB=~;)A&. Substituting the above ansatz into (9) we find the space-time current J$ =epaYa+Xp
,
(11)
and the internal current
&,,(KY)= [A;(X) L’“‘+~~(X)Rm”lg,n,(~) , &vdXY) =&n(Y) 3
(4)
&(X,y)=B,,(X)+[A;(X)L”‘~~(X)R~” L,,“&(X)
R”“]g,,,,,(y)
&(X,y)
= [A;(X)
L”“-&(X)
&,(XY)
=B,,&)
>
,
R”“]g,,(y)
=o,
, (5)
where g,, is the G, x G, invariant metric on G, B,, is given by H mnp=3~[mBnpl =G7np >
(6)
where c,,,,,~= L,,‘L,JLpk~,,k=R,“R,*‘Rpk’C,,,,k. and C,jk=C,‘j’k’ are the Structure Constants Of G. Here L,, ’ and R, I’ are respectively the left and right Killing vectors on G, satisfying [L’, Lj] = cfJkL k, [R”,
RJ’]
= -c”fk.Rk’,
[L’,
R”]
=I).
In fact, we find it more convenient to work in a vielbein formulation, for which &+= z,,,,AeNBLAB. In terms of the corresponding one-forms, the ansatz (4) becomes @“(X,y) =e,“(X) dXP , P(X,y) =e:,(y) dy” -A;(X) _&;,l’a
dXp ,
Lin dX” (7)
where o and a are space-time and internal tangent space indices, respectively. In vielbein formulation the bosonic equation of motion which follows from (2) is a+ J_, +O$;&Jf
J: =0 ,
(12)
The corresponding equations of motion from (8) are then
and for BMN,
-A:(X)
_A’L~~a+X~_A-i’R~‘~a~X~ P P
(8)
(13)
and a+J_,+w:&)JbJC+ +&~:bf,‘R,~‘JhJy+
+A;w:~‘LicJbJ: -F$,L,‘Jfl-J$
~0.
(14)
In terms of the G,_ and G, currents J’ sJ_,L’~,
J; =J,,R”“,
(15)
we have D+JL=Fb,J[J:,
D_J;=&JP+-JY,
(16)
where D+Jt =a+J’_ +C’,kA;d+xgJ”_ is the Yang-Mills covariant derivative, with a similar expression for D_ J'-; . The next step in the derivation of the superstring lagrangian (1) is to specialize to the cases G = Et,x E, or Spin(32)/Z2 for which dim G=496, rank G= 16 and CA= 60. Hence from ( 3) D=506,
d= 10.
(17)
We then make the group decomposition G, + SO( 8)
(18)
with a regular embedding, keeping only the SO(8) potentials & + A,- a$ , a$=
1, .... 8 ,
(19) 445
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PHYSICS
and identify these potentials with the space-time SO( 8) spin connection with torsion A’ @=,,,(+)afl /I li
(20)
.
The identification of spin-connections with gauge potentials [9] is, as discussed in ref. [4], just the field-theoretic realization of the identification of the transverse space-time SO( 8) of the superstring with the diagonal subgroup of the transverse space-time SO(8) of the compactified bosonic string and the internal SO( 8) c G, [ 31. The equations of motion (13) and (16) now become
+tF~,J/“J’:,+tR~~,+‘;‘“J;;_sJP=O,
(21)
and D +J!!=F#JcJ’ ,’
;,
D_J:p=R;,,apJ:Jt,
where we have specialized to the case of an orthogonal group with adjoint indices ZJ. This accounts for the factors of f in (2 1). This covers the case of SO(32) and the SO(16)xSO(16) subgroup of Es x Eg, and allows the theory to be cast into a more familiar form using a non-abelian fermionization of the currents J’i and JCYP. To see this, we now note that the equations of motion (21) and (22) are just equivalent to the equations of motion obtained by varying Lhelerotlcof (1) (excluding the four-fermion term!) provided we make the identifications Jyfl= daip
,
J/J =iy/‘v/J
,
(23)
and provided we take into account the fermion anomaly when computing D_ JTp and D, J’J. It is interesting to note that the “anomaly” terms on the right-hand side of (22) which arise from quantum a-model effects in the fermionized picture [ lo] are present already at the classical level in the bosonized picture. This is why, incidentally, one is able to derive the correct Yang-Mills and Lorentz Chern-Simons terms dH=Tr(Fr,F-R(+)
AR(+))
(24)
in the Kaluza-Klein approach to the heterotic string already at the classical level [ 2,4]. (The generalization of Witten’s non-abelian bosonization to the case of non-vanishing background fields and the 446
LETTERS
B
23 July I987
derivation of the anomalies has also been treated in ref. [ 111. ) Thus we have succeeded in deriving the heterotic string lagrangian (1) up to the four-fermion terms. So far, our analysis has been deficient in two respects: we have ignored the Kaluza-Klein scalars when compactifying the bosonic string on the group manifold and we have failed to account for the fourfermion interaction in the heterotic string lagrangian. We shall now argue that, despite appearances, the four-fermion problem is in fact solved by the inclusion of these Kaluza-Klein scalars. Our argument will be necessarily incomplete because, as we have already discussed, we do not know the exact non-linear ansatz for these scalars I’ and secondly because we do not yet know how to generalize Witten’s non-abelian bosonization to the case of nonvanishing background scalars. Nevertheless, the following argument seems plausible. After compactification, the bosonic string n-model will contain a term [&?,,,(XY)
+&,(~>Y)l~+Y’n~-Yn .
Assuming
a scalar ansatz of the form
&n(0)
(25)
+&&-,Y)
=g,,,(v)
+R,,&)
+L,,‘S”‘(X)
RR” ,
(26)
to describe the scalars S”‘(X) in the (adjoint G,_, adjoint CR) representation [4], we find the current-current interaction J’
S”‘JJ’
+
(27)
.
After the truncation becomes a J’JS’J”p _
of G, to SO( 8) as in (19), this
J@ + >
(28)
or, on using (23) _ t
,/,‘wJS’J*P~“;lP
.
(29)
Comparing (29) with the four-fermion term in (1) we obtain the desired result provided we identify the Kaluza-Klein scalars with the CL Yang-Mills field strength sfJ”p
=
_
2F
IJaP
.
(30)
” It should be borne in mind that the analogous problem for the S’ compactification ofd= 11 supergravity [ 121 took five years tosolve [ 131.
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Although the above argument is only hand-waving, the result of (30) may be substantiated by examining the bosonic string a-model p-functions pAe and Pa;* whose ab components yield the scalar field equation [4]
LETTERS
_ a y”y~/Bs”BV~~V~~
= 2F;,I’
Truncating o:J yields
y6J’+ higher order terms .
G, to SO( 8) and identifying
qSrJaP =2F;J,Ry6@
+ . .. .
.
Since A, Up has been identified we have from (30) S@Va
0s”
23 July 1987
B
=
_
2R
( -
(39) with wi-)@
)&‘a
in (34)
(40)
(31) 2:
with
and hence we recover the Type II cr model
(32)
On the other hand, -2nF’J*”
=2F:J,Ry*”
+ ... ,
(33)
on using the Bianchi identity and field equation for FrJffp. We see that (32) and (33) are entirely consistent with the identification (30). Although we have been focussing our attention on deriving the heterotic string rr-model from the bosonic one, the derivation of the Type II o-model goes through in much the same way. Now, however, we must make the SO( 8) decomposition with both G, and G,: G,-SO(8))
G,-+S0(8))
,L+A,@=W~-)~fi
)
1 d =j)(+mfl & +A,, P
.
The equations
(34)
of motion
(13) and (16) now become
a+J_,+o$;JtJ: +tR~p)VBJysJP++fR~~,I:)Y6JYtsJP
-
(35)
>
and D+J”P=R$;‘apJy
_ J”, ,
D_J”*=R;;‘*PJ:J:,
(36)
where the fermionized JTp = S”l+p ,
currents
J?=iy”t,@
are now
.
(37)
Note that there are still different anomaly terms on the right-hand sides of (36) even in the Type II case, and hence non-zero contributions to the Chern-Simons term [ 4,141 dH=Tr(R(-)
AR(-)
The Kaluza-Klein gian is now
-R(+)
AR(+))
scalar contribution
.
(38)
+$R’-‘“~‘“~“~P,/,VV/~.
(41)
The virtue in deriving the heterotic and Type II gmodels from the bosonic ones, is that the previous derivation of heterotic and Type II p-functions from the bosonic ones to lowest order in CX’[ 2,4], will now carry through to all orders in (Y’ . (Since these j&functions are scheme-dependent, one can expect equivalence only when the same renormalization scheme is used in each case [ 15,161.) In particular, this would resolve a previously outstanding problem concerning the ( curvature)3 terms in the p function. Briefly, the problem was as follows. The /Lfunctions of the bosonic string are known [ 171 to receive contributions cubic in the Riemann tensor (three loops in @, or four loops in p”). For example, the scalar invariants RpupARPAlor R,, flu and RwvA:Rpaar R, p r y are both known to appear. On the other hand, these terms are absent for both the heterotic and Type II strings on the grounds of supersymmetry [ 171. At first sight, this presents a paradox because after Kaluza-Klein compactitication of the bosonic string such terms would be expected to survive even after identifying gauge potentials with spin connections #*. Once we allow the presence of Kaluza-Klein scalars, however, the situation changes. Using the identifications (30) and (40), one sees how the required cancellations could in principle come about. Of course, once the equivalence of the theories at the level of the omodel has been established, there is no need to sweat the details. Finally, nothing in our derivation of superstring omodel from bosonic ones described in this paper prevents us from extending it to dimensions less than
to the lagranp2We are grateful for correspondence
with C. Nlifiez on this point.
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PHYSICS
In particular from (3), we could contemplate compactification on group manifolds for which ten.
rank G=22,
d=4.
(42)
This might, in fact, prove the most efficient way of constructing the non-linear cr-models corresponding to the recently constructed four-dimensional string theories. We strongly suspect that the connection between Kaluza-Klein scalars and four-fermion terms discussed in this paper is related to the work on Thirring strings discussed recently [ 181. We are grateful for conversations
with S. Ferrara.
References [I] P.G.O. Freund, Phys. Lett. B 151 (1985) 387. [ 21 M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Lett. B 163 (1985) 343. [3] A. Casher, F. Englert, H. Nicolai and A. Taormina, Phys. Lett. B 162 (1985) 121. [4] M.J. Duff, B.E.W. Nilsson, C.N. Pope and N.P. Warner, Phys. Lett. B 171 (1986) 170. [ 51 F. Englert, H. Nicolai and A. Schellekens, Nucl. Phys. B 274 (1986) 315. [ 61 W. Lerche, D. Lust and A. Schellekens, Phys. Lett. B 181 (1986) 71.
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23 July 1987
[ 71 E. Witten, Commun. Math. Phys. 92 (1984) 455; D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 54 (1984) 620; P. Goddard and D. Olive, Nucl. Phys. B 257 (1985) 226; S. Jain, R. Shankar and S.R. Wadia, Phys. Rev. D 32 (1985) 2713; E. Bergshoeff, S. Randjbar-Daemi, A. Salam, H. Sarmadi and E. Sezgin, Nucl. Phys. B 269 (1986) 77. [8] M.J. Duff, B.E.W. Nilsson, C.N. Pope and N.P. Warner, Phys. Lett. B 149 (1984) 90. [9] J.M. Charap and M.J. Duff, Phys. Lett. B 69 (1977) 445. [IO] C.M. Hull and E. Witten, Phys. Lett. B 160 (1985) 398. [ 111 K. Kitakaze and J. Masukawa, preprint OUAM 86-5-10 (1986). [ 121 M.J. Duff and C.N. Pope, in: Supersymmetry and supergravity ‘82 (World Scientific, Singapore, 1983). [13]B.deWitandH.Nicolai,Nucl.Phys.B281 (1987)211. [ 141 D.A. Ross, CERN preprint TH. 45 16 (1986). [ 151 M.D. Freeman, C.M. Hull, C.N. Pope and K.S. Stelle, Phys. Lett. B 185 (1987) 351. [ 161 M.D. Freeman, C.N. Pope, M.F. Sohnius and K.S. Stelle, CERN preprint TH. 4632 (1987), in: Proc. Colloq. on Strings and gravity (Meudon, September 1986), to be published. [ 171 R.R. Metsaev and A.A. Tseytlin, Curvature cubed terms in string theory effective actions, Lebedev preprint (1986). [ 181 J. Bagger, D. Nemeschansky, N. Seiberg and S. Yankielowicz, preprint HUTP-86/A088 (1986).
SUPERSTRING
PHYSICS LETTERS B
o-MODELS
23 July 1987
FROM BOSONIC ONES
A. CHAMSEDDINE 1.2,M.J. DUFF 3, B.E.W. NILSSON 4, C.N. POPE and D.A. ROSS 5 CERN, CH-1211 Geneva 23, Switzerland Received S May 1987
We consider the derivation of the background field heterotic u-model starting from the corresponding bosonic a-model compactilied on the group manifold G [G= E8x Es or Spin( 32)/&l. It is necessary not only to identify the SO( 8) c G, gauge potentials with the space-time SO( 8) spin connections, but also to identify the Kaluza-Klein scalars with the G, Yang-Mills field strengths. Similar remarks apply to Type II. This explains the origin of the four-fermion interactions on the world sheet.
It has been suggested that superstrings have their origin in the bosonic string [ l-51. In fact, it has now been proved that the heterotic and Type II superstrings are modular invariant truncations of the bosonic string [ 61. The aim of this paper is to derive the background-field lagrangian for the heterotic string in ten dimensions with background fields g,,, Bpy and A,$ L Heterotic=a+X~a-XY(g~,+B~,)+il”a+~a
G=EsxEs or corresponding to So(32), G= Spin( 32)/Z2 heterotic strings. The derivation of (1) from (2) proceeds as follows. The theory described by eq. (2) is known to be conformally invariant when gMNand BMNare chosen so that the lagrangian reduces to that of a non-linear rr-model on Mdx G with Wess-Zumino term [ 71, where Md is d-dimensional Minkowski space (d= D -dim G) and G is a simply-laced non-abelian group manifold with
+il”(w,“B+5H~ff~)128a+X~+iW’d_V/’ +i$A;Jy/Ja_
.P + 4 FafirJ~“2flv/‘v/J
,
(1)
starting from the lagrangian for the bosonic string in D dimensions with background fields gMNand 8,,
L=a+x”a_XN(g,~N+B,,).
(2)
We shall work in the light-cone formalism so that p, Y= 1, ...) 8 and M, N= 1, .... D-2. In (1) I,J= 1, .... 32, and the background Yang-Mills potentials A: take values in the Lie algebras SO(16) xSO(16) or ’ On leave of absence from the American University of Beirut, Beirut, Lebanon. 2 Present address: Theoretical Physics ETH-Hiinggerberg, CH8093 Zurich, Switzerland. 3 On leave of absence from Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK. 4 Present address: Institute for Theoretical Physics, CTH, S4 1296 Goteborg, Sweden. ’ Permanent address: Department of Physics, The University, Southampton SO9 SNH, UK.
444
D-26=dim
C, G .-2+c,
=dimG
- rankG,
(3)
where C, is the second-order Casimir in the adjoint representation. In the language of Kaluza-Klein, the Einstein-matter field equations for the metric ,&N and the antisymmetric tensor BMN (obtained by required vanishing trace of the world sheet stress tensor) admit a spontaneous compactitication on G from D to d dimensions. The massless states in d dimensions are given by the metric g,,, the antisymmetric tensor B,,, the Yang-Mills gauge potentials (Ah, 1;) of GL x G, and scalars S’j’ in the (adjoint GL, adjoint GR) representation of G,xG, [ 2,4]. Here GL and G, correspond to the left and right actions of G on the group manifold (i, i’ = 1, .... dim G). In principle, we now substitute the Kaluza-Klein ansatz for the above massless states into (2), noting
0370-2693/87/S 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume
193, number
PHYSICS
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that the truncation of the massive Kaluza-Klein modes is, contrary to generic Kaluza-Klein theories [ 81, completely consistent [ 41. Unfortunately, although the ansatz for g,, B,,,,, Al and 2; is known exactly, the ansatz for P’ is known only to linearized order [ 41 in the case where G is non-abelian. To begin with, therefore, we shall see how far we can get by ignoring these Kaluza-Klein scalars and then return to the problem of including them. The ansatz for gMN is then [4] &“(X#) =g,“(X) + [Ah(X) L”‘+&(X) x [A:(X)
L”‘+J$
(X) R”“]g,,,,(y)
Rrnl’]
LETTERS
23 July 1987
B
where we have defined the “current” J$
=CMAak
xM )
(9)
and where O$$& is the spin connection with torsion ? +Ej,,, .
c?& = WABC ? 4Ei,,, ,
(10)
the connection one-form being L;)AB=~;)A&. Substituting the above ansatz into (9) we find the space-time current J$ =epaYa+Xp
,
(11)
and the internal current
&,,(KY)= [A;(X) L’“‘+~~(X)Rm”lg,n,(~) , &vdXY) =&n(Y) 3
(4)
&(X,y)=B,,(X)+[A;(X)L”‘~~(X)R~” L,,“&(X)
R”“]g,,,,,(y)
&(X,y)
= [A;(X)
L”“-&(X)
&,(XY)
=B,,&)
>
,
R”“]g,,(y)
=o,
, (5)
where g,, is the G, x G, invariant metric on G, B,, is given by H mnp=3~[mBnpl =G7np >
(6)
where c,,,,,~= L,,‘L,JLpk~,,k=R,“R,*‘Rpk’C,,,,k. and C,jk=C,‘j’k’ are the Structure Constants Of G. Here L,, ’ and R, I’ are respectively the left and right Killing vectors on G, satisfying [L’, Lj] = cfJkL k, [R”,
RJ’]
= -c”fk.Rk’,
[L’,
R”]
=I).
In fact, we find it more convenient to work in a vielbein formulation, for which &+= z,,,,AeNBLAB. In terms of the corresponding one-forms, the ansatz (4) becomes @“(X,y) =e,“(X) dXP , P(X,y) =e:,(y) dy” -A;(X) _&;,l’a
dXp ,
Lin dX” (7)
where o and a are space-time and internal tangent space indices, respectively. In vielbein formulation the bosonic equation of motion which follows from (2) is a+ J_, +O$;&Jf
J: =0 ,
(12)
The corresponding equations of motion from (8) are then
and for BMN,
-A:(X)
_A’L~~a+X~_A-i’R~‘~a~X~ P P
(8)
(13)
and a+J_,+w:&)JbJC+ +&~:bf,‘R,~‘JhJy+
+A;w:~‘LicJbJ: -F$,L,‘Jfl-J$
~0.
(14)
In terms of the G,_ and G, currents J’ sJ_,L’~,
J; =J,,R”“,
(15)
we have D+JL=Fb,J[J:,
D_J;=&JP+-JY,
(16)
where D+Jt =a+J’_ +C’,kA;d+xgJ”_ is the Yang-Mills covariant derivative, with a similar expression for D_ J'-; . The next step in the derivation of the superstring lagrangian (1) is to specialize to the cases G = Et,x E, or Spin(32)/Z2 for which dim G=496, rank G= 16 and CA= 60. Hence from ( 3) D=506,
d= 10.
(17)
We then make the group decomposition G, + SO( 8)
(18)
with a regular embedding, keeping only the SO(8) potentials & + A,- a$ , a$=
1, .... 8 ,
(19) 445
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PHYSICS
and identify these potentials with the space-time SO( 8) spin connection with torsion A’ @=,,,(+)afl /I li
(20)
.
The identification of spin-connections with gauge potentials [9] is, as discussed in ref. [4], just the field-theoretic realization of the identification of the transverse space-time SO( 8) of the superstring with the diagonal subgroup of the transverse space-time SO(8) of the compactified bosonic string and the internal SO( 8) c G, [ 31. The equations of motion (13) and (16) now become
+tF~,J/“J’:,+tR~~,+‘;‘“J;;_sJP=O,
(21)
and D +J!!=F#JcJ’ ,’
;,
D_J:p=R;,,apJ:Jt,
where we have specialized to the case of an orthogonal group with adjoint indices ZJ. This accounts for the factors of f in (2 1). This covers the case of SO(32) and the SO(16)xSO(16) subgroup of Es x Eg, and allows the theory to be cast into a more familiar form using a non-abelian fermionization of the currents J’i and JCYP. To see this, we now note that the equations of motion (21) and (22) are just equivalent to the equations of motion obtained by varying Lhelerotlcof (1) (excluding the four-fermion term!) provided we make the identifications Jyfl= daip
,
J/J =iy/‘v/J
,
(23)
and provided we take into account the fermion anomaly when computing D_ JTp and D, J’J. It is interesting to note that the “anomaly” terms on the right-hand side of (22) which arise from quantum a-model effects in the fermionized picture [ lo] are present already at the classical level in the bosonized picture. This is why, incidentally, one is able to derive the correct Yang-Mills and Lorentz Chern-Simons terms dH=Tr(Fr,F-R(+)
AR(+))
(24)
in the Kaluza-Klein approach to the heterotic string already at the classical level [ 2,4]. (The generalization of Witten’s non-abelian bosonization to the case of non-vanishing background fields and the 446
LETTERS
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23 July I987
derivation of the anomalies has also been treated in ref. [ 111. ) Thus we have succeeded in deriving the heterotic string lagrangian (1) up to the four-fermion terms. So far, our analysis has been deficient in two respects: we have ignored the Kaluza-Klein scalars when compactifying the bosonic string on the group manifold and we have failed to account for the fourfermion interaction in the heterotic string lagrangian. We shall now argue that, despite appearances, the four-fermion problem is in fact solved by the inclusion of these Kaluza-Klein scalars. Our argument will be necessarily incomplete because, as we have already discussed, we do not know the exact non-linear ansatz for these scalars I’ and secondly because we do not yet know how to generalize Witten’s non-abelian bosonization to the case of nonvanishing background scalars. Nevertheless, the following argument seems plausible. After compactification, the bosonic string n-model will contain a term [&?,,,(XY)
+&,(~>Y)l~+Y’n~-Yn .
Assuming
a scalar ansatz of the form
&n(0)
(25)
+&&-,Y)
=g,,,(v)
+R,,&)
+L,,‘S”‘(X)
RR” ,
(26)
to describe the scalars S”‘(X) in the (adjoint G,_, adjoint CR) representation [4], we find the current-current interaction J’
S”‘JJ’
+
(27)
.
After the truncation becomes a J’JS’J”p _
of G, to SO( 8) as in (19), this
J@ + >
(28)
or, on using (23) _ t
,/,‘wJS’J*P~“;lP
.
(29)
Comparing (29) with the four-fermion term in (1) we obtain the desired result provided we identify the Kaluza-Klein scalars with the CL Yang-Mills field strength sfJ”p
=
_
2F
IJaP
.
(30)
” It should be borne in mind that the analogous problem for the S’ compactification ofd= 11 supergravity [ 121 took five years tosolve [ 131.
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Although the above argument is only hand-waving, the result of (30) may be substantiated by examining the bosonic string a-model p-functions pAe and Pa;* whose ab components yield the scalar field equation [4]
LETTERS
_ a y”y~/Bs”BV~~V~~
= 2F;,I’
Truncating o:J yields
y6J’+ higher order terms .
G, to SO( 8) and identifying
qSrJaP =2F;J,Ry6@
+ . .. .
.
Since A, Up has been identified we have from (30) S@Va
0s”
23 July 1987
B
=
_
2R
( -
(39) with wi-)@
)&‘a
in (34)
(40)
(31) 2:
with
and hence we recover the Type II cr model
(32)
On the other hand, -2nF’J*”
=2F:J,Ry*”
+ ... ,
(33)
on using the Bianchi identity and field equation for FrJffp. We see that (32) and (33) are entirely consistent with the identification (30). Although we have been focussing our attention on deriving the heterotic string rr-model from the bosonic one, the derivation of the Type II o-model goes through in much the same way. Now, however, we must make the SO( 8) decomposition with both G, and G,: G,-SO(8))
G,-+S0(8))
,L+A,@=W~-)~fi
)
1 d =j)(+mfl & +A,, P
.
The equations
(34)
of motion
(13) and (16) now become
a+J_,+o$;JtJ: +tR~p)VBJysJP++fR~~,I:)Y6JYtsJP
-
(35)
>
and D+J”P=R$;‘apJy
_ J”, ,
D_J”*=R;;‘*PJ:J:,
(36)
where the fermionized JTp = S”l+p ,
currents
J?=iy”t,@
are now
.
(37)
Note that there are still different anomaly terms on the right-hand sides of (36) even in the Type II case, and hence non-zero contributions to the Chern-Simons term [ 4,141 dH=Tr(R(-)
AR(-)
The Kaluza-Klein gian is now
-R(+)
AR(+))
scalar contribution
.
(38)
+$R’-‘“~‘“~“~P,/,VV/~.
(41)
The virtue in deriving the heterotic and Type II gmodels from the bosonic ones, is that the previous derivation of heterotic and Type II p-functions from the bosonic ones to lowest order in CX’[ 2,4], will now carry through to all orders in (Y’ . (Since these j&functions are scheme-dependent, one can expect equivalence only when the same renormalization scheme is used in each case [ 15,161.) In particular, this would resolve a previously outstanding problem concerning the ( curvature)3 terms in the p function. Briefly, the problem was as follows. The /Lfunctions of the bosonic string are known [ 171 to receive contributions cubic in the Riemann tensor (three loops in @, or four loops in p”). For example, the scalar invariants RpupARPAlor R,, flu and RwvA:Rpaar R, p r y are both known to appear. On the other hand, these terms are absent for both the heterotic and Type II strings on the grounds of supersymmetry [ 171. At first sight, this presents a paradox because after Kaluza-Klein compactitication of the bosonic string such terms would be expected to survive even after identifying gauge potentials with spin connections #*. Once we allow the presence of Kaluza-Klein scalars, however, the situation changes. Using the identifications (30) and (40), one sees how the required cancellations could in principle come about. Of course, once the equivalence of the theories at the level of the omodel has been established, there is no need to sweat the details. Finally, nothing in our derivation of superstring omodel from bosonic ones described in this paper prevents us from extending it to dimensions less than
to the lagranp2We are grateful for correspondence
with C. Nlifiez on this point.
447
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PHYSICS
In particular from (3), we could contemplate compactification on group manifolds for which ten.
rank G=22,
d=4.
(42)
This might, in fact, prove the most efficient way of constructing the non-linear cr-models corresponding to the recently constructed four-dimensional string theories. We strongly suspect that the connection between Kaluza-Klein scalars and four-fermion terms discussed in this paper is related to the work on Thirring strings discussed recently [ 181. We are grateful for conversations
with S. Ferrara.
References [I] P.G.O. Freund, Phys. Lett. B 151 (1985) 387. [ 21 M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Lett. B 163 (1985) 343. [3] A. Casher, F. Englert, H. Nicolai and A. Taormina, Phys. Lett. B 162 (1985) 121. [4] M.J. Duff, B.E.W. Nilsson, C.N. Pope and N.P. Warner, Phys. Lett. B 171 (1986) 170. [ 51 F. Englert, H. Nicolai and A. Schellekens, Nucl. Phys. B 274 (1986) 315. [ 61 W. Lerche, D. Lust and A. Schellekens, Phys. Lett. B 181 (1986) 71.
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LETTERS
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23 July 1987
[ 71 E. Witten, Commun. Math. Phys. 92 (1984) 455; D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 54 (1984) 620; P. Goddard and D. Olive, Nucl. Phys. B 257 (1985) 226; S. Jain, R. Shankar and S.R. Wadia, Phys. Rev. D 32 (1985) 2713; E. Bergshoeff, S. Randjbar-Daemi, A. Salam, H. Sarmadi and E. Sezgin, Nucl. Phys. B 269 (1986) 77. [8] M.J. Duff, B.E.W. Nilsson, C.N. Pope and N.P. Warner, Phys. Lett. B 149 (1984) 90. [9] J.M. Charap and M.J. Duff, Phys. Lett. B 69 (1977) 445. [IO] C.M. Hull and E. Witten, Phys. Lett. B 160 (1985) 398. [ 111 K. Kitakaze and J. Masukawa, preprint OUAM 86-5-10 (1986). [ 121 M.J. Duff and C.N. Pope, in: Supersymmetry and supergravity ‘82 (World Scientific, Singapore, 1983). [13]B.deWitandH.Nicolai,Nucl.Phys.B281 (1987)211. [ 141 D.A. Ross, CERN preprint TH. 45 16 (1986). [ 151 M.D. Freeman, C.M. Hull, C.N. Pope and K.S. Stelle, Phys. Lett. B 185 (1987) 351. [ 161 M.D. Freeman, C.N. Pope, M.F. Sohnius and K.S. Stelle, CERN preprint TH. 4632 (1987), in: Proc. Colloq. on Strings and gravity (Meudon, September 1986), to be published. [ 171 R.R. Metsaev and A.A. Tseytlin, Curvature cubed terms in string theory effective actions, Lebedev preprint (1986). [ 181 J. Bagger, D. Nemeschansky, N. Seiberg and S. Yankielowicz, preprint HUTP-86/A088 (1986).