Exactly solvable string compactification on Calabi-Yau manifolds in the Green-Schwarz formalism

Volume 224, number 3 PHYSICS LETTERS B 29 June 1989

EXACTLY SOLVABLE STRING C O M P A C T I F I C A T I O N ON CALABI-YAU M A N I F O L D S IN THE GREEN-SCHWARZ F O R M A L I S M

Ashoke SEN Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, India

Received 2 February 1989

We show how the superstring and heterotic string compactification schemes proposed by Gepner using minimal N= 2 superconformal field theories may be formulated in the Green-Schwarz formalism. A modular invariant partition function is constructed in this scheme. All the physical states come from the Ramond sector. Two of the four generators of N= 1 spacetime supersymmetry transformations are given by the generators of world-sheet supersymmetry, the other two are given by free fermion zero-modes.

Most of the recent studies o f type II superstring and the heterotic string theory have been made in the Neveu -Schwarz -Ramond (NSR) formalism [1]. The alternative formulation, known as the Green- Schwarz (GS) formulation [ 2 ], has remained mostly in the background for several reasons. There is no simple manifestly covariant quantum version o f this theory; this makes it very difficult to do perturbation theory in this scheme beyond one loop order. (Re- cently, some progress in this direction has been made [ 3 ]. ) Also it is very complicated to couple this theory to a general background field [ 4 ], unlike the theory in the NSR formulation [ 5 ].

However, it is possible to couple the Green- Schwarz light-cone gauge theory to a special class o f background fields, those which correspond to spacetime supersymmetric compactification of the string theory [6 -10] . Such theories are known to have ex- tended world-sheet superconformal symmetry [ 11,9,12 ]. These include Calabi-Yau manifolds, for example. In this scheme some of the spacetime supersymmetry generators coincide with the world-sheet supersymmetry generators; other spacetime supersymmetry generators are given by free fermion zero- modes [9]. This gives us a simple description of spacetime supersymmetry.

Recently, Gepner [12] has constructed exactly solvable spacetime supersymmetric compactification of type II superstring and heterotic string theo-

ries to four dimensions. He combined several copies o f the minimal N = 2 superconformal field theories [ 13 ] with total central charge 9, with a theory of two free bosons and two free fermions (representing the two transverse directions in the light-cone gauge), and obtained a modular invariant, spacetime supersymmetric spectrum. Subsequently it was realised [ 14- 16 ] that these models describe string compactification on Calabi-Yau manifolds with some specific values o f the moduli. The purpose o f this paper is to show how one can describe string propagation on such manifolds in the light-cone gauge Green-Schwarz formulation.

We start with a brief review of Gepner 's construction of spacetime supersymmetric compactification of heterotic string theories [ 12 ]. In this scheme one combines N copies o f minimal superconformal models o f levels kl, ..., kx to make a theory o f total central charge 9. This gives us the constraint

~ 3k, ,=l k , + ~ =9 . (1)

The characters of an N = 2 superconformal theory [ 12,17 ] o f level k may be characterized by three integers (l, q, s), where O<~l<~k, q is defined modulo 2 ( k + 2 ) , s is defined modulo 4, and l + q + s is re- stricted to be an even integer. Odd s refers to Ramond (R) sector states, and even s to Neveu-Schwarz (NS) sector states. Thus a general partition function o f the

278 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Publishing Divis ion)


combined theory in the light-cone gauge may be written as

y. N{ li, qi,si},s; { ~,Cll,gt },f {h,q~,s, bs: { 7,,qi,g, },g

× (I~Ii )~{l,,q,,st}~,(li,St} )Os,20',2 ' ,l -2 (2)

ignoring the contribution from the free bosons. Here 0,,2 denote level 2 theta functions (see ref. [ 12] for definition), Os.2O*,2 / ] q 12 representing the contribution to the partition function from two real (one complex) fermions, s, Yare defined modulo 4. Again, even s corresponds to the Neveu-Schwarz sector, and odd s to the Ramond sector.

The main problem now is to find the numbers N{h.q,.s,},s;{h,4.~,},s which give a modular invariant partition function, and are consistent with spacetime supersymmetry. In Gepner 's construction, they are given by

Nih,q,,s,},s;{l,,cl,,e,},e= (i=I-Ii Nhh ) M{q,.s&s;{q,.s,}g , (3)

where N¢r is an affine A } ~ ) invariant, and is given by the A - D - E classification [ 18]. A simple choice is N~r=6~r. For consistent spacetime supersymmetric compactification of the type II superstring theory, the matrix M must be chosen so as to give a modular invariant partition function. Furthermore it must van- ish unless q~, s~, s, #,, g,, g satisfy the following conditions:

(i) I f we define the U( 1 ) charges Q (Q) of a state as

o,

i=1 k i + 2

i=l k i + 2

- - - + + ~ + 2 × integer,

- - - + + ~ + 2 × integer, (4)

then Q and Q must be separately odd integers. (ii) Either all si or s are odd, or all s~ and s are even.

Similarly either all g, and g are odd or all g~ and g are even. This implies that separately in the left and the right sector either all states belong to the Ramond sector, or all are in the Neveu-Schwarz sector.

It was shown by Gepner that a standard modular invariant choice for the matrix M may be made as follows. Let us define the vec tor / t a s / t = (qb "., qu,

sl, ..., SN, S ) , and define the inner product of two such vectors as

)ss /=1 2 (ki + 2 ~ + + ~ - . (5)

Let us now consider a set of vectors fl (~), y (k) with the restriction that all entries in p(~), y(k) are integers, y(o.f lo) , fl(i).y(k) and y(k).y(t) are all integers, (flo))2 are odd integers and (y(k))2 are even integers. Let/to be a vector satisfying

/to" fl (i) = integer + ½,

/ to 'Y(k)=integer Vi and k. (6)

Then the following choice of M gives a modular invariant partition function:

M.,,. = E E ( - l) ~''+~"' .#0 {ni,mk,n,,mk }

X (~ij,laO+ nifl(i) + inky(k)81a,laO+flifl(i) + mk~(k), ( 7 )

where the sum over n , ink, fii, r~k runs over integer values that give independent vectors o f the form /to+ n~fl(~) + mk~ (k) and /tO+ ~,fl(') + fftk~' (k), taking into account the fact that the q,'s are defined modulo 2(kg+2) and s, and s are defined modulo 4. The sum over/to runs over all possible values satisfying eq. (6), which are not related by addition of lattice vectors of the form ~f l (~) + rhky (k). Note that if/to satisfies eq. (6), so does any vector of the form /to + ~fl ( o + rhk~, (k). This implies that any/ t and fl for which M,,.~ is non-zero, satisfies

/t" fl (') = integer+ ~,

~.p(i~ = integer+ l,

V i and k.

/t- y (k) = integer,

/~. y (k) = integer

(8)

Gepner found suitable choice of the vectors flu) and y (k) which implements the requirements ( i ) and ( ii ) on M m e n t i o n e d before. His choice was

f l ( l ) = ( 1 , 1 . . . . , 1 ) - - - ~ ~°),

y ( O = ( q r = 0 Vr, s r = 0 Vr¢ i , s ,=2 , s=2)

---o~ (i) (1 ~<i~<N). (9)

One can easily verify that (fl ~ l ) ) z = 1 and (y u) ) 2 = 2. fl(l )./t = integer + ½ implies that

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= odd integer. (10)

This is precisely eq. (4). On the other hand, ? o)./t = integer Vi gives

st + s = e v e n ¥i. ( 11 )

This tells us that i f s is even (odd) then all other s, are also even (odd). Thus either all the states are from the Ramond sector, or all are f rom the Neveu- Schwarz sector. This implements the second constraint mentioned before. Similar constraints are satisfied by )t as well. Thus the partition function described by M given in eq. (7) describes a consistent spacetime supersymmetric superstring compactification in the NSR formulation. The states belonging to the N S - N S or R - R sectors correspond to spacetime bosons, whereas states belonging to R - N S or N S - R sectors correspond to spacetime fermions, and come with negative multiplicity due to the ( - 1 ) x,,+z~, factor in eq. (7).

It was also noted in ref. [ 12 ] that for a given superstring compactification described by the partition function (2), one can construct a heterotic string compactification by replacing the SO (2) characters Os*.z/r/* in the right moving sector by an SO(26) or SO (10) × E8 character with the following rules. The character associated with the vector representation of SO (2) is replaced by that associated with the singlet representation of SO(26) or S O ( 1 0 ) × E 8 and vice versa, whereas the characters associated with the spi- nor representations o f SO (2) are replaced by those associated with the spinor representations of SO (26) or SO (10) × Es with a change in sign. This was shown to give a modular invariant partition function. A state is spacetime fermion or boson depending on whether the left moving part o f the state belongs to the Ra- mond or Neveu-Schwarz sector.

This finishes our brief review of spacetime supersymmetric compactification of type II superstring and heterotic string theories in the NSR formalism. Let us now briefly recall the requirement of spacetime supersymmetric compactification in the Green- Schwarz formalism. The compactification o f heterotic string theory in this formalism was discussed in detail in ref. [ 9 ]. In this case, in order to get a consis-

tent Lorentz invariant theory, we need the compac- tiffed theory to have a (2, 0) supersymmetry algebra generated by the Virasoro generators Lm, two supersymmetry generators G + and the U ( 1 ) generator Jm, all with integer moding. (Note: These generators are associated with the internal part of the theory only, and does not include the part involving the free fields associated with uncompactif ied spacetime dimensions.) This immediately implies that all the states must belong to the Ramond sector. In order to construct the full set of Lorentz generators, however, one also needs two dimension 3 holomorphic fields P (z ) , /5(z) with integral mode expansion around any physical state: P(z)= ~Pn2 - n - 3/2, / 5 ( Z ) = Z/SnT. - n - 3/2,

with integer n, and Pn,/5, are required to satisfy cer- tain commutat ion relations with the generators tm, G + , Jm, and among themselves. These commutat ion relations are automatically satisfied by taking P,/5 of the form ~t,2

P ( z ) = U:exp (m~0 1 Jmz-m-Jolnz):,

/ 5 ( z ) = U * :exp ( -~m~o 1Jmz-m+J°Inz) : ' m (12)

provided there exists a unitary operator U acting on the physical Hilbert space o f the theory, and satisfying

U L m U - 1 = L m ..1_ Jm l + -~C~m,O,

UG,~ U -I =G~,+_I,

U J m U - l = J m -P l C(~m, O. (13)

(These equations differ from those of ref. [ 9] in the choice o f the normalization ofc. The present conven- tion is consistent with that o f ref. [ 12 ] which differs from that of ref. [ 9 ] by a multiplicative factor of 2. ) As can be seen from eq. (12), the requirement of integral mode expansion of P ( z ) immediately restricts us to the space of states carrying half integral U ( 1 ) charge Jo.

~ These fields are identical to the flow generators discussed in ref. [ 15 ]. An additive term of ~ was left out in the expression for {Po, Po} in ref. [9], but this does not change any of the results of this paper.

~2 For string theory in flat ten dimensional spacetime, construction of the generators of anomaly-free super-Poincar6 algebra in terms ofbosonic vertex operators has been discussed in ref. [19l.

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The analysis can easily be generalized to compactification of type II superstring theory as well. In this case the requirement of (2, 0) supersymmetry is replaced by that of (2, 2) supersymmetry, with both the right and the left handed supersymmetry generators required to be integrally moded. Besides the operators P(z) , P(z) , one now needs antiholomorphic

operators/~(g),/5(g), p, /5 have expressions similar to those for P and P in terms of the fight handed U ( 1 ) generators ]m and new operators U, U*.

As in the case of heterotic string compactification, the requirement of integral mode expansion of the superconformal generators and P(z), P(g) restricts us to states belonging to the Ramond sector in both, the left and the right hand sector, and half integral Jo, ]0 charges. Thus in order to find a modular invariant partition function describing compactified superstring theory in the Green-Schwarz formalism the matrix Mj,,~ in eq. (3) must be chosen such that Mu,a vanishes unless /J, fl satisfy the following conditions:

(i): a(int)_ ~ ( qi +2) = i = ] - ki+-----~

= integer+ ½,

( O(int) m i=1 ~ ki+2

=integer+ ½. (14)

Note that Q(int) and Q(int) denote the U( 1 ) charges coming from internal theories only.

(ii): sg=oddVi, s=odd,

g~=odd ¥i, g= odd, (15)

which guarantees that all states belong to the Ramond-Ramond sector.

It turns out that a consistent modular invariant form of M satisfying the requirements (i) and (ii) above may again be found using the "fl-method" de- veloped in ref. [ 12 ]. M is again expressed as in eq. (7) (thereby automatically guaranteeing modular invariance), with the set #(o, ~, (k) chosen as follows:

~(i)= (qr=0 Vr, &=0 Vrv~i, si =2, s = 0 )

___~(i) (1 ~<i~<N),

p(N+ 1)= (qr= 0 Vr, St=0 Vr, s = 2 ) ~(N+ 1),

y(~>= (2, 2, ..., 2 ) - ~ ( ° ) = 20~ <°). (16)

One can easily verify that with the above definitions (fl (,)) 2 = 1, whereas (y ( ~ ) ) 2 ___ 4. The allowed vectors Ito in the sum in eq. (7) now satisfy ~o-flt ')= integer + ½,/to" Y ( ~ > - integer. It is straightforward to see that all the vectors it, fl for which M~,.a is non-zero satisfy similar relations. The first relation immediately gives us eq. (15), while the second relation, combined with the fact that s,g are odd integers gives useq. (14).

This gives us the modular invariant partition function of the compactified heterotic string theory in the Green-Schwarz formalism. In order to show that the theory is SO (3, 1 ) invariant, we still need to show the existence of the operator U ((~) satisfying eq. ( 13 ). Since the (2, 2 ) superconformal generators of the internal conformal field theory may be expressed as sum of the generators of the individual superconformal field theories, eq. ( 13 ) is automatically satisfied if we can find a Usuch that

UL~ ) U - t = L ~ ) + J~) + ~cti)~m,o,

UGh) + - U-~--c:-(i)+- --urn+l, Uj~) U-~ = j ~ ) + ~c(i)O,,,o,

l<~i<~N, (17)

where L ~), etc. denote generators of N = 2 superconformal algebra for the ith theory. Similar relations hold in the right hand sector.

Now the primary states in the Ramond sector of a level k (2, 2) superconformal field theory is labelled by [l, q, s) where O<~l<~k, q and s are defined modulo 2 (k+ 2) and 4 respectively, s is odd, and l+ q+s is even. These states may be divided into two classes, those which are annihilated by GJ- and those which are annihilated by Go. It is easy to see that the second equation of ( 17 )implies that for a primary state II, q, s) satisfying G~ II, q, s) =0, Ull, q, s) is an- other primary state [l', q', s' ) satisfying Gff II', q', s ' ) =0. On the other hand, for a primary state II, q, s) satisfying G6- II, q, s) =0, Ull, q, s) is a secondary state of the form G 7_ t G~- I l', q', s' ), with II', q', s ' ) satisfying G~- II', q', s ' ) =0. Finally, for

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the Ramond sector ground states 1/o, qo, So) which are annihilated by both G~- and G~, U[lo, qo, So) is always a primary state.

It remains to determine I I', q', s ' ) in terms of I l, q, s) . To do this we note that [ 14 ]

Joll, q,s)

( q + s - k + 2 ) = ~ +2×in teger [l,q,s),

Loll, q,s)

( I ( l+2)-q2 Ts2 ) - \ 4 ( k + 2 ) + o +integer II, q,s). (18)

As a result, the following values of l', q', s' are consistent with the first and the third equations in (17) [e.g. U increases the Lo eigenvalue by - - Jo + g C= -- Jo + k~ 2 (k + 2 ), and the Jo eigenvalue by - ~c= - k~ (k+ 2); these follow from the equations U - 1 L m U = L m - J m + l c~m,o, U - I J m U = J m - ~ C ~ m , o ] :

l'=l, q '=q-2 , s '=s-2 . (19)

Thus we get the following action of U acting on the primary state [l, q, s) :

(i) i fG + II, q , s ) = 0 , Ull, q,s)= [l, q - 2 , s - 2 ) ; (ii) ifGff [l, q, s) =0, U[ l, q, s) =NG-_, GC

× I l, q - 2, s - 2 ) , where N is an appropriate normalization constant;

(iii) if II, q, s) denotes a Ramond vacuum state, U[I, q, s) = 1l, q-2, s - 2 ) .

The action of U on the secondary states may be obtained by knowing its action on the primary states and eqs. (17). Note that except for the factor of G- I G6-, the action of U is equivalent to taking a primary state labelled by the vector/ t to a state labelled by / t _ _ ~ ( 0 ) + ~ ( N + l ) with o2(')'s defined as in eq. ( 1 6 ) .

Before we proceed to discuss heterotic string compactification in the Green-Schwarz formalism, we shall discuss a related problem, namely a formulation of compactified type II superstring theory in which the left sector is treated in the Green-Schwarz formalism, but the right sector is treated in the Ramond-Neveu-Schwarz formalism. This is done by the following simple observation. Let us consider the matrix M defined for spaeetime supersymmetric compactification in the NSR formalism. It may be written as

MNS_n . , . = Z Z ( - 1 ) "'+"' #0 nl,nl,{mi,rTti}

X c~,~o+ nta¢(o) +mio~(i)(~l~,itO+~lO¢(o) +mioc(i),

(20)

where/to satisfies

/to- o¢ (°) = integer + ½,

Ito'O~ (i)=integer, 1 <~i<~N. (21)

Using the definition of ~ (i) given in eq. ( 16 ) we get o~(°=o2(°+~ (N+ ~). Using the second of eq. (21), and the fact that all entries in/to are integers, we get

/to "~(') =integer Vi, or integer+ ½ Vi,

l<~i<~N, (22)

depending on whether/to-O'~ ~N+~) is integer or half integer.

But now notice that in the sum in eq. (20) we can replace/to by/to+O~ ~°), since we are summing over all integer values of n~ anyway. This substitution shifts /to-O~ (°) by (~(o))2= 1, and #o "otu) by o¢ ~°).~(~)= ½. Hence by judicious choice of the representative/to in the sum in eq. (20) we can always restrict #o to satisfy

/to "o~°) =integer+ ½,

/to.a2(') =integer+ ½, l<~i<~N. (23)

On the other hand, the matrix M for the spacetime supersymmetric compactification in the Green- Schwarz formalism is given by

GS -- ) ~n, + Yni ,M,,,,~- Z Z ( - 1 liO ml,ml,{ni,& }

X ~/~,/~o+rn,a~(o, +n,&(il(~,&fio+rnl&(O)+,qi&(i,, ( 2 4 )

where ~o satisfies

~o" a~ ~o) = integer,

~o-@(° =integer+ ½, 1 ~<i~<N+ 1. (25)

The first ofeq. (25) gives

/io .0¢(°) =/2o- ½o2 (°) = integer or integer + ½. (26)

Since at (o).o~ ~) = ½ Vi, if/io .a~ (o~ is an integer, we can replace/io in the sum by ~o +o~ ¢°, for any i, thus en- suring that

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/lo" o~ co) = integer + ½,

~o-~ i )= in teger+½, 1 <~i<~N. (27)

Thus we see that both,/~o in eq. (24) and/to in eq. (20) may be taken to satisfy the same set of conditions given by eq. (23) or eq. (27). This still does not imply that in eqs. (20) and (24)/ to and ~o may be taken to run over the same set of values, but we shall now show that this is so. The problem is that in eq. (20) two vectors/to and/ t [ are considered equivalent if they differ by nl~ <°) + rni~ ~), whereas in eq. (24) they are considered equivalent if they differ by ml 2 (0) "~ni ~ ( i ) . Thus we now need to prove that for two vectors/to and/t~ satisfying eq. (23), these two equivalence relations are the same.

Let us take

N / t~=/to-Fnl ~ ° ) + ~ mioc ~i)

i=1

N =/ to+? / l~<° )+ ~ mi(~<i )+~(N+l ) ) . (28)

i = l

Now, since/to and/t~ both satisfy eq. (23), we have

(/t~ - / t o )"~ti)=integer, (29)

which gives

½nl +mi =integer Vi 1 4i<~N. (30)

Thus n~ must be even. Writing n~ = 2~1 we get

N / t ~ = / t O ' ~ ' ~ l ~ ( O ) ' ~ " E m i ( ~ ( i ) + ~ ( N + l ) ) , (31)

i=1

i.e./to and/t~ are equivalent in the sense in which the sum over/~o is performed in eq. (24).

To prove the converse, let

N + I /t 0 ------ /t 0 -FmI0"~(0)-F E n~ ~°)

i=l

N = / t o + 2 m l o ~ ° ) + ~ n ~ <~)

i = l

Again, using eq. (23) for/to and/t~, we get

(#~ -#o) '~<° )= in tege r , (33)

i . e .

2m,+ ~ni+½(nN+~-- i=1 (34)

which gives

N

n N + l - E ni=even. ( 3 5 ) i = l

But since s is only defined modulo 4, 2~ ~x+ 1)= (0, 0 ..... 0, 4) may be taken to be zero. As a result, we may express eq. (32) as

N / t ~ = / t o + 2 m l ~ ( ° ) + ~ niOg (i), (36)

i=l

i.e./to and/t~ are equivalent in the sense in which the sum over/to is performed in eq. (20).

Combining these results we see that the sum over /to in eq. (20) and over/~o in eq. (24) may be taken to be over the same set of values satisfying eq. (23). It is then straightforward to show using the tech- niques of ref. [ 12 ] that a consistent modular invariant choice of M is

gmixed #0 ml,nl,{ni},{mi}

X (~,.O+ml~(O, +ni@(i)(~fl,.O+Jqlo~(O, +rFlio~(i). ( 3 7 )

This choice of M describes the type II superstring compactification in the Green-Schwarz formalism for the left handed sector, and Neveu-Schwarz-Ramond formalism for the right handed sector.

It is now straightforward to generalize the construction to heterotic string compactification. We start with the partition function given by ~/mixed of eq. --- .u..u

(37), and replace the SO (2) character O~2/q* in the right hand sector by the appropriate SO(26) or SO (10) X E8 characters. The modular invariance of the resulting partition function follows from the modular invariance of the superstring partition function exactly in the same way as in the case of ref. [ 12 ].

Before concluding we would like to show the equivalence of the theory described in the Green- Schwarz formalism with that in the Ramond-Neveu- Schwarz formalism. This will be done by comparing the part of the partition function involving spacetime fermions in the two cases, since by spacetime supersymmetry, the partition function involving spacetime bosons is identical to that involving spacetime fermions up to a change of sign. We shall show it for

283


the he te ro t i c str ing theory, bu t s imi la r analysis m a y

be done for supers t r ing theory as well. S ince the right m o v i n g par ts o f M NsR and m mixed

are ident ical , we only need to look at the s u m o v e r

the left m o v i n g part . F o r g iven / to , the sum o v e r the

integers n~ in the R a m o n d - N e v e u - S c h w a r z f o r m u -

la t ion takes the f o r m

( 1) ~+1 - a , , , , ,o + ~ , , , <o, + z ~=, m,,~ I', • ( 3 8 ) {mi},nl

On the o the r hand, in the G r e e n - S c h w a r z fo rmula -

t ion it takes the fo rm

( - 1 ) ~:n'+ la~,~,o+ m,alo,+ z £+,n,a,~. ( 3 9 ) { ni },m I

H e r e the overa l l sign in eqs. (38 ) and (39 ) has been

chosen in such a way that the c o n t r i b u t i o n f r o m the

r ight m o v i n g sector is pos i t ive . The space t ime fer-

m i o n i c con t r i bu t i on ( i den t i f i ed by nega t ive mul t i -

p l ic i ty ) in eq. ( 3 8 ) comes f r o m even va lues o f nl

( - = 2 & ) , whereas that in eq. (39 ) comes f r o m even va lues o f Z N+ 1 ,= ~ n,. In this case, we m a y replace n ~ t ~°)

by ~ ( o ) in eq. ( 3 8 ) and ~f',~=+llni~ (') by ~ N ( i ) ,=1 n~at i n e q . (39 ) [see the d i scuss ion be low eq.

(35 ) ]. Thus the space t ime f e r m i o n i c con t r i bu t i on to

eqs. ( 3 8 ) and (39 ) are ident ical . Th is shows the

equ iva l ence o f the spec t rum o f the G r e e n - S c h w a r z

and the R a m o n d - N e v e u - S c h w a r z desc r ip t ion o f the

he te ro t i c s t r ing compac t i f i ca t ion .

Th is conc ludes our d i scuss ion o f exact ly solvable

str ing c o m p a c t i f i c a t i o n on C a l a b i - Y a u m a n i f o l d s in

the G r e e n - S c h w a r z fo rma l i sm. As has been p o i n t e d

ou t in ref. [ 9 ], in this f o r m a l i s m all the genera to rs o f

the super-Poincar6 algebra may be expressed in t e rms

o f the genera tors o f the (2, 0) [or (2, 2) ] supercon-

fo rmal algebra, and m o d e s o f the f ields P, P ( a n d P,

P).

R e f e r e n c e s

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Exactly solvable string compactification on Calabi-Yau manifolds in the Green-Schwarz formalism