Recent progress in calabi-Yauology

Nuclear Physics B (Proc. Suppl.) 5B (1988) 175-180 North-HoUand, Amsterdam

RECENT

PROGRESS

175

IN CALABI-YAUOLOGY

C.A.Liitken C E R N - T H , CH-1211 Geneva 23, Switzerland

Calabi-Yau manifolds are still the only viable

of polynomials in the homogeneous variables of the the

smooth background spaces on which to compactify the

ambient projective space. Any manifold that can be

heterotic string from ten to four dimensions 1.

The

realized by polynomial constraints in P,~ (or a prod-

number of such spaces is enormous, but a large 'core

uct of projective spaces, which is essentially the same

sample' can be obtained rather easily and analyzed

thing) is called algebraic. From Kodaira's famous em-

in great detail for possible relevance to phenomenol-

bedding theorem and the fact that three dimensional

ogy. They are also useful for testing out general string

Calabi-Yau manifolds never have any nontrivial holo-

theoretical ideas which address the vacuum degener-

morphic 2-forms, it follows that all Calabi-Yau man-

acy problem. These spaces include all complete inter-

ifolds are algebraic.

section Calabi-Yau manifolds (CICY manifolds) which

straint has codimension one (i.e. the dimension of the

are discussed below, their quotients with freely acting

hypersurface is precisely one less than the ambient di-

discrete groups and a few with positive Euler number

mension), the conlmon zero-locus of several constraints

whose construction goes via an orbifold, which corre-

(pl _ 0;p2 = 0 ; . . . ;pro = 0) does not necessarily have

sponds to a particular choice of moduli for which the

codimension an. When it does we have a complete inter-

the metric is singular. We call the latter Z-type man-

section (see Fig.l) and all the discrete topological prop-

ifolds. They have recently received a lot of attention 2

erties of the manifold are determined by the degrees of

and are discussed by P.Nilles elsewhere in these pro-

the defining polynomials alone. We first restrict atten-

ceedings. We first, briefly review the construction of all

tion to this 'best of all worlds' and construct all CICY

(',ICY manifolds, which is purely algebraic and is done

manifolds 3 .

by machine. Tbe first thing to realize about complex manifolds

Although any polynomial con-

The number of CICY manifl~lds is finite because there are only a small number of different ambient

is that they cannot be regarded as submanifolds of C " .

spaces W

We work instead with the complex projective space P,,

This follows from the fact that the multidegrees

which is obtained by including the point at infinity

of the j ' t h constraint in the i'th ambient factor nmst

and 'compactifying' C '~ to P,~. Submanifolds of P~

satisfy the condition for vanishing first Chern class,

are not in general trivial, and easily constructed by intersecting hypersurfaces defined by the vanishing set 0920-5632/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

P,~I x P,~2 × " • P,~P which need be considered 4,

dij

~-~ m

z_,j=l d~j = ni + 1, and that a bilinear constraint in

P1 × P1 is simply a P1. Since a linear constraint in Pn

CA. Liitken / Progress in Calabi-Yauology

176

CD1

p~

~1

133

O~

pS

FIGURE 2

FIGURE 1 A CICY manifold is the common zero-locus of

The construction of all possible CICY manifolds

polynomial constraints in a projective ambient space

reduces to combinatorics. The space represented by

W.

this picture has Euler number - 8 .

is just P,~-1 we also require that the total degree of each

3-dimensional CICY manifolds. Notice that each man-

constaint be at least two. It is convenient to think of

ifold may be represented in m a n y different ways in this

the degree matrix (dij) as a 'pillbox' like the one shown

list, so that we have not classified them by this ex-

in Fig.2, where the n u m b e r of marbles in compartment

plicit construction. Notice also that the degree matri-

( i , j ) is dij. The n u m b e r of columns (constraint equa-

ces contain all the information on the discrete topology

tions) must be m = "~-,~=1ni - D in order to bring the

of these manifolds.

(complex) dimension of the manifold down to D.

Tile most important topological invariant of any

By playing with this picture you can now easily

manifold is tile Euler number, which in two dimen-

convince yourself that the logistics of distributing the

sions simply counts the n u m b e r of holes in the sur-

marbles over a big pillbox while satisfying all the above

face. For manifolds of complete intersection it is very

restrictions are such that it cannot be done when the

simple to calculate in any n u m b e r of dimensions by

size of the box exceeds ( F × ra) = (SD x 6D). The ambi-

using characteristic classes as explained in ref.3.

ent spaces range from PD+a to (i)1) ap × (P2) ~D, with

much more delicate probe of topological structure is

235 possible choices when D = 3. For each of these

provided by the Hodge numbers.

ambient spaces we instructed a computer to sprinkle

direct physical interest because many of the massless

the marbles in all possible (distinct) ways so that each

fields on Calabi-Yau manifolds c,~n be represented by

row sums to nl + 1. After also taking into account a

harmonic forms 1. These are unique representaives of

n u m b e r of algebraic identities the result is that there

the Dolbeault cohomology classes in H ( I ' I ) ( M ) ~nd

are 7868 distiiact degree matrices available to represent

H('%l)(M). H (1'1) is the group of closed modulo exact

A

They are also of

CA. Liitken / Progress in Calabi-Yauology

177

(1, 1)-forms which represent m a t t e r superfields trans-

take values in the bundle EndTM ~_ TM ® T~, i.e. 1-

forming as ( 2 7 , 1 ) (families) under the gauge group

forms which transform as mixed rank-2 tenors (octets)

Es × Es, while H (2'1) is the group of (2,1)-forms which

under SU(3)), whose harmonic forms parametrize scalar

represent the m a t t e r superfields transforming as (27", 1)

zero-modes on Calabi-Yau manitblds.

(anti-families). The number of generations accessible

the computation of the rank of this group is much

in accelerators is h) I - h 21, where h 11 = r k H (1'1) and

harder than the Hodge number calculation, and we

h 21 = r k H ("'1) are the only nontrivial Hodge num-

have so far only been able to pin down the number

bers on a three-dimensional Calabi-Yau manifold. This

of octets for 25 of all the CICY manifolds 7.

number is easily obtained from any of a number of topological indices on the manifold, the simplest being the Euler characteristic X = 2( hll - h21) •

different values ranging from 0 to - 2 0 0 . The extreme cases are uniquely represented by P2

and 0

P4{511200, where the superscripts denote h 11. There is, however, no index in three dimensions which will give the Hodge numbers separately.

Because all CICY manifolds are probably simply connected they cannot by themselves serve as background solutions since we at some point wish to break

For C I C Y manifolds the Euler number takes 70

P2

Unfortunately

Also, the simple ar-

gulnent (popular with physicists) that all (2,1)-forms can be represented by the inequivalent deformations of the defining polynomials, is in general wrong and a more rigorous method must be found s . This is provided by something called 'spectral sequences', which is a standard but somewhat unpredictable tool for computing cohomologies. They have recently been used to show that there are only 264 distinct choices of I-lodge numbers (h 11, h 21 ) possible in the CICY class s, which is therefore a lower bound on the number of distinct Calabi-Yau manifolds.

the gauge group by using the natural Wilson line mechanism.

We therefore search for more realistic vacua

which are constructed by identifying points on a CICY manifold M which are related by a freely acting discrete group Go. The new manifold M/Go not only has Go as fundamental group, but the Euler number (which is a density integrated over the manifold) of the covering space is also reduced. This is important because there is i1o CICY manifold with the Euler number - 6 which is needed for a 3-generation model (we assume here the standard embedding of the spin connection in the gauge connection). To actually exhibit such a group action is usually not easy. First a set of defining polynomials must be chosen for M with the required multidegrees given by the degree matrix and sufficiently general so that the intersection is smooth. This fixes the shape of the manifold which now may posess the required discrete symmetries. If M has Euler umnber - 6 k or - 8 k ,

There is one more cohomology group of immediate

and we can find a freely acting group of order k, then

physical interest. This is HI(EndTM) (1-fornls which

the quotient manifold will have Euler number -6 or -

CA. Liitken / Progress in Calabi-Yauology

178

8, which gives 3 or 4 generations of massless m a t t e r

Since the .vl can be lifted back to the ainbient space

particles after compactification.

where they came from, and integrals over P~ are triv-

We know of no topological criteria that are sufficient for deciding whether such a group action exists for some choice of moduh for a given degree matrix. Necessary conditions are however easy to find, and they turn out to be strong enough to eliminate all but three matrices as representing candidate 3-generation vacua.

ial, we can compute a large number of indices in this way. After testing all these indices of the 7868 CICY manifolds for factors appropriate to bring the Euler number down to - 6 , only one with X = - 1 8 and two with X = - 4 8 survive:

The nmnber of 4-generation survivors is larger, as it Pa

should be since eral different realizations are known,

1

--is

but there are not many s. The tests are generalizations P1 P1 P~ P1 P1

of the criterion discussed above that the Euler number must be divisible by the order of G0. This is because by the index theorem any topological index on the mani-

01

1 1 1 0

1

0

l J -4s

fold can be expressed as an integral over M of a density. The Euler number ()~) is just one of the four classical indices, which count various zeromodes associated with the tangent space TM of the manifold. On Calabi-Yau

1

0

o 2

0 0 0

1 0 O 2 O 0

o

1

1

P1 P1

1 1

P1 P1 Pa

0

2

s

-4s

manifolds the only information is contained in X, but

Tian and Yau 9 showed that the maximaUy symmetric

this changes drastically if we instead compute the in-

choice of polynomials for tile first of these configura-

dices on TM ® V, where V in our case can be built

tions admits a freely acting Za. The complete intersec-

from TM and the normal bundles Ni to the defining

tion of a bilinear and two cubits in P a x Pa is there-

hypersurfaces inside the ambiem space W (provided

fore the covering space of a 3-generation manifold. We

extreme care is excercised). These 'twisted' indices are

have been unable to find freely acting groups of order

given by integrals of characteristic classes and Chern

eight on the other two manifolds despite vigorous ef-

characters.

forts. This leaves the Tian-Yau manifold as the only

In the case of complete intersection they

all reduce to simple polynomials in the 2-forms xl induced on M by the K/ihler forms of the ambient factors.

3-generation model of this simple type 1°. There are, however, several other examples of Calabi-

Consequently, computing twisted indices simply boils

Yau manifolds with Euler number ±6. By dividing the

down to integrating polynomials in the x~ with coef-

C I C Y P= [31 031s i°3

ficients that depend only on the degree matrix (dij).

by a non-freely acting group of

-54

order nine and blowing up the singularities, Schimmrigk 11

CA. Liitken /Progress in Calabi-Yauology

179

has constructed a manifold which is supposed to be dif-

transitions' between the nmltitude of allowed super-

feomorphic to the Tian-Yau manifold 12. On the other

string vacua.

hand, Tian and Yau 13 have shown how to construct

'splitting' they show that the moduli spaces of all CICY

new manifold with Euler number - 6 fronl their old

manifolds are intimately related, and may be connected

one. To see that these manifolds are not diffeomorphic

ill the sense of Reid.

By studying manifolds constructed by

it is enough to exhibit one topological parameter which

'Splitting' was introduced in ref.3 where it was

is sufficiently sensitive to tell them apart as real mani-

used in the actual compilation of the complete list of

folds. This is provided by the 'intersections' of 2-forms

the blind combinatorics described above. It turns out

g

#ijk = /

03i A ~ j A Wk

to be a special case of what mathematicians call smaU

J~ I vi = --

CICY manifolds, since it is much more efficient than

resolutions, which Schoen 18 recently used to construct

Pl /\ c°i,

where Pl is the first Pontrjagin class.

These num-

many new Calabi-Yau manifolds. He found examples

bers are also hard to compute in general, but they

with every (even) Euler number between

8 and 92, as

are important because together with the Euler num-

well as 96 and 168, but his 3-generation models seem to

ber they classify simply connected real spin manifolds

be simply connected and therefore uninteresting fl'om

with torsion free cohonlology 14. ('.alabi-Yau manitolds,

a phenomenological point of view.

however, have a lot more structure than real mani-

Finally we mention a comph, tely different way of

folds (they are K£hler) which is intimately related to

probing Calabi-Yau compactifications which looks ex-

their appearence as supersymmetric vacua, and there-

tremely promising.

fore presumably is of physical importance. The possi-

constructing 2-dimensional superconformaJ field the-

ble choices of K~hler class and complex structure are

ories by tensoring together models fi'om the discrete

parametrized by h 11 + h 21 moduli parameters which

series. There are 228 basic theories of this type cor-

span a complex manifold which is (tangent to) the

responding to the posssible ways of building modular

moduli space is.

Physically these nloduli correspond

invariant partition functions appropriate for the her-

to massless fields with vanishing potential which are

erotic string (this counting includes standard and ex-

called 'blowing-up' modes in orbifoldology.

Very re-

ceptional afline invariants) 2°. Many more models can

cently Green and Hiibsch 16 have presented support-

be constructed by dividing by discrete groups. At least

ing evidence for an exciting conjecture by Reid 17 that

some of these seem to correspond to signla-nlodels with

there exists a 'universal moduli space' for all Calabi-

CMabi-Yau target spaces. In particular, a 3-generation

Yau manifolds, and they suggest that this is the natu-

model exhibited by Gepner 21 seems to have Schimm-

ral framework within which to examine stringy 'phase

rigk's manifold as target space. Not only does this tech-

This is Gepner's 19 technique for

CA. Lfitken / Progress in Calabi-Yauology

180

nique greatly increase the number of available nxodels, it also opens the possibility of analyzing the superstring vacuum in the unifying framework of conformal field theory. Furthermore, in the complementary space-time

Manifolds, CERN preprint, CERN-TH.4933/87 7. ibid, in preparation 8. P.Candelas, C.A.Liitken and R.Schimmrigk, Com-

picture developed above it is extremely difficult to di-

plete Intersection Calabi. Yau Manifolds III: Four Generation Manifolds, in preparation

rectly extract the parameters of crucial importance to

9. S.-T.Yau, with an appendix by G.Tian and S.-

low-energy phenomenology, even in the 'field theory linfit' of the string theory. Perhaps the greatest virtue of the world sheet approach is that all the correlation functions (i.e.

all couplings) of the conformal fields

T.Yau, in PTvceedings of the Argonne Symposium on Anomalies, Geometry and Topology, eds. W.A. Bardeen and R.A.White (World Scientific, Singapore, 1985). Some aspects of the analysis are correefed by B.R.Greene, K.H.Kirklin, P.J.Miron and G.G.Ross, Nucl. Phys. B278 (1986) 667

can be conlputed. This means that the electron mass

model, thus finally confronting string theory with the

10. A similar conclusion was reached by P.S.Aspinwall, B.R.Greene, K.H.Kirklin and P.J.Miron, Oxford preprint (1987), by using an erroneous argument.

facts of life.

11. R.Schimmrigk, Phys. Left. 193B (1987) 175

should be calculable, e.g.

in Gepner's 3-generation

12. B.R.Greene and K.H.Kirklin, Harvard preprint

REFERENCES

HUTP-87/A025

1. P.Candelas, G.Horowitz, A.Strominger and E.Witten, Nucl. Phys. B258 (1985) 46 2. L.Dixon, D.Friedan, E.Martinec and S.Shenker, Nucl. Phys. B282 (1987) 13; M.Cvetic, SLAC-PUB-4325, 4378 ibid, Phys. Rev. Lett. 59 (1!)87) 1795 and 2829 3. P.Candelas,

A.M.Dale,

C.A.Liitken

and

R.Schimmrigk, Complete Intersection Calabi-Yau Manifolds, Nucl. Phys. B298 (1988) 493; P.Candelas, C.A.Liitken and R.Schimmrigk, Complete Intersection Calabi. Yau Manifold~ Ih Three Generation Manifolds, to appear in Nucl. Phys. B

4. P.Green and T.Hfibsch, Comm. Math. Plays. 109 (1987) 99

5. ibid, Polynomial Deformations and Cohomology of Calabi- Yau Manifolds, University of Maryland report UMPP-87-158 6. P.Green, T.Hiibsch and C.A.Liitken, All Hodge Numbers of All Complete Intersection Calabi- Yau

13. G.Tian and S.-T.Yau, Three Dimensional Alge-

braic Manifolds with cl = O and X = -6, University of San Diego preprint 14. C.T.C. Wall, Invent. math. 1 (1966) 355 15. G.Tian, in Mathematical A~pect~ of String Theory, ed. S.-T.Yau (World Scientific, Singapore, 1987) 16. P.S.Green and T.Hiibsch, University of Texas preprints UTTG-04-88 and UTTG-06-88 17. M.Reid, Math. Ann. 278 (1987) 329 18. C.Schoen, Math.Z.197 (1988) 177 19. D. Gepner, Nucl. Phys D296 (1988) 757; ibid, Phys. Left. 199B (1987) 380 20. C.A.Liltken and G.Ross, in preparation 21. D.Gepner, String Theory on Calabi- Yau Manifolds: The Three Generation Case, Princeton preprint, December 1987