View metadata, citation and similar papers at core.ac.uk brought to you by CORE
provided by CERN Document Server
QMW-th/96-4
N=1 Heterotic/M-theory Duality and
Joyce Manifolds
1
B.S. Acharya Queen Mary and West eld Col lege, Mile End Road,
London. E1 4NS
Abstract
We present an ansatz which enables us to construct heterotic/M-
theory dual pairs in four dimensions. It is checked that this ansatz
repro duces previous results and that the massless sp ectra of the pro-
p osed new dual pairs agree. The new dual pairs consist of M -theory
compacti cations on Joyce manifolds of G holonomy and Calabi-Yau
2
compacti cations of heterotic strings. These results are further evi-
dence that M -theory is consistent on orbifolds. Finally,weinterpret
these results in terms of M -theory geometries which are K 3 brations
3
and heterotic geometries which are conjectured to b e T brations.
Even though the new dual pairs are constructed as non-freely acting
orbifolds of existing dual pairs, the adiabatic argument is apparently
not violated.
1
e-mail: [email protected]. Work Supp orted By PPARC 1
1 Orbifolds and M-theory.
The predictions of string dualities [1, 2] have given rise to a fascinating web
of interconnections b etween our most promising candidate descriptions of
nature. An underlying structure is slowly emerging. In particular, it has
now b ecome apparent [2, 21 ] that the strong coupling dynamics of b oth the
Typ eI Ia and E E heterotic string theories can b e understo o d in terms
8 8
of certain one-dimensional compacti cations of M -theory. Sp eci cally the
1
Typ eI Ia theory is related to M -theory on a circle, S , and the heterotic string
1
to M -theory on S =Z , where the Z acts as re ection on the co ordinate of
2 2
1
the S .
A precise de nition of M -theory is yet to b e made; however consistency
with the remarkable evidence that string theories in various dimensions are
connected, not only to each other but to a sup ersymmetric theory in eleven
dimensions (M -theory) gives us some information ab out some of the prop er-
ties of M -theory. In particular, it seems clear that the low energy dynamics of
M -theory are describ ed by eleven-dimensional sup ergravity theory. Secondly,
M -theory must share certain prop erties with string theory.For example, M -
theory on an orbifold must b e a consistent quantum theory [21 , 22]. This is
certainly not a prop erty shared by its low energy cousin. Finally, M -theory
must contain higher dimensional ob jects, p-branes which play a fundamental
role in the duality conjectures. One viewp oint is that fundamental strings
in d<11 arise from closed p-branes in M -theory, and that Dirichlet-branes
(D-branes) in string theory [23] arise from op en p-branes in M -theory [24].
One hop e is that one may b e able to derive al l connections b etween various
string theories in lower dimensions from M -theory.
In [2] evidence was presented that the strong coupling limit of the Typ eI Ia
string theory in ten dimensions is e ectively describ ed by eleven dimensional
sup ergravity compacti ed on a circle. The arguments leading to this conclu-
sion are well known so we do not review them here and instead concentate
on the heterotic string. In [21] evidence was presented that M -theory on an
1
S =Z orbifold gives a description of the strongly coupled E E heterotic
2 8 8
string. The arguments were as follows: The Z kills one of the two sup ersym-
2
metries presentinM-theory on a circle. The two xed p oints which arise in
the orbifold of the theory de ne two xed ten dimensional hyp erplanes, on
which anomaly cancellation requires E E gauge symmetry to b e present
8 8
in the theory. This gauge symmetry was understo o d to have arisen from the 2
twisted sectors of the orbifold, in analogy with string theory.
1
One is then led to the picture that M -theory on X S is asso ciated to the
1
Typ eI Ia theory on X , with X any space. Similarly M -theory on X S =Z
2
is related to the E E heterotic string on X .
8 8
3 1
Consider then, M -theory on T S =Z . One exp ects that this theory is
2
3
related to the heterotic string on T . On the other hand, one exp ects that
3
the strong coupling limit of the heterotic string on T is related to eleven-
dimensional sup ergravityon K3 [2]. This implies various p ossibill iti es; one
3 1
is that whatever M -theory may b e, on T S =Z it is a theory whichatlow
2
energies lo oks like eleven dimensional sup ergravityonK3. A second p ossi-
3 1
billity is that M -theory on two di erent spaces, K 3 and T S =Z are b oth
2
3 2
related to the T compacti ed heterotic string .We will show that a strik-
ingly similar result arises in lower dimensions for dual M -theory/heterotic
compacti cations whichhave less sup ersymmetry. Sp eci cally, for N =2
and N = 1 heterotic string compacti cations to four dimensions it will b e-
come apparent that there are again two M -theory compacti cations which
arise as the duals of these heterotic theories.
The general ambition of this pap er is to nd a heterotic dual for compact-
i cations of M -theory on various Joyce 7-manifolds [13, 14]. We do this by
means of an ansatz, whichwe present shortly. It will later transpire that (at
least lo cally), these Joyce manifolds are K 3 brations. It is thus natural to
exp ect that, if the heterotic dual is the correct one, then we are discussing a
3
bration of the seven-dimensional dualitybetween the T compacti ed het-
erotic string and M -theory on K 3by the brewise application of the seven
dimensional duality in the adiabatic limit [6]. However, if the heterotic dual
is on some space X , then we also exp ect that wehavean M-theory dual com-
1
pacti cation on X S =Z [21 ]. We can naturally interpret this as a bration
2
of the other M -theory/heterotic dualityinseven dimensions.
The aim of the remainder of this section is to use an ansatz to rederive
some of the previously constructed dual pairs, where apart from the Z orb-
2
ifold which de nes the K 3, all other elements of the orbifold group act freely.
4
Consider then M -theory on T .We can takea Z orbifold of this theory
2
in the following way.
2
A related discussion of these compacti cations app eared in [31], where M -theory on
1
K 3S =Z was considered. From one p oint of view this `should' give the heterotic string
2
3 1
on T S =Z which is inconsistent. It was therefore argued that this Z acts also on the
2 2
3
T giving the heterotic string on K 3. 3
:(x ;x ;x ;x )=( x ; x ; x ; x ) (1)
1 2 3 4 1 2 3 4
where (x ::::x ) are the co ordinates of the four torus.
1 4
This orbifold has sixteen xed p oints and de nes a particular orbifold
limit of K 3. From our preceding discussions, one would exp ect that M -
theory on such an orbifold has certain twisted sectors, asso ciated with these
xed p oints, at which extra massless particles may arise. Given the lackof
a de nition of M -theory it is dicult to makeany precise statements ab out
suchtwisted sectors. However, one can draw an analogy again with string
theory whereby blowing up the xed p oints, we should recover the results
of the theory on the resulting smo oth manifold. In this case, the smo oth
manifold is K 3, and we exp ect that in this limit, the theory is dual to the
3
heterotic string on T [2].
Wenow come to the rst and most crucial part of the ansatz which will
enable us to construct dual pairs of heterotic/M -theory compacti cations in
3
lower dimensions. The rst part of the ansatz is to lab el the heterotic T
co ordinates with the same lab els as three of the M -theory K 3 co ordinates.
We will then toroidally compactify b oth theories to four dimensions on a
3 3
further T with co ordinates (x ;x ;x ). Wethus have M -theory on K 3T
5 6 7
6
and the heterotic string on T . The second part of the ansatz is the following:
we will take further Z orbifolds of this M - theory background giving vacua
2
with N = 2 and N = 1 sup ersymmetry in four dimensions. The isometries
de ning these orbifolds will act on the M -theory co ordinates, (x ; :::x ). The
1 7
crucial p oint is that, b ecause of the lab elling choice in the rst part of the
6
ansatz, the six co ordinate lab els of the heterotic string on T are a subset
of the seven co ordinate lab els of the M -theory compacti cation. Thus the
de nition of an orbifold isometry in the M -theory geometry also de nes an
orbifold isometry in the heterotic geometry. In general, if we b egin with
3
M -theory on a K 3T orbifold de ned by , as in eq.(1) and take a further
orbifold of the theory, generated by a group of isometries denoted by , then
b ecause of the lab elling choice in the rst part of our ansatz, M -theory on
7 6
this T =( ; ) orbifold should b e equivalent to the heterotic string on a T =
orbifold.
A priori this lab elling choice may seem like a rather bizarre thing to do;
however, given this as an ansatz and nothing more we will show that sucha
choice of lab elling always gives rise to the correct heterotic/M -theory sp ectra 4
when we orbifold to pro duce new dual pairs in lower dimensions. This only
app ears to work when the only non-freely acting memb ers of the orbifold
group p ossess K 3 orbifold singulariti es. More precisely, this construction only
works when the singular set of , consists solely of K 3 orbifold singularities.
3
For the rest of the pap er wethus cho ose the heterotic T co ordinates as
(x ;x ;x ), where these are the same lab els used to de ne the K 3in M-
1 2 3
theory. A clue that this is the correct choice to makewas given in [31]
and we refer the reader there and to the previous fo otnote for details. The
fact that this strategy app ears to work in all cases strongly suggests that
there is some as yet underlying structure to the way one can construct M -
theory/heterotic dual pairs in dimensions less than seven. Further evidence
of this has emerged in [31].
3
Now let us toroidally compactify the x ;x and x directions. On K 3T ,
5 6 7
6
wethus relate M -theory to the heterotic string on T and to the Typ eI I
2
theory on K 3T . This four dimensional theory has N = 4 sup ersymmetry.
We can further orbifold this theory bya Z isometry which gives N =2
2
sup ersymmetry in four dimensions. This example was considered in [8]. The
action of this isometry is de ned on the seven co ordinates on which M -theory
is compacti ed as follows:
:(x ; :::::x )= ( x +1=2; x ;x ;x +1=2; x ; x ;x +1=2) (2)
1 7 1 2 3 4 5 6 7
Because of the half shifts on the torus de ned by(x ;x ), this Z acts freely.
4 7 2
When combined with the Z de ned by , the blown up Z Z orbifold will
2 2 2
1
give M -theory on CY S ; where the Calabi-Yau manifold CY is self-
11 11
mirror and has h =11 [8]. This M -theory compacti cation is then related
11
to the Typ eI Ia theory on CY . This thus gives rise to an N =2 theory
11
whose massless sp ectrum at generic p oints in the mo duli space of the vector
multiplets is (12; 12), where following [3] (M; N ) denotes an N =2 theory with
N vector multiplets and M hyp ermultiplets.
In [8] and all known examples to date, the action of orbifold isometry
2
groups (which act on the Typ eI Ia theory on K 3T ) on the heterotic string
6
on T were calculated using the connection b etween the lattice of integral
4
cohomology of K 3 and the Narain lattice for the T compacti ed heterotic
string. In other words, all existing dual pairs, constructed as orbifolds, have
b een derived from string-string duality in six dimensions.
Now, let us supp ose that we know nothing ab out the cohomology of
K 3, but that we knowhow to construct consistent heterotic string orbifolds. 5
Then we can ask how can we repro duce the N = 2 (12; 12) sp ectrum from a
heterotic Z orbifold? Because of our ansatz, the action of on the M -theory
2
6
geometry also de nes its action on the T of the heterotic theory, b ecause
6
the T co ordinates wehavechosen are (x ;x ;x ;x ;x ;x ). Thus, all that
1 2 3 5 6 7
remains is to sp ecify the action on the gauge degrees of freedom ie the sixteen
left movers. Given that acts freely on (x ;x ), wehaveaninvarianttwo-
3 7
torus, which will give rise in general to four vector multiplets. Thus, all that
remains is to pro ject out eight of the sixteen p ossible additional vectors which
are asso ciated with the Cartan subalgebra of the ten dimensional heterotic
gauge group. Because the orbifold is of order two, we know that essentially
the only p ossible action is exchanging the two E factors in the gauge group.
8
Finally, in order to achieve mo dular invariance we are forced to include an
3
asymmetric Z shift which is related to the shift on x . This repro duces the
2 7
mo del of [8] without any knowledge of the cohomology of K 3.
In a similar manner, one can consider a further freely acting Z orbifold
2
of the ab ove N = 2 theories whichwas considered in [7] to pro duce dual
theories with N = 1 sup ersymmetry in four dimensions. Again without
using any knowledge ab out the cohomology of K 3 one can repro duce the
result of [7]. We, of course, are not suggesting that the identi cation of the
lattice of integral cohomology of K 3 with the Narain lattice for the heterotic
4
string on T is incorrect. For the examples considered in [7, 8], whichwe
repro duced ab ove, the K 3 orbifold de ned by in equation (1) is the only
element of the orbifold group of this typ e. We make the supp osition that
we know nothing of the cohomology of K 3, in order to pro ceed further and
construct dual pairs when two or more elements of the orbifold group are
of the K 3 orbifold typ e. The identi cation b etween these lattices will b e
seen to hold in the adiabatic limit [6] when, in section four, weinterpret our
results in terms of the ` bration picture' of [6].
In the examples wehave just considered we resolved al l singulariti es b e-
cause it is unclear at presenthow to deal with twisted sectors in M -theory.
3
Sp eci cally, the vacuum energy in the left moving twisted sector is 1=4 whichdoes
not lead to a mo dular invariant orbifold. However if we translate the shift on x in the
7
M -theory background to an asymmetric shift of the x in the heterotic background, then
7
1;1
we can achieve mo dular invariance in the following way[8]: the which corresp onds to
2
x can b e orbifolded by a shift vector of the form =(p;p )=2 with p = 2. Because
7 l r
2
=2= 1=4, the di erence b etween left and rightmoving vacuum energies is zero, and
hence the orbifold is mo dular invariant. 6
Luckily there were no singularities asso ciated with , so there were none to
resolve. However in the more general cases we will consider, we will orbifold
the heterotic theory with isometries that do have singularities and it is natu-
ral to resolveieblow up these as well. The blowing up mo des may naturally
b e asso ciated with the twisted sectors of the heterotic theory.
This suggests the following strategy: (I) Take M -theory on an orbifold.
By analogy with string theory,we can naturally identify the twisted sector
6
states with the blowing up mo des of the orbifold. (I I) The action on the T
of the heterotic string will already b e sp eci ed by the action of the orbifolds
on the geometry of M -theory by our ansatz. (I I I) Then simply pro ject out
the necessary number of vectors from the heterotic string sp ectrum. (IV) If
the heterotic orbifold is a (2; 2) sup erconformal eld theory, in which case
the blowing up mo des are truly mo duli [25] then pro ceed to the smo oth limit
and include the blowing up mo duli in the sp ectrum. In this case one must
cho ose an emb edding of the spin connection in the gauge connection such
that the resulting sp ectrum is correct. In fact, if the orbifold group acts
6;6 6
left-right symmetrically on the of the T compacti ed heterotic string,
then the orbifold has a classical geometric interpretation and one can study
the heterotic string on the blown up orbifold and the mo duli of the smo oth
manifold will in any case app ear as scalar elds of the theory. Because such
a theory is a string theory on a blown up orbifold, the massless sp ectrum is
easy to determine: it is just the untwisted sector at the orbifold limit plus the
mo duli asso ciated with blowing up. (V) Check if the theory is consistent with
mo dular invariance or can b e made so by adding appropriate shift vectors.
In fact, it may b e p ossible to go further than just considering the theories
on blown up orbifolds. We will give strong evidence in some four dimensional
N = 1 examples that b oth the untwisted and twisted sector sp ectra coin-
cide for M -theory on Joyce orbifolds and the heterotic string on Calabi-Yau
orbifolds. In the following sections, we will apply our presented strategy to
prop ose new dual pairs. These constitute examples of dual pairs constructed
as non-freely acting, sup ersymmetry breaking(ie less than N = 4 in 4d) orb-
ifolds of existing dual pairs. Section two discusses an N = 2 example in
detail. In section three we construct some N = 1 examples. In section four
weinterpret our results in terms of the bration picture in the adiabatic
limit [6]. Remarkably,even though our examples are constructed as non-
freely acting orbifolds of existing dual pairs, the adiabatic argument of [6] is
apparently not violated. This is a consequence of the fact that the orbifolds 7
we restrict ourselves to are precisely the ones which preserve the bration
structure. Following this we end with some conclusions and comments.
2 An N =2 Example.
In this section we will consider an N = 2 example rst following the analysis
given in [3] and then following the strategy presented in the last section. In [3]
several examples of p otential dual pairs of N = 2 theories in four dimensions
were constructed. The dual pairs in question were Calabi-Yau compacti -
2
cations of Typ eI I strings and K 3T compacti cations of heterotic strings.
2
The massless sp ectrum of the heterotic string on K 3T is determined from
the exp ectation value of the gauge elds on K 3 and index theory [3, 26 ]. An
N = 2 theory in four dimensions is characterised byvector multiplets and
hyp ermultiplets. The vector multiplets contain adjoint scalar elds, which
in addition to the mo duli hyp ermultiplets of the theory are also mo duli. At
sp ecial p oints in the mo duli space of these scalars, the theory contains mass-
less charged hyp ermultiplets. These b ecome massive at generic p oints in the
mo duli space of the adjoint scalars which corresp ond to the Cartan subalge-
bra of the gauge group. Neutral massless elds will always remain massless
as one moves through the mo duli space of these scalars. After the spin con-
nection has b een emb edded in the gauge connection in sucha way that the
theory is anomaly free, the theory is then characterised by the neutral elds
(the rest of the mo duli) and N vector multiplets, where N is the rank of the
gauge group which survives the emb edding. Kachru and Vafa denoted the
4
theories at these generic p oints by(M; N ), where M is the numb er of mass-
less hyp ermultiplets. There is a universal contribution of 20 to M coming
from the mo duli of K 3 [19 ], hence M 20. Further, b ecause wehave a rank
sixteen gauge group in ten dimensions and a further four U (1)'s coming from
the torus at generic p oints, N 20.
We wish to consider an example where b oth these inequalities are sat-
urated. This places two constraints on p ossible emb eddings of the gauge
bundle on K 3: (i) we are forced to consider giving exp ectation values to
U (1) or pro ducts of U (1) gauge elds on K 3. Fortunately, the sp ectra of
many of these emb eddings has b een calculated in [26 ], although we know not
4
We assume we are at generic p oints in the torus. 8
of any full classi cation; (ii) in order that we obtain 20 mo duli hyp ermulti-
plets we need to nd an example in which all hyp ermultiplets are charged
except the 20 gravitational mo duli.
Examining the sp ectra given in [26], it is not to o dicult to convince
oneself that many of the examples satisfy these criteria, giving a (20; 20)
sp ectrum at generic p oints. For de niteness we consider the mo del with
a single U (1) emb edded in E E ,insucha way that one E is broken to
8 8 8
E U (1). This gives a sp ectrum containing 10 56's of E (half with one U (1)
7 7
charge and half with the opp osite charge) and 46 E singlets(23 each with
7
opp osite U (1) charges). This is the example given in equation (6.1) of [26 ].
Because all matter is charged, wehave a (20; 20) sp ectrum at generic p oints.
As an aside, it is interesting to note that we can connect this example to
the chain of examples considered in section 3 of [3]. Namely,by higgsing the
U (1), we nd 45 additional gauge neutral hyp ermultiplets giving a (65; 19)
sp ectrum. This mo del can b e similarly higgsed [3] several times to give the
5
chain: (20; 20) ! (65; 19) ! (84; 18) ! (101; 17) ! (116; 16).
Wewould now like to nd a Typ eI I dual for this (20; 20) mo del. According
to [5], the Calabi-Yau space describing the background of the Typ eI I theory
should b e a K 3 bration. Further it must b e a self-mirror Calabi-Yau with
h =h =19 b ecause a (M; N ) mo del arises from a Typ eI IA compacti cation
11 21
on a Calabi-Yau with (M = h +1;N = h + 1). Luckily there is a manifold
11 21
which ful lls these requirements. We denote this Calabi-Yau by CY . It can
19
b e constructed as a blown up orbifold as wenow describ e, following Joyce
[14].
1
In [14 ], Joyce constructed CY S as a Z Z blown-up orbifold of the
19 2 2
seven torus. We rep eat the construction here:
De ne the seven-torus co ordinates as (x ; ::::::; x ). Two Z isometries of
1 7 2
7
T are de ned by:
(x ; ::::x )=( x ; x ; x ; x ;x ;x ;x ) (3)
1 7 1 2 3 4 5 6 7
(x ; ::::x )= ( x ;1=2 x ;x ;x ; x ; x ;x ) (4)
1 7 1 2 3 4 5 6 7
Let the Z Z isometry group generated by and b e denoted by . In fact
2 2
it is easy to see that this is precisely the construction of [8], without the extra
5
While this work was in progress, we realised that many of the examples of [26] are
connected via Higgs' and Coulomb branches and whichhaveTyp eI I dual candidates on
K 3- brations [4]. 9
Z shifts which made that construction freely acting. If considered separately,
2
3
each of these Z 's has 16 xed T 's and each one de nes an orbifold limit
2
3 6 7 3
of a particular K 3 T . Hence, the singular set of T = contains 16 T
7
comp onents, as do es the singular set of T = .However, as acts freely on
the 16 xed three tori of , contributes eight three tori to the singular set
7
of T = . Similarly, also contributes eight three tori to the singular set of
7
T = .
The b etti numb ers of the original torus which survive the orbifold pro-
7
jection ie the b etti numb ers of T = are b =1, b = 3 and b = 11. The
1 2 3
blowing up pro cedure is carried out by inserting non-compact Eguchi-Hanson
3
geometries(T ) in each of the singular regions. Each of these adds 1 to
b and 3 to b , giving a seven manifold of SU (3)1 holonomy with b etti
2 3
numb ers: b =1, b = 19 and b = 59. In particular, if we consider the six-
1 2 3
torus de ned by the co ordinates x through x , then the holomorphic three
1 6
form is preserved by the Z Z . The seven manifold thus obtained has the
2 2
1
form CY S . Compacti cation of eleven dimensional sup ergravity on this
19
manifold yields N = 2 sup ergravity with 20 hyp ermultiplets and 19 vector
7
multiplets (not including the graviphoton). The counting go es as follows :
in eleven dimensions, the massless b osonic elds of M -theory are the met-
ric, G , and antisymmetric three-form tensor, A . On compacti cation
1
to four dimensions on CY S , the three form gives rise to b scalars, b
19 3 2
8
vectors and b two forms . In general a higher dimensional metric yields n
1
scalars, where n is the dimension of the mo duli space of the compacti cation
metric. In our case, this is 59 1 = 58. The metric tensor will also yield a
vector in the lower dimensional theory for every continuous isometry of the
1
compactifying manifold. In our case, the S has a U (1) isometry yielding a
U (1) gauge eld in four dimensions.
All in all, for this example we get 118 scalars and 20 vectors (including
graviphoton) plus the graviton. The fermion sp ectrum is implied by sup er-
symmetry and wethus have the (20; 20) mo del as required. This is the same
sp ectrum as the Typ e I Ia/I Ib string on CY .
19
Now let us apply the strategy suggested in the last section and see if it
6 0
In general we de ne the singular set S of M to b e the set of p oints, surfaces and
submanifolds of the manifold M , which are xed under the action of some nite group G.
0
The singular set, S ,ofM=G is then the image of S in M=G.
7
See [18] for a review
8
In four dimensions, two forms are dual to scalars and so may b e counted as scalars. 10
6
gives the correct results. Firstly, the action of on the heterotic string T
2
do es indeed give an orbifold limit of K 3T as b efore. Now, however, we
do not have the extra half shift on x , so this orbifold is not freely acting.
7
Secondly,wewould like a rank 20 gauge group, which means that the action
on the gauge degrees of freedom is trivial. But, this is not the whole story,
for wewould certainly like to preserve mo dular invariance in this orbifold
and the natural choice we make is the standard emb edding. Away from the
2
orbifold limit ie on the smo oth K 3T wemust also sp ecify an anomaly free
background; and, as mentioned at the b eginning of this section this will limit
us to U (1) emb eddings of the spin connection in the gauge connection to give
the required (20; 20) sp ectrum as b efore. Thus it app ears that our strategy
is consistent at least for the rst example wehave considered.
We can provide a further check on whether wehave indeed pro duced
a dual pair, by considering a freely acting orbifold of this dual pair. If
the sp ectra again agree, we will have also pro duced another dual pair. A
simple freely acting orbifold which do es not break any sup ersymmetry is the
following:
(x ;x ; :::x )= (x ;x +1=2;x ;x ;x ;x ;x ) (5)
1 2 7 1 2 3 4 5 6 7
On the M-theory geometry, this isometry has the e ect of halving the number
7
of elements of the singular set of T = . This pro duces an example with a
(12; 12) sp ectrum at generic p oints.
On the heterotic side, the action of corresp onds to exchanging the
two E factors of the gauge group, plus identifying, in eight pairs of two,
8
the sixteen xed p oints asso ciated with which de nes the K 3 orbifold
heterotic background. The shift on x also eliminates massless mo des coming
2
from the twisted sector. In fact the twist by is precisely the one which
was considered in [28]. Because the sixteen xed p oints of on the heterotic
background are asso ciated with sixteen neutral mo duli hyp ermultiplets in the
blowing up limit, the action reduces this numb er to eight. The resulting
sp ectrum is therefore precisely (12; 12) at generic p oints in accord with the
exp ectations of string-string duality.
As a concluding remark to this section, it is useful to p oint out that b e-
cause the resp ective mo duli spaces of these conjectured dual pair of compact-
i cations are constrained by N = 2 sup ersymmetry, many of the imp ortant
results of [8] also apply here. 11
3 N =1 Examples.
Compacti cation of M -theory on a seven-manifold of G holonomy gives rise
2
to N = 1 sup ergravity with b vector multiplets and b chiral multiplets.
2 3
b
2
We denote these manifolds by J . Many examples of such manifolds were
b
3
recently constructed in [13 , 14 ]. A dualitybetween the eleven dimensional
16
theory on J and the heterotic string on a Calabi-Yau with precisely the
39
same Ho dge diamond as CY was conjectured in [12], on the basis of count-
19
ing Betti numb ers and matching the sp ectra. We will see later in this section
that we can derive this result utilising our presented ansatz.
In this section, we pro ceed to apply the strategy of the preceeding sec-
tions to pro duce dual pairs with N = 1 sup ersymmetry in four dimensions.
9
The rst example we consider has b =8 and b =31. This example will b e
2 3
constructed by considering a freely acting orbifold of the (20; 20) N = 2 ex-
ample of the previous section. Evidence for the existence of the N = 1 dual
pair, is then also evidence for the N = 2 dual pair.
3
Consider then the Z orbifold of the seven torus de ned by and
2
of equations (3) and (4); and the third Z de ned as follows:
2
(x ;x ; :::::x )= (1=2 x ;x +1=2; x ;x ;1=2 x ;x ; x ) (6)
1 2 7 1 2 3 4 5 6 7
Because of the half shift on x , acts freely. In fact, takes the sixteen
2
7
elements of the singular set of T =( ; ) and identi es them in eight pairs
7
of two. The b etti numb ers of T = are b =0 and b =7. The singular set
2 3
contains eight elements, the resolution of each of which adds 1 to b and 3 to
2
8
b , giving a Joyce manifold of G holonomy, J .
3 2
31
Thus far, wehave said little ab out the p ossible gauge groups allowed
by string/string/ M-theory duality. The mechanism for gauge symmetry en-
hancement in the Typ eI I theories is a generalisation of that considered in
[20], where p-brane solitons wrap around p-cycles of the compacti cation
space and give rise to massless gauge multiplets (and matter multiplets)
when the cycles degenerate to zero volume. In general, b ecause the singu-
larities corresp onding to the vanishing cycles are of A-D-E typ e, one exp ects
A-D-E symmetries [2, 9]. The singularities wehave b een considering are all
SU (2) orbifold singularities, thus we can at least exp ect an SU (2) factor in
9
To the b est of our knowledge, this Joyce manifold has not b een constructed previously,
even though a manifold with the same b etti numb ers app eared in [14]. 12
the gauge group for each element of the singular set that we blow up. For
example, in the N = 2 (20; 20) example that we constructed, we should ex-
16
p ect an SU (2) factor in the gauge group if we consider M -theory at the
orbifold limit de ned in the previous section, b ecause we resolved sixteen
SU (2) singulariti es on the Typ eI I side to construct CY . So let us break the
19
16
E E gauge symmetry of the heterotic string to SU (2) by orbifolding the
8 8
22;6
Narain lattice for toroidal compacti cation to four dimensions. This is
equivalent to the turning on of Wilson lines. We will orbifold the theory by
three Z shift vectors given by:
2
7 7 5 5
=(1;0 ;1;0 ;1=2;0 )(1=2; 0 );
1
4 4 4 4 4 4
= ((1=2) ; 0 ;(1=2) ; (0) ;0;1=2;(0) )(0; 1=2; (0) )
2
and
2 2 2 2 2 2 2 2 2 3
= ((1=2) ; (0) ; (1=2) ; (0) ;(1=2) ; (0) ; (1=2) ; (0) ; (0) ; 1=2; (0) )
3
2 3
((0) ; 1=2; (0) ).
16
10
This then breaks the E E symmetry to SU (2) as required. The action
8 8
of from equation (4) on the heterotic string is as in the given previous
section, but now acts on the theory with reduced symmetry.
Because each SU (2) factor in the gauge group is asso ciated with an orb-
ifold singularity on the Typ eI I/M-theory side of the duality map, the action
of on the heterotic string gauge group is easily seen from the discussion
8
ab ove to b e the exchange of the two SU (2) factors. The following action on
6
the T co ordinates is given by the ansatz of the preceeding sections.
(x :::::x )= (1=2 x ;x +1=2; x ;1=2 x ;x ; x ) (7)
1 6 1 2 3 5 6 7
In the untwisted sector of the theory the massless sp ectrum is given by eight
N =1 vector multiplets and 15 chiral multiplets. Of the chiral multiplets,
seven are singlet untwisted mo duli multiplets and the other eight are adjoint
multiplets of SU (2). The and twisted sectors pro duce no massless
states. The twisted sector pro duces 64 chiral multiplet SU (2) doublets,
of which only 16 are invariant. Hence the resulting massless sp ectrum is
8
precisely that of M -theory on J .
31
We can mo dify this example slightly and pro duce nine more p otential
N = 1 dual examples. This is done as follows:
(x ; ::::x )=(1=2 x ;x ; x ;x ; x ;x ; x ) (8)
1 7 1 2 3 4 5 6 7
10
This choice of symmetry breaking vectors was considered in [7]. 13
This mo di cation has the following signi cance: (i): is no longer freely
acting on the geometry; (ii):now the element acts trivially on the xed
three tori of . This means that the presence of in this form removes
four elements of the singular set of and of the original N = 2 mo del.
So we de nitely have eightvectors surviving the pro jection. However, we
still need to consider the elements of the singular set induced by . Because
the element acts trivially on the singular set from , must contribute
eight additional elements. However, these elements are di erent to the the
other eight, b ecause when the blowup is p erformed, the additional action
of must b e considered on the blowup itself. It turns out that there are
two top ologicall y distinct ways of considering this action on the blowing up
mo des. These twoways di er by the fact that one preserves the generator
2
of H (X; R) and the other changes its sign, (where X is the Eguchi-Hanson
11
blowing up mo de) .
It follows that these two blowups contribute di erent b etti numb ers to
the Joyce manifold. If the extra Z action was not present then each blowup
2
would add one to b and three to b . When the Z is present, the twochoices
2 3 2
in de ning its action on the blowup has the e ect of splitting these original
b etti numb ers, so that the rst typ e of resolution adds one to b and one
2
to b ; and the second adds zero to b and twoto b . So all in all wehave
3 2 3
eight `standard' blowups and eight for which there are twochoices. The b etti
3
numb ers from the original seven torus which survive the Z isometries are
2
b =b =0 and b =7. The eight `standard' blowups add one to b and three to
1 2 3 2
b , giving b = 8 and b = 31. Of the remaining eight `nonstandard' blowups,
3 2 3
if wecho ose l of them to b e of the rst typ e, then this adds l to b oth b and
2
b . The remaining 8 l add zero to b and 16 2l to b . This means that
3 2 3
the Joyce manifold has
b =8+l; b =47 l; l =0;1; :::8 (9)
2 3
Let us now see if we can nd a family of heterotic duals for this family of
Joyce manifolds.
The rst p oint to note is that if we consider the Z Z orbifold of M -
2 2
7
theory on T which is generated by( ; ) then the resulting N = 2 sp ectrum
1
is precisely (20; 20) ie the manifold is of the form CY S . This is the same
19
sp ectrum we obtained using ( ; ) as the orbifold generators. Hence, the
11
[13, 14] can b e consulted for further details. 14
action of is identical to that of . This symmetry b etween the generators
should b e preserved when we consider the action of and on the heterotic
theory. Because wehave already found that the action of preserved the
rank of the gauge group originating in E E in our (20; 20) N = 2 mo del,
8 8
12
we exp ect to do so also. Further, b ecause the heterotic mo del will have
N = 1 sup ersymmetry in four dimensions, we can exp ect a rank 16 gauge
group. We therefore may exp ect on these general grounds that the heterotic
mo del will b e dual to M -theory on the Joyce manifold with l =8,above,
which has a sp ectrum of 16 vector multiplets and 39 chiral multiplets. We
now construct the heterotic background.
Before considering the action on the gauge degrees of freedom, we rst
sp ecify the action of and on the six-torus co ordinates of the heterotic
string. According to our ansatz, these are as follows:
(x ; ::x )= ( x ;1=2 x ;x ; x ; x ;x ) (10)
1 6 1 2 3 5 6 7
(x ; :::x )=(1=2 x ;x ; x ; x ;x ; x ) (11)
1 6 1 2 3 5 6 7
leaves invariantatwo-torus whichisinverted by ; and leaves invari-
antatwo-torus inverted by .Thus the heterotic background has N =1
spacetime sup ersymmetry as exp ected by duality. In fact it is interesting
to note that if this orbifold is blown up, the resulting smo oth manifold is
none other than the CY that app eared on the Typ eI I side in our N =2
19
example. If wenow consider the heterotic theory on the manifold CY we
19
will nd a non-chiral N = 1 theory with 39 mo duli multiplets. If we also
n
sp ecify a U (1) emb edding of the spin connection in the gauge connection,
then we arrive at the rank sixteen mo del conjectured previously in [12]!
However, we can go one stage further and actually give evidence that M -
theory on the orbifold de ned by( ; ; ) is equivalent to the heterotic string
6
on the orbifold de ned by( ; ). The action of the orbifold on the T piece
of the heterotic theory has b een given. It remains to sp ecify the action on the
gauge degrees of freedom. As already noted, we wish to preserve a symmetry
between and , ie they should give rise to identical sp ectra when considered
separately. This can b e achieved by considering identical emb eddings of
the spin connection in the gauge connection, with the connection in one
12
Note, for the reasons stated ab ove, we again will consider the E E broken down to
8 8
16
SU (2) . 15
8
SU (2) factor and the connection in the other. The choices are restricted
by mo dular invariance. Further, b ecause we exp ect 32 chiral multiplets in
M -theory to arise from the twisted sectors (because 32 harmonic three-forms
arise from blowing up),we can exp ect the same in the heterotic theory.
We nd there are essentially two inequivalentchoices of ab elian emb eddings
which give rise to massless states in the twisted sector. Only one choice,
8
corresp onding to the standard emb edding in each SU (2) factor, gives rise
to the correct numberoftwisted sector multiplets. These are the following:
2 6 8
= ((1=2) ; (0) )((0) ) (12)
8 2 8
= ((0) )((1=2) ; (0) ) (13)
8
where the rst(second) bracket denotes the shift in the rst(second) SU (2)
factor. Let us consider the sp ectrum. In the untwisted sector, the sp ectrum
contains 16 N =1vector multiplets and seven mo duli chiral multiplets
(including dilaton). In fact, with the ansatz wehave alluded to one will
always nd seven mo duli multiplets in the untwisted sector. The analogue of
this statement from the M -theory p oint of view is that any Joyce manifold
3
of G holonomy constructed as a blown up Z orbifold of the seven torus
2 2
3
7
has b (T =Z )=7, corresp onding to seven chiral mo duli multiplets in the
3 2
untwisted sector of M -theory!
Now consider the twisted sectors. We nd 16 SU (2) doublet multiplets
from each of the and sectors resp ectively. This gives a total sp ectrum of
16 vector multiplets and 39 chiral multiplets. This is the same sp ectrum as
the example with l = 8 ab ove, as we initially exp ected. Before commenting
on the examples with l =0;1::7wewould like to make some observations.
Firstly,ifwe examine the Calabi-Yau orbifold of the heterotic theory
de ned by equations (13),(14), we note that if we resolved all the orbifold
singularities then the resulting Calabi-Yau manifold is none other than CY !
19
Wehavethus derived the result presented in [12 ].
Secondly,wehave seen that the untwisted matter content will always
3
agree for heterotic/M -theory duals if the M -theory background is a (Z )
2
7
orbifold of T . In the example wehave just considered, wehave further
observed that the twisted sector sp ectrum in the heterotic theory precisely
repro duces the sp ectrum which arises in M -theory from the blowing up pro-
cedure. This is comp elling evidence that wehave again constructed the
correct heterotic background, dual to M -theory on a Joyce manifold. It is 16
also further evidence that orbifold backgrounds are consistentinM-theory.
In fact similar reasoning also applies to N = 2 dual pairs. The M -theory
background for N = 2 sup ersymmetry in four dimensions will b e of the form
1
CY S , for CY any Calabi-Yau space. This background is equivalentto
the Typ eI Ia theory on CY . If CY is constructed as a Z Z orbifold of
2 2
6
T , then the untwisted matter sp ectrum at generic p oints will always con-
tain four massless hyp ermultiplets. The heterotic dual background will then
4 2
b e of the form T =Z T where the Z de nes a K 3 orbifold. This back-
2 2
ground also contains four hyp ermultiplets in the untwisted sector massless
sp ectrum, and it is yet again tempting to p ostulate that the twisted sector
2
of M -theory on the (Z ) orbifold is identical to that of the heterotic Z
2 2
orbifold. Of course, the heterotic sp ectrum dep ends strongly on the choice of
discrete Wilson lines or shift vectors required for mo dular invariance and it
would b e interesting to identify such degrees of freedom in M -theory. Such
an identi cation was made for the Typ eI I theory recently [27] where it to ok
the form of generalised discrete torsion.
Wehave identi ed a heterotic dual theory for the compacti cation of M -
theory on one of a family of nine Joyce manifolds, parametrised by l . What
can wesay ab out the other memb ers of the family?
16
Consider rst M -theory on the example with l =8 ie J .To make the
39
15
transition to the next memb er of the family, J ,wemust blowdownatwo-
40
cycle and blow up a three-cycle; in other words this is precisely an example
of an M -theory conifold typ e transition, and wehave reasonable grounds to
susp ect that such a transition is physically non-singular [20]. In fact this
is nothing but the Higgs mechanism in an N =1M-theory background.
From the heterotic string p oint of view, wewould need a eld content which
along certain Higgs directions repro duces the eld content of all these Joyce
compacti cations of M -theory. However, wehavechosen the most simple
breaking of E E and have restricted ourselves to ab elian emb eddings of
8 8
the spin connection in the gauge connection. There are of course many other
consistent p ossibill i ties that one may consider and it may certainly b e the
case that we can repro duce the required eld content from the heterotic
compacti cation. We are investigating such p ossibill i ties. Thus the heterotic
analogue of these top ological transitions b etween di erentJoyce manifolds
remains a mystery. This is also a consequence of the fact that the singularities
which allow transitions b etween the di erentJoyce manifolds are certainly
not SU (2) singulariti es, but orbifolds of them. One would certainly need to 17
identify the physical implications of this statement from the M -theory p oint
of view, b efore the heterotic description of the transition could b e made.
4 Interpretation.
Given the apparent success of our ansatz, the question arises as to whether
these results have a more satisfying explanation. It is natural to exp ect that
this should come from some substructure in the Joyce manifolds considered
in this pap er. A clue comes from the ubiquityofK3 brations in string
duality [4, 5 , 6 , 29]. Sp eci cally, given a K 3 for the ve brane of M -theory
1
to wrap around and an S for the dual two-brane to wrap around, one can
derive connections b etween string theories in lower dimensions and M -theory
by b ering the K 3over another space [5, 6, 29 ]. For example N = 2 string-
string duality in four dimensions has an interpretation in terms of N =
2 string- string duality in six dimensions, whereby the geometries of the
4
Typ eI Ia theory (K 3) and heterotic theory (T ) resp ectively are b ered over
1
CP [6, 5 ]. We will now show that (at least lo cally) a similar interpretation
holds here for the results of the preceding sections. More precisely,iftheM-
theory compacti cation is a K 3 bration over some three manifold and the
3
heterotic dual compacti cation is a T bration over the same three manifold,
then in the adiabatic approximation of [6], we can exp ect the duality to hold
between the two theories in four dimensions. The following analysis relies
heavily on that of [13, 14].
A seven manifold of G holonomy has two classes of sp ecial submanifolds.
2
This is essentially a consequence of the fact that the torsion-free G structure
2
ofaJoyce seven-manifold is de ned by sp ecifying a particular closed three
form [13, 14]. This is analagous to the Kahler form in Calabi-Yau spaces.
The three form has a four form Ho dge dual. The three form is naturally
identi ed with the volume form of a sp ecial class of three manifolds which
are submanifolds of the Joyce manifold. These are known as asso ciative
submanifolds. They minimize volume in their homology class. The three
cycle dual to the (minimum) volume form is known as a sup ersymmetric
cycle [17].
Similarly the four form which is dual to the three form can b e used to
de ne sp ecial four dimensional submanifolds of the Joyce manifold. These
are known as coasso ciative submanifolds. It was proven in [16] that the di- 18
mension of the mo duli space of such manifolds is given by the number of
self-dual, harmonic two-forms of the submanifold. Therefore, if this num-
b er is non-zero, then the Joyce manifold is (at least lo cally) b ered by the
coasso ciative submanifold.
Joyce [14] has given a prescription for proving whether or not such sub-
manifolds exist for the manifolds he constructs. The argument pro ceeds in
the following way: givenaJoyce manifold, one can consider further orbifolds
by isometries which preserve the holonomy structure. If the singular set of
the isometry consists of n elements of dimension three (eg three-tori) then
the Joyce manifold contains n asso ciative submanifolds (given by these three
-tori). If the singular set contains n dimension four elements (eg K 3) then
these are coasso ciative submanifolds of the Joyce manifold.
If wenow b egin with the duality in seven dimensions b etween M -theory
3
on K 3 and the heterotic string on T , whichwas our original starting p oint,
then wewould exp ect that b ering b oth of these manifolds over the same
three manifold will lead to a duality in four dimensions b etween these two
theories, at least in the adiabatic approximation [6]. We will show, following
Joyce [13 , 14], that the seven manifolds used for M -theory compacti cation in
this pap er are (at least lo cally) bred by K 3. Further, we will similarly show
3
that the seven manifolds contain T submanifolds, which \pull back" to the
six manifold used for the heterotic compacti cation. Unfortunately,itisnot
yet known whether or not these three-tori b er the M -theory and heterotic
geometries, so we cannot say conclusively that the bration picture [6] holds.
However, given our preceding evidence for dual pairs in four dimensions it
is natural to exp ect that the M -theory (heterotic) compacti cation space is
3
globally a K 3(T ) bration.
Let us consider as an example our N = 2, (20; 20) mo del of section 2,
which is de ned on the M -theory background by equations (3) and (4). This
1
gaveusan M-theory compacti cation on CY S . Consider now the
19
following isometry of this manifold:
(x ::::x )=(x ;x ;x ;x ;1=2 x ;1=2 x ;1=2 x ) (14)
1 7 1 2 3 4 5 6 7
7 4
The singular set of T = contains eight copies of T .Now let us consider the
action on these of and which de ne the manifold. Firstly exchanges
these eight four-tori in four pairs of two, leaving four indep endent elements.
7 4
Thus, the singular set of T =(; ) contains four copies of T asso ciated with 19
.However, the xed p oints of intersect those of , hence the singular set
7 4
of T =( ; ; ) contains four copies of T =Z which are asso ciated with ,
2
where the Z is the action of on the four-tori xed by . It is easily seen
2
that the action of on these four-tori de nes a K 3 orbifold metric for each
1
one. Finally,we can say that the xed set of in CY S is four copies
19
1
of K 3. CY S thus contains at least four K 3 submanifolds, all of which
19
are coasso ciative.
From our discussion ab ove it follows that the dimension of the mo duli
2
space of each of these is b (K 3) = 3. It is therefore true that the manifold
+
1
CY S is at least lo cally b ered by K 3 [14]. In fact it is not dicult
19
to see along similar lines that all of the seven manifolds used for M -theory
compacti cation in this pap er admit lo cal K 3 brations.
Using the same example as wehave just discussed, consider the following
1
isometry of CY S :
19
(x ; :::x )=(1=2 x ;1=2 x ; x ; x ;x ;x ;x ) (15)
1 7 1 2 3 4 5 6 7
1
It is not to o dicult to convince oneself that the xed set of in CY S
19
3
is four copies of T . It follows that these are all asso ciative submanifolds of
the seven manifold. Now, b ecause these three tori are lab elled by(x ;x ;x ),
5 6 7
which are also co ordinates of the heterotic background, the ab ove isometry
also xes precisely four three-tori of the heterotic background. Unfortunately
little app ears to b e known ab out the mo duli space of such submanifolds and
3
so we cannot conclusively say that the six manifold admits a T bration.
However consistency with the dualities presented here and with the general
picture of b ering existing dual pairs to obtain more dual pairs [6] suggests
the following:
(i) That the Joyce seven manifolds used for M -theory compactifcation in this
pap er admit global K 3 brations over some three manifold Y .
(ii) That the six manifolds, used to pro duce the heterotic duals to the ab ove
3
M -theory compacti cations, admit global T brations over the same three
manifold Y .
Wehavethus far given examples of M -theory compacti cations on Joyce
manifolds which admit lo cal K 3 brations. The dual heterotic compacti ca-
tions were found to b e on particular Calabi-Yau spaces, whichwe generically
denote by CY .However, it is also p ossible to inerpret other M -theory du-
als of these heterotic compacti cations, as compacti cations of the eleven 20
1
dimensional theory on CY S =Z [21 ]. This is analogous to the two M -
2
theory duals of the heterotic string in seven dimensions, whichwe discussed
in section (1).
It thus app ears that if the bration picture [6] do es hold for the examples
presented here, then even though wehave constructed new dual pairs with
non-freely acting orbifolds of existing dual pairs (namely heterotic/M -theory
dualityinseven dimensions), the adiabatic argument is not violated. This
is only true b ecause the orbifolds to whichwehave restricted our attention
are precisely those which preserve (at least lo cally) the bration of the seven
manifold. It is thus presumably true that if one can nd examples of orbifolds
with higher order isometry groups which are also of this typ e, then one maybe
able to construct more examples, some of whichmay b e phenomenologically
app ealing.
5 Summary and Conclusions
Summarizing the key p oints of this pap er: For the conjectured dualities b e-
tween the various string theories and M -theory, the theories should yield
3
identical massless sp ectra. We b egan with the 11d-theory on a K 3T orb-
6
ifold and the heterotic string on T .We then orbifolded the seven manifold
on the M -theory side and resolved the singularites by blowing up, rst for
an N = 2 example and then for several N = 1 examples. We presented an
ansatz mapping six of the M -theory co ordinates to the six heterotic co or-
dinates. Because of this ansatz, the action of the orbifold isometry group
on the M -theory geometry also sp eci ed the action on the heterotic geom-
etry. Thus all that remained to sp ecify the heterotic background was the
emb edding of the spin connection in the gauge connection. This was derived
by requiring mo dular invariance at the orbifold limit. Using this ansatz we
rederived previous results in the literature and pro duced new dual pairs in
four dimensions. It therefore app ears that this ansatz is consistent for pro-
ducing dual pairs by orbifolding whenever the non-freely acting elements of
the orbifold group have singularities which are of the K 3 orbifold typ e. This
is in accord with the underlying structure of K 3 brations, [4, 5, 6 ], b ecause,
as was demonstrated in [14 ], and as we showed in the last section , manifolds
which are constructed with such orbifold elements admit (at least lo cally) a
K 3 bration. The duality is then exp ected to hold on general grounds if the 21
3
heterotic geometry is a T bration [6].
Moreover, we also gave further evidence that M -theory is consistenton
orbifolds by comparing the untwisted and twisted matter content in sucha
theory with its heterotic dual. The untwisted matter contents for N =2
and N = 1 dual pairs always agree for the dual pairs constructed according
to our ansatz. Sp eci cally, if the four dimensional M -theory background
1 6
is of the form CY S , where CY is a Z Z orbifold of T , then the dual
2 2
heterotic background will, according to our ansatz, necessarily b e of the form
4 2
T =Z T . Both of these N = 2 theories haveanuntwisted matter content
2
of four hyp ermultiplets. Similarly, M -theory on a Joyce G orbifold of the
2
7 3
form T =Z will always contain an untwisted sector of seven N = 1 mo duli
2
multiplets. Then, according to our ansatz, the dual heterotic background will
6
b e of the form T =Z Z , and will also have seven N = 1 mo duli multiplets
2 2
in the untwisted matter sector. What is even more comp elling is that wewere
able to show that for some simple choices of orbifold gauge emb eddings, the
number of twisted sector multiplets agree also.
Finally we wish to add that this metho d of construcing dual pairs has
b een successfully applied to the construction of N = 1 dual pairs in three
dimensions [30 ]. In fact, using this construction wehave constructed heterotic
duals (on Joyce G manifolds) for M -theory compacti cations on al l known
2
Joyce manifolds of Spin(7) holonomy [15]. This may shed some light on the
mysterious sup ersymmetric theory in twelve dimensions which app ears to b e
required to complete the duality picture [32, 33 ]. As noted in [33], these
theories may b e an explicit realisation of the b eautiful ideas of Witten [34]
whichmay solve some of the long standing problems of theoretical physics
and p ossibly take dualitytowards reality.
Acknowledgements.
The author is extremely indebted to Jerome Gauntlett, Chris Hull, Tomas
Ortin, Bas Peeters, Wa c Sabra, Ashoke Sen and Steven Thomas for discus-
sions. The author would also like to thank PPARC, by whom this work is
supp orted.
References
[1] C.M.Hull and P.Townsend, `Unity of Sup erstring Dualities,' Nucl. Phys.
B438 (1995) 109. 22
[2] E. Witten, `String Theory Dynamics in Various Dimensions,' Nucl.Phys.
B443 (95) 85.
[3] S. Kachru and C. Vafa, `Exact Results for N=2 Compacti cations of
Heterotic Strings,' Nucl.Phys.B450 (95) 69.
[4] A.Klemm, W.Lerche and P.Mayr, `K3 Fibrations and Heterotic- Typ e
I I duality,' Phys.Lett. B357 (95) 313.
[5] P.Aspinwall,J.Louis, `On the Ubiquity of K3 Fibrations in String Dual-
ity.' hep-th 9510234.
[6] C. Vafa and E. Witten,`Dual String Pairs With N=1 and N=2 Sup er-
symmetry in Four Dimensions,' hep-th/9507050.
[7] J.Harvey, D.Lowe and A.Strominger,`N=1 String Duality,' Phys.Lett.
B362 (95) 65.
[8] S. Ferrara, J.Harvey, A.Strominger and C.Vafa,`Second Quantized Mir-
ror Symmetry,' Phys.Lett. B361 (95) 59.
[9] P.Aspinwall,`Enhanced Gauge Symmetries and K3 surfaces,' Phys.Lett.
B357 (1995) 329.
[10] J. Schwarz and A. Sen,`The Typ e I IA Dual of the CHL Compacti ca-
tion,'Phys.Lett. B357 (1995) 323.
[11] C.M.Hull and P.Townsend,`Enhanced Gauge Symmetries In Sup erstring
Theory,' Nucl.Phys.B455 (1995) 525.
[12] G.Papadop oulos and P.Townsend,`Compacti cation of d=11 sup ergrav-
ity on spaces of exceptional holonomy.' Phys.Lett.B356 (1995) 300.
[13] D.D.Joyce,`Compact Riemannian 7-Manifolds with G Holonomy:1' Ox-
2
ford 1994 Preprint, to app ear in J.Di .Geom.
[14] D.D.Joyce,`Compact Riemannian 7-manifolds with G Holonomy:2' Ox-
2
ford 1994 Preprint, to app ear in J.Di .Geom.
[15] D.D.Joyce,`Compact Riemannian 8-manifolds with Holonomy Spin(7)'
Oxford 1994 Preprint, to app ear in Inv.Math. 23
[16] R.Mclean,`Deformations and Mo duli of Calibrated Submanifolds.' Ph.D
thesis, Duke University, (1990).
[17] K.Becker,M.Becker,A.Strominger,`Fivebranes, Membranes and Nonp er-
turbative String Theory.' Nucl.Phys. B456 (1995) 130.
[18] M.Du ,B.Nilsson and C.Pop e,`Kaluza-Klein Sup ergravity.' Phys.Rep
130, 1986, 1-142.
[19] P.Townsend,`A New Anomaly Free Chiral Sup ergravity from Compact-
i cation on K3', Phys.Lett.139B (84) 283.
[20] A.Strominger,`Massless Black holes and Conifolds in String Theory'
Nucl.Phys.B451 (1995) 96, B.Greene,D.Morrison and A.Strominger,
`Black Hole Condensation and the Uni cation of String Vacua' Nucl.
Phys. B451 (1995) 109.
[21] P.Horava, E.Witten,`Heterotic and Typ eI Dynamics from Eleven Di-
mensions.' hepth/9510209.
[22] E.Witten,`Five-branes and M-theory on an Orbifold' hepth 9512219,
K.Dasgupta, S.Mukhi,`Orbifolds of M-theory' hepth 9512196.
[23] J.Polchinski,`Dirichlet Branes and RR charges.' Phys.Rev.Lett.75 (1995)
4724.
[24] P.Townsend,`D
branes from M branes' hepth 9512062, A.Strominger,`Op en P-branes'
hepth 9512059.
[25] L.Dixon,`World Sheet Asp ects of String Compacti cation'. Pro cedings
of 1987 Summer Workshop ICTP,World Scienti c.
[26] M.Green,J.Schwarz,P.West,`Anomaly Free chiral Theories in Six Dimen-
sions' Nucl.Phys.B254 (1985) 327.
[27] P.Aspinwall,`An N=2 Dual Pair and a Phase Transition' hepth 9510142.
[28] S.Chaudhuri,J.Polchinski,`Mo duli Space of CHL Strings' Phys.Rev/D52
(1995) 7168. 24
[29] John Schwarz, `The Power of M Theory' Phys.Lett. B367 (1996) 97.
[30] B.S.Acharya, in preparation.
[31] M.Du , R.Minasian, E.Witten, `Evidence for heterotic-heterotic dual-
ity', hepth/9601036 .
[32] C.M.Hull,`String Dynamics at Strong Coupling',hepth/ 9512181.
[33] C.Vafa,`Evidence for F-theory', hepth/9602022.
[34] E.Witten,`Is Sup ersymmetry Really Broken?' Int.J.Mo d.Phys.A10
(1995) 1247,`Strong Coupling and The Cosmological Constant.'
Mo d.Phys.Lett. A10 (1995) 2153. 25