Evidence for Heterotic/Heterotic Duality
CTP-TAMU-54/95
hep-th/9601036
EVIDENCE FOR HETEROTIC/HETEROTIC
DUALITY
1
M. J. Du
Center for Theoretical Physics, Texas A&M University, Col lege Station,
Texas 77843, U. S. A.
2
R. Minasian
Theory Division, CERN, CH 1211 Geneva 23, Switzerland
and
3
Edward Witten
School of Natural Sciences, Institute for Advanced Study, Olden Lane,
Princeton, NJ 08540, U. S. A.
ABSTRACT
We re-examine the question of heterotic - heterotic string dualityinsix
dimensions and argue that the E E heterotic string, compacti ed on K 3
8 8
with equal instanton numb ers in the two E 's, has a self-duality that inverts
8
the coupling, dualizes the antisymmetric tensor, acts non-trivially on the
hyp ermultiplets, and exchanges gauge elds that can b e seen in p erturbation
theory with gauge elds of a non-p erturbative origin. The sp ecial role of
the symmetric emb edding of the anomaly in the two E 's can b e seen from
8
eld theory considerations or from an eleven-dimensional p oint of view. The
duality can b e deduced by lo oking in two di erentways at eleven-dimensional
1
M -theory compacti ed on K 3 S =Z .
2
1
Research supp orted in part by NSF Grant PHY-9411543.
2
World Lab oratory Fellow.
3
Research supp orted in part by NSF Grant PHY92-45317.
1 Intro duction
Prior to the recent surge of interest in a dualitybetween heterotic and Typ e
I IA strings [1, 2, 3 , 4, 5], it was conjectured (on the basis of D = 10 heterotic
string/ vebrane duality [6, 7]) that in D 6 dimensions there ought to exist
a dualitybetween one heterotic string and another [8, 9, 10 , 11, 12 , 13 , 14 ].
A comparison of the fundamental string solution [15] and the dual solitonic
string solution [10, 11 ] suggests the following D = 6 duality dictionary: the
e e f
dilaton , the string -mo del metric G and 3-form eld strength H of
MN
the dual string are related to those of the fundamental string, , G and
MN
H by the replacements
e
! =
e
G ! G = e G
MN MN MN
f
H ! H = e H (1)
In going from the fundamental string to the dual string, one also inter-
changes the roles of worldsheet and spacetime lo op expansions. Moreover,
since the dilaton enters the dual string equations with the opp osite sign to
the fundamental string, it was argued in [8, 10, 11] that in D = 6 the strong
coupling regime of the string should corresp ond to the weak coupling regime
of the dual string:
=2
e
=
6 6
e
where and are the fundamental string and dual string coupling con-
6 6
stants. Because this dualityinterchanges worldsheet and spacetime lo op
expansions { or b ecause it acts by dualityonH { the duality exchanges the
tree level Chern-Simons contributions to the Bianchi identity
0 2
dH = (2 ) X
4
1
2 2
X = [trR v trF ] (3)
4
2
4(2 )
with the one-lo op Green-Schwarz corrections to the eld equations
0 2
f f
dH = (2 ) X
4
1
2 2
f
e
X = [trR v trF ] (4)
4
2
4(2 ) 2
th
Here F is the eld strength of the comp onent of the gauge group, tr
e
denotes the trace in the fundamental representation, and v ; v are constants.
(As explained in App endix A, wemay, without loss of generality,cho ose the
string tension measured in the string metric and the dual string tension
mesured in the dual string metric to b e equal.) In fact, the Green-Schwarz
anomaly cancellation mechanism in six dimensions requires that the anomaly
eight-form I factorize as a pro duct of four-forms,
8
f
I = X X ; (5)
8 4 4
and a six-dimensional string-string duality with the general features summa-
rized ab ovewould exchange the two factors.
Until now, there has not b een a really convincing example of heterotic-
heterotic duality in six dimensions. In [11], it was prop osed that the D =10
SO(32) heterotic string compacti ed to D =6onK3 might b e dual to
the D =10 SO(32) heterotic vebrane wrapp ed around K 3. However, this
candidate for a heterotic/heterotic dual string pair su ered from the following
drawbacks:
1) The existence of a vebrane carrying the requisite SO(32) quantum
numb ers is still unclear. Even if it exists, its prop erties are not well-understo o d.
2) The anomaly eight-form of this mo del is given by (5) with [19]
1
2 2 2
X = [trR trF 2trF ]
4 SO(28) SU (2)
2
4(2 )
1
2 2 2
f
X = [trR + 2trF 44tr F ]; (6)
4
SO(28) SU (2)
2
4(2 )
and one of the gauge co ecients in the second factor enters with the wrong
sign.
The structure of this equation actually presents a problem that is indep en-
dentofany sp eculation ab out string-string duality.Itwas shown by Sagnotti
[20] that corrections to the Bianchi identities of the typ e (3) and to the eld
equations of the typ e (4) are entirely consistent with sup ersymmetry, with
e
no restrictions on the constants v and v . Moreover, sup ersymmetry relates
these co ecients to the gauge eld kinetic energy. In the Einstein metric
c =2
G = e G , the exact dilaton dep endence of the kinetic energy of
MN MN
the gauge eld F ,is
M N
3
p
(2 )
=2 =2 MN
c
e
L = G v e + v e trF F : (7)
g aug e
M N
0
8 3
e
Positivity of the kinetic energy for all values of thus implies that v and v
should b oth b e non-negative, and at least one should b e p ositive. This fails
for the SO(32) heterotic string, as we see from the formula for the anomaly
eight-form. Some interesting new \phase transition" must o ccur at the value
of at which the SO(28) coupling constant app ears to change sign, and
at least until this phase transition is understo o d, its o ccurrence mightwell
obstruct simple attempts to extrap olate from a string description at large
negative to a dual string description at large p ositive.
In this pap er, we shall attempt to remedy these problems as follows.
0
1 ) It has recently b een recognised that the ten-dimensional E E het-
8 8
10 1
erotic string is related to eleven-dimensional M -theory on R S =Z [16 ],
2
10 1
just as the ten-dimensional Typ e I IA string is related to M -theory on R S
6 1
[17]. By lo oking in two di erentways at M -theory on R K 3 S =Z ,we
2
get a de nite framework for deducing string-string duality. This framework
shows that the gauge group should b e E E , the vacuum gauge bundle
8 8
should have equal instanton numb ers in each E (a situation we will refer to
8
4
as symmetric emb edding ), and the duality acts in a non-trivial fashion on
the hyp ermultiplets.
From this eleven-dimensional p oint of view, one heterotic string comes by
1
wrapping the D = 11 membrane around S =Z and the dual heterotic string
2
1
is obtained by reducing the D =11 vebrane on S =Z and then wrapping
2
around K 3. This is quite similar to the eleven-dimensional derivation of
1 1
heterotic - Typ e I IA duality, which is recovered if we replace S =Z by S in
2
5
the ab ove scenario [18 ].
0
2 )Now let us discuss the anomaly p olynomial. Picking a vacuum on
K 3 with equal instanton numb ers in each E will break E E to a sub-
8 8 8
group. Generically E E is completely broken, so there are no questions of
8 8
whether the gauge contributions to the anomaly eight-form are compatible
with duality. But we also want to understand how the duality acts on vacua
with non-trivial unbroken gauge groups. For instance, in a vacuum in which
the gauge bundle breaks E E to E E (a maximal p ossible unbroken
8 8 7 7
4
Symmetric emb edding entered naturally in [21] in constructing simple examples of
heterotic/Typ e I I duality in four dimensions.
5
The interpretation of the heterotic string as a wrapping of a vebrane around K 3,
or around a K 3 sub-manifold of a Calabi-Yau manifold or Joyce manifold, is presumably
the explanation for the ubiquity in string/string dualityofK3 itself and Calabi-Yau and
Joyce manifolds corresp onding to brations of K 3. 4
subgroup of E E ) the anomaly eight-form is
8 8
1 1 1 1
2 2 2 2
I = [trR trF trF ] [trR ] (8)
8
E E
7 7
2 2
4(2 ) 6 6 4(2 )
e
We see that v = 0, so (i) there is no wrong sign problem and one can
p ossibly extrap olate to strong coupling without meeting a phase transition,
e
but (ii) since v 6= v , there is no manifest self-duality. Qualitatively similar
e
results hold (that is, v = 0) for any other unbroken subgroup of E E .
8 8
This qualitative picture dep ends on having equal instanton numb ers in the
e
two E 's; in any other case, v < 0 for some subgroups of E E , and phase
8 8 8
transitions of some kind are unavoidable.
Because of (ii), it might app ear that duality is imp ossible, but since as
0
in 1 )above there is a systematic framework for deducing the duality,we
are reluctant to accept this interpretation. We are led therefore to assume
that the duality exchanges p erturbative gauge elds, that is gauge elds of
p erturbative origin, with non-p erturbative gauge elds. Despite the name,
non-p erturbative gauge elds, if they app ear at all, app ear no matter how
small the string coupling constantmay b e; in fact, as the dilaton is part
of a tensor multiplet in K 3 compacti cation, the unbroken gauge group is
indep endent of the string coupling constant at least if the low energy world
can b e describ ed by known physics. Non-p erturbative gauge elds, that is
gauge elds that are not seen in p erturbation theory but app ear no matter
howweak the string coupling constantmay b e, can therefore only app ear at
p oints in mo duli space at which p erturbation theory breaks down b ecause of
a kind of singularity.
A prototyp e for non-p erturbative gauge elds in the heterotic string are
the SU (2) gauge elds that arise for the SO(32) heterotic string when an
instanton shrinks to zero size [23]. Such gauge elds have v = 0; this can
b e seen either from (a) the form of the anomaly p olynomial, as computed
in section (4) of [38]; (b) the physical picture of [23] according to which
making the heterotic string dilaton smaller causes the SU (2) gauge multiplet
to app ear \farther down the tub e," without changing its physical prop erties
such as the gauge coupling; or (c) the description in [23 ] in terms of Typ e I
D -branes, where one can explicitly compute the SU (2) gauge coupling and
6
compare to (7). In contrast to non-p erturbative gauge elds whichhave
6
The vact that v = 0 for these non-p erturbative gauge elds was in essence also noted
by V. Kaplunovsky. 5
v = 0, p erturbative gauge elds always have v > 0. In fact, as discussed
in App endix B, v is essentially the Kac-Mo o dy level.
Thus, our prop osal is that heterotic string-string duality, at least in the
case of the symmetric emb edding in E E , exchanges p erturbative gauge
8 8
e
elds of v = 0 with non-p erturbative gauge elds of v =0.Aninteresting
conspiracy of factors makes this p erhaps radical-soundi ng prop osal p ossible.
(A) Since non-p erturbative gauge elds can only arise at particular lo ci in hy-
p ermultiplet mo duli space (where a singularity develops in the K 3 manifold
or its gauge bundle, giving a p ossible breakdown of p erturbation theory as in
[23]), the prop osal is p ossible only b ecause with the symmetric emb edding in
the gauge group, E E is generically completely broken, and p erturbative
8 8
gauge elds only app ear at particular lo ci in hyp ermultiplet mo duli space.
(B) Since the lo ci in hyp ermultiplet mo duli space which are candidates for
nonp erturbative gauge elds (b ecause of a singularity in the manifold or the
gauge bundle) are di erent from the lo ci where symmetry breaking is partly
turned o and unbroken p erturbative gauge elds app ear, the prop osal is
p ossible only b ecause the mechanism for string-string duality alluded to in
0
1 ) gives a duality that acts non-trivially on the hyp ermultiplets. (C) It is
0
e
essential that with the symmetric emb edding promised in 1 ), one has v =0
e
for all p erturbative gauge elds. If indeed one had v < 0 in some case, one
would have to face the issue of the phase transition implied by the wrong
sign gauge kinetic energy. On the other hand, p erturbative gauge elds with
e
v > 0would have to b e dual to p erturbative gauge elds, leading to a con-
tradiction given that one do es not have manifest duality of the p erturbative
gauge elds.
Once one accepts that after exchanging p erturbative and non-p erturbative
e
gauge elds, the v and the v are equal, the equality of the numerical co-
ecents app earing in the gauge kinetic terms (7), and in the eld equations
and Bianchi identities (4), (3) means that there maynow b e a full- edged
self-duality of the D = 6 string extending the symmetry of the low energy
sup ergravity and acting on some of the massless elds by (1):
!
G ! e G
MN MN
H ! e H (9) 6
2 The Fundamental String on K 3 And The
Low Energy Sup ergravity
K 3 compacti cation of a chiral string theory in ten dimensions gives a chiral
six-dimensional theory. K 3 is a four-dimensional compact closed simply-
connected manifold. It is equipp ed with a self-dual metric with holonomy
group SU (2). It was rst considered in a Kaluza-Klein context in [24, 25]
where it was used, in particular, as a way of compactifying D = 11 sup er-
gravityto D= 7 and D = 10 sup ergravityto D= 6. Our interest here is in
K 3 compacti cation of the heterotic string. Because of the SU (2) holonomy,
half the sup ersymmetry survives in K 3 compacti cation, and hence, starting
from N = 1 sup ergravityinD= 10, we get N = 1 sup ergravityinD =6
(which has half as many sup ercharges).
There are four massless N =1;D = 6 sup ermultiplets to consider:
A+
+
S uper g r av ity M ul tipl et G ; ;B
MN MN
M
A
T ensor M ul tipl et B ; ;
MN
a
H y per mul tipl et ;
A+
Y ang M il l s M ul tipl et A ;
M
+
All spinors are symplectic Ma jorana{Weyl. The two-forms B and B
MN MN
have three-form eld strengths that are self-dual and anti-self-dual, resp ec-
tively. Only with precisely one tensor multiplet added to the sup ergravity
multiplet is there a conventional covariant Lagrangian formulation. In K 3
compacti cation, the zero mo des of the sup ergravitymultiplet in D =10
give this combination plus 20 massless matter hyp ermultiplets. The 80
scalars in those multiplets parametrize the coset SO(20; 4)=S O (20) SO(4)
[26, 27 , 28], which is the mo duli space of conformal eld theories on K 3. No
vector multiplets come from the ten-dimensional sup ergravitymultiplet since
K 3 has no isometries and is simply connected.
Six-dimensional vector multiplets and additional hyp ermultiplets come
from reduction on K 3 of the ten-dimensional gauge group SO(32) or E
8
E ,aswas rst analyzed in [29 ]. An imp ortant constraint comes from the
8
P
0 2 2
anomaly cancellation equation dH =( =4) ( trR v trF ) whichwas
discussed in the Intro duction. A global solution for H exists if and only if
P
2 2
the integral over K 3oftrR equals that of v trF . This amounts to the
statement that the vacuum exp ectation value of the SO(32) or E E gauge
8 8 7
elds must b e a con guration with instanton numb er 24. In the E E case,
8 8
it is the sum of the instanton numb ers in the two E 's that must equal 24.
8
We will b e interested mainly in the \symmetric emb edding," the case where
the instanton numberis12ineachE .
8
With instanton number12ineachE, the generic E E instanton
8 8 8
on K 3 completely breaks the gauge symmetry. This will b e explained in
more detail in section (4), together with generalizations. Unbroken gauge
symmetry arises if the vacuum gauge bundle takes values in a subgroup G of
E E , in which case the unbroken subgroup of E E is the commutant G
8 8 8 8
of H , that is the subgroup of E E that commutes with H . The G quantum
8 8
numb ers of the massless hyp ermultiplets can b e determined by decomp osing
the adjoint representation of E E under G H , as in [29].
8 8
The Anomaly Polynomial
Now let us explain some statements made in the Intro duction ab out the
anomaly p olynomials. Let F , i =1;2 b e the eld strengths of the two E 's.
i 8
2
Let TrF b e the traces in the adjoint representations of the two E 's, and
i 8
2 2
let trF =(1=30) Tr F . In ten dimensions, the anomaly twelve-form I
i i 12
f
factorizes as I = X X , with
12 4 8
X
1
2 2
X = [trR trF ] (10)
4 i
2
4(2 )
i
f
and X the more imp osing expression
8
2
3 1
2 2 2 2 2 2
f
X (trF ) + (trF ) trF +trF
8 1 2 1 2
4 4
2
1 1 1
2 2 2 4 2
+ trF +trF : (11) trR trR + trR
1 2
8 8 32
f
The six-dimensional anomaly four-form X is obtained byintegrating
4
f
X over K 3. Also in writing an anomaly four-form in six dimensions, one
8
understands the six-dimensional eld strengths F and F to takevalues in
1 2
the unbroken subgroup of the gauge group, that is the part that commutes
2 2 2
with the gauge bundle on K 3. If we let hR i, hF i, and hF i denote the
1 2
2 2 2
integrals of trR ,trF , and trF over K 3, then
1 2
1 1 1
2 2 2 2
f
X trF hF i hF i hR i
4 1 1 2
2 4 8 8
1 1 1 1 1
2 2 2 2 2 2 2 2
i) : i i i+hF +trF hF hF hR i +trR hR i (hF
2 2 2 1 1
2 4 8 16 8
(12)
The top ological condition on the vacuum gauge bundle that was explained
ab ove amounts to
2 2 2
i hR i =0: (13) i + hF hF
2 1
2
e
With the use of this equation, one sees that the co ecients v of trF and
i 1
2
f
trF in X are equal and opp osite. So the \wrong sign" problem explained
2 4
e
in the Intro duction is avoided if and only if the v vanish. (Otherwise, the
i
problem arises in the E that has smaller instanton numb er.) From the
8
2 2
ab ove formulas, the condition for this is that hF i = hF i, that is the two
1 2
E gauge bundles have equal instanton numb ers. So wehave recovered the
8
statement in the Intro duction that in E E , the sign problem is avoided
8 8
e
only for the symmetric emb edding, for which the v vanish for all subgroups
i
of E E . A similar analysis for SO(32) recovers the anomaly formula
8 8
given in the Intro duction and in particular shows the o ccurrence of the sign
problem for SO(32).
Further Aspects Of The Low Energy Supergravity
As a guide to the kind of dualities one might exp ect in the string theory,
let us lo ok in more detail at the corresp onding N =1;D = 6 sup ergravity
theories. We shall follow [20] but with the following mo di cations: bycho os-
ing just one tensor multiplet wemay write a covariant Lagrangian as well
as covariant eld equations; we use the string -mo del metric so as to em-
phasize the tree-level plus one-lo op nature of the Lagrangia n; we also write
0
the coupling in terms of the string slop e parameter ;we shall also include
Lorentz as well as Yang-Mills corrections to the Bianchi indentities and eld
equations. We shall denote the contribution to the action of L fundamental
e
string lo ops and L dual string lo ops by I . The b osonic part of the action
e
L;L
of the takes the form
I = I + I + I + ::: (14)
00 01 10
where
Z
3
p
(2 )
6 MN
I = d x Ge [R + G @ @
00 G M N
02
1
MQ NR PS
G G G H H ] (15)
MNP QRS
12 9
where M; N =0; :::; 5 are spacetime indices and H is the curl of a 2-form B ,
where
Z
3
p
(2 )
6 MP NQ
d x I = Ge [G G trR R
01 MN PQ
0
8
MP NQ
v G G trF F ] (16)
MN PQ
together with Chern-Simons corrections to H appropriate to (3), and where
Z
3
p
(2 )
6 MP NQ MP NQ
e
d x I = G[G G trR R v G G trF F
10 MN PQ
MN PQ
0
8
Z
1 1
f
e
2 (17) B X + ! !
4 3 3
2 0
(2 ) 3
M
6
f
e e
Here ! and ! ob ey 2d! = X and 2d! = X . This last term ensures that
3 3 3 4 3 4
H ob eys the eld equations appropriate to (4). The metric G is related
MN
c
to the canonical Einstein metric G by
MN
=2 c
G = e G (18)
MN MN
where the D = 6 dilaton. There will also b e couplings to the hyp ermulti-
plets, b oth charged and neutral, whichwe shall not attempt to write down.
They will b elong to some quaternionic manifold, which is probably quite
complicated.
The most obvious dual sup ergravity action is given by a similar expression
obtained by replacing each eld with its dual counterpart according to the
following duality dictionary:
e
=
e
G = e G
MN MN
f
H = e H
e
A = A (19)
M M
where denotes the Ho dge dual. (Since the H equation is conformally in-
variant, it is not necessary to sp ecify which metric is chosen in forming the
e
dual.) The dual metric G is related to the canonical Einstein metric by
MN
=2 c
e
(20) G = e G
MN MN 10
It is also p ossible (and will b e necessary in our application) to combine the
duality just describ ed with a transformation of the hyp ermultiplets and a
p ermutation of the various p ossible factors in the gauge group:
! ( ) (21)
where ( ) is the gauge group into which the duality maps the gauge group
. With or without such a transformation of hyp ermultiplets, the ab ove
dictionary achieves just the rightinterchange of tree-level and one lo op e ects
required by heterotic/heterotic duality, namely
I $ I (22)
10 01
f e
In particular, with H the eld strength of a two-form B , with Chern-Simons
corrections appropriate to (4), this duality exchanges the Bianchi identities
(3) and eld equations (4). When the p ermutation ! ( ) is taken into
account, wehave hop efully
e
v = v (23)
( );
re ecting the discrete symmetry (9).
3 Deduction Of String-String Duality From
Eleven Dimensions
To deduce heterotic-heterotic dualityonK3, we b egin with the eleven-dimensional
6 1
M -theory on R K 3 S =Z . By lo oking at this theory in two di erent
2
ways, we will deduce a dualitybetween heterotic strings. On the one hand,
1
we use the fact that the M -theory on Y S =Z , for any Y , is equivalent
2
to the E E heterotic string on Y , with a string coupling constant that
8 8
1
is small as the radius of the S =Z shrinks. On the other hand, we use the
2
fact that the M -theory on Z K 3, for any Z , is equivalent to the heterotic
3
string on Z T , with a string coupling constant that is small when the K 3
6 1
shrinks. The p oint of starting with W = R K 3 S =Z is that it can
2
1 6
b e written as either Y S =Z , with Y = R K 3, or as Z K 3, with
2
6 1
Z = R S =Z .
2
1 1
If welookatW as Y S =Z , then we deduce that as the S =Z b ecomes
2 2
small, the M -theory on W is equivalenttotheE E heterotic string on
8 8
6
Y = R K 3. 11
A little more subtlety is required if we try to lo ok at W as Z K 3. From
this vantage p oint, it app ears that as the K 3 shrinks, we should get a weakly
3 6 1 3
coupled heterotic string on Z T = R S =Z T . This cannot b e the
2
6 1 3
right answer, as R S =Z T is unorientable, and the parity-violating
2
heterotic string cannot b e formulated on this space. One must note that when
6 1 6 1
one divides R S K 3byZ to get the M -theory on R S =Z K 3,
2 2
the three-form p otential A of the low energy limit of the M -theory is o dd
under the Z . Compacti ed on K 3, the three-form gives 22 vector elds that
2
come from the two-dimensional cohomology of K 3; these are related to the
momentum and winding mo des and Wilson lines of the heterotic string on
3
T .For all the momentum and winding mo des of the heterotic string to b e
3
o dd under the Z means that the Z must act as 1 on the T . So when we
2 2
6 1
shrink the K 3 factor in the M -theory on R S =Z K 3, we get a heterotic
2
6 1 3 6 1 3
string on not R S =Z T but R (S T ) =Z .
2 2
1 3 4
Now roughly sp eaking (S T ) =Z = T =Z is a K 3 orbifold, so we
2 2
have arrived again at a heterotic string on K 3. Actually, there are a few
subtleties hidden here. For one thing, in general we will not really get in this
6 1
waya K3 orbifold. When the M -theory is formulated on R S =Z K 3,
2
6
there are propagating E gauge elds on each copyof R K3 coming from
8
1
a xed p ointintheZ action on S , and a sp eci cation of vacuum requires
2
picking a K 3 instanton on each E .Achoice of such an instanton represents,
8
at least generically, a departure from a strict orbifold vacuum. Since the
6 1
vacuum on R S =Z K 3was not really an orbifold vacuum, the same
2
6 1 3
will b e true by the time we get to a heterotic string on R (S T ) =Z .
2
6 1
The fact that M -theory on R S =Z K 3 turns intoaweakly coupled
2
heterotic string in two di erent limits is a kind of dualitybetween heterotic
strings. From the p oint of view of either one of these limits, the other one
is strongly coupled; we will b e more precise ab out this b elow. An observer
studying one of the two limiting heterotic strings sees a strongly coupled
limit in which there is a weakly coupled description by a di erent heterotic
string; this is heterotic-heterotic duality.
A few p oints should still b e explained: (i) The duality is an electric-
magnetic string-string duality in the sense describ ed in the Intro duction.
(ii) The duality acts non-trivially on the hyp ermultiplets, a fact whose im-
p ortance was explained in the Intro duction. (iii) The construction { which
obviously requires that the heterotic string gauge group b e E E {works
8 8
only for the symmetric emb edding with equal instanton numb ers in the two 12
E 's.
8
7
The rst p oint is a simple consequence of eleven-dimensional facts. We
b egin with the fact that the eleven-dimensional M -theory has two-branes and
ve-branes that are electric-magnetic duals. Consider in general a Kaluza-
Klein vacuum in a theory containing a p-brane; for simplicity consider the
1
illustrative case that the vacuum is Q S for some Q. A p-brane can
1
b e wrapp ed around S , giving a (p 1)-brane on Q.Orap-brane can b e
1
\reduced" on S ,by whichwe mean simply that one takes the p-brane to b e
1 1
lo calized at a p ointonS . This gives a p-brane on Q, with the p osition on S
seen an a massless world-volume mo de. The two op erations of wrapping and
reduction are electric-magnetic duals, so that if one starts with dual p-branes
1
and q -branes, the wrapping of one around S and reduction of the other on
1
S gives dual ob jects on Q.
6 1
Now let us apply this wisdom to M -theory on R S =Z K 3. In eleven
2
dimensions, the M -theory has dual two-branes and ve-branes. When the
1
S =Z shrinks, an e ective heterotic string in ten dimensions is obtained by
2
1
wrapping the two-brane around S =Z , giving a one-brane whichwas seen
2
in [16 ] to have the world-sheet structure of an E E heterotic string. An
8 8
e ective six-dimensional heterotic string is then obtained by reduction on
K 3. On the other hand, when the K 3 shrinks, an e ective heterotic string is
obtained by wrapping the ve-brane around K 3. An e ective six-dimensional
1
heterotic string is then obtained by reduction on S =Z . Since wrapping
2
1
the ve-brane around K 3 and reducing it on S =Z is dual to wrapping
2
1
the membrane around S =Z and reducing it on K 3, the two e ective six-
2
dimensional heterotic strings are electric-magnetic duals to each other in the
sense describ ed in the Intro duction. This provides an answer to question (i)
ab ove. To further con rm our understanding, we compute b elow the six-
dimensional string coupling constants of the two heterotic string theories,
and show that they are inverses of each other, as exp ected for a pair of
six-dimensional dual strings.
Nowwe come to question (ii), whichistoshow that this duality is not the
minimal duality suggested bylow energy sup ergravity, but acts non-trivially
6 1
on the hyp ermultiplets. In fact, b egin with M -theory on R S =Z K 3,
2
with some mo dulus for the K 3 and with a particular choice of E gauge
8
1
bundles at xed p oints. If one shrinks the S =Z one simply gets a heterotic
2
7
The following two paragraphs b ene ted from a discussion with P. Horava. 13
string on the same K 3, with the same E gauge bundles, that one started
8
with. If instead one shrinks the K 3, one has an adventure describ ed ab ove
1 3
involving a non-orbifold vacuum on (S T ) =Z . This is presumably to b e
2
interpreted as a K 3 (in fact, any(0;4) conformal eld theory of the appro-
priate central charge is b elieved to describ e a K 3withavector bundle), but
8
it certainly do es not lo ok like the K 3 that we started with. We take this to
mean that the K 3 asso ciated with the dual string (obtained by wrapping the
ve-brane) is not the same as the K 3we started with in eleven-dimensions,
or di erently put that the duality acts non-trivially on the hyp ermultiplets,
which are the mo duli of K 3 and the vector bundle. In fact, the action on the
hyp ermultiplets lo oks rather complicated, and understanding it b etter would
b e an imp ortant step.
Finally we come to question (iii). As wehave presented the eleven-
dimensional construction so far, the assignment of instanton numb ers to the
two E gauge bundles do es not seem to matter. But we claim that actually,if
8
examined more closely, the construction works only for the symmetric emb ed-
ding, that is for equal instanton numb ers in the two E 's. The three-form
8
p otential of eleven-dimensional sup ergravity has a four-form eld strength
K . M -theory compacti cations on K 3 can b e distinguished according to the
quantized value of the ux [18]
Z
2m
K = ; m = integ er (24)
T
K 3
3
where T is the membrane tension. K 3 compacti cation of M -theory has
3
usually b een discussed only for m =0. In particular, the statement that
3
when a K 3 shrinks, one gets a heterotic string with the K 3 replaced by T
holds for K 3's with m =0.Intuitively, one would exp ect m 6= 0 to change
the b ehavior that o ccurs when one tries to shrink a K 3, b ecause the energy
stored in the trapp ed K eld would resist this shrinking.
8
Wehave not yet given any explicit argument that the dual heterotic string is an
E E theory with symmetric emb edding. That the gauge group of the dual string is
8 8
E E rather than SO(32) we infer from the fact that, if one starts with suitable gauge
8 8
bundles in eleven dimensions, unbroken exceptional gauge groups suchasE E are
7 7
p ossible. Moreover, if one starts with the symmetric emb edding in eleven dimensions {
1
whose necessitywe argue for presently { then there is a symmetry of S =Z that exchanges
2
the two xed p oints and the two E 's. This will carry over in the dual heterotic string
8
theory to a symmetry that exchanges the two E 's, and the existence of this symmetry
8
indicates that the dual heterotic string has equal instanton numb ers in the two E 's.
8 14
One can actually b e more precise. The dual heterotic string that arises
when one shrinks a K 3 comes from a ve-brane wrapp ed around the K 3.
But a ve-brane cannot wrap around a K 3 (or any four-manifold) that has
m 6= 0. The reason for this is that the world-volume sp ectrum of the ve-
brane includes a massless two-form with an anti-self-dual three-form eld
strength T . T do es not ob ey dT = 0, but rather (as one can see from
equation (3.3) of [40]; another argumentisgiven in [41]) it ob eys dT = K .
The existence of a solution for T means that K must b e cohomologically
trivial when restricted to the ve-brane world-volume; that is, the ve-brane
cannot wrap around a four-manifold with m 6=0.
The reason that this is relevant is that, as we will argue momentarily,if
6 1
one compacti es the M -theory on R S =Z K 3 with instanton number
2
k in one E and 24 k in the other E , then the ux of the K eld over K 3
8 8
6 1
(that is, over any K 3 obtained by restricting to a generic p ointinR S =Z )
2
is m = (12 k ). (The sign will b e explained later.) Therefore, m =0 if
and only if k = 12, that is, precisely for the symmetric emb edding. The
eleven-dimensional explanation of heterotic - heterotic dualitythus requires
E E with the symmetric emb edding.
8 8
It remains, then, to explain the relation m = (12 k ). This relation
arises up on writing the anomaly cancellation condition
0
dH = (trR ^ R trF ^ F trF ^ F ) (25)
1 1 2 2
4
in eleven-dimensional terms. The eleven-dimensional version of that equation
must involve the ve-form dK instead of the four-form dH . This requires
incorp orating on the right hand side delta-functions supp orted at the xed
1 11
p oints. If S is parametrized by an angular variable x such that the Z
2
11
xed p oints are at x = 0 and , with F supp orted at the rst and F
1 2
supp orted at the second, then the eleven-dimensional version of (25) is
1
11
dK = dx
2T
3
1 1
11 11
(x ) trR ^ R trF ^ F + (x ) trR ^ R trF ^ F :
1 1 2 2
2 2
(26)
This equation is determined by the following prop erties: dK vanishes ex-
cept at xed p oints, since (in the absence of ve-branes) dK = 0 in the 15
eleven-dimensional theory; F and F contribute only at the appropriate val-
1 2
11
ues of x ; the two xed p oints enter symmetrically; if one integrates over
11
x and interprets H as the part of the zero mo de of K with one index
11
equal to 11, then (26) reduces to (25). Now, let m(x ) b e the function
11
obtained byintegrating T F=2 over K 3 at a given value of x . The Z
3 2
11 11 11
symmetry implies that m( x )= m(x ), and (26) means that m(x )is
11
constant except for jumps at x =0 or , the magnitude of the jump b eing
R R
1 1
2 2
(2=8 ) = (2=8 ) . trR ^ R trF ^ F trR ^ R trF ^ F
1 1 2 2
K 3 K 3
2 2
11
Hence the constantvalue of m(x )away from a xed p oint, whichwe earlier
called m,is
Z
1
2
(1=8 ) : trR ^ R trF ^ F
1 1
2
K 3
This amounts to the statement that m = (12 k ), with k the instanton
11
numb er in the rst E , supp orted at x =0. This con rms the claim
8
made ab ove and so completes our explanation of why the eleven-dimensional
approach to heterotic - heterotic duality requires the symmetric emb edding
as well as requiring gauge group E E .
8 8
Anomaly Cancel lation By Five-Branes
We cannot resist mentioning an application of these ideas that is some-
what outside our main theme. The equation (26) shows that the curvature
tr R ^ R of K 3 gives a magnetic source for the K eld, that is a contribu-
tion to dK supp orted at xed p oints. This is a sort of \anomaly" that must
1
b e canceled, since the integral of dK over the compact space K 3 S =Z
2
will inevitably vanish. The conventional string theory way to cancel this
anomaly is to use E E instantons on K 3, using the fact that tr F ^ F
8 8 i i
also contributes to dK .From this p oint of view, the \magnetic charge" as-
so ciated with the tr R ^ R contribution to dK can b e canceled by24 E E
8 8
instantons.
There is, however, another standard entity that can contribute to dK ; this
is the eleven-dimensional ve-brane, whichischaracterized by the fact that
dK has a quantized delta-function contribution supp orted on the ve-brane
world-volume. This suggests that instead of canceling the total contribution
to dK with 24 E E instantons, one could use 23 instantons on the xed
8 8
1
p oints and one ve-brane at some generic p ointinK3S =Z . More gener-
2
ally, one could use 24 n instantons (distributed as one wishes b etween the
two E 's) and n ve-branes.
8 16
The following is a strong indication that this is correct. Supp orted on the
ve-brane world-volume is one tensor multiplet and one hyp ermultiplet in the
sense of N = 1 sup ergravityin D = 6. (The hyp ermultiplet parametrizes
the p osition of the ve-brane on K 3.) Instead, asso ciated with each E E
8 8
instanton are precisely 30 hyp ermultiplets (a numb er that can b e seen in the
instanton dimension formula with whichwe b egin the next section). Both
the tensor multiplet and the hyp ermultiplet contribute to the irreducible
part of the gravitational anomaly in six dimensions (the part that cannot
b e canceled by a Green-Schwarz mechanism). From equation (118) of [42 ],
one can see that the contribution to this irreducible anomaly of one tensor
multiplet and one hyp ermultiplet equals that of 30 hyp ermultiplets, strongly
suggesting that the M -theory vacua with 24 n instantons and n- vebranes
(and therefore n + 1 tensor multiplets in six dimensions) really do exist.
These cannot b e related to p erturbative heterotic strings, but they might
have limits as Typ e I orientifolds, as in [20]. They are somewhat reminiscent
of the M -theory vacua with wandering ve-branes found in [41 ].
Analysis Of The Couplings
6 1
Wenow return to the M -theory on R S =Z K 3, with the intention of
2
examining somewhat more quantitatively the two limits in which it is related
1
to a heterotic string { the limits in which the S =Z or the K 3 shrinks to
2
small volume. Wewanttoshow that in either such limit, the heterotic string
that emerges is weakly coupled. In fact, part of the meaning of any claim
that the M -theory turns into a string theory in a particular limit should
b e that the resulting string theory is weakly coupled. We will also, more
precisely, show that the coupling constant of the heterotic string obtained
1
by shrinking the S =Z is the inverse of the coupling of the heterotic string
2
obtained by shrinking the K 3. This is what one would exp ect given that
these strings are electric - magnetic duals. The calculations we will need are
quite straightforward given formulas in [2]. In these computations we will
not keep track of some absolute constants.
1
We let R b e the radius of S =Z , measured with resp ect to the metric
2
of eleven-dimensional sup ergravity, and we let V b e the volume of the K 3,
likewise measured in eleven-dimensional terms. According to [16, 2], as R
go es to zero with large V , one gets a heterotic string with the ten-dimensional
3=2
string coupling constant b eing = R . Also, the string metric di ers from
10
the eleven-dimensional metric byaWeyl rescaling, such that the K 3volume 17
2
measured in the string metric is V = VR . The six-dimensional string
st
2 2
coupling constant ob eys the standard relation 1= = V = ,sowe get
6 st
6 10
2
R
10
2
= = : (27)
6
V V
st
This shows that the six-dimensionl string coupling constant is small if R<<
V.
Nowwe consider the opp osite limit in which the K 3 shrinks. Then one
6 1
gets in seven dimensions a dual heterotic string R S =Z with (according
2
3=4
e
to formulas in [2]) a seven-dimensional string coupling constant = V .
7
Also, the string metric of the dual heterotic string theory di ers bya Weyl
rescaling from the eleven-dimensional metric, such that the radius of the
1 1=2
e
S =Z in the dual string metric is R = V R. The six-dimensional dual
2 st
string coupling constant ob eys
2
e
V
2 7
e
= : (28) =
6
R R
st
e
Putting these formulas together, we get =1= , as exp ected from the fact
6 6
1
that the string obtained by wrapping the two-brane around S =Z is dual to
2
the string obtained by wrapping the ve-brane around K 3.
Behavior Under Further Compacti cation
It is interesting to consider further toroidal compacti cation to four di-
6 4 2
mensions, replacing R by R T . Starting with a K 3vacuum in which
the E E gauge symmetry is completely Higgsed, the toroidal compacti -
8 8
cation to four dimensions gives an N = 2 theory with the usual three vector
multiplets S , T and U related to the four-dimensional heterotic string cou-
2
pling constant and the area and shap e of the T . When reduced to four
dimensions, the six-dimensional string-string duality (9) b ecomes [14] an op-
2
eration that exchanges S and T . Since the heterotic string on T K 3 also
has R ! 1=R symmetries that exchange T and U , this is an example with
complete S T U triality symmetry, as discussed in [5].
Kachru and Vafa [21 ] made a prop osal for a Typ e I I dual of the E E
8 8
2
heterotic string on T K 3 with this precise vacuum, that is equal instanton
numb ers in the two E 's and complete Higgsing. Some evidence for the S T
8
interchange symmetry has app eared in subsequent study of this example
[35, 36 , 37]. 18
4 The Duals Of Some Unbroken Gauge Groups
The mo duli space M (E )ofE instantons on K 3 with instanton number k
k 8 8
has a dimension (predicted from the index formula) whichis
dim M (E ) = 120k 992 (29)
k 8
if k is suciently big. We will make frequent use of the sp ecial case
dim M (E ) = 448: (30)
12 8
The formula for dim M actually gives the the correct dimension of the
k
mo duli space if k is large enough that a K 3 instanton of instanton number k
can completely break the E gauge symmetry. A necessary condition for this
8
to b e p ossible is that the right hand side of (29) must b e p ositive, restricting
us to k 9. Wehavechecked that complete Higgsing is p ossible for k 10
9
and do not know if it is p ossible for k =9. We note that complete Higgsing
may b e p ossible for k = 9 in conformal eld theory even if it do es not o ccur
10
in classical geometry.
The generalization of (29) for an arbitrary simple Lie group G with dual
Coxeter number h and dimension dim G is
dim M (G)=4hk 4 dim G; (31)
k
9
The check for k = 10 can b e made for instance by starting with instanton number10in
an SU (2) subgroup of E (a con guration that is p ossible by standard existence theorems),
8
breaking E to E and giving a low energy sp ectrum that consists of six 56's of E . (Note
8 7 7
that as the 56 is pseudoreal, it is p ossible for E to act on 28 hyp ermultiplets transforming
7
in the 56 of E . The sp ectrum consists of six copies of this.) Sequential Higgsing, turning
7
on the exp ectation values of successive 56's, can then b e seen to completely break E .
7
For k = 9, a similar construction gives ve 56's of E , and sequential Higgsing now leads
7
toavacuum with an unbroken level one SU (3). Sequential Higgsing do es not always give
all the p ossible vacua, as shown in an explicit example in [43], and there may b e other
branches, but at any rate there is one branch of the k = 9 mo duli space in which E is
8
generically broken to SU (3).
10
Six-dimensional sup ersymmetry p ermits \phase transitions" in which branches of the
mo duli space of vacua with di erent generic unbroken gauge groups meet at a p ointof
enhanced gauge symmetry.For example, according to [43], a branch with generic unbroken
SU (3) and a branch with generic complete Higgsing can meet at a p oint at which the
unbroken gauge symmetry is SU (6), with a hyp ermultiplet sp ectrum consisting of six 6's
of SU (6): The necessary enhanced gauge symmetry mightwell o ccur in conformal eld
theory but not in classical geometry. 19
valid whenever complete Higgsing is p ossible and in particular whenever k is
suciently big.
For our problem of E instantons with instanton numb er 12, complete
8
Higgsing is p ossible, and the gauge group is generically completely broken.
On suitable lo ci in mo duli space, with the prop erty that the instantons t
into a subgroup H of E , a subgroup of E is restored { namely the subgroup
8 8
G that commutes with H , known as the commutantofH. When the vacuum
gauge eld reaches such a lo cus, p erturbative G gauge elds will app ear. Ac-
cording to our discussion in the last section, the un-Higgsing or restoration of
G will b e dual to the app earance, on some other lo cus, of a non-p erturbative
gauge invariance with a gauge group isomorphic to G.
Non-p erturbative gauge elds, that is gauge elds that are not seen in
conformal eld theory but app ear no matter howweak the string coupling
constantmay b e, can only arise when the K 3 or the vacuum gauge bundle
develops a singularity, causing string p erturbation theory not to b e uniformly
valid for all states. Wewould like to nd plausible examples of how this works
in practice. That is, for suitable groups G,wewould like to identify the K 3
or gauge singularity that generates non-p erturbative gauge invariance with
gauge group G.We will not b e able to do this for all G, but we will nd
what seem like comp elling candidates for some simple cases. Our discussion
is necessarily incomplete, and no substitute for actually understanding the
map of hyp ermultiplet mo duli space that app eared in the last section.
It seems natural to rst consider singularities of the gauge bundle only,
keeping the K 3 smo oth. On a smo oth K 3, a singularity of the gauge bundle
(keeping the E completely broken) necessarily consists of a certain number
8
of instantons shrinking to p oints. For example, consider the basic case in
which a single instanton shrinks to a p oint. The e ective k of the remaining
gauge bundle diminishes by 1, so according to (29) the dimension of the
E instanton mo duli space drops by 120. However, one is left with four
8
parameters for the p osition of the p oint instanton, so actually only 120 4=
116 parameters must b e adjusted to make a single instanton shrink.
For the SO(32) heterotic string, a single p oint instanton gives [23] a non-
p erturbative SU (2) gauge symmetry. There is no general derivation of this
for E E , and it seems doubtful that it is true in general (as illustrated,
8 8
among other things, by the sp ecial role that k = 12 is ab out to play). But
fortune sometimes favors the brave, and let us ask whether in our particular
case, the collapse of an instanton might b e dual to an unbroken p erturbative 20
SU (2) gauge symmetry.
E has a maximal subgroup SU (2) E . The SU (2) app earing in suchan
8 7
SU (2) E is a minimal or (in conformal eld theory language) \level one"
7
emb edding of SU (2) in E , and its commutantis E .To get an unbroken
8 7
level one p erturbative SU (2) from one of the E 's, the vacuum gauge bundle
8
must t into an E subgroup. As E has h = 18 and dimension 133, (31)
7 7
gives dim M (E )=72k 532. In particular, dim M (E ) = 332. Using
k 7 12 7
also (30) and the fact that 448 332 = 116, one must adjust 116 parameters
to get a p erturbativeunbroken SU (2) gauge symmetry. The fact that this is
the numb er of parameters needed to get a p oint instanton strongly suggests
that at k = 12 the shrinking of an instanton to a p oint really is dual to the
un-Higgsing of an SU (2).
Forti ed by this result, let us consider the p ossible o ccurrence of two
p oint instantons, rst considering the generic case that they are placed at
distinct p oints on K 3. The numb er of parameters that must b e adjusted
to get two p oint instantons is 2 116 = 232. If one p oint instanton gives
an SU (2) gauge symmetry, then two disjoint p oint instantons should very
plausibly give SU (2) SU (2). The dual should involve p erturbative un-
Higgsing of SU (2) SU (2). A level one emb edding of SU (2) SU (2) in
11
E has commutant SO(12). As SO(12) has h = 10 and dimension 66,
8
we get dim M (SO(12)) = 216. Since 448 216 = 232, the exp ected 232
12
parameters are needed to restore an SU (2) SU (2) subgroup of E .
8
Let us now consider the case of two p oint instantons at the same p oint.
Since one must adjust four more parameters to make the p ositions of the
two p oint instantons in K 3 coincide, the numb er of parameters required is
now2116 + 4 = 236. Two coincident p oint instantons must give a gauge
group that contains SU (2) SU (2). For the SO(32) heterotic string, two
coincident p oint instantons give gauge group Sp(2) = SO(5); let us make
the ansatz that that is true here also. A level one emb edding of SO(5) in
E has commutant SO(11) (with SO(5) SO(11) SO(16) E ). As
8 8
SO(11) has h = 9 and dimension 55, one gets dim M (SO(11)) = 212.
12
With 448 212 = 236, the exp ected 236 parameters must b e adjusted to see
an unbroken SO(5) subgroup of E .
8
11
Recall that E has a maximal subgroup SO(16), which contains SO(12) SO(4) =
8
SO(12) SU (2) SU (2). Actually, SO(16) admits another inequivalent level one em-
b edding of SU (2) SU (2) (the commutant b eing SO(8) SU (2) SU (2)), but the two
emb eddings are conjugate in E .
8 21
Nowwemove on to consider the case of three small instantons. The
numb er of parameters that must b e adjusted to create three small instantons
at distinct p oints is 3 116 = 348. (Here we are on shakier grounds, as we
will assume that there is a branchofthek= 9 mo duli space with complete
Higgsing, and as explained ab ovewe do not know this to b e true. This
uncertainty will a ect many of the observations b elow.) The generic non-
3
p erturbative gauge group for three small instantons should b e SU (2) . A
3
level one emb edding of SU (2) in E has commutant SO(8) SU (2) (with
8
3 2
(SO(8) SU (2)) SU (2) = SO(8) SO(4) SO(16) E ). Here
8
there is the new feature that SO(8) SU (2) is not simple; we can place
k instantons in SO(8) and k in SU (2), with k + k = 12. Since SO(8)
1 2 1 2
has h = 6 and dimension 28 while SU (2) has h = 2 and dimension 3, the
dimension of the SO(8) SU (2) mo duli space is 24k +8k 124. We
1 2
will make the ansatz of picking k and k to make this as large as p ossible
1 2
sub ject to the constraint that bona de SU (2) instantons on K 3 of the given
12
k actually exist. That latter constraint forces k 4, so to maximize the
2 2
dimension of the mo duli space, we take(k ;k )=(8;4), and then we nd
1 2
that dim M (SO(8) SU (2)) = 100. As 448 100 = 348, one must adjust
(8;4)
3
the exp ected 348 parameters to restore a level one SU (2) subgroup of E .
8
0
In future, when we meet instantons in a group H = H SU (2), we will
always set the SU (2) instanton numb er to b e 4, as in the calculation just
p erformed.
One can similarly consider the case of three collapsed instantons that
are not at distinct p oints. For instance, two collapsed instantons at one
p oint and one at a third p oint should give a non-p erturbative gauge group
Sp(2) SU (2) = SO(5) SU (2). To obtain such a con guration, one must
adjust 3 116 + 4 = 352 parameters (3 116 to make three instantons collapse
and 4 to maketwo of the collapsed instantons app ear at the same p oint). The
commutant of a level one emb edding of SO(5) SU (2) in E is SO(7) SU (2)
8
(via SO(7) SU (2) SO(5) SU (2) SO(16) E ). Using the fact that
8
SO(7) has h = 5 and dimension 21, along with facts already exploited, we get
12
For SU (2), (31) gives dim M (SU (2)) = 8k 12. k p oint instantons would have
k
a4kdimensional mo duli space. Honest SU (2) instantons as opp osed to collapsed p oint
instantons exist only when dim M (SU (2)) exceeds the dimension of the mo duli space of
k
collapsed instantons, that is when 8k 12 > 4k or k>3 The dimension-counting alone
do es not give a rigorous argument here, but a rigorous argument can b e made using the
index theorem for the Dirac op erator in the instanton eld and a vanishing argument. 22
dim M (SO(7) SU (2)) = 96. With 448 96 = 352, one must adjust the
(8;4)
exp ected 352 parameters to restore a p erturbative SO(5) SU (2) subgroup.
The last example of this kind is the case of three collapsed instantons all
at the same p ointinK3. 3 116+24 = 356 parameters must b e adjusted to
achieve this situation. Based on the result of [23 ] for SO(32), wemay guess
that the non-p erturbative gauge group for three coincident small instantons
will b e Sp(3). The commutantofalevel one emb edding of Sp(3) in E is
8
G SU (2). (This was found byemb edding Sp(3) SU (6) SO(12)
2
SO(16) E and then, by hand, determining that the commutantof Sp(3)
8
was G SU (2) { a metho d that also gave the decomp osition used later of
2
the E Lie algebra under Sp(3) G SU (2).) With G having h = 4 and
8 2 2
dimension 14, one nds that dim M (G SU (2)) = 92; as 448 92 = 356;
(8;4) 2
one must adjust the exp ected 356 parameters to restore an Sp(3) subgroup
of E .
8
For more than three small instantons, the residual E instanton would
8
have instanton number 8, so that the right hand side of (29) would b e
negative. This actually means that the residual instanton cannot completely
break the E symmetry, so that p erturbativeaswell as non-p erturbative
8
gauge elds will app ear. Wehave not b een successful in nding duals of
con gurations with p erturbativeaswell as non-p erturbative gauge elds,
and will not discuss this here.
Let us now lo ok in somewhat more detail at the example with three p oint
instantons all at the same p ointinK3, giving a non-p erturbative Sp(3)
subgroup. Sp(3) has many subgroups that wehave seen, suchas Sp(2)
3
SU (2), SU (2) , etc. In the non-p erturbative description, one sees these
subgroups of Sp(3) by p erturbing the very exceptional con guration with
three coincident p oint instantons to a less exceptional { but still singular {
con guration in which the instantons are not all small or all coincident. In
the p erturbative description, an Sp(3) which has b een restored or un-Higgsed
can b e broken down to a subgroup by turning back on the ordinary Higgs
mechanism.
But when one thinks ab out doing so, a puzzle presents itself. Apart
from subgroups of Sp(3) that wehave seen, there are other subgroups of
Sp(3), suchas SU (3), U (1), U (1) SU (2), that do not app ear anywhere
on the non-p erturbative side (in any deformation of the con guration with
the three coincident p oint instantons). What prevents Higgsing of Sp(3) to
SU (3) or other unwanted groups? 23
To answer this question on the p erturbative side, we need to know the
Sp(3) quantum numb ers of the massless hyp ermultiplets that app ear when
the Sp(3) is un-Higgsed. The adjoint representation of E decomp oses under
8
G SU (2) Sp(3) as
2
0
(7; 1; 14) (1; 2; 14 ) (7; 2; 6); (32)
plus the adjointofG SU (2) Sp(3); here 1 is a trivial representation,
2
7, 2, and 6 are the de ning representations of G , SU (2), and Sp(3), of
2
0
the indicated dimensions, and 14 and 14 are the two fourteen dimensional
representations of Sp(3) { the 14 is the traceless second rank antisymmetric
0
tensor, and the 14 is a traceless third rank antisymmetric tensor. The
decomp osition in (32) means that when a level one p erturbative Sp(3) is
unbroken, the massless hyp ermultiplets will transform as a certain number of
0
14's, 14 's, and 6's, determined by index theorems. No other representations
of Sp(3) can app ear, as they are absent in the adjoint representation of E .
8
For instance, the number of 14's will b e the index of the Dirac op erator
on K 3 with values in the (7; 1)of G SU (2), and similarly for the other
2
representations.
Now, the 14's and 6's are the desired representations that can Higgs
Sp(3) to the groups that one actually sees on the non-p erturbative side, and
nothing else. For instance, with 6's alone, one can Higgs only down to Sp(2)
or Sp(1) = SU (2) { two of the groups that we actually encountered ab ove.
Using also the 14 one can get the other desired subgroups of Sp(3). But the
0
14 would make it p ossible to Higgs down to unwanted subgroups of Sp(3)
suchas SU (3) or U (1).
At this p ointwemust recall that there are several lo ci on which restoration
of a level one Sp(3) o ccurs. They are lab eled by the G SU (2) instanton
2
numb ers (k ;k ) with k + k = 12. For general (k ;k ), the unwanted
1 2 1 2 1 2
representation will app ear. But we found the situation of three coincident
p oint instantons to b e dual to un-Higgsed Sp(3) with (k ;k )=(8;4). When
1 2
and only when the instanton numb er is 4, the Dirac index with values in
13 0
the (1; 2) of G SU (2) vanishes, and the dreaded 14 do es not arise.
2
13
A short-cut to deduce this directly from (31) without having to go back to the general
index theorem is as follows. From (31), dim M (SU (3)) = 20 but also dim M (SU (2)
4 4
U (1)) = 20 (in the latter case one understands that the instantons are all in the SU (2),
and that h = 0 for U (1)). As these numb ers are equal, for instanton numb er four an 24
Computing the other indices, one nds that the hyp ermultiplet sp ectrum
consists of one 14 and thirty-two 6's. (For unclear reasons, but surely not
coincidentally, this is the sp ectrum that app ears in the ADHM construction
4
of instantons with instanton numb er three with gauge group SO(32) on R .)
With this sp ectrum, the p erturbative Sp(3) can b e Higgsed down precisely
to those subgroups that we found on the non-p erturbative side.
Inclusion Of K3 Singularities
To go farther, since wehave exhausted the list of singularities of a com-
pletely Higgsed gauge bundle on a generic K 3, a natural step is to lo ok for
non-p erturbative gauge elds whose origin involves a K 3 singularity. This is
a p otentially vast sub ject (unless one has a systematic p oint of view), with
many p ossibiliti es to consider. We will p oint out a few candidates.
Since we found the dual to an un-Higgsed Sp(3), let us lo ok for the dual
to a relatively small group containing Sp(3), namely SU (6). The commu-
tantofalevel one SU (6) is SU (3) SU (2) (E has a maximal subgroup
8
SU (6) SU (3) SU (2)). The SU (3) SU (2) instanton mo duli space with
instanton numb ers (8; 4) has dimension 84 (using the fact that SU (3) has
h = 3 and dimension eight); as 448 84 = 364, 364 parameters must b e
adjusted to restore a p erturbative SU (6) gauge symmetry.We should try
to interpret this numb er in terms of a singular K 3, since wehave exhausted
what can b e done with non-singular K 3's. The simplest K 3 singularityis
an A singularity; in classical geometry one must adjust three parameters to
1
make such a singularity, but to make the conformal eld theory singular, one
must adjust also a theta angle [39], making four. Since one must adjust 120
parameters to make an instanton collapse and x its p osition, we are tempted
to write 364 = 3 120 + 4 and to prop ose that the dual of a p erturbative
SU (6) is a K 3 with an A singularity at which there are also three p oint
1
instantons.
Tocheck out this idea further, we note that in general, to get the A
1
singularity with k p oint instantons sitting on top of it, one should exp ect to
have to adjust
w = 120k +4 (33)
k
SU (3) instanton on K 3 automatically takes values in an SU (2) U (1) subgroup, so that
there are no zero mo des in the 2 of SU (2) by which one could deform to an irreducible
SU (3) instanton. 25
parameters. Wehave already considered the k = 3 case. Let us lo ok at
k =2. Here wehavetwo coincident p oint instantons at the singularity.
The gauge group is at least Sp(2), whichwewould get with two coincident
instantons without the A singularity. A relatively small group containing
1
Sp(2) is SU (4); let us ask if this is the right group for two p oint instantons
on top of an A singularity. In fact, the commutantof SU (4) is SO(10)
1
(think of the chain SU (4) SO(10) = SO(6) SO(10) SO(16) E ).
8
SO(10) has h = 8 and dimension 45, and dim M (SO(10)) = 204. Thus
12
448 204 = 244 parameters must b e adjusted to restore a p erturbative SU (4)
symmetry, in agreement with the k = 2 case of (33).
One might ask ab out the k = 1 case of (33). Wewere not able to nd
a group whose restoration involves adjusting 124 parameters. However, note
that the results so far can b e expressed by stating that an A singularity that
1
captures k p oint instantons gives an SU (2k ) gauge group. If we supp ose that
that holds also for k = 1, then the A singularity with one p oint instanton
1
gives an SU (2) gauge group, which is the same gauge group we obtained for
one p oint instanton without the A singularity.Sowe p ostulate that with
1
only one p oint instanton, the coincidence with an A singularity leads to no
1
enhancement of the gauge group. In essence, for each k , the k coincident
p oint instantons give SU (2k )orSp(k) dep ending on whether or not they lie
at an A singularity, but the fact that SU (2) = Sp(1) means that for k =1
1
the A singularity gives no enhanced gauge symmetry.
1
If we feel b old, we can observe that since in classical geometry an A
1
singularity is the same as a Z orbifold singularity, it is p ossible for an A
2 1
singularity to capture a half-integral numb er of p oint instantons. Why not
then try to use (33) for half-integral k ? This turns out to work p erfectly,
though we nd this somewhat puzzling for a reason stated b elow. Since k
p oint instantons on the A singularitygavean SU (2k ) gauge group at least
1
for k =2;3, we compare the k =5=2 and k =3=2 cases to SU (5) and
SU (3), resp ectively. The commutantof SU (5) is SU (5) (E has a maxi-
8
mal subgroup SU (5) SU (5)). Since SU (5) has h = 5 and dimension 24,
dim M (SU (5)) = 144; as 448 144 = 304, 304 parameters must b e ad-
12
justed to restore a p erturbative SU (5), in agreement with the k =5=2 case
of (33). Likewise, SU (3) has commutant E , which has h = 12 and dimen-
6
sion 78, so dim M (E ) = 264. With 448 264 = 184, one must adjust
12 6
184 parameters to restore an SU (3) gauge symmetry, in agreement with the
k =3=2 case of (33). Now, however, wemust confess to what is unsettling 26
ab out these \successes" for SU (3) and SU (5): in classical geometry (33) is
not valid for half-integral k (there is a correction involving an eta-invariant),
and we do not knowwhy this simple formula seems to work in string theory.
Wehave encountered many but not all subgroups of SU (6) in this discus-
sion. Just as in the discussion of Sp(3), we should ask on the p erturbative
side to what subgroups the SU (6) can b e Higgsed. For this we need the
fact that under SU (3) SU (2) SU (6) the adjointofE decomp oses as
8
(3; 2; 6) (3; 2; 6) (3; 1; 15) (3 ; 1; 15) (1; 2; 20) (plus the adjoint). The
SU (6) content will consist of a certain number of 20's as well as 6's and
15's and their complex conjugates. The 20's are unwanted as they would
enable Higgsing to subgroups of SU (6) that would not haveaninterpretation
in terms of a p erturbation of an A singularity with three p oint instantons.
1
Happily, b ecause the instanton numb er in the SU (2) is four, the Dirac index
with values in the (1; 2)ofSU (3) SU (2) vanishes, the same lucky fact we
used in the Sp(3) discussion, so again the unwanted representation do es not
o ccur.
The actual sp ectrum thus consists only of 6's, 15's, and their conjugates.
With these representations, one can break SU (6) only to groups that have
natural interpretations using the ab ove ideas. (SU (5) and SU (3) do o ccur {
they can b e reached by Higgsing with 6's { so the k =5=2 and k =3=2 cases
of (33) are needed.) For instance, with the 15 one can Higgs down to SU (4)
SU (2), which corresp onds to two p oint instantons at an A singularity and
1
one somewhere else. The commutantofSU (4) SU (2) is again SU (4)
SU (2). As dim M (SU (4) SU (2)) = 88, one must adjust 448 88 = 360
(8;4)
parameters to see a p erturbative SU (4) SU (2) gauge symmetry.We write
360 = 244 + 116 and prop ose that in the dual interpretation, one adjusts 244
parameters to get two p oint instantons at an A singularity and 116 to get
1
one p oint instanton somewhere else. Similarly,by using also a 6, one can
Higgs SU (6) to SU (3) SU (2), whose commutantin E is another SU (6).
8
As dim M (SU (6)) = 148, one must adjust 448 148 = 300 parameters
12
to see a p erturbative SU (3) SU (2). On the non-p erturbative side, we
write 300 = 184 + 116 and interpret this as 184 parameters to get an A
1
singularity that absorbs 3=2 p oint instantons, and 116 to get a p oint instanton
somewhere else.
One more example of a similar kind, though not obtained by Higgsing of
this particular SU (6), is to lo ok for an unbroken level one SU (3) SU (3)
subgroup of E . The commutantof SU (3) SU (3) is another SU (3) SU (3)
8 27
4
(E has a subgroup SU (3) ). As dim M (SU (3) SU (3)) = 80 (for any
8 k ;k
1 2
k ;k with k + k = 12), one must adjust 448 80 = 368 parameters to
1 2 1 2
get an unbroken p erturbative SU (3) SU (3). Writing 368 = 164 + 164, we
prop ose that the dual of this consists of two disjoint A singularities eachof
1
which has captured 3=2 p oint instantons.
We should note that similar examples with co dimension very close to
448, like the k =7=2 case of (33), do not seem to work. We susp ect that
this is b ecause in these cases complete Higgsing to the exp ected group is not
p ossible.
5 Higher Lo ops
In this section, we will compare the lo op expansion of the fundamental string
to that of the dual string. The lo op expansion of the fundamental string for
any given physical observable is an expansion of that observable in p owers of
e ,valid as as aymptotic expansion near = 1. The expansion takes the
P
n
b e where n dep ends on what observable is considered. general form
n 0
nn
0
(In the Einstein metric, the exp onents may not b e integers.) The p erturba-
tion expansion of the dual string is an analogous asymptotic expansion in
P
m
powers of e ,valid near = +1. The general form is c e .
m
mm
0
Wewould liketomake a term-wise comparison of these expansions, but in
general such a term-wise comparison of asymptotic expansions of a func-
tion ab out two di erent p oints is not valid. One situation in which sucha
termwise comparison is valid is the case that each series has only nitely
P P
n m
n m
1 1
many terms; thus given an equality b e = c e between
n m
n=n m=m
0 0
nite sums, the exp onents and co ecients must b e equal.
An imp ortant reason for such a series to have only nitely many terms
is that sup ersymmetry may allow only nitely many terms in the expan-
sion of a given observable in p owers of e .For instance, in this pap er we
have exploited the fact that low energy gauge eld kinetic energy has a -
dep endence with only two p ossible terms. In what follows, we will work out
the consequences of a term-wise comparison of the two p erturbation expan-
sions in any situation in which such a comparison is valid, including but not
limited to the case in which sup ersymmetry allows only nitely many terms.
The fundamental string involves two kinds of lo op expansion: quantum
0
D = 6 string lo ops (L) with lo op expansion parameter e and classical 2- 28
0
dimensional -mo del lo ops with lo op expansion parameter .Following [8],
let us consider the purely gravitational contribution to the resulting e ective
action, using the string -mo del metric:
!
L
3
p
(2 )
0 0 m n
L = a Ge e R ; (34)
L;L+m L;L+m
02
n
where R is symb olic for a scalar contribution of n Riemann tensors eachof
dimension two. One could also include covariant derivatives of R (or other
interactions with a known transformation law under duality), but for our
purp oses (34) will b e sucient. The a are numerical co ecients, not
L;L+m
involving . The subscripts L and L + m have b een chosen in anticipation
e e
of the relation L + m = L, with L the numb er of lo ops in the dual theory.
Since
0
[L ]= 6; [ ]= 2; (35)
L;L+m
wehave, on dimensional grounds,
m = n 1 2L: (36)
Likewise, the theory of the dual string involves quantum D = 6 dual string
0
e
lo ops (L) with lo op expansion parameter e and classical 2-dimensional
0
-mo del lo ops with lo op expansion parameter . The corresp onding La-
grangian using the dual -mo del metric is
q
e
3
L
(2 )
n 0 me 0
e e e
e
L = a R : (37)
e Ge
e e e
L+m;e L L+m;Le
02
Again, on dimensional grounds,
e
f
m = n 1 2L: (38)
e
Our fundamental assumption is that L and L are related by duality which
implies, in particular, that the purely gravitational contributions should b e
identical when written in the same variables. So from (18) and (20) and
transforming to the canonical metric, but dropping the c sup erscript, we
nd:
3
p
(2 )
0L+m m=2 n
L = a e GR ; (39)
L;L+m L;L+m
02
29
3
p
(2 )
e
0 L+me me =2 n
e
e
L = a e GR (40)
e e
e e
+m;e L +m;e L
L L
02
where wehave also dropp ed the dilaton derivative terms. We nd that L
e
and L do coincide provided
f
m + m =0; (41)
i.e from (36) and (38), provided
e
f
m = L L = m; (42)
e
n = L + L +1; (43)
with
e
a = a : (44)
e e
L;L L;L
Hence the total Lagrangian can b e elegantly written
X
L = L :
e
L;L
e
L;L
p
e e e
3 0L+L 2 (L L)=2 L+L +1
L = a (2 ) e GR : (45)
e e
L;L L;L
Thus we see that under heterotic string/string duality, the worldsheet lo op
expansion of one string corresp onds to the spacetime lo op expansion of the
other. Moreover, (45) gives rise to an in nite numb er of non-renormalization
theorems. (As explained at the b eginning of this section, these theorems hold
if it is known a priori that each p erturbation expansion has only nitely many
terms, and p erhaps under wider but presently unknown hyp otheses.) The
p
0
rst of these is the absence of a cosmological term GR . The second
p
1
e
states that GR app ears only at (L =0;L= 0) and hence the tree level
p
2
action do es not get renormalized. The third states that GR app ears
e e
only at (L =0;L= 1) and (L =1;L= 0), and so on. Since F has the same
dimension as R, similar restrictions will apply to the pure Yang-Mills and
e e
mixed gravity-Yang Mills Lagrangian s. The (L =0;L= 0), (L =0;L=1)
e
and (L =1;L = 0) terms corresp ond resp ectively to the I , I and I of the
00 01 10
previous section. Self-duality, given by (9), imp oses the further constraint
a = a (46)
e e
L;L L;L 30
and means that the spacetime and worldsheet lo op expansions are in fact
identical.
In the situation in which the term-by-term comparison of the two expan-
sions is justi ed by a sup ersymmetry argument showing that each expansion
has only nitely many terms, it would b e interesting to compare the re-
strictions on exp onents following from sup ersymmetry with those that follow
from duality. In principle, duality might give a restriction more severe or less
severe than the one that follows from duality, but in the few familiar cases
the two restrictions agree.
6 Acknowledgements
MJD and RM have b ene tted from conversations with Joachim Rahmfeld
and Jim Liu.
A The String Tension And The Dual String
Tension
We will here explore from an eleven-dimensional p oint of view what is entailed
in setting the fundamental and dual string tensions equal. Since the string is
1
obtained by wrapping a membrane of tension T around S =Z with radius R
3 2
e
and the dual string is obtained by wrapping a vebrane of tension T around
6
^
e
^
K 3ofvolume V , the string tension T and dual string tension T , measured
in the D = 11 metric are given by
^
T = RT
3
^
e e
T = V T (47)
6
^
Let us denote length scales measured in the D = 11 metric G, D = 6 string
e e
^
metric G and D = 6 dual string metric G as L, L and L, resp ectively. Since
1
^
G = R G
1
e
^
G = V G (48) 31
they are related by
2 2
^
L = RL
2 2
e
^
L = V L (49)
Since the string tension measured in the string metric, T , and the dual
e
string tension measured in the dual string metric, T , b oth have dimensions
2
(l eng th) , they are given by
1
^
T = R T = T
3
^
1
e e e
T = V T = T (50)
6
and are therefore b oth indep endentof R and V . In fact, since [18]
2
e
(51) T T
6 3
e
wemay, without loss of generality,cho ose units for which T and T are equal.
e
B A Note On The v 's and v 's
An imp ortant feature of the heterotic/heterotic duality conjecture concerns
f
e
the numerical co ecients v app earing in X and v app earing in X . Here
4 4
we wish to make a remark on the precise normalization of these co ecients.
In string p erturbation theory each v is given by [33]
n
2 2
v trF = TrF (52)
h
where n is the level of the Kac-Mo o dy algebra, h is the dual Coxeter numb er,
tr denotes the trace in the fundamental representation and Tr is the trace in
the adjoint representation. For example,
2 2
h = N TrF = 2N tr F
SU (N ) SU (N ) SU (N )
2 2
h = N 2 TrF = (N 2)tr F
SO(N ) SO(N ) SO(N )
2 2
= (2N + 2)trF h = N +1 TrF
Sp(N ) Sp(N ) Sp(N )
2 2
h = 4 TrF = 4tr F
G G G
2 2 2
(53)
2 2
h = 9 TrF = 3tr F
F F F
4 4 4
2 2
h = 12 TrF = 4tr F
E E E
6 6 6
2 2
h = 18 TrF = 3tr F
E E E
7 7 7
2 2
h = 30 TrF = 30tr F
E E E
8 8 8 32
The values of h were used in section (4).
So, for G = E ;E;SO(10);SU(5) one nds v = n=6;n=3;n;2n,
7 6 G
resp ectively. In the case of SO(32) with an k =24emb edding, one nds
e e
from (6) that (n =1;n = 1) but (n = 2; n = 22) and
SO(28) SU (2) SO(28) SU (2)
14
hence we encounter the \wrong-sign problem" . A similar problem arises
for E E except in the case of symmetric emb edding where b oth k = 12.
8 8 i
For generic emb eddings one nds that n = 1 but
i
1
e
n = (k 12) (54)
i i
2
Since one is limited to k + k = 24, one factor will always have the wrong
1 2
sign except when for the k = 12 case discussed in this pap er, for which the
e
n b oth vanish.
i
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