arXiv:0704.0777v1  [hep-th]  5 Apr 2007
CALT-68-2636
DAMTP-2007-25
UT-07-11
Decoupling Supergravity from the Superstring
Michael B. Green,1 Hirosi Ooguri,2,3 and John H. Schwarz2
1Department of Applied Mathematics and Theoretical Physics
Cambridge University, Cambridge CB3 0WA, UK
2California Institute of Technology, Pasadena, CA 91125, USA
3Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
Abstract
We consider the conditions necessary for obtaining perturbative maximal supergrav-
ity in d dimensions as a decoupling limit of type II superstring theory compactified on
a (10 − d)-torus. For dimensions d = 2 and d = 3 it is possible to define a limit in
which the only finite-mass states are the 256 massless states of maximal supergravity.
However, in dimensions d ≥ 4 there are infinite towers of additional massless and finite-
mass states. These correspond to Kaluza–Klein charges, wound strings, Kaluza–Klein
monopoles or branes wrapping around cycles of the toroidal extra dimensions. We con-
clude that perturbative supergravity cannot be decoupled from string theory in dimensions
≥ 4. In particular, we conjecture that pure N = 8 supergravity in four dimensions is in
the Swampland.
March, 2007
http://arxiv.org/abs/0704.0777v1
There has recently has been some speculation that four-dimensional N = 8 super-
gravity might be ultraviolet finite to all orders in perturbation theory [1,2,3]. If true, this
would raise the question of whether N = 8 supergravity might be a consistent theory that
is decoupled from its string theory extension. A related issue is whether N = 8 supergrav-
ity can be obtained as a well-defined limit of superstring theory. Here we argue that such
a supergravity limit of string theory does not exist in four or more dimensions, irrespective
of whether or not the perturbative approximation is free of ultraviolet divergences.
In this paper, we will study limits of Type IIA superstring theory on a (10 − d)-
dimensional torus T 10−d for various d. One may regard the following analysis as analogous
to the study of the decoupling limit on Dp-branes (the limit where field theories on branes
decouple from closed string degrees freedom in the bulk) for various p [4,5]. The decoupling
limit on Dp-branes is known to exist for p ≤ 5. On the other hand, subtleties have been
found for p ≥ 6, where infinitely many new world-volume degrees of freedom appear in the
limit. This has been regarded as a sign that a field theory decoupled from the bulk does
not exist on Dp-branes for p ≥ 6. We will find similar subtleties for Type IIA theory on
T 10−d ×Rd for d ≥ 4.
It will be sufficient for our purposes to consider the torus T 10−d to be the product of
(10− d) circles, each of which has radius R. Numerical factors, such as powers of 2π, are
irrelevant to the discussion that follows and therefore will be dropped. In ten dimensions,
Newton’s constant is given by
G10 = g
2ℓ8s ,
where ℓs is the string scale and g is the string coupling constant. Thus, the effective Newton
constant in d dimensions is given by
Gd ≡ ℓd−2d =
R10−d
g2ℓ8s
R10−d
, (1)
where ℓd is the d-dimensional Planck length, so that
. (2)
We are interested in whether there is a limit of string theory that reduces to maximal
supergravity, which is defined purely in terms of the dynamics of the 256 states in the
massless supermultiplet. In other words, we are interested in the limit in which all the
excited string states, together with the Kaluza–Klein excitations and string winding states
associated with the (10− d)-torus, decouple. A necessary condition for this to happen is
that these states are all infinitely massive compared to the d-dimensional Planck scale ℓd.
This is achieved by taking
, and
, (3)
with ℓd fixed. This is compatible with keeping g fixed for d < 6. If the extra states do
decouple then the surviving states are the 256 massless states of maximal supergravity,
which is N = 8 supergravity when d = 4.
Let us now consider the spectrum of nonperturbative superstring excitations in this
limit. First consider a Dp-brane wrapping a p cycle of the torus. The mass of such a state
in d dimensions is
· ℓ1−
. (4)
When d ≤ 5, we also need to consider a NS5-brane wrapping a 5 cycle. This has a mass
given by
MNS5 =
g2ℓ6s
· ℓ2−d
. (5)
In order to obtain the pure supergravity theory with 32 supercharges in d dimensions, these
nonperturbative states also need to decouple, so their masses must satisfy Mp,MNS5 ≫
1/ℓd. In the case of d = 4 the nonperturbative BPS particle spectrum also includes
Kaluza–Klein monopoles, which are discussed in the next paragraph.
Before studying the limit in any dimension, d, we will discuss what to expect on
general ground. A Kaluza–Klein momentum state and a wrapped string state have masses
1/R and R/ℓ2s , respectively, and they are half-BPS objects that carry a single unit of
a conserved charge. In d-dimensions, their magnetic duals are (d − 4)-branes. The BPS
saturation condition together with the Dirac quantization condition implies quite generally
that the mass m of a BPS particle and the tension T of its magnetic dual (d − 4)-brane
are related by
mT ∼ 1
. (6)
1 In this limit, the string length ℓs provides a regularization scale for supergravity. Thus, if
string amplitudes depend sensitively on ℓs, it can be taken as evidence for ultraviolet divergences
in supergravity. This is seen explicitly, for example, in the one-loop four graviton amplitude,
which is ultraviolet divergent in nine dimensions. The corresponding string expression is finite
and its low-energy limit is sensitive to the presence of these massive states with momenta ∼ 1/ℓs.
Applying this to d = 4, we immediately conclude that there is no limit in four dimensions
where we can keep all BPS particles heavier than the Planck scale. In particular, magnetic
duals of Kaluza–Klein excitations, which are the well-known Kaluza–Klein monopoles, are
BPS states with masses ∼ R/ℓ24 → 0.2 Similarly, magnetic duals of wrapped strings are
NS5-branes wrapping 5-cycles of T 6, and their masses go as ℓ2s/Rℓ
4 → 0. Later, we will
discuss implications of these light states. When d ≥ 5, at least a subset of the BPS branes
become tensionless in the limit (3).
By contrast, in three dimensions it is possible to define a limit where all BPS particles
become infinitely massive simultaneously. In this case, magnetic duals of BPS particles
are (−1)-branes, namely instantons, and their Euclidean actions vanish in the limit. Thus,
one would expect nonperturbative effects to be very large in three dimensions even though
no singularity is apparent from the spectrum.
In two dimensions, there are no magnetic duals of BPS particles, and we expect that
there is a smooth limit where all BPS particles are massive and instanton actions remain
non-vanishing.
Now, let us look at each case in more detail. When d = 2, the conditions we want to
impose are
and MNS5 =
→ ∞. (7)
On the other hand, the string coupling constant is given by
. (8)
Thus, the desired limit can be taken by sending R → 0 while keeping the string coupling
constant finite. In this limit, all particle masses are much higher than the Planck mass,
except for the massless two-dimensionalN = 16 supergravity states [6]. However, Dp-brane
and NS5-brane instantons wrapping T 8 have Euclidean actions proportional to (ℓs/R)
3−p ∼
8 and (ℓs/R)
2 ∼ g− 14 , respectively. Though the actions all remain finite and non-zero in
the limit, their effects are not uniformly suppressed for small g. Thus, the resulting theory
may not have a weak coupling limit that is dominated by the perturbative contribution.
When d = 3, the conditions we need to impose are
and MNS5 =
→ ∞. (9)
2 If the torus has six independent radii Ri, the Kaluza–Klein monopole mass spectrum has the
form M2 =
(niRi/ℓ
Since we now have
· ℓ3, (10)
we can rewrite (9) as
and MNS5 =
→ ∞. (11)
Since p = 0, 2, 4, 6 in Type IIA theory, this can again be arranged by taking R → 0
keeping g finite.3 This is also compatible with the limit (3). Thus, all particle states
develop large masses and may decouple, except for those in three-dimensional N = 16
supergravity theory [7]. However, Dp-brane and NS5-brane instanton actions, which are
given by g
4 (R/ℓ3)
8 and g−
2 (R/ℓ3)
4 , vanish in the limit R → 0 for any finite value of
g. This means that nonperturbative effects are strong and it may be difficult to determine
the properties of the resulting three-dimensional supergravity.
In view of these observations, it is interesting that gravity theories formulated in
terms of a finite number of fields are known to exist in two and three dimensions. In
three dimensions, the relation with Chern-Simons gauge theory [8] suggests that pure
Einstein gravity is finite to all orders in perturbation theory. However, this theory has
no propagating degrees of freedom, and it is not known whether there is a finite quantum
gravity theory in three dimensions that includes propagating (scalar or spin-1/2) degrees
of freedom. Such degrees of freedom are present, of course, in the examples considered
here. The fact that we find limits of string theory compactifications with a finite number
of such propagating degrees of freedom in these dimensions may be encouraging, though
the implications of the nonperturbative instanton contributions need to be understood.
When d = 4, the conditions, (3), necessary for the extra modes to have infinite masses
and MNS5 =
→ ∞. (12)
Clearly, this cannot be realized simultaneously for all p = 0, 2, 4, 6. This is in accord with
the general argument given earlier, since a wrapped Dp-brane and a wrapped D(6−p)-brane
3 Note that, in the Type IIB theory, a wrapped D7-brane cannot be made heavy unless g ≫ 1.
This is not in contradiction with T-duality since g transforms under T-duality in such a way that
ℓp given by (1) remains invariant. T-duality along one of the circles on T
10−d transforms the
coupling g → gℓs/R so it diverges in the limit R → 0 with the original coupling constant, given
by (10), kept finite.
are electric–magnetic duals. Similarly, the magnetic duals of Kaluza–Klein excitations and
wrapped strings are Kaluza–Klein monopoles and wrapped NS5-branes, whose masses
behave as R/ℓ24 and ℓ
4, respectively. There are infinitely many such states since they
have arbitrary integer charges. In the limit R, ℓ2s/R → 0, there is no mass gap and the
spectrum becomes continuous.
To understand the implications of these infinitely many light states, we note that
among the elements of the four-dimensional U-duality group E7(Z) is the four-dimensional
S-duality transformation that interchanges the 28 types of electric charge with the corre-
sponding magnetic charges [9,10]. This duality is described by the following transforma-
tions of the moduli,
S : R → R̃ = ℓ
and ℓs → ℓ̃s =
. (13)
Note that this transformation inverts the radius R in four-dimensional Planck units (in
contrast to T-duality, which inverts R in string units). Since g is related to R and ℓs by
(2), this transformation acts as the inversion g → g̃ = 1/g, which maps BPS states into
each other. For example, a wrapped Dp-brane is interchanged with a wrapped D(6 − p)-
brane. Similarly, a Kaluza–Klein excitation is interchanged with a Kaluza–Klein monopole
(whereas T-duality would relate it to a wrapped F-string). Thus, in the dual frame in
which the compactification scale R̃ → ∞, the six-torus is decompactified. This explains
the continuous spectrum in the limit (3). The fact that an infinite set of states from the
nonperturbative sector become massless shows that the limit of interest does not result
in pure N = 8 supergravity in four dimensions. Rather, it results in 10-dimensional
decompactified string theory with the string coupling constant inverted. This is true in
both the type IIA and type IIB cases. The only way of avoiding this would be to relax (3),
in which case there would instead be extra finite-mass Kaluza–Klein or winding number
states, which would therefore not decouple.
One may regard our results on the limit of superstring compactification on T 10−d as
examples illustrating the conjectures formulated in [11,12] on the geometry of continuous
moduli parameterizing the string landscape. The conjectures concern consistent quantum
gravity theories with finite Planck scale in four or more dimensions. Among the conjectures
are the statements that, if a theory has continuous moduli, there are points in the moduli
space that are infinitely far away from each other, and an infinite tower of modes becomes
massless as a point at infinity is approached [12]. Since the limit considered in this paper
corresponds to a point in the moduli space of string compactifications at infinite distance
from a generic point in the middle of moduli space, the conjectures predict than an infinite
number of particles become massless in the limit. For d = 4, we have found that among
such particles are Kaluza–Klein monopoles, i.e., Kaluza–Klein modes on T 6 in the dual
frame in the limit R̃ → ∞. On the other hand, the moduli space of pureN = 8 supergravity
also contains infinite distance points, but it does not take account of new light particles
appear near these points. If the BPS particles required by string theory were included
one would have string theory and not N = 8 supergravity.4 Thus, the conjectures of
[12] imply that the N = 8 supergravity is in the Swampland. Similarly, there are many
superstring compactifications with N < 8 supersymmetry, and discarding stringy states in
these compactifications results in further supergravity theories in the Swampland.
It is interesting to see how scattering amplitudes behave in the limit (3). Consider
a four-dimensional graviton scattering amplitude where the graviton momenta are below
the four-dimensional Planck scale. According to (1) and (2), the ten-dimensional Planck
length, ℓ10, is given by
ℓ10 = g
4 ℓs = R
. (14)
After the S-duality transformation (13), the limit R → 0 turns into R̃ → ∞. Thus, we have
ℓ̃10 = R̃
→ ∞ in ten dimensions. Since ℓ̃10 ≪ R̃, the extra dimensions decompactify
and the theory is effectively ten-dimensional. Furthermore, if we take this limit keeping
the graviton momenta fixed (in units of the four-dimensional Planck mass), the scattering
process becomes trans-Planckian. Generically, we expect that it will involve formation and
evaporation of virtual black holes in ten dimensions.
The original motivation of this work was to investigate the relation between super-
string theory and N = 8 supergravity to see, in particular, under what conditions super-
gravity might be ultraviolet finite. What we have found is that in four or more dimensions
(d ≥ 4) there is no limit of compactified superstring theory in which the stringy effects
decouple and only the 256 massless supergravity fields survive below the four-dimensional
Planck scale. This is true whether or not there are ultraviolet divergences in supergravity
perturbation theory. Of course, there is a well-defined procedure for extracting UV finite
four-dimensional scattering amplitudes from perturbative string theory. This involves tak-
ing g → 0 first, before taking the limit (3). However, this procedure does not keep ℓ4 fixed,
and therefore it does not correspond to the limit considered in this paper.
4 One can imagine an alternative history in which type II superstring theory and M-theory
were discovered by properly interpreting the BPS solitons of N = 8 supergravity.
It might be instructive to compare the situation to that of the conifold limit of Calabi–
Yau compactified type II superstring theory studied by Strominger [13]. In that case,
certain terms in the low-energy effective theory that are independent of the string coupling
constant g, due to the decoupling of vector and hypermultiplet fields, can be computed in
string perturbation theory. One can estimate the singularity of these terms using the fact
that a brane wrapping a vanishing cycle describes a nonperturbative BPS particle that
becomes massless in the conifold limit. If one could identify analogous terms in N = 8
supergravity, one could transform the Feynman diagram computation in four-dimensional
supergravity into a corresponding computation in ten dimensions, which might give insight
into the question of ultraviolet finiteness.
The situation is qualitatively different in two and three dimensions (d = 2, 3), where
all non-supergravity states develop masses larger than the Planck scale in the limit (3),
and therefore they can decouple. In these cases only the 256 massless supergravity states
survive, and a self-contained quantum gravity theory may well exist decoupled from string
theory. We have found, however, that in the d = 3 case there are instantons with zero
action, which give rise to large nonperturbative contributions. In the d = 2 case the
instanton actions do not vanish in the limit (3), but not all of them are small when g is
small. Therefore the amplitudes may not be dominated by the perturbative contribution
in this case, too.
Acknowledgments
We thank Z. Bern, N. Dorey, C. Hull, J. Russo, N. Seiberg, A. Sen, M. Shigemori,
Y. Tachikawa, D. Tong, P. Vanhove and E. Witten for discussions. H.O. thanks the
hospitality of the particle theory group of the University of Tokyo.
H.O. and J.H.S. are supported in part by the DOE grant DE-FG03-92-ER40701. The
research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st
Century COE Program at the University of Tokyo.
References
[1] M. B. Green, J. G. Russo and P. Vanhove, “Non-renormalisation conditions in type
II string theory and maximal supergravity,” JHEP 0702, 099 (2007) [arXiv:hep-
th/0610299].
[2] Z. Bern, L. J. Dixon and R. Roiban, “Is N = 8 supergravity ultraviolet finite?,” Phys.
Lett. B 644, 265 (2007) [arXiv:hep-th/0611086].
[3] Z. Bern, J. J. Carrasco, L. J. Dixon, H. Johansson, D. A. Kosower and R. Roiban,
“Three-loop superfiniteness of N = 8 supergravity,” arXiv:hep-th/0702112.
[4] A. Sen, “D0-branes on Tn and matrix theory,” Adv. Theor. Math. Phys. 2, 51 (1998)
[arXiv:hep-th/9709220].
[5] N. Seiberg, “Why is the matrix model correct?,” Phys. Rev. Lett. 79, 3577 (1997)
[arXiv:hep-th/9710009].
[6] H. Nicolai and N. P. Warner, “The structure of N = 16 supergravity in two dimen-
sions,” Commun. Math. Phys. 125, 369 (1989).
[7] N. Marcus and J. H. Schwarz, “Three-dimensional supergravity theories,” Nucl. Phys.
B 228, 145 (1983).
[8] E. Witten, “(2+1)-dimensional gravity as an exactly soluble system,” Nucl. Phys. B
311, 46 (1988).
[9] C. M. Hull and P. K. Townsend, “Unity of superstring dualities,” Nucl. Phys. B 438,
109 (1995) [arXiv:hep-th/9410167].
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[arXiv:hep-th/9512181].
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[12] H. Ooguri and C. Vafa, “On the geometry of the string landscape and the swampland,”
[arXiv:hep-th/0605264].
[13] A. Strominger, “Massless black holes and conifolds in string theory,” Nucl. Phys. B
451, 96 (1995) [arXiv:hep-th/9504090].
http://arxiv.org/abs/hep-th/0610299
http://arxiv.org/abs/hep-th/0610299
http://arxiv.org/abs/hep-th/0611086
http://arxiv.org/abs/hep-th/0702112
http://arxiv.org/abs/hep-th/9709220
http://arxiv.org/abs/hep-th/9710009
http://arxiv.org/abs/hep-th/9410167
http://arxiv.org/abs/hep-th/9512181
http://arxiv.org/abs/hep-th/0509212
http://arxiv.org/abs/hep-th/0605264
http://arxiv.org/abs/hep-th/9504090

  We consider the conditions necessary for obtaining perturbative maximal
supergravity in d dimensions as a decoupling limit of type II superstring
theory compactified on a (10 -- d)-torus. For dimensions d = 2 and d = 3 it is
possible to define a limit in which the only finite-mass states are the 256
massless states of maximal supergravity. However, in dimensions d > 3 there are
infinite towers of additional massless and finite-mass states. These correspond
to Kaluza--Klein charges, wound strings, Kaluza--Klein monopoles or branes
wrapping around cycles of the toroidal extra dimensions. We conclude that
perturbative supergravity cannot be decoupled from string theory in dimensions
d > 3. In particular, we conjecture that pure N = 8 supergravity in four
dimensions is in the Swampland.

arXiv:0704.0777v1  [hep-th]  5 Apr 2007
CALT-68-2636
DAMTP-2007-25
UT-07-11
Decoupling Supergravity from the Superstring
Michael B. Green,1 Hirosi Ooguri,2,3 and John H. Schwarz2
1Department of Applied Mathematics and Theoretical Physics
Cambridge University, Cambridge CB3 0WA, UK
2California Institute of Technology, Pasadena, CA 91125, USA
3Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
Abstract
We consider the conditions necessary for obtaining perturbative maximal supergrav-
ity in d dimensions as a decoupling limit of type II superstring theory compactified on
a (10 − d)-torus. For dimensions d = 2 and d = 3 it is possible to define a limit in
which the only finite-mass states are the 256 massless states of maximal supergravity.
However, in dimensions d ≥ 4 there are infinite towers of additional massless and finite-
mass states. These correspond to Kaluza–Klein charges, wound strings, Kaluza–Klein
monopoles or branes wrapping around cycles of the toroidal extra dimensions. We con-
clude that perturbative supergravity cannot be decoupled from string theory in dimensions
≥ 4. In particular, we conjecture that pure N = 8 supergravity in four dimensions is in
the Swampland.
March, 2007
http://arxiv.org/abs/0704.0777v1
There has recently has been some speculation that four-dimensional N = 8 super-
gravity might be ultraviolet finite to all orders in perturbation theory [1,2,3]. If true, this
would raise the question of whether N = 8 supergravity might be a consistent theory that
is decoupled from its string theory extension. A related issue is whether N = 8 supergrav-
ity can be obtained as a well-defined limit of superstring theory. Here we argue that such
a supergravity limit of string theory does not exist in four or more dimensions, irrespective
of whether or not the perturbative approximation is free of ultraviolet divergences.
In this paper, we will study limits of Type IIA superstring theory on a (10 − d)-
dimensional torus T 10−d for various d. One may regard the following analysis as analogous
to the study of the decoupling limit on Dp-branes (the limit where field theories on branes
decouple from closed string degrees freedom in the bulk) for various p [4,5]. The decoupling
limit on Dp-branes is known to exist for p ≤ 5. On the other hand, subtleties have been
found for p ≥ 6, where infinitely many new world-volume degrees of freedom appear in the
limit. This has been regarded as a sign that a field theory decoupled from the bulk does
not exist on Dp-branes for p ≥ 6. We will find similar subtleties for Type IIA theory on
T 10−d ×Rd for d ≥ 4.
It will be sufficient for our purposes to consider the torus T 10−d to be the product of
(10− d) circles, each of which has radius R. Numerical factors, such as powers of 2π, are
irrelevant to the discussion that follows and therefore will be dropped. In ten dimensions,
Newton’s constant is given by
G10 = g
2ℓ8s ,
where ℓs is the string scale and g is the string coupling constant. Thus, the effective Newton
constant in d dimensions is given by
Gd ≡ ℓd−2d =
R10−d
g2ℓ8s
R10−d
, (1)
where ℓd is the d-dimensional Planck length, so that
. (2)
We are interested in whether there is a limit of string theory that reduces to maximal
supergravity, which is defined purely in terms of the dynamics of the 256 states in the
massless supermultiplet. In other words, we are interested in the limit in which all the
excited string states, together with the Kaluza–Klein excitations and string winding states
associated with the (10− d)-torus, decouple. A necessary condition for this to happen is
that these states are all infinitely massive compared to the d-dimensional Planck scale ℓd.
This is achieved by taking
, and
, (3)
with ℓd fixed. This is compatible with keeping g fixed for d < 6. If the extra states do
decouple then the surviving states are the 256 massless states of maximal supergravity,
which is N = 8 supergravity when d = 4.
Let us now consider the spectrum of nonperturbative superstring excitations in this
limit. First consider a Dp-brane wrapping a p cycle of the torus. The mass of such a state
in d dimensions is
· ℓ1−
. (4)
When d ≤ 5, we also need to consider a NS5-brane wrapping a 5 cycle. This has a mass
given by
MNS5 =
g2ℓ6s
· ℓ2−d
. (5)
In order to obtain the pure supergravity theory with 32 supercharges in d dimensions, these
nonperturbative states also need to decouple, so their masses must satisfy Mp,MNS5 ≫
1/ℓd. In the case of d = 4 the nonperturbative BPS particle spectrum also includes
Kaluza–Klein monopoles, which are discussed in the next paragraph.
Before studying the limit in any dimension, d, we will discuss what to expect on
general ground. A Kaluza–Klein momentum state and a wrapped string state have masses
1/R and R/ℓ2s , respectively, and they are half-BPS objects that carry a single unit of
a conserved charge. In d-dimensions, their magnetic duals are (d − 4)-branes. The BPS
saturation condition together with the Dirac quantization condition implies quite generally
that the mass m of a BPS particle and the tension T of its magnetic dual (d − 4)-brane
are related by
mT ∼ 1
. (6)
1 In this limit, the string length ℓs provides a regularization scale for supergravity. Thus, if
string amplitudes depend sensitively on ℓs, it can be taken as evidence for ultraviolet divergences
in supergravity. This is seen explicitly, for example, in the one-loop four graviton amplitude,
which is ultraviolet divergent in nine dimensions. The corresponding string expression is finite
and its low-energy limit is sensitive to the presence of these massive states with momenta ∼ 1/ℓs.
Applying this to d = 4, we immediately conclude that there is no limit in four dimensions
where we can keep all BPS particles heavier than the Planck scale. In particular, magnetic
duals of Kaluza–Klein excitations, which are the well-known Kaluza–Klein monopoles, are
BPS states with masses ∼ R/ℓ24 → 0.2 Similarly, magnetic duals of wrapped strings are
NS5-branes wrapping 5-cycles of T 6, and their masses go as ℓ2s/Rℓ
4 → 0. Later, we will
discuss implications of these light states. When d ≥ 5, at least a subset of the BPS branes
become tensionless in the limit (3).
By contrast, in three dimensions it is possible to define a limit where all BPS particles
become infinitely massive simultaneously. In this case, magnetic duals of BPS particles
are (−1)-branes, namely instantons, and their Euclidean actions vanish in the limit. Thus,
one would expect nonperturbative effects to be very large in three dimensions even though
no singularity is apparent from the spectrum.
In two dimensions, there are no magnetic duals of BPS particles, and we expect that
there is a smooth limit where all BPS particles are massive and instanton actions remain
non-vanishing.
Now, let us look at each case in more detail. When d = 2, the conditions we want to
impose are
and MNS5 =
→ ∞. (7)
On the other hand, the string coupling constant is given by
. (8)
Thus, the desired limit can be taken by sending R → 0 while keeping the string coupling
constant finite. In this limit, all particle masses are much higher than the Planck mass,
except for the massless two-dimensionalN = 16 supergravity states [6]. However, Dp-brane
and NS5-brane instantons wrapping T 8 have Euclidean actions proportional to (ℓs/R)
3−p ∼
8 and (ℓs/R)
2 ∼ g− 14 , respectively. Though the actions all remain finite and non-zero in
the limit, their effects are not uniformly suppressed for small g. Thus, the resulting theory
may not have a weak coupling limit that is dominated by the perturbative contribution.
When d = 3, the conditions we need to impose are
and MNS5 =
→ ∞. (9)
2 If the torus has six independent radii Ri, the Kaluza–Klein monopole mass spectrum has the
form M2 =
(niRi/ℓ
Since we now have
· ℓ3, (10)
we can rewrite (9) as
and MNS5 =
→ ∞. (11)
Since p = 0, 2, 4, 6 in Type IIA theory, this can again be arranged by taking R → 0
keeping g finite.3 This is also compatible with the limit (3). Thus, all particle states
develop large masses and may decouple, except for those in three-dimensional N = 16
supergravity theory [7]. However, Dp-brane and NS5-brane instanton actions, which are
given by g
4 (R/ℓ3)
8 and g−
2 (R/ℓ3)
4 , vanish in the limit R → 0 for any finite value of
g. This means that nonperturbative effects are strong and it may be difficult to determine
the properties of the resulting three-dimensional supergravity.
In view of these observations, it is interesting that gravity theories formulated in
terms of a finite number of fields are known to exist in two and three dimensions. In
three dimensions, the relation with Chern-Simons gauge theory [8] suggests that pure
Einstein gravity is finite to all orders in perturbation theory. However, this theory has
no propagating degrees of freedom, and it is not known whether there is a finite quantum
gravity theory in three dimensions that includes propagating (scalar or spin-1/2) degrees
of freedom. Such degrees of freedom are present, of course, in the examples considered
here. The fact that we find limits of string theory compactifications with a finite number
of such propagating degrees of freedom in these dimensions may be encouraging, though
the implications of the nonperturbative instanton contributions need to be understood.
When d = 4, the conditions, (3), necessary for the extra modes to have infinite masses
and MNS5 =
→ ∞. (12)
Clearly, this cannot be realized simultaneously for all p = 0, 2, 4, 6. This is in accord with
the general argument given earlier, since a wrapped Dp-brane and a wrapped D(6−p)-brane
3 Note that, in the Type IIB theory, a wrapped D7-brane cannot be made heavy unless g ≫ 1.
This is not in contradiction with T-duality since g transforms under T-duality in such a way that
ℓp given by (1) remains invariant. T-duality along one of the circles on T
10−d transforms the
coupling g → gℓs/R so it diverges in the limit R → 0 with the original coupling constant, given
by (10), kept finite.
are electric–magnetic duals. Similarly, the magnetic duals of Kaluza–Klein excitations and
wrapped strings are Kaluza–Klein monopoles and wrapped NS5-branes, whose masses
behave as R/ℓ24 and ℓ
4, respectively. There are infinitely many such states since they
have arbitrary integer charges. In the limit R, ℓ2s/R → 0, there is no mass gap and the
spectrum becomes continuous.
To understand the implications of these infinitely many light states, we note that
among the elements of the four-dimensional U-duality group E7(Z) is the four-dimensional
S-duality transformation that interchanges the 28 types of electric charge with the corre-
sponding magnetic charges [9,10]. This duality is described by the following transforma-
tions of the moduli,
S : R → R̃ = ℓ
and ℓs → ℓ̃s =
. (13)
Note that this transformation inverts the radius R in four-dimensional Planck units (in
contrast to T-duality, which inverts R in string units). Since g is related to R and ℓs by
(2), this transformation acts as the inversion g → g̃ = 1/g, which maps BPS states into
each other. For example, a wrapped Dp-brane is interchanged with a wrapped D(6 − p)-
brane. Similarly, a Kaluza–Klein excitation is interchanged with a Kaluza–Klein monopole
(whereas T-duality would relate it to a wrapped F-string). Thus, in the dual frame in
which the compactification scale R̃ → ∞, the six-torus is decompactified. This explains
the continuous spectrum in the limit (3). The fact that an infinite set of states from the
nonperturbative sector become massless shows that the limit of interest does not result
in pure N = 8 supergravity in four dimensions. Rather, it results in 10-dimensional
decompactified string theory with the string coupling constant inverted. This is true in
both the type IIA and type IIB cases. The only way of avoiding this would be to relax (3),
in which case there would instead be extra finite-mass Kaluza–Klein or winding number
states, which would therefore not decouple.
One may regard our results on the limit of superstring compactification on T 10−d as
examples illustrating the conjectures formulated in [11,12] on the geometry of continuous
moduli parameterizing the string landscape. The conjectures concern consistent quantum
gravity theories with finite Planck scale in four or more dimensions. Among the conjectures
are the statements that, if a theory has continuous moduli, there are points in the moduli
space that are infinitely far away from each other, and an infinite tower of modes becomes
massless as a point at infinity is approached [12]. Since the limit considered in this paper
corresponds to a point in the moduli space of string compactifications at infinite distance
from a generic point in the middle of moduli space, the conjectures predict than an infinite
number of particles become massless in the limit. For d = 4, we have found that among
such particles are Kaluza–Klein monopoles, i.e., Kaluza–Klein modes on T 6 in the dual
frame in the limit R̃ → ∞. On the other hand, the moduli space of pureN = 8 supergravity
also contains infinite distance points, but it does not take account of new light particles
appear near these points. If the BPS particles required by string theory were included
one would have string theory and not N = 8 supergravity.4 Thus, the conjectures of
[12] imply that the N = 8 supergravity is in the Swampland. Similarly, there are many
superstring compactifications with N < 8 supersymmetry, and discarding stringy states in
these compactifications results in further supergravity theories in the Swampland.
It is interesting to see how scattering amplitudes behave in the limit (3). Consider
a four-dimensional graviton scattering amplitude where the graviton momenta are below
the four-dimensional Planck scale. According to (1) and (2), the ten-dimensional Planck
length, ℓ10, is given by
ℓ10 = g
4 ℓs = R
. (14)
After the S-duality transformation (13), the limit R → 0 turns into R̃ → ∞. Thus, we have
ℓ̃10 = R̃
→ ∞ in ten dimensions. Since ℓ̃10 ≪ R̃, the extra dimensions decompactify
and the theory is effectively ten-dimensional. Furthermore, if we take this limit keeping
the graviton momenta fixed (in units of the four-dimensional Planck mass), the scattering
process becomes trans-Planckian. Generically, we expect that it will involve formation and
evaporation of virtual black holes in ten dimensions.
The original motivation of this work was to investigate the relation between super-
string theory and N = 8 supergravity to see, in particular, under what conditions super-
gravity might be ultraviolet finite. What we have found is that in four or more dimensions
(d ≥ 4) there is no limit of compactified superstring theory in which the stringy effects
decouple and only the 256 massless supergravity fields survive below the four-dimensional
Planck scale. This is true whether or not there are ultraviolet divergences in supergravity
perturbation theory. Of course, there is a well-defined procedure for extracting UV finite
four-dimensional scattering amplitudes from perturbative string theory. This involves tak-
ing g → 0 first, before taking the limit (3). However, this procedure does not keep ℓ4 fixed,
and therefore it does not correspond to the limit considered in this paper.
4 One can imagine an alternative history in which type II superstring theory and M-theory
were discovered by properly interpreting the BPS solitons of N = 8 supergravity.
It might be instructive to compare the situation to that of the conifold limit of Calabi–
Yau compactified type II superstring theory studied by Strominger [13]. In that case,
certain terms in the low-energy effective theory that are independent of the string coupling
constant g, due to the decoupling of vector and hypermultiplet fields, can be computed in
string perturbation theory. One can estimate the singularity of these terms using the fact
that a brane wrapping a vanishing cycle describes a nonperturbative BPS particle that
becomes massless in the conifold limit. If one could identify analogous terms in N = 8
supergravity, one could transform the Feynman diagram computation in four-dimensional
supergravity into a corresponding computation in ten dimensions, which might give insight
into the question of ultraviolet finiteness.
The situation is qualitatively different in two and three dimensions (d = 2, 3), where
all non-supergravity states develop masses larger than the Planck scale in the limit (3),
and therefore they can decouple. In these cases only the 256 massless supergravity states
survive, and a self-contained quantum gravity theory may well exist decoupled from string
theory. We have found, however, that in the d = 3 case there are instantons with zero
action, which give rise to large nonperturbative contributions. In the d = 2 case the
instanton actions do not vanish in the limit (3), but not all of them are small when g is
small. Therefore the amplitudes may not be dominated by the perturbative contribution
in this case, too.
Acknowledgments
We thank Z. Bern, N. Dorey, C. Hull, J. Russo, N. Seiberg, A. Sen, M. Shigemori,
Y. Tachikawa, D. Tong, P. Vanhove and E. Witten for discussions. H.O. thanks the
hospitality of the particle theory group of the University of Tokyo.
H.O. and J.H.S. are supported in part by the DOE grant DE-FG03-92-ER40701. The
research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st
Century COE Program at the University of Tokyo.
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