# Effective potential and warp factor dynamics

Michael R. Douglas1,&
1Simons Center for Geometry and Physics
Stony Brook University
Stony Brook, NY 11794 USA

&I.H.E.S., Le Bois-Marie, Bures-sur-Yvette, 91440 France
<EMAIL_ADDRESS>
###### Abstract:

We define an effective potential describing all massless and massive modes in
the supergravity limit of string/M theory compactification which is valid off-
shell, i.e. without imposing the equations of motion. If we neglect the warp
factor, it is unbounded below, as is the case for the action in Euclidean
quantum gravity. By study of the constraint which determines the warp factor,
we solve this problem, obtaining a physically satisfying and tractable
description of the dynamics of the warp factor.

##  1 Introduction

If our universe is described by string/M theory, there exist six or seven
extra dimensions of space, not yet detected by experiment. This is possible
because the extra dimensions can take the form of a small, compact manifold
$X$. A basic question in string/M theory is to know what types of manifold are
allowed, and to find general relations between their geometry and physics. In
the present state of the art, this is generally done by solving the equations
of motion for the supergravity theory which describes the low energy limit,
and then taking into account various stringy and quantum effects.

The first works on string compactification assumed that some supersymmetry is
preserved at the compactification scale, for good physical and mathematical
reasons. Well known physical arguments suggest low energy supersymmetry;
supersymmetry favors stability; and supersymmetry places strong constraints on
the local geometry of $X$. In the best studied cases, $X$ is a complex Kähler
manifold, so powerful techniques of algebraic geometry are available for the
analysis.

More recently, there has been a shift towards techniques which do not assume
supersymmetry. After all, supersymmetry is broken in our universe, and we
don’t know at what scale it is broken. String/M theory suggests many other
solutions to the hierarchy problem, such as large extra dimensions [4] or
warped hierarchies [29]. Or, the hierarchy might simply be a chance property,
following from a lucky tuning of parameters in a small subset of vacua. Given
a measure factor (a probability distribution over vacua), presumably emerging
from early cosmology, it might turn out that this class of compactifications
outweighs those with low energy supersymmetry [14, 33].

Another reason to be interested in compactifications without supersymmetry is
to get models of inflation, because the required positive vacuum energy breaks
supersymmetry. Indeed this is a reason to study not just vacua, meaning long-
lived compactifications to maximally symmetric space-times, but the dynamics
on the larger configuration space which contains the vacua.

At present the main technique for addressing any of these questions is to
start from a class of compactifications, say with a specific choice of
topology for the compact manifold $X$, and derive a four (or $d$) dimensional
effective field theory which describes theories within this class. One begins
by identifying various “pseudo-moduli fields” such as metric moduli, brane
positions and the like, which parameterize the vacua within this class. One
then derives an effective potential, which is a function of the these fields.
This is usually done by combining various exact results for related
supersymmetric compactifications, brane world-volume theories, and
supersymmetric quantum field theories, in a first approximation by adding
them, and then considering corrections.

Having derived an effective potential, one then looks for its local minima.
The first issue is that, since the potential always goes to zero at large
volume and weak coupling [13, 22], one must find effects that produce a
barrier to this runaway. Having done this, the easiest way to argue that local
minima exist at finite moduli is to show that, to a good approximation, the
effective potential in the regime under study is a sum of many different,
loosely correlated, contributions from different sources: fluxes, branes,
quantum effects, curvature, and so on. By approximately balancing two or three
of these contributions and then tuning their precise strengths, one can obtain
potentials with local minima.

One important conclusion from this work is that there is no particular
obstacle within string theory to realizing de Sitter space-time [25], and thus
the small observed dark energy could be a cosmological constant. Indeed, given
that one can obtain local minima with a small negative vacuum energy, compared
to the individual contributions, it would seem that only an incredible
conspiracy between the different effects would prevent one from obtaining
similar vacua with small positive vacuum energy. In relatively simple and
controllable models such as the explicit KKLT realizations of [10], one could
in principle get enough control to rule out such conspiracies. In practice,
one brings in some physical intuition at this point, using arguments such as
parametric separation of energy scales of different effects (so they cannot
always cancel) or spatial separation of effects from different regions in $X$.
Combining these arguments with the lack of any proposed mechanism for the
supposed conspiracy, the conclusion seems well enough justified, though better
arguments would certainly be welcome.

This general approach and many examples are reviewed in [15]. It is very
effective in determining general structure and relations between parameters
such as masses and coupling constants; for example the general differences
between IIb, IIa, large volume and heterotic compactifications are all readily
understandable in these terms. One can also show that certain classes of
models cannot contain vacua, or cannot realize slow-roll inflation [24],
because of runaways to large volume or weak coupling. Other reasons to develop
this approach are that it could in principle be generalized to nongeometric
compactifications, and finally it is the best we can hope to do for the vast
majority of compactifications.

Once one grants the validity of the effective action, the question of the
existence of vacua with positive vacuum energy, and even general
nonsupersymmetric vacua, becomes in principle straightforward to answer. It
also leads to a simple picture of dynamics. The best studied example is the
dynamics of inflation, which can be realized by slowly rolling down a nearly
flat potential. More recently, models have been suggested which take advantage
of other structure, such as in the kinetic terms [3]. Although one expects the
effective action picture to break down at high energies, in the best case at
the (higher dimensional) Planck scale, this can still be well above the energy
scale of slow-roll inflation.

One problem with the effective potential approach is that string theory and
even simple Kaluza-Klein reduction involve an infinite set of fields, while
the usual truncation to the pseudo-moduli fields is somewhat ad hoc. Although
there is a simple argument that one can solve the equations of motion for
massive fields in terms of light fields, this ignores the possibility of
tachyonic modes, which will destabilize vacua. A candidate vacuum for present-
day physics must be tachyon-free, while configurations with tachyons are
important for cosmology. In dynamical situations in which fields undergo large
variations, of course the splitting of fields into “massless” and “massive”
can change drastically.

There are simple arguments that massive modes are under control, but these
tend to use supersymmetry. For example, one often builds up an effective
potential by combining sectors which individually respect different
supersymmetries. Another simple argument is that, even after supersymmetry
breaking, the effective potential is approximately a sum of squares, up to
corrections of order $F/M_{Planck}^{2}$. Such arguments seem believable given
a hierarchy between the supersymmetry breaking scale and the masses of the new
modes, but are not convincing otherwise.

A second level of analysis in which one can see the massive modes is to solve
the full $9+1$ or $10+1$-dimensional supergravity field equations, or the beta
function conditions for a conformal world-sheet theory, or perhaps even the
full string field theory equations of motion. These various approaches include
more and more modes at the cost of eventual intractability. Of course, one
does not need an exact solution for all the modes; even the analysis of
linearized fluctuations of massive modes around a solution would go a long way
towards answering the stability question.

A problem with these approaches is that they are classical (from the point of
view of space-time), while the existing constructions of vacua usually rely on
quantum corrections for stability and supersymmetry breaking. To try to
address this problem, one can note that, despite being formulated in higher
dimensions or with string fields, we can still think of these approaches as
each based on an effective potential, essentially defined as the higher
dimensional part of the action, but now taken as a function of all of the
massless and massive modes. The higher dimensional equations of motion
correspond to solving the condition $\partial V/\partial\phi^{i}=0$, while the
linearized stability analysis simply corresponds to computing the mass matrix
$\partial^{2}V/\partial\phi^{i}\partial\phi^{j}$. To the extent that one can
make this idea precise, one can then try to deal with quantum corrections by
the same prescription which was used before, namely to simply add them in, or
make other corresponding adjustments of the effective potential. We could
refer to this entire class of approaches as “semiclassical,” to be contrasted
with a fully nonperturbative approach such as gauge-gravity duality, which
unfortunately is not known to exist for theories with positive vacuum energy.

Since in the supergravity limit, a vacuum is a solution of well understood
higher dimensional equations of motion, the problem of constructing an
effective potential which takes massive modes into account is in principle
just one of isolating the appropriate terms in the original action. Following
up on works such as [34, 21, 9], in [23] Giddings and Maharana proposed an
effective potential, based on reinterpreting the constraint in the Einstein
equations. They went on to show that it worked in examples such as that of
[21].

While we did not know about this proposal during the course of our work, and
only found out about it after having distributed our work as a preprint, in
general we followed a similar approach, and obtained an effective potential
consistent with that of [23], but with many differences in our discussion. We
show explicitly that a critical point of the potential solves all the
equations of motion, and we have eliminated the implicit assumption that
space-time is Minkowski made at various points in [23]. Another difference is
that the two proposals use different conventions for the warp factor. The
conventions followed here have the great advantage that the constraint becomes
linear, enabling us to go on and better understand the physics in §3 and §4.

On further consideration of the resulting effective potential, one realizes
that there is a significant problem with its interpretation, which we now
explain.

###  1.1 Questions addressed in this work

In this work we develop the analysis of compactifications starting from the
equations of motion of general relativity coupled to the type of matter
sectors which appear in maximal supergravity theories, including possible
stringy corrections. We believe that the same ideas would apply to any of the
semiclassical approaches, if we understood the relevant configuration spaces.

Some issues which we try to shed light on include:

  * •

The definition of the four (or $d$) dimensional effective potential. While it
is in principle clear how to define this for a compactified field theory (not
including gravity), this is not the case once gravity is included. Indeed,
there are well-known difficulties in making a global definition of energy in
general relativity [37]. The usual response in this context (as in [15]) is to
work in approximately asymptotically flat backgrounds, and appeal to the
standard definition for this case. This is obviously not satisfactory when
doing cosmology.

One particularly interesting contribution to the effective potential is minus
the integral of the scalar curvature of the compactification manifold,

$V_{eff}=-\frac{1}{2}\int\sqrt{g}R^{(k)}.$ (1)

This term can be obtained by the general procedure we just discussed, of
regarding the higher dimensional part of the action as an effective potential,
as we will review in section §2 (see Eq. (12)). At least naively, it is
responsible for anticorrelating the curvature of the compactification manifold
with that of space-time, as in the well-known ${AdS}_{p}\times S^{q}$
solutions. While there are additional terms in the Einstein equation and this
is oversimplified, at first sight it seems reasonable to think of this as one
of the many terms in the effective potential.

However, a serious problem with this interpretation of Eq. (1) is that the
integral of the scalar curvature can be made arbitrarily large and positive,
by making a rapidly varying conformal transformation of the metric. Thus, Eq.
(1) is unbounded below, and does not even have local minima. This problem is
closely related to the fact that the action in Euclidean gravity is unbounded
below, and to the “wrong sign kinetic term” for the conformal factor which may
be familiar from minisuperspace treatments of gravity. As such, one would
expect that it is solved (at least classically) by imposing the appropriate
constraints. In particular, in the canonical formulation, the Hamiltonian
constraint must be imposed at each point in space, and determines the
conformal factor in terms of the other fields. At least in asymptotically flat
space-times, the resulting energy would be non-negative, by the positive
energy theorem [31, 38].

However, it is not so clear how the Hamiltonian constraint solves this
problem, because it directly determines only the conformal factor on
$D-1$-dimensional space, leaving the conformal factor on $X$ free to vary. It
corresponds to the “warp factor” constraint in the existing analyses, which is
a clue to the physical interpretation.

Is the potential Eq. (1) bounded below, and if so why? If it is, what
determines the correct conformal factor with which to evaluate it? Can we
solve for this variable to simplify the potential?

  * •

The nature of the warp factor constraint. In doing Kaluza-Klein reduction, the
$d$-dimensional part of the Einstein equation (in fact, the Hamiltonian
constraint) turns into a partial differential equation for the warp factor. By
analysis of this equation, in [11, 27] it was shown that one cannot realize de
Sitter space-time in pure supergravity, i.e. without stringy corrections or
singularities. Later it was argued [21, 12] that this could be evaded using
orientifolds and/or “$0$-form flux” (the Romans mass term in IIa
supergravity), and this is no longer considered a major obstacle.

As the warp factor plays such an important role in the physics, perhaps even
solving the hierarchy problem, it would be nice to have a better conceptual
understanding of it. Are there other conditions for this constraint to have a
solution? What if the various sources to it are large, or widely separated on
$X$ ? What if the sources evolve in time?

  * •

The possibility of compactification on negatively curved manifolds, and the
nature of flux or other effects needed to stabilize negative curvature.
According to Eq. (1), negative scalar curvature makes a positive contribution
to the vacuum energy, which could make de Sitter space easier to realize. This
point has been particularly emphasized by Silverstein, who has proposed a
variety of constructions of this type [30, 32]. There are many other works on
the subject, including [8].

Now, the no-go theorems of [11, 27] do not make any assumption about the
curvature, and would hold in this case as well. Thus, even with negative
curvature, one still needs to call on stringy effects to get de Sitter
compactifications. Is negative curvature actually relevant for this, and if so
why? Perhaps consistent compactifications of this type are always dual to
others of more familiar types?

The possibility of compactification on negatively curved manifolds raises
another point, which is that there are far more of these than manifolds of
zero or positive curvature. This is illustrated by the familiar example of
Riemann surfaces, for which the curvature is proportional to the Euler
characteristic $\chi=2-2g$, and is true in higher dimensions as well. If such
manifolds could be used to get compactifications with negative or small
positive vacuum energy, it seems almost inevitable that they would dominate
the landscape.

In this work, we will give an explicit expression for the effective potential
in supergravity compactifications, Eq. (47), mostly answering the first two
questions, and making a start on the third.

Besides understanding solutions, another important application of an effective
potential is to study time dependence. In particular, slow-roll inflation can
be described as gradient descent. With this potential, this leads to equations
similar to the Ricci flow equations, as we will discuss elsewhere. We might
also hope that this will shed light on the deeper questions regarding the
existence of the effective potential raised by Banks in [5], or on the old
problem of the conformal factor in Euclidean quantum gravity.

##  2 Effective potential

###  2.1 General discussion

We consider $D$-dimensional solutions of general relativity coupled to matter,
with an action such as

$S=\int\sqrt{-g}\left(R^{(D)}-\frac{1}{2}\sum_{p}|F^{(p)}|^{2}\right).$ (2)

We take metric signature $-+++\ldots$, and define units so that the
fundamental Planck scale $M_{Pl,D}=1$.

The equations of motion are the Einstein equation,

$R_{MN}-\frac{1}{2}g_{MN}R=T_{MN}$ (3)

coupled to $p$-form gauge field strengths, with equation of motion $d*F=0$ and
stress-energy

$T_{MN}=pF_{MI_{1}\cdots I_{p-1}}F_{N}^{I_{1}\cdots
I_{p-1}}-\frac{1}{2}g_{MN}F^{2}.$ (4)

We choose a nonstandard normalization so that the case $p=0$ can be treated
uniformly.

At least in the absence of Chern-Simons terms, one can choose to use an action
in terms of either $F^{(p)}$ or its dual $F^{(D-p)}$. In the following, we
will use this freedom to consider all background fluxes as magnetic fluxes
(so, $F_{0\ldots}=0$), just to simplify the discussion.

We consider compactification on a $k=D-d$-dimensional compact manifold $M$ to
$d$-dimensional maximally symmetric space-time (Minkowski, AdS, dS). Whenever
there is any ambiguity, we superscript the metric and curvatures with the
dimension of space-time $D,k$ or $d$. We superscript forms with their rank –
since forms can be pulled back, generally there is no ambiguity.

We make a Kaluza-Klein warped metric ansatz,

$ds^{2}=\eta_{\mu\nu}e^{2A(y)}dx^{\mu}dx^{\nu}+g_{ij}(y)dy^{i}dy^{j}.$ (5)

Here $\eta_{\mu\nu}$ is a maximally symmetric metric in $d$ dimensions, which
could be dS, AdS or Minkowski, and $g_{ij}$ is a metric on $M$.

Our goal is to write a $d$-dimensional effective action,111 We absorb the
customary $1/16\pi$ into our definition of $G_{N}$.

$S^{(d)}=\int\sqrt{-g^{(d)}}\left(\frac{1}{G_{N}}R^{(d)}-\frac{1}{2}G_{ab}(\phi)\partial\phi^{a}\partial\phi^{b}-2V_{eff}(\phi)+\ldots\right),$
(6)

whose equations of motion agree with the KK reduction of the $D$-dimensional
Einstein equations. By $\ldots$ we denote gauge field actions and other terms
we will not treat in detail here. Thus, the $d$-dimensional Einstein equation
is

$R^{(d)}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{(d)}=G_{N}T^{(d)}_{\mu\nu}$ (7)

with

$T^{(d)}_{\mu\nu}=-\frac{1}{2}G_{ab}(\phi)\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{b}-\frac{1}{2}g_{\mu\nu}\left(\frac{1}{2}G_{ab}(\phi)\partial\phi^{a}\partial\phi^{b}+2V_{eff}(\phi)\right);$
(8)

in particular $T_{00}=V_{eff}+\ldots$.

In principle, we now want to rewrite the $D$-dimensional fields using a mode
decomposition, as appears in [17] and many other works. We would then
substitute these expressions into the $k$-dimensional part of the action, do
the integral over $X$ and reinterpret the results as terms in Eq. (6). Any
term with no space-time derivatives would become part of the effective
potential.

The main difficulty in doing explicit mode expansions is to diagonalize the
various Laplacians on $X$ which appear as kinetic terms. In a mathematical
sense this step is well understood, and is not directly relevant for our
purposes. Since we do not need to diagonalize the metric $G_{ab}(\phi)$ on
field space, any complete basis for functions on $X$ would suffice. Thus, we
can regard the $D$-dimensional fields as generating functions for the totality
of massive modes in the $d$-dimensional compactified theory.

###  2.2 Einstein equations

The subtleties we are concerned with appear elsewhere, and can be seen by
reviewing the standard discussion of compactification based on the
$D$-dimensional Einstein equations. Since these look rather different in $d$
and $k$ dimensions, we consider the two components separately. The
$d$-dimensional components can be written

$R^{(d)}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R^{(d)}=T^{(d)}_{\mu\nu}+\frac{1}{2}g_{\mu\nu}R^{(k)}$
(9)

and we would like to interpret the right-hand side as an effective
$d$-dimensional stress tensor. Since the $d$ dimensions have maximal symmetry,
we lose nothing by taking the trace,

$\displaystyle-\frac{d-2}{2}R^{(d)}$ $\displaystyle=$ $\displaystyle
T^{(d)}+\frac{d}{2}R^{(k)}$ (10) $\displaystyle=$
$\displaystyle-d\cdot\Lambda$ (11)

The right hand side of this equation defines an effective cosmological
constant,

$\Lambda=T_{00}-\frac{1}{2}R^{(k)},$ (12)

with a contribution from the curvature of $X$. To get a $d$-dimensional
equation, we then integrate over $X$, producing Eq. (1). More precisely, all
three terms depend on the volume of $X$, which will lead to the dependence on
Newton’s constant $1/G_{N}$ in the result.

For standard (classical) sources of energy, except for negative potential
energy (which is not present in the string/M theory case), $T_{00}\geq 0$, so
naively one might expect de Sitter to be easy to realize. On the other hand,
the scalar curvature also contributes, so one needs to decide which effect is
more important.

The $k$-dimensional curvature can be determined by using the $D$-dimensional
trace of the Einstein equation,

$\left(1-\frac{D}{2}\right)R=T^{(d)}+T^{(k)},$ (13)

to solve for $R$ in Eq. (3), obtaining

$R^{(k)}_{ij}=T^{(k)}_{ij}-\frac{1}{D-2}g_{ij}\left(T^{(d)}+T^{(k)}\right).$
(14)

Taking the trace, one sees that the sign of $R^{(k)}$ is correlated to that of
$T^{(k)}$, while the $T^{(d)}=-dT_{00}$ contribution now favors positive
internal curvature. Thus, the effects of $T^{(d)}$ in the two equations
conflict with each other.

Solving for $R^{(k)}$ and substituting back into Eq. (12), we get

$\displaystyle\Lambda$ $\displaystyle=$ $\displaystyle
T_{00}-\frac{1}{2}\left(\frac{d-2}{D-2}T^{(k)}-\frac{k}{D-2}T^{(d)}\right)$
(15) $\displaystyle=$
$\displaystyle-\frac{(d-2)(k-2)}{2(D-2)}T_{00}-\frac{d-2}{2(D-2)}T^{(k)}.$
(16)

Evidently the total effect of $T_{00}$ always favors AdS, with the only hope
for dS being to have $T^{(k)}$ very negative compared to $T_{00}$, more
precisely

$T^{(k)}<-(k-2)T_{00}.$ (17)

Now, taking magnetic flux $F_{0\ldots}=0$ in Eq. (4), we have

$T^{(k)}=(p-\frac{k}{2})F^{2}=(2p-k)T_{00}.$ (18)

Thus, this condition will never hold for any $p\geq 1$, or sums of such
stress-tensors.

Thus, de Sitter space-time can only be realized if the matter stress tensor
violates the strong energy condition [37], which is

$0\leq R_{00}=T_{00}-\frac{1}{D-2}g_{00}T.$ (19)

Terms which violate this condition include positive potential energy, certain
string/M theory corrections, and the $p=0$ flux which appears in massive IIa
supergravity. Conceptually, this is the statement that the strong energy
condition is preserved under restriction; if it holds on $M\times X$ then it
will hold on $M$ [19, 20]. This no-go theorem was independently rediscovered
by deWit-Hari Dass-Smit [11] and Maldacena-Nunez [27].

Coming back to Eq. (12), while it might seem that we have justified Eq. (1),
there is an issue. We needed to use the $D$-dimensional Einstein equation Eq.
(3) in order to determine $R^{(k)}$. On the other hand, the effective
potential was supposed to be defined independently of solving the $d$ or
$D$-dimensional Einstein equation, so it is not clear that Eq. (1) can be
interpreted in this way.

Certainly, we cannot interpret Eq. (16) as an effective potential. This is
clear from the nontrivial factor in the relation between $T_{00}$ and
$\Lambda$, which even has the wrong sign. This factor arose because we solved
various equations of motion, including Eq. (3), and substituted the solutions.
While the final results are correct, identifying terms in partial results with
terms in an effective action is usually not.

We could correctly identify the curvature term in Eq. (12) with a term Eq. (1)
in the $d$-dimensional effective potential, if we could do this without
solving any equations of motion. However, we cannot simply relax Eq. (3), as
part of it (the $G_{00}$ component) is in fact a constraint. As we mentioned
in the introduction, if we do not enforce this constraint, the Einstein action
and thus the potential Eq. (1) is unbounded below and has no local minima. To
see this, evaluate Eq. (1) (using the results in the appendix) for the
conformally transformed metrics

${\tilde{g}}_{ij}=e^{2B}g_{ij},$ (20)

obtaining

$V_{eff}=\frac{1}{2}\int\sqrt{g}e^{(k-2)B}\left(-R^{(k)}[g]-(k-2)(k-1)(\nabla
B)^{2}\right),$ (21)

with the wrong sign for the derivative term.

Now it is true that the potential in gravity and supergravity is often
unbounded below, but only to one or a few directions in field space. On the
other hand, with a negative definite spatial derivative term, every short
wavelength perturbation of the conformal factor is tachyonic, so the theory
would be completely unstable. Somehow, this problem must be fixed by
incorporating the constraints of the $D$-dimensional theory.

Having seen the problem, it is not hard to come up with a strategy for dealing
with it. The constraints have two parts, a “zero mode” part and a “non-zero
mode” part. The former become constraints in the $d$-dimensional theory, while
the latter need to be solved in the process of dimensional reduction.

###  2.3 Incorporating the warp factor

We now study the dependence of the action on the warp factor, and consider the
general warped ansatz

$ds^{2}\equiv{\tilde{g}}_{AB}dx^{A}dx^{B}=e^{2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+e^{2B(y)}g_{ij}dy^{i}dy^{j}.$
(22)

We denote the metric with $A=B=0$ as $g_{AB}$, and the metric Eq. (22) as
${\tilde{g}}_{AB}$.

As $B$ is redundant with $g_{ij}$, for now we will take $B=0$, until we are
ready to discuss this part of the problem. Using the results in the appendix,
the $k$-dimensional part of the Einstein equations becomes

$R^{(k)}_{ij}-\frac{1}{2}g_{ij}R^{(k)}=T^{(k)}_{ij}+d\nabla_{i}\nabla_{j}A+d\nabla_{i}A\nabla_{j}A+g_{ij}\left(\frac{1}{2}e^{-2A}R^{(d)}-d\nabla^{2}A-\frac{d(d+1)}{2}(\nabla
A)^{2}.\right)$ (23)

The $d$-dimensional trace Eq. (10) becomes

$-\frac{d-2}{2}e^{-2A}R^{(d)}+d(d-1)\nabla^{2}A+\frac{d^{2}(d-1)}{2}(\nabla
A)^{2}=T^{(d)}+\frac{d}{2}R^{(k)}.$ (24)

with

$T^{(d)}=-\frac{d}{2}\sum_{p}|F^{(p)}|^{2}+T^{(d)}_{string}$ (25)

At this point we have added the term $T^{(d)}_{string}$ which represents the
non-classical sources present in superstring theory. We will not use its
detailed form, only the fact that it allows violating the inequality
$T^{(d)}\leq 0$.

Note that, except for $R^{(d)}$, every term in Eq. (24) comes with the same
weight $e^{\alpha A}$ with $\alpha=0$.222 See for example equations (2.12) and
(2.13) of [11]. This dependence has only two sources: the overall
$\sqrt{g^{(d)}}$, which sits in front of every term, and explicit factors of
the $d$-dimensional metric $g_{\mu\nu}$ or its inverse. However, since every
term is a scalar in space-time, and does not contain $d$-dimensional
derivatives, the $d$-dimensional metric cannot appear. This also includes
sources in $T^{(d)}_{string}$ which are space-time scalars. While the argument
we just gave does not cover Chern-Simons terms or electric field strengths,
they also have the same weight, as we argue in §2.7.

Eq. (24) can be dramatically simplified by the change of variable

$u\equiv e^{dA/2},$ (26)

and becomes

$-\frac{d-2}{2}R^{(d)}u^{1-4/d}=-{2(d-1)}\nabla^{2}u+\left(\frac{d}{2}R^{(k)}+T^{(d)}\right)u.$
(27)

This change of variables is used to great advantage in related mathematical
work, on the Yamabe problem [26] and in Perelman’s entropy functional [28].
Readers who have looked at this, or at Tseytlin’s discussion of an entropy
functional for sigma models [35] will recognize many ingredients of the
following discussion, for example Eq. (60).

###  2.4 Effective potential

We seek a functional of $u$ and $g_{ij}$ (and matter fields) whose variation
produces the two Einstein equations. Let us try direct substitution of the
ansatz into the $D$-dimensional action. We continue to take $B=0$, then

$\displaystyle S_{eff,R}$ $\displaystyle=$
$\displaystyle\int\sqrt{{\tilde{g}}}\left({\tilde{R}}^{(D)}-\frac{1}{2}|F^{(p)}|^{2}\right)$
$\displaystyle=$
$\displaystyle\int\sqrt{g}\left(u^{2-4/d}R^{(d)}+u^{2}R^{(k)}+\frac{4(d-1)}{d}(\nabla
u)^{2}-\frac{u^{2}}{2}|F^{(p)}|^{2}\right).$

The similarity to the string action with the dilaton is not coincidental, and
is because one can also obtain the dilaton by dimensional reduction.

Varying this with respect to $g_{ij}^{(k)}$, we get

$\displaystyle
u^{2}\left(R^{(k)}_{ij}-\frac{1}{2}g_{ij}R^{(k)}\right)-\nabla_{i}\nabla_{j}u^{2}+g_{ij}\nabla^{2}u^{2}$
$\displaystyle=$ $\displaystyle
u^{2}T^{(k)}_{ij}-\frac{4(d-1)}{d}\nabla_{i}u\nabla_{j}u$
$\displaystyle-\frac{1}{2}g_{ij}\left(-\frac{4(d-1)}{d}(\nabla
u)^{2}-R^{(d)}u^{2-4/d}\right)$

which can be checked to be equivalent to Eq. (23) with the substitution Eq.
(26). Here $T^{(k)}_{ij}$ is defined to be the variation of the matter action
with respect to $g_{ij}^{(k)}$.

Since $e^{2A}=u^{4/d}$, the variation $\delta g^{\mu\nu}=g^{\mu\nu}$ which
leads to Eq. (24), should be equivalent to varying with respect to $u$ and
multiplying by $-d/4$, and it is.

If we now try to identify the effective potential by direct comparison of Eq.
(2) and Eq. (6), we find

$\int\sqrt{g}\frac{u^{2}}{2}\left(-R^{(k)}+\frac{1}{2}\sum_{p}|F^{(p)}|^{2}-\frac{2}{d}T^{(d)}_{string}\right)-\frac{2(d-1)}{d}(\nabla
u)^{2}$ (30)

which is close but not quite right, because the $R^{(d)}$ term is missing from
the equations of motion. However, we cannot simply add it back, as we are
trying to define an effective potential which is purely a functional of the
$k$-dimensional fields.

Rather, we implement the strategy described at the end of §2.2. The equation
$0=\delta S/\delta u$ is a constraint equation, so we need to enforce it in
the definition of $V_{eff}$, except for a zero mode part. The zero mode part
is exactly the part sourced by the $R^{(d)}$ term, and thus we can do this by
replacing $R^{(d)}$ with an undetermined constant $C$. If the $d$-dimensional
equations equate this to $R^{(d)}$, we will get the correct $k$-dimensional
equations of motion.

On the other hand, from the point of view of the $d$-dimensional equations Eq.
(7), we are adding an extra term to $V_{eff}$. To reconcile the various
equations, we need to subtract the same term without the $u$ dependence.

While this may seem a bit ad hoc, there is another way to justify it. The term
in Eq. (2.4) which becomes the $d$-dimensional Einstein term is

$\int\sqrt{g}u^{2-4/d}\,R^{(d)}.$ (31)

Thus we identify the $d$-dimensional Newton’s constant as

$\frac{1}{G_{N}}=M_{Planck,d}^{d-2}=\int\sqrt{g}e^{(d-2)A}=\int\sqrt{g}u^{2-4/d},$
(32)

so we can interpret the constant $C$ as a Lagrange multiplier which enforces
this definition. This leads to

$V_{eff}=\frac{1}{2}\int\sqrt{g}\left[u^{2}\left(-R^{(k)}+\frac{1}{2}\sum_{p}|F^{(p)}|^{2}-\frac{2}{d}T^{(d)}_{string}\right)-\frac{4(d-1)}{d}(\nabla
u)^{2}\right]+\frac{1}{2}C\left(\frac{1}{G_{N}}-\int\sqrt{g}u^{2-4/d}\right)$
(33)

With this definition, the $\delta V_{eff}/\delta u=0$ constraint becomes

$-\frac{d-2}{2}Cu^{1-4/d}=-{2(d-1)}\nabla^{2}u+\left(T^{(d)}+\frac{d}{2}R^{(k)}\right)u,$
(34)

Given $C$, this is a sensible constraint, but we need to explain how to choose
$C$, and how this eventually implies Eq. (27). Before explaining this, let us
look at how we would solve Eq. (34).

###  2.5 Solving the constraint

To begin, let us set $C=0$, and instead add a term $\lambda u$ on the left
hand side. We get a Schrödinger-type equation on $X$,333 This equation (with
$T^{(d)}=0$) appears in [28, 35], although not with the interpretation of $u$
as a warp factor. It is also the $d\rightarrow\infty$ limit of Eq. (34), a
relation used in the math literature. Also, on replacing $d$ with $2-k$, Eq.
(34) with $T^{(d)}=0$ becomes the Yamabe equation [26].

$\lambda u=-\nabla^{2}u+\frac{d}{2(d-1)}U\cdot u.$ (35)

with a “local potential”

$U\equiv\frac{1}{2}R^{(k)}+\frac{1}{d}T^{(d)}.$ (36)

As is very familiar, such an equation has solutions $u_{i}$ for a discrete
spectrum of eigenvalues $\lambda_{i}$, and the $u_{i}$ form a complete basis
for functions on $X$.

This is the relevant equation for Minkowski space-time with $R^{(d)}=0$, so
let us discuss this case first. Since $u>0$ by its definition Eq. (26), the
only acceptable solution is the ground state $u_{0}$. However, the eigenvalue
$\lambda_{0}$ typically will not be zero. For this to be true, the integral of
the potential (against $u$) must be zero, which is the “warp factor
constraint” of [6, 21]. For compactification with $X$ Ricci flat and nonzero
flux, one can only satisfy this by adding sources with $T^{(d)}_{string}>0$,
such as orientifolds or certain $\alpha^{\prime}$ corrections. The same will
be true if we try to realize Minkowski space-time using an $X$ with negative
total scalar curvature $\int\sqrt{g}R^{(k)}<0$.

Keeping in mind our Schrödinger equation intuition, we return to the actual
constraint equation Eq. (34). Integrating over $X$, we derive the warp factor
constraint,

$-\frac{d-2}{2}C\int\sqrt{g}u^{1-4/d}=\int\sqrt{g}u\left(T^{(d)}+\frac{d}{2}R^{(k)}\right).$
(37)

While this is still a necessary condition, once we allow $C\neq 0$ it can
always be solved, and relates $C$ to the scale of $u$.

This is not yet the constraint of the no-go theorems [11, 27], which (as we
discuss in §5.1) uses the $k$-dimensional Einstein equation to control the
sign of the right-hand-side. If one knows that the local potential Eq. (36)
has a definite sign, clearly $C$ (and ultimately $R^{(d)}$) must have the
opposite sign. However, the local potential $U$ need not have a definite sign,
in which case one needs to know the warp factor $u$ to find the sign of the
integral in Eq. (37). Even when it has a definite sign, we need this
information to estimate $C$.

Rather fortuitously, for $d=4$, the equation Eq. (34) is linear inhomogeneous,
so we can easily get more information. Given the normalized eigenfunctions
$u_{i}$, satisfying Eq. (35) with eigenvalues $\lambda_{i}$ and

$\int\sqrt{g}u_{i}u_{j}=\delta_{ij},$ (38)

we can write the solution as

$u(y)=-\frac{1}{6}C\sum_{i}u_{i}(y)\frac{1}{\lambda_{i}}\int_{X}\sqrt{g}\,u_{i}.$
(39)

If there are any modes with $\lambda_{i}=0$ (normally there will not be), they
can be left out of the sum, and enter the solution with undetermined
coefficients.

To summarize, although we cannot solve the constraint Eq. (27) before knowing
the $d=4$-dimensional curvature $R^{(d)}$, we can solve it up to an overall
coefficient. The equation Eq. (34) obtained by varying Eq. (33) is equivalent
to Eq. (27); we just need to relate the coefficient $C$ to $R^{(d)}$.

The same idea can be applied in $d\neq 4$, and by rescaling $u$ one can set
$C=1$ in Eq. (34), restoring it with the relation $u\propto C^{d/4}$. Of
course, one would have to solve a nonlinear equation. At first sight, the
cases $d>4$ would appear similar to $d=4$, as the nonlinearity is mild. On the
other hand, for $d=3$ the source term blows up as $u\rightarrow 0$, which
looks significant; however we leave the analysis of this for subsequent work.

###  2.6 Interpreting the constraint

To summarize the discussion so far, we seek a $k$-dimensional functional whose
variation leads to the Einstein equations for maximally symmetric
compactifications. By direct substitution of the ansatz, we obtain Eq. (2.4)
which has this property, but it is not purely $k$-dimensional, since it
depends on the $d$-dimensional curvature $R^{(d)}$.

Usually (this is a convention), the $d$-dimensional effective action is
defined in Einstein frame, i.e. with fixed Newton’s constant $G_{N}$. But this
is easy to obtain from Eq. (2.4), because $R^{(d)}$ and $G_{N}$ are conjugate
variables. Thus the effective potential is the Legendre transform of Eq.
(2.4),

$V_{eff}=\frac{1}{2G_{N}}R^{(d)}-\frac{1}{2}S_{eff,R}\,\bigg{|}_{\delta
V_{eff}/\delta R^{(d)}=0}\\\ $ (40)

with $R^{(d)}$ relabeled $C$, and the factor $\frac{1}{2}$ introduced to agree
with the conventions of Eq. (6). This reproduces Eq. (33).

The condition $\delta V_{eff}/\delta R^{(d)}=0$ defining the Newton constant
is

$\frac{1}{G_{N}}=M_{Planck,d}^{d-2}=\int\sqrt{g}e^{(d-2)A}=\int\sqrt{g}u^{2-4/d}.$
(41)

This is an independent condition on $u$ and can also be used to determine the
overall normalization of the warp factor. Thus, we choose the coefficient $C$
in order to satisfy this condition.

Substituting the solution of the constraint Eq. (34) into the effective
potential $V_{eff}$ defined in Eq. (33), and using the definition Eq. (41), we
get

$V_{eff}=\frac{d-2}{2d}\frac{C}{G_{N}}.$ (42)

Thus, if we use this effective potential in Eq. (6), and impose the Einstein
equation following from this action, we obtain Eq. (10) with $C=R^{(d)}$, and
have reproduced the discussion of §2. We stress that we do not need to know
$R^{(d)}$ to compute it, rather this is done by solving Eq. (34), and then
imposing Eq. (41). Nor does one need to impose the $k$-dimensional Einstein
equations; it is defined for general metric and field configurations.

Substituting Eq. (39) into Eq. (41) (for $d=4$), we find that

$\frac{1}{C}=-\frac{1}{6}G_{N}\sum_{i}\frac{1}{\lambda_{i}}\bigg{|}\int\sqrt{g}u_{i}\bigg{|}^{2}.$
(43)

and

$\frac{1}{G_{N}^{2}V_{eff}}=-\frac{2}{3}\sum_{i}\frac{1}{\lambda_{i}}\bigg{|}\int\sqrt{g}u_{i}\bigg{|}^{2},$
(44)

an explicit expression for $V_{eff}$ in terms of $k$-dimensional quantities.

###  2.7 Incorporating the dilaton and other fields

Essentially the same results apply to very general matter theories coupled to
Einstein gravity; in particular supergravity, because every term in the
effective potential will have the same dependence on the warp factor. A
general argument to this effect is as follows. Since the effective potential
is a scalar, whose integral makes a contribution to the $d$-dimensional action
proportional to the space-time volume, the same factor of the $d$-dimensional
space-time volume element must appear in every term. Then, using the
definition made in Eq. (5), the dependence on the warp factor is the same as
the dependence on this volume element.

For example, the NS sector of the type II string actions in string frame leads
to

$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\int\sqrt{g^{(k)}}e^{-2\Phi}\left[u^{2}\left(-R^{(k)}-4(\nabla\Phi)^{2}+\frac{1}{2}|H^{(3)}|^{2}-T^{(d)}_{string}\right)-\frac{4(d-1)}{d}(\nabla
u)^{2}\right]$ (45)
$\displaystyle+C\left(\frac{1}{G_{N}}-\int\sqrt{g}e^{-2\Phi}u^{2-4/d}\right).$

In this form, it is not manifest that the dilaton contribution to the
potential is bounded below. One could make this manifest either by going to
Einstein frame in $k$ dimensions, or (for IIa theory) going to an M theory
description in $k+1$ dimensions. Our later arguments that the potential is
bounded below will assume that this has been done.

Two cases in which the warp factor dependence may not be immediately obvious
are contributions from electric flux, and contributions from the Chern-Simons
terms. The easy way to argue in the first case is to use duality to reexpress
the action in terms of magnetic flux. Thus, a $p$-form electric flux behaves
in expressions such as Eq. (18) like a $D-p$-form magnetic flux.

A more direct argument to this effect uses the fact that a background electric
flux compatible with maximal space-time symmetry must have $p\geq d$, and
transforms like the $d$-dimensional volume form multiplied by a $p-d$-form on
$X$. Then, it is quantized in units of the $d$-dimensional volume form [7].
Taking into account the space-time metric in the term $|F^{(p)}|^{2}$, this
term is independent of the warp factor, so again the entire dependence comes
from the volume form $\sqrt{g^{(d)}}$.

As for the Chern-Simons terms, these will only contribute to the effective
potential in the presence of a background electric field; for example in M
theory we can have

$\int C^{(3)}\wedge G^{(4)}\wedge G^{(4)}$ (46)

and a background $G^{(4)}=Gdx^{0}dx^{1}dx^{2}dx^{3}$ in $d=4$. Again, the
quantization condition on the electric field will force the same warp factor
dependence.

##  3 Physical discussion

Let us recap the final expression for the effective potential. We have
eliminated all dependence on $d$-dimensional physics except the number $d$,
which we now set to $d=4$. The metric, curvature, fluxes and warp factor $u$
are all defined on $X$, as is the additional “stringy source”
$T^{(d)}_{string}$.

$V_{eff}=\frac{1}{2}\int\sqrt{g}\left[u^{2}\left(-R+\frac{1}{2}\sum_{p}|F^{(p)}|^{2}-\frac{1}{2}T^{(d)}_{string}\right)-3(\nabla
u)^{2}\right]+\frac{1}{2}C\left(\frac{1}{G_{N}}-\int\sqrt{g}\,u\right).$ (47)

To use it, one enforces the warp factor constraint $\delta V_{eff}/\delta
u=0$, which in $d=4$ is linear,

$-\frac{1}{6}C=\left(-\nabla^{2}+\frac{2}{3}U\right)u;\qquad
U=\frac{1}{2}R-\frac{1}{4}\sum_{p}|F^{(p)}|^{2}+\frac{1}{4}T^{(d)}_{string}$
(48)

thus determining $C$ and $u$ up to an overall normalization, and the warped
volume constraint

$\frac{1}{G_{N}}=\int\sqrt{g}u,$

thus determining the normalization. Due to the simple form of Eq. (47), its
final value on substituting $u$ is simply $C/4G_{N}$, as in Eq. (42).

###  3.1 Physical regimes

With the main result in hand, let us discuss some of its physics in $d=4$.
First, we consider a constant warp factor,

$u=\frac{1}{G_{N}\mbox{Vol}X},$ (49)

so Eq. (47) reduces to

$V_{eff,unwarped}=-\frac{\int\sqrt{g}\,U}{(G_{N}\mbox{Vol}X)^{2}}.$ (50)

In this case, the original intuition leading to Eq. (1) is correct. This will
be exact if $U$ is constant.

Before we continue, there are two general conventions we could take for
$G_{N}$. When discussing fixing of moduli, such as $\mbox{Vol}X$, we want to
exhibit the dependence of different terms of the potential on the moduli, so
we should choose a fixed $G_{N}$ in fundamental units. In this case we have a
universal $1/(\mbox{Vol}X)^{2}$ (in $d=4$) prefactor for the potential, as in
[22].

As the simplest possible example, taking $X=S^{k}$, with $\mbox{Vol}X=L^{k}$,
and turning on flux $F^{(p)}$ with $p=k$, we have

$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\sqrt{g}u^{2}(-R+\frac{1}{2}||F||^{2})$ (51)
$\displaystyle=$
$\displaystyle\frac{1}{2G_{N}^{2}L^{2k}}\left(-R\,L^{k-2}+\frac{F^{2}}{2L^{k}}\right)$
(52)

where $R>0$ is the curvature of a diameter $1$ sphere. This potential has a
unique negative minimum. In this approximation, one recovers many of the
effective potentials found in the string compactification literature.

Once the volume is fixed, one is better off taking $G_{N}\propto
1/\mbox{Vol}X$, in which case $u\sim 1$. This is the convention we will follow
below.

As we now explain, there appear to be two general regimes, distinguished by
whether the warp factor is slowly varying or not. Since the new physics of the
warp factor has to do with its variation, the first might also be called “weak
warping,” or some more euphonious version of this.

###  3.2 Slowly varying warp factor

If the variation of $U$ is small, we can try to ignore the derivatives in Eq.
(48), to find

$u=-\frac{C}{4U}=-\frac{G_{N}V_{eff}}{U}.$ (53)

One necessary condition for this to make sense is that $U$ must have a
definite sign. Treating the derivatives as a perturbation, and solving to
first order, another condition for this approximation to be good is

$\bigg{|}\nabla^{2}\left(\frac{1}{U}\right)\bigg{|}<<1.$ (54)

While these conditions can be satisfied, for example in AdS compactifications,
they are quite restrictive. Localized sources like branes and orientifold
planes will violate them. Even without these, if $U$ varies on a scale $L$ (in
fundamental units), say set by the size of $X$, then we require

$L^{k+2}\cdot|G_{N}^{2}V_{eff}|>>1.$ (55)

Thus one requires large vacuum energy, large volume or both.

In this approximation, $V_{eff}$ is always smaller in magnitude than Eq. (50).
To see this, we begin by substituting Eq. (53) into Eq. (47), to get an
explicit expression for $V_{eff}$ in terms of $U$,

$\frac{1}{G_{N}^{2}V_{eff}}=\pm\left\|{\frac{1}{U}}\right\|_{L^{2-4/d}},$ (56)

where the sign is opposite to that of $U$, and we use the mathematical
notation

$\left\|{f}\right\|_{L^{\alpha}}=\left(\int\sqrt{g}\,|f|^{\alpha}\right)^{1/\alpha}.$
(57)

Then, we use the Cauchy-Schwarz inequality

$\left\|{f}\right\|\,\left\|{g}\right\|\geq(f,g)^{2},$ (58)

applied to $f=U^{1-2/d}$ and $g=U^{2/d-1}$.

Thus, in this regime warping increases the effective potential for AdS, and
would decrease it for dS.

###  3.3 The ground state approximation

When the condition Eq. (54) is not satisfied, we need to take the derivatives
into account. Let us consider the idea that, to a good approximation, the warp
factor will be proportional to the ground state wave function $u_{0}$. When it
is, we can drop the other terms in Eq. (39), to find

$u\sim cu_{0};\qquad\frac{1}{c}\equiv G_{N}\int\sqrt{g}u_{0};\qquad
C\sim-6G_{N}c^{2}\lambda_{0}$ (59)

and

$V_{eff}=-\frac{3}{2}c^{2}\lambda_{0}.$ (60)

Here $u_{0}\sim 1/\sqrt{\mbox{Vol}X}$ (since it is normalized), so
$c\sim\sqrt{\mbox{Vol}X}$ and $u\sim 1$.

Of course, a constant warp factor is the ground state for constant $U$, so
there will some nearby regime in which this is a good approximation. Here are
various arguments that this regime is large:

  * •

Out of all the source terms $\int\sqrt{g}u_{i}$ in Eq. (39), only $i=0$ has a
positive definite integrand. For the other eigenmodes, the integral will have
cancellations, typically making it much smaller than $1/G_{N}$.

  * •

Since the other eigenfunctions always have nodes and are negative in some part
of $X$, if they come in with significant coefficients, the solution is in
danger of violating the consistency condition that $u>0$ at each point.

  * •

Normally, the $1/\lambda_{i}$ factor will be maximized for $i=0$. This is
clear if the spectrum is positive. If $\lambda_{0}<0$, there are good reasons
(which we discuss below) to expect that the other $\lambda_{i}>0$, in which
case all but a few will have $\lambda_{i}>>|\lambda_{0}|$.

In §3.5, we will further discuss the validity of this approximation. To
oversimplify a bit, it will be good in the opposite regime from Eq. (55),

$L^{k+2}\cdot|G_{N}^{2}V_{eff}|\lesssim 1.$ (61)

In this approximation, warping significantly increases the effective
potential:

$V_{eff}=-\frac{3}{2}c^{2}\lambda_{0}>V_{eff,unwarped}=-\frac{\int\sqrt{g}\,U}{(G_{N}\mbox{Vol}X)^{2}}.$
(62)

This follows from the variational bound for the wave function $u={\rm
constant}$,

$-\lambda_{0}\geq-\frac{\int\sqrt{g}\,U}{\mbox{Vol}X},$ (63)

the Cauchy-Schwarz inequality

$\frac{1}{c^{2}}=G_{N}^{2}(\int\sqrt{g}u_{0})^{2})\leq G_{N}^{2}\mbox{Vol}X,$
(64)

and finally $3/2>1$.

###  3.4 Basic picture

The two approximations we discussed do not cover all the possibilities, and
one might study others, or develop a mixed picture by using each in different
regions and patching them together. However the general picture is similar
enough in both that we leave this for subsequent work, and instead try to
outline the physical picture.

As is familiar from quantum mechanics, the ground state wave function will
have most of its support in potential wells. This is also true (although less
so) for Eq. (53) with $U>0$.

Here, the local potential is minus the energy density $T_{00}$, so the warp
factor will be concentrated in regions of positive energy. Conversely, regions
with negative energy will have this energy warped (or redshifted) away.

Most sources, such as fluxes, make positive contributions to the energy
density. This also includes negative curvature. On the other hand, positive
curvature, and some stringy sources like orientifolds, make negative
contributions to the energy density.

Physically, the warp factor relates energy scales on $X$ to those in space-
time. In the language of AdS/CFT (see for example [36]), one says that $u$ is
large in the “UV” and small in the “IR.” The usual AdS solutions are supported
by $X$ with positive curvature. This contributes a negative energy density
which outweighs the flux energy, and sends $u\rightarrow 0$, consistent with
the picture we just gave. Conversely, the parts of $X$ with positive energy
density, either from flux or from negative curvature, are the UV.

Thus, the basic physics is that, while the vacuum energy density is a sum of
effects from different regions of the manifold, each given by minus the local
potential $U$, we must take into account the local warp factor in adding
energy densities, which we do with the weighing $u^{2}\cdot U$. This favors
the UV region, with positive energy, and thus raises $V_{eff}$.

This gives us a physical answer to the puzzle raised in the introduction, that
varying the conformal factor might send the effective potential to arbitrarily
negative values. Such variations produce positive scalar curvature, so
decrease the warp factor, and redshift away the negative energy. In §4 we will
try to demonstrate this from the equations.

In an actual vacuum, the local potential $U$ is determined by the
$k$-dimensional Einstein equation in a way we will discuss below. If one can
find solutions in which it is negative in some regions and positive in others,
since the warp factor weighs the negative regions more heavily, taking it into
account should make it easier to find de Sitter solutions. On the other hand,
the no-go theorems show that negative $U$ is not so easy to obtain.

###  3.5 Going through $\Lambda=0$

What happens if one varies parameters in $k$ dimensions (say pseudo-moduli or
fluxes), so that $V_{eff}$ crosses zero? While in conventional field theory,
there is nothing special about zero energy, in gravity there definitely is.
Can we see any sign of this in $k$ dimensions?

As we recalled in the introduction, the constructions of de Sitter vacua using
the effective potential, implicitly or explicitly rely on precisely such a
continuation of parameters. For example, one style of argument [25] is to
combine sources of vacuum energy which are computable in a supersymmetric AdS
vacuum, with a single additional supersymmetry breaking energy which “uplifts”
the vacuum to dS. This might be justified by continuing the parameters from a
better controlled nonsupersymmetric AdS vacuum, or conversely some flaw in the
reasoning might appear at this step. Thus, it is crucial to understand this
point.

At first sight, the series solution Eq. (39) for the warp factor looks
singular as an eigenvalue $\lambda$ passes through zero. We can get a simple
model for this by considering the family of local potentials obtained by a
simple shift $a$ of the energy,

$U_{a}\equiv U+a.$ (65)

In massive IIa theory, the $(F^{(0)})^{2}$ source has precisely this effect
(with $a<0$).

Clearly the eigenfunctions $u_{i}$ are independent of $a$, while the
eigenvalues simply become $\lambda_{i}+a$. Thus the series solution becomes

$u(y)=-\frac{C}{6}\sum_{i}u_{i}(y)\frac{1}{\lambda_{i}+a}\int\sqrt{g}u_{i}.$
(66)

As we approach the resonance, it should be a good approximation to keep only
the resonant term in the sum. Now, there are two cases:

  * •

An excited state, i.e. $i>0$, passes through zero, $\lambda_{i}+a\sim 0$.
Assuming that the matrix element $\int\sqrt{g}u_{i}\neq 0$, which (in the
absence of some special consideration such as symmetry) will almost surely be
true, we will have $u\sim u_{i}$ near the resonance. But since the excited
state wave functions $u_{i}$ all have nodes, this is inconsistent with
positivity of $u$. Before we reach this point, we will find $u=0$ somewhere on
$X$, so our assumptions must break down.

  * •

On the other hand, if the ground state energy passes through zero, we are
simply doing the same truncation to the ground state that we discussed in
§3.3. All of the approximate results given there, such as Eq. (59) and Eq.
(60), have sensible continuations through $\lambda_{0}=0$.

Thus, we see no obstacle to continuing $V_{eff}$ through zero to a de Sitter
vacuum with $\lambda_{0}<0$, up to the point where $\lambda_{1}$ crosses zero.
Somewhere before that point, at a vacuum energy

$\Lambda\sim M_{Planck,4}^{4}(\lambda_{1}-\lambda_{0}),$ (67)

there will be some sort of transition. It is not clear whether this can be
described using supergravity. If it can, perhaps the factorized form Eq. (5)
of the metric breaks down. If not, then assuming $u$ is analytic (as will be
true for $U$ analytic) and vanishes at a point $y_{0}$, the metric near this
point will look like (in $d=4$)

$ds^{2}=\alpha|y-y_{0}|\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}+\ldots,$ (68)

in other words an event horizon. The precise meaning of this depends on the
metric on $X$, and since the $k$-dimensional Einstein equation has source
terms which blow up as $u\rightarrow 0$, we need to look at details of the
example to work this out.

One could get good estimates for this gap using general results on Schrödinger
operators. For example, if the wavefunction is localized to a potential well
$U\sim m^{2}x^{2}/2$, there will be excited states with $\Delta\lambda\sim m$.
Again, the best way to do this probably depends on details of the example at
hand.

A simplified picture is to grant that

$\lambda_{1}-\lambda_{0}\sim L^{-2},$ (69)

which leads to the bound Eq. (61).

###  3.6 Cosmological application

In subsequent work we will try to develop a picture of early cosmological
dynamics based on these results. The basic idea is that the equations of
motion for slow roll inflation,

$3H\frac{\partial\phi^{a}}{\partial t}=-G^{ab}\frac{\partial
V_{eff}}{\partial\phi^{b}},$ (70)

can be computed using Eq. (47), to get a flow on the space of metrics and
other fields on $X$ similar to Ricci flow. This is somewhat like Perelman’s
treatment of Ricci flow as a gradient flow [28].

As is manifest from its definition, the effective potential $V_{eff}$
decreases under the flow; thus one expects that the local potential $U$
becomes more positive, and the eigenvalues $\lambda_{i}$ each increase, in a
way we can roughly model by taking Eq. (65) with $a\sim t$. In this situation
the consistency condition we just discussed might be a significant constraint
on the compactifications which lead to a sensible cosmology.

##  4 Stability

As we saw in §3, we expect the problem raised in the introduction, that the
effective potential is unbounded below under varying the conformal factor, to
be solved by the warp factor. Such variations produce positive scalar
curvature, so decrease the warp factor, and redshift away the negative energy.

In this section we look at how one might show this from the equations. In
fact, at this point we have a precise mathematical

###### Conjecture 1

Consider a conformal class of metrics $\tilde{g}=e^{2B}g$ on a $k$-dimensional
manifold $X$; then the functional Eq. (47) evaluated at its critical point
$\delta V/\delta u=0$, considered as a function on the space of all conformal
factors $B$ with fixed warped volume and volume

$\int\sqrt{\tilde{g}}u^{2-4/d}=C_{1};\qquad\int\sqrt{\tilde{g}}=C_{2};$ (71)

and all $F$, is bounded below.

To be precise, the volumes $C_{1}$ and $C_{2}$ should be defined in the
$k$-dimensional Einstein frame.444 By “Einstein frame,” one means conventions
in which $V_{eff}\sim-\int\sqrt{g}R$ without any field-dependent prefactor, of
the type which appears (for example) in Eq. (45). Such a prefactor can be
removed by a field-dependent conformal transformation.

Although this is not the same claim as the positive energy theorem (there are
AdS vacua), clearly it is related, and perhaps it can proven using some
variation of the arguments used there [31, 38]. Here we make some nonrigorous
but suggestive arguments.

###  4.1 Linearized stability

This was already checked for various explicit solutions in the early Kaluza-
Klein literature. Let us check it at short distances, for which one can simply
take a flat background metric. Thus, we allow a general conformal factor $B$
as in Eq. (22), turning Eq. (2.4) into

$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\int-\sqrt{g}e^{dA+(k-2)B}\left(R^{(k)}-2(k-1)\nabla^{2}B-(k-2)(k-1)(\nabla
B)^{2}+d(d-1)(\nabla A)^{2}\right)$ $\displaystyle=$
$\displaystyle-\int\sqrt{g}e^{dA+(k-2)B}\left(R^{(k)}+2d(k-1)\nabla A\nabla
B+(k-2)(k-1)(\nabla B)^{2}+d(d-1)(\nabla A)^{2}\right)$

While every term individually is negative definite, the determinant of the
$2\times 2$ matrix of kinetic terms is

$\det=(k-1)(k-2)d(d-1)-d^{2}(k-1)^{2}=-d(k-1)(D-2)<0$ (73)

so it has signature $(+1,-1)$. As we expect, the conformal factor is the only
mode with a wrong sign kinetic term. Solving the constraint and substituting
back in, can in principle produce a positive definite effective potential. We
check that this is true at the linearized level for short wavelength
fluctuations of $B$; the linearization of Eq. (24) is

$p^{2}A=-\nabla^{2}A=\frac{k-1}{d-1}\nabla^{2}B$

and the prefactor of the kinetic term becomes

$2d(k-1)\frac{k-1}{d-1}-(k-2)(k-1)-d(d-1)\left(\frac{k-1}{d-1}\right)^{2}=\frac{D-2}{(k-1)(d-1)}.$
(74)

###  4.2 WKB analysis

Next we do a WKB analysis of the effect of short length variations of the
conformal factor. We insert the ansatz

$u\propto e^{dA/2\hbar}$ (75)

into Eq. (48), and take the formal $\hbar\rightarrow 0$ limit, obtaining

$\frac{d}{4}g^{ij}(\nabla_{i}A)\nabla_{j}(dA+2(k-2)\hbar
B)=\frac{\hbar^{2}}{6}U+{\cal O}(\hbar^{2}),$ (76)

where the extra term on the left hand side comes from the connection. Using
the standard formulas, the curvature contribution to $U$ will go as

$U\sim\frac{(k-1)(k-2)}{2}e^{-2B}(\nabla B)^{2},$ (77)

so by taking $B\sim 1/\hbar$ we keep the growing term in the curvature, while
$C$ and the background value of $U$ can be neglected. Thus we are effectively
in the “ground state” regime of §3.3. This also makes the $e^{-2B}$ factor
scale in the same way as the metric $g^{ij}\sim e^{-2A}$.

The WKB estimate for this contribution to the effective potential is then

$V_{eff}\sim-2\int\sqrt{g}e^{dA}U$ (78)

While the integrand looks exponentially small, we should be careful as the
conformal factor can give us exponentially large $\sqrt{g}$. In a small
region, we can approximate $A$ and $B$ by their linear parts $A\sim a\cdot x$
and $B\sim b\cdot x$; then

$da(da+2(k-2)b)=\frac{(k-1)(k-2)}{3}b^{2},$ (79)

while $\sqrt{g}\sim e^{kB}$. We choose $b>0$, while out of the two solutions
for $a$, we choose the one with $a<0$ by the WKB ansatz. One can then show
that the overall exponent $da+(k-2)b$ is negative for all $k>2$.

While the computation looks similar to the linearized analysis, it is not the
same because it takes into account the exponentials in the integrand, and is
thus nonlinear.

###  4.3 Radially symmetric ansatz

Another nonlinear test at short distances can be done by considering a
radially symmetric conformal factor, say

$e^{2B}=(a^{2}+r^{2})^{\gamma},$ (80)

For $-2<\gamma<-1$, the conformal factor becomes large at $r\sim a$, producing
a region of large volume and large positive curvature. On the other hand, it
becomes small for $r>>a$, so one can patch such a region (or “bubble”) into a
general manifold $X$. This type of construction is used in the study of the
Yamabe problem [26].

While we omit the details here, one can show that the effective potential Eq.
(47) remains bounded below in these metrics as well.

To summarize this section, a variety of simple modifications to the conformal
factor have the full nonlinear energy bounded below. Since this would seem to
be necessary for string/M theory compactification on $X$ to be well defined,
we conjecture that it is always true.

From a physics point of view, it would be even more interesting if this worked
for some $X$ and failed for others, as this could give us a consistency
condition and cut down on the plethora of vacua. The only evidence for this we
see so far is that the case $d=3$ might be special, since the source in Eq.
(34) has a negative power of $u$ in this case.

While one might go on to conjecture that $V$ is bounded below for all metrics
on a given manifold $X$, there are some mathematical reasons to doubt this
[2]. Physically it is not required, as long as $V$ has local minima, for which
the tunnelling rates to lower minima are very small (as in [25]).

When $V$ is bounded below, the bound can be regarded as a topological
invariant of $X$, directly analogous to the Yamabe invariant [26]. Of course,
it might be the same invariant, as turned out to be the case for Perelman’s
entropy [2].

##  5 Solutions

So far, we avoided using the $k$-dimensional Einstein equation Eq. (2.4), so
that we could derive an effective potential which is valid off-shell. In this
section we generally assume we are expanding around a solution.

###  5.1 Solving the $k$-dimensional equations

We now bring in the Einstein equation, reinterpreted as above:

$\displaystyle
u^{2}\left(R^{(k)}_{ij}-\frac{1}{2}g_{ij}R^{(k)}\right)-\nabla_{i}\nabla_{j}u^{2}+g_{ij}\nabla^{2}u^{2}$
$\displaystyle=$ $\displaystyle
u^{2}T^{(k)}_{ij}-\frac{4(d-1)}{d}\nabla_{i}u\nabla_{j}u$
$\displaystyle-\frac{1}{2}g_{ij}\left(-\frac{4(d-1)}{d}(\nabla
u)^{2}-Cu^{2-4/d}\right)$

Note that $T^{(d)}_{string}$ does not appear in this formula, however since
normally it depends on the metric on $X$, terms from it will appear in
$T^{(k)}_{ij}$.

Its trace is

$\frac{k-2}{2}R^{(k)}=2(k-1)u^{-1}\nabla^{2}u+\frac{2(D-2)}{d}(u^{-1}\nabla
u)^{2}-\frac{k}{2}u^{-4/d}C-T^{(k)}.$ (82)

Finally, the flux equations of motion are

$d*(u^{2}F)=dF=0.$ (83)

It might seem tempting to eliminate $C$ using the constraint Eq. (24), to get
an equation in terms of $k$-dimensional quantities, equivalent to Eq. (14).
However this is not a good idea, because we would lose the knowledge that $C$
is constant on $X$.

Rather, we can use Eq. (24) to eliminate either $R^{(k)}$ or $-\nabla^{2}u$.
If we do the first, we get

$-(D-2)\nabla^{2}u^{2}+\left(dT^{(k)}-(k-2)T^{(d)}\right)u^{2}=(D-2)Cu^{2-4/d}.$
(84)

Integrating this equation over the manifold gives

$(D-2)C=-G_{N}\int\sqrt{g}\left(dT^{(k)}-(k-2)T^{(d)}\right)u^{2},$ (85)

which is the constraint appearing in the no-go theorems of [11, 27]. Writing
it in terms of $u^{2}$, it becomes a Schrödinger equation with a nonlinear
source.

If we do the other elimination, we get

$R^{(k)}=-\frac{4(d-1)}{d}(u^{-1}\nabla
u)^{2}+u^{-4/d}C+\frac{2(d-1)}{D-2}T^{(k)}-\frac{2(k-1)}{D-2}T^{(d)}.$ (86)

Both are interesting equations. The combinations of flux stress-tensors which
appear are

$\alpha T^{(k)}+\beta
T^{(d)}=\frac{1}{2}\sum_{p}((2p-k)\alpha-d\beta)|F^{(p)}|^{2}.$ (87)

In the original trace equation Eq. (82), we have

$-T^{(k)}-\frac{k}{2}T^{(d)}=\frac{1}{4}\sum_{p}(-4p+k(d+2))|F^{(p)}|^{2},$
(88)

which for $k>4$ is always positive. For Eq. (84), we have

$dT^{(k)}-(k-2)T^{(d)}=d\sum_{p}(p-1)|F^{(p)}|^{2}.$ (89)

As in the no-go theorems, this is non-negative except for $p=0$. Finally, in
Eq. (86) we have

$2(d-1)T^{(k)}-2(k-1)T^{(d)}=\sum_{p}((d-1)2p+k-d)|F^{(p)}|^{2}$ (90)

Again, for $k>d$ this is always positive.

To summarize, Eq. (82) and Eq. (86) make the point that flux favors positive
scalar curvature on $X$, while stringy corrections are needed to get negative
scalar curvature. We can now go on to consider how the curvature is
distributed in the internal dimensions, how this impacts the warp factor, and
whether the consistency condition $\lambda_{1}>0$ of the previous section is
significant or not. We will look at this in examples in future work.

###  5.2 Conformal factor

Next, we introduce the conformal factor. Fixing all the other fields, we would
like to find the conformal factor which minimizes $V_{eff}$. Granting that
this is bounded below, there should be a minimum satisfying $\delta
V_{eff}/\delta B=0$ (of course, there could be other critical points). We need
to fix the volume modulus to have a minimum; rather than do this physically we
simply impose $\mbox{Vol}(X)=\int\sqrt{g}v^{2k/(k-2)}$ with a Lagrange
multiplier $D$.

Taking $v=e^{(k-2)B/2}$, and $F=0$, we can rewrite Eq. (47) as

$\displaystyle V_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\sqrt{g^{(k)}}\bigg{[}-u^{2}v^{2}R-\frac{4(k-1)}{k-2}(\nabla
v)\nabla(u^{2}v)-\frac{4(d-1)}{d}v^{2}(\nabla u)^{2}$ (93)
$\displaystyle\qquad\qquad+\frac{u^{2}}{2}\sum_{p}v^{2(k-2p)/(k-2)}|F^{(p)}|^{2}\bigg{]}$
$\displaystyle+C\left(\frac{1}{G_{N}}-\int\sqrt{g}v^{2+4/(k-2)}u^{2-4/d}\right)+D\left(\mbox{Vol}(X)-\int\sqrt{g}v^{2+4/(k-2)}\right).$

The various $p$-form flux energies come with different powers of the conformal
factor $v$, given by the $L$ scaling of §3, while Chern-Simons terms do not
depend on the conformal factor. We will not quote these in the equations
below, but they can be easily added.

We expect that $V_{eff}$ is minimized at the simultaneous critical point

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\delta V_{eff}}{\delta
u}=-uv^{2}R+\frac{4(k-1)}{k-2}\,uv\,\nabla^{2}v+\frac{4(d-1)}{d}\nabla(v^{2}\nabla
u)-\frac{d-2}{d}Cv^{2+4/(k-2)}u^{1-4/d}$ $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\delta V_{eff}}{\delta
v}=-u^{2}vR+\frac{2(k-1)}{k-2}\nabla^{2}(u^{2}v)+\frac{2(k-1)}{k-2}\nabla(u^{2}\nabla
v)$ $\displaystyle\qquad\qquad-\frac{4(d-1)}{d}v(\nabla
u)^{2}-\frac{k}{k-2}v^{1+4/(k-2)}\left(Cu^{2-4/d}+D\right).$

We can simplify the $u$ constraint a bit by the change of variables $u=w/v$,
to get

$\displaystyle 0$ $\displaystyle=$ $\displaystyle-
wvR-\left(\frac{4(d-1)}{d}-\frac{4(k-1)}{k-2}\right)\,w\,\nabla^{2}v+\frac{4(d-1)}{d}v\nabla^{2}w-\frac{d-2}{d}Cv^{2+4/(k-2)}(w/v)^{1-4/d}$
(94) $\displaystyle 0$ $\displaystyle=$
$\displaystyle-w^{2}R+\frac{2(k-1)}{k-2}\,v\nabla
v^{-1}\nabla(w^{2})-\frac{4(d-1)}{d}(\nabla w-w\nabla\log
v)^{2}-\frac{k}{k-2}v^{2+4/(k-2)}\left(C(w/v)^{2-4/d}+D\right).$

Except for the source term, the constraint is now linear in $v$ and $w$
separately.

###  5.3 Comparison with supersymmetric ansatzes

Any supersymmetric ansatz should be using the conformal factor which minimizes
the potential, so let us sketch how this works in two well-studied examples.

In [6], K. and M. Becker constructed M theory compactifications on Calabi-Yau
fourfolds, thus $d=3$ and $k=8$. They take $B=-A/2$ (see their Eqs. (2.11) and
(2.30)), so $v=u^{-1}$. Their construction involves both electric and magnetic
four-form flux, so in the magnetic notation uses $p=4$ and $p=7$. Stringy
ingredients include the Chern-Simons term, and an $R^{4}$ anomaly term.

In [21], Giddings, Kachru and Polchinski developed supersymmetric type IIb
compactification on Calabi-Yau threefolds with flux. Here $d=4$ and $k=6$. We
will consider the special case in which the dilaton is constant. They take
$B=-A$, so again $v=u^{-1}$. This construction uses $p=3$ and $p=5$ flux, a
Chern-Simons term $F\wedge H\wedge C^{(4)}$, and orientifold sources to
$T^{(d)}_{string}$.

The two constructions are similar enough that we can make a unified
discussion. Both are based on Calabi-Yau manifolds, so $R=0$. Both lead to
Minkowski space-time, so we can set $C=0$. We note in passing that the ground
state approximation of §3.3 is always exact in this case.

Both constructions use conformally invariant flux with $2p=k$ (call this
$F_{a}$), and an additional $p>k/2$ magnetic flux (call this $F_{b}$) which is
determined by the warp factor, $F_{b}=*^{(D)}\epsilon^{(d)}\wedge d(u^{2})$,
where $*^{(D)}$ is the $D$-dimensional Hodge star in the warped metric.555This
follows from the cancellation between warped tension and potential energy for
supersymmetric space-time filling M2 or D3 branes. Substituting the ansatz, in
both this becomes $F_{b}=*^{(k)}d(v^{2})$.

In both cases, the function $w$ of Eq. (94) is set to a constant (say $1$), so
this equation becomes linear. For Minkowski space-time, $C=0$ and one has an
integral constraint on $T^{(d)}$, which in some sense is a supersymmetric
partner to a topological constraint on $F_{a}^{2}-T^{(d)}_{string}$ (the M2 or
D3 tadpole condition).

Let us derive the equations of motion by re-expressing the effective potential
Eq. (93) in terms of $v$ and $w$,

$V_{eff}=\frac{1}{2}\int\sqrt{g^{(k)}}\bigg{[}-\frac{4(k-1)}{k-2}(\nabla
v)\nabla(w^{2}/v)-\frac{4(d-1)}{d}v^{2}(\nabla(w/v))^{2}+\frac{w^{2}}{2v^{2}}(F_{a}^{2}+v^{-2}F_{b}^{2})\bigg{]}$
(95)

There is also a term $\frac{w^{2}}{2v^{2}}T^{(d)}_{string}$, but since this
always comes with $F_{a}^{2}$ and has the same dependence on $v$ and $w$, we
leave it out until the end.

After varying $V_{eff}$, we set $w=1$, to find

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\delta V_{eff}}{\delta
w}=\alpha\frac{1}{v}\nabla^{2}v+\frac{1}{2v^{2}}(F_{a}^{2}+v^{-2}F_{b}^{2})$
(96) $\displaystyle 0$ $\displaystyle=$ $\displaystyle\frac{\delta
V_{eff}}{\delta v}=-\alpha(v^{-1}\nabla^{2}v-v^{-2}(\nabla
v)^{2})-\frac{1}{2v^{2}}(F_{a}^{2}+2v^{-2}F_{b}^{2})$ (97)

where

$\alpha\equiv\left(\frac{4(k-1)}{k-2}-\frac{4(d-1)}{d}\right)>0.$

Subtracting these equations, we find

$\alpha(\nabla v)^{2}=\frac{1}{2v^{2}}F_{b}^{2},$ (98)

which is satisfied by $F_{b}=\sqrt{\alpha/2}\nabla v^{2}$. Substituting in Eq.
(96), and restoring $T^{(d)}_{string}$, we find

$-\alpha\nabla^{2}v^{2}=F_{a}^{2}-T^{(d)}_{string},$

which is the warp factor equation in both cases.

Two points worth observing are the supersymmetry relation between the warp
factor and $F_{b}$, and how the positive curvature required by Eq. (82) is
provided by the variation of the conformal factor $v$.

It would be very interesting to find a derivation of the effective
superpotential along the lines of §2, starting with the $D$-dimensional
supergravity Lagrangian and an unbroken supersymmetry, and producing a
functional of the $k$-dimensional fields.

Acknowledgements

This work grew out of ongoing discussions with Renata Kallosh on
compactification using negatively curved manifolds, to whom we give special
thanks. Also influential were many discussions over the years with Tom Banks
on problems with the effective potential in gravity and string theory. We also
thank Michael Anderson, Xiuxiong Chen, Gary Gibbons, Marcus Khuri, Luca
Mazzucato, John Morgan, Joe Polchinski, Dennis Sullivan, Peter van
Nieuwenhuizen and Edward Witten for valuable discussions. Finally, we thank
Steve Giddings for pointing out [23] and subsequent discussions.

This research was supported in part by DOE grant DE-FG02-92ER40697.

##  Appendix A Conventions

Supergravity considerations will be accurate when all geometric length scales,
such as the diameter and volume of $X$ and curvature lengths, are greater than
$1$ in fundamental units (we ignore the string coupling), so we take
$M_{Planck,D}=1$ to make this condition evident. We furthermore take the
coordinates on $X$ to range over order $1$, so a geometric length scale $L$
goes as $L^{2}\sim g_{ij}$. Then $\mbox{Vol}X\sim\sqrt{g}\sim L^{k}$,
derivatives $\nabla\sim 1$, and the scalar curvature $R\sim 1/L^{2}$. The
integrals of the $p$-form fluxes over homology cycles $\Sigma^{(p)}$ are
quantized as

$N_{p}\sim\int_{\Sigma^{(p)}}F^{(p)}\sim F\mbox{Vol}\Sigma^{(p)}$ (99)

Thus $\int\sqrt{g}|F^{(p)}|^{2}\sim N_{p}^{2}L^{k-2p}$ in fundamental units,
and $T_{string}\sim 1$.

##  Appendix B Computations of curvature

Following Wald appendix D, we write $\nabla_{A}$ for the covariant derivative
adapted to the metric $g$, and ${\tilde{\nabla}}_{A}$ for that adapted to
${\tilde{g}}$, so

$C^{c}_{~{}ab}\equiv\nabla_{a}\omega_{b}-{\tilde{\nabla}}_{a}\omega_{b}.$
(100)

satisfies

$C^{c}_{~{}ab}=\frac{1}{2}{\tilde{g}}^{cd}\left(\nabla_{a}{\tilde{g}}_{bd}+\nabla_{b}{\tilde{g}}_{ad}-\nabla_{d}{\tilde{g}}_{ab}\right).$
(101)

The curvature of ${\tilde{g}}$ is then (Wald 7.5.8)

${\tilde{R}}_{abc}^{~{}~{}~{}d}=R_{abc}^{~{}~{}~{}d}-2\nabla_{[a}C^{d}_{~{}b]c}+2C^{e}_{~{}c[a}C^{d}_{~{}b]e}.$
(102)

Note that Eq. (22) has a $\mathbb{Z}_{2}$ symmetry under
$e^{\mu}\rightarrow-e^{\nu}$, which constrains the connection and curvature
coefficients. Using $\nabla_{a}g_{bc}=R_{ij\mu}^{~{}~{}~{}\nu}=0$, we find

$\displaystyle C^{i}_{~{}jk}$ $\displaystyle=$
$\displaystyle\delta^{i}_{j}\partial_{k}B+\delta^{i}_{k}\partial_{j}B-g_{jk}\nabla^{i}B$
(103) $\displaystyle C^{\mu}_{~{}\nu i}$ $\displaystyle=$
$\displaystyle\delta^{\mu}_{\nu}\partial_{i}A$ (104) $\displaystyle
C^{i}_{~{}\mu\nu}$ $\displaystyle=$
$\displaystyle-e^{2A-2B}\eta_{\mu\nu}\nabla^{i}A$ (105)
$\displaystyle{\tilde{R}}_{ijk}^{~{}~{}~{}l}$ $\displaystyle=$ $\displaystyle
R_{ijk}^{~{}~{}~{}l}+2\delta^{l}_{[i}\nabla_{j]}\nabla_{k}B-2g_{k[i}\nabla_{j]}\nabla^{l}B$
$\displaystyle+2(\nabla_{[i}B)\delta^{l}_{j]}\nabla_{k}B-2(\nabla_{[i}B)g_{j]k}\nabla^{l}B-2g_{k[i}\delta^{l}_{j]}|\nabla
B|^{2}$ $\displaystyle{\tilde{R}}_{ij\mu}^{~{}~{}~{}\nu}$ $\displaystyle=$
$\displaystyle-2\nabla_{[i}C^{\nu}_{~{}j]\mu}+2C^{\lambda}_{~{}\mu[i}C^{\nu}_{~{}j]\lambda}$
$\displaystyle=$ $\displaystyle
0\qquad(\sim\nabla_{[i}\nabla_{j]}A+\nabla_{[i}A\nabla_{j]}A)$
$\displaystyle{\tilde{R}}_{i\mu j}^{~{}~{}~{}\nu}$ $\displaystyle=$
$\displaystyle-\nabla_{i}C^{\nu}_{~{}\mu j}+C^{k}_{~{}ji}C^{\nu}_{~{}\mu
k}-C^{\lambda}_{~{}j\mu}C^{\nu}_{~{}i\lambda}$ $\displaystyle=$
$\displaystyle\delta^{\nu}_{\mu}\left(-\nabla_{i}\nabla_{j}A+C^{k}_{ij}\nabla_{k}A-(\nabla_{i}A)(\nabla_{j}A)\right)$
$\displaystyle=$
$\displaystyle\delta^{\nu}_{\mu}\left(-\nabla_{i}\nabla_{j}A-\nabla_{i}(A-B)\nabla_{j}(A-B)+\nabla_{i}B\nabla_{j}B-g_{ij}(\nabla
A\cdot\nabla B)\right)$
$\displaystyle{\tilde{R}}_{\mu\nu\lambda}^{~{}~{}~{}~{}\rho}$ $\displaystyle=$
$\displaystyle
R_{\mu\nu\lambda}^{~{}~{}~{}~{}\rho}+2C^{i}_{~{}\lambda[\mu}C^{\rho}_{~{}\nu]i}$
$\displaystyle=$ $\displaystyle
R_{\mu\nu\lambda}^{~{}~{}~{}~{}\rho}-2e^{2A-2B}\eta_{\lambda[\mu}\delta^{~{}\rho}_{\nu]}(\nabla
A)^{2}$

The other mixed components can be obtained using the antisymmetry
$R_{[ab][cd]}$. Here ${\tilde{R}}_{ijk}^{~{}~{}~{}l}$ is the same as for a
$k$-dimensional conformal transformation (Wald D.7).

One check is the case $A=B$. The only nontrivial one is

${\tilde{R}}_{i\mu
j}^{~{}~{}~{}\nu}=\delta^{\nu}_{\mu}\left(-\nabla_{i}\nabla_{j}B+\nabla_{i}B\nabla_{j}B-g_{ij}|\nabla
B|^{2}\right)$ (110)

from the standard formula, which agrees with Eq. (B) at $A=B$.

Contracting indices, we find

$\displaystyle{\tilde{R}}_{ik}=R_{ik}$
$\displaystyle-(k-2)\nabla_{i}\nabla_{k}B-g_{ik}\nabla^{2}B+(k-2)\nabla_{i}B\nabla_{k}B-(k-2)g_{ik}(\nabla
B)^{2}$
$\displaystyle+d\left(-\nabla_{i}\nabla_{k}A-\nabla_{i}(A-B)\nabla_{k}(A-B)+\nabla_{i}B\nabla_{k}B-g_{ik}(\nabla
A\cdot\nabla B)\right)$ $\displaystyle{\tilde{R}}_{\mu\lambda}=R_{\mu\lambda}$
$\displaystyle+e^{2A-2B}\eta_{\mu\lambda}\left(-d(\nabla
A)^{2}-\nabla^{2}A-(k-2)(\nabla A\cdot\nabla B)\right)$ (112)

Again, this agrees with the standard formula for $A=B$.

Finally, the scalar curvature is a sum of two terms obtained by tracing these
two contributions. Let us define the Laplacian with respect to the conformally
transformed 6d metric,

$\displaystyle{\tilde{\Delta}}$
$\displaystyle\equiv-\frac{1}{\sqrt{{\tilde{g}}}}\partial_{i}\sqrt{{\tilde{g}}}{\tilde{g}}^{ij}\partial_{j}$
(113) $\displaystyle=$ $\displaystyle-\nabla^{2}-(k-2)\nabla B\cdot\nabla,$
(114)

then

$\displaystyle{\tilde{R}}^{(k)}$ $\displaystyle=$ $\displaystyle
e^{-2B}\left(R^{(k)}-(2k-2)\nabla^{2}B-(k-2)(k-1)(\nabla
B)^{2}+d({\tilde{\Delta}}A-(\nabla A)^{2})\right)$ (115)
$\displaystyle{\tilde{R}}^{(d)}$ $\displaystyle=$ $\displaystyle
e^{-2A}R^{(d)}+de^{-2B}\left({\tilde{\Delta}}A-d(\nabla A)^{2}\right)$ (116)

Note that the term ${\tilde{\Delta}}A-d(\nabla A)^{2}$ is precisely the scalar
Laplacian associated to the metric Eq. (22), restricted to functions
independent of space-time. Thus, it is a total derivative, and its
contribution to $\int\sqrt{{\tilde{g}}}{\tilde{R}}^{(d)}$ vanishes (of course
the other term still depends on $A$).

Finally, the conformally invariant Laplacian is $\Delta_{c}=\Delta+\alpha R$
with

$s=-\frac{k-2}{2};\qquad\alpha=\frac{k-2}{4(k-1)};$ (117)

such that

${\tilde{\Delta}}_{c}e^{sB}\phi=e^{(s-2)B}\Delta_{c}\phi.$ (118)

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