CPHT-RR087.122022, December 2022

A unimodular-like string effective description

Alex<EMAIL_ADDRESS>Hervé
<EMAIL_ADDRESS>and Nicolaos
<EMAIL_ADDRESS>
1 Department of Physics, School of Natural Sciences, NTUA
15780 Athens, Greece

2 CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris,
91120 Palaiseau, France

3 Department of Physics, University of Cyprus,
Nicosia 1678, Cyprus

Abstract

> We show that vanishing of the sigma-model beta-functions for the metric
> $g_{\mu\nu}$, antisymmetric tensor $B_{\mu\nu}$ and dilaton $\Phi$ of any
> string-theory model in $D$ dimensions at tree level can be obtained from a
> universal effective action, provided that the field combination
> $\sqrt{-g}e^{-2\Phi}$ is non-dynamical. The central-charge deficit term is
> omitted from the action and appears as an integration constant at the level
> of the field equations. The analysis is done at leading order in
> $\alpha^{\prime}$, i.e. at the two-derivative level in the effective
> description. It can be extended to incorporate higher order string-loop
> corrections, Ramond-Ramond forms and moduli fields. In all cases, the non-
> dynamical field combination turns out to be $\sqrt{-g}\,U$, where $U$ is the
> effective potential. When certain fluxes are fixed and the moduli fields are
> stabilized, no cosmological term will arise in the action. Instead, the
> cosmological constant will appear as an integration constant as in
> unimodular modifications of general relativity.

##  1 Introduction

Although the General Theory of Relativity has been tremendously successful in
describing the gravitational interactions at the classical level, it presents
various unresolved issues like for instance the cosmological constant problem.
See e.g. Refs. [1, 2] for review. Despite considerable effort over the years,
the discrepancy between theory and experiment for the value of the
cosmological constant, which amounts to 120 orders of magnitude, still remains
unresolved.

An interesting direction to approach the cosmological constant problem is to
modify the theory of classical gravity so that the cosmological constant
appears as an integration constant and not as a fundamental parameter in the
Lagrangian of the theory [1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In
particular, recall that if the determinant of the metric is treated as a non-
dynamical field, the variations in the metric should be restricted to satisfy
$g_{\mu\nu}\delta g^{\mu\nu}=0$ [1]. As a result, the cosmological term can be
omitted from the action and the equations of motion reduce to the traceless
part of the Einstein field equations,

$R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R=8\pi
G\Big{(}T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T^{\lambda}_{\lambda}\Big{)}.$ (1.1)

This traceless part of the equations of motion does not involve a cosmological
constant term.

Naively we seem to loose one combination of the equations of motion. However,
the Bianchi identity implies that [1]

$\partial_{\mu}\Big{(}R+8\pi GT^{\lambda}_{\lambda}\Big{)}=0,$ (1.2)

leading to

$R+8\pi GT^{\lambda}_{\lambda}=-4\Lambda=\rm{constant}.$ (1.3)

Combining with Eq. (1.1), we obtain Einstein’s equations in the usual form,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu},$ (1.4)

but with the cosmological constant arising as an integration constant. Notice
in particular that in the absence of matter and radiation, all de Sitter
backgrounds, irrespective of curvature, are solutions of Eq. (1.1). Among
these solutions, one must choose the one that is compatible with the
observational data. Of course, still it would be desirable to have a selection
mechanism among the solutions based solely on theoretical considerations and
compare with the experiment.

It is important to stress that in the approach outlined above, the formalism
retains the full general coordinate covariance, except that the determinant of
the metric $\sqrt{-g}$ is treated as a non-dynamical background field
($g_{\mu\nu}\delta g^{\mu\nu}=0$). We are however always free to use the gauge
freedom of the theory to work in a frame where $\sqrt{-g}=1$. In such a frame,
the gauge symmetry is not fixed completely. The residual gauge freedom amounts
to the group of volume-preserving diffeomorphisms. As Weinberg emphasized [1],
the above point of view, where general coordinate covariance is a symmetry,
differs from other “unimodular” formulations of gravity, e.g. Refs. [4, 5, 6],
in which the gauge symmetry of the theory is restricted to the group of
volume-preserving diffeomorphisms. In such unimodular versions of gravity, the
determinant of the metric is dynamical and behaves like a scalar field since
it is invariant under volume-preserving diffeomorphisms. As a result, there is
freedom to introduce arbitrary functions of this determinant in the action.

In the framework of string theory, we use the perturbative, on-shell
formulation of the theory, as well as symmetries and string dualities, to
construct the low energy effective action. Indeed, at the moment, a complete
off-shell formulation in terms of a string field theory action is lacking.
Therefore, we cannot derive the low energy effective action directly by taking
a suitable low energy or a classical limit. Rather, we construct the effective
action for the massless fields in order to reproduce the string-sigma model
equations, which follow from conformal invariance, and the string S-matrix.

At the two-derivative level, however, the vanishing of the string sigma-model
beta functions does not uniquely specify the form of the low energy effective
action. As we will see, the alternative formulation of general relativity
described above can be readily extended to the string theory context, with a
certain combination of fields treated as a non-dynamical background. More
specifically, we may treat a particular combination of the dilaton $\Phi$ and
the determinant of the spacetime metric as a non-dynamical field and reproduce
the string sigma-model equations. At tree-level, this combination turns out to
be $\sqrt{-g}e^{-2\Phi}$, where $g$ is the determinant of the string-frame
metric $g_{\mu\nu}$. The beta-function equations associated with
$g_{\mu\nu}(x)$ and the antisymmetric tensor $B_{\mu\nu}(x)$ field follow by
extremizing the low energy effective action. The Bianchi identity can then be
used to show that the beta function associated with the dilaton field
$\Phi(x)$ must be a constant: $\beta^{\Phi}=\rm constant$. To recover
conformal invariance, or select a proper string background, we set this
integration constant to be zero.

This formalism allows us to omit non-universal terms associated with the
central charge deficit from the action, in the same way the cosmological term
can be omitted from the action in the case of general relativity. As a result,
we may use a universal form for the low energy effective action in string
frame to describe allowable string backgrounds. In the Einstein frame, the
term associated with the central charge deficit appears as a dilaton-tadpole
term in the action. The formalism that we develop allows us to obtain the
central-charge deficit as an integration constant and omit the corresponding
tadpole term from the low energy effective action.

At the two-derivative level, the analysis can also be generalized in various
ways. First, it can incorporate quantum corrections at higher order in the
string coupling $e^{\Phi}$. As we will see, it is possible to adjust the non-
dynamical combination of the dilaton field and the metric determinant, order
by order in the string coupling, with the low-energy effective action
retaining a universal form. In fact, the action will not involve any dilaton-
potential term. Nevertheless, the string equations of motion will be fully
reproducible after properly incorporating the Bianchi identity and determining
the field combination that must be treated as non-dynamical. The dilaton
potential term, which can be derived from higher genus string amplitudes,
appears instead in the field combination that is treated as non-dynamical.

Second, extra degrees of freedom can be included in the effective description,
such as Ramond-Ramond (RR) forms. Additional scalar fields arising from the
NSNS of RR sectors can also be incorporated, including those that are massless
at tree level, i.e. the moduli. Denoting by $U$ the perturbative string-frame
effective potential that depends on the scalar fields, the non-dynamical
combination of fields in this more general case turns out to be
$\sqrt{-g}\,U$. In fact, our analysis can be extended to the non-perturbative
level, where string dualities and/or supersymmetry can be used to determine
$U$ and the form of the low energy effective action. Since the potential term
$U$ does not appear in the effective action, even when certain fluxes are
fixed and the moduli stabilized, no cosmological term will arise in the
action. Rather the cosmological constant can be treated as an integration
constant, as in the case of unimodular modifications of general relativity.

In Sect. 2, we show that at leading order in $\alpha^{\prime}$, the sigma-
model beta-functions associated with the metric, antisymmetric tensor and
dilaton fields can be recovered from a universal action, while
$\sqrt{-g}e^{-2\Phi}$ is non-dynamical and the central-charge deficit appears
as an integration constant at the level of the field equations. Sect. 3
generalizes the analyses to include RR forms, while higher-order loop
corrections in the string coupling are incorporated in Sect. 4. Extra scalars,
and in particular the moduli fields, are included in Sect. 5. In Sect. 6, we
show that the number of propagating degrees of freedom is not reduced even if
a field combination is non-dynamical. We conclude with a discussion and
present open questions in Sect. 7.

##  2 Another approach to the string effective action

Let us briefly review the string effective action approach to worldsheet
conformal invariance in string theory. We consider a string with worldsheet
coordinates $(\tau,\sigma)$ moving in a $D$-dimensional spacetime. The
embedding fields are denoted by $X^{\mu}(\tau,\sigma)$,
$\mu\in\\{0,1,\dots,D-1\\}$, the spacetime metric by $g_{\mu\nu}(X)$, the
antisymmetric-tensor field by $B_{\mu\nu}(X)$ and the dilaton by $\Phi(X)$.
The condition for conformal invariance amounts to the vanishing of the string
sigma-model $\beta$-functions [16, 17, 18, 20, 19, 21, 22, 23]. The latter are
then viewed as the field equations of motion associated with a spacetime
effective action, which takes the form

$\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int{\rm
d}^{D}x\sqrt{-g}e^{-2\Phi}\left(R+4\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{\lambda}{\alpha^{\prime}}\right),$
(2.5)

where $\alpha^{\prime}$ is the Regge slope and $\lambda=2\delta c/3$, with
$\delta c$ the central charge deficit ($\delta c=26-D$ for the example of the
bosonic string). The variation of the action (2.5) yields

$\displaystyle\delta\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int{\rm d}^{D}x$
$\displaystyle\sqrt{-g}e^{-2\Phi}\left\\{\delta
g^{\mu\nu}\left(R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi-\frac{1}{4}H_{\mu\rho\sigma}{H_{\nu}}^{\rho\sigma}\right)+\frac{1}{2}\delta
B_{\mu\nu}e^{2\Phi}\nabla_{\rho}\left(e^{-2\Phi}H^{\mu\nu\rho}\right)\right.$
$\displaystyle\left.-\left(\frac{1}{2}g_{\mu\nu}\delta
g^{\mu\nu}+2\delta\Phi\right)\left(R+4\nabla^{2}\Phi-4\partial_{\rho}\Phi\partial^{\rho}\Phi-\frac{1}{12}H_{\rho\sigma\tau}H^{\rho\sigma\tau}+\frac{\lambda}{\alpha^{\prime}}\right)\right\\}.$
(2.6)

Extremizing the action, $\delta\mathcal{S}=0$, leads to the vanishing of the
string sigma-model $\beta$-functions

$\displaystyle\frac{\beta^{g}_{\mu\nu}}{\alpha^{\prime}}$
$\displaystyle=R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi-\frac{1}{4}H_{\mu\rho\sigma}{H_{\nu}}^{\rho\sigma}=0,$
(2.7) $\displaystyle\frac{\beta^{B}_{\mu\nu}}{\alpha^{\prime}}$
$\displaystyle=-\frac{1}{2}\nabla^{\rho}\left(e^{-2\Phi}H_{\mu\nu\rho}\right)=0,$
(2.8) $\displaystyle-\frac{2\beta^{\Phi}}{3\alpha^{\prime}}$
$\displaystyle=R+4\nabla^{2}\Phi-4\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{\lambda}{\alpha^{\prime}}=0.$
(2.9)

We would like to stress two points concerning the string effective action in
Eq. (2.5):

  * •

In Eq. (2.6), we see that the contributions to $\delta{\cal S}$ arising from
the variations $\delta\Phi$ and $\frac{1}{2}g_{\mu\nu}\delta
g^{\mu\nu}=-\delta\sqrt{-g}/\sqrt{-g}$ are proportional. Therefore, once we
impose the dilaton field equation $\delta{\cal S}/\delta\Phi=0$, the variation
of $\sqrt{-g}$ does not induce any contribution to $\delta{\cal S}$ or
$\beta_{\mu\nu}^{g}$. As a result, there is a possibility to restrict the
variation of the determinant of the metric to be equal to the variation of a
generic functional $A$ of all fields,

$\delta\sqrt{-g}=\delta\Big{[}A(\Phi,g_{\mu\nu},B_{\mu\nu})\Big{]}.$ (2.10)

Upon imposing such an extra constraint, we still obtain the same field
equations (2.7), (2.8) and (2.9).

  * •

Another observation is that for any particular model, we have a different
effective action since each model is associated with its own central-charge
deficit. Therefore, there is no “universal” string effective action that
describes in a unified way all string models: Each string model has his own
low energy effective action, the field equations of which lead to conformal
invariance when satisfied.

The comments above seem to point towards the conclusion that the string
effective action is not unique, and the question arises if we can use a single
string effective action to describe the possible models.

To answer this question, let us observe from Eq. (2.6) that there is another
way to achieve $\delta\mathcal{S}=0$. We can keep equations (2.7) and (2.8)
but instead of Eq. (2.9), we may impose

$\frac{1}{2}g_{\mu\nu}\delta g^{\mu\nu}+2\delta\Phi=0,$ (2.11)

which is equivalent to111This condition has already been used in [24].

$\delta\left(\sqrt{-g}e^{-2\Phi}\right)=0.$ (2.12)

That is, the field $\sqrt{-g}e^{-2\Phi}$ is treated to be non-dynamical with
vanishing variation. Notice that even though $\sqrt{-g}e^{-2\Phi}$ is treated
as a background field, the theory retains the full diffeomorphism invariance.
This is similar to Weinberg’s approach outlined in the introduction for the
cosmological constant problem, where the determinant of the Einstein-frame
metric is treated to be non-dynamical [1]. In the case at hand, one can use
the gauge freedom of the theory to set $\sqrt{-g}e^{-2\Phi}=1$. This is simply
a gauge choice. This condition does not fix the gauge freedom completely.
There is a residual gauge symmetry comprising volume-preserving (transverse)
diffeomorphisms. Indeed, the condition $\sqrt{-g}e^{-2\Phi}=1$ is invariant
under diffeomorphisms $x^{\mu}\to x^{\mu}+\xi^{\mu}$ that satisfy
$\partial_{\mu}\xi^{\mu}=1$.

To recap, in the above approach, the equations of motion to be satisfied are
those associated with the metric $g_{\mu\nu}$ and the antisymmetric tensor
$B_{\mu\nu}$,

$\displaystyle
R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi-\frac{1}{4}H_{\mu\nu\rho}H^{\mu\nu\rho}=0,$
(2.13) $\displaystyle\nabla^{\rho}\left(e^{-2\Phi}H_{\mu\nu\rho}\right)=0,$
(2.14)

together with condition (2.12). Observe that the central charge deficit does
not appear in Eqs. (2.13), (2.14) and (2.12). As a result, we are free to
start with the universal action

$\widehat{\mathcal{S}}=\frac{1}{2\kappa_{0}^{2}}\int{\rm
d}^{D}x\sqrt{-g}e^{-2\Phi}\left(R+4\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}\right),$
(2.15)

subject to (2.12). A question then arises: Do the extrema of $\widehat{\cal
S}$ describe conformal string backgrounds? For this to be true, we need
$\beta^{\Phi}=0$. However, it is known that the vanishing of
$\beta^{g}_{\mu\nu}$ and $\beta^{B}_{\mu\nu}$ leads to [18]

$\beta_{\mu\nu}^{g}=\beta_{\mu\nu}^{B}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\nabla^{\mu}\beta^{\Phi}=0.$ (2.16)

To see this, recall that by using the Bianchi identity, we obtain

$0=\nabla^{\mu}\beta^{g}_{\mu\nu}=-2\nabla_{\nu}\left(-\frac{1}{2}\nabla^{2}\Phi+\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{24}H_{\mu\nu\rho}H^{\mu\nu\rho}\right)$
(2.17)

and so

$-\,\frac{1}{2}\nabla^{2}\Phi+\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{24}H_{\mu\nu\rho}H^{\mu\nu\rho}=c_{0},$
(2.18)

where $c_{0}$ is a constant. Choosing appropriately $c_{0}=\delta
c/6\alpha^{\prime}$, this is equivalent to Eq. (2.9) i.e.

$\beta^{\Phi}=0.$ (2.19)

To conclude this subsection, let us stress again that the central-charge
deficit does not appear in the string effective action (2.15). Instead, it
arises as an integration constant, as in the case of the cosmological constant
in unimodular gravity.

###  2.1 Generalization

To check whether the string effective action (2.15) is special, let us
consider the most general diff invariant action for the metric and dilaton
(ignoring the antisymmetric tensor for simplicity), where we keep terms with
two derivatives only and no dilaton potential. Without loss of generality, it
can be written as

$\widehat{\cal S}=\frac{1}{2\kappa_{0}^{2}}\int{\rm
d}^{D}x\sqrt{-g}F(\Phi)\bigg{(}R+4\partial_{\mu}\Phi\partial^{\mu}\Phi\bigg{)},$
(2.20)

where $F(\Phi)$ is an arbitrary function of $\Phi$. Indeed, even if we start
with an action involving a generic sigma-model kinetic term
$G(\Phi)\partial_{\mu}\Phi\partial^{\mu}\Phi$, where $G$ and $F$ are different
functions, we can make a field redefinition of $\Phi$ to bring the action to
the form 2.20.222This is true for a one-dimensional sigma-model. We would like
to determine under which condition the contribution to $\delta\widehat{S}$
from the variation of $\sqrt{-g}$ vanishes, once the equation of motion for
$\Phi$ is satisfied. Indeed, when this occurs, the constraint that
$\delta\sqrt{-g}$ should be proportional to $\delta\Phi$, as in equation
(2.11), is harmless, since the equation of motion for $g_{\mu\nu}$ will not be
affected.

The variation of the action is

$\displaystyle\delta\widehat{\cal S}=\frac{1}{2\kappa_{0}^{2}}\int{\rm
d}^{D}x\sqrt{-g}F(\Phi)$ $\displaystyle\bigg{\\{}\delta
g^{\mu\nu}\left(R_{\mu\nu}+4\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{\nabla_{\mu}\nabla_{\nu}F}{F}\right)$
$\displaystyle-\frac{1}{2}g_{\mu\nu}\delta
g^{\mu\nu}\bigg{(}R+4\partial_{\rho}\Phi\partial^{\rho}\Phi-2\,\frac{\nabla^{2}F}{F}\bigg{)}$
$\displaystyle+\frac{F^{\prime}}{F}\delta\Phi\bigg{(}R+4\partial_{\mu}\Phi\partial^{\mu}\Phi-8\,\frac{\partial_{\mu}\big{(}F\partial^{\mu}\Phi\big{)}}{F^{\prime}}\bigg{)}\bigg{\\}},$
(2.21)

where $F^{\prime}=dF/d\Phi$. From the last two lines of Eq. (2.21), we see
that in order for the contribution proportional to $g_{\mu\nu}\delta
g^{\mu\nu}$ to vanish when the contribution proportional to $\delta\Phi$ is
zero, we must have

$-2\,\frac{\nabla^{2}F}{F}=-8\,\frac{\partial_{\mu}\big{(}F\partial^{\mu}\Phi\big{)}}{F^{\prime}}.$
(2.22)

This is satisfied when

$\displaystyle\frac{F^{\prime}}{F}=4\,\frac{F}{F^{\prime}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\frac{F^{\prime\prime}}{F}=4,$ (2.23)

leading to

$F(\Phi)=e^{\pm 2\Phi}.$ (2.24)

Choosing the minus sign, we get (2.15) (for the metric and dilaton). Of
course, the choice of plus sign is also possible, but it is equivalent under
the trivial field redefinition $\Phi\to-\Phi$. In other words, the string
effective action (2.15) is the unique action of the form (2.20), for which the
contribution to $\delta\widehat{S}$ from the variation of the determinant of
the metric vanishes when the dilaton equation holds.

###  2.2 An application

In the case of vanishing antisymmetric tensor field ($B_{\mu\nu}=0$), we have
seen that conformal invariance is guaranteed when

$R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi=0,$ (2.25)

subject to the condition of Eq.(2.12). As we have already remarked, we may fix
the gauge

$\sqrt{-g}e^{-2\Phi}=e^{-2\Phi_{0}},$ (2.26)

where $\Phi_{0}$ is an arbitrary constant. In that case, Eq. (2.25) takes the
form

$R_{\mu\nu}+\nabla_{\mu}\bigg{(}\frac{1}{\sqrt{-g}}\nabla_{\nu}\sqrt{-g}\bigg{)}=0.$
(2.27)

Using the definition of the Ricci tensor in terms of Christoffel symbols, it
is then straightforward to verify that the above equation can be written as

$K_{\mu\nu}=0,\qquad\mbox{where}\qquad
K_{\mu\nu}=\partial_{\kappa}\Gamma^{\kappa}_{\mu\nu}-\Gamma^{\kappa}_{\nu\lambda}\Gamma^{\lambda}_{\kappa\mu}.$
(2.28)

One may worry about the fact that Eq. (2.28) is not covariant. Indeed, under a
general coordinate transformation $x^{\mu}\to\tilde{x}^{\mu}(x^{\nu})$,
$K_{\mu\nu}$ transforms as

$K_{\mu\nu}\longrightarrow K_{\mu^{\prime}\nu^{\prime}}=\frac{\partial
x^{\mu}}{\partial x^{\mu^{\prime}}}\frac{\partial x^{\nu}}{\partial
x^{\nu^{\prime}}}K_{\mu\nu}+\frac{\partial x^{\mu}}{\partial
x^{\mu^{\prime}}}\frac{\partial x^{\nu}}{\partial
x^{\nu^{\prime}}}\nabla_{\mu}\partial_{\nu}\ln\left(\det\frac{\partial
x^{\alpha}}{\partial x^{\alpha^{\prime}}}\right).$ (2.29)

Therefore, although $K_{\mu\nu}$ is not a tensor under the full diffeomorphism
group, it is a tensor under volume-preserving diffeomorphisms ($\det\partial
x^{\alpha}/\partial x^{\alpha^{\prime}}=1)$, as expected, since the gauge we
have chosen has the residual volume-preserving diffeomorphism invariance.

Let us now study an example. We will consider a two-dimensional Euclidean
space, with metric

${\rm d}s^{2}=f(r)\Big{(}{\rm d}r^{2}+r^{2}{\rm d}\theta^{2}\Big{)},$ (2.30)

where $f(r)$ is to be determined. In that case, Eq. (2.27) reduces to

$\frac{1}{2}f^{\prime\prime}-\frac{f^{\prime}{}^{2}}{f}-\frac{f^{\prime}}{r}-\frac{f}{r^{2}}=0,$
(2.31)

whose solution is

$f(r)=\frac{f_{0}}{r^{2}}\,\frac{1}{1-\dfrac{M}{r}},$ (2.32)

where $f_{0}$ and $M$ are constants. Using Eq. (2.26), the dilaton is given by

$\Phi=\frac{1}{2}\ln\sqrt{g}+\Phi_{0}=\frac{1}{2}\ln\Big{(}rf(r)\Big{)}+\Phi_{0}.$
(2.33)

The constant $f_{0}$ determines the associated two-dimensional CFT. In
particular, the metric (2.30), (2.32) and the dilaton (2.33) describe the
Witten black hole if $k=8f_{0}$ is identified with the level $k$ of the
$SU(2)$ current algebra, since it can be readily verified that $\Phi$
satisfies

$-\frac{1}{2}\nabla^{2}\Phi+\nabla_{\mu}\Phi\nabla^{\mu}\Phi=\frac{1}{4f_{0}}.$
(2.34)

Interestingly, in our approach, the level $k$ appears in the equations as an
integration constant. Since it is not present in the action (2.15 subject to
the constraint (2.12), a universal effective description is used to encompass
the string solutions irrespectively of $k$.

##  3 Inclusion of RR fields

In type II string theory, one should include RR-fields in the effective action
(2.5). We will omit the NS-NS two-form and consider only the NS-NS metric and
dilaton, along with a RR one-form for simplicity:

$\mathcal{S}=\int\mathrm{d}^{D}x\sqrt{-g}\left\\{\frac{1}{2\kappa_{0}^{2}}e^{-2\Phi}\left(R+4\partial_{\mu}\Phi\partial^{\mu}\Phi+\frac{\lambda}{\alpha^{\prime}}\right)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right\\}.$
(3.35)

The variation of this action turns out to be

$\displaystyle\delta\mathcal{S}=\int\mathrm{d}^{D}x\sqrt{-g}\left\\{\frac{1}{2\kappa_{0}^{2}}e^{-2\Phi}\right.$
$\displaystyle\bigg{[}\delta
g^{\mu\nu}\Big{(}R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi\Big{)}$
$\displaystyle-\left(\frac{1}{2}g_{\mu\nu}\delta
g^{\mu\nu}+2\delta\Phi\right)\left(R+4\nabla^{2}\Phi-4\partial_{\rho}\Phi\partial^{\rho}\Phi+{\lambda\over\alpha^{\prime}}\right)\bigg{]}$
$\displaystyle\left.-\;\frac{1}{2}\delta
g^{\mu\nu}\left(F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\kappa\lambda}F^{\kappa\lambda}\right)+\delta
A_{\nu}\nabla_{\mu}F^{\mu\nu}\right\\},$ (3.36)

leading to the following equations of motion:

$\displaystyle
R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi=\kappa_{0}^{2}e^{2\Phi}\left(F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\kappa\lambda}F^{\kappa\lambda}\right)$
(3.37) $\displaystyle\nabla_{\mu}F^{\mu\nu}=0,$ (3.38) $\displaystyle
R+4\nabla^{2}\Phi-4\partial_{\mu}\Phi\partial^{\mu}\Phi+\frac{\lambda}{\alpha^{\prime}}=0.$
(3.39)

Let us now consider the alternative action

$\widehat{\cal
S}=\int\mathrm{d}^{D}x\sqrt{-g}\left\\{\frac{1}{2\kappa_{0}^{2}}e^{-2\Phi}\Big{(}R+4\partial_{\mu}\Phi\partial^{\mu}\Phi\Big{)}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right\\},$
(3.40)

where there is no reference to $\lambda$. Imposing the constraint of Eq.
(2.11), the only field equations we have to consider are Eqs. (3.37) and
(3.38),

$\displaystyle
R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi=\kappa_{0}^{2}e^{2\Phi}\left(F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\kappa\lambda}F^{\kappa\lambda}\right),$
(3.41) $\displaystyle\nabla_{\mu}F^{\mu\nu}=0.$ (3.42)

Hence, it seems again that Eq. (3.39) for the dilaton is missing. However, we
will show below that the dilaton equation of motion follows from Eqs. (3.41)
and (3.42), and that the central-charge deficit, which is proportional to
$\lambda$, appears as an integration constant.

To see this, let us note that from Eq. (3.41) we find that

$\nabla^{\mu}\Big{(}R_{\mu\nu}+2\nabla_{\mu}\nabla_{\nu}\Phi\Big{)}=\kappa_{0}^{2}T_{\mu\nu}\nabla^{\mu}e^{2\Phi}+\kappa_{0}^{2}e^{2\Phi}\nabla^{\mu}T_{\mu\nu},$
(3.43)

where

$T_{\mu\nu}=F_{\mu\rho}F_{\nu}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\kappa\lambda}F^{\kappa\lambda}.$
(3.44)

Using Eq. (3.42) (or, equivalently, invariance under $\delta
g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$), we get that
$T_{\mu\nu}$ is conserved: $\nabla^{\mu}T_{\mu\nu}=0$. Then, by using the
identity

$\nabla^{\mu}\nabla_{\nu}\nabla_{\mu}\Phi=\nabla_{\nu}\nabla^{2}\Phi+R_{\mu\nu}\nabla^{\mu}\Phi,$
(3.45)

we find that Eq. (3.43) can be written as follows

$\nabla_{\nu}\left(\frac{1}{2}R+2\nabla^{2}\Phi-2\partial_{\mu}\Phi\partial^{\mu}\Phi\right)=0.$
(3.46)

Therefore, we obtain the desired equation

$R+4\nabla^{2}\Phi-4\nabla_{\mu}\Phi\nabla^{\mu}\Phi+\frac{\lambda}{\alpha^{\prime}}=0,$
(3.47)

where $\lambda$ now appears as an integration constant. In other words, the
action (3.40), where the field combination $\sqrt{-g}e^{-2\Phi}$ is non-
dynamical (${1\over 2}g_{\mu\nu}\delta g^{\mu\nu}+2\delta\Phi=0$), yields the
same equations of motion as the original string effective action (3.35).

##  4 Extension to higher string-loops

After showing that $\sqrt{-g}e^{-2\Phi}$ can be non-dynamical including RR
fields (which are string one-loop effects compared to the tree-level string
action), one may wonder if the analysis can be extended to higher-order string
loops. In other words, we would like to investigate if it is possible to keep
a field combination of $\sqrt{-g}$ and $\Phi$ non-dynamical when higher string
loops are included, or if this feature is a particularity of the tree-level
string effective action. One may expect that the combination
$\sqrt{-g}e^{-2\Phi}$ may need to be corrected by higher-order contributions
in string coupling. To see what is going on, let us incorporate string-loop
corrections in the string effective action at the two-derivative level. The
action takes the general form

$\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-g}\bigg{(}F(\Phi)R+G(\Phi)\partial_{\mu}\Phi\partial^{\mu}\Phi-U(\Phi)\bigg{)},$
(4.48)

where $F(\Phi)$ and $G(\Phi)$ admit a string loop expansion of the form

$\displaystyle F(\Phi)$
$\displaystyle=e^{-2\Phi}+c_{0}+c_{1}e^{2\Phi}+c_{2}e^{4\Phi}+\cdots$
$\displaystyle G(\Phi)$
$\displaystyle=4e^{-2\Phi}+d_{0}+d_{1}e^{2\Phi}+d_{2}e^{4\Phi}+\cdots\,.$
(4.49)

We have also included a dilaton potential, which can be expanded in a similar
way

$U(\Phi)=-\frac{\lambda}{\alpha^{\prime}}e^{-2\Phi}+u_{0}+u_{1}e^{2\Phi}+u_{2}e^{4\Phi}+\cdots\,.$
(4.50)

It is convenient at this point to transform the above action to the Einstein
frame via the following conformal transformation,

$g_{\mu\nu}=F^{-\frac{2}{D-2}}\bar{g}_{\mu\nu},$ (4.51)

which leads to

$\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}\bigg{(}\bar{R}+H(\Phi)\partial_{\mu}\Phi\partial^{\mu}\Phi-\bar{U}(\Phi)\bigg{)}.$
(4.52)

In this formula, we have defined

$H(\Phi)=-\frac{D-1}{D-2}\left(\frac{F^{\prime}}{F}\right)^{2}+\frac{G}{F},\qquad\bar{U}(\Phi)=F^{-\frac{D}{D-2}}\,U$
(4.53)

and a prime denotes derivative with respect to the dilaton $\Phi$. The
variation of the action (4.52) turns out to be

$\displaystyle\delta\mathcal{S}$
$\displaystyle=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}\bigg{[}\delta\bar{g}^{\mu\nu}\left(\bar{R}_{\mu\nu}+H\partial_{\mu}\Phi\partial_{\nu}\Phi\right)-\frac{1}{2}\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu}\left(\bar{R}+H(\partial\Phi)^{2}-\bar{U}\right)$
$\displaystyle\hskip
85.35826pt+\delta\Phi\left(H^{\prime}(\partial\Phi)^{2}-2\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-\bar{U}^{\prime}\right)\bigg{]},$
(4.54)

where $\bar{\nabla}_{\mu}$ stands for the Levi-Civita connection associated
with the Einstein-frame metric. Therefore, the equations of motion are

$\displaystyle\bar{R}_{\mu\nu}+H\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{2}\bar{g}_{\mu\nu}\left(\bar{R}+H(\partial\Phi)^{2}-\bar{U}\right)=0,$
(4.55) $\displaystyle
H^{\prime}(\partial\Phi)^{2}-2\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-\bar{U}^{\prime}=0.$
(4.56)

We look for a string effective action without any dilaton potential, which
takes the following form in the Einstein frame

$\widehat{\cal
S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}\bigg{(}\bar{R}+H(\Phi)\partial_{\mu}\Phi\partial^{\mu}\Phi\bigg{)},$
(4.57)

subject to the condition

$K(\Phi)^{-1}\delta\Phi=\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu},$ (4.58)

for some function $K(\Phi)$. The latter should admit a perturbative expansion
of the form

$K(\Phi)=-{1\over 4}+k_{1}e^{2\Phi}+k_{2}e^{4\Phi}+\cdots\,.$ (4.59)

Given condition (4.58), we find that the variation of the action is now given
by

$\displaystyle\delta\widehat{\cal
S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}$
$\displaystyle\bigg{[}\delta\bar{g}^{\mu\nu}\left(\bar{R}_{\mu\nu}+H\partial_{\mu}\Phi\partial_{\nu}\Phi\right)$
$\displaystyle-\frac{1}{2}\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu}\left(\bar{R}+H(\partial\Phi)^{2}+2K\Big{(}2\bar{\nabla}_{\rho}(H\partial^{\rho}\Phi)-H^{\prime}(\partial\Phi)^{2}\Big{)}\right)\bigg{]},$
(4.60)

leading to a single equation of motion

$R_{\mu\nu}+H\partial_{\mu}\Phi\partial_{\nu}\Phi-\frac{1}{2}\bar{g}_{\mu\nu}\left(\bar{R}+H(\partial\Phi)^{2}+2K\Big{(}2\bar{\nabla}_{\rho}(H\partial^{\rho}\Phi)-H^{\prime}(\partial\Phi)^{2}\Big{)}\right)=0.$
(4.61)

We would now like, by appropriately choosing $K(\Phi)$ and utilizing the
Bianchi identity, to recover the dilaton equation (4.56). Acting with
$\nabla^{\mu}$ on both sides of Eq. (4.61), we obtain

$\bar{\nabla}^{\mu}\Big{(}H\partial_{\mu}\Phi\partial_{\nu}\Phi\Big{)}-\frac{1}{2}\bar{\nabla}_{\nu}\bigg{\\{}H(\partial\Phi)^{2}+2K\Big{(}2\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-H^{\prime}(\partial\Phi)^{2}\Big{)}\bigg{\\}}=0,$
(4.62)

leading to

$\bigg{\\{}\bar{\nabla}^{\mu}\Big{(}H\partial_{\mu}\Phi\Big{)}-\frac{1}{2}H^{\prime}(\partial\Phi)^{2}\bigg{\\}}\partial_{\nu}\Phi-\bar{\nabla}_{\nu}\bigg{\\{}2KV\,V^{-1}\Big{(}\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-\frac{1}{2}H^{\prime}(\partial\Phi)^{2}\Big{)}\bigg{\\}}=0,$
(4.63)

where we have introduced a function $V(\Phi)$ to be chosen suitably. Indeed,
the above equation can be written as

$\displaystyle\bigg{\\{}\bar{\nabla}^{\mu}\Big{(}H\partial_{\mu}\Phi\Big{)}-\frac{1}{2}H^{\prime}(\partial\Phi)^{2}\bigg{\\}}\bigg{(}\partial_{\nu}\Phi-$
$\displaystyle 2V^{-1}\partial_{\nu}(KV)\bigg{)}$
$\displaystyle-2KV\bar{\nabla}_{\nu}\bigg{\\{}V^{-1}\Big{(}\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-\frac{1}{2}H^{\prime}(\partial\Phi)^{2}\Big{)}\bigg{\\}}=0$
(4.64)

and for

$\partial_{\nu}\Phi-2V^{-1}\partial_{\nu}(KV)=0,$ (4.65)

it simplifies to

$\bar{\nabla}_{\nu}\bigg{\\{}V^{-1}\Big{(}\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-\frac{1}{2}H^{\prime}(\partial\Phi)^{2}\Big{)}\bigg{\\}}=0,$
(4.66)

or

$\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)-\frac{1}{2}H^{\prime}(\partial\Phi)^{2}=\nu
V,$ (4.67)

where $\nu$ is an integration constant. Comparing Eqs. (4.67) and (4.56), we
see that they coincide if

$V=-\frac{1}{2\nu}\bar{U}^{\prime}.$ (4.68)

Then Eq. (4.65) can be written as

$\partial_{\nu}\left(\bar{U}-2K\bar{U}^{\prime}\right)=0,$ (4.69)

which specifies $K$ to be

$K=\frac{\bar{U}+\kappa}{2\,\bar{U}^{\prime}},$ (4.70)

where $\kappa$ is an integration constant. However, using this result and the
expression of $V$, Eq. (4.67) becomes

$2K\Big{[}H^{\prime}(\partial\Phi)^{2}-2\bar{\nabla}_{\mu}(H\partial^{\mu}\Phi)\Big{]}\\!=\bar{U}+\kappa.$
(4.71)

As a result, for Eq. (4.61) to also be identical to the metric equation of
motion (4.55), we must have $\kappa=0$. This allows to write Eq. (4.58) as

${\bar{U}^{\prime}\over\bar{U}}\,\delta\Phi={1\over
2}\,\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu},$ (4.72)

which leads to

$\delta\Big{[}\bar{U}(\Phi)\,\sqrt{-\bar{g}}\Big{]}\\!=0.$ (4.73)

Using the relations between the string-frame and Einstein-frame metrics and
potentials, Eqs. (4.51) and (4.53), this is equivalent to having

$\delta\Big{[}U(\Phi)\,\sqrt{-g}\Big{]}\\!=0,$ (4.74)

which generalizes the tree-level condition in Eq. (2.12).

##  5 Inclusion of scalar fields

At tree level, string theory contains moduli fields that need to be included
in the effective description at low energies. In general, the moduli acquire a
potential at higher-loop levels, which may lead to their stabilization.
However, the alternative action (4.57) we have considered for the metric and
dilaton field does not contain any scalar potential. So it is not clear how
the extra moduli may be stabilized. In the following, we show that our
effective description captures correctly the dynamics of moduli fields and,
more generally, the dynamics of scalar fields with tree-level potential.

To begin with, let us consider the standard string effective action for the
metric and $N+1$ scalars fields $\Phi\equiv(\Phi^{0},\dots,\Phi^{N})$, where
$\Phi^{0}$ stands for the dilaton. We assume non-trivial sigma-model kinetic
terms and a potential,

$\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-g}\bigg{(}F(\Phi)R+G_{ij}(\Phi)\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-U(\Phi)\bigg{)},$
(5.75)

where $i\in\\{0,\dots,N\\}$ and

$\displaystyle F(\Phi)$
$\displaystyle=e^{-2\Phi^{0}}+c_{0}(\Phi^{1},\dots,\Phi^{N})+c_{1}(\Phi^{1},\dots,\Phi^{N})e^{2\Phi^{0}}+\cdots$
$\displaystyle G_{ij}(\Phi)$
$\displaystyle=d_{ij,-1}(\Phi^{1},\dots,\Phi^{N})e^{-2\Phi^{0}}+d_{ij,0}(\Phi^{1},\dots,\Phi^{N})+d_{ij,1}(\Phi^{1},\dots,\Phi^{N})e^{2\Phi^{0}}+\cdots$
$\displaystyle U(\Phi)$
$\displaystyle=u_{-1}(\Phi^{1},\dots,\Phi^{N})e^{-2\Phi^{0}}+u_{0}(\Phi^{1},\dots,\Phi^{N})+u_{1}(\Phi^{1},\dots,\Phi^{N})e^{2\Phi}+\cdots\leavevmode\nobreak\
.$ (5.76)

In terms of the Einstein metric $\bar{g}_{\mu\nu}$, given by

$g_{\mu\nu}=F^{-\frac{2}{D-2}}\bar{g}_{\mu\nu},$ (5.77)

the action becomes

$\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}\bigg{(}\bar{R}+H_{ij}(\Phi)\partial_{\mu}\Phi\partial^{\mu}\Phi-\bar{U}(\Phi)\bigg{)},$
(5.78)

where

$H_{ij}(\Phi)=-\frac{D-1}{D-2}\,\frac{\partial_{i}F}{F}\,\frac{\partial_{j}F}{F}+\frac{G_{ij}}{F},\qquad\bar{U}(\Phi)=F^{-\frac{D}{D-2}}\,U,$
(5.79)

and $\partial_{i}$ denotes derivative with respect to $\Phi^{i}$. Varying the
action, we obtain

$\displaystyle\delta\mathcal{S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}$
$\displaystyle\bigg{[}\delta\bar{g}^{\mu\nu}\left(\bar{R}_{\mu\nu}+H_{ij}\partial_{\mu}\Phi^{i}\partial_{\nu}\Phi^{j}\right)-\frac{1}{2}\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu}\left(\bar{R}+H_{ij}\partial_{\rho}\Phi^{i}\partial^{\rho}\Phi^{j}-\bar{U}\right)$
$\displaystyle+\delta\Phi^{k}\left(\partial_{k}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-2\bar{\nabla}_{\mu}(H_{ki}\partial^{\mu}\Phi^{i})-\partial_{k}\bar{U}\right)\bigg{]},$
(5.80)

from which we find the equations of motion

$\displaystyle\bar{R}_{\mu\nu}+H_{ij}\partial_{\mu}\Phi^{i}\partial_{\nu}\Phi^{j}-\frac{1}{2}\bar{g}_{\mu\nu}\left(\bar{R}+H_{ij}\partial_{\rho}\Phi^{i}\partial^{\rho}\Phi^{j}-\bar{U}\right)=0,$
(5.81)
$\displaystyle\partial_{k}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-2\bar{\nabla}_{\mu}(H_{ki}\partial^{\mu}\Phi^{i})-\partial_{k}\bar{U}=0.$
(5.82)

Let us now look for an alternative effective description, where the string-
frame action is identical to ${\cal S}$ in Eq. (5.75) but without any scalar
potential. In the Einstein frame, the action takes the form

$\widehat{\cal
S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}\bigg{(}\bar{R}+H_{ij}(\Phi)\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\bigg{)}.$
(5.83)

As before, the latter has to be subjected to some condition on the fields,
which at first order in fluctuations takes a linear form

$L_{i}(\Phi)\delta\Phi^{i}=\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu}.$ (5.84)

This is the most general ansatz compatible with diff invariance. The functions
$L_{i}(\Phi)$ to be determined are expected to admit expansions of the form

$L_{i}(\Phi)=\ell_{i,0}(\Phi^{1},\dots,\Phi^{N})+\ell_{i,1}(\Phi^{1},\dots,\Phi^{N})e^{2\Phi}+\cdots\,.$
(5.85)

Using the constraint (5.84), the variation of ${\cal S}$ can be written in
terms of the variations of the metric and of only $N$ scalar fields among the
$N+1$ available. For instance, eliminating the variation of $\Phi^{i_{0}}$ for
some $i_{0}\in\\{0,\dots,N\\}$, one way of writing the variation of the action
is

$\displaystyle\delta\widehat{\cal S}=$
$\displaystyle\,\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-\bar{g}}\bigg{[}\delta\bar{g}^{\mu\nu}\left(\bar{R}_{\mu\nu}+H_{ij}\partial_{\mu}\Phi^{i}\partial_{\nu}\Phi^{j}\right)$
$\displaystyle-\frac{1}{2}\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu}\left(\bar{R}+H_{ij}\partial_{\rho}\Phi^{i}\partial^{\rho}\Phi^{j}+2L^{-1}_{i_{0}}\Big{(}2\bar{\nabla}_{\rho}(H_{i_{0}i}\partial^{\rho}\Phi^{i})-\partial_{i_{0}}H_{ij}\partial_{\rho}\Phi^{i}\partial^{\rho}\Phi^{j}\Big{)}\right)$
$\displaystyle+\delta\Phi^{\hat{\imath}}\Big{[}\partial_{\hat{\imath}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-2\bar{\nabla}_{\mu}(H_{{\hat{\imath}}i}\partial^{\mu}\Phi^{i})-{L_{\hat{\imath}}\over
L_{i_{0}}}\Big{(}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-2\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})\Big{)}\Big{]}\bigg{]},$
(5.86)

where ${\hat{\imath}}\in\\{0,\dots,N\\}\backslash\\{i_{0}\\}$. In that case,
the equations of motion for the metric and scalar fields $\Phi^{\hat{\imath}}$
read

$\displaystyle
R_{\mu\nu}+H_{ij}\partial_{\mu}\Phi^{i}\partial_{\nu}\Phi^{j}-\frac{1}{2}\bar{g}_{\mu\nu}\bigg{(}\bar{R}+H_{ij}\partial_{\rho}\Phi^{i}\partial^{\rho}\Phi^{j}$
$\displaystyle\hskip
147.95424pt+2L^{-1}_{i_{0}}\Big{(}2\bar{\nabla}_{\rho}(H_{i_{0}i}\partial^{\rho}\Phi^{i})-\partial_{i_{0}}H_{ij}\partial_{\rho}\Phi^{i}\partial^{\rho}\Phi^{j}\Big{)}\bigg{)}=0,$
(5.87)
$\displaystyle\partial_{\hat{\imath}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-2\bar{\nabla}_{\mu}(H_{{\hat{\imath}}i}\partial^{\mu}\Phi^{i})-{L_{\hat{\imath}}\over
L_{i_{0}}}\Big{(}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}-2\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})\Big{)}=0.$
(5.88)

Let us first derive from Eqs. (5.87) and (5.88) the equation for
$\Phi^{i_{0}}$, which is Eq. (5.82) for $k=i_{0}$. Applying the covariant
derivative $\bar{\nabla}^{\mu}$ on Eq. (5.87), we obtain

$\displaystyle\bar{\nabla}^{\mu}\Big{(}H_{ij}$
$\displaystyle\partial_{\mu}\Phi^{i}\partial_{\nu}\Phi^{j}\Big{)}$
$\displaystyle-\frac{1}{2}\bar{\nabla}_{\nu}\bigg{\\{}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}+2L^{-1}_{i_{0}}\Big{(}2\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\Big{)}\bigg{\\}}=0,$
(5.89)

which can be rewritten as

$\displaystyle\bigg{\\{}\bar{\nabla}^{\mu}\Big{(}H_{ki}\partial_{\mu}\Phi^{i}\Big{)}$
$\displaystyle-\frac{1}{2}\partial_{k}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\bigg{\\}}\partial_{\nu}\Phi^{k}$
$\displaystyle-\bar{\nabla}_{\nu}\bigg{\\{}2L^{-1}_{i_{0}}V_{i_{0}}\,V^{-1}_{i_{0}}\Big{(}\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\Big{)}\bigg{\\}}=0,$
(5.90)

where the function $V_{i_{0}}(\Phi)$ can be chosen arbitrarily. This yields

$\displaystyle\bigg{\\{}\bar{\nabla}^{\mu}\Big{(}H_{i_{0}i}\partial_{\mu}\Phi^{i}\Big{)}$
$\displaystyle-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\bigg{\\}}\bigg{(}\partial_{\nu}\Phi^{i_{0}}-2V_{i_{0}}^{-1}\partial_{\nu}(L_{i_{0}}^{-1}V_{i_{0}})\bigg{)}$
$\displaystyle+\bigg{\\{}\bar{\nabla}^{\mu}\Big{(}H_{{\hat{\imath}}i}\partial_{\mu}\Phi^{i}\Big{)}-\frac{1}{2}\partial_{{\hat{\imath}}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\bigg{\\}}\partial_{\nu}\Phi^{\hat{\imath}}$
$\displaystyle-2L_{i_{0}}^{-1}V_{i_{0}}\bar{\nabla}_{\nu}\bigg{\\{}V_{i_{0}}^{-1}\Big{(}\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\Big{)}\bigg{\\}}=0.$
(5.91)

Using Eq. (5.88), the second and first lines can be combined to obtain

$\displaystyle\bigg{\\{}\bar{\nabla}^{\mu}\Big{(}H_{i_{0}i}\partial_{\mu}\Phi^{i}\Big{)}$
$\displaystyle-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\bigg{\\}}\bigg{(}\partial_{\nu}\Phi^{i_{0}}+L_{i_{0}}^{-1}L_{\hat{\imath}}\partial_{\nu}\Phi^{\hat{\imath}}-2V_{i_{0}}^{-1}\partial_{\nu}(L_{i_{0}}^{-1}V_{i_{0}})\bigg{)}$
$\displaystyle-2L_{i_{0}}^{-1}V_{i_{0}}\bar{\nabla}_{\nu}\bigg{\\{}V_{i_{0}}^{-1}\Big{(}\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\Big{)}\bigg{\\}}=0.$
(5.92)

Hence, choosing $L_{i_{0}}(\Phi)$ such that

$\partial_{\nu}\Phi^{i_{0}}+L_{i_{0}}^{-1}L_{\hat{\imath}}\partial_{\nu}\Phi^{\hat{\imath}}-2V_{i_{0}}^{-1}\partial_{\nu}(L_{i_{0}}^{-1}V_{i_{0}})=0,$
(5.93)

Eq. (5.92) reduces to

$\displaystyle\bar{\nabla}_{\nu}\bigg{\\{}V_{i_{0}}^{-1}\Big{(}\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\Big{)}\bigg{\\}}=0,$
(5.94)

which yields upon integration

$\displaystyle\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\frac{1}{2}\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}=\nu_{i_{0}}V_{i_{0}},$
(5.95)

where $\nu_{i_{0}}$ is a constant. For this result to coincide with Eq. (5.82)
for $k=i_{0}$, we take

$V_{i_{0}}=-\frac{1}{2\nu_{i_{0}}}\partial_{i_{0}}\bar{U}.$ (5.96)

In that case, Eq. (5.95) implies

$-2L_{i_{0}}^{-1}\Big{(}2\bar{\nabla}_{\mu}(H_{i_{0}i}\partial^{\mu}\Phi^{i})-\partial_{i_{0}}H_{ij}\partial_{\mu}\Phi^{i}\partial^{\mu}\Phi^{j}\Big{)}=2L_{i_{0}}^{-1}\partial_{i_{0}}\bar{U},$
(5.97)

whose r.h.s. should equal $\bar{U}$, for Eq. (5.87) to match with the metric
equation of motion (5.81). As a result, we necessarily have

$L_{i_{0}}=2\,{\partial_{i_{0}}\bar{U}\over\bar{U}}.$ (5.98)

Since $i_{0}$ is chosen arbitrarily, we conclude that

$L_{i}=2\,{\partial_{i}\bar{U}\over\bar{U}}$ (5.99)

is required for all scalar equations of motion (5.82) to be satisfied, where
$i\in\\{0,\dots,N\\}$.

To conclude that we succeeded in reproducing all equations of motions (5.81)
and (5.82), we still have to check if Eq. (5.93) and its counterparts where
$i_{0}$ is replaced by any $k$ are satisfied. In fact, using the expressions
of $V_{i_{0}}$ and $L_{i}$, Eq. (5.93) becomes

$\partial_{i}\bar{U}\partial_{\nu}\Phi^{i}=\partial_{\nu}\bar{U},$ (5.100)

which is independent of $i_{0}$ and turns out to be true.

As a result, Eq. (5.84) becomes

${\partial_{i}\bar{U}\over\bar{U}}\,\delta\Phi^{i}={1\over
2}\,\bar{g}_{\mu\nu}\delta\bar{g}^{\mu\nu}\quad\Longleftrightarrow\quad\delta\Big{[}\bar{U}(\Phi)\,\sqrt{-\bar{g}}\Big{]}\\!=0,$
(5.101)

which yields in string-fame

$\delta\Big{[}U(\Phi)\,\sqrt{-g}\Big{]}\\!=0.$ (5.102)

##  6 Spectrum

Since in our approach a field combination is always non-dynamical, one is
tempted to question if the number of propagating degrees of freedom is reduced
or not. In the following, we answer this question by explicitly showing that
imposing $\sqrt{-g}e^{-2\Phi}$ to be non-dynamical does not eliminate the
dilaton degree of freedom appearing in the action

$\widehat{\cal
S}=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\sqrt{-g}e^{-2\Phi}\Big{(}R+4\partial_{\mu}\Phi\partial^{\mu}\Phi\Big{)}.$
(6.103)

Indeed, a massless graviton and a scalar degree of freedom are still
propagating.

For definiteness, let us consider a flat Minkowski vacuum with a constant
dilaton, i.e.

$g_{\mu\nu}=\eta_{\mu\nu},\qquad\Phi=\Phi_{0}={\rm constant}.$ (6.104)

We can expand the action (6.103) to second order in the fluctuations about
this background as

$\widehat{\cal S}\left[\eta+\delta g,\Phi_{0}+\delta\Phi\right]=\widehat{\cal
S}\left[\eta,\Phi_{0}\right]+\frac{1}{2}\delta^{2}\widehat{\cal
S}\left[\eta,\Phi_{0}\right]+\cdots\,.$ (6.105)

The second-order variation is given by

$\delta^{2}\widehat{\cal
S}\left[\eta,\Phi_{0}\right]=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\,e^{-2\Phi_{0}}\left\\{h^{\mu\nu}\left(\partial_{\mu}\partial^{\rho}h_{\nu\rho}-\frac{1}{2}\square
h_{\mu\nu}\right)\right\\},\leavevmode\nobreak\
\quad\mbox{where}\quad\leavevmode\nobreak\ h_{\mu\nu}=\delta g_{\mu\nu}.$
(6.106)

To obtain this result, we have implemented the condition ${1\over
2}g_{\mu\nu}\delta g^{\mu\nu}+2\delta\Phi=0$ and set

$\delta\Phi=\frac{1}{4}h,\qquad h=h_{\mu}^{\mu}.$ (6.107)

Using the standard decomposition of a symmetric tensor

$h_{\mu\nu}=h_{\mu\nu}^{TT}+\partial_{\mu}A_{\nu}^{T}+\partial_{\nu}A_{\mu}^{T}+\left(\partial_{\mu}\partial_{\nu}-\frac{1}{D}\eta_{\mu\nu}\square\right)B+\frac{1}{D}\eta_{\mu\nu}h,$
(6.108)

where the fields $h_{\mu\nu}^{TT}$ and $A_{\mu}^{T}$ are respectively
transverse-traceless and transverse,

$h_{\mu}^{TT}=\partial^{\nu}h_{\nu\mu}^{TT}=0,\qquad\partial^{\mu}A_{\mu}^{T}=0,$
(6.109)

we find that the second-order variation becomes

$\displaystyle\delta^{2}\widehat{\cal S}\left[\eta,\Phi_{0}\right]$
$\displaystyle=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}x\,e^{-2\Phi_{0}}\left\\{-h^{TT\mu\nu}\square
h_{\mu\nu}^{TT}+\frac{(D-1)(D-2)}{D^{2}}\,\Psi\square\Psi\right.$ (6.110)
$\displaystyle\hskip
101.00737pt\left.+4\,\frac{D-1}{D^{2}}\,h\square\Psi-\frac{D-2}{D^{2}}\,h\square
h\right\\}$ (6.111)
$\displaystyle=\frac{1}{2\kappa_{0}^{2}}\int\mathrm{d}^{D}xe^{-2\Phi_{0}}\left\\{-h^{TT\mu\nu}\square
h_{\mu\nu}^{TT}-S\square S+\frac{D-1}{D-2}\,\Psi\square\Psi\right\\},$ (6.112)

where

$\Psi=\square B,\qquad
S=\frac{\sqrt{D-2}}{D}\left(h-2\,\frac{D-1}{D-2}\Psi\right).$ (6.113)

Therefore, the spectrum comprises a helicity-2 state $h_{\mu\nu}^{TT}$, a
spin-0 state $S$ and a ghost state $\Psi$. The latter corresponds to the usual
conformal ghost mode of general relativity, which here manifests itself as the
longitudinal part of the spin-1 part of the metric fluctuations $h_{\mu\nu}$.

##  7 Discussion and conclusion

Consistency of a string model at tree level requires the vanishing of the
sigma-model beta-functions on the worldsheet. We have seen that these
equations for the metric, the antisymmetric tensor and the dilaton can be
recovered by extremizing a universal effective action. To achieve this, we
need to impose that the field combination $\sqrt{-g}e^{-2\Phi}$ is non-
dynamical. Therefore, the model-dependent central-charge deficit term
$\sqrt{-g}e^{-2\Phi}\delta c$ in the Lagrangian no longer contributes and can
be omitted. In fact, the central-charge deficit arises as an integration
constant, when solving the equations of motion. On may think that a drawback
of this approach is that all values of the integration constant are allowed,
regardless of the existence or not of a string-theory model. For instance, in
the description of the Witten black hole of Sect. 2.2, the Kac-Moody level
$k=8f_{0}$ should be integer. However, at the two-derivative level, only large
values of $k$ should be considered, spanning effectively a continuum.

At tree level, the string models admit moduli fields parametrizing moduli
spaces. To incorporate them in the effective description, we must add to the
Lagrangian a sigma-model kinetic term that involves the moduli-space metric.
Since the latter is model dependent (in particular, in models that are not
maximally supersymmetric at tree level), the effective action is no longer
universal, and the moduli-space metric has to be determined case by case at
the string level. Moreover, at higher genus, the moduli-space metric admits
quantum corrections and an effective potential $U$ can be generated. In the
alternative approach we outline in this work, the latter is not included in
the Lagrangian. Instead, it appears in the combination of fields constrained
to be non-dynamical, namely $\sqrt{-g}\,U$, which eventually leads to the
appearance of the gradient of $U$ in the scalar equations of motion. The
moduli-space metric and effective potential must therefore be predetermined at
the string level in order to obtain the field equations and the solutions.
However, the key point is that once the dynamics stabilizes all moduli fields
at a minimum of the effective potential, the sigma-model kinetic term vanishes
and the potential $U$ reduces to a cosmological term that in this alternative
description appears as an integration constant, as in [1].

We emphasize that the fact that $\sqrt{-g}\,U$ is non-dynamical means that its
variation vanishes ($\delta[\sqrt{-g}\,U]=0$) when one extremizes the action
to obtain the independent equations of motion. This condition respects the
full diffeomorphism invariance, and so does our effective description.
However, as in general relativity, nothing prevents us to use a particular
frame such that $\sqrt{-g}\,U=\varphi(x)$, where $\varphi(x)$ is a fixed
background. When we choose to work in such a frame, the residual gauge
symmetry is the group of volume-preserving diffeomorphisms. Moreover, even
though the combination $\sqrt{-g}\,U$ is non-dynamical, we have shown that the
number of the propagating degrees of freedom is not reduced.

Beyond tree level, our derivations do not rely on the fact that the Lagrangian
and the non-dynamical field combination admit an expansion in powers of the
string coupling $e^{\Phi}$. Therefore, all conclusions should remain valid at
the non-perturbative level. On the other hand, our analysis was based on beta
functions derived at leading order in $\alpha^{\prime}$, thus at the two
derivative level. As part of future work, it would be interesting to see how
higher-derivative terms in the effective action may be incorporated.

We have shown that besides the standard approach for describing the low energy
physics of string theory, there exists an alternative effective field theory
description, without any effective potential in the action but with a non-
dynamical field combination instead. Since both approaches yield the same
field equations for all the degrees of freedom, it would be interesting to
know if other criteria can distinguish them. Such criteria may be experimental
or theoretical, at the level of string theory. In fact, the question already
arises when one wants to decide whether the general theory of relativity or a
version of unimodular gravity is preferable for describing the real world [1,
12]. In particular, even if general relativity and unimodular gravity are
equivalent at the classical level, they yield distinct Wheeler-DeWitt
equations at the quantum level [25, 26] and thus different predictions
concerning the wavefunction of the universe.

Another question concerns the dimensional reduction of the unimodular-like
string effective action. In certain cases (e.g. toroidal compactifications
without fluxes), the minimal unimodular description for the compactified
theory, where the effective potential does not appear in the action but in a
non-dynamical field combination instead, can be directly obtained via the
dimensional reduction of the corresponding unimodular description in higher
dimensions. This is also true for the precise form of the non-dynamical field
combination involving the determinant of the metric and the scalar fields: The
non-dynamical field combination for the compactified theory can be directly
obtained from the non-dynamical field combination in the “parent” theory.
However, this direct relationship between the two minimal unimodular
descriptions will not hold in the generic cases. In particular, turning on
non-trivial fluxes associated with the antisymmetric RR fields may lead to the
appearance of a potential for the scalar fields in the action for the
compactified theory. So it seems that the passage to the minimal unimodular
description, with no potential in the action, will have to be implemented at a
new step.

Let us however emphasize that the situation is similar in the conventional
description. For us the starting point is the string model and the sigma model
action associated with the compactification manifold. In the compactified
theory, there are two kinds of fields. Those that can be seen as components of
the higher dimensional fields and those that emerge at special points of the
moduli space, and are associated with stringy states and wrapped branes
becoming light. For example, after compactifying a ten-dimensional theory on a
Calabi-Yau manifold, the metric decomposes in an obvious notation as
$(g_{\mu\nu},g_{i\bar{\jmath}},g_{ij})$333Gauge fields of the form $g_{\mu
i},g_{\mu\bar{\imath}}$ also appear. and therefore, we will have a four-
dimensional graviton $g_{\mu\nu}$, $h_{(2,1)}$ complex scalars and $h_{(1,1)}$
real scalars. In addition, extra states may emerge at special points of the
moduli space, which do not exist in the original higher dimensional effective
action. The first class of fields are obviously accommodated in the unimodular
restricted string effective action (after all, consistent dimensional
reduction requires solving the higher dimensional equations, here the
$\beta$-function equations). On the other hand, the new fields, as well as
fluxes associated with compact cycles, which arise after compactification and
do not appear in the effective action for the uncompactified theory, cannot be
directly incorporated via a dimensional reduction of the effective action. One
must resort to the full string model and the moduli space to obtain the
effective description. This was also the case in the conventional description,
where extra fields must be incorporated in the action to account for new
stringy states becoming light. A full analysis concerning the inclusion of
such fields, as well as fluxes in our new setting, would require a full in
depth analysis that goes beyond the scope of the present work.

Finally, we may ask whether the effective field theory with non-dynamical
field combination we consider can be embedded in a general framework similar
to supergravity [27, 28]. In other words, are such constraints compatible with
local supersymmetry? This is a non-trivial question since the fact that
$\sqrt{-g}\,U$ is non-dynamical should have consequences for the superpartners
of the graviton and scalar fields.

## Acknowledgements

A.K. and H.P. would like to thank the University of Cyprus for hospitality,
while A.K and N.T. acknowledge hospitality by the Ecole Polytechnique where
early stages of this work have been done. The work of A.K. is partially
supported by the NTUA basic research grant no 65228100. The work of H.P. and
N.T. is partially supported by the Cyprus Research and Innovation Foundation
grant EXCELLENCE/0421/0362.

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