Alexandros Kehagias
email: kehagias@central.ntua.gr

Hervé Partouche
email: herve.partouche@polytechnique.edu

The Casimir Effect in String Theory

Keywords: Vacuum Energy, Casimir Force, Partition Function, Supersymmetry, Supersymmetry Breaking PACS numbers: 11.25.-w, 11.10.-z, 11.30.Pb, 11.30.Qc

We discuss the Casimir effect in heterotic string theory. This is done by considering a Z 2 twist acting on one external compact direction and three internal coordinates. The hyperplanes fixed by the orbifold generator G realize the two infinite parallel plates. For the latter to behave as "conducting material", we implement in a modular invariant way the projection (1 -G)/2 on the spectrum running in the vacuum-to-vacuum amplitude at one-loop. Hence, the relevant projector to account for the Casimir effect is orthogonal to that commonly used in string orbifold models, which is (1 + G)/2. We find that this setup yields the same net force acting on the plates in the context of quantum field theory and string theory. However, when supersymmetry is not present from the onset, finiteness of the resultant force in field theory is reached by adding formally infinite forces acting on either side of each plate, while in string theory both contributions are finite. On the contrary, when supersymmetry is spontaneously broken à la Scherk-Schwarz, finiteness of each contribution is fulfilled in field and string theory.

Introduction

The prototypical form of the Casimir force is the attraction felt by two flat, closely spaced, parallel mirrors originating from the vacuum energy of the electromagnetic field between the plates [START_REF] Bordag | New developments in the Casimir effect[END_REF][START_REF] Lamoreaux | The Casimir force: Background, experiments, and applications[END_REF]. Precise measurements of the Casimir force become technically possible only about twenty years ago thanks to the pioneering experimental work of Lamoreaux [START_REF] Lamoreaux | Demonstration of the Casimir force in the 0.6 to 6 micrometers range[END_REF]. It is also interesting to notice that it took almost half a century for the first prediction of this force to be measured at the level of per cent accuracy. The force F exerted between two parallel conducting plates of area A and at a distance L turns out to satisfy

F ≡ F A = - cπ 2 240L 4 .
(1.1)

This force arises due to the structure of the electromagnetic modes (with zero-point energy ω/2 each) between the two plates, as compared to free space without the plates. The reason is that in quantum field theory, in the absence of gravity, only differences in energy have a physical meaning. Therefore, one should compare the energy of the two-plates configuration with some reference background, which is flat empty Minkowski space here. Of course, things are quite different when gravity is turned on. In this case, vacuum energy contributes to the energy-momentum tensor as a cosmological constant and therefore to the spacetime curvature through Einstein equations. In other words, vacuum energy gravitates as all forms of energy. However, the cosmological constant turns out to be infinite in quantum field theory and some regularization is needed, which shifts the cosmological constant to the cutoff scale.

In string theory, on the other hand, there are no infinities in the calculation of the vacuum energy and therefore no need for regularization. The cosmological constant is of the order of the supersymmetry breaking scale, however in conflict with observations.

Nevertheless, the Casimir effect is a remarkable result as it was the first indication of the vacuum energy, although it can be formulated and computed with no reference to any zero-point energy. In the latter approach, the Casimir force is regarded as the retarded relativistic van der Waals force between the metal plates [START_REF] Jaffe | The Casimir effect and the quantum vacuum[END_REF]. In other words, this point of view suggests that the Casimir effect does not really support the existence of vacuum energy of quantum fields more than any other one-loop effect in quantum field theory. In that case, the Casimir force should vanish as the fine structure constant goes to zero, as all one-loop effects in quantum electrodynamics [START_REF] Jaffe | The Casimir effect and the quantum vacuum[END_REF]. However, here we will follow the traditional route and calculate the Casimir force by summing up zero-point energies in field and string theory in spacetime dimension d.

To describe a universe comprised between two hyperplanes, we consider spacetime to be R 1,d-1 × S 1 (R d-1 )/Z 2 , where the Z 2 generator G acts as a twist x d-1 → -x d-1 on the circle of radius R d-1 . As a result, infinite parallel plates are located at the fixed points

x d-1 = 0 and x d-1 = πR d-1 . However, this setup is not quite what we are interested in for describing the Casimir effect. The reason is that the plates should be "conducting", which means that all modes propagating between the hyperplanes should have nodes at x d-1 = 0 and x d-1 = πR d-1 . However, all degrees of freedom in the background above are even rather than odd under the twist. If the orbifold projector (1 + G)/2 is common (at least in string theory) to project the spectrum on even modes, it turns out that the correct orbifold projector for restricting to odd modes is the orthogonal one, (1 -G)/2.

The above result is shown in quantum field theory in Sect. 2. Actually, the force acting on the plates is equal, whether even or odd modes are selected, as the vacuum amplitude with G inserted projects on modes non-propagating along x d-1 and thus not contributing to the transverse force. We analyse the case where supersymmetry is not present from the onset and recover the fact that the Casimir force can eventually be interpreted as the finite resultant of two formally infinite forces exerted on either side of each plate: One derived from the vacuum between the hyperplanes, and one derived from the vacuum in an infinite half space. However, we show that when supersymmetry is present in a spontaneously broken phase, these forces are individually finite.

In Sect. 3, we generalize the orbifold prescription to the heterotic string theory. For consistency arising from modular invariance, G also acts on three internal directions we take toroidal. Moreover, a twisted sector arises, which has no counterpart in field theory. However, the twisted spectrum being localized on the fixed hyperplanes, it does not contribute to the transverse force. Therefore, string theory reproduces in great details the field theory derivation of the forces acting of either side of each plate, which are individually finite when supersymmetry is spontaneously broken. The outcomes in both frameworks match up to contributions arising from heavy string modes, which turn out to be irrelevant as they are exponentially suppressed.

In Sect 4, we consider the SO(16) × SO(16) heterotic string [START_REF] Alvarez-Gaumé | An O(16) × O(16) heterotic string[END_REF], which is known to be explicitly non-supersymmetric. In this case, finiteness of each individual force remains valid, due to the natural regularization of the vacuum amplitude arising from the infinite spectrum with unbounded masses. In other words, when supersymmetry is broken in a hard way, not only the heavy states contribute significantly, since they are even rendering the vacuum energy finite in every case, whether the plates are at finite or infinite distance. This is the great advantage of string theory over field theory where differences of energies can only be considered, for the infinite sums over zero-point energies to cancel out. However, considering the energy gap between the configurations where the plates are either at finite distance, or infinite distance, the field theory and string theory outcomes are in perfect agreement, up to exponentially suppressed corrections arising from heavy string states.

Our conclusion can be found in Sect. 5, and we mention that throughout this work, we use string units, α = 1, and the notations of Ref. [START_REF] Kiritsis | String theory in a nutshell[END_REF].

Casimir effect in quantum field theory

In this section, our aim is to review and then develop different approaches for computing the Casimir force in field theory, when supersymmetry is not present from the onset, or when it is realized in a spontaneously broken phase. In particular, we will introduce an orbifold point of view suitable for later generalization in the framework of string theory. The field theory analysis will be presented in d dimensions for a real scalar field, or for a single degree of freedom, fermionic or bosonic.

Second quantized formalism

Our basic consideration is the path integral computation of the partition function of a real bosonic free field. In dimension d and for a mass M B , the partition function is

Z B = Dφ e -i d d x 1 2 (∂µφ∂ µ φ+M 2 B φ 2 ) = Dφ e -d d x E 1 2 φ(-∂µ∂µ+M 2 B )φ .
(2.1)

The former expression uses Lorentzian metric (-1, 1, . . . , 1), while the latter is obtained by Wick rotation x 0 E = -ix 0 . We use a compact version E d of the Euclidean spacetime, with finite volume V 0,...,d-1 ,

E d = d-1 µ=0 S 1 (R µ ), V 0,...,d-1 = d-1 µ=0 2πR µ , (2.2)
where all directions are circles of radii R µ . Imposing the field to have periodic boundary conditions, its mode expansion in orthonormal modes satisfies

φ(x E ) = 1 V 0,...,d-1 m∈Z d φ m e i mµ Rµ x µ E , φ -m = φ * m .
(2.3) By using the above expression, the path integral can be written as

Z B = Dφ e -1 2
m∈Z d ( mµ Rµ ) 2 + M 2 B ) |φ m | 2 = dφ 0 √ 2π e -1 2 M 2 B φ 2 0 m∈Z d + dRe φ m √ π dIm φ m √ π e -( mµ Rµ ) 2 + M 2 B [(Re φm) 2 +(Im φm) 2 ] , (2.4)
where Z d + is the set of non-vanishing d-tuples whose first nonzero entry is positive. The integration over the real and imaginary parts of the modes φ m being Gaussian, we find the well know result

ln Z B = - 1 2 m∈Z d ln m µ R µ 2 + M 2 B .
(2.5)

Splitting even and odd modes: In Minkowski spacetime, Casimir effect arises between two parallel, infinite hyperplanes, we choose to be located at x d-1 = 0 and

x d-1 = πR d-1 .
A field between the hyperplanes has nodes on the "conducting" boundary plates. To make contact with the unconstrained field φ periodic along x d-1 , we define

∀x E ∈ R d-1 , ∀x d-1 ∈ R, φ e (x E , x d-1 ) = 1 2 φ(x E , x d-1 ) + φ(x E , -x d-1 ) , φ o (x E , x d-1 ) = 1 2 φ(x E , x d-1 ) -φ(x E , -x d-1 ) , (2.6)
so that φ = φ e + φ o , where φ e , φ o are respectively even and odd under x d-1 → -x d-1 and 2πR d-1 -periodic. Actually, φ e contains all modes cos(m d-

1 x d-1 /R d-1 ), m d-1 ≥ 0, while φ o contains all modes sin(m d-1 x d-1 /R d-1 ), m d-1 ≥ 1.
We can now define path integrals Z e B , Z o B for only even and odd fields, respectively as

Z e B = Dφ e e -d d x E 1 2 φ e (-∂µ∂µ+M 2 B )φ e , Z o B = Dφ o e -d d x E 1 2 φ o (-∂µ∂µ+M 2 B )φ o , (2.7)
where the integrals in the actions cover the whole circle,1 -πR d-1 ≤ x d-1 ≤ πR d-1 . In these notations, the partition function in Eq.(2.4) can be written as

Z B = Dφ e Dφ o e -d d x E 1 2 (φ e +φ o )(-∂µ∂µ+M 2 B )(φ e +φ o ) = Z e B Z o B , (2.8)
where the contributions of the cross products in the action vanish due to the distinct parities of the fields. In fact, it is straightforward to verify that ln Z e B = -

1 2 m∈Z d-1 m d-1 ≥0 ln m µ R µ 2 + M 2 B , ln Z o B = - 1 2 m∈Z d-1 m d-1 ≥1 ln m µ R µ 2 + M 2 B .
(2.9)

Several problems can then be analyzed:

• We can keep the whole set of modes m ∈ Z d , in order to describe a universe circular along x d-1 .

• We can also restrict to the cosine modes m d-1 ≥ 0, to describe a universe with an orbifolded direction S 1 (R d-1 )/Z 2 , where the Z 2 generator G acts as x d-1 → -x d-1 and where all fields are even.

• Finally, we can restrict to the sine modes m d-1 ≥ 1, in order to describe a "Casimir-like universe", i.e. with orbifolded direction but with all fields odd under the generator G, i.e.

with nodes on the hyperplanes.

To express these remarks in a more formal way, we can associate to the modes used in the expansion (2.3) where ln Z e B = -

1 2 Tr 1 + G 2 ln m µ R µ 2 + M 2 B , ln Z o B = - 1 2 Tr 1 -G 2 ln m µ R µ 2 + M 2 B , (2.13)
which will turn out to be a suitable form for comparison with the string theory analysis to be presented in Sects 3 and 4.

Statistical physics interpretation:

To present shortly a usual derivation of the Casimir effect, we can interpret the above results from a statistical physics viewpoint. We are interested in the large volume limit along the directions x 1 , . . . , x d-2 , where the discrete modes become continuous. In this case, the path integral expression (2.5) reads

ln Z B = - 1 2 m 0 V 1,...,d-2 (2π) d-2 d d-2 k m d-1 ln m 0 R 0 2 + ω 2 k where ω k = d-2 µ=1 k 2 µ + m d-1 R d-1 2 + M 2 B .
(2.14)

To make contact with thermodynamics, we write

m 0 ln m 0 R 0 2 + ω 2 k = 2 ln 2 sinh(πR 0 ω k ) + 2 ln 1 2πR 0 m 0 ≥1 m 0 R 0 2 , (2.15)
by employing the identity

sinh(x) = x m 0 ≥1 1 + x 2 π 2 m 2 0 .
(2.16)

Using the ζ-regularisation formulas

m 0 ≥1 1 R 2 0 = 1 R 2ζ(0) 0 = R 0 , m 0 ≥1 m 0 = e -ζ (0) = √ 2π, (2.17)
the second logarithm in the r.h.s. of Eq. (2.15) vanishes and we obtain

- ln Z B 2πR 0 = V 1,...,d-2 (2π) d-2 d d-2 k m d-1 ω k 2 + 1 2πR 0 V 1,...,d-2 (2π) d-2 d d-2 k m d-1 ln 1 -e -2πR 0 ω k . (2.18)
This result is nothing but the Helmholtz free energy of a perfect gas at finite temperature T = 1/(2πR 0 ). Note that the corresponding expression for a fermionic degree of freedom could have been obtained similarly if we had imposed an antiperiodic boundary condition along S 1 (R 0 ). Thus, the zero-temperature limit we are interested in is attained by taking R 0 → ∞. In this limit, the second line vanishes and we are left with the vacuum energy E, whose density reads

V = E V 1,...,d-1 ≡ - ln Z B V 0,...,d-1 = 1 2πR d-1 d d-2 k (2π) d-2 m d-1 ω k 2 . (2.19)
For the orbifold cases, we may restrict to the cosine or sine modes and consider the length of the direction x d-1 to be πR d-1 . Defining V half 1,...,d-1 = V 1,...,d-1 /2, the respective energy densities are

V e = E e V half 1,...,d-1 ≡ - ln Z e B V half 0,...,d-1 = 1 πR d-1 d d-2 k (2π) d-2 m d-1 ≥0 ω k 2 , V o = E o V half 1,...,d-1 ≡ - ln Z o B V half 0,...,d-1 = 1 πR d-1 d d-2 k (2π) d-2 m d-1 ≥1 ω k 2 , (2.20)
which obviously satisfy 2V = V e + V o . Moreover, using the expression of ln Z e B in Eq. (2.13), we have

V e = V + V tw , where V tw = 1 2 Tr G ln mµ Rµ 2 + M B V 0,...,d-1 . (2.21)
However, evaluating the trace in the original orthonormal basis |m , m ∈ Z d , all contributions with m d-1 = 0 cancel out of V tw , implying V tw to be inversely proportional to R d-1 . In total, we obtain the final result

V e = V + λ tw R d-1 , V o = V - λ tw R d-1 , λ tw = 1 2
Z B = - 1 2 m d-1 V 1,...,d-2 (2π) d-2 d d-2 k m 0 ln m d-1 R d-1 2 + Ω 2 k E
where

Ω k E = d-2 µ=1 k 2 µ + m 0 R 0 2 + M 2 B .
(2.23)

Following identical steps, we obtain the counterpart of Eq. (2.18),

- ln Z B 2πR d-1 = V 1,...,d-2 (2π) d-2 d d-2 k m 0 Ω k E 2 + 1 2πR d-1 V 1,...,d-2 (2π) d-2 d d-2 k m 0 ln 1 -e -2πR d-1 Ω k E .
(2.24)

In the continuous limit i.e. R 0 → +∞, the density (2.19) takes the new form

V ≡ - ln Z B V 0,...,d-1 = d d-1 k E (2π) d-1 Ω k E 2 + 1 2πR d-1 d d-1 k E (2π) d-1 ln 1 -e -2πR d-1 Ω k E , where Ω k E = k 2 E + M 2 B , k E ≡ (k E0 , . . . , k d-2 ).
(2.25)

Notice that the second integral, which encodes the dependence on R d-1 , is convergent. Hence, the ill-defined part of V is the first integral, which is somehow an "infinite constant". As an example, we find for a vanishing mass,

M B = 0 : V = d d-1 k E (2π) d-1 ||k E || 2 - v d R d d-1
, where

v d = ζ(d) (2π) d Γ d 2 π d 2
.

(2.26)

Casimir force: Let us now consider the odd modes between the hyperplanes and compute the Casimir force in a usual way [START_REF] Itzykson | Quantum field theory[END_REF]. This can be done by considering the energy density between the plates when R d-1 is finite, and when R d-1 is very large (continuous limit). Using

Eq. (2.20), they are respectively

V o = 1 πR d-1 d d-2 k (2π) d-2 m d-1 ≥1 1 2 d-2 µ=1 k 2 µ + m d-1 R d-1 2 + M 2 B , V o ∞ = 1 πR d-1 d d-2 k (2π) d-2 R d-1 +∞ 0 dk d-1 1 2 d-2 µ=1 k 2 µ + k 2 d-1 + M 2 B .
(2.27)

Clearly, both are ill-defined due to the discrete sum and integrations over infinitely large momenta. However, we may consider their difference,

πR d-1 V o -(πR d-1 V o ) ∞ = 1 2 d d-2 k (2π) d-2 ×    m d-1 ≥1 d-2 µ=1 k 2 µ + m d-1 R d-1 2 + M 2 B -R d-1 +∞ 0 dk d-1 d-2 µ=1 k 2 µ + k 2 d-1 + M 2 B    , (2.28)
which can be evaluated by introducing a UV cutoff, and then sending the latter to infinity, which yields a finite result [START_REF] Itzykson | Quantum field theory[END_REF].

However, the final answer can be obtained more easily by using our previous results.

From Eq. (2.22), we may write either

V o -V o ∞ = V - λ tw R d-1 -V ∞ or πR d-1 V o -(πR d-1 V o ) ∞ = πR d-1 V -(πR d-1 V) ∞ , (2.29)
depending on which quantity, V o or πR d-1 V o , we choose to take the R d-1 → +∞ limit.

However, the second expression is more suitable, since λ tw is actually infinite. Hence, we obtain for vanishing mass from Eq. (2.26),

M B = 0 : πR d-1 V o -(πR d-1 V o ) ∞ = - πv d R d-1 d-1 , (2.30)
which yields the force per unit area of the plates, i.e. a pressure,

F ≡ - ∂ R d-1 V o -(R d-1 V o ) ∞ ∂R d-1 = -(d -1) v d R d d-1 . (2.31)
This outcome is the Casimir force per unit area acting on each plate. In particular, it reproduces Eq. (1.1) after it is multiplied by 2 to account for the 2 degrees of freedom of the electromagnetic field in d = 4, and for L = πR d-1 . To understand why, let us note that on the hyperplane located at

x d-1 = πR d-1
, there is a force arising from the vacuum comprised in the range 0 < x d-1 < πR d-1 , to which we must add a force induced by the vacuum in the infinite half space x d-1 > πR d-1 . Altogether, the resulting force per unit area acting on the plate is therefore

F int + F ext = - ∂(R d-1 V o ) ∂R d-1 + ∂(R d-1 V o ∞ ) ∂R d-1 ≡ F. (2.32)
Hence, the Casimir force in field theory is the resultant of two formally infinite forces acting on each side of the plate, leaving a finite result. As we have just seen, this is equivalent to saying that the total force acting on a plate follows from the difference between the vacuum energies of two configurations, one in the presence of the plates at finite distance, and one with the second plate sent to infinity. This has the effect of removing the infinities encountered in the calculation of each configuration individually and is totally meaningful, as only differences in energies have a physical meaning when gravity is turned off. However, in presence of gravity, the situation is different, as the vacuum energy is nothing but the cosmological constant, and the latter cannot be infinite in a UV-complete theory.

First quantized formalism with spontaneous or explicit supersymmetry breaking

To get closer to the string derivations to be given in the next sections, we may translate the above results in first quantized formalism. From Eq. (2.5), and inspired by the following equality

- 1 2 ln C = +∞ 0 dτ 2 2τ 2 (e -πτ 2 C -e -πτ 2 ), ∀C > 0, (2.33)
we may define the regularized quantity

ln Z reg B = +∞ dτ 2 2τ 2 m∈Z d e -πτ 2 mµ Rµ 2 +M 2 B . (2.34)
In the above definition, the C-independent term of the integrand in Eq (2.33) is not included, because empty of any information about the physical system. The dummy variable τ 2 is the Schwinger parameter, which is the proper time of the particle along its trajectory, whose topology is that of a circle. In order to regulate the UV region τ 2 → 0, a cutoff > 0 is introduced, and should be sent to 0 only at the end of any sensible computation. Notice that in this formulation, the discrete sum appearing in ln Z reg B can be taken over infinitely large momenta. Of course, the counterpart of in second quantized formalism, Eq. (2.5),

is a cutoff Λ 2 co > d-1 µ=0 (m µ /R µ ) 2 .
As mentioned in the computation of the Casimir force in Eq. (2.28), the latter should also be sent to infinity at the final step of a derivation. Hence, whether the first or second quantized formalism is used, only differences between two regulated energies associated with different configurations should be considered, in order to get finite (and equal) answers once the limits → 0 or Λ co → +∞ are taken.

However, compared to what we have discussed so far in field theory, the string theory case discussed shortly will be different in two respects: It involves bosonic and fermionic degrees of freedom, and full towers of KK modes arising from a Scherk-Schwarz mechanism [START_REF] Scherk | Spontaneous breaking of supersymmetry through dimensional reduction[END_REF][START_REF] Scherk | How to get masses from extra dimensions[END_REF][START_REF] Rohm | Spontaneous supersymmetry breaking in supersymmetric string theories[END_REF][START_REF] Kounnas | Spontaneous supersymmetry breaking in string theory[END_REF][START_REF] Ferrara | Superstring solutions with spontaneously broken four-dimensional supersymmetry[END_REF][START_REF] Ferrara | Superstrings with spontaneously broken supersymmetry and their effective theories[END_REF][START_REF] Kounnas | Coordinate-dependent compactifications and discrete symmetries[END_REF].

For these reasons, we will not make use of the definition (2.34) and will follow another route, in our presentation of the first quantized formalism in quantum field theory.

Scherk-Schwarz breaking of supersymmetry: By considering the framework used in Sect. 2.1 with one more circle denoted S 1 (R 9 ), and along which the quantum field has periodic or antiperiodic boundary conditions [START_REF] Scherk | Spontaneous breaking of supersymmetry through dimensional reduction[END_REF][START_REF] Scherk | How to get masses from extra dimensions[END_REF],

∀x E ∈ R d , ∀x 9 ∈ R, φ(x E , x 9 ) = (-1) ξ φ(x E , x 9 + 2πR 9 )
, where ξ = 0 or 1, (2.35) the partition function Eq. (2.5) in the massless case is generalized to

ln Z B = - 1 2 m∈Z d m 9 ln m µ R µ 2 + m 9 + ξ 2 R 9 2 .
(2.36)

In fact, we can also see the above formula as resulting from an infinite tower of KK modes

with masses M B = |m 9 + ξ 2 | R 9 .
To achieve the Scherk-Schwarz mechanism along S 1 (R 9 ), we can consider a massless fermionic free field with reversed boundary conditions along S 1 (R 9 ). The logarithm of the associated path integral is dressed with an overall minus sign,

ln Z F = + 1 2 m∈Z d m 9 ln m µ R µ 2 + m 9 + 1-ξ 2 R 9 2 .
(2.37)

Note that the masses of the fermionic KK states are

M F = |m 9 + 1-ξ 2 | R 9
so that the mass splitting between bosons and fermions at each level m 9 is identified to be

M = 1 2R 9 , (2.38)
which can be interpreted as a scale of supersymmetry breaking.

Using Eq. (2.33) for the couple of such bosonic and fermionic fields, we obtain

ln Z B+F = V 0,...,d-2 (2π) d-1 d d-1 k E m d-1 +∞ 0 dτ 2 2τ 2 m 9 e -πτ 2 k 2 E + m d-1 R d-1 2 + m 9 + ξ 2 R 9 2 -e -πτ 2 k 2 E + m d-1 R d-1 2 + m 9 + 1-ξ 2 R 9 2 , (2.39)
where we have considered the continuous limit in the Euclidean directions x 0 E , . . . , x d-2 . Note that the C-independent term in the integrand of Eq. (2.33) drops by itself. Integrating over the continuous momenta, and applying Poisson resummations over the momenta m d-1 , m 9 , we obtain

ln Z B+F = V 0,...,d-2 (2π) d-1 R d-1 R 9 md-1 +∞ 0 dτ 2 2τ 1+ d+1 2 2 m9 e -π τ 2 R 2 d-1 m2 d-1 +R 2 9 m2 9 (-1) ξ m9 1 -(-1) m9 , (2.40)
where the last bracket imposes m9 to be odd, in order to contribute. As a result, the integral over τ 2 is convergent, even in presence of the dangerous UV regime τ 2 → 0, because of the exponential factor in the integrand. There is no need to introduce a cutoff of any kind, and the result can be written as

V B+F = - ln Z B+F V 0,...,d-1 = -(-1) ξ f R d-1 R 9 1 R d 9 , (2.41)
where we have defined the function 2

f (u) = Γ d+1 2 (2π) d π d+1 2 md-1 , k9 1 (2 k9 + 1) 2 + m2 d-1 u 2 d+1 2 . (2.42)
Of course the fact that we find a finite result without having to subtract anything to the energy is a consequence of the fact that we have considered the spontaneous breaking of a supersymmetric theory. In the decompactification limit R 9 → +∞, supersymmetry is

restored in d + 1 dimensions.
Notice that by integrating over the continuous momenta, Eq. (2.39) can be formally written as

ln Z B+F = R 0 . . . R d-2 m d-1 +∞ 0 dτ 2 2τ 1+ d-1 2 2 Str e -πτ 2 m d-1 R d-1 2 +M 2 , (2.43)
where the supertrace is over the bosonic and fermionic KK towers of states propagating along S 1 (R 9 ), and M is the associated KK mass operator. In this form, it is straightforward to split the set of modes m d-1 ∈ Z relevant for a circular universe into those surviving the orbifold projection S 1 (R d-1 )/Z 2 , and those involved in a Casimir-like universe, as explained 2 Such functions have already appeared in the literature in the expression of the effective potential at finite temperature derived in string theory models [START_REF] Catelin-Jullien | Thermal/quantum effects and induced superstring cosmologies[END_REF]. The latter describe cosmological evolutions which are attractors of the dynamics [START_REF] Bourliot | Attraction to a radiation-like era in early superstring cosmologies[END_REF][START_REF] Bourliot | Cosmological phases of the string thermal effective potential[END_REF][START_REF] Estes | Superstring cosmology for N 4 = 1 → 0 superstring vacua[END_REF][START_REF] Heurtier | Spontaneous dark-matter mass generation along cosmological attractors in string theory[END_REF].

below Eq. (2.9). In terms of projector and orthogonal projector, we obtain respectively

ln Z e B+F = R 0 . . . R d-2 m d-1 +∞ 0 dτ 2 2τ 1+ d-1 2 2 Str 1 + G 2 e -πτ 2 m d-1 R d-1 2 +M 2 , ln Z o B+F = R 0 . . . R d-2 m d-1 +∞ 0 dτ 2 2τ 1+ d-1 2 2 Str 1 -G 2 e -πτ 2 m d-1 R d-1 2 +M 2 , (2.44)
which are the analog of what will be computed in string theory. Proceeding as in the derivation of Eq. (2.22) in second quantized formalism, we obtain

V e B+F = V B+F + λ tw,B+F R d-1 , V o B+F = V B+F - λ tw,B+F R d-1 ,
where

λ tw,B+F = 1 (2π) d +∞ 0 dτ 2 2τ 1+ d-1 2 2 Str e -πτ 2 M 2 .
(2.45)

These quantities are all finite, and so are the forces per unit area F int and F ext exerted on either side of the plate located at

x d-1 = πR d-1 , F int = - ∂(R d-1 V o B+F ) ∂R d-1 , F ext = - ∂(R d-1 V o B+F ) ∞ ∂R d-1 , (2.46)
those resultant F = F int + F ext characterizes the Casimir effect. We will come back to its explicit expression when we discuss the string theory point of view.

Hard breaking of supersymmetry: In order to construct a model realizing a "hard breaking" of supersymmetry, we are going to take the limit R 9 → 0, which sends M → +∞.

This is an arbitrary approach from the field theory point of view, which is efficient in the context of string theory, for further comparison. Starting from Eq. (2.39), we integrate as before over the continuous momenta and Poisson resum over m d-1 . However, we Poisson resum over m 9 only when md-1 = 0, and keep the sum over m 9 as it is when md-1 = 0. As a result, the contribution md-1 = 0 is as in Eqs (2.41), (2.42) and we obtain

V B+F = -(-1) ξ ξ d R d 9 - 1 (2π) d md-1 =0 +∞ 0 dτ 2 2τ 1+ d 2 2 e -π τ 2 R 2 d-1 m2 d-1 m 9 e -πτ 2 m 9 + ξ 2 R 9 2 -e -πτ 2 m 9 + 1-ξ 2 R 9 2 , (2.47)
where

ξ d = Γ d+1 2 (2π) d π d+1 2 k9 1 |2 k9 + 1| d+1 . (2.48)
Clearly, the first term in Eq. (2.47), which is independent of R d-1 , is singular in the limit R 9 → 0. On the contrary, in the R d-1 -dependent part, the only contribution surviving the limit is that associated to the KK mode m 9 + ξ 2 = 0 or m 9 + 1-ξ 2 = 0, and it is finite. Hence, we may write, by abuse of notations,

V B+F = -(-1) ξ ξ d R d 9 -(-1) ξ v d R d d-1
, where

v d = 1 2(2π) d Γ d 2 π d 2 md-1 =0 1 | md-1 | d , (2.49)
and where it is understood that R 9 is formally vanishing, making V B+F and -(-1) ξ ξ d /R d 9 individually infinite. Alternatively, we can write both diverging quantities in the l.h.s. for their difference to be finite in the R 9 → 0 limit, In fact, in the limit R 9 → 0, all massive KK modes propagating along S 1 (R 9 ) decouple and we are left with a single massless boson (for ξ = 0) or fermion (for ξ = 1), with associated KK excitations along x d-1 , contributing to the energy density, however in an infinite way in V B+F . There is no left notion of superpartner, thus realizing a hard, or explicit breaking of supersymmetry. Thus, the index B + F is no more justified and should be replaced by either B of F. Notice that the l.h.s. of Eq. (2.50) amounts precisely to subtract the energy density at infinite R d-1 to the result at finite R d-1 , to obtain a finite answer. Moreover, notice that for a bosonic tower of KK modes, the "infinite constant" in Eq. (2.26) is positive, while in Eq. (2.49) it is negative (ξ = 0). This is another occasion to stress that these equations are understood to make sense only when they are applied to evaluate the energy gap between two configurations. Under no circumstances, any of these equations should be used to evaluate some absolute energy, even in presence of a cutoff. As we will see in Sect. 4, this is where string theory is useful.

V B+F -V B+F,∞ ≡ V B+F + (-1) ξ ξ d R d 9 = -(-1) ξ v d R d d-1 . ( 2
Having discussed the case of a hard breaking of supersymmetry in a universe circular along x d-1 , we can deduce the result once the circular direction is modded by Z 2 , and the modes are imposed to have nodes on the hyperplanes. From Eq. (2.45), we can evaluate λ tw,B+F in the limit R 9 → 0, which yields an infinite result (as in Eq. (2.22) for λ tw ),

λ tw,B+F = (-1) ξ (2π) d +∞ 0 dτ 2 2τ 1+ d-1 2 2
.

(2.51) Therefore, applying to V o B+F the second of the prescriptions given in Eq. (2.29), we obtain

πR d-1 V o B+F -(πR d-1 V o B+F ) ∞ = -(-1) ξ πv d R d-1 d-1 , (2.52)
which yields the force per unit area

F ≡ - ∂ R d-1 V o B+F -(R d-1 V o B+F ) ∞ ∂R d-1 = -(-1) ξ (d -1) v d R d d-1 , (2.53)
in agreement with Eq (2.31), for ξ = 0

Casimir effect in string theory

The discussion of the Casimir effect in string theory we will present below, follows the logic we have developed in field theory, in first quantized formalism. In this section, we consider the case of a theory where supersymmetry is spontaneously broken à la Scherk-Schwarz, while the analysis where supersymmetry is explicitly broken will be presented in Sect. 4.

Orbifold realization and Scherk-Schwarz mechanism

An appropriate framework for describing the case of a universe with one orbifolded direction S 1 (R d-1 )/Z 2 , where the modding acts as x d-1 → -x d-1 , while all degrees of freedom are imposed to the even under the transformation, is provided by the following background in heterotic string,

R 1,d-2 × S 1 (R d-1 ) × T 3 Z 2 × T 6-d × S 1 (R 9 ). (3.1)
In order for the vacuum energy not to be trivial, we implement a coordinate-dependent compactification along S 1 (R 9 ), which is nothing but a stringy version [START_REF] Rohm | Spontaneous supersymmetry breaking in supersymmetric string theories[END_REF][START_REF] Kounnas | Spontaneous supersymmetry breaking in string theory[END_REF][START_REF] Ferrara | Superstring solutions with spontaneously broken four-dimensional supersymmetry[END_REF][START_REF] Ferrara | Superstrings with spontaneously broken supersymmetry and their effective theories[END_REF][START_REF] Kounnas | Coordinate-dependent compactifications and discrete symmetries[END_REF] of the Scherk-Schwarz mechanism [START_REF] Scherk | Spontaneous breaking of supersymmetry through dimensional reduction[END_REF][START_REF] Scherk | How to get masses from extra dimensions[END_REF] responsible for the total spontaneous breaking of supersymmetry.

In string theory, the Z 2 orbifold generator G acts as a twist on S 1 (R d-1 ) × T 3 , namely

G : X d-1 , X d , X d+1 , X d+2 -→ -X d-1 , X d , X d+1 , X d+2 , (3.2)
where the X's are the worldsheet bosonic fields realizing spatial directions. Unless specified, all formulas to come are valid for arbitrary radii R d-1 , R 9 , and arbitrary T 3 × T 6-d torii moduli, provided

R 9 > R H = √ 2 + 1 √ 2 or R 9 < √ 2 -1 √ 2 = 1 2R H , (3.3)
for Hagedorn-like tachyonic instabilities not to occur. In practice, the directions of T 3 × T 6-d × S 1 (R 9 ) should be seen as internal, while R 1,d-2 × S 1 (R d-1 ) should be interpreted as spacetime. However, due to the Z 2 -twist, the 10-dimensional background cannot be seen as a Cartesian product of internal and external spaces. The (d -1)-dimensional external space is anisotropic, being restricted to sit between the hyperplanes invariant under G, i.e.

located at X d-1 = 0 and X d-1 = πR d-1 .
Putting the non-compact directions of R 1,d-2 in a "box", the regularized volume of spacetime is

V half 0,...,d-1 = (2πR 0 ) • • • (2πR d-2 ) × πR d-1 . (3.4)
Following the prescription as shown in the first line of Eq. (2.44), the one-loop effective potential V e can be expressed in terms of the partition function in light-cone gauge, integrated over the fundamental domain F of the modular group,

-V half 0,...,d-1 V e = F dτ 1 dτ 2 2τ 2 R 0 R 1 τ 2 R 2 • • • R d-2 ( √ τ 2 η η) d-3 1 2 H,G Z (4,4) H G Γ (6-d,6-d) (η η) 6-d R 9 √ τ 2 η η n 9 , m9 e - πR 2 9 τ 2 | m9 +n 9 τ | 2 1 2 a,b (-1) a+b+ab θ[ a b ] 2 θ a+H b+G θ a-H b-G η 4 1 2 γ,δ θ[ γ δ ] 6 θ γ+H δ+G θ γ-H δ-G η8 1 2 γ ,δ θ γ δ 8 η8
(-1) m9 a+n 9 b+ m9 n 9 (-1) ξ( m9 γ+n 9 δ+ m9 n 9 ) (-1) ξ ( m9 γ +n 9 δ + m9 n 9 ) . (3.5)

Our notations are as follows [START_REF] Kiritsis | String theory in a nutshell[END_REF]:

• τ = τ 1 + iτ 2
is the Teichmüller parameter of the genus-1 Riemann surface, while η(τ ) and θ[ α β ](τ ) are the Dedekind and Jacobi modular forms.

• The spin structures a, b on the worldsheet, as well as the indices γ, δ and γ , δ take values 0 or 1.

• The lattice of zero modes associated to S 1 (R 9 ) is presented in Lagrangian form, with n 9 , m9 ∈ Z, while that for a torus T n is denoted Γ (n,n) .

• The conformal block associated to the twisted coordinates of (S

1 (R d-1 ) × T 3 )/Z 2 is Z 4,4 H G =            R d-1 √ τ 2 n d-1 , md-1 e - πR 2 d-1 τ 2 | md-1 +n d-1 τ | Γ 3,3 η 4 η4 if (H, G) = (0, 0), 2 4 η 2 η2 θ 1-H 1-G 2 θ 1-H 1-G 2 if (H, G) = (0, 0), (3.6)
where H, G take values 0 or 1. In particular, string theory contains a twisted sector H = 1 which has no counterpart in field theory. In the sector (H, G) = (0, 0), the lattice of zero modes associated to S 1 (R d-1 ) is presented in Lagrangian form, with n d-1 , md-1 ∈ Z.

• In the last line of Eq. (3.5), the first sign depends on the spin structures a, b and induces the super-Higgs mechanism [START_REF] Kounnas | Coordinate-dependent compactifications and discrete symmetries[END_REF]. The mass M of the gravitini is as given in Eq. (2.38), which is the scale of spontaneous supersymmetry breaking. In the second and third signs, we have introduced discrete parameters ξ and ξ equal to 0 or 1, which implement a Higgs mechanism of the E 8 ×E 8 gauge symmetry when they are not both vanishing. Choosing ξ = 1 (ξ = 1) enforces the E 8 → SO( 16) spontaneous breaking of the first (second) E 8 gauge group factor. Together, super-Higgs and Higgs mechanisms combine to yield 4 different patterns of breakings: For ξ = ξ = 0, all initially massless fermions of the parent supersymmetric model (obtained when the sign (-1) m9 a+n 9 b+ m9 n 9 is omitted) acquire a mass 1/(2R 9 ), while their bosonic superpartners remain massless. However, a non-trivial ξ and/or ξ results in reversing the roles of bosons and fermions in supermultiplets, thus giving a mass to some bosons while maintaining their fermionic partners massless [START_REF] Catelin-Jullien | Thermal/quantum effects and induced superstring cosmologies[END_REF][START_REF] Kounnas | Super no-scale models in string theory[END_REF][START_REF] Kounnas | N = 2 → 0 super no-scale models and moduli quantum stability[END_REF].

By noticing that in V e , the sectors (H, G) = (0, 0) are inversely proportional to R d-1 , we may split the potential energy density as

V e = V(R d-1 ) + λ tw R d-1 . (3.7)
The first contribution,

V = - R 9 2(2π) d F dτ 1 dτ 2 τ 1+ d+1 2 2 n d-1 , md-1 e - πR 2 d-1 τ 2 | md-1 +n d-1 τ | n 9 , m9 e - πR 2 9 τ 2 | m9 +n 9 τ | 2 1 2 a,b (-1) a+b+ab θ[ a b ] 4 (-1) m9 a+n 9 b+ m9 n 9 (3.8) 1 2 γ,δ θ γ δ 8 (-1) ξ( m9 γ+n 9 δ+ m9 n 9 ) 1 2 γ ,δ θ γ δ 8 (-1) ξ ( m9 γ +n 9 δ + m9 n 9 ) Γ (3,3) Γ (6-d,6-d) η 12 η24
, is actually the effective potential arising in the model when no orbifold twist is implemented.

Its expression is that found in the d-dimensional background periodic along

x d-1 , R 1,d-2 × S 1 (R d-1 ) × T 3 × T 6-d × S 1 (R 9 ), (3.9)
with Scherk-Schwarz mechanism implemented along S 1 (R 9 ). The second term in Eq. (3.7) is given by

λ tw = - R 9 2(2π) d F dτ 1 dτ 2 τ 1+ d 2 2 1 2 a,b (-1) a+b+ab n 9 , m9 e - πR 2 9 τ 2 | m9 +n 9 τ | 2 (-1) m9 a+n 9 b+ m9 n 9 1 2 γ,δ (-1) ξ( m9 γ+n 9 δ+ m9 n 9 ) 1 2 γ ,δ (-1) ξ ( m9 γ +n 9 δ + m9 n 9 ) Γ (6-d,6-d) (H,G) =(0,0) ω H,a,γ,γ G, b, δ, δ , (3.10)
where we have defined

ω H,a,γ,γ G, b, δ, δ = 1 η 8 η20 2 4 η 2 η2 θ 1-H 1-G 2 θ 1-H 1-G 2 θ[ a b ] 2 θ a+H b+G θ a-H b-G θ[ γ δ ] 6 θ γ+H δ+G θ γ-H δ-G θ γ δ 8 . (3.11)
To describe the case of a universe with one orbifolded direction S 1 (R d-1 )/Z 2 , while all degrees of freedom are imposed to the odd under the Z 2 transformation, we may consider the string theory orbifold model based on the orthogonal projector, as compared to the above described case,

1 -G 2 = 1 - 1 + G 2 , (3.12)
exactly as developed in the quantum field theory framework, Eq. (2.44). To write the associated effective potential V o , notice that the flipped sign in front of G implies the sector (H, G) = (0, 1) to appear with an opposite sign, as compared to the case discussed so far.

Given that the sectors (H, G) = (0, 0) realize a modular orbit of SL(2, Z), we conclude that all of them show up with an opposite sign. As a result, we obtain

V o = V(R d-1 ) - λ tw R d-1 , (3.13)
which can also be seen as a direct consequence of the equality in Eq. (3.12).

The remaining contribution (H, G) = (0, 0) being modular invariant by itself, we may "unfold" in the expression of V the domain of integration F into the "upper-half strip", by using the following result [START_REF] Mcclain | Modular invariance for interacting bosonic strings at finite temperature[END_REF][START_REF] O'brien | Modular invariance of thermopartition function and global phase structure of heterotic string[END_REF]: For any set of functions v (n, m) (τ, τ ) satisfying

∀M = a b c d ∈ SL(2, Z), v (n, m) aτ + b cτ + d , aτ + b cτ + d = v (n, m)M (τ, τ ), (3.14)
the sum n, m v (n, m) is modular invariant and, provided the discrete series are absolutely convergent, for the exchange of the discrete sum and integration to be legitimate, we have

F dτ 1 dτ 2 τ 2 2 n, m v (n, m) (τ, τ ) = F d 2 τ τ 2 2 v (0,0) (τ, τ ) + 1 2 -1 2 dτ 1 +∞ 0 dτ 2 τ 2 2 m =0 v (0, m) (τ, τ ). (3.15)
In our case at hand, this equality turns out to be valid for the series of functions labeled by (n d-1 , md-1 ) when R d-1 > 1, which we know is satisfied since x d-1 is an external space direction. As result, we obtain

V o = V 0 + V * (R d-1 ) - λ tw R d-1 , (3.16)
where the first contribution is independent of R d-1 ,

V 0 = - R 9 2(2π) d F dτ 1 dτ 2 τ 1+ d+1 2 2 n 9 , m9 e - πR 2 9 τ 2 | m9 +n 9 τ | 2 1 2 a,b (-1) a+b+ab θ[ a b ] 4 (-1) m9 a+n 9 b+ m9 n 9 1 2 γ,δ θ γ δ 8 (-1) ξ( m9 γ+n 9 δ+ m9 n 9 ) 1 2 γ ,δ θ γ δ 8 (-1) ξ ( m9 γ +n 9 δ + m9 n 9 ) Γ (3,3) Γ (6-d,6-d) η 12 η24 , (3.17)
whereas the second depends non-trivially on R d-1 ,

V * = - R 9 2(2π) d 1 2 -1 2 dτ 1 +∞ 0 dτ 2 τ 1+ d+1 2 2 md-1 =0 e - πR 2 d-1 τ 2 m2 d-1 n 9 , m9 e - πR 2 9 τ 2 | m9 +n 9 τ | 2 1 2 a,b (-1) a+b+ab θ[ a b ] 4 (-1) m9 a+n 9 b+ m9 n 9 (3.18) 1 2 γ,δ θ γ δ 8 (-1) ξ( m9 γ+n 9 δ+ m9 n 9 ) 1 2 γ ,δ θ γ δ 8 (-1) ξ ( m9 γ +n 9 δ + m9 n 9 ) Γ (3,3) Γ (6-d,6-d) η 12 η24 .
Due to the integration over τ 1 , V * involves only the level-matched spectrum of the string.

Notice that it is also finite, due to the presence of the factor exp(-πR

2 d-1 m2 d-1 /τ 2 )
, which allows the integration even in the dangerous UV region τ 2 → 0.

As a result, the force per unit area arising from the vacuum comprised in the range 0 < x d-1 < πR d-1 and acting on the boundary hyperplane located at

x d-1 = πR d-1 is F int = - ∂(R d-1 V o ) ∂R d-1 = -V 0 - ∂(R d-1 V * ) ∂R d-1 . (3.19)
This is the end of the story if we want to truly interpret the hyperplanes as the boundaries of the universe. However, applying the result to describe the Casimir effect, we need to add the force per unit area arising from the infinite half space x d-1 > πR d-1 . Up to a sign, it is equal to F int evaluated in the limit R d-1 → +∞, as the second plate is sent to infinity.

Since V * vanishes in this limit, we have

F ext = V 0 , (3.20)
and the Casimir force per unit area is

F ≡ F int + F ext = - ∂(R d-1 V * ) ∂R d-1 . (3.21)
It is independent of the twisted sector, H = 1, of the string spectrum. This is not a surprise since the latter is restricted to live on the hyperplanes fixed by the orbifold projection.

Having no modes propagating in the transverse direction x d-1 , they cannot contribute to the force.

Comparing the result with that found for a single degree of freedom in field theory, Eqs (2.32) or (2.53), we see that in the string theory framework, the Casimir force is not derived by subtracting two infinite quantities to end up with a finite result. Actually, both

V * and V 0 are finite, where V 0 is the analogue of what was denoted V o ∞ (or V o B+F,∞
) in field theory. However, one may think that the finiteness of V 0 arises only because we considered a string theory model where supersymmetry is spontaneously broken à la Scherk-Schwarz, since we have shown in Eq. (2.41) that finiteness is also encountered in field theory for each pair of non-degenerate superpartners with gap M between their masses. To clarify this issue, we now compare in details the two frameworks when they both realize a spontaneous breaking of supersymmetry, and will show in Sect. 4 that finiteness survives in string theory even when supersymmetry is explicitly broken.

String theory versus quantum field theory

Our aim is to figure out the differences between the finite field theory result V B+F given in Eq. (2.41), and that found in string theory, V 0 + V * , which are both associated with configurations where supersymmetry breaking is spontaneously broken via a Scherk-Schwarz mechanism. To be specific, for the notion of spontaneous breaking to be valid in the string theory model, let us assume in this subsection R 9 > R H in Eq. (3.3), in order for R 9 to be allowed to take very large values and possibly restore supersymmetry (in d + 1 dimensions). the series of functions labeled by (n 9 , m9 ). This can be done since absolute convergence is valid for R 9 > R H . The contribution for (n 9 , m9 ) = (0, 0), which is integrated over the fundamental domain F , vanishes due to the exact supersymmetry of the integrand. For the same reason, the contributions (n 9 , m9 ) = (0, 2 k9 ), k9 ∈ Z * , which are integrated over the upper-half strip, vanish as well. Thus, we obtain

V 0 = - R 9 2(2π) d 1 2 -1 2 dτ 1 +∞ 0 dτ 2 τ 1+ d+1 2 2 k9 e - πR 2 9 τ 2 (2 k9 +1) 2 1 2 a,b (-1) a+b+ab θ[ a b ] 4 (-1) a 1 2 γ,δ θ γ δ 8 (-1) ξγ 1 2 γ ,δ θ γ δ 8 (-1) ξ γ Γ (3,3) Γ (6-d,6-d) η 12 η24 , (3.22)
which reduces to an expression involving the level-matched spectrum only, due to the integration over τ 1 .

The field theory we have considered in Sect. 2.2 involves one massless boson (fermion)

and one fermion (boson) of mass M , together with their KK towers of states propagating along the internal circle S 1 (R 9 ). However, beside such KK towers, the physical spectrum contributing to V 0 also contains states that cannot be organized as pure KK towers of modes associated to S 1 (R 9 ). The latter involve momentum and/or winding modes along all internal directions and/or from string oscillators. To make contact with the field theory result, we thus restrict from now on to the case where M is much lower than all other mass scales present in the model. In particular, we have R 9 1, for the string scale to be much heavier than M . In practice, the lightest states of the model are therefore nothing but the KK modes propagating along S 1 (R 9 ) and they will dominate in the expression of V 0 . To see this, we expand 1 2

a,b

(-1) a+b+ab θ[ a b ] 4 η 12 (-1) a = θ 1 0 4 η 12 = 16 1 + O(q) , 1 2 γ,δ θ γ δ 8 (-1) ξγ = 1 + 112q + (-1) ξ 128q + O(q 2 ), 1 η24 = 1 q 1 + 24q + O(q 2 ) , Γ (3,3) Γ (6-d,6-d) = 1 + d en q + O(e -πc 2 τ 2 ), (3.23)
where q = e 2iπτ . In the last line, we take into account the possibility that the internal metric and antisymmetric tensor appearing in the Γ [START_REF] Lamoreaux | Demonstration of the Casimir force in the 0.6 to 6 micrometers range[END_REF][START_REF] Lamoreaux | Demonstration of the Casimir force in the 0.6 to 6 micrometers range[END_REF] and Γ (6-d,6-d) lattices sit at an enhanced symmetry point in moduli space, thus yielding d en additional massless vector multiplets in the supersymmetric parent theory. Moreover, a moduli-dependent constant c is introduced to account for the fact that the mass scales introduced by the lattices are heavier than the supersymmetry breaking scale, i.e. that 1 > c 1/R 9 . By using the following result, valid for K, M > 0,

+∞ 0 dτ 2 τ 1+ d+1 2 2 e -π τ 2 K 2 e -πτ 2 M 2 ∼ M d 2 K 1+ d 2 e -2πKM when KM 1, (3.24)
we obtain Notice that the dressing coefficient ξ d in Eq. (3.25) reflects the fact that their entire KK towers of states associated to the large circle S 1 (R 9 ) contribute significantly. On the contrary, all other string states yield exponentially suppressed contributions to V 0 . Note that the sign of V 0 can be positive or negative: (ξ, ξ ) = (1, 1) implies n F -n B < 0, while (ξ, ξ ) = (1, 1),

V 0 = (n F -n B ) ξ d R d 9 + O c R 9 d 2 e -2πcR
d en ≤ 8 yields n F -n B ≥ 0.
The expression of V * in Eq. (3.18) can also be simplified. Applying a Poisson resummation over m9 , the lattice of zero modes associated to the Scherk-Schwarz circle can be written in Hamiltonian form, ellipses are positive, one concludes that since R 9 1, the contributions for n 9 = 0 are exponentially suppressed, as compared to those arising for n 9 = 0. In that case, we may write

R 9 √ τ 2 m9 e - πR 2 9 τ 2 | m9 +n 9 τ | 2 (-1) m9 [a+ξγ+ξ γ +n 9 (1+ξγ+ξ γ )] = m 9 q 1 4 M 9 R 9 +n 9 R 9 2 q 1 4 M 9 R 9 -n 9 R 9 2 , where M 9 = m 9 + 1 2 a + ξγ + ξ γ + n 9 (1 + ξ + ξ ) . ( 3
V * = - R 9 2(2π) d 1 2 -1 2 dτ 1 +∞ 0 dτ 2 τ 1+ d+1 2 2 md-1 =0 e - πR 2 d-1 τ 2 m2 d-1 k9 e - πR 2 9 τ 2 (2 k9 +1) 2 1 2 a,b (-1) a+b+ab θ[ a b ] 4 (-1) a 1 2 γ,δ θ γ δ 8 (-1) ξγ 1 2 γ ,δ θ γ δ 8 (-1) ξ γ Γ (3,3) Γ (6-d,6-d) η 12 η24 + O e -2πR d-1 R 9 , (3.28)
where we have used the fact that only odd values of m9 = 2 k9 + 1 contribute, due to supersymmetry. In fact, V 0 given in Eq. (3.22) is nothing but the "missing" md-1 = 0 term of the above dominant contribution of V * . By proceeding exactly as we did for evaluating V 0 , we find

V * = (n F -n B ) f * R d-1 R 9 1 R d 9 + O c R 9 d 2 e -2πc √ R 2 9 +R 2 d-1 , (3.29)
where we have defined

f * (u) = Γ d+1 2 2 d π 3d+1 2 md-1 =0, k9 1
(2 k9 + 1) 2 + m2 d-1 u 2 d+1 2 . (3.30)
Combining Eqs (3.25) and (3.29), the final expression in string theory of the energy density in a universe circular along x d-1 turns out to be in agreement with the field theory analysis, when supersymmetry is spontaneously broken by the Scherk-Schwarz mechanism at a low scale. The finite result is,3

V 0 + V * = (n F -n B ) f R d-1 R 9 1 R d 9 + O c R 9 d 2 e -2πc √ R 2 9 +R 2 d-1 , (3.31)
which matches with Eq. (2.41) for n B and n F massless bosonic and fermionic degrees of freedom and their KK towers of modes, up to exponentially suppressed contributions arising from heavy string states not present in field theory. Hence, the Casimir force can be derived in both frameworks from Eqs. (3.21) and (3.29).

Casimir effect and hard breaking of supersymmetry

So far, we have analyzed the Casimir effect in the case of a spontaneous breaking of supersymmetry. We observed that in string and field theory, the force acting on the hyperplane located at x d-1 = 0 and generated by the vacuum energy between the two plates is finite, and so is the force induced by the vacuum in the infinite space x d-1 < 0. In the limit of low supersymmetry breaking scale, as compared to the other scales arising in the string model (Higgs-like or string scale), the two frameworks yield identical results. Therefore, string theory is of limited interest in this case, since the field theory approach is technically simpler. However, the situation may be drastically different in models where supersymmetry is explicitly broken, due to the illness of the definition of the energy density in field theory in these conditions. As seen in Eq. (2.29) (or in Eqs (2.30) and (2.50) for the massless case) a prescription consisting in subtracting the energies associated to two configurations, though individually divergent, is required to make sense.

To analyse the string theory picture when supersymmetry breaking is hard, we note that Eq. (3.16) is valid for both ranges defined in Eq. (3.3), which are related by T-duality R 9 → 1/(2R 9 ). Thus, instead of considering the model of the previous section at large R 9 (and even recovering exact supersymmetry in d + 1 dimensions as R 9 → +∞), we may consider the opposite limit, R 9 → 0, where the scale of supersymmetry breaking and actually the mass of the gravitinos are sent to infinity. The gravitini being decoupled from the rest of the spectrum, explicit and consistent models in d + 1 spacetime dimensions with hard breaking of supersymmetry may be found this way, provided no tachyons are generated in taking the limit [START_REF] Alvarez-Gaumé | An O(16) × O(16) heterotic string[END_REF].

In the expressions of V 0 , V * and λ tw given in Eqs (3.17 (-1) m9 a+n 9 b+ m9 n 9 (-1) ξ( m9 γ+n 9 δ+ m9 n 9 ) (-1) ξ ( m9 γ +n 9 δ + m9 n

9 ) = 1 2 h,g 2R 9 l 9 , k9 e - π(2R 9 ) 2 τ 2 | k9 + g 2 +(l 9 + h 2 )τ | 2 (-1) g(a+ξγ+ξ γ ) (-1) h(b+ξδ+ξ δ ) (-1) gh(1+ξ+ξ ) (4.1) = 1 2R 9 k 9 , l9 e - π (2R 9 ) 2 τ 2 | l9 +k 9 τ | 2 1 2 h,g
(-1) gk 9 +h l9 (-1) g(a+ξγ+ξ γ ) (-1) h(b+ξδ+ξ δ ) (-1) gh(1+ξ+ξ ) .

In the limit R 9 → 0, the only surviving contribution is for (k 9 , l9 ) = (0, 0), which yield

1 2R 9 h δ a+ξγ+ξ γ +h(1+ξ+ξ ),0 mod 2 (-1) h(b+ξδ+ξ δ ) ≡ 1 2R 9 Φ ξξ a,γ,γ b, δ, δ , (4.2)
where Φ ξξ satisfies Φ 10 a,γ,γ b, δ, δ = 2δ aγ δ bδ , Φ ξξ a,γ,γ b, δ, δ = (-1) ab (-1) ξ[a(δ+δ )+b(γ+γ )+(δ+δ )(γ+γ )] .

(4.3) Hence, the total energy in the universe is better understood from a d + 1-dimensional point of view, since

d d x V o ∼ R 9 →0 d d x 2π 1 2R 9 Vo ≡ d d+1 x Vo , (4.4)
where we have defined

Vo = V0 + V * (R d-1 ) - λtw R d-1 . (4.5)
In our notations, the first contribution in Vo is independent of R d-1 , while the second does not, Notice that without Φ ξξ included, the above conformal block appears in the partition function of the parent supersymmetric E 8 × E 8 heterotic string. Finally, the third contribution

V0 = - 1 2(2π) d+1 F dτ 1 dτ 2 τ 1+ d+1 2 2 Γ (3,3) Γ (6-d,6-d) 1 η 8 η8 Ẑ, V * = - 1 2(2π) d+1 1 2 -1 2 dτ 1 +∞ 0 dτ 2 τ 1+ d+1 2 2 md-1 =0 e - πR 2 d-1 τ 2 m2 d-1 Γ (3,3) Γ (6-d,6-d) 1 η 8 η8 Ẑ,
involves λtw = - 1 2(2π) d+1 F dτ 1 dτ 2 τ 1+ d 2 2 Γ (6-d,6-d) 1 2 a,b (-1) a+b+ab 1 2 γ,δ 1 2 γ ,δ (H,G) =(0,0) ω H,a,γ,γ G, b, δ, δ Φ ξξ a,γ,γ b, δ, δ . (4.8)
To describe the particle content of the conformal block Ẑ which appears in V0 + V * , we introduce the O(2n) affine characters,

O 2n = θ 0 0 n + θ 0 1 n 2η n , V 2n = θ 0 0 n -θ 0 1 n 2η n , S 2n = θ 1 0 n + (-i) n θ 1 1 n 2η n , C 2n = θ 1 0 n -(-i) n θ 1 1 n 2η n . ( 4.9)
• For ξ = ξ = 0, the only effect of Φ 00 is to reverse spacetime chirality, S 8 ↔ C 8 , so that

Ẑ = (V 8 -C 8 )( Ō16 + S16 )( Ō 16 + S 16 ), (4.10)
where "primed" characters refer to those arising from the sums over γ , δ . In fact, this shows that the model based on background (3.1) is supersymmetric in both T-dual decompactifications limits, R 9 → +∞ and 1/(2R 9 ) → +∞, with E 8 × E 8 gauge symmetry. While the light gravitinos present at large R 9 become infinitely massive and decouple in the 2R 9 → 0 limit, other gravitinos arising from states winding S 1 (R 9 ) are becoming light. As a result, this case does not yield the hard breaking of supersymmetry we are looking for.

• For ξ = 1, ξ = 0, the presence of Φ

η4 = 1 q 1 2 q 1 2 + • • • . (4.12)
As a result, the model does not define a true vacuum, and the integral form of the energy density V0 is divergent, which does not meet our objectives.

• For ξ = ξ = 1, supersymmetry is broken and the gauge symmetry is SO( It turns out that V0 is of order 1 and positive when so are the internal moduli (c = O(1) in Eq. (3.23)) [START_REF] Alvarez-Gaumé | An O(16) × O(16) heterotic string[END_REF][START_REF] Kounnas | N = 2 → 0 super no-scale models and moduli quantum stability[END_REF]. Moreover, expanding in q, q the integrand of V * , we find that when the compact (though external) direction x d-1 is large, namely c 1/R d-1 , the contributions of the KK towers of modes propagating along x d-1 dominate over all other string states, which are heavier. To be specific, we find

V * = (n F -n B ) v d+1 R d+1 d-1 + O c d 2 R d 2 +1 d-1
e -2πcR d-1 , where n F -n B = 8(264 -d en ), (4.14) and where v d was defined in field theory in Eq. (2.49). In the above expressions, n F , n B denote the numbers of massless fermionic and bosonic degrees of freedom, while v d+1 captures the contributions of their KK towers of states propagating in the (Z 2 -modded) spacetime direction x d-1 .

Notice that V = V0 + V * reproduces the field theory answer, which is written in Eq. (2.26) for a single bosonic degree of freedom, or in Eq. (2.49) for a single bosonic or fermionic degree of freedom, but with three differences:

• The string theory answer contains exponentially suppressed contributions arising from the heavy string modes which are not included in field theory.

• The string model with hard breaking we have constructed is in d + 1 dimensions, rather than d dimensions in field theory. Actually, starting from a d-dimensional model and then sending to zero the radius R 9 of an internal direction, we end up in string theory with d + 1 spacetime coordinates because of the reconstruction of an external direction from the towers of winding modes becoming massless. On the contrary, taking R 9 → 0 in field theory makes completely disappear an internal direction already invisible at small but finite R 9 .

• This is the analogue of the field theory result, Eqs (2.31), (2.32), where the forces per unit area acting on either side of the plate were infinite, albeit with finite sum. Therefore, string theory provides a more rigorous derivation of the resulting Casimir force when supersymmetry is not present at all (explicit breaking), to the extent that the forces on either side of each plate are individually finite.

Conclusion

We have shown that at weak coupling, string theory and quantum field theory yield similar results for the net force exerted on parallel plates on which all degrees of freedom in the bulk are imposed vanishing boundary conditions (Dirichlet boundary conditions). To be specific, the answers are equal, up to exponentially suppressed corrections arising in the string computation from the heavy string states, which are absent in field theory. The net force can be derived by subtracting the energies associated with the vacuum comprised between the two plates, when they are at finite or infinite distance.

In configurations where supersymmetry is spontaneously broken à la Scherk-Schwarz, both frameworks yield a finite energy density between the two plates, whatever is their distance. There is no need to subtract energies associated with two configurations to obtain consistent results. Moreover, if this fact turns out to remain true in string theory when supersymmetry is explicitly broken, it is not valid anymore in quantum field theory. Therefore, it is in this case that the infinite spectrum of the string, with unbounded masses, plays a major role, as it provides the necessary UV regularization responsible for an absolute energy to make sense. Thus, it is in this case that string theory provides a more robust derivation of the Casimir effect, as compared to the usual quantum field theory analysis.

We have derived these results by using a non-standard orbifold action. Whereas Z 2 twists of four internal directions are very common in string theory for reducing the number of supersymmetries, we have considered a Z 2 generator G that twists one external compact direction and three internal coordinates. Moreover, for the two hyperplanes of the external space fixed by G to behave as "conductive plates", we have implemented in a way consistent with modular invariance the projection (1 -G)/2 on the spectrum running in the vacuumto-vacuum amplitude at one-loop, which is orthogonal to the usual projector.

  .50) Note that v d encodes the contributions of the KK excitations along x d-1 . Since the discrete sum appearing in v d is 2ζ(d), the result (2.49) matches with Eq. (2.26), where the interpretation of v d was obscure.

  .27) Applying Eq. (3.24) for K = R d-1 | md-1 | and any mass M = (n 9 R 9 ) 2 + • • •, where the

9 n 9 , m9 e - πR 2 9 τ 2 |

 992 ),(3.18) and (3.10), let us redefine m9 ≡ 2 k9 + g and n 9 ≡ 2l 9 + h, where g, h ∈ {0, 1}, and perform Poisson resummations over k9 and l 9 . Including the signs responsible for super-Higgs and the Higgs mechanisms, the relevant terms are R m9 +n 9 τ | 2

  V0 replaces (in d + 1 dimensions) the divergent integral over the Euclidean momentum in Eq. (2.26) (derived in second quantized formalism), or equivalently the term proportional to ξ d /R d 9 which diverges when R 9 → 0 in Eq. (2.49) (derived in first quantized formalism). In the Casimir effect, as shown in Eqs (3.19) and (3.20), the forces per unit area exerted on the plate at x d-1 = πR d-1 from the vacuum located on each sides are Fint = -∂(R d-1 Vo )∂R d-1 = -V0 -∂(R d-1 V * ) ∂R d-1 , Fext = V0 ,(4.15)and they add to give the final answerF ≡ Fint + Fext = (n F -n B )

  G|m 0 , . . . , m d-2 , 0 = |m 0 , . . . , m d-2 , 0 , G|m 0 , . . . , m d-2 , m d-1 ; ± = ±|m 0 , . . . , m d-2 , m d-1 ; ± ,

	Because this new basis diagonalizes G,
				(2.12)
	we may translate Eq. (2.9) into
		ln Z B = ln Z e B + ln Z o B ,
	the orthonormal basis of "bra" |m ≡ |m 0 , . . . , m d-1 , m ∈ Z d , which
	satisfy
	G|m 0 , . . . , m d-1 = |m 0 , . . . , -m d-1 .	(2.10)
	In these notations, it is convenient to define a second orthonormal basis, where m ∈ Z d but
	with m d-1 ≥ 1:
	|m 0 , . . . , m d-2 , 0 ,
	|m 0 , . . . , m d-2 , m d-1 ; ± ≡	1 √ 2	|m 0 , . . . , m d-2 , m d-1 ± |m 0 , . . . , m d-2 , -m d-1 .	(2.11)

  9 ,(3.25) where ξ d is defined in Eq. (2.48). In the above expression, n F and n B are the numbers massless fermionic and bosonic degrees of freedom, and they satisfy n F -n B = 8 (-1) ξ+1 128 + (-1) ξ +1 128 -248 -d en .

	(3.26)

  [START_REF] Rohm | Spontaneous supersymmetry breaking in supersymmetric string theories[END_REF] breaks supersymmetry and preserves an SO(16) × E 8

	gauge symmetry,
	Ẑ = (O 8 V16 + V 8 Ō16 -S 8 S16 -C 8 C16 )( Ō 16 + S 16 ).	(4.11)
	However, it also has a dramatic consequence, due to the following sector which contains a
	level-matched tachyon,	O 8	V16	Ō 16
		η 8	η4

We may choose to integrate only over 0 ≤ x d-1 ≤ πR d-1 . This is a matter of convention, since this may be compensated by a rescaling of the fields and path integral measure.

Such expressions also appear in the context of cosmological solutions[START_REF] Catelin-Jullien | Thermal/quantum effects and induced superstring cosmologies[END_REF][START_REF] Catelin-Jullien | Induced superstring cosmologies and moduli stabilization[END_REF][START_REF] Bourliot | Attraction to a radiation-like era in early superstring cosmologies[END_REF] or in models with vanishing effective potential at one-loop[START_REF] Kounnas | Super no-scale models in string theory[END_REF][START_REF] Kounnas | N = 2 → 0 super no-scale models and moduli quantum stability[END_REF].

Acknowledgement

We are grateful to Guillaume Bossard for fruitful discussions. A.K. is supported by the GSRT under the EDEIL/67108600 and by CNRS. The work of H.P. is partially supported by the Royal Society International Cost Share Award. A.K. would like to thank the CPHT of Ecole Polytechnique for hospitality.


Hervé Partouche
email: herve.partouche@polytechnique.edu

S Abel

E Dudas

D Lewis

Quantum stability in open string theory with broken supersymmetry 1

We consider the 1-loop effective potential in type I string theory compactified on a torus, with supersymmetry broken by the Scherk-Schwarz mechanism. At fixed supersymmetry breaking scale M , and up to exponentially suppressed terms, we show that the potential admits local minima of arbitrary sign, in dimension d ≤ 5. While the open string Wilson lines are massive, the closed string moduli are flat directions. In a T-dual picture, the relevant backgrounds involve isolated 1 2 -branes, whose positions are frozen on orientifold planes, thus decreasing the rank of the gauge group, and introducing massless fermions in fundamental representations.

Introduction

In string theory, when supersymmetry is spontaneously broken in flat space at a scale M moderately smaller than the string scale M s , the effective potential simplifies greatly and the questions of its sign, magnitude and stability at extrema can be addressed is a systematic way. In this work, we focus on the type IIB orientifold theory compactified on a torus T 10-d of metric G IJ , with supersymmetry spontaneously broken à la Scherk-Schwarz [START_REF] Blum | Strong / weak coupling duality relations for nonsupersymmetric string theories[END_REF] along the internal direction X 9 . In this case, M = M s √ G 99 M s , where G IJ = G -1 IJ . Note that because of the underlying extended supersymmetry (with 16 supercharges), all moduli fields can be interpreted as Wilson lines (WLs), but we find convenient to use this denomination for all moduli fields except the so-called no-scale modulus M [START_REF] Cremmer | Naturally vanishing cosmological constant in N = 1 supergravity[END_REF].

The dominant contribution of the 1-loop effective potential V arises from the lightest states of the spectrum. Assuming that the background does not contain mass scales between 0 and M , the latter are n F and n B massless fermionic and bosonic degrees of freedom, with their Kaluza-Klein (KK) states propagating along the large direction X 9 . The result takes the form

V (n F -n B ) ξ d M d , (1.1)
where ξ d > 0 captures the contributions of the KK modes [START_REF] Abel | Towards a non-supersymmetric string phenomenology[END_REF]. At such a point in moduli space, the potential is also critical with respect to the WLs, as follows from the enhancement of the massless spectrum (no Higgs-like scale between 0 and M ). However, even if all tadpoles vanish (except for M when n F = n B ), the stability of the background is guaranteed only if the WLs are non-tachyonic, when V is expanded at quadratic order in small marginal deformations. The point is that the higher V is (due to the presence of massless fermions), the more unstable the background is, as follows from dangerous contributions arising from the massless fermions charged under the gauge group i.e. coupled to the WLs. In the present work, we review the fact that stable backgrounds exist, while satisfying V ≥ 0 [START_REF] Abel | Stability and vacuum energy in open string models with broken supersymmetry[END_REF].

Note that such models may be relevant in several respects. Nearly vanishing potentials may be relevant for achieving the goal of describing a small (and positive) cosmological term.

Moreover, flat cosmological evolutions of models with positive potentials have been shown

to be attracted to a "Quantum No-Scale Regime" describing an expanding universe, where the no-scale structure, which is exact classically, is restored at the quantum level. On the contrary, when the potential can reach negative values, the universe eventually collapses, 1 unless the initial conditions are tuned in a tiny region of the phase space [START_REF] Coudarchet | Quantum no-scale regimes in string theory[END_REF][START_REF] Coudarchet | Quantum no-scale regimes and moduli dynamics[END_REF].

We describe the models in a geometric picture obtained by T-dualizing all internal directions. 1 2 -branes with positions frozen on orientifold planes play a crucial role, since (i) they do not yield marginal deformations that may be tachyonic, (ii) they decrease the dimension of the gauge group, thus lowering n B , (iii) and they yield massless fermionic strings stretched between them and other branes, due to the interplay between WL deformations and Scherk-Schwarz mechanism.

Wilson line stability in 9 dimensions

The main ideas to raise minima of the effective potential can be understood in 9 dimensions.

In type I theory, the gauge group arising from open strings is actually O(32) rather than SO(32). Hence, two disconnected open string WLs moduli spaces can be considered, which are disconnected to one another, because parametrized by WL matrices of determinant 1 or -1 [START_REF] Schwarz | Some properties of type I' string theory[END_REF], W = diag (e 2iπaα , α = 1, . . . , 32)

≡ diag (e 2iπa 1 , e -2iπa 1 , . . . , e 2iπa 16 , e -2iπa 16 ) or diag (e 2iπa 1 , e -2iπa 1 , . . . , e 2iπa 15 , e -2iπa 15 , 1, -1).

(2.1)

In both cases, the open strings having Chan-Paton charges at their ends, their KK momentum along S 1 (R 9 ) takes the form

m 9 + F 2 + a α -a β R 9 M s , (2.2)
where m 9 ∈ Z, F is the fermionic number and R 9 = √ G 99 1, where in our conventions all moduli fields are dimensionless. It is convenient to switch to the type I' picture obtained by T-dualizing R 9 → 1/R 9 = R9 . The internal space becomes S 1 ( R9 )/Z 2 , with two orientifold O8-planes located at the fixed points X9 = 0 and X9 = π R9 . In this framework, the WLs are the positions X9 = 2πa α of 32 1 2 -D8-branes. When two 1 2 -branes are located at 2πa α and -2πa α on the double cover, they are actually coincident in S 1 ( R9 )/Z 2 and give rise to a plain D8-brane, whose position a α is a modulus free to vary. On the contrary, when a single 1 2 -brane is not paired with a mirror object, a fact that can only happen when a α = 0 or 1 2 , it is frozen on one of the O8-planes and we are left with only 15 independent WL deformations, as indicated in the last line of Eq. (2.1). When there are p 1 1 2 -branes located 2 at a = 0, p 2 at a = 1 2 , and stacks of r σ 1 2 -branes at a σ ∈ )0, 1 2 ( with their mirrors at -a σ , the gauge symmetry is

U (1) G × U (1) C × SO(p 1 ) × SO(p 2 ) × σ U (r σ ), where U (1) G , U (1) C arise
from the dimensionally reduced metric G 9µ and antisymmetric tensor C 9µ . As announced in the introduction, Eq. ( 2.2) is telling us that Higgs and super Higgs mechanisms cancel each other for F = 1, a α = 1 2 , a β = 0 i.e. for fermionic states stretched between the stacks of p 1 and p 2 1 2 -branes, which are in the bifundamental representation of SO(p 1 ) × SO(p 2 ). Notice that a α = 1 4 = -a β also yields massless fermions. Hence, we expect Eq. ( 1.1) to be valid when all WLs belong to {0, 1 2 , ± 1 4 }, even if there is a priori no reason to believe that a = ± 1 4 corresponds in general to critical points of V.

The effective potential can be evaluated at 1-loop. When M M s , it takes the form [START_REF] Abel | Stability and vacuum energy in open string models with broken supersymmetry[END_REF] V = Γ( 5)

π 14 M 9 n 9 N 2n 9 +1 (W) (2n 9 + 1) 10 + O (M s M ) 9 2 e -π Ms M , (2.3)
where N 2n 9 +1 arises from contributions of the torus, Klein bottle, annulus and Möbius strip amplitudes,

N 2n 9 +1 (W) = 4 -16 -0 -(tr W 2n 9 +1 ) 2 + tr (W 2(2n 9 +1) ) = -16 N r,s=1 r =s cos 2π(2n 9 + 1)a r cos 2π(2n 9 + 1)a s + N -4 , (2.4)
and where N = 16 or 15 is the number of independent, dynamical WLs a r . When all WLs are in {0, 1 2 , ± 1 4 }, we find as expected N 2n 9 +1 = n F -n B , which yields Eq. (1.1). Moreover, the WLs in the neighborhood of a = ± 1 4 have vanishing tadpoles only when p 1 = p 2 i.e. when the configuration of 1 2 -branes is symmetric with respect to the transformation a → 1 2 -a. However, restricting to the tachyon free configurations, WLs at ± 1 4 are excluded and only two solutions are found, namely (p 1 , p 2 ) = (32, 0) and (31, 1) (up to the exchange p 1 ↔ p 2 ). It turns out that both SO(32) and SO(31)×SO( 1) configurations (where the inert SO(1) reminds the presence of an isolated frozen 1 2 -brane and fermions in the fundamental representation of SO(p 1 )) yield a negative minimum of V. However, we stress that the SO(31) × SO(1) solution has an energy slightly raised, as compared to the SO(32) one.

Wilson line stability and positive potential in d dimensions

By generalizing the above considerations to lower dimensions, the hope is that we may freeze more 1 2 -branes at orientifold planes and obtain a non-negative potential. The type IIB orientifold theory compactified on T 10-d can be analyzed in the "most geometrical picture" obtained by T-dualizing all of the internal directions. The internal space becomes a "(10-d)dimensional box" of metric GIJ = G IJ , with one orientifold O(d-1)-plane at each of its 2 10-d corners. Moreover, the initial D9-branes turn into 32 From the results in 9 dimensions, we expect configurations where all 1 2 -branes are coincident with O-planes to yield stable critical points of the potential, with respect to all WLs. Let us denote p A the number of 1 2 -branes sitting on the A-th O-plane. By convention, we label the corners so that the coordinates of the (2A -1)-th and 2A-th differ only along the supersymmetry breaking direction X9 . In such a background, the numbers of massless bosonic and fermionic degrees of freedom turn out to be Notice that if Eq. (1.1) holds when all 1 2 -branes are sitting on orientifold planes, this expression is only the first term of a Taylor expansion in WLs. However, the linear term, when 1 2 -branes move slightly, vanishes because the WLs are dressed by the charges of the states running in the virtual loop, and states with opposite charges can always be paired. At the next order, the mass terms of the WLs are proportional to squared charges. In general, WLs associated to a gauge group factor G κ are non-tachyonic if [START_REF] Coudarchet | Quantum no-scale regimes and moduli dynamics[END_REF][START_REF] Kounnas | Super no-scale models in string theory[END_REF] T F of G κ that are realized by the n B and n F states. In our case, the above condition yields for A = 1, . . . , In order to write the 1-loop potential [START_REF] Abel | Stability and vacuum energy in open string models with broken supersymmetry[END_REF], it is convenient to split the WLs as background values plus deviations,

n B = 8 8 + 2 10-d A=1 p A (p A -1) 2 , n F = 8
R (κ) B -T R (κ) F ≥ 0, ( 3
2 10-d /2 p 2A-1 -2 -p 2A ≥ 0, for the SO(p 2A-1 ) WLs, if p 2A-1 ≥ 2, p 2A -2 -p 2A-1 ≥ 0, for the SO(p 2A ) WLs, if p 2A ≥ 2, ( 3
a I α = a I α + ε I α , where a I α ∈ 0, 1 2 , I = d, . . . , 9, α = 1, . . . , 32. (3.6)
Note that the ε I α 's can be arbitrary, and their magnitudes are not supposed to be small. However, assuming that all mass scales in the undeformed background are greater than M , the NSNS metric is bounded in the following sense,

G 99 |G ij | G 99 , |G 9j | G 99 , i, j = d, . . . , 8, G 99 1. (3.7)
The remaining moduli arise from the RR closed string sector and are the internal components C IJ , I, J = d, . . . , 9, of the antisymmetric tensor. In these conditions, the potential reads

V = Γ d+1 2 π 3d+1 2 M d l 9
N2l 9 +1 (ε, G)

|2l 9 + 1| d+1 + O (M s M ) d 2 e -2πc Ms M , (3.8)
where c = O( 1) is positive. The dominant contribution is expressed in terms of

N2l 9 +1 (ε, G) = 4 -16 -0 - (α,β)∈L (-1) F cos 2π(2l 9 + 1) ε 9 α -ε 9 β + G 9i G 99 (ε i α -ε i β ) × Hd+1 2 π|2l 9 + 1| (ε i α -ε i β ) Ĝij (ε j α -ε j β ) 1 2 √ G 99 (3.9) + α cos 4π(2l 9 + 1) ε 9 α + G 9i G 99 ε i α H d+1 2 2π|2l 9 + 1| ε i α Ĝij ε j α 1 2 √ G 99 ,
which arises from the torus, Klein bottle, annulus and Möbius strip amplitudes. In our notations, L is the set of pairs (α, β) such that α and β label 1 2 -branes in the neighborhood of corners 2A -1 and 2A, for some A = 1, . . . , 2 10-d /2. Hence, for any (α, β) ∈ L, massless strings stretched between these 1 2 -branes contribute, and they generate the bosonic adjoint and fermionic bifundamental representations of SO(p 2A-1 ) × SO(p 2A ). Moreover, we have

defined Ĝij = G ij - G i9 G 99 G 99 G 9j G 99 , i, j = d, . . . , 8, H ν (z) = 1 Γ(ν) +∞ 0 dx x 1+ν e -1 x -z 2 x = 2 Γ(ν) z ν K ν (2z).
(3.10) Some remarks are in order:

• Denoting the true dynamical degrees of freedom of the open string WLs as ε I r , I = d, . . . , 9, r = 1, . . . , 2 10-d A=1 p A /2 , N2l 9 +1 (ε, G) can be expanded to quadratic order to find, as expected, .11) In this expression, A(r) denotes the corner around which the brane r vary, while Ã(r) is the partner corner along the Scherk-Schwarz direction X9 , while

V = n F -n B ξ d M d + 1 2 ξ d M d r p A(r) -p Ã(r) 2 -1 ε I r ∆IJ ε J s + O(ε 4 ) + O (M s M ) d 2 e -2πc Ms M . ( 3
ξ d = Γ( d+1 2 ) π 3d+1 2 n 9 1 |2n 9 + 1| d+1 , ξ d = Γ( d+1 2 ) π 3d+1 2 n 9 128π 2 |2n 9 + 1| d-1 , ∆IJ = 1 d -1 G IJ G 99 + (d -2) G I9 G 99 G 9J G 99 .
(3.12) It is not difficult to show that all eigenvalues of ∆IJ are strictly positive, so that the conditions (3.3) for tachyons not to arise are recovered.

• In particular, the WLs associated to SO( 2) and [SO(3) × SO( 1)] gauge group factors are massless. However, it turns out the dominant term of the full 1-loop potential Eqs (3.8), (3.9) is totally independent of these WLs, which are therefore flat directions (up to the exponentially suppressed terms).

• When Eq. (3.4) holds, keeping M fixed, V is at a local minimum when all massive open string WLs are set to 0, while all massless ones are arbitrary. Thus, Eq. (1.1) remains valid in this more general case, since [SO(3) × SO [START_REF] Abel | Stability and vacuum energy in open string models with broken supersymmetry[END_REF]] can be broken by its WLs, with Higgs masses lower than M . Furthermore, the minima are independent of the NSNS moduli Ĝij , G i9 , i, j = d, . . . 8, and of the RR ones C IJ , I, J = d, . . . , 9. Therefore, all moduli arising from the closed string sector are flat directions, except M (unless n F -n B = 0).

Conclusion and remarks

We have seen that at fixed supersymmetry breaking scale M , and up to exponentially sup- can be elements of SO(32), or in O(32) but not in SO(32). Many of the configurations we have described correspond to the former case, and heterotic dual descriptions should exist.

1 2 -

 2 D(d -1)-branes, whose positions are XI = 2πa I α GII , I = d, . . . , 9, α = 1, . . . , 32.

n

  B contains 8 × 8 states arising from the closed string sector, which correspond to the 10-dimensional dilaton φ and metric G M N in the NSNS sector, and the RR 2-form C M N , M, N = 0, . . . 9, which are dimensionally reduced. It also contain the bosonic parts of vector multiplets in the adjoint representations of the SO(p A )'s gauge group factors, A = 1, . . . , 2 10-d . On the other hand, n F contains the fermionic parts of vector multiplets in the bifundamental representation of SO(p 2A-1 ) × SO(p 2A ), A = 1, . . . , 2 10-d /2. They arise from the cancellation of the Scherk-Schwarz and WL shifts of the momenta along the direction X 9 (in the original type I picture).

. 3 ) 1 2

 31 whose compatibility implies, up to exchanges p 2A-1 ↔ p 2A , ∀A = 1, . . . , 2 10-d /2, (p 2A-1 , p 2A ) to be of the form (p, 0), p ≥ 0 or (p, 1), p ≥ 3 or p = 1.(3.4)Without surprise, the SO(32) configuration corresponding to all 1 2 -branes coincident on a single orientifold plane yields the lowest value n F -n B = -8 × 504, and it is stable with respect to all open string WLs in arbitrary dimension d. The number of stable configurations and the rank of the gauge group fall, as we freeze 1 the largest allowed values of p A 's are SO(5) × SO(1) × SO(1) × SO(1) 13 , SO(4) × SO(1) × SO(1) 14 , (3.5) where the brackets indicate that the 1 2 -branes are located on corners separated along X9 only. The maximal value n F -n B = 8 × 8 is reached when there is no left gauge symmetry arising from the open string sector, [SO(1) × SO(1)] 16 for d ≤ 5. However, in all cases the models admit a U (1) 10-d G × U (1) 10-d C gauge symmetry arsing from the closed string sector, G Iµ , C Iµ , I = d, .. . , 9. The above results can also be derived by computing explicitly the 1-loop effective potential. Moreover, having discussed so far non-tachyonic distributions of -branes, the potential will allow to conclude whether massless WLs introduce instabilities from interactions (still at 1-loop), an issue which arises for the SO(2) and [SO(3) × SO(1)] gauge group factors.

  pressed corrections, local minima of the 1-loop effective potential of arbitrary sign exist in dimension d ≤ 5, in type I string theory. In a T-dual picture, they are realized by freezing isolated 1 2 -branes on orientifold planes, whose effect is to reduce the rank of the gauge symmetry and to introduce massless fermions in the fundamental representations of SO(p) gauge group factors. Therefore, the WL matrices W I = diag (e 2iπa I α , α = 1, . . . , 32), I = d, . . . ,

-branes on O-planes in order to raise n F -n B .[START_REF] Abel | Stability and vacuum energy in open string models with broken supersymmetry[END_REF] 2 -branes distributions yielding n F -n B = 0 exist for d ≤ 5[START_REF] Abel | Stability and vacuum energy in open string models with broken supersymmetry[END_REF], with rank of the open string gauge group at most equal to 4. Among them, the simplest solutions involving

Acknowledgement

This work was partially supported by the Royal Society International Cost Share Award.

However, when WL matrices are not in SO(32), no heterotic dual description can be defined and inconsistencies at the non-perturbative level are expected to arise [START_REF] Witten | D-branes and K theory[END_REF].