Supersymmetry breaking and stability in string vacua

Abstract

We review some aspects of the dramatic consequences of supersymmetry breaking on string vacua. In particular, we focus on the issue of vacuum stability in ten-dimensional string models with broken, or without, supersymmetry, whose perturbative spectra are free of tachyons. After formulating the models at stake, we introduce their unified low-energy effective description and present a number of vacuum solutions to the classical equations of motion. In addition, we present a generalization of previous no-go results for de Sitter vacua in warped flux compactifications. Then we analyze the classical and quantum stability of these vacua, studying linearized field fluctuations and bubble nucleation. Then, we describe how the resulting instabilities can be framed in terms of brane dynamics, examining in particular brane interactions, back-reacted geometries and commenting on a brane-world string construction along the lines of a recent proposal. After providing a summary, we conclude with some perspectives on possible future developments.

Tensor spherical harmonics: a primer

In this appendix we review some results that were needed for our stability analysis in Sect. 4, starting from an ambient Euclidean space. In Sect. A.1 we build scalar spherical harmonics, and in Sect. A.2 we extend our considerations to tensors of higher rank. The results agree with the constructions presented in [179, 180].⁸¹

1.1 Scalar spherical harmonics

Let $Y^1,\dots Y^{n+1}$ be Cartesian coordinates of ${\mathbb {R}}^{n+1}$, so that the unit sphere ${\mathbb {S}}^{n}$ is described by the constraint

$$\begin{aligned} \begin{aligned} \delta _{IJ} \, Y^I \, Y^J = r^2 \end{aligned}\end{aligned}$$

(A.1)

on the radial coordinate r, solved by spherical coordinates $y^i$ according to

$$\begin{aligned} \begin{aligned} Y^I = r \, {\widehat{Y}}^I(y) \, . \end{aligned}\end{aligned}$$

(A.2)

The scalar spherical harmonics on ${\mathbb {S}}^{n}$ can be conveniently constructed starting from harmonic polynomials of degree $\ell$ in the ambient Euclidean space ${\mathbb {R}}^{n+1}$. A harmonic polynomial of degree $\ell$ takes the form

$$\begin{aligned} \begin{aligned} H_{(n)}^\ell (Y) = \alpha _{I_1 \dots I_\ell } \, Y^{I_1} \dots Y^{I_\ell } \, , \end{aligned}\end{aligned}$$

(A.3)

and is therefore determined by a completely symmetric and trace-less tensor $\alpha _{I_1\dots I_\ell }$ of rank $\ell$, as can be clearly seen applying to it the Euclidean Laplacian

$$\begin{aligned} \begin{aligned} \nabla ^2_{n+1} = \sum _{I=1}^{n+1} \frac{\partial ^2}{\partial Y_I^2} \, . \end{aligned}\end{aligned}$$

(A.4)

The scalar spherical harmonics $\mathcal{Y}_{(n)}^{I_1\dots I_\ell }$ are then defined restricting the $H_{(n)}^{\ell }(Y)$ to the unit sphere $S^n$, or equivalently as

$$\begin{aligned} \begin{aligned} H_{(n)}^\ell ({\widehat{Y}}(y)) = r^\ell \, \alpha _{I_1 \dots I_\ell } \, {\mathscr {Y}}_{(n)}^{I_1 \dots I_\ell }(y) \, . \end{aligned}\end{aligned}$$

(A.5)

As a result, the Euclidean metric can be recast as

$$\begin{aligned} \begin{aligned} ds^2_{n+1} = dr^2 + r^2 \, d\varOmega _n^2 \, , \end{aligned}\end{aligned}$$

(A.6)

and the scalar Laplacian decomposes according to

$$\begin{aligned} \begin{aligned} 0 = \nabla ^2_{n+1} H_{(n)}^\ell (Y) = \frac{1}{r^n} \, \frac{\partial }{\partial r} \left( r^n \, \frac{\partial H_{(n)}^\ell (Y)}{\partial r} \right) + \frac{1}{r^2} \, \nabla ^2_{{\mathbb {S}}^n} \, H_{(n)}^\ell (Y) \, , \end{aligned}\end{aligned}$$

(A.7)

where

$$\begin{aligned} \begin{aligned} \frac{\partial H_{(n)}^\ell (Y)}{\partial r} = \frac{\ell }{r} \, H_{(n)}^\ell (Y) \end{aligned}\end{aligned}$$

(A.8)

for the homogeneous polynomials $H_{(n)}^{\ell }(Y)$. All in all

and the degeneracy of the scalar spherical harmonics for any given $\ell$ is the number of independent components of a corresponding completely symmetric and trace-less tensor, namely

$$\begin{aligned} \begin{aligned} \frac{\left( n + 2\ell - 1 \right) \left( n + \ell - 2 \right) !}{\ell ! \left( n-1 \right) !} \, . \end{aligned}\end{aligned}$$

(A.10)

1.2 Spherical harmonics of higher rank

In discussing more general tensor harmonics, it is convenient to notice that, in the coordinate system of Eq. (A.6), the non-vanishing Christoffel symbols ${\widetilde{\varGamma }}_{IJ}^K$ for the ambient Euclidean space read

$$\begin{aligned} \begin{aligned} {\widetilde{\varGamma }}_{ij}^r = - \, r \, g_{ij} \, , \qquad {\widetilde{\varGamma }}_{jr}^i = \frac{1}{r} \, \delta _i^j \, , \qquad {\widetilde{\varGamma }}_{ij}^k = \varGamma _{ij}^k \, , \end{aligned}\end{aligned}$$

(A.11)

where the labels i, j, k refer, as above, to the n-sphere, whose Christoffel symbols are denoted by $\varGamma _{ij}^k$.

The construction extends nicely to tensor spherical harmonics, which can be defined starting from generalized harmonic polynomials, with one proviso. The relation in Eq. (A.2) and its differentials imply that the actual spherical components of tensors carry additional factors of r, one for each covariant tensor index, with respect to those naïvely inherited from the Cartesian coordinates of the Euclidean ambient space, as we shall now see in detail. To begin with, vector spherical harmonics arise from one-forms in ambient space, built from harmonic polynomials of the type

$$\begin{aligned} \begin{aligned} H_{(n) \, J}^\ell (Y) = \alpha _{I_1 \dots I_\ell \, , \, J} \, Y^{I_1} \dots Y^{I_\ell } \, , \end{aligned}\end{aligned}$$

(A.12)

where the coefficients $\alpha _{I_1 \dots I_\ell \,,\,J}$ are completely symmetric and trace-less in any pair of the first $\ell$ indices. They are also subject to the condition

$$\begin{aligned} \begin{aligned} Y^J \, H_{(n) \, J}^\ell (Y) = 0 \, , \end{aligned}\end{aligned}$$

(A.13)

since the radial component, which does not pertain to the sphere $S^n$, ought to vanish. This implies that the complete symmetrization of the coefficients vanishes identically,

$$\begin{aligned} \begin{aligned} \alpha _{(I_1 \dots I_\ell \, , \, J)} = 0 \, , \end{aligned}\end{aligned}$$

(A.14)

and on account of the symmetry in the first $\ell$ indices. As a result, $H_{n\,,\,J}^{\ell }(Y)$ is thus transverse in the ambient space,

$$\begin{aligned} \begin{aligned} \partial ^J H_{(n) \, J}^\ell (Y) = 0 \, . \end{aligned}\end{aligned}$$

(A.15)

Moreover, any Euclidean vector V such that $V_I\, Y^I = 0$ couples with differentials according to the general rule inherited from Eq. (A.2),

$$\begin{aligned} \begin{aligned} V_I \, dY^I = V_I \, r \, d{\widehat{Y}}^I \, , \end{aligned}\end{aligned}$$

(A.16)

so that the actual sphere components, which are associated to $d {\widehat{Y}}^I$, include an additional power of r, and the vector spherical harmonics ${\mathscr {Y}}_{(n)\, i}^{I_1\dots I_\ell \, , \, J}$ are thus obtained from

$$\begin{aligned} \begin{aligned} r^{\ell + 1} \, {\mathscr {Y}}_{(n) \, i}^{I_1 \dots I_\ell \, , \, J} \, \alpha _{I_1 \dots I_\ell \, , \, J} \, dy^i = r \, H_{(n) \, J}^\ell (Y) \, d{\widehat{Y}}^J \, . \end{aligned}\end{aligned}$$

(A.17)

Therefore,

$$\begin{aligned} \begin{aligned} \nabla _r \nabla _r \left( r \, H_{(n) \, J}^\ell (Y) \right) = \left( \frac{\partial }{\partial r} - \frac{1}{r}\right) ^2 \left( r \, H_{(n) \, J}^\ell (Y) \right) = \frac{\ell \left( \ell - 1 \right) }{r} \, H_{(n) \, J}^\ell (Y) \, , \end{aligned}\end{aligned}$$

(A.18)

while the remaining contributions to the Laplacian give

$$\begin{aligned} \begin{aligned} \frac{1}{r^2} \, \nabla ^2_{{\mathbb {S}}^n} \left( r \, H_{(n) \, J}^\ell (Y)\right) + \frac{n \left( \ell + 1\right) - n - 1}{r} \left( r \, H_{(n) \, J}^\ell (Y)\right) \, , \end{aligned}\end{aligned}$$

(A.19)

taking into account the Christoffel symbols in Eq. (A.11). Since the total Euclidean Laplacian vanishes by construction, adding Eqs. (A.18) and (A.19) finally results in

with $\ell \ge 1$.

In a similar fashion, the spherical harmonics ${\mathscr {Y}}_{(n)\, i_1\dots i_p}^{I_1\dots I_\ell \, , \, J_1 \dots J_p}$, corresponding to generic higher-rank transverse tensors which are also trace-less in any pair of symmetric I-indices, can be described starting from harmonic polynomials of the type $H_{(n) \, J_1\dots J_p}^{\ell }(Y)$, and satisfy

with $\ell \ge p$.

In Young tableaux language, the scalar harmonics correspond to trace-less single-row diagrams of the type

(A.22)

while the independent vectors associated to vector harmonics correspond to two-row trace-less hooked diagrams of the type

(A.23)

as we have explained. Similarly, the independent tensor perturbations of the metric in the internal space correspond to trace-less diagrams of the type

(A.24)

while the independent perturbations associated to a $(p+1)$-form gauge field in the internal space correspond, in general, to multi-row diagrams of the type

(A.25)

The degeneracies of these representations can be related to the corresponding Young tableaux, as in [182]. The structure of the various types of harmonics, which are genuinely different for large enough values of n, reflects nicely the generic absence of mixings between different classes of perturbations.