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We present an overview of both older and recent developments concerning scale separation in string theory. We focus on parametric scale separation obtained at the classical level in flux compactifications down to AdS vacua. We review the scenarios that have been proposed to achieve a hierarchy of scales between spacetime and the internal space, built from a low-dimensional perspective. We then discuss how they have been understood to arise from proper higher-dimensional descriptions. Eventually, limitations of these constructions as well as Swampland and holographic arguments addressing the question of scale separation in string theory are discussed. The purpose of the review is to draw an accurate picture of the state of the art of the subject at the moment. [ABSTRACT FROM AUTHOR]

Massive type IIA flux compactifications of the form AdS4 × X6, where X6 admits a Calabi-Yau metric and O6-planes wrapping three-cycles, display families of vacua with parametric scale separation between the compactification scale and the AdS4 radius, generated by an overall rescaling of internal four-form fluxes. For toroidal orbifolds one can perform two T-dualities and map this background to an orientifold of massless type IIA compactified on an SU(3)-structure manifold with fluxes. Via a 4d EFT analysis, we generalise this last construction and embed it into new branches of supersymmetric and non-supersymmetric vacua with similar features. We apply our results to propose new infinite families of vacua based on elliptic fibrations with metric fluxes. Parametric scale separation is achieved by an asymmetric flux rescaling which, however, in general is not a simple symmetry of the 4d equations of motion. At this level of approximation the vacua are stable but, unlike in the Calabi-Yau case, they display a non-universal mass spectrum of light fields. [ABSTRACT FROM AUTHOR]

The analysis of type IIB flux vacua on warped Calabi-Yau orientifolds becomes considerably involved for a large number of complex structure fields. We however show that, for a quadratic flux superpotential, one can devise simplifying schemes which effectively reduce the large number of equations down to a few. This can be achieved by imposing the vanishing of certain flux quanta in the large complex structure regime, and then choosing the remaining quanta to respect the symmetries of the underlying prepotential. One can then implement an algorithm to find large families of flux vacua with a fixed flux tadpole, independently of the number of fields. We illustrate this approach in a Calabi-Yau manifold with 51 complex structure moduli, where several reduction schemes can be implemented in order to explicitly solve the vacuum equations for that sector. Our findings display a flux-tadpole-to-stabilized-moduli ratio that is marginally above the bound proposed by the Tadpole Conjecture, and we discuss several effects that would take us below such a bound. [ABSTRACT FROM AUTHOR]

We analyze the tree-level potential of type IIB flux compactifications in warped Calabi-Yau orientifolds, in regions of weak coupling and moderately large complex structure. In this regime, one may approximate the flux-induced superpotential W by a polynomial on the axio-dilaton and complex structure fields, and a significant fraction of vacua corresponds to a quadratic W. In this quadratic case, we argue that vacua fall into three classes, for which one can push the analytic description of their features. In particular, we provide analytic expressions for the vacuum expectation values and flux-induced masses of the axio-dilaton and complex structure fields in a large subclass of vacua, independently of the Calabi-Yau and the number of moduli. We show that supersymmetric vacua always contain flat directions, at least at this level of approximation. Our findings allow to generate vast ensembles of flux vacua in specific Calabi-Yau geometries, as we illustrate in a particular example. [ABSTRACT FROM AUTHOR]

We consider an open-string realisation of N = 2 → N = 0 spontaneous breaking of supersymmetry in four-dimensional Minkowski spacetime. It is based on type IIB orientifold theory compactified on T 2 × T 4 / Z 2 , with Scherk–Schwarz supersymmetry breaking implemented along T 2. We show that in the regions of moduli space where the supersymmetry breaking scale is lower than the other scales, there exist configurations with minima that have massless Bose-Fermi degeneracy and hence vanishing one-loop effective potential, up to exponentially suppressed corrections. These backgrounds describe non-Abelian gauge theories, with all open-string moduli and blowing up modes of T 4 / Z 2 stabilized, while all untwisted closed-string moduli remain flat directions. Other backgrounds with strictly positive effective potentials exist, where the only instabilities arising at one loop are associated with the supersymmetry breaking scale, which runs away. All of these backgrounds are consistent non-perturbatively. [ABSTRACT FROM AUTHOR]

In this paper, we compute at one loop the two-point functions of massless scalars in the Neumann–Dirichlet open-string sector of the type IIB orientifold compactified on T 2 × T 4 / ℤ 2 , when = 2 supersymmetry is spontaneously broken. This is done by evaluating correlation functions of "boundary-changing vertex operators" which are analogous to correlators of twist fields for closed strings. We use our results to compute the mass developed at one loop by the moduli fields arising in the Neumann–Dirichlet sector. [ABSTRACT FROM AUTHOR]

We derive the masses acquired at one loop by massless scalars in the Neumann-Dirichlet sector of open strings, when supersymmetry is spontaneously broken. It is done by computing two-point functions of “boundary-changing vertex operators” inserted on the boundaries of the annulus and Möbius strip. This requires the evaluation of correlators of “excited boundary-changing fields,” which are analogous to excited twist fields for closed strings. We work in the type IIB orientifold theory compactified on T2× T4/ℤ2, where N = 2 supersymmetry is broken to N = 0 by the Scherk-Schwarz mechanism implemented along T2. Even though the full expression of the squared masses is complicated, it reduces to a very simple form when the lowest scale of the background is the supersymmetry breaking scale M3/2. We use our results to analyze in this regime the stability at the quantum level of the moduli fields arising in the Neumann-Dirichlet sector. This completes the study of ref. [32], where the quantum masses of all other types of moduli arising in the open- or closed-string sectors are derived. Ultimately, we identify all brane configurations that produce backgrounds without tachyons at one loop and yield an effective potential exponentially suppressed, or strictly positive with runaway behavior of M3/2. [ABSTRACT FROM AUTHOR]

Starting from a peculiar orientifold projection proposed long ago by Angelantonj and Cardella, we elaborate on a novel perturbative scenario that involves only D-branes, together with the two types of orientifold planes O± and anti-orientifold planes O ¯ ± . We elucidate the microscopic ingredients of such models, connecting them to a novel realization of brane supersymmetry breaking. Depending on the position of the D-branes in the internal space, supersymmetry can be broken at the string scale on branes, or alternatively only at the massive level. The main novelty of this construction is that it features no NS-NS disk tadpoles, while avoiding open-string instabilities. The one-loop potential, which depends on the positions of the D-branes, is minimized for maximally broken, non-linearly realized supersymmetry. The orientifold projection and the effective field theory description reveal a soft breaking of supersymmetry in the closed-string sector. In such models it is possible to decouple the gravitino mass from the value of the scalar potential, while avoiding brane instabilities. [ABSTRACT FROM AUTHOR]

We analyze quantum no-scale regimes (QNSR) in perturbative heterotic string compactified on tori, with total spontaneous breaking of supersymmetry. We show that for marginal deformations initially at any point in moduli space, the dynamics of a flat, homogeneous and isotropic universe can always be attracted to a QNSR. This happens independently of the characteristics of the 1-loop effective potential V 1-loop , which can be initially positive, negative or vanishing, and maximal, minimal or at a saddle point. In all cases, the classical no-scale structure is restored at the quantum level, during the cosmological evolution. This is shown analytically by considering moduli evolutions entirely in the vicinity of their initial values. Global attractor mechanisms are analyzed numerically and depend drastically on the sign of V 1-loop . We find that all initially expanding cosmological evolutions along which V 1-loop is positive are attracted to the QNSR describing a flat, ever-expanding universe. On the contrary, when V 1-loop can reach negative values, the expansion comes to a halt and the universe eventually collapses into a Big Crunch, unless the initial conditions are tuned in a tiny region of the phase space. This suggests that flat, ever-expanding universes with positive potentials are way more natural than their counterparts with negative potentials. [ABSTRACT FROM AUTHOR]

We show that in generic no-scale models in string theory, the flat, expanding cosmological evolutions found at the quantum level can be attracted to a “quantum no-scale regime”, where the no-scale structure is restored asymptotically. In this regime, the quantum effective potential is dominated by the classical kinetic energies of the no-scale modulus and dilaton. We find that this natural preservation of the classical no-scale structure at the quantum level occurs when the initial conditions of the evolutions sit in a subcritical region of their space. On the contrary, supercritical initial conditions yield solutions that have no analogue at the classical level. The associated intrinsically quantum universes are sentenced to collapse and their histories last finite cosmic times. Our analysis is done at 1-loop, in perturbative heterotic string compactified on tori, with spontaneous supersymmetry breaking implemented by a stringy version of the Scherk–Schwarz mechanism. [ABSTRACT FROM AUTHOR]

We consider an open-string realisation of N = 2 → N = 0 spontaneous breaking of supersymmetry in four-dimensional Minkowski spacetime. It is based on type IIB orientifold theory compactified on T 2 × T 4 / Z 2 , with Scherk–Schwarz supersymmetry breaking implemented along T 2. We show that in the regions of moduli space where the supersymmetry breaking scale is lower than the other scales, there exist configurations with minima that have massless Bose-Fermi degeneracy and hence vanishing one-loop effective potential, up to exponentially suppressed corrections. These backgrounds describe non-Abelian gauge theories, with all open-string moduli and blowing up modes of T 4 / Z 2 stabilized, while all untwisted closed-string moduli remain flat directions. Other backgrounds with strictly positive effective potentials exist, where the only instabilities arising at one loop are associated with the supersymmetry breaking scale, which runs away. All of these backgrounds are consistent non-perturbatively. [ABSTRACT FROM AUTHOR]