Flux Compactifications Research Articles

Stabilization of a twisted modulus on a mirror of rigid Calabi-Yau manifold

We study the stabilization of a twisted modulus in Type IIB flux compactifications on a mirror of the rigid Calabi-Yau threefold. By analyzing the effective action of twisted and untwisted moduli, we find that three-form fluxes satisfying the tadpole cancellation conditions lead to supersymmetric AdS vacua. We also investigate swampland conjectures on this non-geometric background.

Almost classical de Sitter?

The classical-dS scenario in the type II string theories proposes to search for dS vacua of orientifold flux compactifications in a regime where string corrections to the compactified effective field theory are negligible. We study a minimal extension of this scenario in which the leading string corrections to the O-plane/D-brane actions at the 4-derivative order are included but higher orders as well as string corrections in the bulk are self-consistently neglected. Our proposal is motivated by a recent debate about dS solutions with O8-planes which circumvent a classical no-go theorem due to unusual sources leading to so-called permissive boundary conditions for the 10D supergravity fields. We argue that such sources do not arise in classical supergravity but ask whether including the 4-derivative corrections leads to sources that have a similar effect. However, we find that the 4-derivative corrections do not allow meta-stable dS in a class of models with O8-planes and/or D8-branes we consider. We also study related models which in addition contain O6-planes/D6-branes and find that again no meta-stable dS is allowed, both classically and including the 4-derivative corrections. While some of the arguments in this work require the backreaction of the O-plane/D-brane sources to be small, others are valid including the full backreaction.

Spontaneous CP violation and partially broken modular flavor symmetries

We study the realization of spontaneous CP violation through moduli stabilization. In modular flavor models, the source of CP violation is the vacuum expectation values of the complex structure moduli of toroidal compact space. We demonstrate that the combined effects of Type IIB flux compactifications with modular invariant couplings between the moduli and matter fields can induce spontaneous CP violation without or with supersymmetry breaking. Furthermore, some general properties of CP and modular invariant scalar potentials are presented. It is found that certain modifications or partial breakings of modular symmetry are useful in generating spontaneous CP violation.

Stabilizing massless fields with fluxes in Landau-Ginzburg models

Recent work on flux compactifications suggests that the tadpole constraint generically allows only a limited number of complex structure moduli to become massive, i.e., be stabilized at quadratic order in the spacetime superpotential. We study the effects of higher-order terms systematically around the Fermat point in the 19 Landau-Ginzburg model. This model lives at strong coupling and features no Kähler moduli. We show that indeed massless fields can be stabilized in this fashion. We observe that, depending on the flux, this mechanism is more effective when the number of initially massless fields is large. These findings are compatible with both the tadpole conjecture and the massless Minkowski conjecture. Along the way, we complete the classification of integral flux vectors with small tadpole contribution. Thereby we are closing in on a future complete understanding of all possible flux configurations in the 19 Landau-Ginzburg model.

Finiteness theorems and counting conjectures for the flux landscape

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.

On the absence of supergravity solutions for localized, intersecting sources

For decades intersecting D-branes and O-planes have been playing a very important role in string phenomenology in the context of particle physics model building and in the context of flux compactifications. The corresponding supergravity equations are hard to solve so generically solutions only exist in a so-called smeared limit where the delta function sources are replaced by constants. We are showing here that supergravity solutions for two perpendicularly intersecting localized sources in flat space do not exist for a generic diagonal metric Ansatz. We show this for two intersecting sources with p = 1, 2, 3, 4, 5, 6 spatial dimensions that preserve 8 supercharges, and we allow for fully generic fluxes.

W0_sample = np.random.normal(0,1)?

In this note, we explore the distribution of vacuum expectation values of the superpotential W0 in explicit Type IIB flux compactifications. We show that the distribution can be approximated universally across geometries by a two-dimensional Gaussian with a model-dependent standard deviation. We identify this behaviour in 20 Calabi-Yau orientifold compactifications with between two and five complex structure moduli by constructing a total of O(107) flux vacua. We observe a characteristic scaling behaviour of the width σ of our distributions with respect to the D3-charge contributions Nflux from fluxes which can be approximated by σ∼Nflux. This W0 distribution implies that locating small values of |W0| as a preferred regime associated with classes of string theory solutions typically featuring hierarchies, simplifies to the basic statement of finding small Euclidean norms of normally distributed values. We also identify small modifications to this Gaussian behaviour in our samples, which might be seen as indications for the breakdown of the continuous flux approximation commonly used in the context of statistical analyses of the flux landscape.

A worldsheet description of flux compactifications

We demonstrate how recent developments in string field theory provide a framework to systematically study type II flux compactifications with non-trivial Ramond-Ramond profiles. We present an explicit example where physical observables can be computed order by order in a small parameter which can be effectively viewed as string coupling constant. We obtain the corresponding background solution of the string field equations of motions up to the second order in the expansion. Along the way, we show how the tadpole cancellations of the string field equations lead to the minimization of the F-term potential of the low energy supergravity description. String field action expanded around the obtained background solution furnishes a “worldsheet” description of the flux compactifications.

Weak Scale Supersymmetry Emergent from the String Landscape.

Superstring flux compactifications can stabilize all moduli while leading to an enormous number of vacua solutions, each leading to different 4-d laws of physics. While the string landscape provides at present the only plausible explanation for the size of the cosmological constant, it may also predict the form of weak scale supersymmetry which is expected to emerge. Rather general arguments suggest a power-law draw to large soft terms, but these are subject to an anthropic selection of a not-too-large value for the weak scale. The combined selection allows one to compute relative probabilities for the emergence of supersymmetric models from the landscape. Models with weak scale naturalness appear most likely to emerge since they have the largest parameter space on the landscape. For finetuned models such as high-scale SUSY or split SUSY, the required weak scale finetuning shrinks their parameter space to tiny volumes, making them much less likely to appear compared to natural models. Probability distributions for sparticle and Higgs masses from natural models show a preference for Higgs mass mh∼125 GeV, with sparticles typically beyond the present LHC limits, in accord with data. From these considerations, we briefly describe how natural SUSY is expected to be revealed at future LHC upgrades. This article is a contribution to the Special Edition of the journal Entropy, honoring Paul Frampton on his 80th birthday.

Bubbles of nothing:the tunneling potential approach

Bubbles of nothing (BoNs) describe the decay of spacetimes with compact dimensions and are thus of fundamental importance for many higher dimensional theories proposed beyond the Standard Model. BoNs admit a 4-dimensional description in terms of a singular Coleman-de Luccia (CdL) instanton involving the size modulus field, stabilized by some potential V(ϕ). Using the so-called tunneling potential (Vt ) approach, we study which types of BoNs are possible and for which potentials V(ϕ) can they be present. We identify four different types of BoN, characterized by different asymptotic behaviours at the BoN core and corresponding to different classes of higher dimensional theories, which we also classify. Combining numerous analytical and numerical examples, we study the interplay of BoN decays with other standard decay channels, identify the possible types of quenching of BoN decays and show how BoNs for flux compactifications can also be described in 4 dimensions by a multifield Vt . The use of the Vt approach greatly aids our analyses and offers a very simple picture of BoNs which are treated in the same language as any other standard vacuum decays.

Bounds on field range for slowly varying positive potentials

In the context of quantum gravitational systems, we place bounds on regions in field space with slowly varying positive potentials. Using the fact that V<Λs2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ V<{\\Lambda}_s^2 $$\\end{document}, where Λs(ϕ) is the species scale, and the emergent string conjecture, we show this places a bound on the maximum diameter of such regions in field space: ∆ϕ ≤ a log(1/V) + b in Planck units, where a ≤ d−1d−2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{\\left(d-1\\right)\\left(d-2\\right)} $$\\end{document}, and b is an \U0001d4aa(1) number and expected to be negative. The coefficient of the logarithmic term has previously been derived using TCC, providing further confirmation. For type II string flux compactifications on Calabi-Yau threefolds, using the recent results on the moduli dependence of the species scale, we can check the above relation and determine the constant b, which we verify is \U0001d4aa(1) and negative in all the examples we studied.

De Sitter-eating O-planes in supercritical string theory

It has been proposed that flux compactifications of supercritical string theories (i.e., with spacetime dimension D > 10) have dS vacua, with large D acting as a control parameter for corrections to the classical spacetime effective action. In this paper, we provide a detailed analysis of the self-consistency of such models, focussing on α′ and backreaction corrections. We first show that all supercritical AdS, Minkowski and dS vacua in this setting have gtrsim mathcal{O}(1) curvature and/or field strengths in the string frame. This may be in tension with suppressing α′ corrections unless the coefficients of the higher-derivative terms have a sufficiently strong large-D suppression. We then argue that an additional and more severe problem arises in the dS case due to the backreaction of O-planes. In particular, we argue using a combination of geometric bounds and string-theory constraints that the O-plane backreaction is large in supercritical dS models. This implies that a large part of the naive classical geometry is eaten up by singular holes and thus indicates a breakdown of the classical description. Our finding resonates with several other recent results suggesting that string theory does not admit dS vacua in regimes where string and backreaction corrections are under control. As byproducts of our analysis, we derive a number of technical results that are useful beyond the specific applications in this paper. In particular, we compute the leading backreaction corrections to the smeared solution in a general flux compactification from D to d dimensions for an arbitrary distribution of O-planes and D-branes. We further argue for a general estimate for Green’s functions on compact manifolds (and therefore for the backreaction corrections) in terms of their diameter, volume and dimension.

New non-supersymmetric flux vacua in string theory

In this note we construct large ensembles of supersymmetry breaking solutions arising in the context of flux compactifications of type IIB string theory. This class of solutions was previously proposed in [1] for which we provide the first explicit examples in Calabi-Yau orientifold compactifications with discrete fluxes below their respective tadpole constraint. As a proof of concept, we study the degree 18 hypersurface in weighted projective space CP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbbm{CP} $$\\end{document}1,1,1,6,9. Furthermore, we look at 10 additional orientifolds with h1,2 = 2, 3. We find several flux vacua with hierarchical suppression of the vacuum energy with respect to the gravitino mass. These solutions provide a crucial stepping stone for the construction of explicit de Sitter vacua in string theory. Lastly, we also report the difference in the distribution of W0 between supersymmetric and non-supersymmetric minima.

Flavor, CP and metaplectic modular symmetries in Type IIB chiral flux vacua

We examine symmetries of chiral four-dimensional vacua of Type IIB flux compactifications with vanishing superpotential W = 0. We find that the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric MSSM-like and Pati-Salam vacua possess enhanced discrete symmetries in the effective action below the mass scale of stabilized complex structure moduli and dilaton. Furthermore, a generation number of quarks/leptons is small on these vacua where the flavor, CP and metaplectic modular symmetries are described in the framework of eclectic flavor symmetry.

D3¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{\ extrm{D}3} $$\\end{document}-branes and gaugino condensation

Anti-D3 branes at the bottom of warped throats, commonly used to uplift the cosmological constant in String-Theory de Sitter proposals, source a plethora of supersymmetry-breaking fluxes, that can interact nontrivially with other ingredients of the flux compactification. In this paper we perform a complex-structure decomposition of these fluxes, and compute the effect of the (0,3) flux component on the stabilization of Kähler moduli via D7-branes gaugino condensation. This allows us to obtain a new constraint on the validity of this stabilization mechanism. This effect does not appear hard to satisfy in de Sitter construction proposals that use long warped throats, but may be problematic in proposals where the warping is small.

Bounds on tower mass scales in the presence of throats of different warping

In Type IIB flux compactification realizing the metastable de Sitter (dS) vacuum, the uplift potential can be generated by D3‾-branes at the tip of Klebanov-Strassler throat. Then the uplift potential obeys the scaling law with respect to the tower mass scale msc, which can be the Kaluza-Klein (KK) mass scale associated with the throat containing D3‾-branes or the bulk tower mass scales, depending on the warping of the throat. On the other hand, in the presence of another throat of stronger warping, the KK mass scale associated with this throat is lower than msc. Nevertheless, the Higuchi bound and the condition that the tower mass scale is higher than the gravitino mass provide the upper bound on msc determined by the lowest tower mass scale (or gravitino mass). This bound also can be interpreted as the lower bound on the lowest tower mass scale determined by msc. We investigate this bound in detail when the throat containing D3‾-branes is strongly and weakly warped, respectively.

New families of scale separated vacua

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.

De Sitter vacua — when are ‘subleading corrections’ really subleading?

We consider various string-loop, warping and curvature corrections that are expected to appear in type IIB moduli stabilisation scenarios. It has recently been argued, in the context of strongly-warped LVS de Sitter vacua, that it is impossible to achieve parametric suppression in all of these corrections simultaneously [1]. We investigate corrections in the context of the recently discovered weakly-warped LVS de Sitter vacua, which represent a distinct branch of solutions in type IIB flux compactifications. The weakly-warped solution is supported by small conifold flux numbers MK ≲ 32, but still requires a large flux contribution to the D3-tadpole, now from the bulk. Warping corrections become less problematic, and some corrections even help to reach the weakly-warped regime of parameter space. Other corrections continue to be dangerous and would require numerical coefficients to be computed — and found to be small — in order not to destroy the consistency of the weakly-warped LVS de Sitter solution. We motivate why this may be possible.

Chiral Matter Multiplicities and Resolution-Independent Structure in 4D F-Theory Models

Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our approach is the conjecturally resolution-independent intersection pairing of the vertical part of the integer middle cohomology of smooth elliptic Calabi–Yau fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua. We illustrate our approach by analyzing vertical flux backgrounds for F-theory models with simple, simply-laced gauge groups and generic matter content, as well as models with U(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {U}}(1)$$\\end{document} gauge factors. We explicitly analyze resolutions of these F-theory models in which the elliptic fiber is realized as a cubic in P2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^2$$\\end{document} over an arbitrary (e.g., not necessarily toric) smooth base, and confirm the independence of the intersection pairing of the vertical part of the middle cohomology for the resolutions we study. In each model, we find that vertical flux backgrounds can produce nonzero multiplicities for a spanning set of anomaly-free chiral matter field combinations, suggesting that F-theory geometry imposes no additional linear constraints on allowed matter representations beyond those implied by 4D anomaly cancellation.

Brane nucleation in supersymmetric models

This paper explores the process of vacuum decay in supersymmetric models related to flux compactifications. In particular, we describe these instabilities within supersymmetric Lagrangians for a single three-form multiplet. This multiplet combines scalar fields, representing the moduli fields in four dimensions, with 3-form fields that influence the potential for these moduli via the integer flux of their associated 4-form field strength. Furthermore, using supersymmetry as a guide we obtain the form of the couplings of these fields to the membranes that act as sources to the 3-form potentials. Adding small supersymmetry breaking terms to these Lagrangians one can obtain instanton solutions describing the decay of the vacua in these models by the formation of a membrane bubble. These instantons combine the usual Coleman-de Luccia and the Brown-Teitelboim formalisms in a single unified model. We study simple numerical examples of theories with and without gravity in this new framework and generalize known Euclidean methods to accomodate the simulataneous inclusion of scalar fields and charged membranes to these instanton solutions. Moreover, we show explicitly in these examples how one recovers the static supersymmetric solutions in the limiting case where the supersymmetry breaking terms vanish. In this limit, the bubble becomes infinite and flat and represents a hybrid between the usual supersymmetric domain walls of field theory models and the brane solutions interpolating between the supersymmetric vacua; a sort of dressed supermembrane BPS solution. Finally, we briefly comment on the implications of these solutions in cosmological models based on the String Theory Landscape where these type of 4d effective theories could be relevant in inflationary scenarios.