Flux Superpotential Research Articles

Engineering small flux superpotentials and mass hierarchies

We study the stabilization of complex structure moduli in Type IIB flux compactifications by using recent general results about the form of the superpotential and Kähler potential near the boundaries of the moduli space. In this process we show how vacua with an exponentially small vacuum superpotential can be realized systematically and understood conceptually within asymptotic Hodge theory. We distinguish two types of vacua realizing such superpotentials that differ by the mass scales of the stabilized moduli. Masses polynomially depending on the moduli arise if the superpotential contains exponential corrections whose existence is required to ensure the non-degeneracy of the moduli space metric. We use the fact that such essential corrections are controlled by asymptotic Hodge theory and have recently been constructed for all one- and two-moduli asymptotic regimes. These insights allow us to obtain new vacua near boundaries in complex structure moduli space that include Seiberg-Witten points. In these examples we find that the scale of the vacuum superpotential can be bounded from below through the exponential of the negative D3-brane tadpole.

Conifold Vacua with Small Flux Superpotential

AbstractWe introduce a method for finding flux vacua of type IIB string theory in which the flux superpotential is exponentially small and at the same time one or more complex structure moduli are stabilized exponentially near to conifold points.

Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold

AbstractWe generalize the recently proposed mechanism by Demirtas, Kim, McAllister and Moritz[1] for the explicit construction of type IIB flux vacua with to the region close to the conifold locus in the complex structure moduli space. For that purpose tools are developed to determine the periods and the resulting prepotential close to such a codimension one locus with all the remaining moduli still in the large complex structure regime. As a proof of principle we present a working example for the Calabi‐Yau manifold .[24]

Vacua with Small Flux Superpotential.

We describe a method for finding flux vacua of type IIB string theory in which the Gukov-Vafa-Witten superpotential is exponentially small. We present an example with W_{0}≈2×10^{-8} on an orientifold of a Calabi-Yau hypersurface with (h^{1,1},h^{2,1})=(2,272), at large complex structure and weak string coupling.

α′-Corrections and de Sitter vacua - A mirage?

In this work we analyze the role of α′-corrections to type IIB orientifold compactifications in Kähler moduli stabilization and inflation. In particular, we propose a model independent scenario to achieve non-supersymmetric Minkowski and de Sitter vacua for geometric backgrounds with positive Euler-characteristic and generic number of Kähler moduli. The vacua are obtained by a tuning of the flux superpotential. Moreover, in the one-modulus case we argue for a mechanism to achieve model independent slow-roll.

The type IIA flux potential, 4-forms and Freed-Witten anomalies

We compute the full classical 4d scalar potential of type IIA Calabi-Yau orientifolds in the presence of fluxes and D6-branes. We show that it can be written as a bilinear form V = ZABρAρB, where the ρA are in one-to-one correspondence with the 4-form fluxes of the 4d effective theory. The ρA only depend on the internal fluxes, the axions and the topological data of the compactification, and are fully determined by the Freed-Witten anomalies of branes that appear as 4d string defects. The quadratic form ZAB only depends on the saxionic partners of these axions. In general, the ρA can be seen as the basic invariants under the discrete shift symmetries of the 4d effective theory, and therefore the building blocks of any flux-dependent quantity. All these polynomials may be obtained by derivation from one of them, associated to a universal 4-form. The standard mathcal{N}=1 supergravity flux superpotential is uniquely determined from this master polynomial, and vice versa.

D7‐brane moduli space in axion monodromy and fluxbrane inflation

AbstractWe analyze the quantum‐corrected moduli space of D7‐brane position moduli with special emphasis on inflationary model building. D7‐brane deformation moduli are key players in two recently proposed inflationary scenarios: The first, D7‐brane chaotic inflation, is a variant of axion monodromy inflation which allows for an effective 4d supergravity description. The second, fluxbrane inflation, is a stringy version of D‐term hybrid inflation. Both proposals rely on the fact that D7‐brane coordinates enjoy a shift‐symmetric Kähler potential at large complex structure of the Calabi‐Yau threefold, making them naturally lighter than other fields. This shift symmetry is inherited from the mirror‐dual Type IIA Wilson line on a D6‐brane at large volume. The inflaton mass can be provided by a tree‐level term in the flux superpotential. It induces a monodromy and, if tuned to a sufficiently small value, can give rise to a large‐field model of inflation. Alternatively, by a sensible flux choice one can completely avoid a tree‐level mass term, in which case the inflaton potential is induced via loop corrections. The positive vacuum energy can then be provided by a D‐term, leading to a small‐field model of hybrid natural inflation. In the present paper, we continue to develop a detailed understanding of the D7‐brane moduli space focusing among others on shift‐symmetry‐preserving flux choices, flux‐induced superpotential in Type IIB/F‐theory language, and loop corrections. While the inflationary applications represent our main physics motivation, we expect that some of our findings will be useful for other phenomenological issues involving 7‐branes in Type IIB/F‐theory constructions.

Type II/F-theory superpotentials and Ooguri–Vafa invariants of compact Calabi–Yau threefolds with three deformations

We calculate the D-brane superpotentials for two Calabi–Yau manifolds with three deformations by the generalized hypergeometric GKZ systems, which give rise to the flux superpotentials 𝒲 GVW of the dual F-theory compactification on the relevant Calabi–Yau fourfolds in the weak decoupling limit. We also compute the Ooguri–Vafa invariants from A-model expansion with mirror symmetry, which are related to the open Gromov–Witten invariants.

Holomorphic couplings in non‐perturbative string compactifications

AbstractIn this review article we present an analysis of several aspects of four‐dimensional, non‐perturbative 𝒩 = 1 compactifications of string theory. Our study focuses on brane dynamics and their effective physics as encoded in the holomorphic couplings of the low‐energy 𝒩 = 1 effective action, most prominently the superpotential W. This article is divided into three parts. In part one we derive the effective action of a spacetime‐filling D5‐brane in generic Type IIB Calabi‐Yau orientifold compactifications. In the second part we invoke tools from string dualities, namely from F‐theory, heterotic/F‐theory duality and mirror symmetry, for a more elaborate study of the dynamics of (p,q) 7‐branes and heterotic five‐branes. In this context we perform exact computations of the complete perturbative effective superpotential, both due to branes and background fluxes. Finally, in the third part we present a novel geometric description of five‐branes in Type IIB and heterotic M‐theory Calabi‐Yau compactifications via a non‐Calabi‐Yau threefold $\hat {Z}_3$, that is canonically constructed from the original five‐brane and the Calabi‐Yau threefold Z3 via a blow‐up. We use the blow‐up threefold $\hat {Z}_3$ to derive open‐closed Picard‐Fuchs differential equations, that govern the complete effective brane and flux superpotential. In addition, we present first evidence to interpret $\hat {Z}_3$ as a flux compactification geometrically dual to the original five‐brane by defining an SU(3)‐structure on $\hat {Z}_3$, that is generated dynamically by the five‐brane backreaction.

Five-brane superpotentials and heterotic/F-theory duality

Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branes is mapped into the geometry of an F-theory compactification manifold. In four-dimensional compactifications this identifies a five-brane wrapped on a curve in the base of an elliptically fibered Calabi–Yau threefold with a specific F-theory Calabi–Yau fourfold containing the blow-up of the five-brane curve. We argue that this duality can be reformulated by first constructing a non-Calabi–Yau heterotic threefold by blowing up the curve of the five-brane into a divisor with five-brane flux. Employing heterotic/F-theory duality this leads us to the construction of a Calabi–Yau fourfold and four-form flux. Moreover, we obtain an explicit map between the five-brane superpotential and an F-theory flux superpotential. The map of the open–closed deformation problem of a five-brane in a compact Calabi–Yau threefold into a deformation problem of complex structures on a dual Calabi–Yau fourfold with four-form flux provides a powerful tool to explicitly compute the five-brane superpotential.

Computing brane and flux superpotentials in F-theory compactifications

In four-dimensional F-theory compactifications with N=1 supersymmetry the fields describing the dynamics of space-time filling 7-branes are part of the complex structure moduli space of the internal Calabi-Yau fourfold. We explicitly compute the flux superpotential in F-theory depending on all complex structure moduli, including the 7-brane deformations and the field corresponding to the dilaton-axion. Since fluxes on the 7-branes induce 5-brane charge, a local limit allows to effectively match the F-theory results to a D5-brane in a non-compact Calabi-Yau threefold with threeform fluxes. We analyze the classical and instanton contributions to the F-theory superpotential using mirror symmetry for Calabi-Yau fourfolds. The F-theory compactifications under consideration also admit heterotic dual descriptions and we discuss the identification of the moduli in this non-perturbative duality.

Minkowski vacuum transitions in (nongeometric) flux compactifications

In this work we study the generalization of twisted homology to geometric and nongeometric backgrounds. In the process, we describe the necessary conditions to wrap a network of D-branes on twisted cycles. If the cycle is localized in time, we show how by an instantonic brane mediation, some D-branes transform into fluxes on different backgrounds, including nongeometric fluxes. As a consequence, we show that in the case of a IIB six-dimensional torus compactification on a simple orientifold, the flux superpotential is not invariant by this brane-flux transition, allowing the connection among different Minkowski vacuum solutions. For the case in which nongeometric fluxes are turned on, we also discuss some topological restrictions for the transition to occur. In this context, we show that there are some vacuum solutions protected to change by a brane-flux transition.

The D5-brane effective action and superpotential in [formula omitted] compactifications

The four-dimensional effective action for D5-branes in generic compact Calabi–Yau orientifolds is computed by performing a Kaluza–Klein reduction. The N = 1 Kähler potential, the superpotential, the gauge–kinetic coupling function and the D-terms are derived in terms of the geometric data of the internal space and of the two-cycle wrapped by the D5-brane. In particular, we obtain the D5-brane and flux superpotential by integrating out four-dimensional three-forms which couple via the Chern–Simons action. Also the infinitesimal complex structure deformations of the two-cycle induced by the deformations of the ambient space contribute to the F-terms. The superpotential can be expressed in terms of relative periods depending on both the open and closed moduli. To analyze this dependence we blow up along the two-cycle and obtain a rigid divisor in an auxiliary compact threefold with negative first Chern class. The variation of the mixed Hodge structure on this blown-up geometry is equivalent to the original deformation problem and can be analyzed by Picard–Fuchs equations. We exemplify the blow-up procedure for a non-compact Calabi–Yau threefold given by the canonical bundle over del Pezzo surfaces.

Geometric transitions and dynamical SUSY breaking

We show that the physics of D-brane theories that exhibit dynamical SUSY breaking due to stringy instanton effects is well captured by geometric transitions, which recast the nonperturbative superpotential as a classical flux superpotential. This allows for simple engineering of Fayet, Polonyi, O'Raifeartaigh, and other canonical models of supersymmetry breaking in which an exponentially small scale of breaking can be understood either as coming from stringy instantons or as arising from the classical dynamics of fluxes.

Runaway in the landscape

We consider flux compactifications of type IIB string theory on the mirror of a rigid Calabi-Yau. In special cases, these models are dual to the type IIA flux vacua with runaway direction in flux space. We show that new weak coupling AdS solutions can be found for large complex structure, while Minkowski solutions with all moduli stabilized are confined to be at strong coupling. The existence of these solutions, as found in a previous work, is nevertheless guaranteed by a nonrenormalization theorem of the type IIB flux superpotential. Based on our results, we are led to the conjecture that supersymmetric runaway directions in flux space are always accompanied by a spectrum of moduli masses reaching down to the AdS scale. This could be violated in a nonsupersymmetric situation.

Non-perturbative and flux superpotentials for type I strings on the [formula omitted] orbifold

Non-perturbative effects are studied for type I strings on the Z 3 orbifold with Chan–Paton symmetry broken by Wilson lines. Generalizing previous analyses that have focussed on (bi)fundamentals, it is argued that (anti)symmetric representations of the resulting gauge group play a decisive role in generating an ADS-like superpotential in this and related cases. Non-perturbative corrections in the closed string moduli are only allowed if properly dressed with open string fields charged under the anomalous U ( 1 ) of the orbifold. A general discussion of instanton effects in SYM and string theories is given. Non-perturbative superpotentials induced by both ED5's and ED1's are analyzed. The superpotential generated by closed string fluxes, viz. Scherk–Schwarz shifts (torsion), R–R 3-form flux as well as non-geometric fluxes is derived. Some preliminary comments on the compatibility of the two kinds of superpotentials and the issue of moduli stabilization are presented.

Nonrenormalization of flux superpotentials in string theory

Recent progress in understanding modulus stabilization in string theory relies on the existence of a non-renormalization theorem for the 4D compactifications of Type IIB supergravity which preserve N=1 supersymmetry. We provide a simple proof of this non-renormalization theorem for a broad class of Type IIB vacua using the known symmetries of these compactifications, thereby putting them on a similar footing as the better-known non-renormalization theorems of heterotic vacua without fluxes. The explicit dependence of the tree-level flux superpotential on the dilaton field makes the proof more subtle than in the absence of fluxes.

De Sitter vacua from heterotic M-theory

It is shown how metastable de Sitter vacua might arise from heterotic M-theory. The balancing of its two non-perturbative effects, open membrane instantons against gaugino condensation on the hidden boundary, which act with opposing forces on the interval length, is used to stabilize the orbifold modulus (dilaton) and other moduli. The non-perturbative effects break supersymmetry spontaneously through F-terms which leads to a positive vacuum energy density. In contrast to the situation for the weakly coupled heterotic string, the charged scalar matter fields receive non-vanishing vacuum expectation values and therefore masses in a phenomenologically relevant regime. It is important that in order to obtain these de Sitter vacua we are not relying on exotic effects or fine-tuning of parameters. Vacua with more realistic supersymmetry breaking scales and gravitino masses are obtained by breaking the hidden E 8 gauge group down to groups of smaller rank. Also small values for the open membrane instanton Pfaffian are favored in this respect. Finally we outline how the incorporation of additional flux superpotentials can be used to stabilize the remaining moduli.