Type IIB Compactifications Research Articles

Flux vacua in type IIB compactifications on orbifolds: their finiteness and minimal string coupling

We perform a detailed study of (supersymmetric) moduli stabilisation in type IIB toroidal orientifolds with fluxes. We provide strong evidence towards exhaustion of the finite number of inequivalent vacua for a given total 3-form flux charge Nflux. We also find that the minimal value of the string coupling gs is given in terms of Nflux and present strong evidence for the asymptotic relation gs, min ∼ c/Nfluxα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ c/{N}_{\ extrm{flux}}^{\\alpha } $$\\end{document}, with α = 1, 2, valid for not too small Nflux, where c is an order 1 coefficient and α depends on the orbifold. Imposing tadpole cancellation, Nflux is bounded from the number of orientifold O3-planes. Combined with the flux quantisation, this forbids orientifold vacua without anti-brane charge. On the other hand, the presence of negative D3-brane charge induced by magnetised D7-branes breaks supersymmetry and relaxes the bound, allowing significantly smaller values for gs.

Loop blow-up inflation

We present a new model of string inflation driven by a blow-up Kähler modulus of type IIb compactifications with a potential generated by string loops. Slow-roll is naturally realized thanks to the fact that the blow-up mode is a leading-order flat direction lifted by string loops which are unavoidable and generate a plateau at large field values. We check that throughout the whole inflationary dynamics the effective field theory is under control. We perform a phenomenological analysis determining the exact number of efoldings by studying the post-inflationary evolution. We determine the values of the microscopic parameters which lead to agreement with CMB data, together with the prediction of a tensor-to-scalar ratio of order r ∼ 10−5.

Swampland constraints on the symmetry topological field theory of supergravity

We consider string/M-theory reductions on a compact space X=Xloc∪X∘, where Xloc contains the singular locus and X∘ is its complement. For the resulting supergravity theories, we construct a suitable symmetry topological field theory (SymTFT) associated with the boundary ∂Xloc∐∂X∘. We propose that boundary conditions for different Bogomol’nyi-Prasad-Sommerfield branes wrapping the same boundary cycles must be correlated for the SymTFT to yield an absolute theory consistent with quantum gravity. Using heterotic/M-theory duality, this constraint can be translated into a field-theoretic statement, which restricts the global structure of d≥7, N=1 supergravity theories to reproduce precisely the landscape of untwisted toroidal heterotic compactifications. Furthermore, for six-dimensional (2, 0) theories, we utilize a subtle interplay between gauged 0-, 2-, and 4-form symmetries to provide a bottom-up explanation of the correlated boundary conditions in K3 compactifications of type IIB. Published by the American Physical Society 2024

Galois groups of uplifted de Sitter vacua

We compute the Galois group of a polynomial whose roots are determined by the critical points of a scalar potential in type IIB compactifications. We focus our study on certain perturbative models where it is feasible to construct a de Sitter vacuum within the effective theory by introducing non-geometric fluxes, D-branes or non-BPS states. Our findings clearly show that all de Sitter vacua derived from lifting AdS stable vacua are associated with an unsolvable Galois group. This suggests a deeper connection between the fundamental principles of Galois theory and its applications in the construction of dS vacua.

On the intermediate Jacobian of M5-branes

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise a method to analyze the Hodge structure (and hence the dimension of the intermediate Jacobian) of vertical divisors in these fourfolds, using only the data available from a type IIB compactification on the O3/O7 Calabi-Yau orientifold. Our method utilizes simple combinatorial formulae (that we prove) for the equivariant Hodge numbers of the Calabi-Yau orientifolds and their prime toric divisors, along with a formula for the Euler characteristic of vertical divisors in the corresponding elliptic Calabi-Yau fourfold. Our formula for the Euler characteristic includes a conjectured correction term that accounts for the contributions of pointlike terminal ℤ2 singularities corresponding to perturbative O3-planes. We check our conjecture in a number of explicit examples and find perfect agreement with the results of direct computations.

A 4D IIB flux vacuum and supersymmetry breaking. Part II. Bosonic spectrum and stability

We recently constructed type-IIB compactifications to four dimensions depending on a single additional coordinate, where a five-form flux Φ on an internal torus leads to a constant string coupling. Supersymmetry is fully broken when the internal manifold includes a finite interval of length ℓ, which is spanned by a conformal coordinate in a finite range 0 < z < zm. Here we examine the low-lying bosonic spectra and their classical stability, paying special attention to self-adjoint boundary conditions. Special boundary conditions result in the emergence of zero modes, which are determined exactly by first-order equations. The different sectors of the spectrum can be related to Schrödinger operators on a finite interval, characterized by pairs of real constants μ and μ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\\mu } $$\\end{document}, with μ equal to 1/3 or 2/3 in all cases and different values of μ~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\\mu } $$\\end{document}. The potentials behave as μ2−1/4z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{\\mu^2-1/4}{z^2} $$\\end{document} and μ~2−1/4zm−z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{{\\overset{\\sim }{\\mu}}^2-1/4}{{\\left({z}_m-z\\right)}^2} $$\\end{document} near the ends and can be closely approximated by exactly solvable trigonometric ones. With vanishing internal momenta, one can thus identify a wide range of boundary conditions granting perturbative stability, despite the intricacies that emerge in some sectors. For the Kaluza-Klein excitations of non-singlet vectors and scalars the Schrödinger systems couple pairs of fields, and the stability regions, which depend on the background, widen as the ratio Φ/ℓ4 decreases.

Open strings in IIB orientifold reductions

We consider type IIB compactifications on a general 4D group manifold with different types of possible spacetime filling O-planes and the corresponding D-branes parallel to them. Once fluxes allowed by the associated orientifold projection are included, a 6D \U0001d4a9 = (1, 1) gauged supergravity is obtained. In this paper we show how the consistent coupling to dynamical open strings living on the spacetime filling D-branes may be captured by the inclusion of extra vector multiplets and extra embedding tensor deformations on the gauged supergravity side. As a result, the quadratic constraints on the embedding tensor consistently reproduce the source corrected 10D Bianchi identities. Furthermore, the field strength modifications induced by the open string sector could potentially be understood as U-dual versions of the Green-Schwarz terms. Finally, the entire scalar potential of the theory exactly matches the one obtained from reduction of the bulk action plus the source contributions.

Smearing and unsmearing KKLT AdS vacua

Gaugino condensation on D-branes wrapping internal cycles gives a mechanism to stabilize the associated moduli. According to the effective field theory, this gives rise, when combined with fluxes, to supersymmetric AdS4 solutions. In this paper we provide a ten-dimensional description of these vacua. We first find the supersymmetry equations for type II AdS4 vacua with gaugino condensates on D-branes, in the framework of generalized complex geometry. We then solve them for type IIB compactifications with gaugino condensates on smeared D7-branes. We show that supersymmetry requires a (conformal) Calabi-Yau manifold and imaginary self-dual three-form fluxes with an additional (0,3) component. The latter is proportional to the cosmological constant, whose magnitude is determined by the expectation value of the gaugino condensate and the stabilized volume of the cycle wrapped by the branes. This confirms, qualitatively and quantitatively, the results obtained using effective field theory. We find that exponential separation between the AdS and the KK scales seems possible as long as the three-form fluxes are such that their (0,3) component is exponentially suppressed. As for the localized solution, it requires going beyond SU(3)-structure internal manifolds. Nevertheless, we show that the action can be evaluated on-shell without relying on the details of such complicated configuration. We find that no “perfect square” structure occurs, and the result is divergent. We compute the four-fermion contributions, including a counterterm, needed to cancel these divergences.

D-instanton superpotential in string theory

We study the non-perturbative superpotential generated by D(-1)-branes in type IIB compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. To compute the D-instanton superpotential, we study F-theory compactification on toric complete intersection elliptic Calabi-Yau fourfolds. We take the Sen-limit, but with finite gs, in F-theory compactifications with the restriction that all D7-branes are carrying SO(8) gauge groups, which we call the global Sen-limit. In the global Sen-limit, the axio-dilaton is not varying in the compactification manifold. We compute the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, and show that the Picard-Fuchs equations of the elliptic fourfolds split into that of the underlying Calabi-Yau threefolds and of the elliptic fiber. We then demonstrate that this splitting property of the Picard-Fuchs equation implies that the fourform period of the elliptic Calabi-Yau fourfolds in the global Sen-limit does not contain exponentially suppressed terms mathcal{O}left({e}^{-pi /{g}_s}right) . With this result, we finally show that in the global Sen-limit, the superpotential of the underlying type IIB compactification does not receive D(-1)-instanton contributions.

The tadpole problem

We examine the mechanism of moduli stabilization by fluxes in the limit of a large number of moduli. We conjecture that one cannot stabilize all complex-structure moduli in F-theory at a generic point in moduli space (away from singularities) by fluxes that satisfy the bound imposed by the tadpole cancellation condition. More precisely, while the tadpole bound in the limit of a large number of complex-structure moduli goes like 1/4 of the number of moduli, we conjecture that the amount of charge induced by fluxes stabilizing all moduli grows faster than this, and is therefore larger than the allowed amount. Our conjecture is supported by two examples: K3 × K3 compactifications, where by using evolutionary algorithms we find that moduli stabilization needs fluxes whose induced charge is 44% of the number of moduli, and Type IIB compactifications on mathbbm{CP} 3, where the induced charge of the fluxes needed to stabilize the D7-brane moduli is also 44% of the number of these moduli. Proving our conjecture would rule out de Sitter vacua obtained via antibrane uplift in long warped throats with a hierarchically small supersymmetry breaking scale, which require a large tadpole.

Tackling the SDC in AdS with CFTs

We study the Swampland Distance Conjecture for supersymmetric theories with AdS5 backgrounds and fixed radius through their mathcal{N} = 2 SCFT holographic duals. By the Maldacena-Zhiboedov theorem, around a large class of infinite-distance points there must exist a tower of exponentially massless higher-spin fields in the bulk, for which we find bounds on the decay rate in terms of the conformal data. We discuss the origin of this tower in the gravity side for type IIB compactification on S5 and its orbifolds, and comment about more general cases.

Dynamical tadpoles, stringy cobordism, and the SM from spontaneous compactification

We consider string theory vacua with tadpoles for dynamical fields and uncover universal features of the resulting spacetime-dependent solutions. We argue that the solutions can extend only a finite distance ∆ away in the spacetime dimensions over which the fields vary, scaling as ∆n∼ mathcal{T} with the strength of the tadpole mathcal{T} . We show that naive singularities arising at this distance scale are physically replaced by ends of spacetime, related to the cobordism defects of the swampland cobordism conjecture and involving stringy ingredients like orientifold planes and branes, or exotic variants thereof. We illustrate these phenomena in large classes of examples, including AdS5×T1,1 with 3-form fluxes, 10d massive IIA, M-theory on K3, the 10d non-supersymmetric USp(32) strings, and type IIB compactifications with 3-form fluxes and/or magnetized D-branes. We also describe a 6d string model whose tadpole triggers spontaneous compactification to a semirealistic 3-family MSSM-like particle physics model.

String (In)Stability Issues with Broken Supersymmetry

We review the main results of our investigations motivated by the tadpole potentials of ten--dimensional strings with broken supersymmetry. While these are at best partial indications, it is hard to resist the feeling that they do capture some lessons of String Theory. For example, these very tadpole potentials lead to weak-string-coupling cosmologies that appear to provide clues on the onset of the inflation from an initial fast roll. The transition, if accessible to us, would offer a natural explanation for the lack of power manifested by the CMB at large angular scales. In addition, the same tadpole potentials can drive spontaneous compactifications to lower--dimensional Minkowski spaces at corresponding length scales. Furthermore, the cosmological solutions exhibit an intriguing ``instability of isotropy that, if taken at face value, would point to an accidental origin of compactification. Finally, symmetric static \(AdS \times S\) solutions driven by the tadpole potentials also exist, but they are unstable due to mixings induced by their internal fluxes. On the other hand, the original Dudas--Mourad solution is perturbatively stable, and we have gathered some detailed evidence that instabilities induced by internal fluxes can be held under control in a similar class of weak--coupling type-IIB compactifications to Minkowski space.

Testing swampland conjectures with machine learning

We consider Type IIB compactifications on an isotropic torus T^6 threaded by geometric and non geometric fluxes. For this particular setup we apply supervised machine learning techniques, namely an artificial neural network coupled to a genetic algorithm, in order to obtain more than sixty thousand flux configurations yielding to a scalar potential with at least one critical point. We observe that both stable AdS vacua with large moduli masses and small vacuum energy as well as unstable dS vacua with small tachyonic mass and large energy are absent, in accordance to the refined de Sitter conjecture. Moreover, by considering a hierarchy among fluxes, we observe that perturbative solutions with small values for the vacuum energy and moduli masses are favored, as well as scenarios in which the lightest modulus mass is much smaller than the corresponding AdS vacuum scale. Finally we apply some results on random matrix theory to conclude that the most probable mass spectrum derived from this string setup is that satisfying the Refined de Sitter and AdS scale conjectures.

On the Euler characteristic of a relative hypersurface

We derive a general formula for the Euler characteristic of a fibration of projective hypersurfaces in terms of invariants of an arbitrary base variety. When the general fiber is an elliptic curve, such formulas have appeared in the physics literature in the context of calculating D-brane charge for M-/F-theory and type-IIB compactifications of string vacua. While there are various methods for computing Euler characteristics of algebraic varieties, we prove a base-independent pushforward formula which reduces the computation of the Euler characteristic of relative hypersurfaces to simple algebraic manipulations of rational expressions determined by its divisor class in a projective bundle. We illustrate our methods by applying them to an explicit family of relative hypersurfaces whose fibers are of arbitrary dimension and degree.

High U(1) charges in type IIB models and their F-theory lift

We construct models with U(1) gauge group and matter with charges up to 6, in the context of type IIB compactifications. We show explicitly that models with charges up to 4 can be derived from corresponding models in F-theory by applying the Sen weak coupling limit. We derive which type IIB models should be the limit of charge 5 and 6 F-theory models. Explicit six dimensional type IIB models with maximal charge 5 and 6 are constructed on an algebraic K3 surface that is the double cover of ℂℙ2. By using type IIB results we are also able to rediscover the F-theory charge 4 model in a straightforward way.

RG flows with supersymmetry enhancement and geometric engineering

In this paper we study a class of mathcal{N}=2 SCFTs with ADE global symmetry defined via Type IIB compactification on a class of hypersurfaces in ℂ3 × ℂ*. These can also be constructed by compactifying the 6d (2,0) theory of type ADE on a sphere with an irregular and a full punctures. When we couple to the ADE moment map a chiral multiplet in the adjoint representation and turn on a (principal) nilpotent vev for it, all the theories in this family display enhancement of supersymmetry in the infrared. We observe that all known examples of theories which flow, upon the same type of deformation, to strongly coupled mathcal{N}=2 theories fit naturally in our framework, thus providing a new perspective on this topic. We propose an infrared equivalence between this RG flow and a manifestly mathcal{N}=2 preserving one and, as a byproduct, we extract a precise prescription to relate the SW curves describing the UV and IR fixed points for all theories with A or D global symmetry. We also find, for a certain subclass, a simple relation between UV and IR theories at the level of chiral algebras.

Revisiting arithmetic solutions to theW=0condition

The gravitino mass is expected not to be much smaller than the Planck scale for a large fraction of vacua in flux compactifications. There is no continuous parameter to tune even by hand, and it seems that the gravitino mass can be small only as a result of accidental cancellation among period integrals weighted by integer-valued flux quanta. DeWolfe et.al. (2005) proposed to pay close attention to vacua where the Hodge decomposition is possible within a number field, so that the precise cancellation takes place as a result of algebra. We focus on a subclass of those vacua---those with complex multiplications---and explore more on the idea in this article. It turns out, in Type IIB compactifications, that those vacua admit non-trivial supersymmetric flux configurations if and only if the reflex field of the Weil intermediate Jacobian is isomorphic to the quadratic imaginary field generated by the axidilaton vacuum expectation value. We also found that flux statistics is highly enriched on such vacua, as F-term conditions become linearly dependent.

Kähler moduli stabilization in semirealistic magnetized orbifold models

We study K\"ahler moduli stabilizations in semi-realistic magnetized D-brane models based on $ Z_2\times Z_2'$ toroidal orbifolds. In type IIB compactifications, 3-form fluxes can stabilize the dilaton and complex structure moduli fields, but there remain some massless closed string moduli fields, K\"ahler moduli. The magnetic fluxes generate Fayet-Iliopoulos terms, which can fix ratios of K\"ahler moduli. On top of that, we consider D-brane instanton effects to stabilize them in concrete D-brane models and investigate the brane configurations to confirm that the moduli fields can be stabilized successfully. In this paper, we treat two types of D-brane models. One is based on D9-brane systems respecting the Pati-Salam model. The other is realized in a D7-brane system breaking the Pati-Salam gauge group. We find suitable configurations where the D-brane instantons can stabilize the moduli fields within both types of D-brane models, explaining an origin of a small constant term of the superpotential which is a key ingredient for successful moduli stabilizations.

The gravitational sector of 2d (0, 2) F-theory vacua

F-theory compactifications on Calabi-Yau fivefolds give rise to two-dimensional N = (0, 2) supersymmetric field theories coupled to gravity. We explore the dilaton supergravity defined by the moduli sector of such compactifications. The massless moduli spectrum is found by uplifting Type IIB compactifications on Calabi-Yau fourfolds. This spectrum matches expectations from duality with M-theory on the same elliptic fibration. The latter defines an N = 2 Supersymmetric Quantum Mechanics related to the 2d (0, 2) F-theory supergravity via circle reduction. Using our recent results on the gravitational anomalies of duality twisted D3-branes wrapping curves in Calabi-Yau fivefolds we show that the F-theory spectrum is anomaly free. We match the classical Chern-Simons terms of the M-theory Super Quantum Mechanics to one-loop contributions to the effective action by S1 reduction of the dual F-theory.