Calabi-Yau Compactifications Research Articles

The dimensional reduction and Kähler metric of forms in flux and warping

We present a first-principles derivation of the Kähler metric for axion-like moduli of conformally Calabi-Yau compactifications of IIB string theory with imaginary self-dual 3-form flux at the classical level. We find that the warp factor and flux modify the moduli space metric and therefore Kähler potential even in classical supergravity, with the modifications scaling as (volume)−2/3 in the large-volume limit. Our derivation emphasizes the role of constraints from 10D gauge symmetries and highlights metric formality as a geometric property that protects the moduli space of highly supersymmetric toroidal orientifolds. Our results have important quantitative implications for nonperturbative moduli stabilization, phenomenology, and cosmology in flux compactifications.

Constraints on 2d CFT partition functions

Modular invariance is known to constrain the spectrum of 2d conformal field theories. We investigate this constraint systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We also consider generalized partition functions of N = (2,2) superconformal theories and discuss the application of our results to Calabi-Yau compactifications. For Calabi-Yau threefolds with no enhanced symmetry we find that there must always be non-BPS primary states of weight 0.6 or less.

Heterotic Calabi-Yau compactifications with flux

Compactifications of the heterotic string with NS flux normally require non Calabi-Yau internal spaces which are complex but no longer K\"ahler. We point out that this conclusion rests on the assumption of a maximally symmetric four-dimensional space-time and can be avoided if this assumption is relaxed. Specifically, it is shown that an internal Calabi-Yau manifold is consistent with the presence of NS flux provided four-dimensional space-time is taken to be a domain wall. These Calabi-Yau domain wall solutions can still be associated with a covariant four-dimensional N=1 supergravity. In this four-dimensional context, the domain wall arises as the "simplest" solution to the effective supergravity due to the presence of a flux potential with a runaway direction. Our main message is that NS flux is a legitimate ingredient for moduli stabilization in heterotic Calabi-Yau models. Ultimately, the success of such models depends on the ability to stabilize the runaway direction and thereby "lift" the domain wall to a maximally supersymmetric vacuum.

Vacuum varieties, holomorphic bundles and complex structure stabilization in heterotic theories

We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a complicated landscape of vacua in complex structure moduli space. We develop methods to systematically map out this multi-branched vacuum space, in a computable and explicit manner. In analysing the resulting vacua, it is found that the associated Calabi-Yau three-folds are sometimes stabilized at a value of complex structure resulting in a singular compactification manifold. We describe how it is possible to resolve these singularities, in some cases, while maintaining computational control over the moduli stabilization mechanism. The discussion is illustrated throughout the paper with explicit worked examples.

Finding all flux vacua in an explicit example

AbstractWe explicitly construct all supersymmetric flux vacua of a particular Calabi-Yau compactification of type IIB string theory for a small number of flux carrying cycles and a given D3-brane tadpole. The analysis is performed in the large complex structure region by using the polynomial homotopy continuation method, which allows to find all stationary points of the polynomial equations that characterize the supersymmetric vacuum solutions. The number of vacua as a function of the D3 tadpole is in agreement with statistical studies in the literature. We calculate the available tuning of the cosmological constant from fluxes and extrapolate to scenarios with a larger number of flux carrying cycles. We also verify the range of scales for the moduli and gravitino masses recently found for a single explicit flux choice giving a Kähler uplifted de Sitter vacuum in the same construction.

The web of D-branes at singularities in compact Calabi-Yau manifolds

We present novel continuous supersymmetric transitions which take place among different chiral configurations of D3/D7 branes at singularities in the context of type IIB Calabi-Yau compactifications. We find that distinct local models which admit a consistent global embedding can actually be connected to each other along flat directions by means of transitions of bulk-to-flavour branes. This has interesting interpretations in terms of brane recombination/splitting and brane/anti-brane creation/annihilation. These transitions give rise to a large web of quiver gauge theories parametrised by splitting/recombination modes of bulk branes which are not present in the non-compact case. We illustrate our results in concrete global embeddings of chiral models at a dP_0 singularity.

Finite time singularities for Lagrangian mean curvature flow

Given any embedded Lagrangian on a four-dimensional compact Calabi-Yau, we find another Lagrangian in the same Hamiltonian isotopy class that develops a finite time singularity under mean curvature flow. This contradicts a weaker version of the Thomas-Yau conjecture regarding long time existence and convergence of Lagrangian mean curvature flow

Quantum black holes in Type-IIA String Theory

We study black hole solutions of Type-IIA Calabi-Yau compactifications in the presence of quantum perturbative corrections. We define a class of black holes that only exist in the presence of quantum corrections and that, consequently, can be considered as purely quantum black holes. The regularity conditions of the solutions impose the topological constraint h 1,1 > h 2,1 on the Calabi-Yau manifold, defining a class of admissible compactifications, which we prove to be non-empty for h 1,1 = 3 by explicitly constructing the corresponding Calabi-Yau manifolds, new in the literature.

Geodesics on Calabi-Yau manifolds and winding states in non-linear sigma models

We conjecture that a non-flat $D$-real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4, has locally length minimizing closed geodesics, and that the number of these with length less than L grows asymptotically as L^{D}. We also outline the physical arguments behind this conjecture, which involve the claim that all states in a nonlinear sigma model can be identified as "momentum" and "winding" states in the large volume limit.

Moduli stabilization in type II Calabi-Yau compactifications at finite temperature

Abstract We consider the type II superstring compactified on Calabi-Yau threefolds, at finite temperature. The latter is implemented at the string level by a free action on the Euclidean time circle. We show that all Kähler and complex structure moduli involved in the gauge theories geometrically engineered in the vicinity of singular loci are lifted by the stringy thermal effective potential. The analysis is based on the effective gauged super-gravity at low energy, without integrating out the non-perturbative BPS states becoming massless at the singular loci. The universal form of the action in the weak coupling regime and at low enough temperature is determined in two cases. Namely, the conifold locus, as well as a locus where the internal space develops a genus-g curve of A N −1 singularities, thus realizing an SU(N ) gauge theory coupled to g hypermultiplets in the adjoint. In general, we argue that the favored points of stabilization sit at the intersection of several such loci. As a result, the entire vector multiplet moduli space is expected to be lifted, together with hypermultiplet moduli. The scalars are dynamically stabilized during the cosmological evolution induced by the back-reaction of the thermal effective potential on the originally static background. When the universe expands and the temperature T drops, the scalars converge to minima, with damped oscillations. Moreover, they store an energy density that scales as T 4, which never dominates over radiation. The reason for this is that the mass they acquire at one-loop is of order the temperature scale, which is time-dependent rather than constant. As an example, we analyze the type IIA compactification on a hy-persurface $$ \mathbb{P}_{{\left( {1,1,2,2,6} \right)}}^4 $$ [12], with Hodge numbers h 11 = 2 and h 12 = 128. In this case, both Kähler moduli are stabilized at a point, where the internal space develops a node and an enhanced SU(2) gauge theory coupled to 2 adjoint hypermultiplets. This shows that in the dual thermal heterotic picture on K3 × T 2, the torus modulus and the axio-dilaton are stabilized, though in a strong coupling heterotic regime.

Global embeddings for branes at toric singularities

Abstract We describe how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D-branes, including non-compact flavor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. As an illustration of our discussion we focus on D3 and D7-branes on (the partially resolved) (dP 0)3 singularity, its embedding in a specific Calabi-Yau manifold as a hypersurface in a toric variety, the related type IIB orientifold compactification, as well as the corresponding F-theory uplift. Our techniques generalize naturally to complete intersections, and to a large class of F-theory backgrounds with singularities.

Heterotic line bundle standard models

Abstract In a previous publication, arXiv:1106.4804, we have found 200 models from heterotic Calabi-Yau compactifications with line bundles, which lead to standard models after taking appropriate quotients by a discrete symmetry and introducing Wilson lines. In this paper, we construct the resulting standard models explicitly, compute their spectrum including Higgs multiplets, and analyze some of their basic properties. After removing redundancies we find about 400 downstairs models, each with the precise matter spectrum of the supersymmetric standard model, with one, two or three pairs of Higgs doublets and no exotics of any kind. In addition to the standard model gauge group, up to four Green-Schwarz anomalous U(1) symmetries are present in these models, which constrain the llowed operators in the four-dimensional effective supergravity. The vector bosons associated to these anomalous U(1) symmetries are massive. We explicitly compute the spectrum of allowed operators for each model and present the results, together with the defining data of the models, in a database of standard models accessible here. Based on these results we analyze elementary phenomenological properties. For example, for about 200 models all dimension four and five proton decay violating operators are forbidden by the additional U(1) symmetries.

A note on poly-instanton effects in type IIB orientifolds on Calabi-Yau threefolds

The zero mode structure for the generation of poly-instanton corrections for Euclidian D3-branes wrapping complex surfaces in Type IIB orientifolds with O7- and O3-planes is analyzed. Working examples of such surfaces and explicit embeddings into compact Calabi-Yau threefolds are presented, with special emphasis on geometries capable of realizing the LARGE volume scenario.

Automorphic instanton partition functions on Calabi-Yau threefolds

We survey recent results on quantum corrections to the hypermultiplet moduli space ℳ in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X × S1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that when the theory in three dimensions exhibits an arithmetic "U-duality" symmetry G(ℤ) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. In the case of four-dimensional \U0001d4a9 = 2 theories on ℝ × S1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function is a holomorphic section on the twistor space over ℳ.

Toric K3-fibred Calabi-Yau manifolds with del Pezzo divisors for string compactifications

AbstractWe analyse several explicit toric examples of compact K3-fibred Calabi-Yau three-folds. These manifolds can be used for the study of string dualities and are crucial ingredients for the construction of LARGE Volume type IIB vacua with promising applications to cosmology and particle phenomenology. In order to build a phenomenologically viable model, on top of the two moduli corresponding to the base and the K3 fibre, we demand also the existence of two additional rigid divisors: the first supporting the non-perturbative effects needed to achieve moduli stabilisation, and the second allowing the presence of chiral matter on wrapped D-branes. We clarify the topology of these rigid divisors by discussing the interplay between a diagonal structure of the Calabi-Yau volume and D-terms. Del Pezzo divisors appearing in the volume form in a completely diagonal way are natural candidates for supporting non-perturbative effects and for quiver constructions, while ‘non-diagonal’ del Pezzo and rigid but not del Pezzo divisors are particularly interesting for model building in the geometric regime. Searching through the existing list of four dimensional reflexive lattice polytopes, we find 158 examples admitting a Calabi-Yau hypersurface with a K3 fibration and four Kähler moduli where at least one of the toric divisors is a ‘diagonal’ del Pezzo. We work out explicitly the topological details of a few examples showing how, in the case of simplicial polytopes, all the del Pezzo divisors are ‘diagonal’, while ‘non-diagonal’ ones appear only in the case of non-simplicial polytopes. A companion paper will use these results in the study of moduli stabilisation for globally consistent explicit Calabi-Yau compactifications with the local presence of chirality.

A sufficient condition for de Sitter vacua in type IIB string theory

We derive a sufficient condition for realizing meta-stable de Sitter vacua with small positive cosmological constant within type IIB string theory flux compactifications with spontaneously broken supersymmetry. There are a number of `lamp post' constructions of de Sitter vacua in type IIB string theory and supergravity. We show that one of them -- the method of `K\"ahler uplifting' by F-terms from an interplay between non-perturbative effects and the leading $\alpha'$-correction -- allows for a more general parametric understanding of the existence of de Sitter vacua. The result is a condition on the values of the flux induced superpotential and the topological data of the Calabi-Yau compactification, which guarantees the existence of a meta-stable de Sitter vacuum if met. Our analysis explicitly includes the stabilization of all moduli, i.e. the K\"ahler, dilaton and complex structure moduli, by the interplay of the leading perturbative and non-perturbative effects at parametrically large volume.

Spinflation with angular potentials

We investigate in detail the cosmological consequences of realistic angular dependent potentials in the brane inflation scenario. Embedding a warped throat into a compact Calabi-Yau space with all moduli stabilized breaks the no-scale structure and induces angular dependence in the potential of the probe D3-brane. We solve the equations of motion from the DBI action in the warped deformed conifold including linearized perturbations around the imaginary self-dual solution. Our numerical solutions show that angular dependence is a next to leading order correction to the dominant radial motion of the brane, however, just as angular motion typically increases the amount of inflation (spinflation), having additional angular dependence also increases the amount of inflation. We also derive an analytic approximation for the number of e-foldings along the DBI trajectory in terms of the compactification parameters.

Poly-instanton inflation

We propose a new inflationary scenarioin type IIB Calabi-Yau compactifications,where the inflaton is a Kähler modulus parameterising the volume of an internal four-cycle.The inflaton potential is generated via poly-instanton corrections to the superpotential whichgive rise to a naturally flat direction due to their double exponential suppression.Given that the volume mode is kept stable during inflation,all the inflaton-dependent higher dimensional operators are suppressed.Moreover, string loop effects can be shownto be negligible throughout all the inflationarydynamics for natural values of the underlying parameters.The model is characterised by a reheating temperatureof the order Trh ≃ 106 GeV which requires Ne ≃ 54 e-foldings of inflation.All the inflationary observables are compatible with current observationssince the spectral index is ns ≃ 0.96,while the tensor-to-scalar ratio is r ≃ 10−5.The volume of the Calabi-Yau is of order 103 in string units,corresponding to an inflationary scale around 1015 GeV.

A toolkit for perturbing flux compactifications

We develop a perturbative expansion scheme for solving general boundary value problems in a broad class of type IIB flux compactifications. The background solution is any conformally Calabi-Yau compactification with imaginary self-dual (ISD) fluxes. Upon expanding in small deviations from the ISD solution, the equations of motion simplify dramatically: we find a simple basis in which the n-th order equations take a triangular form. This structure implies that the system can be solved iteratively whenever the individual, uncoupled equations can be solved. We go on to demonstrate the solution of the system for a general warped Calabi-Yau cone: we present an algorithm that yields an explicit Green's function solution for all the supergravity fields, to any desired order, in terms of the harmonic functions on the base of the cone. Our results provide a systematic procedure for obtaining the corrections to a warped throat geometry induced by attachment to a compact bulk. We also present a simple method for determining the sizes of physical effects mediated through warped geometries.

The geometry of D-brane superpotentials

The disk partition function of the open topological string computes the spacetime superpotential for D-branes wrapping cycles of a compact Calabi-Yau threefold. We use string duality to show that when appropriately formulated, the problem admits a natural geometrization in terms of a non-compact Calabi-Yau fourfold without D-branes. The duality relates the D-brane superpotential to a flux superpotential on the fourfold. This sheds light on several features of superpotential computations appearing in the literature, in particular on the observation that Calabi-Yau fourfold geometry enters the problem. In one of our examples, we show that the geometry of fourfolds also reproduces the D-brane superpotentials obtained from matrix factorization methods.