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Gottfried Curio

Publications and source records attributed to Gottfried Curio.

At least 19 recordsLinked to original sources
arXiv2012

Complete Bundle Moduli Reduction in Heterotic String Compactifications

A major problem in discussing heterotic string models is the stabilisation of the many vector bundle moduli via the superpotential generated by world-sheet instantons. In arXiv:1110.6315 we have discussed the method to make a discrete twist in a large and much discussed class of vector bundles such that the generation number gets new contributions (which can be tuned suitably) and at the same time the space of bundle moduli of the new, twisted bundle is a proper subspace (where the 'new', non-generic twist class exists) of the original bundle moduli space; one thus gets a model, closely related to the original model one started with, but with enhanced flexibility in the generation number and where on the other hand the number of bundle moduli is {\em somewhat} reduced. Whereas in the previous paper the emphasis was on examples for the new flexibility in the generation number we here classify and describe explicitly the twists and give the precise reduction formula (for the number of moduli) for SU(5) bundles leading to an SU(5) GUT group in four dimensions. Finally we give various examples where the bundle moduli space is reduced {\em completely}: the superpotential for such {\em rigid} bundles becomes a function of the complex structure moduli alone (besides the exponential Kahler moduli contribution).

hep-th↗
arXiv2012

Moduli restriction and Chiral Matter in Heterotic String Compactifications

Supersymmetric heterotic string models, built from a stable holomorphic vector bundle $V$ on a Calabi-Yau threefold $X$, usually come with many vector bundle moduli whose stabilisation is a difficult and complex task. It is therefore of interest to look for bundle constructions which, from the outset, have as few as possible bundle moduli. One way to reach such a set-up is to start from a generic construction and to make discrete modifications of it which are available only over a subset of the bundle moduli space. Turning on such discrete 'twists' constrains the moduli to the corresponding subset of their moduli space: the twisted bundle has less parametric freedom. We give an example of a set-up where this idea can be considered concretely. Such non-generic twists lead also to new contributions of chiral matter (which greatly enhances the flexibility in model building); their computation constitutes the main issue of this note.

hep-th↗
arXiv2012

Higgs Multiplets in Heterotic GUT Models

For supersymmetric GUT models from heterotic string theory, built from a stable holomorphic SU(n) vector bundle $V$ on a Calabi-Yau threefold $X$, the net amount of chiral matter can be computed by a Chern class computation. Corresponding computations for the number $N_H$ of Higgses lead for the phenomenologically relevant cases of GUT group SU(5) or SO(10) to consideration of the bundle $\La^2 V$. In a class of bundles where everything can be computed explicitly (spectral bundles on elliptic $X$) we find that the computation for $N_H$ gives a result which is in conflict with expectations. We argue that this discrepancy has its origin in the subtle geometry of the spectral data for $\La^2 V$ and that taking this subtlety into account properly should resolve the problem.

hep-th↗
arXiv2010

On possible Chern Classes of stable Bundles on Calabi-Yau threefolds

Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$ endowed with a stable vector bundle $V$, usually lead to an anomaly mismatch between $c_2(V)$ and $c_2(X)$; this leads to the question whether the difference can be realized by a further bundle in the hidden sector. In math.AG/0604597 a conjecture is stated which gives sufficient conditions on cohomology classes on $X$ to be realized as the Chern classes of a stable reflexive sheaf $V$; a weak version of this conjecture predicts the existence of such a $V$ if $c_2(V)$ is of a certain form. In this note we prove that on elliptically fibered $X$ infinitely many cohomology classes $c\in H^4(X, {\bf Z})$ exist which are of this form and for each of them a stable SU(n) vector bundle with $c=c_2(V)$ exists.

hep-th↗
arXiv2010

Spectral Bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$ endowed with a stable vector bundle $V$, usually start from a phenomenologically motivated choice of a bundle $V_v$ in the visible sector, the spectral cover construction on an elliptically fibered $X$ being a prominent example. The ensuing anomaly mismatch between $c_2(V_v)$ and $c_2(X)$, or rather the corresponding differential forms, is often 'solved', on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The 'DRY'-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on $X$ to be realized as the Chern classes of a stable sheaf. In arXiv:1010.1644 we showed that infinitely many classes on $X$ exist for which the conjecture ist true. In this note we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in arXiv:1011.6246 we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.

hep-th↗
arXiv2010

On the Heterotic World-sheet Instanton Superpotential and its individual Contributions

For supersymmetric heterotic string compactifications on a Calabi-Yau threefold $X$ endowed with a vector bundle $V$ the world-sheet superpotential $W$ is a sum of contributions from isolated rational curves $\C$ in $X$; the individual contribution is given by an exponential in the Kähler class of the curve times a prefactor given essentially by the Pfaffian which depends on the moduli of $V$ and the complex structure moduli of $X$. Solutions of $DW=0$ (or even of $DW=W=0$) can arise either by nontrivial cancellations between the individual terms in the summation over all contributing curves or because each of these terms is zero already individually. Concerning the latter case conditions on the moduli making a single Pfaffian vanish (for special moduli values) have been investigated. However, even if corresponding moduli - fulfilling these constraints - for the individual contribution of one curve are known it is not at all clear whether {\em one} choice of moduli exists which fulfills the corresponding constraints {\em for all contributing curves simultaneously}. Clearly this will in general happen only if the conditions on the 'individual zeroes' had already a conceptual origin which allows them to fit together consistently. We show that this happens for a class of cases. In the special case of spectral cover bundles we show that a relevant solution set has an interesting location in moduli space and is related to transitions which change the generation number.

hep-th↗
arXiv2009

World-sheet Instanton Superpotentials in Heterotic String theory and their Moduli Dependence

To understand in detail the contribution of a world-sheet instanton to the superpotential in a heterotic string compactification, one has to understand the moduli dependence (bundle and complex structure moduli) of the one-loop determinants from the fluctuations, which accompany the classical exponential contribution (involving Kähler moduli) when evaluating the world-volume partition function. Here we use techniques to describe geometrically these Pfaffians for spectral bundles over rational base curves in elliptically fibered Calabi-Yau threefolds, and provide a (partially exhaustive) list of cases involving {\em factorising} (or vanishing) superpotential. This gives a conceptual explanation and generalisation of the few previously known cases which were obtained just experimentally by a numerical computation.

hep-th↗
arXiv2009

Perspectives on Pfaffians of Heterotic World-sheet Instantons

To fix the bundle moduli of a heterotic compactification one has to understand the Pfaffian one-loop prefactor of the classical instanton contribution. For compactifications on elliptically fibered Calabi-Yau spaces X this can be made explicit for spectral bundles and world-sheet instantons supported on rational base curves b: one can express the Pfaffian in a closed algebraic form as a polynomial, or it may be understood as a theta-function expression. We elucidate the connection between these two points of view via the respective perception of the relevant spectral curve, related to its extrinsic geometry in the ambient space (the elliptic surface in X over b) or to its intrinsic geometry as abstract Riemann surface. We identify, within a conceptual description, general vanishing loci of the Pfaffian, and derive bounds on the vanishing order, relevant to solutions of W=dW=0.

hep-th↗
arXiv2009

From Local to Global in F-Theory Model Building

When locally engineering F-theory models some D7-branes for the gauge group factors are specified and matter is localized on the intersection curves of the compact parts of the world-volumes. In this note we discuss to what extent one can draw conclusions about F-theory models by just restricting the attention locally to a particular seven-brane. Globally the possible D7-branes are not independent from each other and the (compact part of the) D7-brane can have unavoidable intrinsic singularities. Many special intersecting loci which were not chosen by hand occur inevitably, notably codimension three loci which are not intersections of matter curves. We describe these complications specifically in a global SU(5) model and also their impact on the tadpole cancellation condition.

hep-th↗
arXiv2007

S-Track Stabilization of Heterotic de Sitter Vacua

We present a new mechanism, the S-Track, to stabilize the volume modulus S in heterotic M-theory flux compactifications along with the orbifold-size T besides complex structure and vector bundle moduli stabilization. The key dynamical ingredient which makes the volume modulus stabilization possible, is M5-instantons arising from M5-branes wrapping the whole Calabi-Yau slice. These are natural in heterotic M-theory where the warping shrinks the Calabi-Yau volume along S^1/Z_2. Combined with H-flux, open M2-instantons and hidden sector gaugino condensation it leads to a superpotential W which stabilizes S similar like a racetrack but without the need for multi gaugino condensation. Moreover, W contains two competing non-perturbative effects which stabilize T. We analyze the potential and superpotentials to show that it leads to heterotic de Sitter vacua with broken supersymmetry through non-vanishing F-terms.

hep-th↗
arXiv2007

Deformations of Bundles and the Standard Model

We modify a recently proposed heterotic model hep-th/0703210, giving three net-generations of standard model fermions, to get rid of an additional U(1) factor in the gauge group. The method employs a stable SU(5) bundle on a Calabi-Yau threefold admitting a free involution. The bundle has to be built as a deformation of the direct sum of a stable SU(4) bundle and the trivial line bundle.

hep-th↗
arXiv2007

Extension Bundles and the Standard Model

We construct a new class of stable vector bundles suitable for heterotic string compactifications. Using these we describe a novel way to derive the fermionic matter content of the Standard Model from the heterotic string. For this we compactify on an elliptically fibered Calabi-Yau threefold X with two sections, the B-fibration, a variant of the ordinary Weierstrass fibration, which allows X to carry a free involution. We construct rank five vector bundles, invariant under this involution, such that turning on a Wilson line we obtain the Standard Model gauge group and find various three generation models. This rank five bundle is derived from a stable rank four bundle that arises as an extension of bundles pulled-back from the base and twisted by suitable line bundles. We also give an account of various previous results and put the present construction into perspective.

hep-th↗
arXiv2007

Invariant Bundles on $B$-fibered Calabi-Yau Spaces and the Standard Model

We derive the Standard model gauge group together with chiral fermion generations from the heterotic string by turning on a Wilson line on a non-simply connected Calabi-Yau threefold with an SU(5) gauge group. For this we construct stable ${\bf Z_2}$-invariant $SU(4)\times U(1)$ bundles on an elliptically fibered cover Calabi-Yau threefold of special fibration type (the $B$-fibration). The construction makes use of a modified spectral cover approach giving just invariant bundles.

hep-th↗
arXiv2007

Heterotic Models without Fivebranes

After discussing some general problems for heterotic compactifications involving fivebranes we construct bundles, built as extensions, over an elliptically fibered Calabi-Yau threefold. For these we show that it is possible to satisfy the anomaly cancellation topologically without any fivebranes. The search for a specific Standard model or GUT gauge group motivates the choice of an Enriques surface or certain other surfaces as base manifold. The burden of this construction is to show the stability of these bundles. Here we give an outline of the construction and its physical relevance. The mathematical details, in particular the proof that the bundles are stable in a specific region of the Kähler cone, are given in the mathematical companion paper math.AG/0611762.

hep-th↗
arXiv2007

Stable bundle extensions on elliptic Calabi-Yau threefolds

We construct stable bundle extensions on elliptically fibered Calabi-Yau threefolds. We show that these bundles can solve the topological anomaly constraint in heterotic string theory without the need of invoking background fivebranes.

math.AG↗
arXiv2006

On the modified KKLT procedure: a case study for the P_{11169}[18] model

We probe the existence of supersymmetric vacua of the type IIB orientifold of the elliptic Calabi-Yau space P_{11169}[18] where generically two complex structure moduli z_i, the dilaton tau and the two Kähler moduli T_i are stabilized by fluxes and gaugino condensates. The usual KKLT procedure, which integrates out the complex structure moduli and the dilaton, actually has to be modified, such that one keeps the dependence on tau. We derive explicitely the resulting effective superpotential W_{eff}(tau) for the dilaton for various flux combinations. As this is actually a non-holomorphic quantity one must properly work with the G-function. The remaining SUSY equations for tau and the T_i can be resolved explicitely.

hep-th↗
arXiv2006

Standard Model bundles of the heterotic string

We show how to construct supersymmetric three-generation models with gauge group and matter content of the Standard Model in the framework of non-simply-connected elliptically fibered Calabi-Yau manifolds Z. The elliptic fibration on a cover Calabi-Yau, where the model has 6 generations of SU(5) and the bundle is given via the spectral cover description, has a second section leading to the needed free involution. The relevant involution on the defining spectral data of the bundle is identified for a general Calabi-Yau of this type and invariant bundles are generally constructible.

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