Gottfried Curio

From local to global in F-theory model building

Abstract When locally engineering F-theory models some D 7 -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 D 7 -branes are not independent from each other and the (compact part of the) D 7 -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 S U ( 5 ) model and also their impact on the tadpole cancellation condition.

Stable bundle extensions on elliptic Calabi–Yau threefolds

Abstract 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 for invoking background five-branes.

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. More precisely, we can get either the Standard Model gauge group GSM times an additional U(1), or just GSM but with additional exotic matter. 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.

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 Kahler 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 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.

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 θ-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 the equations W = dW = 0 for the superpotential.

Heterotic models without fivebranes

Abstract 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 Kahler cone, are given in the mathematical companion paper math.AG/0611762 .

Spectral bundles and the DRY-Conjecture

Abstract 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 1010.1644 [hep-th], we showed that infinitely many classes on X exist for which the conjecture is 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 1011.6246 [hep-th], we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.

On possible Chern classes of stable bundles on Calabi–Yau threefolds

Abstract 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 [M.R. Douglas, R. Reinbacher, S.-T. Yau, Branes, Bundles and Attractors: Bogomolov and Beyond, 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 ∈ H 4 ( X , Z ) exist which are of this form and for each of them a stable S U ( n ) vector bundle with c = c 2 ( V ) exists.

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 sumof contributions from isolated rational curves

Moduli restriction and chiral matter in heterotic string compactifications

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