We describe blowups of C n /Z n orbifolds as complex line bundles over CP n−1 . We construct some gauge bundles on these resolutions. Apart from the standard embedding, we describe U(1) bundles and an SU(n−1) bundle. Both blowups and their gauge bundles are given explicitly. We investigate ten dimensional SO(32) super Yang-Mills theory coupled to supergravity on these backgrounds. The integrated Bianchi identity implies that there are only a finite number of U(1) bundle models. We describe how the orbifold gauge shift vector can be read off from the gauge background. In this way we can assert that in the blow down limit these models correspond to heterotic C 2 /Z 2 and C 3 /Z 3 orbifold models. (Only the Z 3 model with unbroken gauge group SO(32) cannot be reconstructed in blowup without torsion.) This is confirmed by computing the charged chiral spectra on the resolutions. The construction of these blowup models implies that the mismatch between type-I and heterotic models on T 6 /Z 3 does not signal a complication of S-duality, but rather a problem of type-I model building itself: The standard type-I orbifold model building only allows for a single model on this orbifold, while the blowup models give five different models in blow down.

Abstract:We clarify the relation between six-dimensional Abelian orbifold compactifications of the heterotic string and smooth heterotic K3 compactifications with line bundles for both SO(32) and E 8 × E 8 gauge groups. The T 4 /Z N cases for N = 2, 3, 4 are treated exhaustively, and for N = 6 some examples are given. While all T 4 /Z 2 and nearly all T 4 /Z 3 models have a simple smooth match involving one line bundle only, this is only true for some T 4 /Z 4 and T 4 /Z 6 cases. We comment on possible matchings with more than one line bundle for the remaining cases. The matching is provided by comparisons of the massless spectra and their anomalies as well as a field theoretic analysis of the blow-ups.

Heterotic orbifolds provide promising constructions of MSSM-like models in string theory. We investigate the connection of such orbifold models with smooth Calabi-Yau compactifications by examining resolutions of the T 6 / 6-II orbifold (which are far from unique) with Abelian gauge fluxes. These gauge backgrounds are topologically characterized by weight vectors of twisted states; one per fixed point or fixed line. The VEV's of these states generate the blowup from the orbifold perspective, and they reappear as axions on the blowup. We explain methods to solve the 24 resolution dependent Bianchi identities and present an explicit solution. Despite that a solution may contain the MSSM particle spectrum, the hypercharge turns out to be anomalous: Since all heterotic MSSM orbifolds analyzed so far have fixed points where only SM charged states appear, its gauge group can only be preserved provided that those singularities are not blown up. Going beyond the comparison of purely topological quantities (e.g. anomalous U(1) masses) may be hampered by the fact that in the orbifold limit the supergravity approximation to lowest order in α ′ is breaking down.

We construct an MSSM with three generations from the heterotic string compactified on a smooth 6D internal manifold using Abelian gauge fluxes only. The compactification space is obtained as a resolution of the T 6 / 2 × 2 × 2,free orbifold. The 2,free involution of such a resolution breaks the SU(5) GUT group down to the SM gauge group using a suitably chosen (freely acting) Wilson line. Surprisingly, the spectrum on a given resolution is larger than the one on the corresponding orbifold taking into account the branching and Higgsing due to the blow-up modes. The existence of extra resolution states is closely related to the fact that the resolution procedure is not unique. Rather, the various resolutions are connected to each other by flop transitions.

It is well-known that heterotic string compactifications have, in spite of their conceptual simplicity and aesthetic appeal, a serious problem with precision gauge coupling unification in the perturbative regime of string theory. Using both a duality-based and a field-theoretic definition of the boundary of the perturbative regime, we reevaluate the situation in a quantitative manner. We conclude that the simplest and most promising situations are those where some of the compactification radii are exceptionally large, corresponding to highly anisotropic orbifold models. Thus, one is led to consider constructions which are known to the effective field-theorist as higher-dimensional or orbifold grand unified theories (orbifold GUTs). In particular, if the discrete symmetry used to break the GUT group acts freely, a non-local breaking in the larger compact dimensions can be realized, leading to a precise gauge coupling unification as expected on the basis of the MSSM particle spectrum. Furthermore, a somewhat more model dependent but nevertheless very promising scenario arises if the GUT breaking is restricted to certain singular points within the manifold spanned by the larger compactification radii.

We investigate resolutions of heterotic orbifolds using toric geometry. Our starting point is provided by the recently constructed heterotic models on explicit blowup of C n /Z n singularities. We show that the values of the relevant integrals, computed there, can be obtained as integrals of divisors (complex codimension one hypersurfaces) interpreted as (1, 1)-forms in toric geometry. Motivated by this we give a self contained introduction to toric geometry for non-experts, focusing on those issues relevant for the construction of heterotic models on toric orbifold resolutions. We illustrate the methods by building heterotic models on the resolutions of C 2 /Z 3 , C 3 /Z 4 and C 3 /Z 2 × Z ′ 2 . We are able to obtain a direct identification between them and the known orbifold models. In the C 3 /Z 2 × Z ′ 2 case we observe that, in spite of the existence of two inequivalent resolutions, fully consistent blowup models of heterotic orbifolds can only be constructed on one of them.

We examine the large volume compactification of Type IIB string theory or its F theory limit and the associated supersymmetry breakdown and soft terms. It is crucial to incorporate the loop-induced moduli mixing, originating from radiative corrections to the Kähler potential.We show that in the presence of moduli mixing, soft scalar masses generically receive a D-term contribution of the order of the gravitino mass m 3/2 when the visible sector cycle is stabilized by the D-term potential of an anomalous U(1) gauge symmetry, while the moduli-mediated gaugino masses and A-parameters tend to be of the order of m 3/2 /8π 2 . It is noticed also that a too large moduli mixing can destabilize the large volume solution by making it a saddle point. * In this picture, the uplifting potential is exponentially small as it arises from a SUSY breakdown at the tip of warped throat, and then small W 0 is required to tune the cosmological constant to a nearly vanishing value.Planck discussed in [8]. We also stress the importance of the D-terms along the visible sector 4-cycle in the LVS-models. They tend to dominate the soft scalar mass terms and give a contribution of the order of the gravitino mass m 3/2 . Gaugino masses and A-parameters do not receive D-term contributions, and generically tend to be loopsuppressed, being of the order of O(m 3/2 /8π 2 ). With these contributions from moduli mixings, if the gravitino mass were of the order of 10 11 GeV as conjectured in [8] (in order to accommodate the unification scale M GU T ∼ 10 16 GeV with W 0 ∼ O(1)), severe fine-tuning of the Kähler potential at the multi-loop level would be required to keep the soft terms in the TeV range.We would therefore argue that the gravitino mass should not exceed the (multi)-TeV range. This paper is organized as follows. In section 2, we revisit the LVS-scheme while incorporating the moduli redefinition discussed in [9]. We also include a discussion of the stability of the large volume solution in the presence of moduli mixing. Section 3 discusses the D-term stabilization of the visible sector cycle with an explicit scheme to stabilize the remained D-flat direction which is parameterized in this case by U(1) A -charged (but MSSM singlet) matter fields breaking a global Peccei-Quinn symmetry spontaneously. This scheme naturally generates an intermediate axion scale in LVS, and can be implemented in other scenarii with a high string scale close to the Planck scale. Section 4 is devoted to the discussion of supersymmtry breakdown and resulting soft terms, and conclusions and outlook will be given in section 5.

We present a 2 × 2 orbifold compactification of the E 8 × E 8 heterotic string which gives rise to the exact chiral MSSM spectrum. The GUT breaking SU(5) → SU(3) C × SU(2) L × U(1) Y is realized by modding out a freely acting symmetry. This ensures precision gauge coupling unification. Further, it allows us to break the GUT group without switching on flux in hypercharge direction, such that the standard model gauge bosons can remain massless when the orbifold singularities are blown up. The model has vacuum configurations with matter parity, a large top Yukawa coupling and other phenomenologically appealing features.