Abstract
AbstractWe determine threshold corrections to the gauge couplings in local models of$ \mathcal{N}=2 $smooth heterotic compactifications with torsion, given by the direct product of a warped Eguchi-Hanson space and a two-torus, together with a line bundle. Using the worldsheet cft description previously found and by suitably regularising the infinite target space volume divergence, we show that threshold corrections to the various gauge factors are governed by the non-holomorphic completion of the Appell-Lerch sum. While its holomorphic Mock-modular component captures the contribution of states that localise on the blown-up two-cycle, the non-holomorphic correction originates from non-localised bulk states. We infer from this analysis universality properties for$ \mathcal{N}=2 $heterotic local models with flux, based on target space modular invariance and the presence of such non-localised states. We finally determine the explicit dependence of these one-loop gauge threshold corrections on the moduli of the two-torus, and by S-duality we extract the corresponding string-loop and E1-instanton corrections to the Kähler potential and gauge kinetic functions of the dual type i model. In both cases, the presence of non-localised bulk states brings about novel perturbative and non-perturbative corrections, some features of which can be interpreted in the light of analogous corrections to the effective theory in compact models.
Highlights
Supersymmetric compactifications of the heterotic string [1] were soon recognised as a very successful approach to string phenomenology
These compactifications with torsion were explored in the early days of the heterotic string [3, 4]. Their analysis is quite involved, as generically the compactification manifold is not even conformally Kahler. In view of this complexity, it is usefull to describe more quantitatively such flux compactifications with non-compact geometries that can be viewed as local models thereof
In order to be able to make sensible phenomenological predictions, one should properly engineer the gluing of sixteen of these heterotic local models into a T 2 × K3 compactification, which would give us a proper effective field theory understanding of bulk state contributions. This could be of particular interest, on the dual type i side, to clarify the role of these novel bulk state contributions we find in E1-instanton corrections, which include an infinite sum over descendants of the modified elliptic genus, as functions of the induced T 2 moduli
Summary
Supersymmetric compactifications of the heterotic string [1] were soon recognised as a very successful approach to string phenomenology. A deeper physical insight into the shadow function featured in the bulk state contribution is achieved by investigating the l = 1 model, whose interacting part enjoys an enhanced (4, 4) worldsheet superconformal symmetry We observe in this particular case that localised effects splits on the one hand into 4/χ(K3) of the gauge threshold corrections for a T 2 ×K3 model, for which there is a rich literature both in heterotic and type i theories [64,65,66,67,68,69,70,71,72], and on the other hand into a Mock modular form F (τ ) encoding the presence of warping due to nsns flux threading the geometry. Some material about superconformal characters, modular form, and some lengthy computations are given in the various appendices
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