## On $$\mathcal{N} = 1$$ 4d effective couplings for $$F$$-theory and heterotic vacua.(English)Zbl 1251.81076

The authors show how the combination of the methods of mirror symmetry and Hodge theory enable one to compute effective couplings of some heterotic/type II compactfications, including the superpotential and the Kähler potential. After describing the involvement of Hodge theory in two steps, they explain that the four-fold geometry represents the compactification manifold of a dual $$F$$-theory or type IIA compactification. They show that the four-fold result agrees with the three-fold result when it should, but gives more general results, including the case when the heterotic three-fold is not a Calabi-Yau three-fold. More specificly, the heterotic case includes a class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten. Indeed, the result obtained from an $$F$$-theory/type IIA compactification on the dual four-fold differs from the three-fold result. These deviations represent physical corrections to the dual type II/heterotic compactification from perturbative and instanton effects and describe how Hodge theory and mirror symmetry on the four-fold provides a powerful tool to determine these perturbative and non-perturbative contributions. In effect, mirror symmetry of four-folds computes non-perturbative corrections to mirror symmetry on the three-folds, including $$D$$-instanton corrections. The authors also discuss the type II/heterotic duality in the context of non-compact four-folds that arise as two-dimensional ALE fibration. Finally, the authors conjecture an extension of an observation due to Warner which relates the deformation superpotential of matrix factorizations of minimal models to the flux superpotential of local four-folds near an ADE singularity.

### MSC:

 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)

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