Non-perturbative aspects of string theory from elliptic curves

Abstract

We consider two examples for non-perturbative aspects of string theory involving elliptic curves. First, we discuss F-theory on genus-one fibered Calabi-Yau manifolds with the fiber being a hypersurface in a toric fano variety. We discuss in detail the fiber geometry in order to find the gauge groups, matter content and Yukawa couplings of the corresponding supergravity theories for the four examples leading to gauge groups SU(3)xSU(2)xU(1), SU(4)xSU(2)xSU(2)/Z₂, U(1) and Z₃. The theories are connected by Higgsings on the field theory side and conifold transitions on the geometry side. We extend the discussion to the network of Higgsings relating all theories stemming from the 16 hypersurface fibrations. <br /> For the models leading to gauge groups SU(3)xSU(2)xU(1), SU(4)xSU(2)xSU(2)/Z₂ and U(1) we discuss the construction of vertical G4 fluxes. Via the D3-brane tadpole cancelation condition we can restrict the minimal number of families in the first two of these models to be at least three.<br /> As a second example for non-perturbative aspects of string theory we discuss a proposal for a non-perturbative completion of topological string theory on local B-model geometries. We discuss in detail the computation of quantum periods for the examples of local F₁, local IF₂ and the resolution of C³/Z₅. The quantum corrections are calculated order by order using second order differential operators acting on the classical periods. Using quantum geometry we calculate the refined free energies in the Nekrasov-Shatashvili limit. Finally we check the non-perturbative completion of topological string theory for the geometry of local F₂ against numerical calculations.