D-instanton superpotential in string theory

We study the non-perturbative superpotential generated by D(-1)-branes in type IIB compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. To compute the D-instanton superpotential, we study F-theory compactification on toric complete intersection elliptic Calabi-Yau fourfolds. We take the Sen-limit, but with finite g(s), in F-theory compactifications with the restriction that all D7-branes are carrying SO(8) gauge groups, which we call the global Sen-limit. In the global Sen-limit, the axio-dilaton is not varying in the compactification manifold. We compute the Picard-Fuchs equations of elliptic Calabi-Yau fourfolds in the global Sen-limit, and show that the Picard-Fuchs equations of the elliptic fourfolds split into that of the underlying Calabi-Yau threefolds and of the elliptic fiber. We then demonstrate that this splitting property of the Picard-Fuchs equation implies that the fourform period of the elliptic Calabi-Yau fourfolds in the global Sen-limit does not contain exponentially suppressed terms O(e())(-pi/gs). With this result, we finally show that in the global Sen-limit, the superpotential of the underlying type IIB compactification does not receive D(-1)-instanton contributions.