Compactifications of F-theory on Calabi-Yau threefolds at constant coupling

Generalizing the work of Sen, we analyze special points in the moduli space of the compactification of the F-theory on elliptically fibered Calabi-Yau threefolds where the coupling remains constant. These contain points where they can be realized as orbifolds of six torus T⁶ by Z_m × Z_n(m,n = 2,3,4,6). At various types of intersection points of singularities, we find that the enhancement of gauge symmetries arises from the intersection of two kinds of singularities. We also argue that when we take the Hirzebruch surface as a base for the Calabi-Yau threefold, the condition for constant coupling corresponds to the case where the point like instantons coalesce, giving rise to enhanced gauge group of Sp(k).