Low-energy analysis of M and F theories on Calabi-Yau threefolds

We elucidate the interplay between gauge and supersymmetry anomalies in six-dimensional N=1 supergravity with generalized couplings between tensor and vector multiplets. We derive the structure of the five-dimensional supergravity resulting from the S_1 reduction of these models and give the constraints on Chern-Simons couplings that follow from duality to M theory compactified on a Calabi-Yau threefold. The duality is supported only on a restricted class of Calabi-Yau threefolds and requires a special type of intersection form. We derive five-dimensional central-charge formulas and discuss briefly the associated phase transitions. Finally, we exhibit connections with F-theory compactifications on Calabi-Yau manifolds that admit elliptic fibrations. This analysis suggests that F theory unifies Type-IIb three branes and M-theory five branes.