  The six-dimensional effective action of F-theory compactified on a singular
elliptically fibred Calabi-Yau threefold is determined by using an M-theory
lift. The low-energy data are derived by comparing a circle reduction of a
general six-dimensional (1,0) gauged supergravity theory with the effective
action of M-theory on the resolved Calabi-Yau threefold. The derivation
includes six-dimensional tensor multiplets for which the (anti-) self-duality
constraints are imposed on the level of the five-dimensional action. The vector
sector of the reduced theory is encoded by a non-standard potential due to the
Green-Schwarz term in six dimensions. This Green-Schwarz term also contains
higher curvature couplings which are considered to establish the full map
between anomaly coefficients and geometry. F-/M-theory duality is exploited by
moving to the five-dimensional Coulomb branch after circle reduction and
integrating out massive vector multiplets and matter hypermultiplets. The
associated fermions then generate additional Chern-Simons couplings at
one-loop. Further couplings involving the graviphoton are induced by quantum
corrections due to excited Kaluza-Klein modes. On the M-theory side integrating
out massive fields corresponds to resolving the singularities of the Calabi-Yau
threefold, and yields intriguing relations between six-dimensional anomalies
and classical topology.
