M-Theory and Topological Strings--I

The $R^2 F^{2g-2}$ terms of Type IIA strings on Calabi-Yau 3-folds, which are given by the corresponding topological string amplitudes (a worldsheet instanton sum for all genera), are shown to have a simple M-theory interpretation. In particular, a Schwinger one-loop computation in M-theory with wrapped M2 branes and Kaluza-Klein modes going around the loop reproduces the all genus string contributions from constant maps and worldsheet instanton corrections. In the simplest case of an isolated M2 brane with the topology of the sphere, we obtain the contributions of small worldsheet instantons (sphere ``bubblings'') which extends the results known or conjectured for low genera. Surprisingly, the 't Hooft expansion of large $N$ Chern-Simons theory on $S^3$ can also be used in a novel way to compute these gravitational terms at least in special cases.