  We prove a recent conjecture that the partition function of N=(2, 2) gauge
theories on the two-sphere which flow to Calabi-Yau sigma models in the
infrared computes the exact Kahler potential on the quantum Kahler moduli space
of the corresponding Calabi-Yau. This establishes the two-sphere partition
function as a new method of computation of worldsheet instantons and
Gromov-Witten invariants. We also calculate the exact two-sphere partition
function for N=(2,2) Landau-Ginzburg models with an arbitrary twisted
superpotential W. These results are used to demonstrate that arbitrary abelian
gauge theories and their associated mirror Landau-Ginzburg models have
identical two-sphere partition functions. We further show that the partition
function of non-abelian gauge theories can be rewritten as the partition
function of mirror Landau-Ginzburg models.
