  We compute the exact two-sphere partition function and matrix of two-point
functions of operators in the chiral ring with their complex conjugates in
two-dimensional supersymmetric gauge theories. For gauge theories that flow in
the infrared to a Calabi-Yau nonlinear sigma model, these renormalization group
invariant observables determine the exact Kahler potential and associated
Zamolodchikov metric in the complex structure moduli space of the Calabi-Yau
manifold.
