  Applying advances in exact computations of supersymmetric gauge theories, we
study the structure of correlation functions in two-dimensional N=(2,2) Abelian
and non-Abelian gauge theories. We determine universal relations among
correlation functions, which yield differential equations governing the
dependence of the gauge theory ground state on the Fayet-Iliopoulos parameters
of the gauge theory. For gauge theories with a non-trivial infrared N=(2,2)
superconformal fixed point, these differential equations become the
Picard-Fuchs operators governing the moduli-dependent vacuum ground state in a
Hilbert space interpretation. For gauge theories with geometric target spaces,
a quadratic expression in the Givental I-function generates the analyzed
correlators. This gives a geometric interpretation for the correlators, their
relations, and the differential equations. For classes of Calabi-Yau target
spaces, such as threefolds with up to two Kahler moduli and fourfolds with a
single Kahler modulus, we give general and universally applicable expressions
for Picard-Fuchs operators in terms of correlators. We illustrate our results
with representative examples of two-dimensional N=(2,2) gauge theories.
