  We study the physics of two-dimensional N=(2,2) gauged linear sigma models
(GLSMs) via the two-sphere partition function. We show that the classical phase
boundaries separating distinct GLSM phases, which are described by the
secondary fan construction for abelian GLSMs, are completely encoded in the
analytic structure of the partition function. The partition function of a
non-abelian GLSM can be obtained as a limit from an abelian theory; we utilize
this fact to show that the phases of non-abelian GLSMs can be obtained from the
secondary fan of the associated abelian GLSM. We prove that the partition
function of any abelian GLSM satisfies a set of linear differential equations;
these reduce to the familiar A-hypergeometric system of Gel'fand, Kapranov, and
Zelevinski for GLSMs describing complete intersections in toric varieties. We
develop a set of conditions that are necessary for a GLSM phase to admit an
interpretation as the low-energy limit of a non-linear sigma model with a
Calabi-Yau threefold target space. Through the application of these criteria we
discover a class of GLSMs with novel geometric phases corresponding to
Calabi-Yau manifolds that are branched double-covers of Fano threefolds. These
criteria provide a promising approach for constructing new Calabi-Yau
geometries.
