New Methods for Characterizing Phases of 2D Supersymmetric Gauge Theories

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.