  In this work we study the phase structure of skew symplectic sigma models,
which are a certain class of two-dimensional N = (2,2) non-Abelian gauged
linear sigma models. At low energies some of them flow to non-linear sigma
models with Calabi-Yau target spaces, which emerge from non-Abelian strong
coupling dynamics. The observed phase structure results in a non-trivial
duality proposal among skew symplectic sigma models and connects non-complete
intersection Calabi-Yau threefolds, that are non-birational among another, in a
common quantum Kahler moduli space. As a consequence we find non-trivial
identifications of spectra of topological B-branes, which from a modern
algebraic geometry perspective imply derived equivalences among Calabi-Yau
varieties. To further support our proposals, we calculate the two sphere
partition function of skew symplectic sigma models to determine geometric
invariants, which confirm the anticipated Calabi-Yau threefold phases. We show
that the two sphere partition functions of a pair of dual skew symplectic sigma
models agree in a non-trivial fashion. To carry out these calculations, we
develop a systematic approach to study higher-dimensional Mellin-Barnes type
integrals. In particular, these techniques admit the evaluation of two sphere
partition functions for gauged linear sigma models with higher rank gauge
groups, but are applicable in other contexts as well.
